COSMOS-e -soft Higgsotic attractors

TIFR/TH/16-50

COSMOS-e0- soft Higgsotic attractors

arXiv:1703.01750v1 [hep-th] 6 Mar 2017

Sayantan Choudhury

1a

a Department

of Theoretical Physics, Tata Institute of Fundamental Research, Colaba, Mumbai 400005, India

E-mail: [email protected] Abstract: In this work, we have developed an elegant algorithm to study the cosmological consequences from a huge class of quantum field theories (i.e. superstring theory, supergravity, extra dimensional theory, modified gravity etc.), which are equivalently described by soft attractors in the effective field theory framework. In this description we have restricted our analysis for two scalar fields - dilaton and Higgsotic fields minimally coupled with Einstein gravity, which can be generalized for any arbitrary number of scalar field contents with generalized non-canonical and non-minimal interactions. We have explicitly used R2 gravity, from which we have studied the attractor and non-attractor phase by exactly computing two point, three point and four point correlation functions from scalar fluctuations using In-In (Schwinger-Keldysh) and δN formalism. We have also presented theoretical bounds on the amplitude, tilt and running of the primordial power spectrum, various shapes (equilateral, squeezed, folded kite or counter collinear) of the amplitude as obtained from three and four point scalar functions, which are consistent with observed data. Also the results from two point tensor fluctuations and field excursion formula are explicitly presented for attractor and non-attractor phase. Further, reheating constraints, scale dependent behaviour of the couplings and the dynamical solution for the dilaton and Higgsotic fields are also presented. New sets of consistency relations between two, three and four point observables are also presented, which shows significant deviation from canonical slow roll models. Additionally, three possible theoretical proposals have presented to overcome the tachyonic instability at the time of late time acceleration. Finally, we have also provided the bulk interpretation from the three and four point scalar correlation functions for completeness. Keywords: Cosmology beyond the standard model, De Sitter space, String Cosmology, Modified gravity.

1

Presently working as a Visiting (Post-Doctoral) fellow at DTP, TIFR, Mumbai, Alternative E-mail: [email protected].

Contents 1 Introduction

1

2 Model building from scale free gravity

12

3 Soft attractor: A two field approach 3.1 Case I: Power-law solution 3.2 Case II: Exponential solution

19 22 23

4 Constraints on inflation with soft attractors 4.1 Number of e-foldings 4.2 Primordial density perturbation 4.2.1 Two point function 4.2.2 Present observables 4.3 Primordial tensor modes and future observables 4.4 Reheating

35 35 36 36 38 44 48

5 Cosmological solutions from soft attractors 5.1 Solutions for inflaton 5.2 Solutions for field dependent coupling λ(Ψ)

50 50 52

6 Beyond soft attractor: A single field approach

53

7 Constraints on inflation beyond soft attractor 7.1 Number of e-foldings 7.2 Primordial density perturbation 7.2.1 Two point function 7.2.2 Present observables 7.3 Primordial tensor modes 7.3.1 Two point function 7.3.2 Future observables 7.4 Reheating

55 55 57 59 62 64 64 66 71

8 Future probe: Primordial Non-Gaussianity 8.1 Three point function 8.1.1 Using In-In formalism 8.1.2 Using δN formalism 8.2 Four point function 8.2.1 Using In-In formalism 8.2.2 Using δN formalism

73 73 73 86 122 122 169

9 Conclusion

186

–i–

10 Appendix 10.1 Effective Higgsotic models for generalized P (X, φ) theory 10.2 Dynamical dilaton at late times 10.3 Two field approach from dilaton-Higgsotic portal 10.3.1 Flow functions in Einstein frame 10.3.2 Two field perturbation setup 10.3.3 Two point function and related primordial observables 10.3.4 Late time power spectra 10.3.5 Running of the spectral tilt at late time 10.3.6 Scalar three point function using δN formalism

1

191 191 197 200 200 201 205 207 208 209

Introduction

The inflationary paradigm is a theoretical proposal which attempts to solve various longstanding issues with standard Big Bang Cosmology and has been studied earlier in various works [1–12]. But apart from the success of the this theoretical framework it is important to note that there is no single model exists till now using which one can explain the complete evolution history of the universe and also unable to break the degeneracy between various cosmological parameters computed from various models of inflation [13–23, 25–27, 61]. It is important to note that, the vacuum energy contribution generated by the trapped Higgs field in a metastable vacuum state which mimics the role of effective cosmological constant in effective theory. At the later stages of Universe such vacuum contribution dominates over other contents and correspondingly Universe expands in a exponential fashion. But using such metastable vacuum state it is not possible to explain the tunneling phenomena and also impossible to explain the end of inflation. To serve both of the purposes shape of the effective potential for inflation should have flat structure. Due to such specific structure effective potential for inflation satisfy flatness or slow-roll condition using which one can easily determine the field value corresponding to the end of inflation. There are various classes of models exists in cosmological literature where one can derive such specific structure of inflation [14, 28–33]. For an example, the Coleman-Weinberg effective potential serves this purpose [34, 35]. Now if we consider the finite temperature contributions in the effective potential [36, 37] then such thermal effects need to localize the inflaton field to small expectation values at the beginning of inflation. The flat structure of the effective potential for inflation is such that the scalar inflaton field slowly rolls down in the valley of potential during which the scale factor varies exponentially and then infation ends when the scalar inflaton field goes to the non slow-rolling region by violating the flatness condition. At this epoch inflaton field evolves to the true minimum very fast and then it couples to the matter content of the Universe and reheats our Universe via subsequent oscillations about the minimum of the slowly varying effective potential for inflation. These class of models are very successful theoretical probe through which it is possible to explain the characteristic and amplitude of the spectrum of density fluctuations with high statistical accuracy (2σ CL from Planck 2015 data [38–40]) and at late times this perturbations act

–1–

as the seeds for the large scale structure formation, which we observe at the present epoch. Apart from this huge success of inflationary paradigm in slowly varying regime it is important to mention that, these density fluctuations generated from various class of successful models were unfortunately large enough to explain the physics of standard Grand Unified Theory (GUT) with well known theoretical frameworks and also it is not possible to explain the observed isotropy of the Cosmic Microwave Background Radiation (CMBR) at small scales during inflationary epoch. The only physical possibility is that the self interactions of the inflaton field and the associated couplings to other matter field contents would be sufficiently small for which it is possible to satisfy these cosmological and particle physics constraints. But the prime theoretical challenge at this point is that for such setup it is impossible to achieve thermal equilibrium at the end of inflation. Consequently, it is not at all possible to localize the scalar inflaton field near zero Vacuum Expectation Value (VEV), hφi = h0|φ|0i = 0, where |0i is the corresponding vacuum state in quasi de Sitter space time. Therefore, a sufficient amount of expansion will not be obtained from this prescribed setup. Here it is important to note that, for a broad category of effective potentials the inflaton field evolves with time very slowly compared to the Hubble scale following slow-roll conditions and satisfies all of the observational constraints [38–40] computable from various inflationary observables from this setup. However, apart from the success of slow roll inflationary paradigm the density fluctuations or more precisely the scalar component of the metric perturbations restricts the coupling parameters to be sufficiently small enough and allows huge fine-tuning in the theoretical set up. This is obviously a not recommendable prescription from model builder’s point of view. Additionally, all of these class of models are not ruled out completely by the present observed data (Planck 2015 and other joint data sets [38–41]), as they are degenerate in terms of determination of inflationary observables and associated cosmological parameters in precision cosmology. There are various ideas exist in cosmological literature which can drive inflation. These are appended below: • Category I: In this class of models, inflation drives through a field theory which involves a very high energy physics phenomena. Example: string theory and its supergravity extensions [13, 15–17, 19, 22, 42–77], various supersymmetric models [14, 28–33] etc. • Category II: In this case, inflation is driven by changing the mathematical structure of the gravitational sector. This can be done using the following ways: 1. Introducing higher derivative terms of the form of f (R), where R is the Ricci scalar [78–81]. Example: Starobinsky inflationary framework which is governed by the model [78], f (R) = aR + bR2 , (1.1) where the coefficients a and b are given by, a = Mp2 and b = 1/6M 2 . If we set a = 0 and b = 1/6M 2 = α then we can get back the theory of scale free gravity in this context. In this paper will we explore the cosmological consequences from scale free theory of gravity.

–2–

2. Introducing higher derivative terms of the form of Gauss Bonnet gravity coupled with scalar field in non-minimal fashion, where the contribution in the effective action can be expressed as [82, 83], Z   √ SGB = d4 x −g f (φ) Rµναβ Rµναβ − 4Rµν Rµν + R2 , (1.2) where f (φ) is the inflaton dependent coupling which can be treated as the nonminimal coupling in the present context. This is also an interesting possibility which we have not explored in this paper. Here one cannot consider the Gauss Bonnet term in the gravity sector in 4D without coupling to other matter fields as in 4D Gauss Bonnet term is topological surface term. 3. Another possibility is to incorporate the effect of non-minimal coupling of inflaton field and the gravity sector [84–86]. The simplest example is, f (φ)R gravity theory. For Higgs inflation [84]:
f (φ) = 1 + ξφ2 , (1.3) where ξ is the non-minimal coupling of the Higgs field. Here one can consider more complicated possibility as well by considering a non-canonical interaction between inflaton and f (R) gravity by allowing f (φ)f (R) term in the 4D effective action [87]. For the construction of effective potential we have considered this possibility. 4. One can also consider the other possibility, where higher derivative non-local terms can be incorporated in the gravity sector [88–93]. For example one can consider the following possibilities: (a) (b) (c) (d) (e) (f)

Rf1 (2)R Rµν f2 (2)Rµν Rµναβ f3 (2)Rµναβ Rf4 (2)∇µ ∇ν ∇α ∇β Rµναβ Rµναβ f5 (2)∇α ∇β ∇ν ∇ρ ∇λ ∇γ Rµρλγ Rµναβ f6 (2)∇α ∇β ∇ν ∇µ ∇λ ∇γ ∇η ∇ξ Rλγηξ

where 2 is defined as: √  1 2 = g µν ∇µ ∇ν = √ ∂µ −g g µν ∂ν −g

(1.4)

is the d’Alembertian operator in 4D and the fi (2)∀i = 1, 2, · · · , 6 are analytic entire functions containing higher derivatives up to infinite order. This is itself a very complicated possibility which we have not explored in this paper. • Category III: In this case, inflation is driven by changing the mathematical structure of both the gravitational and matter sector of the effective theory. One of the examples is to use Jordan-Brans-Dicke (JBD) gravity theory [94, 95] along with extended inflationary

–3–

models which includes non-canonical interactions. By adjusting the value of BransDicke parameters one can study the observational consequences from this setup. Instead of Jordan-Brans-Dicke (JBD) gravity theory one can also use non local gravity or many other complicated possibilities. In this paper, we consider the possibility of soft inflationary paradigm in Einstein frame, where a chaotic Higgsotic inflationary effective potential is coupled to a dilaton via exponential type of potential, which is appearing through the conformal transformation from Jordan to Einstein frame in the metric within the framework of scale free αR2 gravity. This methodology of frame transformation from Jordan to Einstein frame can be applicable to any arbitrary class of f (R) gravity models and a theory with non-minimal coupling between gravity and matter sector. Also for N = 1 supergravity theory one can write down an effective action in Einstein frame using frame transformation from Jordan to Einstein frame. Here it is important to mention that, in case of soft inflationary model, the dilaton exponential potential is multiplied by an coupling constant of the Higgsotic theory which mimics the role of an effective coupling constant and its value always decreases with the field value. One can generalize this idea for any arbitrary matter interactions which is also described by generalized P (X, φ) theory [96, 97](see appendix 10.1 for more details). In this context of discussion also it is important to specify that, one can treat the field dependent couplings in the simple effective potentials or may be in a generalized P (X, φ) functionals contains a decaying behaviour with dilaton field value as it contains an overall exponential factor which is coming from the mathematical structure of the dilaton potential itself in Einstein frame. This is a very interesting feature from the point of view of renormalization group flow in quantum field theory as the effective field dependent coupling in Einstein frame captures the effect of field flow or more precisely the behaviour of the effective coupling in all energy scales. In this context instead of solving directly the renormalization group equation 1 for the effective coupling we solve the dynamical equations for the fields (inflation and dilaton) and the effective coupling for power law and exponential attractors and give an analytical expression for the field dependent coupling at any arbitrary energy scale. Due to the similarities in the technical structures in both of the techniques here one can arrive at the conclusion that in cosmology solving a dynamical attractor problem in presence of effective coupling in Einstein frame mimics the role of solving renormalization group equation in the context of quantum field theory. Thus due to the exponential suppression in the effective coupling in Einstein frame it is naturally expected from the prescribed framework that for suitable choices of the model parameters soft cosmological constraints can be obtained [98, 99]. As in this prescribed framework dilaton exponential coupling plays very significant role, one can ask a very crucial question about its theoretical origin. Obviously there are various sources exist from which one can derive exponential effective couplings or more precisely the effective potentials for dilaton. These possibilities are appended bellow: • Source I: One of the source for dilaton exponential potential is string theory, which is appearing in the Category I. Specifically, superstring theory and low energy supergravity models are the theoretical possibilities in string theory [100–109] where dilaton exponential potential is appearing in the gravity part of the action in Jordan frame and 1

Here by renormalization group equation we want to mean Calan Symazik equations.

–4–

Scale free 𝑹𝟐 gravity + Higgsotic scalar field (in Jordan frame) Two,three and four point (using δN) function for inflation +Reheating constraints

Conformal Transformation

Attractor phase Power law (stable) attractor

Consistent with observation (Planck and other joint data)

Einstein gravity+ Higgsotic dilaton coupled two-field theory (in Einstein frame)

Dynamical Attractor solutions Exponential (unstable) attractor

Stability from String Theory, non-minimal coupling and contribution from mass

Consistent with reheating and late time acceleration

Figure 1. Diagrammatic representation of attractor phase of soft Higgsotic inflation. In this representative diagram we have shown the steps followed during the computation.

after conformal transformation in Einstein frame such dilaton effective potential is coupled with the matter sector. • Source II: Another possible source of dilaton exponential potential is coming from modified gravity theory framework such as, f (R) gravity [78–81], f (φ)f (R) gravity [84–87] and Jordan-Brans-Dicke theory [94, 95] in Jordan frame, which are appearing in the Category II (1& 3) and Category III. After transforming the theory in the Einstein frame via conformal transformation one can derive dilaton exponential potential. In fig (1), fig (2) and fig (3), we have shown the diagrammatic representation of attractor and non-attractor phase of soft Higgsotic inflation. In these representative diagrams we have shown the steps followed during the computation. In this work we have addressed the following important points through which it is possible to understand the underlying cosmological consequences from the proposed setup. These issues are:

–5–

Scale free 𝑹𝟐 gravity + Higgsotic scalar field (in Jordan frame) Two, three and four point (using Schwinger Keldysh/In-In) function for inflation +Reheating constraints

Conformal Transformation

Non attractor phase

Consistent with observation (Planck and other joint data)

Einstein gravity+ Higgsotic dilaton coupled two-field theory (in Einstein frame)

Assume dilaton is heavy field Freezing dilaton at UV cut-off of Effective theory

Consistent with reheating and late time acceleration

Dynamical non attractor solution

Figure 2. Diagrammatic representation of non-attractor phase of soft Higgsotic inflation. In this representative diagram we have shown the steps followed during the computation.

• Transition from scale free gravity model to (Planck) scale dependent gravity model have discussed via conformal transformation in the metric. • Specific role of inflaton φ and an additional scalar field Ψ have discussed in the attractor and non-attractor regime of inflation. • Explicit calculation of δN formalism is presented by considering the effect up to second order perturbation in the solution of the field equation in attractor regime. • Deviation in the consistency relation between the non-Gaussian amplitude for four point and three point scalar correlation function aka Suyama Yamaguchi relation is presented to explicitly show the consequences from attractor and non-attractor phase. • Additionally, new sets of consistency relations in terms of tensor-to-scalar ratio, scalar and tensor spectral tilt, running and running of running of the tilt are presented in attractor (two field case but the solution of the fields are connected with each other)

–6–

Dynamical dilaton

Attractor phase

Inflation from soft Higgsotic sector coupled with dilaton (in Einstein frame)

Fixed dilaton

Non attractor phase

Old consistency relation for NG+ New consistency relation for PGW

New consistency relations for NG+ PGW for two attractors

Figure 3. Diagrammatic representation of attractor and non-attractor dynamical phase of soft Higgsotic inflation which is coupled with dilaton in Einstein frame.

and non-attractor (single field case where the additional dilaton field is heavy and fixed in a Planckian field value) phase of inflation to explicitly show the deviation from the results obtained from canonical single field slow roll model. • Numerical bound on the Higgsotic coupling and parameter of the scale free theory of gravity is given by considering all the constraints obtained from Planck 2015 data on inflationary observables and reheating. • Detailed numerical estimations are given for all the inflationary observables for attractor and non-attractor phase of inflation which confronts well Planck 2015 data. Additionally, constraints on reheating is also presented for attractor and non-attractor phase. • Bulk interpretation are given explicitly in terms of S, T and U chanel contribution for all the individual terms obtained from four point correlation function computation.

–7–

• Scale dependent or more precisely the field dependent behaviour of the non-minimal coupling between inflaton field and additional dilaton field obtained through conformal transformation in Einstein frame is computed for power law and exponential type of attractor solution. Further we have presented the expression for the couplings in terms of the number of e-foldings which lying within, 50 < Ncmb < 70. • Three possible theoretical proposals have presented to overcome the tachyonic instability [110–114] at the time of late time acceleration in Jordan frame and due to this fact the structure of the effective potentials changes in Einstein frame as well. These proposals are inspired from: – I. Non-BPS D-brane in superstring theory [22, 115–120], – II. An alternative situation where we switch on the effects of additional quadratic mass term in the effective potential, – III. Also we have considered a third option where we switch on the effect of non-minimal coupling between scale free αR2 gravity and the inflaton field. Now before going to the further technical details let us clearly mention the underlying assumptions to understand the background physical setup of this paper: 1. We have restricted our analysis up to monomial φ4 model and due to the structural similarity with Higgs potential at the scale of inflation we have identified monomial φ4 model as Higgsotic model in the present context. 2. To investigate the role of scale free theory of gravity, as an example we have used αR2 gravity. But the present analysis can be generalized to any class of f (R) gravity models. 3. In the matter sector we allow only simplest canonical kinetic term which are minimally coupled with αR2 gravity sector. For such canonical slow roll models the effective sound speed cS = 1. But for more completeness one can consider most generalized version of P (X, φ) models, where 1 X = − g µν ∂µ φ∂ν φ 2

(1.5)

and the effective sound speed cS < 1 for such models. For an example one can consider following structure [50, 96]: P (X, φ) = −

1 p 1 1 − 2Xf (φ) + − V (φ), f (φ) f (φ)

(1.6)

which is exactly similar to DBI model. But here one can implement our effective Higgsotic models in V (φ) instead of choosing the fixed structure of the DBI potential in UV and IR regime. Here one can choose [50]: f (φ) ≈

–8–

g , φ4

(1.7)

which is known as throat factor in string theory. In string theory g is the parameter which depends on the flux number. But other choices for f (φ) are also allowed for general class of P (X, φ) theories which follows the above structure. Similarly one can consider the following structure of P (X, φ) which are given for tachyon and Gtachyon models given by [22, 121]: For Tachyon : For GTachyon :

p P (X, φ) = −V (φ) 1 − 2Xα0 ,   0 q P (X, φ) = −V (φ) 1 − 2Xα

(1.8) (1/2 < q < 2), (1.9)

0

where α is the Regge slope. Here we consider the most simplest canonical form, P (X, φ) = X − V (φ),

(1.10)

where V (φ) is the effective potential for monomial φ4 model considered here for our computation. 4. As a choice of initial condition or precisely as the choice of vaccum state we restrict our analysis using Bunch Davies vacuum. If we relax this assumption, then one can generalize the results for α vacua as well. 5. During our computation we have restricted upto the minimal interaction between the αR2 gravity and matter sector. Here one can consider the possibility of non-minimal interaction between αR2 gravity and matter sector. 6. During the implementation of In-In formalism [2] to compute three and four point correlation function we have use the fact that the additional dilaton field Ψ is fixed at Planckian field value to get the non attrator behavior of the present setup. One can relax this assumption and can redo the analysis of In-In formalism to compute three and four point correlation function without freezing the dilaton field Ψ and also use the attractor behaviour of the model to simplify the results. 7. During the computation of correlation functions using semi classical method, via δN formalism [22, 122–126], we have restricted up to second order contributions in the solution of the field equation in FLRW background and also neglected the contributions from the back reaction for all type of effective Higgsotic models derived in Einstein frame. For more completeness, one can relax these assumptions and redo the analysis by taking care of all such contributions. 8. In this work we have neglected the contribution from the loop effects (radiative corrections) in all of the effective Higgsotic interactions (specifically in the self couplings) derived in the Einstein frame. After switching on all such effects one can investigate the numerical contribution of such terms and comment on the effects of such terms in precision cosmology measurement.

–9–

9. We have also neglected the interactions between gauge fields and Higgsotic scalar field in this paper. One can consider such interactions by breaking conformal invariance of the U (1) gauge field in presence of time dependent coupling f (φ(η)) to study the features of primordial magnetic field through inflationary magnetogenesis [127–129]. The plan of this paper is as follows: • In sec 2, we start our discussion with a specific class of modified theory of gravity, aka f (R) gravity where a single matter (scalar field) is minimally coupled with the gravity sector and contains only canonical kinetic term. To build effective potential from this toy setup of modified gravity in 4D we choose a simplest example, f (R) = αR2 , where α is a dimensionless parameter and describes a scale free modified gravity setup. Next in the matter sector we choose a very simple monomial model of potential, V (φ) = λ4 φ4 . This type of potential can be treated as a Higgs like potential as at the scale of inflation specific contribution from the VEV of Higgs almost negligible and consequently one can recast the Higgs potential in the monomial form, V (φ) ≈ λ4 φ4 . Here the only difference is in case of Higgs, λ is Yukawa coupling and in case of general monomial model, λ is a free parameter of the theory. • Further, in sec 3, we provide the field equations in Jordan frame written in spatially flat FLRW background in presence of αR2 scale free gravity. To simplify the structure of the derived equations and to analyze the cosmological solutions in detail next we perform a conformal transformation in the metric to the Einstein frame and introduce a new dilaton field. In this new frame we have scale dependent Einstein gravity with a scale Mp , which is the reduced Planck mass. Next, we derive the field equations in Einsein frame with spatially flat FLRW background and try to solve them for two dynamical attractor features as given by-Power law solution, and Exponential solution. However, the second case give rise to tachyonic behaviour which can be resolved in various ways. To solve this problem we consider three physical situation which are inspired from- I. non-BPS D-brane in superstring theory, II. via switching on the effect of mass like quadratic term in the effective potential and III. by introducing a non-minimal coupling between matter and αR2 gravity sector. • Next, in sec 4, using two dynamical attractors, Power law and Exponential solution we study the cosmological constraints in presence of two fields. We study the constraints from primordial density perturbation, by deriving the expressions for two point function and the present inflationary observables in sec 4.2. Further, we repeat the analysis for tensor modes and also comment on the future observables-amplitude of the tensor fluctuations and tensor-to-scalar ratio in sec 4.3. Here we also provide a modified formula for the field excursion in terms of tensor-to-scalar ratio, scale of inflation and the number of e-foldings. Further, we confront our model with Planck 2015 data and constrain the parameters of the model. Additionally, in sec 4.4, we study the constraint for reheating temperature and using this we have further constrain the model parameters. Finally, in sec 5.1 and sec 5.2, we derive the expression for inflaton φ and the non minimal coupling parameter λ(Ψ) at horizon crossing, during reheating and at the onset of inflation for two above mentioned dynamical cosmological attractors.

– 10 –

• Further, in sec 6, we have explored the cosmological solutions beyond attractor regime. Here we restrict ourselves at spatially flat FLRW background and made cosmological predictions from this setup in sec 7.1. To serve this purpose we have used ADM formalism using which we perturbed the FLRW metric and choose a preferred gauge and compute two point function and associated present inflationary observables using Bunch Davies initial condition for scalar fluctuations in sec 7.2.1 and sec 7.2.2. Further, in sec 7.3.1 and sec 7.3.2, we repeat the procedure for tensor fluctuations as well where we have compute two point function and the associated future observables. In the non-attractor regime, we also derive a modified version of field excursion formula in terms of tensor-to-scalar ratio, scale of inflation and the number of e-foldings. We also derive few sets of consistency relations in this context which are different from the usual single field slow roll models. Further, in sec 7.4, we derive the constraints on reheating temperature in terms of inflationary observables and number of e-foldings. • Next, in sec 8.1.1 and sec 8.1.2, as a future probe we compute the expression for three point function and as well as the bispectrum of scalar fluctuations using In-In formalism for non attractor case and δN formalism for the attractor case. Further, we derive the result for non-Gaussian amplitude fNlocL for equilateral limit and squeezed limit triangular shape configuration. For more completeness we give a bulk (gravity) interpretation of each of the momentum dependent terms appearing in the expression for the three point scalar correlation function in terms of S, T and U channel contributions. Further, for the consistency check we freeze the additional field Ψ in Planck scale and redo the analysis of δN formalism. Here we show that the expression for the three point non-Gaussian amplititude is slightly different as expected for single field case. Further, in sec 8.1.1 and sec 8.1.2, we compare the results obtained from In-In formalism and δN formalism for the non attractor phase, where the additional field Ψ is fixed in Planck scale. Finally, we give a theoretical bound on the scalar three point non-Gaussian amplitude computed from equilateral and squeezed limit configurations by constraining the parameter space of the prescribed theory. • Finally, in sec 8.2.1 and sec 8.2.2, as an additional future probe we have also computed the expression for four point function and as well as the trispectrum of scalar fluctuations using In-In formalism for non attractor case and δN formalism for the atloc loc tractor case. Further, we derive the results for non-Gaussian amplitude gN L and τN L for equilateral limit, counter collinier or folded kite limit and squeezed limit shape configuration from In-In formalism. For more completeness we give a bulk (gravity) interpretation of each of the momentum dependent terms appearing in the expression for the four point scalar correlation function in terms of S, T and U channel contributions. In the attractor phase following the prescription of δN formalism we also loc loc derive the expressions for the four point non-Gaussian amplitude gN L and τN L . Next we have shown that the consistency relation connecting three and four point nonGaussian amplitude aka Suyama Yamaguchi relation is modified in attractor phase and further given an estimate of the amount of deviation for super Planckian and sub-Planckian field range. Further, for the consistency check we freeze the additional field Ψ in Planck scale and redo the analysis of δN formalism. Here we show that

– 11 –

the four point non-Gaussian amplitude is slightly different as expected for single field case. Finally, we give a theoretical bound on the scalar four point non-Gaussian amplitude computed from equilateral, folded kite and squeezed limit configurations by constraining the parameter space of the prescribed theory.

2

Model building from scale free gravity

To described the theoretical setup let us start with the total action of f (R) gravity coupled minimally along with a scalar inflaton field φ, as given by:   Z √ g µν 4 (∂µ φ) (∂ν φ) − V (φ) (2.1) S= d x −g f (R) − 2 where in general f (R) can be arbitrary function of Ricci scalar R. For an example one can choose a generic form given by [130]: f (R) =

∞ X

an R n ,

(2.2)

n=1

where an ∀n are the expansion coefficients for the above mentioned generic expansion. Here one can note down the following features of this generic choice of the expansion: 1. If we set a1 = Mp2 /2, an = 0∀n > 1,

(2.3) (2.4)

then one can get back well known Einstein Hilbert action (GR) in Joradn frame as given by: Mp2 f (R) = R. (2.5) 2 In this particular case Jordan frame and Einstein frame is exactly same because the conformal factor for the frame transformation is unity. This directly implies that no dilaton potential is appearing due to the frame transformation from Jordan to Einstein frame. But since in this paper we are specifically interested in the effects of modified gravity sector, the higher powers of R is more significant in the above mentioned generic expansion of f (R) gravity. 2. If we set, a1 = a = Mp2 /2, a2 = b = α, an = 0∀n > 2,

(2.6) (2.7) (2.8)

then one can get back the specific structure of the very well known Starobinsky model as given by: Mp2 f (R) = aR + bR2 = R + αR2 . (2.9) 2

– 12 –

Here one can treat the αR2 term as an additional quantum correction to the Einstein gravity. 3. One can also set, a1 = a = Mp2 /2, an = α∀n ≥ 2,

(2.10) (2.11)

then one can get back the following specific structure: f (R) =

Mp2 R + αRn , 2

(2.12)

which describes the situation where the Einstein Hilbert gravity action is modified by the monomial powers of R. Here also one can treat the αRn term as an additional quantum correction to the Einstein gravity. 4. In our computation we set, a1 = a = 0, a2 = b = α, an = 0∀n > 2,

(2.13) (2.14) (2.15)

which is known as scale free gravity in Jordan frame as given by: f (R) = αR2 ,

(2.16)

where α is a dimensionless scale free coefficient. For this type of theory if er perform the conformal transformation from Jordan to Einstein frame then it will induce a constant term in the effective potential and can be interpreted as the 4D cosmological constant using which one can fix the scale of the theory for early and late universe. But in our computation we introduce an additional scalar field in the action in Jordan frame, which we identified to be the inflaton. After conformal transformation in Einstein frame we get an effective potential which is coming from the interaction between the dilaton exponential potential and the inflationary potential as appearing in Jordan frame. We will show that here the two fields- dilaton and inflaton forms dynamical attractors using which one can very easily solve this two field complicated model in the context of cosmology. Next we will discuss about the structure of the inflational as appearing in Eq (2.1). Generically in 4D effective theory the effective potential can be expressed as: V (φ) =

Vren (φ) | {z }

Renormalizable part

+

∞ X

φδ Jδ (g) δ−4 Mp |δ=5 {z }

=

∞ X δ=0

Cδ (g)

φδ , Mpδ−4

(2.17)

Non−renormalizable part

where Jδ (g) and Cδ (g) are the Wilson coeeficients in effective theory. Here g stands for the scale of theory and the dependences of the Wilson coefficients on the scale can be

– 13 –

exactly computed for a full UV complete theory using renormalization group equations. In this paper the similar scale dependence on the couplings we will calculate using dynamical attractor method in Einstein frame, which exactly mimics the role of solving renormalization group equations in the context of cosmology. As written here, the total effective potential is made by renormalizable (relevant operators) and non-renormalizable (irrelevant operators) part, which can be obtained by heavy degrees of freedom from a known UV complete theory. In our computation we just concentrate on the renormalizable part of the action, which can be recast as: 4 X φδ (2.18) V (φ) = Cδ (g) δ−4 , Mp δ=0 Next to get the Higgslike monomial structure of the potential we set C3 (g) = 0,

(2.19)

as in this paper our prime motivation is to look into only Higgsotic potentials. Consequently we get: V (φ) = C0 + C2 (g)Mp2 φ2 + C4 (g)φ4 . (2.20) To get the Higgsotic structure of the potential one should set, λ 4 v , 4 λ C2 (g) = − v 2 , 2 λ C4 (g) = . 4

C0 (g) =

(2.21) (2.22) (2.23)

Here v is the VEV of the field φ. Consequently, one can write the potential in the following simplified form: λ V (φ) = (φ2 − v 2 )2 . (2.24) 4 Now we consider a situation where scale of inflation as well as the field value are very very larger than the VEV of the field. This assumption is pretty consistent with inflation with Higgs field. Consequently, in our case the final simplified monomial form of the Higgsotic potential is given by: V (φ) =

λ 4 φ. 4

(2.25)

Further varying Eq. (2.1) with respect to the metric and using Eq (2.16) and Eq (2.25) eqn of motion (modified Einstein eqn) for the αR2 scale free gravity can be written as: h gµν 2 i ˜ Gµν : = α 2 {Rµν + (gµν 2 − ∇µ ∇ν )} R − R = Tµν , 2 ˜ µν : = α [{Rµν + 2 (gµν 2 − ∇µ ∇ν )} + Gµν ] R = Tµν =⇒ G (2.26) where the D’Alembertian operator is defined as:
√ 1 2 = g αβ ∇α ∇β = g αβ ∇α ∂β = √ ∂α −gg αβ ∂β −g

– 14 –

(2.27)

and the energy-momentum stress tensor can be expressed as:   √ 2 δ ( −gLM ) λ 4 1 αβ Tµν = − √ g ∂α φ∂β φ + φ = ∂µ φ∂ν φ − gµν −g δg µν 2 4

(2.28)

Here it is important to note that the Einstein tensor is defined as: g µν R. 2

(2.29)

T 6α

(2.30)

Gµν := Rµν − Now after taking the trace of Eq. (2.26) we get: R2R =

where the trace of energy momentum tensor is characterized by the symbol T = Tµµ .

(2.31)

In this modified gravity picture we have: ˜ µν = 4α [∇µ , 2] R 6= 0 ∇µ G

(2.32)

where we use the following result: ∇µ Rµν =

g µν µ ∇ R 2

(2.33)

dirctly follows from the Bianchi identity ∇µ Gµν = 0.

(2.34)

Now varying Eq (2.1) with respect to the field φ we get the following eqn of motion in curved spacetime:
√ 1 0 2φ = −V (φ) = −λφ3 =⇒ √ ∂α −gg αβ ∂β φ = −λφ3 . −g

(2.35)

Further assuming the flat (k = 0) FLRW background metric the Friedmann Equations can be written from Eq. (2.26) as: !  2 ˙ a ˙ ρ R R φ H2 = = + − H, (2.36) a 6αR 2 R !    2 ˙ ¨ R R a ¨ a ˙ p R φ 2H˙ + 3H 2 = 2 + =− −2 H− + (2.37) a a 2αR R R 4 where we have assumed the energy-momentum tensor can be described by perfect fluid as: Tνµ = diag (−ρφ , pφ , pφ , pφ )

– 15 –

(2.38)

where the energy density ρφ and the pressure density pφ can be expressed for scalar field φ as: φ˙ 2 λ 4 + φ, 2 4 2 ˙ φ λ pφ = − φ4 . 2 4

ρφ =

(2.39) (2.40)

Similarly the field eqn for the scalar field φ in the flat (k = 0) FLRW background can be recast as: φ¨ + 3H φ˙ + λφ3 = 0 In the flat (k = 0) FLRW background we have the following expressions:   R = 6 H˙ + 2H 2 ,   ˙ ¨ ˙ R = 6 H + 4H H ,  ...  ¨=6 H ¨ . R + 4H˙ 2 + 4H H

(2.41)

(2.42) (2.43) (2.44)

Substituting these results in Eq (2.36) and Eq (10.4) the Friedmann eqns can be recast in the Jordan frame as:   ρφ ¨ ˙ 2H H + 3H H − H˙ 2 = , (2.45) 18α   ... pφ ¨ +H 9H˙ H˙ + H 2 + 6H H =− . (2.46) 6α In the slow-roll regime (φ˙ 2 /2 0 with effective potential: √

2 2 Ψ ˜ (φ, Ψ) ≈ λ(Ψ) φ4 = λ e− √3 Mp φ4 W 4 4

(for Case I).

(3.29)

Consequently the field equations can be recast as: √

˜ dφ + λe− √3 Mp φ3 = 0, 3H dt˜ √ 4 2 2 Ψ ˜ dΨ − √λφ e− √3 Mp = 0, 3H dt˜ 6Mp √ λ − 2√32 MΨp 4 2 ˜ H = e φ. 12Mp2 2

2 Ψ

(3.30) (3.31) (3.32)

This is the case where the cosmological constant V0 or more precisely the parameter α will not appear in the final solution. The cosmological solutions of Eq. (3.30-3.32) are given

– 22 –

by 4 : Case I Ψ − Ψ0 ≈ = = a≈ N (φ) − N (φ0 ) =

√   2 2Mp a √ ln a0 3 √   3M t √ p ln t0 2
9 − √ φ2 − φ20 , 2 6Mp  3/4 t a0 , t0 Z φ 1 V˜ (φ) dφ Mp2 φ0 ∂φ V˜ (φ)

(3.33) (3.34)

φ2 − φ20 8Mp2   a 5 ≈ − ln 9 a 0  5 t = − ln , 12 t0 =

3.2

(3.35)

Case II: Exponential solution

We consider small α, large V0 with λ < 0 with effective potential: ˜ (φ, Ψ) ≈ W

Mp4 λ(Ψ) 4 Mp4 λ − 2√√2 MΨ 4 + φ = − e 3 pφ 8α 4 8α 4

(for Case II).

(3.36)

Here to aviod any confusion we have taken out the signature of the coupling λ outside in the expression for the effective potential for λ < 0 case. Finally the field equations can be expressed as: √ 2 2 Ψ

˜ dφ − λe− √3 Mp φ3 = 0 3H dt˜ √ dΨ λφ4 − 2√32 MΨp ˜ √ 3H + e = 0, dt˜ 6Mp √ 2 2 Ψ M2 ˜ 2 = p − λ e− √3 Mp φ4 , H 24α 12Mp2 The cosmological solutions of Eq. (3.37-3.39) are given by: 4

Throughout the paper the subscript ‘00 is used to describe the inflationary epoch.

– 23 –

(3.37) (3.38) (3.39)

Case II Ψ − Ψ0 ≈ = = a≈ N (φ) − N (φ0 ) = = =

≈ ≈

√   2 2Mp a √ ln a0 3 2 Mp √ (t − t0 ) 3 α
1 − √ φ2 − φ20 , 2 6Mp   Mp a0 exp √ (t − t0 ) , 2 6α Z φ 1 V˜ (φ) dφ Mp2 φ0 ∂φ V˜ (φ)   Mp2 1 1 − − 16αλ(Ψ) φ20 φ2   2 Mp 1 1 −   − 2 2 8M p 16αλ(Ψ)φ0 1 − φ2 ln aa0 0   Mp4 a ln 4 2αλ(Ψ)φ0 a0 5 Mp √ (t − t0 ) 4α3/2 λ(Ψ) 6φ40

(3.40) (3.41)

(3.42)

This is the specific case where the cosmological constant is explicitly appearing in the potential. To end inflation we need to fulfill an extra requirement that λ < 0 and this will finally led to massless tachyonic solution. In In fig. 4(a) and fig. 4(b) we have shown the behaviour of the inflationary potential for the following two cases: 1. V0 ≈ 0 and λ > 0, 2. V0 6= 0 and λ < 0. Fig. 4(a) implies that the inflaton rolls down from a large field value and inflation ends at φf ≈ 1.09 Mp .

(3.43)

Also the potential has a global minimum at φ = 0,

(3.44)

around which field is start to oscillate and take part in reheating. On the other hand in fig. (4(b)) the inflaton field rolls down from a small field value and the inflation ends at the field value φf = 2.88 α1/8 Mp , (3.45)

– 24 –

Potential for Case I

VHΦLHin M p4L

1. ´ 10-8 8. ´ 10-9 6. ´ 10-9 4. ´ 10-9 Φ f = 1.09 M p 2. ´ 10-9

0 0.2

0.4

Φ Hin M p L

0.6

0.8

1.0

1.2

1.4

(a) Case I : Power-law behavior.

ΑVHΦLHin M p4L

Potential for Case II

0.15

0.10 Φ f ‘ Α18 = 2.88 M p 0.05

0.00 0.0

0.5

1.0

1.5

2.0

֐Α18 Hin M p L

2.5

3.0

3.5

(b) Case II : Tachyonic behaviour.

Figure 4. Behaviour of the inflationary potential for 4(a) V0 ≈ 0 and λ > 0 (Case I) and 4(b) V0 6= 0 and λ < 0 (Case II). In fig. 4(a) the inflaton rolls down from a large field value and inflation ends at φf ≈ 1.09 Mp . On the other hand in fig. (4(b)) the inflaton field rolls down from a small field value and the inflation ends at the field value φf = 2.88 α1/8 Mp , where the lower bound on the parameter α is: α ≥ 2.51 × 107 , which is consistent with Planck 2015 data [38–40].

– 25 –

where the lower bound on the parameter α is: α ≥ 2.51 × 107 ,

(3.46)

which is consistent with Planck 2015 data [38–40]. Within this prescription it is possible to completely destroy the effect of cosmological constant at the end of inflationary epoch. But within this setup to explain the particle production during reheating and also explain the late time acceleration of our universe we need additional features in the total effective potential in scale free αR2 gravity theory. It is general notion that , the reheating phenomena can only be explained if the effective potential have a local minimum and a remnant contribution (vacuum energy or equivalent to cosmological constant) in the total effective potential finally produce the observed energy density at the present epoch as given by 5 : ρnow ≈ 10−47 GeV4 ,

(3.48)

which is necessarily required to explain the late time acceleration of the universe. Now here one can ask a very relevant question that if we include some additional features to the effective Higgsotic potential, which is also can be treated as a massless tachyonic potential, then how one can interpret the justifiability as well as the behaviour of effective field theory framework around the minimum of the potential which will significantly control the dynamical behaviour in the context of cosmology. The most probable answer to this very significant question can be described in various ways. In the present context to get a stable minimum (vacuum) of the derived effective Higgsotic potential in Einstein frame here we discuss few physical possibilities which are appended in following points: • Choice I: The first possible solution of the mentioned problem is motivated from non-BPS Dbrane in superstring theory. In this prescription the effective potential have a pair of global extrima at the field value: φextrema = φ = ±φV

(3.49)

for the non-BPS D-brane within the framework of superstring theory [22, 115–120]. Additionally it is important to note that, here a one parameter (γ) family of global extrima exists at the field value: φ = φV eiγ (3.50) for the brane-antibrane system. Here φV is identified to be the field value where reheating phenomena occurs. At this specified field value of the minimum the brane tension of the D-brane configuration which is exactly canceled by the negative contribution as appearing in the expression for effective potential in Einstein frame. Here for the sake of simplicity we little bit relax the constraints as appearing exactly in Case II. To explore the behaviour of the derived effective potential here we have 5

For Einstein gravity one can write the observed energy density at the present epoch in the following form: ρnow ≈ 3H02 Mp2 , (3.47) where H0 is the Hubble parameter at the present epoch.

– 26 –

allowed both of the signatures of the coupling parameter λ. This directly implies the following constraint condition: √

λ − 2√ 2 Ψ (for λ < 0), (3.51) − e 3 Mp φ4V + Θp = 0 4 √ λ − 2√32 MΨp 4 (for λ > 0), (3.52) e φ V + Θp = 0 4 where Θp is the above mentioned additional contribution and in the context of superstring theory this is given by:  √    2(2π)−p gs−1 for non-BPS Dp-brane Θp = (3.53)   −p −1 ¯ brane pair.  2(2π) gs for non-BPS Dp-Dp with string coupling constant gs . This implies that the inflaton energy density vanishes at the minimum of the tachyon type of derived effective potential and in this connection the remnant energy contribution is given by: Mp4 (3.54) 8α which serves the explicit role of cosmological constant in the context of late time acceleration of the universe. In this case concedering the additional contribution as mentioned above the total effective potential can be modified as: V0 =


Mp4 λ − 2√√2 MΨ − e 3 p φ4 − φ4V (for λ < 0), (3.55) 8α 4
Mp4 λ − 2√√2 MΨ ˜ + e 3 p φ4 − φ4V v2 : W (φ, Ψ) = (for λ > 0). (3.56) 8α 4 Here to aviod any confusion we have taken out the signature of the coupling λ outside in the expression for the effective potential for λ < 0 case. v1 :

˜ (φ, Ψ) = W

In the present context the field equations can be expressed as: 2

For v1 :

√

˜ dφ − λe− √3 Mp φ3 = 0, 3H dt˜ √ 4 dΨ λ (φ − φ4V ) − 2√32 MΨp ˜ 3H + √ e = 0, dt˜ 6Mp √ 2 2 Ψ
M2 ˜ 2 = p − λ e− √3 Mp φ4 − φ4 . H V 24α 12Mp2 2 Ψ

(3.57) (3.58) (3.59)

and 2

For v2 : ˜ 3H ˜2 = H

dΨ dt˜

Mp2 24α

√

˜ dφ + λe− √3 Mp φ3 = 0, 3H dt˜ √ λ (φ4 − φ4V ) − 2√32 MΨp − √ e = 0, 6Mp √
λ − 2√32 MΨp 4 + e φ − φ4V . 2 12Mp

– 27 –

2 Ψ

(3.60) (3.61) (3.62)

The solutions of Eq. (3.57-3.62) are given by: Choice I(v1) √   Mp2 2 2Mp a = √ (t − t0 ) Ψ − Ψ0 ≈ √ ln a0 3 α 3   
1 1 1 2 2 4 , φ − φ0 + φV =− √ − φ2 φ20 2 6Mp   Mp a ≈ a0 exp √ (t − t0 ) , 2 6α Z φ 1 V˜ (φ) N (φ) − N (φ0 ) = 2 dφ Mp φ0 ∂φ V˜ (φ)    Mp2 φ4V 1 1 + − ≈− 16αλ(Ψ) 8Mp2 φ20 φ2     Mp4 φ4V a + 2 ln , ≈ 2αλ(Ψ) φ0 a0

Choice I(v2) Ψ − Ψ0 ≈ = = a≈ N (φ) − N (φ0 ) = ≈ ≈

√   a 2 2Mp √ ln a0 3 2 Mp √ (t − t0 ) 3 α   
1 1 1 2 2 4 − , − √ φ − φ0 + φV φ2 φ20 2 6Mp   Mp a0 exp √ (t − t0 ) , 2 6α Z φ 1 V˜ (φ) dφ Mp2 φ0 ∂φ V˜ (φ)    Mp2 φ4V 1 1 − − 16αλ(Ψ) 8Mp2 φ20 φ2     Mp4 φ4V a − 2 ln , 2αλ(Ψ) φ0 a0

(3.63) (3.64)

(3.65)

(3.66) (3.67)

(3.68)

In fig. (5(a)) and Fig. (5(b)) we have shown the variation of the potential with respect to the inflaton field for both the cases. For fig. (5(a)) the inflaton can roll down in both ways. Firstly, this can roll down to a global minimum at the field value: φV = 0

(3.69)

from higher to lower field value and take part in particle production procedure during reheating. On the other hand, in the same picture the inflaton can also roll down to

– 28 –

higher to lower field value in a opposite fashion. In that case the inflaton goes up to the zero energy level of the effective potential and cannot able to explain the thermal history of the early universe in a proper sense. It is also important to note that, in this picture the position of the maximum of the effective potential in Einstein frame is at around the field value: φV = 0.42 Mp . (3.70) Fig. (5(b)) is the case where the signature of the coupling λ is positive. Also the behavior of the effective potential is completely opposite compared to the situation arising in fig. (5(a)). In this case the inflaton field can be able to roll down to higher to lower field value or lower to higher field value. But in both the cases the inflaton field settle down to a local minimum at φmin = φV = 0.42 Mp

(3.71)

and within the vicinity of this point it will produce particles via reheating. In both of the situations the lower bound on the parameter α is fixed at: α ≥ 2.51 × 107 ,

(3.72)

which is perfectly consistent with Planck 2015 data [38–40]. • Choice II: It is possible to explain the reheating as well as the lite time cosmic acceleration once we switch on the effect of mass like quadratic term in the effective potential. In such a case the modified effective potential in Einstein frame can be written as:  2  √ Mp4 m λ − 2√ 2 Ψ c 2 4 ˜ (φ, Ψ) = + φ − φ e 3 Mp (for m2c > 0, λ < 0), (3.73) v1 : W 8α 2 4   √ Mp4 m2c 2 λ 4 − 2√32 MΨp ˜ v2 : W (φ, Ψ) = − φ − φ e (for m2c < 0, λ > 0). (3.74) 8α 2 4 Here to avoid any confusion we have taken out the signature of the coupling λ outside in the expression for the effective potential for λ < 0 case. Here during inflation the inflaton field satisfies the constraint s 2 |mc |. φ >> |λ| After inflation when reheating starts then the field satisfies s 2 φ 0.

Figure 5. Behaviour of the modified effective potential for case II with 5(a) Choice I(v1) : V0 6= 0, λ < 0, 5(b) Choice I(v2) : V0 6= 0, λ > 0, where Mp = 2.43 × 1018 GeV .

– 30 –

the remnant energy V0 =

Mp4 8α

(3.78)

serves the purpose of explaining the late time acceleration of the universe. In the present context the field equations can be expressed as: For v1 :


− 2√√2 Ψ dφ 3 2 ˜ + m φ − λφ e 3 Mp = 0, 3H dt˜  c √ m2 2 λ 4  √ 2 2 2c φ − 4 φ dΨ − 2√ 2 Ψ ˜ √ 3H − e 3 Mp = 0, dt˜ 3Mp  2  mc 2 λ 4 √ 2 φ − φ Ψ Mp 2 4 − 2√ 2 M 2 ˜ + H = e 3 p. 2 24α 3Mp

(3.79) (3.80)

(3.81)

and √

For v2 :


− 2√ 2 Ψ dφ − m2c φ − λφ3 e 3 Mp = 0, dt˜  √ m2 2 λ 4  √ 2 2 2c φ − 4 φ Ψ dΨ − 2√ 2 M ˜ √ + e 3 p = 0, 3H dt˜ 3Mp  2  mc 2 λ 4 √ 2 φ − 4φ Ψ Mp 2 − 2√ 2 M 2 ˜ 3 p. − e H = 24α 3Mp2 ˜ 3H

(3.82) (3.83)

(3.84)

The solutions of Eq. (3.79-3.84) are given by: Choice II(v1) √   Mp2 2 2Mp a Ψ − Ψ0 ≈ √ ln = √ (t − t0 ) a0 3 α 3   2 
m2c 1 mc − λφ2 2 2 =− √ φ − φ0 + ln , (3.85) λ m2c − λφ20 2 6Mp   Mp a ≈ a0 exp √ (t − t0 ) , (3.86) 2 6α Z φ 1 V˜ (φ) N (φ) − N (φ0 ) = 2 dφ Mp φ0 ∂φ V˜ (φ)  2 2  Mp4 φ (mc − λφ20 ) = ln 16m2c (Ψ)α φ20 (m2c − λφ2 )  i  h 8Mp2 a 2 2 4 1 − ln (m − λφ ) 2 c 0 Mp a0 φ0 h  i  , (3.87) ≈ ln   2 2 8M 16mc (Ψ)α m2 − λφ2 1 − 2p ln a c

– 31 –

0

φ0

a0

Modified potential for Case II with choice IIHv1L

ΑVHΦLHin M p4L

0.18

0.16

0.14

0.12

0.10 0.0

0.5

1.0

1.5

2.0

2.5

֐Α18 Hin M p L

3.0

3.5

(a) Choice II(v1) : Modified potential with mass m2c > 0, λ < 0.

Modified potential for Case II with choice IIHv2L 0.18

ΑVHΦLHin M p4L

0.16 0.14 0.12 0.10 0.08 0.06 0.0

0.5

1.0

1.5

2.0

2.5

֐Α18 Hin M p L

3.0

3.5

(b) Choice II(v2) : Modified potential with mass m2c < 0, λ > 0.

Figure 6. Behaviour of the modified effective potential for case II with 6(a) Choice II(v1) : q 2 V0 6= 0, λ < 0, m2c > 0 and φ 0, m2c < 0 and q 2 φ 0, ξ = Mp−2 (red), 10−8 Mp−2 (blue), where Mp = 2.43 × 1018 GeV .

√

˜ 3H

dφ λφ (1 + ξφ2V ) (φ2 − φ2V ) − 2√32 MΨp + e = 0, dt˜ (1 + ξφ2 )3 2

˜ 3H

√

λ (φ2 − φ2V ) dΨ − 2√ 2 Ψ −√ e 3 Mp = 0, 2 dt˜ 6Mp (1 + ξφ2 ) ˜2 = H

2 √ λ Mp2 (φ2 − φ2V ) − 2√ 2 MΨ 4 + e 3 p. 24α 3Mp2 (1 + ξφ2 )2

– 34 –

(3.97) (3.98) (3.99)

The solutions of Eq. (3.97-3.99) are given by: Choice III Ψ − Ψ0 ≈ =

= a≈ N (φ) − N (φ0 ) = =

=

√   a 2 2Mp √ ln a0 3 2 Mp √ (t − t0 ) 3 α h  i
ξ 2 2 2 2 2 2 (φ − φ0 ) 1 + 2 (φ + φ0 − 2φV ) + 2φV ln φφ0 1 − √ , (3.100) (1 + ξφ2V ) 2 6Mp   Mp a0 exp √ (t − t0 ) , (3.101) 2 6α Z φ V˜ (φ) 1 dφ Mp2 φ0 ∂φ V˜ (φ)  2 2  Mp2 φ0 (φ − φ2V ) ln 16φ2V αλ(Ψ) (1 + ξφ2V ) φ2 (φ20 − φ2V )   i  h 8Mp2 a 2 2 2 1 − φ ln − φ 0 V Mp a0 φ20  , (3.102)  i ln  h 2 2 2 8M 16φV αλ(Ψ) (1 + ξφV ) 1 − 2p ln a (φ2 − φ2 ) φ0

a0

0

V

In fig. (3.2), we have shown the behavior of the effective potential with respect to inflaton field in presence of non-minimal coupling parameter, ξ = Mp−2

(3.103)

ξ = 10−8 Mp−2

(3.104)

and depicted by red and blue colored curves respectively. For both of the cases we have taken the self interacting coupling parameter λ > 0. Also it is important to mention here that, if we decrease the strength of the non-minimal coupling parameter then the effective potential become more steeper. For both the situations the inflaton field can roll-down from higher to lower or lower to higher field values and finally settle down to a local minimum at φV = Mp . (3.105)

4

Constraints on inflation with soft attractors

Here we require the following constraints to study inflationary paradigm in the attractor regime: 4.1

Number of e-foldings

To get sufficient amount of inflation from the proposed setup (for both the Case I and Case II) it necessarily requires:   af |N (φ0 ) − N (φf )| ≈ ln & 50 − 70. (4.1) a0

– 35 –

which is a necessary quantity that can able to solve horizon problem associated with standard big-bang cosmology. The subscripts ‘f’ and ‘0’ physically signify the final and initial values of the inflationary epoch. Further using Eq (3.35) and Eq (3.42) the field value at the end of inflation can be explicitly computed for the above mentioned two cases as:   2 1/2   φ 1 − 480Mp  for Case I   0 φ20 (4.2) φf ∼ φ0   . for Case II . h i    1 + 960αλ(Ψf )φ20 1/2 M2 p

Here it is important to mention the following facts: • For the Case I the expression for the field associated with the end of inflation φf is completely fixed by the value initial field value φ0 . Here no information for the field dependent coupling λ(ψf ) = λ(Ψ = Ψf ) is required for this case as the expression for φf is independent of the dilaton field dependent coupling. • For the Case II the expression for the field associated with the end of inflation φf is fixed by the value initial field value φ0 as well as by the field dependent coupling λ(ψf ) = λ(Ψ = Ψf ). 4.2 4.2.1

Primordial density perturbation Two point function

The next observational constraint comes from the imprints of density perturbations through scalar fluctuations. Such fluctuations in CMB map directly implies that 6 :   p δρ δρ < = AS ∼ 10−5 (4.3) ρ ρ cr measured on the horizon crossing scales, where δρ is the perturbation in the density ρ. Additionally it is important to note that, AS , represents the amplitude of the scalar power spectrum. Also in the present context for both the cases one can write:     δρ δρ σ = σ (4.4) ρ t1 ρ t2 where the parameter σ is the parameter in the present context, which can be expressed in terms of equation parameter as: σ = 1+ w=

2 , 3(1 + w)

(4.5)

p . ρ

6

Here one equivalent notation for the amplitude of the scalar perturbation used as which we have used in the non attracor case.

– 36 –

(4.6) √

Pcmb =

p

P(Ncmb )

It is important to note that, (t1 , t2 ) represent the times when the perturbation first left and re-entered the horizon, respectively. At time t1 , Eq (3.21) and Eq (3.22) perfectly hold good in the present context. On the other hand at time t = t2 the representative parameter σ take the value, σ = 3/2 and σ = 5/3 during radiation and matter dominated epoch respectively. For the potential dominated inflationary epoch, w ≈ −1 and consequently one can write the following constraint condition:      1 δρ δρ ≈ 1− . (4.7) ρ t2 σ ρ t1 Further using Eq (3.21) and Eq (3.22) and approximated equation of motion in slow-roll regime of fluctuation in the total energy density or equivalently in the scalar modes can be written as:     ˙ ˙ ˙ ˙ ˙ ˙ ˜ ˙ ˜ ˙ δρ = φδ φ + Ψδ Ψ − 3H φδφ + ΨδΨ ≈ −2H φδφ + ΨδΨ . (4.8) where we use the symbol as, ˙ ≡ d/dt˜ and one can write down the following simplified expressions: δ φ˙ ˙ δΨ δφ δΨ

≈ ≈ ≈ ≈

˜ Hδφ, ˜ HδΨ, ˜ H, ˜ H,

(4.9) (4.10) (4.11) (4.12)

and finally the fractional density contrast can be expressed as:       ˙ + |Ψ| ˙ ˜ 2 |φ| H δρ C = ˙2 ρ t2 φ˙ 2 + Ψ

(4.13)

t1

with the following constraint on the parameter C as given by: C ∼ O(1)

(4.14)

and it serves the purpose of a normalization constant in this context. Then we get the two physically acceptable situations for both of the cases which can be written as: p ˜h ˜2 δρ H W ˙ < |Ψ| ˙ ⇒ |φ| ≈ ≈ √ , (4.15) Region I : ˙ ρ 2 2Mp2 |Ψ| ˜2 ˜ 3/2 δρ H W ˙ ˙ h  . Region II : |φ| > |Ψ| ⇒ ≈ ≈ (4.16) ˙ ρ 3 ˜ |φ| M ∂ W p

φ

h

Here one can interpret the results as: • In the Region I, the amplitude of the density fluctuation at the horizon crossing is only controlled by the scale of inflation and the magnitude of the dilaton dependent effective coupling parameter λ(Φh ).

– 37 –

• In the Region II, the amplitude of the density fluctuation at the horizon crossing is given by:     δρ 2 δρ = √
. (4.17) ρ Region II W˜ h ρ Region I This implies that that contribution from the first slow roll parameter as given by: ! ˜ Mp2 ∂φ W W˜ = , (4.18) ˜ 2 W controls the magnitude of the amplitude of density perturbation apart from the effect from the scale of inflation and the magnitude of the dilaton dependent effective coupling parameter λ(Φh ). 4.2.2

Present observables

Further using the approximate equations of motion the fractional density contrast for the above mentioned two cases can be written as: " #  r √ 2  φ0 2 6Mp λ(Ψh )   1− (Ψh − Ψ0 ) for Region I  2  4Mp 2 9φ20 δρ " #3/2 Case I : ∼ (4.19) p √ 3  ρ φ λ(Ψ ) 2 6M  h p 0   1− (Ψh − Ψ0 ) for Region II.  8Mp3 9φ20

Case II :

  ( )2 1/2 √  2  1 2 6Mp 2φ αλ(Ψ )    √ 1 + 0 4 h 1 − (Ψh − Ψ0 )  for Region I  8 α Mp φ20

δρ ∼  ρ      

Mp3 λ(Ψh )(8α)3/2 φ30

h

1−

√ 2 6Mp φ20

i3/2 (Ψh − Ψ0 )

for Region II.

(4.20) Here one can interpret the results as: • In the Region I and Region II of Case I, the amplitude of the density fluctuation at the horizon crossing are related as: " #1/2 √     δρ φ0 δρ 2 6Mp = √ 1− (Ψh − Ψ0 ) ρ Region II 9φ20 2Mp ρ Region I   φ0 δρ ≈ √ . (4.21) 2Mp ρ Region I This imples that if we know the field value at the starting point of inflation then one can directly quantify the amplitude of density perturbation. Most importantly, if inflation starts from the vicinity of the Planck scale i.e. √ φ0 ∼ 2Mp ∼ O(Mp ) (4.22)

– 38 –

then by evaluating the amplitude of the density perturbation in the Region I one can easily quantify the amplitude of the density perturbation in the Region II. In this setup within the range 50 < Nf /h < 70, we get:     δρ δρ ∼ ∼ 2.2 × 10−9 , (4.23) ρ Region I ρ Region II which is consistent with Planck 2015 data. But if inflation starts at the following field value: √ (4.24) φ0 = 2∆ Mp , where the parameter ∆ ≷ 1 then one ca write the following relationship between the amplitude of the density perturbation in the Region I and Region II as: 

δρ ρ



 =∆

Region II

 ≈∆

δρ ρ



δρ ρ



√

#1/2 6 1− (Ψh − Ψ0 ) 9∆2 Mp

" Region I

.

(4.25)

Region I

This implies that for ∆ ≷ 1 we get:   δρ ρ Region

 ≷ II

In this case for the Region I we get:   δρ ρ Region then for the Region II we get:   δρ ρ Region

δρ ρ

 .

(4.26)

Region I

∼ 2.2 × 10−9 ,

(4.27)

≷ 2.2 × 10−9 .

(4.28)

I

I

This implies that for ∆ ≷ 1 in Region II we get tightly constrained result for the amplitude for the density perturbation. • In the Region I and Region II of Case II, the amplitude of the density fluctuation at the horizon crossing are related as:     Mp3 δρ δρ ≈√ . (4.29) 3 ρ Region II 8αλ(Ψh )φ0 ρ Region I This implies that if we know the field value at the starting point of inflation, the dilaton field dependent coupling at the horizon crossing λ(Ψh ) and the coupling of scale free gravity α, then one can directly quantify the amplitude of density perturbation. Most importantly, if inflation starts from the vicinity of the Planck scale i.e. φ0 ∼ O(Mp )

– 39 –

(4.30)

and we have an additional constraint: 1 λ(Ψh ) ∼ √ , 8α

(4.31)

then by evaluating the amplitude of the density perturbation in the Region I one can easily quantify the amplitude of the density perturbation in the Region II. Here one can also consider an equivalent constraint:  1/3 1 φ0 ∼ √ Mp . (4.32) 8αλ(Ψh ) For both the situations in the present setup within the range 50 < Nf /h < 70, we get:     δρ δρ ∼ ∼ 2.3 × 10−9 , (4.33) ρ Region I ρ Region II which is also consistent with Planck 2015 data. But if inflation starts at the following field value: φ0 = ∆ Mp , (4.34) where the parameter ∆ ≷ 1 and we define:   1 , Γ= √ 8αλ(Ψh )∆3

(4.35)

where the parameter Γ ≷ 1 and then one can write the following relationship between the amplitude of the density perturbation in the Region I and Region II as:     δρ δρ =Γ . (4.36) ρ Region II ρ Region I This implies that for ∆ ≷ 1 and Γ ≷ 1 we get:     δρ δρ ≷ . ρ Region II ρ Region I In this case for the Region I we get:   δρ ρ Region then for the Region II we get:   δρ ρ Region

(4.37)

∼ 2.3 × 10−9 ,

(4.38)

≷ 2.3 × 10−9 .

(4.39)

I

I

This implies that for ∆ ≷ 1 and Γ ≷ 1 in Region II we get tightly constrained result for the amplitude for the density perturbation.

– 40 –

In this context the scalar spectral tilt can be written at the horizon crossing as 7 :   3    
for Case I − d ln AS Nf /h + 1 nS − 1 = ≈ (4.41)  df 3  h  for Case II. − Nf /h Further using Eq (4.41) the running and running of the running of scalar spectral tilt can be computed as:  3   for Case I   
2  dnS Nf /h + 1 ≈ (4.42) βS = df h  3   for Case II  2 Nf /h 7

Here we use a new symbol Nf /h , which is defined as,   af = |N (φh ) − N (φf )| ∼ 50 − 70. Nf /h = ln ah

(4.40)

Figure 8. Plot for running of the spectral index βS = dnS /d ln k vs spectral index nS for scalar modes. Here for Case I and Case II we have drawn green and pink colored lines. We also draw the background of confidence contours obtained from various joint constraints [38–40].

– 41 –

and  κS =

dβS df

 h

    −

6

Nf /h + 1 ≈  6   − 3 Nf /h


3

for Case I

(4.43) for Case II.

Finally combining Eq (4.41), Eq (4.42) and Eq (4.43) we get the following consistency relation for both Case I and Case II we get:  κ 2/3 (nS − 1)2 S =3 − βS = . 3 6

(4.44)

This is obviously a new a consistency relation for the present Higgsotic model of inflation and is also consistent with Planck 2015 data [38–40]. In table (1) we have shown the numerical estimations of the inflationary observables for the Higgsotic attractors depicted in Case I and Case II within the range 50 < Nf /h < 70. In fig. (8), we have plotted running of the spectral tilt for scalar perturbation (βS = dnS /d ln k) vs spectral tilt for scalar perturbation (nS ) in the light of Planck data along

Figure 9. Plot for running of the running of spectral index κS = d2 nS /d2 ln k vs running of the spectral index βS = dnS /d ln k for scalar modes. Here for Case I and Case II we have drawn green and pink colored lines. We also draw the background of confidence contours obtained from various joint constraints [38–40].

– 42 –

with various joint constraints. Here it is important to note that, for Case I and Case II the Higgsotic models are shown by the green and pink colored lines. Also the big circle, intermediate size circle and small circle represent the representative points in (r, nS ) 2D plane for the number of e-foldings, Nf /h = 70, Nf /h = 60 and Nf /h = 50 respectively. To represent the present status as well as statistical significance of the Higgsotic model for the dynamical attractors as depicted in Case I and Case II, we have drawn the 1σ and 2σ confidence contours from Planck 2013 data, Planck TT+low P and Planck TT,TE,EE+low P joint data sets [38–40]. It is clearly visualized from the fig. (8) that, for Case I we cover the range, 0.941 < nS < 0.958 (4.45) and

dnS < 1.16 × 10−3 d ln k in the (βS , nS ) 2D plane. Similarly for Case II we cover the range, 0.59 × 10−3 < βS =

0.940 < nS < 0.957 and 0.62 × 10−3 < βS =

dnS < 1.20 × 10−3 d ln k

(4.46)

(4.47)

(4.48)

in the (βS , nS ) 2D plane. In fig. (9), we have plotted running of the running of spectral tilt for scalar perturbation (κS = d2 nS /d2 ln k) vs spectral tilt for scalar perturbation (nS ) in the light of Planck data along with various joint constraints. Here it is important to note that, for Case I and Case II the Higgsotic models are shown by the green and pink colored lines. Also the big circle, intermediate size circle and small circle represent the representative points in (κS , nS ) 2D plane for the number of e-foldings, Nf /h = 70, Nf /h = 60 and Nf /h = 50 respectively. To represent the present status as well as statistical significance of the Higgsotic model for the dynamical attractors as depicted in Case I and Case II, we have drawn the 1σ and 2σ confidence contours from Planck+WMAP+BAO joint data sets [38–40]. It is clearly visualized from the fig. (12) that, for Case I we cover the range, 0.59 × 10−3 < βS =

dnS < 1.16 × 10−3 d ln k

(4.49)

and

d2 nS > −4.56 × 10−5 d2 ln k in the (κS , βS ) 2D plane. Similarly for Case II we cover the range, − 1.65 × 10−5 > κS =

0.62 × 10−3 < βS = and −5

− 1.78 × 10

dnS < 1.20 × 10−3 d ln k

d2 nS > κS = 2 > −4.80 × 10−5 d ln k

in the (κS , βS ) 2D plane.

– 43 –

(4.50)

(4.51)

(4.52)

Nf /h

AS

nS

(×10−9 ) Case I

Case II

50 60

2.2

2.3

70

βS

κS

(×10−3 )

(×10−5 )

Case I

Case II

Case I

Case II

Case I

Case II

0.941

0.940

1.16

1.20

-4.56

-4.80

0.951

0.950

0.80

0.83

-2.61

-2.76

0.958

0.957

0.59

0.62

-1.65

-1.78

Table 1. Inflationary observables and model constraints in the light of Planck 2015 data [38–40] for the dynamical attractors considered in Case I and Case II.

4.3

Primordial tensor modes and future observables

In terms of the number of e-foldings (N ) the the most useful parametrization of the primordial scalar and tensor power spectrum or equivalently for tensor-to-scalar ratio can be written near the horizon crossing Nh = N (φh ) as:  2 dφ 8 = r(Nh )e(N −Nh ){Ah +Bh (N −Nh )} (4.53) r(N ) = 2 Mp dN where in the slow-roll regime of inflation the tensor-to-scalar ratio r(Nh ) can be written in terms of the inflationary potential as:   128Mp2   0 2  for Case I  Vh φ2h 2 r = r(Nh ) ≈ 8Mp = (4.54)  Vh 512α2 λ2 (Ψh )φ6h   for Case II.  Mp6 and the symbols Ah , Bh and Ch are expressed in terms of the inflationary observables at horizon crossing as: Ah = nT − nS + 1, 1 Bh = (βT − βS ) . 2

(4.55) (4.56)

In the above parametrization Ah >> Bh ⇒ βS − 2(nS − 1) >> βT − 2nT ,

(4.57)

is always required for convergence of the Taylor expansion. Using this assumption the relationship between field excursion ∆φ = φh − φf

– 44 –

(4.58)

r vs λ(Ψh ) 0.10

r

0.08 0.06 0.04 0.02 0.00 0

2.× 10-15

4.× 10-15

6.× 10-15

8.× 10-15

1.× 10-14

λ(Ψh ) (a) Case I : r vs λ(Ψh ).

r vs α 0.12 0.10

r

0.08 0.06 0.04 0.02 0.00 0

1 × 108

2 × 108

3 × 108

4 × 108

α (b) Case II : r vs α.

Figure 10. Variation of tensor-to-scalar ratio r with respect to 10(a) coupling parameter λ(Ψh ) (Case I) and 10(b) scale free parameter α (Case II). For both the plots dotted region is disfavoured by Planck 2015 data along with BICEP2+Keck Array joint constraint [38–40].

– 45 –

(a) r vs nS polt for Case I and Case II in the background of confidence contours obtained from Planck TT+low P, Planck TT+low P+BKP, Planck TT+low P+BKP+BAO joint data sets.

(b) r vs nS polt for Case I and Case II in the background of confidence contours obtained from Planck 2013, Planck TT+low P, Planck TT,TE,EE+low P joint data sets.

Figure 11. r vs nS plot for Case I and Case I in the background of confidence contours obtained from various joint constraints [38–40].

– 46 –

and tensor-to-scalar ratio r(Nh ) can be computed as: ! r r r   A2 h Ah |∆φ| r(Nh ) − 2B 2π Bh Ah − erfi √ − ≈ e h Nf /h . erfi √ Mp 8 Bh 8 2Bh 2Bh

(4.59)

Now the scale of infation is connected with the tensor-to-scalar ratio in the following fashion: 1/4 Vh

 =

1/4  1/4 3 2 r(fh ) −3 π AS r(fh ) Mp ∼ 7.9 × 10 Mp × . 2 0.11

(4.60)

Substituting Eq (4.60) in Eq (7.102) we compute the relationship between field excursion and the scale of inflation as: ! s r   A2 h |∆φ| Vh A A B − 2B h h h h erfi √ √ ≈ e N (4.61) − erfi − f /h . Mp 6πMp4 AS Bh 8 2Bh 2Bh Also using Eq (4.60) the tensor-to-scalar ratio can be written as:   λ(Ψh )φ4h   for Case I  (2 × 10−2 Mp )4 r = r(Nh ) =  1   . for Case II.  α (2.4 × 10−2 )4

(4.62)

Further using Eq (4.54) and Eq (8.73) we get the following constraints from primordial tensor perturbation:  0.17Mp    for Case I p 6 λ(Ψh ) (4.63) φh =  4.25M p   . for Case II.  1/2 p α 3 λ(Ψh ) Consequently the model parameters of the prescribed theory can be recast in terms of tensor-to-scalar ratio as:  r 3 −15 λ(Ψh ) = 9.358 × 10 × for Case I (4.64) 0.11  r −1 α = 2.740 × 107 × . for Case II. (4.65) 0.11 To satisfy the upeer bound of tensor-to-scalar ratio as obtained from Planck 2015+ BICEP2 + Keck Array i.e. r ∼ 0.11 [38–40], Eq (4.64) and Eq (4.65), gives the upper bound of the model parameters λ(Ψh ) and α respectively. In fig. (10(a)) and figu. (10(b)), we have shown the variation of tensor-to-scalar ratio r with respect to field dependent coupling parameter λ(Ψh ) for Case I and scale free parameter α for Case II. For both the plots dotted region is disfavoured by Planck 2015 data along with BICEP2+Keck Array joint constraint.

– 47 –

In fig. (11(a)) and fig. (11(b)), we have plotted tensor-to-scalar ratio (r) vs spectral tilt for scalar perturbation (nS ) in the light of Planck data along with various joint constraints. Here it is important to note that, for Case I and Case II the Higgsotic models are shown by the green and pink colored lines. Also the big circle, intermediate size circle and small circle represent the representative points in (r, nS ) 2D plane for the number of e-foldings, Nf /h = 70, Nf /h = 60 and Nf /h = 50 respectively. To represent the present status as well as statistical significance of the Higgsotic model for the dynamical attractors as depicted in Case I and Case II, we have drawn the 1σ and 2σ confidence contours from Planck TT+low P, Planck TT+low P+BKP, Planck TT+low P+BKP+BAO joint data sets [38– 40] in fig. (11(a)) and Planck 2013 data, Planck TT+low P and Planck TT,TE,EE+low P joint data sets [38–40] in fig. (11(b)). It is clearly visualized from the fig. (11(a)) and fig. (11(b)) that, for Case I we cover the range, 0.941 < nS < 0.958

(4.66)

0.07 < r < 0.11

(4.67)

and in the (r, nS ) 2D plane for the effective field dependent coupling constrained within the window, 5.956 × 10−15 < λ(Ψh ) < 9.358 × 10−15 . (4.68) Similarly for Case II we cover the range, 0.940 < nS < 0.957

(4.69)

0.056 < r < 0.09,

(4.70)

and in the (r, nS ) 2D plane for the effective scale of the Higgsotic potential constrained within the window, 5.382 × 107 > α > 3.349 × 107 . (4.71) Additionally it is important to mention here that, the area bounded by the parallel vertical green lines and the pink lines represent the allowed parameter space in the (r, nS ) 2D plane for the two Higgsotic dynamical attractors as depicted in Case I and Case II. 4.4

Reheating

To get successful amount of reheating from the proposed setup we consider the fact that reheating to commence at the end of slow rolling of the inflaton, ˙ ˜ φ. φ¨ ≈ 3H

(4.72)

This condition translates into the following physical constraint for Case I and Case II as given by: ∂φφ

p

˜ W



=

3 p˜ W. 2Mp2

– 48 –

(4.73)

Further using this constraint in Eq (4.2) the inflaton field value at the end of inflation can be computed for Case I and Case II as: r  6   for Case I   5 Mp φf ∼ (4.74) φ0  . for Case II.  h i  1/2   1 + 270φ2 20 M p

Additionally it is important to mention here that the slow-roll approximation in Eq (6.3), Eq (6.4) and Eq (6.5) is only valid when the following constraint is satisfied: ˜ φW ˜ ≥ ∂√ Mp . W 6

(4.75)

For our present setup this condition translates into the following constraint for Case I and Case II: √ 2 6 Case I : Mp , (4.76) φinf ≥ 3 √ !1/3 4 2 √ Case II : φinf ≤ Mp , (4.77) 3 3 where φinf represent the field value of the inflaton during inflation. Here it is important to note that, once the φ˙ term become dominant, then the slow-roll condition is not valid. In such case we need to solve the equation of motions with large inflaton kinetic terms where ˙ ∼ O(1). ¨ H ˜ φ| |φ/3

(4.78)

This also implies that when φ˙ 2 contribution is dominant in the energy density, the slow-roll approximation for inflaton field completely breaks down and reheating starts. Further we assume that to occur successful reheating in the proposed framework it is important to convert energy from the potential energy density to radiation. Consequently one can write the following expression for reheating temperature as given by: !1/4 1/4  ˜f 30W 30ρf > TRH,min ∼ 1010 GeV (4.79) TRH = ≈ 2 2 π g∗ π g∗ where g∗ is the effective number of particle species. Equivalently Eq (4.79) can be translated for Case I and Case II as:  2  √ Ψ 5π g∗ − 2√ 2 Mf 3 p Case I : λ(Ψf ) = λe > × (4.12 × 10−9 )4 , (4.80) 54   15 Case II : α< × (4.12 × 10−9 )−4 . (4.81) 4π 2 g∗ One can interpret the results obtained in this section as:

– 49 –

• For Case I we get a lower bound on the field dependent coupling λ(Ψf ) which is expressed in terms of the effective number of particle g∗ .

• Similarly for Case II we get a upper bound on the scale free gravity coupling α which is also expressed in terms of the effective number of particle g∗ .

• For a given value of g∗ one can explicitly determine the respective bounds on the coupling parameters. For an example one can fix g∗ ∼ 100 in the present context.

5

5.1

Cosmological solutions from soft attractors

Solutions for inflaton

To study the inflationary constraints and the cosmological consequences from our proposed setup here we first express the the value of the inflaton field at the onset of inflation, the horizon, reheating and including the density perturbation conditions as given by:

Case I  1/2 6 5 φ0 > φh = Mp 1 + Nf /h , (5.1) 5 27  2 1/4 π g∗ λ(Ψ0 ) φ0 < φRH = TRH,min × , (5.2) 120  "  s #1/2  2   δρ 2 φ 3 20  0  + for Region I − − Nf /h   2Mp ρ cr λ(Ψh ) 2Mp 10 9 φ0 > φD = (5.3) "  #1/2  2 2/3   δρ 1 φ 3 20 0   + − − Nf /h for Region II.  2Mp  ρ cr (λ(Ψh ))3/2 2Mp 10 9 r

– 50 –

Case II   φ0 > φh = Mp 



φ0 Mp

2



φ0 Mp

1/2  2 + 8Nf /h 

,

1 + 270 s " #1/4  4 2 4 π g∗ TRH,min 1 φ0 > φRH = Mp 4 − , λ(Ψ0 ) 30 Mp 8α   ( )1/2  2   1 δρ   Mp  p 64α −1    ρ cr 2αλ(Ψ )  h   1/2   4   φ0   270 Mp     +  2 − 16Nf /h     φ0  1 + 270 M p  φ0 > φD =  −1   δρ 1      M  p  φ0 ρ cr   (8α)3/2 λ(Ψh ) M  p   1/2  4   φ  0  270 Mp     +   2 − 16Nf /h    φ0  1 + 270 M p

(5.4)

(5.5)

for Region I

(5.6)

for Region II.

The physical interpretation of the obtained results are given bellow: • For Case I the field value at the horizon crossing is completely specified by the number of e-foldings, which is lying within the window, 50 < Nf /h = Ncmb < 70. On the other hand for Case II field value at the horizon crossing is specified by two parametersA. number of e-foldings and B. the field value at the starting point of inflation (initial condition). • For Case I the field value during the time of reheating is specified by three parametersA. minimum value of the reheating temperature, B. value of the field dependent coupling parameter at the starting point of inflation, and C. the effective number of degrees of freedom g∗ . For Case II to determine the field value at the time of reheating we need to know additionally the numerical value of the scale free gravity parameter α. • For Case I the field value during the density perturbation is specified by four parametersA. value of the density contrast or more precisely the amplitude of the scalar perturbation, B. value of the field dependent coupling parameter at the horizon crossing and at the starting point of inflation, and C. number of e-foldings. For Case II to determine the field value during the density perturbation we need to know additionally the numerical value of the scale free gravity parameter α. For the Case I one can express

– 51 –

the solution for Region II in terms of Region I as: "  2/3  2/3 2 #1/2  φ0 δρ 1 (φD )Region I √ . (5.7) (φD )Region II = 2Mp + (λ(Ψh ))3/2 ρ Region I 2Mp 2Mp Similarly for the Case II one can express the solution for Region II in terms of Region I as: "  −1 1 φ20 δρ (φD )Region II = 2Mp 8α1/2 Mp2 ρ Region I ( )#1/2   2 2 (φD )Region I δρ 1 + 64α −p −1 . (5.8) Mp ρ Region I 2αλ(Ψh ) 5.2

Solutions for field dependent coupling λ(Ψ)

Now let us describe the behaviour of the running or the scale dependence of the field dependent coupling λ(Ψ) in the above mentioned two cases as: Case I λ(Ψ) = λ(Ψ0 )e

√ 2 (Ψ−Ψ0 ) 3Mp

− √2

= λ(Ψ0 )e

1 2 3Mp

(φ2 −φ20 )

 2 t0 = λ(Ψ0 ) , t

(5.9)

Case II λ(Ψ) = λ(Ψ0 )e

√ 2 (Ψ−Ψ0 ) 3Mp

− √2

= λ(Ψ0 )e

3 2 Mp

(φ2 −φ20 )

= λ(Ψ0 )e

−

√ 2 2Mp √ (t−t0 ) 3 3α

.

(5.10)

Further using Eq (4.63), Eq (5.1) and Eq (5.4) we get the following constraint on the field dependent coupling λ(Ψh ) at horizon crossing:  √ Ψ  1.4 × 10−5 − 2√ 2 Mh   3 p = λe for Case I   3  5  1 + α  f /h 27  √ Ψ 77 Mp − 2√ 2 Mh λ(Ψh ) = (5.11) −λe 3 p = − for Case II.  "  2 #3/2 .   φ0   Mp 3/2    + 8αf /h α  φ0 2  1+270

Mp

Similarly using Eq (5.11) the field dependent coupling λ(Ψ0 ) can be expressed in terms of the number of e-foldings as:  5 √ − 52 [1+ 27 αf /h ] φ20 −5 Ψ  1.4 × 10 × e − 2√ 2 M0 2   λe 3 p = e 3Mp for Case I   3  5  1 + α  27 f /h       φ0 2   M   p − 13    +8αf /h  φ0 2 λ(Ψ0 ) = 1+270 √ φ2 Mp 0 Ψ  77 Mp × e − 2√ 2 M0  2 3Mp  3 p −λe = − e . for Case II.  # "    3/2  φ0 2   Mp    + 8αf /h  α3/2  φ0 2 1+270 Mp

(5.12)

– 52 –

Additionally, we get the following constraint condition on the ratio of the couplings at the horizon crossing and at the starting point of inflation as given by:   2 5   e 5 [1+ 27 αf /h ] for Case I   φ2    2 λ(Ψh ) 3M0p2 φ0 (5.13) e = Mp  1  λ(Ψ0 )  2 +8αf /h    3 φ  0   e 1+270 Mp . for Case II.

6

Beyond soft attractor: A single field approach

In this section our prime objective is to analyze the non attractor phase of inflation. To serve this purpose let us start with the Klien-Gordon field equations for inflaton field φ and dilaton field Ψ, which can be written in the flat (k = 0) FLRW background as: d2 φ ˜ dφ + ∂φ W ˜ (φ, Ψ) = 0 ⇒ + 3H 2 dt˜ dt˜ d2 Ψ ˜ dΨ + ∂Ψ W ˜ (φ, Ψ) = 0 ⇒ + 3H 2 ˜ dt dt˜

d2 φ ˜ dφ + λ(Ψ)φ3 = 0 + 3H 2 dt˜ dt˜ 2 4 dΨ ˜ dΨ − λ(Ψ)φ √ + 3 H = 0. dt˜2 dt˜ 6Mp

(6.1) (6.2)

Now in the slow-roll approximated regime the field equations are approximated as: dφ + λ(Ψ)φ3 = 0 dt˜ 4 ˜ dΨ − λ(Ψ)φ √ 3H = 0, dt˜ 6Mp   ˜ (φ, Ψ) W V 2αλ(Ψ) 0 2 4 ˜ = H = φ , 1+ 3Mp2 3Mp2 Mp4 ˜ 3H

(6.3) (6.4) (6.5)

During the non-attractor phase of inflation we assume that the φ field is the only dynamical field controlling the scenario and at that time the Ψ field freezes at the Planck scale. On the other hand, at late times the dynamical contribution comes from the Ψ field and the inflaton field φ freezes at Planck scale. Assuming this fact the general behavior during inflationary epoch are governed by: " √     #
3π φ φ 1 i φ5 − φ5b − , (6.6) a = ai exp − ¯ Erf √3Mp − Erf √3Mp 20Mp5 16αλ "√  #     ¯
3 3π φ φi αλ − Erf √ + φ5 − φ5b . (6.7) t − ti ≈ − √ Erf √ ¯ 2α 2 5Mp5 2λ 3Mp 3Mp where ‘i0 subscript is used to describe the boundary/initial condition within the prescribed ¯ which signifies setup. It is important to note that in Eq (6.6,6.7) we introduce new symbol λ, the value of the self coupling at the freezing value of dilaton field Ψ ∼ O(Mp )

– 53 –

(6.8)

during inflation i.e.

√ # 2 ¯ = λ(Ψ) = λ exp − √ 2 . λ 3 "

(6.9)

On the other hand at late time inflaton field get its VEV at φ ∼ O(Mp ) and correspondingly ˆ at late time is defined as: the self coupling λ ˆ ≡ λ φˆ4 ∼ λ M 4 . λ (6.10) 4 4 p Here the obtained results can be interpreted as following: • Solution for the scale factor a(t) and inflaton field φ(t) admits quasi de-Sitter behaviour in presence of additional contribution coming from error functions. • For the large value of the product αλ one can further neglect the contributions from the error function. In that case one can get back the exact de-Sitter behaviour in the present context. For further analysis let us introduce the following Hubble flow functions in Einstein frame: ˜ 1 dH , (6.11) ˜H = − ˜ 2 dt˜ H ! d2 φ 1 dt˜2 . (6.12) η˜H = − dφ ˜ H dt˜ The flow functions in the Einstein frame can be expressed in terms of the Jordan frame Hubble flow functions as:   2  1 d2 ln Ω2 1 d ln Ω2 1 d ln Ω2 1 − 2H 2 H dt2 + HH dt − 4 dt ˜H = H , (6.13)   1 d ln Ω2 2 1 + 2H dt h i 1 d ln Ω2 1 + 2HηH dt  . η˜H = ηH  (6.14) 1 d ln Ω2 1 + 2H dt Further we introduce potential flow-functions in Einstein frame for the Higgs field φ as: 2 ¯ 2 φ6 Mp2  32α2 λ ˜ (φ, Ψ) = ˜W˜ = (6.15) ∂φ ln W h i , ¯ 4 2 2 2αλ 6 Mp 1 + M 4 φ p

¯ ˜ (φ, Ψ) ∂φφ W 24αλφ h i, = ¯ 4 ˜ (φ, Ψ) 2αλ 2 W Mp 1 + M 4 φ p    ˜ (φ, Ψ) ∂φφφ W ˜ (φ, Ψ) ∂φ ln W ¯ 2 φ4 384α2 λ 2 4 ξ˜W = M =   h i ˜ p 2 ¯ 4 2 2αλ 4 ˜ W (φ, Ψ) Mp 1 + M 4 φ p  2   ˜ (φ, Ψ) ˜ (φ, Ψ) ∂φ ln W ∂φφφφ W ¯ 3 φ6 3072α3 λ 3 6 σ ˜W˜ = Mp =  3 h i . ¯ 4 3 λ ˜ (φ, Ψ) W Mp6 1 + 2α φ M4 2

η˜W˜ = Mp2

p

– 54 –

(6.16)

(6.17)

(6.18)

where we assume that the dilaton field Ψ freezes at the field value Ψ during inflation. For further numerical estimation during inflation we fix the freezing value of dilaton field Ψ ∼ O(Mp ). During inflation potential is characterized by:   ¯ λ 2αλ(Ψ) 4 ˜ ˜ ˜ = V + φ4 . (6.19) W (φ, Ψ) = U (Ψ) + V (φ) = V0 1 + φ 0 Mp4 4 On the other hand in Einstein frame the Potential and Hubble flow functions are connected through the following relations: ˜H ≈ ˜W˜ =

¯ 2 φ6 32α2 λ i , h ¯ 4 2 2αλ 6 Mp 1 + M 4 φ

(6.20)

p

η˜H ≈ η˜W˜ − ˜W˜ =

¯ 2 φ6 ¯ 2 24αλφ 32α2 λ h i− i . h ¯ 4 ¯ 4 2 λ 2αλ 6 Mp2 1 + 2α φ Mp 1 + M 4 φ Mp4

(6.21)

p

7 7.1

Constraints on inflation beyond soft attractor Number of e-foldings

In the present context the total number of e-foldings is defined as: Z te H dt = Ncmb + ∆N Ntotal = N (te , ti ) =

(7.1)

ti

where Ncmb and ∆N are defined as: Z

te

H dt, Ncmb = N (te , tcmb ) = tcmb Z tcmb ∆N = N (tcmb , ti ) = H dt.

(7.2) (7.3)

ti

Further substituting the explicit form of the potential preseneted in this paper, the number of e-foldings can be recast as: Z φe ˜ 1 W (φ, Ψ) Ntotal ≈ − 2 dφ ˜ (φ, Ψ) Mp φi ∂φ W  
Mp2 1 1 1 2 2 = − − φ − φ , (7.4) e i 2 2 2 ¯ φe φi 8Mp 16αλ Z φe ˜ 1 W (φ, Ψ) Ncmb ≈ − 2 dφ ˜ (φ, Ψ) Mp φcmb ∂φ W  
Mp2 1 1 1 = − 2 − φ2e − φ2cmb , (7.5) 2 2 ¯ 8Mp 16αλ φe φcmb where superscript e, cmb and i denote the values of the inflaton field evaluated at the end of inflation, horizon crossing and starting point of inflation respectively.

– 55 –

In the present context the field value of the inflaton at the inflation is determined from the following condition: W˜ (φe ) = 1 = |ηW˜ (φe )|.

(7.6)

Further substituting Eq (6.15,6.16) in Eq (7.6) the inflaton field value at the end of inflation can be computed as, √ φe = 3 2 Mp . (7.7) Now using Eq (7.7) in Eq (7.5) the expression for the inflaton field value at the horizon crossing is given by: s #1 " r ¯ 2 8αλ Mp Acmb φcmb = ¯ 1 + 1 + A2 2 αλ cmb r p Mp 2Acmb ≈ Ncmb (7.8) ≈ 2M p ¯ 2 αλ where we use the following constraint condition: s ¯ ¯ 8αλ 4αλ 1 + 2 ≈ 1 + 2 + · · · ∼ O(1) (7.9) Acmb Acmb ¯ 2 > B(Ncmb ) >> C(Ncmb )

(7.101)

is always required for convergence of the Taylor expansion. For the time being to make the computation simpler let us assume that the term involving the co-efficient of the quadratic term B(Ncmb ) and cubic term C(Ncmb ) is negligibly small compared to the leading order term A(Ncmb ) as appearing in the exponent of the above mentioned parametrization. Using this assumption the relation between field excursion and tensor-to-scalar ratio can be computed as: r r    A(Ncmb ) 2 r(Ncmb ) r(Ncmb ) |∆φ| −∆N 2 ≈ 1−e ≈ ∆N (7.102) Mp A(Ncmb ) 8 8 and finally using the above relation from our R2 gravity model we get:  r(Ncmb ) ≈ 8

|∆φ| Mp ∆N

2 ≈ 32

 !2 √ √ Ntotal − Ncmb 32 = √ 2 . √ ∆N Ntotal + Ncmb

(7.103)

For our prescribed model |∆φ| ≈ 1.2 Mp

(7.104)

∆N = 10

(7.105)

and

– 69 –

is fixed by Planck 2015 observation. Substituting these values in the relation stated in Eq (7.103), the upper bound of tensor-to-scalar ratio at the scale of horizon crossing computed from our setup as: r(Ncmb ) . 0.11.

(7.106)

Now in the present context using Eq (7.60) we can express the number of e-foldings at the horizon crossing as: Ncmb =

3 1 − nS (Ncmb )

(7.107)

and substituing in Eq (7.61,7.90) we get the following sets of consistency relations for scalar modes from our analysis: (1 − nS (Ncmb ))2 , 3 2 (1 − nS (Ncmb ))3 κS (Ncmb ) ≈ − 9

αS (Ncmb ) ≈

(7.108) (7.109)

and combining Eq (7.108) and Eq (7.109) we finally get: 1 − nS (Ncmb ) +

3κS (Ncmb ) ≈ 0. 2αS (Ncmb )

(7.110)

Similarly from tensor modes we get the following sets of consistency relations from our model: ¯2 884736α2 λ r(Ncmb ) ≈ 16∗W˜ = h i2 ¯ λ (1 − nS (Ncmb ))3 1 + (1−n288α (N ))2 S

cmb

2 ¯2

=

884736α λ h i2 ¯ λ (3αS (Ncmb ))3/2 1 + αS96α (Ncmb )

¯2 196608α2 λ  2 , ¯ 288αλ κS (Ncmb ) 1 + 9 2/3 {− 2 κS (Ncmb )} 24n2T (Ncmb ) = . 1 − nS (Ncmb ) =−

r(Ncmb ) ≈ 16∗W˜

(7.111)

(7.112)

It is important mention here that in the resent context the usual consistency relation for single field slow-roll inflation, r(Ncmb ) = −8nT (Ncmb )

(7.113)

violates and after doing the anaysis we found a completely new consistency relation as presented in Eq (7.112). In case of usual slow-roll single field inflationary setup the tensor spectral tilt nT (Ncmb ) < 0 (7.114)

– 70 –

always. But for prescribed setup Eq (7.88), Eq (7.95) and Eq (7.96) suggests that, ¯ > 0, λ α>0

(7.115) (7.116)

nT (Ncmb ) > 0.

(7.117)

always imples Further using Eq (7.103) in Eq (7.112) we get the following constraint relationship: i hp p |∆φ| Ntotal − Ncmb = ≈2 Mp 7.4

s

  3 3 nT (Ncmb ) Ntotal − . (7.118) 1 − nS (Ncmb ) 1 − nS (Ncmb )

Reheating

The above results provide limits on the reheating temperature Treh , defined as the initial temperature of the homogeneous radiation dominated universe. In general, the reheating temperature Treh is related to energy density ρreh through the following expression: ρreh

π2 4 ⇒ Treh = = geff (Treh )Treh 30

 

≈

30 2 π geff (Treh )

1/4

30 2 π geff (Treh )

1/4

1/4

ρreh 1/4

Vreh

(7.119)

where geff (Treh ) is the effective number of relativistic degrees of freedom present in the thermal bath at the temperature T = Treh and Vreh represents the scale of reheating at φ = φreh given by the following expression:   ¯ 2αλ 4 (7.120) Vreh = V (φ = φreh ) = V0 1 + 4 φreh . Mp Counting all degrees of freedom of the Standard Model and the dilaton degrees of freedom, one has geff (Treh ) = 107.75. (7.121) To find the reheating constraint from our prescribed setup let us introduce the number of e-foldings at the time of reheating defined as: Z

te

Nreh =

H dt treh

= Ntotal − ∆N Z φe ˜ 1 W (φ, Ψ) ≈− 2 dφ ˜ (φ, Ψ) Mp φreh ∂φ W  
Mp2 1 1 1 = − 2 − φ2e − φ2reh . 2 2 ¯ 8Mp 16αλ φe φreh

– 71 –

(7.122)

For the sake of clarity let us express the interval ∆N as: Z

te

Z

te

H dt −

∆N =

H dt treh

t Z itreh

H dt

= ti

= Ntotal − Nreh = Ntotal − Ncmb + Ncmb − Nreh = ∆N − (Nreh − Ncmb )  
Mp2 1 1 1 2 2 ⇒ ∆N − ∆N = Nreh − Ncmb = − − φ − φ . (7.123) cmb reh ¯ φ2 φ2reh 8Mp2 16αλ cmb Finally using the last step of Eq (7.123) the field value during reheating can be expressed as:

φreh

Mp = 2

r

"

Mreh ¯ 1+ αλ

s

¯ 8αλ 1+ M2reh

# 21

r

Mreh ¯ αλ q
≈ 2Mp ∆N − ∆N p ≈ 2Mp Nreh − Ncmb Mp ≈ 2

(7.124)

where 
 ¯ 16 ∆N − ∆N + 8Ncmb − Mreh = αλ


1 ¯ ∆N − ∆N . ≈ 16αλ 4Ncmb

(7.125)

Substituting Eq (7.125) in Eq (7.120) finally we get the scale of reheating in terms of the number of e-foldings as: h
i   ¯ ∆N − ∆N 2 = V0 1 + 32αλ ¯ (Nreh − Ncmb )2 . Vreh ≈ V0 1 + 32αλ

(7.126)

Further substituting Eq (7.126) in Eq (7.119) the reheating temperature can be expressed in terms of the number of e-foldings and scale of inflation in the context of our proposed model as:  Treh ≈

30 2 π geff (Treh )

1/4

1/4 V0

h
2 i1/4 ¯ 1 + 32αλ ∆N − ∆N

1/4 1 2 ¯ (Nreh − Ncmb ) = + 4λ Mp 8α " ( )#1/4  1/2 45π 2 PS (Ncmb )r(Ncmb ) r(Ncmb ) 1+2 (Nreh − Ncmb )2 Mp . = 3 geff (Treh ) 128Ncmb 

30 2 π geff (Treh )

1/4 

– 72 –

(7.127)

8

Future probe: Primordial Non-Gaussianity

8.1 8.1.1

Three point function Using In-In formalism

Here we discuss about the constraint on the primordial three point scalar correlation function in the non attractor regime of soft inflation. In general one can write down the following expressions for the three point function of the scalar fluctuation as [2, 136–147]: hζ(k1 )ζ(k2 )ζ(k3 )i = (2π)3 δ (3) (k1 + k2 + k3 )B(k1 , k2 , k3 ).

(8.1)

In our computation we choose Bunch-davies vacuum state and for single field soft slow-roll inflation we get the following expression for the bispectrum: " 3 X 1 H4 ∗ ∗ 2(2 − η ) ki3 B(k1 , k2 , k3 ) = H H 32(∗H )2 Mp4 (k1 k2 k3 )3 i=1 !# 3 3 3 X X X 8 +∗H − ki kj2 + ki3 + ki2 kj2 K i=1 i,j=1,i>j i,j=1,i6=j " 3 X ˜ 2 (φcmb , Ψ) 1 W ∗ ∗ 2(3W˜ − ηW˜ ) ki3 ≈ 288(∗W˜ )2 Mp6 (k1 k2 k3 )3 i=1 !# 3 3 3 X X X 8 ki kj2 + +∗W˜ − ki3 + ki2 kj2 , (8.2) K i=1 i,j=1,i>j i,j=1,i6=j where K = k1 + k2 + k3 =

3 X

ki ,

(8.3)

i=1

and the potential flow-functions in Einstein frame can be expressed in terms of number of e-foldings Ncmb as:  2 2  Mp ˜ ∂φ ln W (φ, Ψ) ˜W˜ = 2 φ=φcmb 2 ¯2 6 ¯2N 3 2048α2 λ 32α λ φcmb cmb = (8.4) 2 , h i2 =  2 ¯ ¯ 2α λ 4 1 + 32α λN 6 cmb Mp 1 + M 4 φcmb p " # ˜ 2 ∂φφ W (φ, Ψ) η˜W˜ = Mp ˜ (φ, Ψ) W φ=φ cmb

¯ 2 ¯ cmb 24αλφ 96αλN cmb h i=  = ¯ 4 ¯ 2 . 2αλ 1 + 32α λN 2 cmb Mp 1 + M 4 φcmb

(8.5)

p

In the present context one can parameterize non-Gaussianity phenomenologically via a nonlinear correction to a Gaussian perturbation ζg in position space as [2]:   3 ζ(x) = ζg (x) + fNlocL ζg2 (x) − hζg2 (x)i + · · · , (8.6) 5

– 73 –

where · · · represent higher order non-Gaussian contributions. This definition is local in real space and therefore called local non-Gaussianity. In case local non-Gaussianity amplitude of the bispectrum from the three point function is defined as [2]: fNlocL (k1 , k2 , k3 ) =

5 B(k1 , k2 , k3 ) . 6 [Pζ (k1 )Pζ (k2 ) + Pζ (k2 )Pζ (k3 ) + Pζ (k1 )Pζ (k3 )]

(8.7)

Further substituting the expression for bispectrum and power spectrum the non-Gaussianity amplitude can be expressed as: " 3 X 1 5 ∗ ∗ loc ki3 fN L (k1 , k2 , k3 ) = 2(2H − ηH ) P3 3 12 i=1 ki i=1 !# 3 3 3 X X 8 X 2 2 ∗ 3 2 +H − ki + ki kj + k k K i,j=1,i>j i j i=1 i,j=1,i6=j " 3 X 5 1 ∗ ∗ 2(3 − η ) ki3 ≈ P3 ˜ ˜ W W 12 i=1 ki3 i=1 !# 3 3 3 X X X 8 ki kj2 + +∗W˜ − ki3 + ki2 kj2 . (8.8) K i=1 i,j=1,i>j i,j=1,i6=j To give the bulk interpretation of the obtained results for scalar three point correlation function here we start with the graviton propagator which can be computed from the secornd order fluctuation in δgµν for the canonical scalar field action minimally coupled with Einstein gravity. In this context we choose a guage δgzz = 0 = δgzi which is equivalent to choosing N i = 0, N = 1 in ADM formalism. After choosing this gauge we get: Z Z J 3 (qz1 )J 3 (qz2 ) d3 k ik.(y1 −y2 ) ∞ 2 e qdq ∆ijkl , (8.9) Gij;kl (z1 , y1 ; z2 , y2 ) = √ 2 (2π)3 2 z1 z2 0 where ∆ijkl is defined as: ∆ijkl = Pik Pjl + Pil Pjk − Pij Pkl . Here Pij is the projection operator in momentum space, which is defined as:   ki kj Pij = δij + 2 . q

(8.10)

(8.11)

Further one can also write down the expression for the transverse part of the graviton propagator from this compution: Z Z ∞ 3 J 3 (qz1 )J 3 (qz2 ) d k ik.(y −y ) 2 1 2 ¯ ij;kl (z1 , y1 ; z2 , y2 ) = ¯ ijkl , G e qdq ∆ (8.12) √ 2 (2π)3 2 z1 z2 0 ¯ ijkl is defined as: where ∆ ¯ ijkl = P¯ik P¯jl + P¯il P¯jk − P¯ij P¯kl . ∆

– 74 –

(8.13)

(a) S channel diagram.

(b) T channel diagram.

(c) U channel diagram.

Figure 15. Representative S, T and U channel Feynman-Witten diagram for bulk interpretation of three point scalar correlation function in presence of graviton exchange contribution. In all the diagrams graviton is propagating on the bulk and the end point of scalars are attached with the boundary at z = 0. More precisely the wavy line denotes the bulk-to-bulk graviton propagator, the solid lines represent the bulk-to-boundary propagators for the scalar field and the dashed line ¯ denotes background represented by, ∂z φ.

– 75 –

Here P¯ij is the transverse part of the projection operator in momentum space, which is defined as:   ki kj ¯ Pij = δij − 2 . (8.14) q Similarly the longitudinal part of the graviton propagator can be expressed as: ˆ ij;kl (z1 , y1 ; z2 , y2 ) = Gij;kl (z1 , y1 ; z2 , y2 ) − G ¯ ij;kl (z1 , y1 ; z2 , y2 ). G

(8.15)

Consequently in the bulk the onshell action can be written as a sum of transverse and longitudinal contribution as: Bulk Son−shell

3Mp2 =− (Ztr + Zlong ), 2Λ

(8.16)

where Λ is the cosmological constant. In this context the transverse contribution Ztr and longitudinal contribution Zlong are defined as: Z ¯ mn;kl (z1 , y1 ; z2 , y2 )Tkl (z1 , y1 ), Ztr = dz1 dz2 d3 y1 d3 y2 Tmn (z1 , y1 )G (8.17) Z Z h z dz 3 −2 d y T (z, y)∂ T (z, y) + ∂k Tzk (z, y)∂ −2 Tzz (z, y) Zlong = − zk zk z2 2  1 −2 2 (8.18) + ∂k Tzk (z, y)(∂ ) ∂m Tzm (z, y) . 4 Finally one can write down the follwing simplified expression: Bulk Son−shell =−

3 Y 3Mp2 √ φ0 (kn ) 2(2π)3 δ (3) (k1 + k2 + k3 ) 2Λ n=1 " 3 # 3 3 X X X 8 ki kj2 + × − ki3 + ki2 kj2 . K i,j=1,i>j i=1 i,j=1,i6=j

(8.19)

further taking the derivatives with respect to the background field value φ0 and choosing the following gauge H (8.20) ζ = − δφ, φ˙ one can write down the following expression for the scalr three point function: " 3 4 X H 1 3 (3) ∗ ∗ hζ(k1 )ζ(k2 )ζ(k3 )i = (2π) δ (k1 + k2 + k3 ) 2(2H − ηH ) ki3 ∗ 2 4 3 32(H ) Mp (k1 k2 k3 ) i=1 !# 3 3 3 X X 8 X 2 2 +∗H − ki3 + ki kj2 + k k . (8.21) K i,j=1,i>j i j i=1 i,j=1,i6=j The representative S, T and U channel Feynman Witten diagrams for bulk interpretation of the three point scalar correlation function in presence of graviton exchange is shown in

– 76 –

shown in fig. (15(a)), fig. (15(b)) and fig. (15(c)). In these diagrams we have explicitly shown that, graviton is propagating on the bulk and the end point of scalars are attached with the boundary at z = 0. Additionally it is important to note that, the dashed line represents background denoted by, ∂z φ¯ in all of the representative diagrams. Here φ¯ is the background field value. More precisely the wavy line denotes the bulk-to-bulk graviton propagator, the solid lines represent the bulk-to-boundary propagators for the scalar field. In our computation all the representative diagrams are important to explain the total three point scalar correlation function. To analyze the shape of the bispectrum here we further consider two limiting configurationsequilateral limit and squeezed limit. In these limits, the final simplified results are appended below: 1. Equilateral limit configuration: For this case we have |k1 | = |k2 | = |k3 | = k

(8.22)

and the bispectrum for scalar fluctuation can be written as: 1 H4 ∗ [23∗H − 6ηH ] ∗ 2 4 6 32(H ) Mp k ˜ 2 (φcmb , Ψ) 1   W ∗ ∗ ≈ 29 − 6η . ˜ ˜ W W 288(∗W˜ )2 Mp6 k 6

B(k, k, k) =

(8.23)

In this case the non-Gaussian amplitude for bispectrum can be expressed as: 5 ∗ [23∗H − 6ηH ] 36  5  ∗ ∗ ≈ 29W˜ − 6ηW ˜ . 36

loc fNequil L = fN L (k, k, k) =

2. Squeezed limit configuration: For this case we have k1 ≈ k2 (= kL ) >> k3 (= kS ),

(8.24)

(8.25)

where ki = |ki |∀i = 1, 2, 3.

(8.26)

Here kL and kS represent momentum for long and short modes respectively. Consequently the bispectrum for scalar fluctuation for arbitrary vacuum can be expressed as: "  2  3 # k kS H4 1 S ∗ ∗ ∗ ∗ ∗ B(kL , kL , kS ) = 4(3 − η ) + 10 − ( + 2η ) H H H H H ∗ 2 3 3 32(H ) Mp4 kL kS kL kL ≈

˜ 2 (φcmb , Ψ) 1  W ∗ 4(4∗W˜ − ηW ˜) 288(∗W˜ )2 Mp6 kL3 kS3  2  3 # k kS S ∗ ∗ +10∗W˜ − (2ηW . ˜ − W ˜) kL kL

– 77 –

(8.27)

Equilateral non-Gaussianity

Equilateral non-Gaussianity 0.08

0.10

0.06

equil

0.06

fNL

fNL

equil

0.08

0.04

0.04

Ncmb =50

Ncmb =50

Ncmb =60 0.02

Ncmb =60

0.02

Ncmb =70

Ncmb =70 0.00

0.00 0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0000

0.00002

0.00004

αλ

0.00006

0.00008

0.0001

αλ

(a) Range I.

(b) Range II.

Equilateral non-Gaussianity

Equilateral non-Gaussianity 0.0000

0.020

Ncmb =50

Ncmb =50

Ncmb =60 0.015

Ncmb =60

-0.0005

Ncmb =70

Ncmb =70 equil

0.010

fNL

fNL

equil

-0.0010

-0.0015

0.005 -0.0020

0.000

-0.0025

-0.005 0

2.×10-6

4.×10-6

6.×10-6

8.×10-6

0.00001

αλ

-0.0030 0

2.×10-7

4.×10-7

6.×10-7

8.×10-7

1.×10-6

αλ

(c) Rangle III.

(d) Range IV.

Figure 16. Representative diagram for equilateral non-Gaussian three point amplitude vs product ¯ in four different region for Ncmb = 50 (red), Ncmb = 60 (blue) and Ncmb = 70 of the parameters αλ (green).

– 78 –

(a) Angle I.

(b) Angle II.

Figure 17. Representative 3D diagram for equilateral non-Gaussian three point amplitude vs the ¯ for Ncmb = 60 in two differenent angular views. model parameters α and λ

– 79 –

Squeezed non-Gaussianity

Squeezed non-Gaussianity 0.12

0.15 0.10 0.08 sq

fNL

sq

fNL

0.10

Ncmb =50

0.05

0.06

Ncmb =50

0.04

Ncmb =60

Ncmb =60

0.02

Ncmb =70

Ncmb =70

0.00

0.00 0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0000

0.00002

0.00004

αλ

0.00006

0.00008

0.0001

αλ

(a) Range I.

(b) Range II.

Squeezed non-Gaussianity

Squeezed non-Gaussianity 0.000

Ncmb =50

Ncmb =50 -0.001

Ncmb =60

0.02

Ncmb =70

-0.002 fNL

0.01

sq

sq

fNL

Ncmb =70

Ncmb =60

-0.003 -0.004

0.00 -0.005 -0.006

-0.01 0

2.×10-6

4.×10-6

6.×10-6

8.×10-6

0.00001

αλ

0

2.×10-7

4.×10-7

6.×10-7

8.×10-7

1.×10-6

αλ

(c) Rangle III.

(d) Range IV.

Figure 18. Representative diagram for squeezed non-Gaussian three point amplitude vs product ¯ in four different region for Ncmb = 50 (red), Ncmb = 60 (blue) and Ncmb = 70 of the parameters αλ (green).

– 80 –

(a) Angle I.

(b) Angle II.

Figure 19. Representative 3D diagram for squeezed non-Gaussian three point amplitude vs the ¯ for Ncmb = 60 in two differenent angular views. model parameters α and λ

– 81 –

Scanning Region

¯ Bound on αλ

equil fN L

sq fN L

I

¯ < 0.001 0.0001 < αλ

equil 0.06 < fN L < 0.11

sq 0.09 < fN L < 0.16

II

¯ < 0.0001 0.00001 < αλ

equil 0.01 < fN L < 0.08

sq 0.01 < fN L < 0.12

III

¯ < 0.00001 0.000001 < αλ

equil −0.004 < fN L < 0.02

sq −0.01 < fN L < 0.03

IV

¯ < 0.000001 0.0000001 < αλ

equil −0.0001 < fN L < −0.0029

sq −0.0005 < fN L < −0.006

I+II+III+IV

¯ < 0.001 0.0000001 < αλ

equil −0.0001 < fN L < 0.11

sq −0.0005 < fN L < 0.16

Table 4. Contraint on scalar three point non-Gaussian amplitude from equilateral and squeezed configuration.

In this case the non-Gaussian amplitude for bispectrum can be expressed as: "  2  3 # k kS 5 S ∗ ∗ fNsqL = fNlocL (kL , kL , kS ) ≈ 4(3∗H − ηH ) + 10∗H − (∗H + 2ηH ) 12 kL kL "  2  3 # k kS 5 S ∗ ∗ ∗ ∗ 4(4∗W˜ − ηW − (2ηW . ≈ ˜ ) + 10W ˜ ˜ − W ˜) 12 kL kL In table. (4), we give the numerical estimates and constraints on the three point nonGaussian amplitude from equilateral and squeezed configuration. Here all the obtained results are consistent with the two point constraints as well as with the Planck 2015 data. In fig. (16), we have shown the features of non-Gaussian amplitude from three point function in equilateral limit configuration in four different scanning region of product of the ¯ in the (f equil , αλ) ¯ 2D plane for the number of e-foldings 50 < Ncmb < 70. two parameters αλ NL Physical explanation of the obtained features are appended following:• Region I: Here for the parameter space ¯ < 0.001 0.0001 < αλ

(8.29)

the non-Gaussian amplitude lying within the window 0.06 < fNequil L < 0.11.

(8.30)

¯ then the magnitude of the nonFurther if we increase the numerical value of αλ, Gaussian amplitude saturates and we get maximum value for Ncmb = 50: |fNequil L |max ∼ 0.11.

– 82 –

(8.31)

(8.28)

• Region II Here for the parameter space ¯ < 0.0001 0.00001 < αλ

(8.32)

the non-Gaussian amplitude lying within the window 0.01 < fNequil L < 0.08.

(8.33)

In this region we get maximum value for Ncmb = 50: |fNequil L |max ∼ 0.08.

(8.34)

¯ = 0.00004 the lines Additionally it is important to note that, in this case for αλ obtained for Ncmb = 50, Ncmb = 60 and Ncmb = 70 cross each other. • Region III Here for the parameter space ¯ < 0.00001 0.000001 < αλ

(8.35)

the non-Gaussian amplitude lying within the window − 0.004 < fNequil L < 0.02.

(8.36)

In this region we get maximum value for Ncmb = 70: |fNequil L |max ∼ 0.02.

(8.37)

Additionally it is important to note that, in this case for ¯ ≤ 0.000006 0.000003 ≤ αλ

(8.38)

the lines obtained for Ncmb = 50, Ncmb = 60 and Ncmb = 70 cross the zero line of non-Gaussian amplitude and transition takes place from negative to positive values of fNequil L . • Region IV Here for the parameter space ¯ < 0.000001 0.0000001 < αλ

(8.39)

the non-Gaussian amplitude lying within the window − 0.0001 < fNequil L < −0.0029.

(8.40)

In this region we get maximum value for Ncmb = 60: |fNequil L |max ∼ 0.0029.

– 83 –

(8.41)

Further combining the contribution from Region I, Region II, Region III and Region IV we finally get the following constraint on the three point non-Gaussian amplitude in the equilateral limit configuration: Region I + Region II + Region III + Region IV : − 0.0001 < fNequil L < 0.11

(8.42)

for the following parameter space: Region I + Region II + Region III + Region IV :

¯ < 0.001. 0.0000001 < αλ

(8.43)

In this analysis we get the following maximum value of the three point non-Gaussian amplitude in the equilateral limit configuration as given by: |fNequil L |max ∼ 0.11.

(8.44)

¯ To visualize these constraints more clearly we have also presented (fNequil L , α, λ) 3D plot in fig. (17(a)) and fig. (17(b)), for two different angular orientations as given by Angle I and Angle II. From the the representative surfaces it is clearly observed the behavior of three point non-Gaussian amplitude in the equilateral limit for the variation of two fold parameter ¯ and the results are consistent with the obtained constraints in 2D analysis. Here α and λ all the obtained results are consistent with the two point constraints and the Planck 2015 data. In fig. (18), we have shown the features of non-Gaussian amplitude from three point function in squeezed limit configuration in four different scanning region of product of the ¯ in the (f sq , αλ) ¯ 2D plane for the number of e-foldings 50 < Ncmb < 70. two parameters αλ NL Physical explanation of the obtained features are appended following:• Region I: Here for the parameter space ¯ < 0.001 0.0001 < αλ

(8.45)

the non-Gaussian amplitude lying within the window 0.09 < fNsqL < 0.16.

(8.46)

¯ then the magnitude of the nonFurther if we increase the numerical value of αλ, Gaussian amplitude saturates and we get maximum value for Ncmb = 50: |fNsqL |max ∼ 0.16.

(8.47)

¯ < 0.0001 0.00001 < αλ

(8.48)

• Region II Here for the parameter space

the non-Gausiian amplitude lying within the window 0.01 < fNsqL < 0.12.

– 84 –

(8.49)

In this region we get maximum value for Ncmb = 50: |fNsqL |max ∼ 0.12.

(8.50)

¯ = 0.00004 the lines Additionally it is important to note that, in this case for αλ obtained for Ncmb = 50, Ncmb = 60 and Ncmb = 70 cross each other. • Region III Here for the parameter space ¯ < 0.00001 0.000001 < αλ

(8.51)

the non-Gaussian amplitude lying within the window − 0.01 < fNsqL < 0.03.

(8.52)

In this region we get maximum value for Ncmb = 70: |fNsqL |max ∼ 0.03.

(8.53)

Additionally it is important to note that, in this case for ¯ ≤ 0.000006 0.000003 ≤ αλ

(8.54)

the lines obtained for Ncmb = 50, Ncmb = 60 and Ncmb = 70 cross the zero line of non-Gaussian amplitude and transition takes place from negative to positive values of fNsqL . • Region IV Here for the parameter space ¯ < 0.000001 0.0000001 < αλ

(8.55)

the non-Gaussian amplitude lying within the window − 0.0005 < fNsqL < −0.006.

(8.56)

In this region we get maximum value for Ncmb = 60: |fNsqL |max ∼ 0.006.

(8.57)

Further combining the contribution from Region I, Region II, Region III and Region IV we finally get the following constraint on the three point non-Gaussian amplitude in the squeezed limit configuration: Region I + Region II + Region III + Region IV : − 0.0005 < fNsqL < 0.16

(8.58)

for the following parameter space: Region I + Region II + Region III + Region IV :

– 85 –

¯ < 0.001. 0.0000001 < αλ

(8.59)

¯ 3D plot in To visualize these constraints more clearly we have also presented (fNsqL , α, λ) fig. (19(a)) and fig. (19(b)), for two different angular orientations as given by Angle I and Angle II. From the the representative surfaces it is clearly observed the behavior of three point non-Gaussian amplitude in the squeezed limit for the variation of two fold parameter ¯ and the results are consistent with the obtained constraints in 2D analysis. Here α and λ all the obtained results are consistent with the two point constraints and the Planck 2015 data. 8.1.2

Using δN formalism

A. Basic methodology:

Time arrow

Final uniform energy density hypersurface

δt

Initial flat hypersurface

Figure 20. Diagrammatic representation of δN formalism.

In this section our prime objective is to use δN formalism to compute the three point and four point correlation functions in the attractor regime. Here N signifies the number of e-foldings as we have defined earlier. In this formalism the dominant contribution comes from only on the perturbations of the scalar field trajectories with respect to the field value ˙ Ψ. ˙ This can be realized by providing at the initial hypersurface φ, Ψ and the velocity φ, two initial conditions on both of them on the initial hypersurface. More specifically, in the present context, we have assumed that the evolution of the universe is governed by a unique fashion after the value of the scalar field achieved at φ = φ∗ and Ψ = Ψ∗ , where it is mimicking the role of standard clock in inflationary cosmology. Here the value of its ˙ ∗ is completely insignificant. Let us mention that only in this case δN is velocity φ˙ ∗ and Ψ equal to the final value of the comoving curvature perturbation ζ which is conserved at the epoch t ≥ t∗ . In figure (20), we have shown the schematic diagram of δN formalism. For further computation we assume that on large scales the dynamical behaviour which permit us to ignore time derivatives appearing in the cosmological perturbation theory, the

– 86 –

horizon volume will evolve in such a way that it were a perfectly self contained universe. As a result the scalar curvature perturbation can be expressed beyond liner order in cosmological perturbation theory as: ζ = δN = [N,φ δφ + N,Ψ δΨ] +

1 [N,φφ δφδφ + (N,φΨ + N,Ψφ ) δφδΨ + N,ΨΨ δΨδΨ] 2!

1 [N,φφφ δφδφδφ + (N,φΨΨ + N,ΨφΨ + N,ΨΨφ ) δφδΨδΨ 3! + (N,φφΨ + N,φΨφ + N,Ψφφ ) δφδφδΨ + N,ΨΨΨ δΨδΨδΨ] + · · · , +

(8.60)

where we use the following notations for simplicity: N,φ = ∂φ N , N,Ψ = ∂Ψ N , N,φφ = ∂φ2 N , N,ΨΨ = ∂Ψ2 N , N,φΨ = ∂φ ∂Ψ N, N,Ψφ = ∂Ψ ∂φ N, N,φφφ N,φφΨ N,Ψφφ N,ΨφΨ

= = = =

∂φ3 N , N,ΨΨΨ = ∂Ψ3 N , ∂φ ∂φ ∂Ψ N , N,φΨφ = ∂φ ∂Ψ ∂φ N , ∂Ψ ∂φ ∂φ N , N,φΨΨ = ∂φ ∂Ψ ∂Ψ N , ∂Ψ ∂φ ∂Ψ N , N,ΨΨφ = ∂Ψ ∂Ψ ∂φ N .

(8.61) (8.62) (8.63) (8.64) (8.65) (8.66)

Here we use the notation, ∂φ = ∂/∂φ and ∂Ψ = ∂/∂Ψ to denote the partial derivatives. But here we have to point that in attractor regime both the fields φ and Ψ are connected with each other, which we have already pointed earlier in this paper. To simplify the calculation further let us consider the all of these possibilities to write down the infinitesimal change in Ψ field in terms of the inflaton field φ: Case I : Case II : Case II + Choice I(v1&v2) : Case II + Choice II(v1&v2) : Case II + Choice III :

9φ δΨ = − √ 6Mp φ δΨ = − √ 6Mp φ δΨ = − √ 6Mp φ δΨ = − √ 6Mp φ δΨ = − √ 6Mp

δφ,

(8.67)

δφ,  δφ 1 −  δφ 1 −  δφ 1 +

(8.68)  φ4V , φ4

(8.69)

 m2c , (8.70) (m2c − λφ2 ) ξ 2 (φ + φ20 − 2φ2V ) 2  ξ 2 φ2V 2 + (φ − φ0 ) + 2 . (8.71) 2 φ

Combining all of these possibilities one can write the following expression: δΨ = V(φ) δφ,

– 87 –

(8.72)

where we introduce a function V(φ), which can be written as:    9          1.           1− φ V(φ) = − √ ×  6Mp   1−           1+         

for Case I for Case II

φ4V . φ4  m2c . (m2c − λφ2 ) ξ 2 (φ + φ20 − 2φ2V ) 2  ξ 2 φ2V 2 + (φ − φ0 ) + 2 . 2 φ 

for Case II+Choice I(v1& v2) for Case II+Choice II(v1& v2)

for Case II+Choice III.

(8.73) This additionally implies that one can write down the following differential operator for the Ψ field: ∂Ψ = ∂Ψ2 = ∂Ψ3 = ∂φ ∂Ψ = ∂Ψ ∂φ = ∂φ ∂φ ∂Ψ = ∂φ ∂Ψ ∂φ = ∂Ψ ∂φ ∂φ = ∂φ ∂Ψ ∂Ψ = ∂Ψ ∂φ ∂Ψ = ∂Ψ ∂Ψ ∂φ =

1 ∂φ , V(φ)   0 V (φ) 1 2 ∂ − ∂φ , V 2 (φ) φ V 3 (φ)   0 0 1 V (φ) 2 V 2 (φ) 3 ∂ −3 4 ∂ +3 5 ∂φ , V 3 (φ) φ V (φ) φ V (φ)   0 1 V (φ) 2 ∂ − ∂φ , V(φ) φ V 2 (φ) 1 ∂2, V(φ) φ   00   0 0 1 V (φ) 2 V (φ) V 2 (φ) 3 ∂ −2 2 ∂ − −2 3 ∂φ , V(φ) φ V (φ) φ V 2 (φ) V (φ)   0 1 V (φ) 2 3 ∂ − ∂ , V(φ) φ V 2 (φ) φ 1 ∂3, V(φ) φ   0 0 1 V (φ) 2 V 2 (φ) 3 ∂ −3 3 ∂ +3 4 ∂φ , V 2 (φ) φ V (φ) φ V (φ)   0 0 1 V (φ) 2 V 2 (φ) 3 ∂ −2 3 ∂ +2 4 ∂φ , V 2 (φ) φ V (φ) φ V (φ)   0 1 V (φ) 2 3 ∂ − ∂ , V 2 (φ) φ V 3 (φ) φ

where 0 is defined as the partial derivative with respect to the field φ i.e. 0 = ∂φ .

– 88 –

(8.74) (8.75) (8.76) (8.77) (8.78) (8.79) (8.80) (8.81) (8.82) (8.83) (8.84)

Consequently one can write: N,Ψ = N,ΨΨ = = N,φΨ = = N,Ψφ = N,ΨΨΨ = =

1 1 ∂φ N = N,φ , V(φ) V(φ)   0 V (φ) 1 2 ∂ − ∂φ N V 2 (φ) φ V 3 (φ)   0 1 V (φ) N,φφ − 3 N,φ , V 2 (φ) V (φ)   0 V (φ) 1 2 ∂φ N ∂ − V(φ) φ V 2 (φ)   0 1 V (φ) N,φφ − 2 N,φ , V(φ) V (φ) 1 1 ∂φ2 N = N,φφ V(φ) V(φ)   0 0 1 V (φ) 2 V 2 (φ) 3 ∂ −3 4 ∂ +3 5 ∂φ N , V 3 (φ) φ V (φ) φ V (φ)   0 0 V (φ) V 2 (φ) 1 N,φφφ − 3 4 N,φφ + 3 5 N,φ , V 3 (φ) V (φ) V (φ)

   00 0 0 V 2 (φ) V (φ) 2 V (φ) 1 3 ∂ −2 2 ∂ − −2 3 ∂φ N , V(φ) φ V (φ) φ V 2 (φ) V (φ)   00   0 0 1 V (φ) V (φ) V 2 (φ) N,φφφ − 2 2 N,φφ − −2 3 N,φ , V(φ) V (φ) V 2 (φ) V (φ)   0 V (φ) 2 1 3 ∂ − ∂ N V(φ) φ V 2 (φ) φ   0 1 V (φ) N,φφφ − 2 N,φφ , V(φ) V (φ) 1 1 ∂φ3 N = N,φφφ , V(φ) V(φ)   0 0 1 V (φ) 2 V 2 (φ) 3 ∂ −3 3 ∂ +3 4 ∂φ N , V 2 (φ) φ V (φ) φ V (φ)   0 0 1 V (φ) V 2 (φ) N,φφφ − 3 3 N,φφ + 3 4 N,φ , V 2 (φ) V (φ) V (φ)   0 0 1 V (φ) 2 V 2 (φ) 3 ∂ −2 3 ∂ +2 4 ∂φ N , V 2 (φ) φ V (φ) φ V (φ)   0 0 1 V (φ) V 2 (φ) N,φφφ − 2 3 N,φφ + 2 4 N,φ , V 2 (φ) V (φ) V (φ)     0 0 1 V (φ) 2 1 V (φ) 3 ∂ − ∂ N = N,φφφ − 3 N,φφ . V 2 (φ) φ V 3 (φ) φ V 2 (φ) V (φ)

(8.85)

(8.86)

(8.87) (8.88)

(8.89)



N,φφΨ = = N,φΨφ = = N,Ψφφ = N,φΨΨ = = N,ΨφΨ = = N,ΨΨφ =

– 89 –

(8.90)

(8.91) (8.92)

(8.93)

(8.94) (8.95)

Further the curvature perturbation can be recast as: ζ = δN   0 V (φ) N,φ δφδφ = 2N,φ δφ + 2N,φφ − V(φ)   02   0 00 4 V (φ) 5 V (φ) 1 V (φ) + N,φφφ − 2 N,φφ + − N,φ δφδφδφ + · · · , (8.96) 3 V(φ) 3 V 2 (φ) 6 V(φ) which implies that if we compute N,φ , N,φφ and N,φφφ , then one can determine the curvature perturbation and also compute the three and four point functions using Eq (8.96). B. Generalized convention for field solution: In δN formalism to compute N,φ , N,φφ and N,φφφ we start with the background equation of motion for the φ field: ˜ φ˙ + ∂φ W ˜ (φ, Ψ) = 0, φ¨ + 3H ˜ (φ, Ψ) is given by: where the effective potential W  √ λ − 2√32 MΨp 4   e φ    4            Mp4 λ − 2√√2 MΨ 4    − e 3 pφ .   8α 4           
Mp4 λ − 2√√2 MΨ  4 4  3 p  − e φ − φ  V .  8α 4            Mp4 λ − 2√√2 Ψ
˜ + e 3 Mp φ4 − φ4V . W (φ, Ψ) = 8α 4           2   √  Mp4  mc 2 λ 4 − 2√32 MΨp   + φ − φ e .   8α 2 4            2   √  Mp4  mc 2 λ 4 − 2√32 MΨp   − φ − φ e .   8α 2 4           2  √ λ  Mp4 (φ2 − φ2V ) − 2√ 2 MΨ  4   + e 3 p. 8α (1 + ξφ2 )2

– 90 –

(8.97)

for Case I

for Case II

for Case II+Choice I(v1)

for Case II+Choice I(v2)

for Case II+Choice II(v1)

for Case II+Choice II(v2)

for Case II+Choice III.

(8.98)

Here it is important to note that the exact connecting relations between the Ψ field and the inflaton field φ is given by: 
   9 φ2 − φ20 . for Case I     
   φ2 − φ20 . for Case II        
 1 1   . for Case II+Choice φ2 − φ20 + φ4V −  2 2  φ φ  0      
1 1  2 2 4   . for Case II+Choice  φ − φ0 + φV φ2 − φ2 1 0   2 Ψ − Ψ0 = − √ × 
m2c mc − λφ2 2 6Mp  2 2  φ − φ0 + . for Case II+Choice ln    λ m2c − λφ20    2    2
m2c  m − λφ  c 2 2  φ − φ0 + ln . for Case II+Choice   λ m2c − λφ20       

 1 ξ 2  2 2 2 2  φ − φ0 1 + φ + φ0 − 2φV   (1 + ξφ2V ) 2        φ 2   + 2φV ln . for Case II+Choice φ0 (8.99) It is obvious from the structural form of the effective potential for all of these cases that the general analytical solution for the inflaton field φ is too much complicated. To simplify the job here we consider a particular solution of the following form: φ = φL ∝ exp(YHt) (i.e. φ = φL (N ) = φ∗ exp(−YN )).

(8.100)

Here we assume that Y is a time independent quantity. Further our prime motivation is to obtain a more generalized version of the solution for FLRW cosmological background up to the consistent second order in cosmological perturbations around the prescribed particular solution. During our computation we also assume that the boundary between the attractor phase and the non attractor phase is determined by the field value φ = φ∗ = φ(Ncmb ) = φcmb , which in cosmological literature identified to be the field value associated at the pivot scale. To proceed further here we define a theoretical perturbative parameter which accounts the deviation from the actual inflaton field value compared to the field value after perturbation: ∆part ≡ φ − φ0 − φL =

∞ X

∆n ,

(8.101)

n=1

where in general φ0 is the VEV of the inflaton field φ. Here we assume that the parameter can take into account the difference between the true FLRW background solution and the proposed reference solution to solve the background Eq (8.97) in the physical domian where cosmological perturbation theory is valid. Additionally we claim that to validate the cosmological perturbation theory in the preferred physical domain, the infinite series sum should be convergent. Consequently in the present context we only look into ∆1 and ∆2 ,

– 91 –

I(v1) I(v2) II(v1) II(v2)

III.

which are the general linearized and second order solution within cosmological perturbation theory for the background field equations. We also neglect all the higher order contribution in the perturbative regime of solution as they are very small. C. Linearized perturbative solution: Before proceed further in this section let us clearly mention that, here we use the following ansatz to derive the results for linearized solution in the perturbative regime. In this case we assume that at the equation of motion level in the linear regime of perturbation theory there is no contribution from effective potential which contains quadratic structure or more complicated than that in terms of field φ. In our calculation we treat all such contributions to be the back reactions and in the linear perturbative regime of solution our claim is such effects are small and largely suppressed. In this paper the derived effective potentials for all of these cases are also complicated and to get a preferred analytical solution in the linearized perturbative regime we use this Ansatz. Here we get: h i ˜ ∂φ W ≈ 0, (8.102) φ=φ0 +φL +∆1

which is valid for all types of derived potentials in the present context. Now let us consider the linearized perturbative solution ∆1 in this section. Consequently in the leading order of cosmological perturbation the background linearized version of the equation of motion takes the following form using the prescribed Ansatz:  ¨ 1 + 3H ∆ ˙ 1 + φL Y 2 H 2 (1 − tHH )2 − 2YH 2 H + 2tYH 3 H ηH ∆ (8.103) + 3H 2 Y(1 − tHH ) = 0, where H and ηH are the Hubble slow-roll parameters as given by: H = −

H˙ , H2

ηH = H −

˙H . 2H

(8.104) (8.105)

Here it is important to point that in the equation of motion we have taken the cubic order terms in the slow-roll parameter H . The exact analytical solution of the Eq (8.103) is given by:   1 φ∗ ∆1 = C2 − C1 e−3Ht + eHYt Y(3 + Y)3 3 3H Y(3 + Y)
(8.106) + (3 + Y)H 9 − 6η + 6Y − 4ηH Y + 2Y 2 + H(2ηH − Y − 3)Y(3 + Y)t  + 2H (18 + Y(3 + Y) (6 + Ht(−6 + Y(−4 + Ht(3 + Y))))) . Here C1 and C2 are dimensionful arbitrary integration constants which can be determined by imposing the appropriate boundary condition. Additionally it is important to note that,

– 92 –

in the present context this solution is valid in case quasi de-Sitter case also where the Hubble parameter H is not exactly constant. Additionally it is important to note that during the computation we assume that the H and ηH are the slowly varying time dependent functions. Now for the sake simplicity if we use that fact that the all of the higher order slow-roll corrections are small and sufficiently suppressed, then one can recast the linearized version of the equation of motion takes the following form using the prescribed Ansatz:  ¨ 1 + 3H ∆ ˙ 1 + φL Y 2 H 2 (1 − 2tHH ) − 2YH 2 H + 3H 2 Y(1 − tHH ) = 0, ∆

(8.107)

The exact analytical solution of the Eq (8.107) is given by: ∆1 = D2 −

 1 1 HYt D1 e−3Ht + φ e −Y(3 + Y)2 ∗ 2 3H Y(3 + Y)

(8.108)

+ H (−9 + Y(3 + Y) {−2 + H(3 + 2Y)t})] . Here also D1 and D2 are dimensionful arbitrary integration constants which can be determined by imposing the appropriate boundary condition. Here it is also valid for quasi de-Sitter case. However if we want to write down the result for exact deSitter case then we have to consider the fact that the Hubble parameter is constant and in that case no slow-time variation is allowed. Consequently in that case the linearized version of the equation of motion takes the following form using the prescribed Ansatz: ¨ 1 + 3H ∆ ˙ 1 + φL YH 2 (Y + 3) = 0, ∆

(8.109)

The exact analytical solution of the Eq (8.109) is given by: ∆1 = E2 −

1 E1 e−3Ht − φ∗ eHYt . 3H

(8.110)

Here E1 and E2 are dimensionful arbitrary integration constants which can be determined by imposing the appropriate boundary condition. D. Second-order perturbative solution: Here we have considered the effect from the second order cosmological perturbation, ∆2 . It is important to note that during the computation here we also follow the same Ansatz, which we have already introduced in the last section. As a result including the contribution from slow-roll correction, the perturbative second order background equation of motion takes the following simplified form:  ¨ 2 + 3H ∆ ˙ 2 + φL Y 2 H 2 (1 − tHH )2 − 2YH 2 H + 2tYH 3 H ηH ∆ + 3H 2 Y(1 − tHH ) = ΣS . (8.111)

– 93 –

If we take only the leading order contributions then equation of motion for ∆2 takes the following simplified form:  ¨ 2 + 3H ∆ ˙ 2 + φL Y 2 H 2 (1 − 2tHH ) − 2YH 2 H + 3H 2 Y(1 − tHH ) = ΣS . ∆

(8.112)

Finally, if we neglect all the contributions from the slow-roll corrections then equation of motion for ∆2 can be recast as: ¨ 1 + 3H ∆ ˙ 1 + φL YH 2 (Y + 3) = ΣS . ∆

(8.113)

Here it is important to note that in Eq (8.111), Eq (8.112) and Eq (8.113), ΣS is the source contribution which is commping from the linear order perturbation ∆1 . In this paper ΣS can be expressed for all derived effective potentials as:  2 3((∆1 +φL ) −φ2 0)   2  Mp  (∆1 + φL )4 for Case I Λc e    2 2   ((∆1 +φL ) −φ0 )  Mp3  2 3Mp   − Λ e (∆1 + φL )4 . for Case II c   8α !# "   1 1  ((∆1 +φL )2 −φ20 )+φ4V (∆ +φ  2 − φ2  3 1 0 L)  Mp
 2 3Mp  − Λ e (∆1 + φL )4 − φ4V . for Case II+Choice  c   8α " !#   1 1  ((∆1 +φL )2 −φ20 )+φ4V (∆ +φ 2 − φ2  3  1 0 L)
M  p 2 4  4 3Mp  + Λ e (∆ + φ ) − φ . for Case II+Choice  c 1 L V  8α      Mp3 m2c   + (∆1 + φL )2   2Mp  8α " 2 !# m2 m2 c −λ(∆1 +φL ) ΣS = (∆1 +φL )2 −φ20 )+ λc ln ( 2 2 mc −λφ0   2 4
 3Mp  − Λ (∆ + φ ) e . for Case II+Choice c 1 L        Mp3 m2c   − (∆1 + φL )2   8α 2Mp  "  2 !#  m2 m2 c −λ(∆1 +φL )  (∆1 +φL )2 −φ20 )+ λc ln  ( 2 2  mc −λφ0   2 4
3Mp  − Λ (∆ + φ ) e . for Case II+Choice  c 1 L   
2   Mp3 Λc (∆1 + φL )2 − φ2V    +   2
2 8α  1 + ξ (∆ + φ )  1 L " !#    L)  ((∆1 +φL )2 −φ20 )(1+ 2ξ ((∆1 +φL )2 +φ20 −2φ2V ))+2φ2V ln (∆1φ+φ  0   2 1+ξφ2  3Mp ( V)  ×e . for Case II+Choice (8.114) where we define a new parameter: √

Λc =

Ψ0 λ − 2√32 M p. e 4Mp

– 94 –

(8.115)

I(v1)

I(v2)

II(v1)

II(v2)

III.

From the complicated mathematical structure of the source function ΣS it is clear that using it it is not possible to solve second order perturbation equations. To solve this problem one can simplify the the source function in the following way:     ∆1  4  Λc φL 1 + 4 for Case I    φL       ∆1   . for Case II β − Λc φ4L 1 + 4   φL          ∆1   − φ4V . for Case II+Choice I(v1) β − Λc φ4L 1 + 4   φL        ∆ ΣS ≈ β + Λc φ4L 1 + 4 1 − φ4V . for Case II+Choice I(v2)  φ  L         Mc 2 ∆1 ∆1  4   β+ φ 1+2 − Λc φL 1 + 4 . for Case II+Choice II(v1)   2 L φL φL          ∆1 Mc 2 ∆1  4  β− φ 1+2 − Λc φL 1 + 4 . for Case II+Choice II(v2)    2 L φL φL       ∆ 1   β + Γξ 1 + Θ ξ . for Case II+Choice III.  φL (8.116) where β, Mc and Γc is defined as: Mp3 , 8α m2 Mc = c , Mp

(8.117)

β=

(8.118)


2
Γξ = Λc φ2L − φ2V 1 + 2ξφ2L ,   1 2 . Θξ = 4φL ξ + 2 φL − φ2V

(8.119) (8.120)

The representative solutions of Eq (8.111) four various sources are given by: For Case I : 
1 27Hφ∗ eHYt  ∆2 = C4 + − Y(3 + Y)3 4Λc φ3L + H 2 Y(3 + Y) 3 2 4 27H Y (3 + Y)
+ 2H 4Λc φ3L (54 + Y(3 + Y)(20 + Ht(−12 + Y(−8 + Ht(3 + Y))))) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2 + HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (18 − Y(3 + Y)(−6 + H(3 + Y)t) + 2ηH (−6 + Y(−4 + H(3 + Y)t)))))} + 9H 2 Λc φ3L t(φL + 4C2 ) + e−3Ht 4Λc φ3L (1 + 3Ht)C1 − 9H 2 C3




– 95 –

.

(8.121)

For Case II : 
1 27Hφ∗ eHYt  ∆2 = C4 + − Y(3 + Y)3 −4Λc φ3L + H 2 Y(3 + Y) 3 2 4 27H Y (3 + Y)
+ 2H −4Λc φ3L (54 + Y(3 + Y)(20 + Ht(−12 + Y(−8 + Ht(3 + Y))))) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2

(8.122)

+ HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (2(ηH (6 + 2Y) − 3(3 + Y(3 + Y))) + HY(3 + Y)(3 − 2ηH + 2Y)t)))} + 9H 2 t(β − Λc φ3L (φL + 4C2 )) − e−3Ht 4Λc φ3L (1 + 3Ht)C1 + 9H 2 C3




.

For Case II + Choice I(v1) : 
27Hφ∗ eHYt  1 − Y(3 + Y)3 −4Λc φ3L + H 2 Y(3 + Y) ∆2 = C4 + 3 2 4 27H Y (3 + Y)
+ 2H −4Λc φ3L (54 + Y(3 + Y)(20 + Ht(−12 + Y(−8 + Ht(3 + Y))))) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2

(8.123)

+ HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (2(ηH (6 + 2Y) − 3(3 + Y(3 + Y))) + HY(3 + Y)(3 − 2ηH + 2Y)t)))} + 9H 2 t(β + Λc φ4V − Λc φ3L (φL + 4C2 )) − e−3Ht 4Λc φ3L (1 + 3Ht)C1 + 9H 2 C3




.

For Case II + Choice I(v2) : 
1 27Hφ∗ eHYt  ∆2 = C4 + − Y(3 + Y)3 4Λc φ3L + H 2 Y(3 + Y) 3 2 4 27H Y (3 + Y)
+ 2H 4Λc φ3L (54 + Y(3 + Y)(20 + Ht(−12 + Y(−8 + Ht(3 + Y)))) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2 + HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (18 − Y(3 + Y)(−6 + H(3 + Y)t) + 2ηH (−6 + Y(−4 + H(3 + Y)t)))))} + 9H 2 t(β − Λc φ4V + Λc φ3L (φL + 4C2 )) + e−3Ht 4Λc φ3L (1 + 3Ht)C1 − 9H 2 C3




. (8.124)

– 96 –

For Case II + Choice II(v1) : 
1 54Hφ∗ eHYt  ∆2 = C4 + − Y(3 + Y)3 Mc φL − 4Λc φ3L + H 2 Y(3 + Y) 3 2 4 54H Y (3 + Y) + H (3 + Y) H 2 Y(3 + Y)(9 − 6ηH + 6Y − 4ηH Y + 2Y 2 + HtY(3 + Y)(−3 + 2ηH − 2Y)) + 4Λc φ3L (2(ηH (6 + 4Y) − 3(3 + Y(3 + Y))) + HtY(3 + Y)(3 − 2ηH + 2Y)) + Mc φL (18 − Y(3 + Y)(−6 + H(3 + Y)t) + 2ηH (−6 + Y(−4 + H(3 + Y)t)))) + 2H ((Mc φL − 4Λc φ3L )(54 + Y(3 + Y)(20 + Ht(−12 + Y(−8 + Ht(3 + Y)))))
+ H 2 Y(3 + Y)(18 + Y(3 + Y)(6 + Ht(−6 + Y(−4 + Ht(3 + Y)))) + 9H 2 t(2β + φL (Mc (φL + 2C2 ) − 2Λc φ2L (φL + 4C2 ))) + e−3Ht 2φL (Mc − 4Λc φ2L )(1 + 3Ht)C1 − 18H 2 C3




. (8.125)

For Case II + Choice II(v2) :  1 54Hφ∗ eHYt − × ∆2 = C4 + 54H 3 Y 2 (3 + Y)4 
Y(3 + Y)3 −Mc φL + 4Λc φ3L + H 2 Y(3 + Y) + H (3 + Y) H 2 Y(3 + Y)(9 − 6ηH + 6Y − 4ηH Y + 2Y 2 + HtY(3 + Y)(−3 + 2ηH − 2Y)) − 4Λc φ3L (2(ηH (6 + 4Y) − 3(3 + Y(3 + Y))) + HtY(3 + Y)(3 − 2ηH + 2Y)) − Mc φL (18 − Y(3 + Y)(−6 + H(3 + Y)t) + 2ηH (−6 + Y(−4 + H(3 + Y)t)))) + 2H ((4Λc φ3L − Mc φL )(54 + Y(3 + Y)(20 + Ht(−12 + Y(−8 + Ht(3 + Y)))))
+ H 2 Y(3 + Y)(18 + Y(3 + Y)(6 + Ht(−6 + Y(−4 + Ht(3 + Y)))) + 9H 2 t(2β + φL (−Mc (φL + 2C2 ) + 2Λc φ2L (φL + 4C2 ))) − e−3Ht 2φL (Mc − 4Λc φ2L )(1 + 3Ht)C1 + 18H 2 C3




. (8.126)

– 97 –

For Case III :  1 27Hφ∗ eHYt ∆2 = C4 + − × 27H 3 φL Y 2 (3 + Y)4  Y(3 + Y)3 (Γξ Θξ + H 2 φL Y(3 + Y)) + H (3 + Y)(H 2 φL Y(3 + Y)(9 − 6ηH + 6Y − 4ηH Y + 2Y 2 + Ht(−3 + 2ηH − 2Y)Y(3 + Y)) + Γξ Θξ (18 − Y(3 + Y)(−6 + Ht(3 + 2Y)) + 2ηH (−6 + Y(−4 + Ht(3 + Y))))) + 2H (Γξ Θξ (54 + Y(3 + Y)(20 + Ht(−12 + Y(−8 + Ht(3 + Y)))))
+ H 2 φL Y(3 + Y)(18 + Y(3 + Y)(6 + Ht(−6 + Y(−4 + Ht(3 + Y))))) + 9H 2 t(φL (β + Γξ ) + Γξ Θξ C2 ) − e−3Ht Γξ Θξ (1 + 3Ht)C1 − 9H 2 φL C3




.

(8.127) Here C3 and C4 are dimensionful arbitrary integration constants which can be determined by imposing the appropriate boundary condition. Now if we negelect the quadratic slow-roll corrections then the solution of Eq (8.112) takes the following form for the all different cases considered here: For Case I : 
1 27φ∗ HeHYt  ∆2 = D4 + −Y(3 + Y)2 4Λc φ3L + H 2 Y(3 + Y) 3 2 3 27H Y (3 + Y) (8.128)

+ H 4Λc φ3L (−18 + Y(3 + Y)(−6 + Ht(3 + 2Y)))
+ H 2 Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))) + 9H 2 Λc φ3L t (φL + 4D2 ) + e−3Ht 4Λc φ3L (1 + 3Ht)D1 − 9H 2 D3




.

For Case II :  1 27φ∗ HeHYt ∆2 = D4 + × 27H 3 Y 2 (3 + Y)3
 −Y(3 + Y)2 −4Λc φ3L + H 2 Y(3 + Y) + H

−4Λc φ3L (−18

+ Y(3 + Y)(−6 + Ht(3 + 2Y)))


+ H 2 Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y)))

+ 9H 2 t β − Λc φ3L (φL + 4D2 ) − e−3Ht 4Λc φ3L (1 + 3Ht)D1 + 9H 2 D3 .

– 98 –

(8.129)

For Case II + Choice I(v1) :  1 27φ∗ HeHYt ∆2 = D4 + × 27H 3 Y 2 (3 + Y)3
 −Y(3 + Y)2 −4Λc φ3L + H 2 Y(3 + Y) + H −4Λc φ3L (−18 + Y(3 + Y)(−6 + Ht(3 + 2Y)))
+ H 2 Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y)))

+ 9H 2 t β + Λc φ4V − Λc φ3L (φL + 4D2 ) − e−3Ht 4Λc φ3L (1 + 3Ht)D1 + 9H 2 D3 . (8.130)

For Case II + Choice I(v2) :  27φ∗ HeHYt 1 × ∆2 = D4 + 27H 3 Y 2 (3 + Y)3 
−Y(3 + Y)2 4Λc φ3L + H 2 Y(3 + Y) + H 4Λc φ3L (−18 + Y(3 + Y)(−6 + Ht(3 + 2Y)))
+ H 2 Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y)))

 + 9H 2 t β − Λc φ4V + Λc φ3L (φL + 4D2 ) + e−3Ht 4Λc φ3L (1 + 3Ht)D1 − 9H 2 D3 . (8.131)

For Case II + Choice II(v1) :  1 54φ∗ HeHYt ∆2 = D4 + × 54H 3 Y 2 (3 + Y)3 
−Y(3 + Y)2 Mc φL − 4Λc φ3L + H 2 Y(3 + Y) + H (4Λc φ3L − Mc φL )(18 − Y(3 + Y)(−6 + Ht(3 + 2Y)))
+ H 2 Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))) + 9H 2 t (2β + φL (Mc (φL + 2D2 ) − Λc φ2L (φL + 4D2 ))) + e−3Ht 2φL (Mc − 4Λc φ2L )(1 + 3Ht)D1 − 18H 2 D3

– 99 –




.

(8.132)

For Case II + Choice II(v2) :  1 54φ∗ HeHYt ∆2 = D4 + × 54H 3 Y 2 (3 + Y)3
 −Y(3 + Y)2 −Mc φL + 4Λc φ3L + H 2 Y(3 + Y) (8.133)

+ H (−4Λc φ3L + Mc φL )(18 − Y(3 + Y)(−6 + Ht(3 + 2Y)))
+ H 2 Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))) + 9H 2 t (2β + φL (−Mc (φL + 2D2 ) + Λc φ2L (φL + 4D2 ))) − e−3Ht 2φL (Mc − 4Λc φ2L )(1 + 3Ht)D1 + 18H 2 D3




.

For Case III :  1 27φ∗ HeHYt ∆2 = D4 + × 27H 3 φL Y 2 (3 + Y)3 
−Y(3 + Y)2 Γξ Θξ + H 2 φL Y(3 + Y)

(8.134)

+ H (Γξ Θξ (−18 + Y(3 + Y)(−6 + Ht(3 + 2Y)))
+ H 2 φL Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))) + 9H 2 t (φL (β + Γξ ) + Γξ Θξ D2 ) + e−3Ht Γξ Θξ (1 + 3Ht)D1 − 9H 2 φL D3




.

Here D3 and D4 are dimensionful arbitrary integration constants which can be determined by imposing the appropriate boundary condition. Finally in the exact de Sitter case the contribution from all the slow-roll roll corrections completely vanishes and in that case the solution of Eq (8.113) takes the following form for the all different cases considered here: For Case I : ∆2 = E4 −

 Λc φ3L t φ∗ eHYt  3 2 4Λ φ + H Y(3 + Y) + (φL + 4E2 ) c L H 2 Y(3 + Y) 3H  e−3Ht  2 3 (1 + 3Ht)E − 9H E . + 4Λ φ C 1 3 L 27H 3

(8.135)

For Case II : ∆2 = E4 −

  φ∗ eHYt  t  −4Λc φ3L + H 2 Y(3 + Y) + β − Λc φ3L (φL + 4E2 ) 2 H Y(3 + Y) 3H −3Ht   e − 4ΛC φ3L (1 + 3Ht)E1 + 9H 2 E3 . 3 27H

– 100 –

(8.136)

For Case II + Choice I(v1) : ∆2 = E4 −

  φ∗ eHYt  t  −4Λc φ3L + H 2 Y(3 + Y) + β + Λc φ4V − Λc φ3L (φL + 4E2 ) 2 H Y(3 + Y) 3H −3Ht   e 2 3 (1 + 3Ht)E + 9H E . − 4Λ φ 1 3 C L 27H 3 (8.137)

For Case II + Choice I(v2) : ∆2 = E4 −

  t  φ∗ eHYt  2 3 3 4 (φ + 4E ) + H Y(3 + Y) + + Λ φ 4Λ φ β − Λ φ L 2 c c c L L V H 2 Y(3 + Y) 3H  e−3Ht  3 2 + 4Λ φ (1 + 3Ht)E − 9H E . C 1 3 L 27H 3 (8.138)

For Case II + Choice II(v1) : ∆2 = E4 −

 φ∗ eHYt  3 2 M φ − 4Λ φ + H Y(3 + Y) c L c L H 2 Y(3 + Y)  t  + 2β + φL (Mc (φL + 2C2 ) − Λc φ2L (φL + 4E2 )) 6H  e−3Ht  2 2 2φ (M − 4Λ φ )(1 + 3Ht)E − 18H E . + L c c 1 3 L 54H 3 (8.139)

For Case II + Choice II(v2) : ∆2 = E4 −

 φ∗ eHYt  3 2 −M φ + 4Λ φ + H Y(3 + Y) c L c L H 2 Y(3 + Y)  t  + 2β + φL (−Mc (φL + 2C2 ) + Λc φ2L (φL + 4E2 )) 6H  e−3Ht  2 2 2φ (M − 4Λ φ )(1 + 3Ht)E + 18H E − . L c c 1 3 L 54H 3 (8.140)

For Case III :   φ∗ eHYt t 2 Γ Θ + H Y(3 + Y) + [φL (β + Γξ ) + Γξ Θξ E2 ] ξ ξ 2 H Y(3 + Y)φL 3HφL  e−3Ht  2 − Γ Θ (1 + 3Ht)E − 9H φ E . ξ ξ 1 L 3 27H 3 φL (8.141) Here E3 and E4 are dimensionful arbitrary integration constants which can be determined by imposing the appropriate boundary condition. ∆2 = E4 −

– 101 –

D. Implementation of δN at the final hypersurface: Using the results derived in the previous two sections here our prime objective is to explicitly compute the expression for the cosmological scalar perturbations in terms of the number of e-folds, δN , which we have already introduced earlier. In the present conetxt the truncated version of the background solution of the inflaton field φ corrected upto the second order cosmological perturbations around the reference trajectory, φL ∝ e−YN or φL ∝ eYHt , is generically given by for all the various physical cases are: φ(N ) = φ0 +

φ∗



ˆ 1 (N = 0) + ∆ ˆ 2 (N = 0) 1+∆

 ˆ 1 (N ) + ∆ ˆ 2 (N ) , e−YN + ∆

(8.142)

 ˆ ˆ + ∆1 (t) + ∆2 (t) .

(8.143)

or equivalently one can write: φ(t) = φ0 +

φ∗



ˆ 1 (t = 0) + ∆ ˆ 2 (t = 0) 1+∆

YHt

e

But for the sake of simplicity we use Eq (8.142) as we want to implement the methodology of δN formalism. Additionally, it is important to mention that the symbolˆis introduced in the present context to rescale the integration constants and the perturbative solutions by the field value φ∗ i.e. ˆ 1, ∆1 = φ∗ ∆ ˆ 2. ∆2 = φ∗ ∆

(8.144) (8.145)

Considering the contribution from quadratic slow-roll corrections expression for ∆1 (N = 0) we get:  ∆1 (N = 0) = C2 −

 φ∗ 1 C1 + Y(3 + Y)3 3H Y(3 + Y)3


 + (3 + Y)H 9 − 6η + 6Y − 4ηH Y + 2Y 2 + 2H (18 + 6Y(3 + Y)) . (8.146) Further negelecting the contribution from the qudratic slow roll term and taking upto linear order term in slow-roll we get the following result:   1 1 2 φ −Y(3 + Y) − 2 (−9 + Y(3 + Y)) . D1 + ∗ H 3H Y(3 + Y)2 (8.147) Finally, in exact de Sitter case we get the following simplified expression: ∆1 (N = 0) = D2 −

∆1 (N = 0) = E2 −

– 102 –

1 E1 − φ∗ . 3H

(8.148)

Similarly including the contribution from quadratic slow-roll corrections representative solutions of ∆2 (N = 0) four various sources are given by: For Case I : 
1 27Hφ∗  ∆2 (N = 0) = C4 + − 2 Y(3 + Y)3 4Λc φ3L + H 2 Y(3 + Y) 3 4 27H Y (3 + Y)
+ 2H 4Λc φ3L (54 + 20Y(3 + Y)) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2 + HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (18 + 6Y(3 + Y) − 2ηH (6 + 4Y))))} + (4Λc φ3L C1 − 9H 2 C3 )] . (8.149) For Case II : 
27Hφ∗  1 3 3 2 Y(3 + Y) − −4Λ φ + H Y(3 + Y) ∆2 (N = 0) = C4 + c L 27H 3 Y 2 (3 + Y)4
+ 2H −4Λc φ3L (54 + 20Y(3 + Y)) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2

+ HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (2(ηH (6 + 2Y) − 3(3 + Y(3 + Y))))) − (4Λc φ3L C1 + 9H 2 C3 )] . (8.150) For Case II + Choice I(v1) : 
1 27Hφ∗  3 3 2 ∆2 (N = 0) = C4 + − Y(3 + Y) −4Λ φ + H Y(3 + Y) c L 27H 3 Y 2 (3 + Y)4
+ 2H −4Λc φ3L (54 + 20Y(3 + Y)) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2

+ HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (2(ηH (6 + 2Y) − 3(3 + Y(3 + Y)))) − 4Λc φ3L C1 + 9H 2 C3




. (8.151)

For Case II + Choice I(v2) : 
1 27Hφ∗  3 3 2 − Y(3 + Y) 4Λ φ + H Y(3 + Y) ∆2 (N = 0) = C4 + c L 27H 3 Y 2 (3 + Y)4
+ 2H 4Λc φ3L (54 + 20Y(3 + Y) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2

 +HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (18 + 6Y(3 + Y) − 2ηH (6 + 4Y)) + (4Λc φ3L C1 − 9H 2 C3 ) . (8.152)

– 103 –

For Case II + Choice II(v1) : 
1 54Hφ∗  ∆2 (N = 0) = C4 + − 2 Y(3 + Y)3 Mc φL − 4Λc φ3L + H 2 Y(3 + Y) 3 4 54H Y (3 + Y) + H (3 + Y) H 2 Y(3 + Y)(9 − 6ηH + 6Y − 4ηH Y + 2Y 2 ) + 4Λc φ3L (2(ηH (6 + 4Y) − 3(3 + Y(3 + Y)))) + Mc φL (18 + 6Y(3 + Y) − 2ηH (6 + 4Y))) + 2H ((Mc φL − 4Λc φ3L )(54 + 20Y(3 + Y)) + H 2 Y(3 + Y)(18 + 6Y(3 + Y))} + 2φL (Mc − 4Λc φ2L )C1 − 18H 2 C3




. (8.153)

For Case II + Choice II(v2) : 
1 54Hφ∗  3 3 2 ∆2 (N = 0) = C4 + − Y(3 + Y) −M φ + 4Λ φ + H Y(3 + Y) c L c L 54H 3 Y 2 (3 + Y)4 + H (3 + Y) H 2 Y(3 + Y)(9 − 6ηH + 6Y − 4ηH Y + 2Y 2 ) − 4Λc φ3L (2(ηH (6 + 4Y) − 3(3 + Y(3 + Y)))) − Mc φL (18 + 6Y(3 + Y) − 2ηH (6 + 4Y))) + 2H ((4Λc φ3L − Mc φL )(54 + 20Y(3 + Y)) + H 2 Y(3 + Y)(18 + 6Y(3 + Y))} − 2φL (Mc − 4Λc φ2L )(1 + 3Ht)C1 + 18H 2 C3




. (8.154)

For Case III :  1 27Hφ∗  ∆2 (N = 0) = C4 + − Y(3 + Y)3 (Γξ Θξ + H 2 φL Y(3 + Y)) 3 2 4 27H φL Y (3 + Y) +H (3 + Y)(H 2 φL Y(3 + Y)(9 − 6ηH + 6Y − 4ηH Y + 2Y 2

(8.155)

+ Γξ Θξ (18 + 6Y(3 + Y) − 2ηH (6 + 4Y)))
+ 2H Γξ Θξ (54 + 20Y(3 + Y)) + H 2 φL Y(3 + Y)(18 + 6Y(3 + Y)) − Γξ Θξ C1 − 9H 2 φL C3




.

Now if we negelect the quadratic slow-roll corrections then the solution of ∆2 (N = 0) takes the following form for the all different cases considered here: ∆1 (N = 0) = E2 −

– 104 –

1 E1 − φ∗ . 3H

(8.156)

For Case I : 
1 27φ∗ H  ∆2 (N = 0) = D4 + −Y(3 + Y)2 4Λc φ3L + H 2 Y(3 + Y) 3 2 3 27H Y (3 + Y)
 +H −4Λc φ3L (18 + 6Y(3 + Y)) − H 2 Y(3 + Y)(9 + 2Y(3 + Y)) + (4Λc φ3L D1 − 9H 2 D3 ) . (8.157)

For Case II : 
1 27φ∗ H  2 3 2 ∆2 (N = 0) = D4 + −Y(3 + Y) −4Λ φ + H Y(3 + Y) c L 27H 3 Y 2 (3 + Y)3
 +H 4Λc φ3L (18 + 6Y(3 + Y)) − H 2 Y(3 + Y)(9 + 2Y(3 + Y)) − (4Λc φ3L D1 + 9H 2 D3 ) . (8.158)

For Case II + Choice I(v1) : 
1 27φ∗ H  ∆2 (N = 0) = D4 + −Y(3 + Y)2 −4Λc φ3L + H 2 Y(3 + Y) 3 2 3 27H Y (3 + Y) 
+H 4Λc φ3L (18 + 6Y(3 + Y)) − H 2 Y(3 + Y)(9 + 2Y(3 + Y)) − (4Λc φ3L D1 + 9H 2 D3 ) . (8.159)

For Case II + Choice I(v2) : 
1 27φ∗ H  2 3 2 ∆2 (N = 0) = D4 + −Y(3 + Y) 4Λ φ + H Y(3 + Y) c L 27H 3 Y 2 (3 + Y)3
 +H −4Λc φ3L (18 + 6Y(3 + Y)) − H 2 Y(3 + Y)(9 + 2Y(3 + Y)) + (4Λc φ3L D1 − 9H 2 D3 ) . (8.160)

For Case II + Choice II(v1) : 
1 54φ∗ H  2 3 2 −Y(3 + Y) M φ − 4Λ φ + H Y(3 + Y) ∆2 (N = 0) = D4 + c L c L 54H 3 Y 2 (3 + Y)3 + H (4Λc φ3L − Mc φL )(18 + 6Y(3 + Y))
 − H 2 Y(3 + Y)(9 + 2Y(3 + Y))) + (2φL (Mc − 4Λc φ2L )D1 − 18H 2 D3 ) . (8.161)

– 105 –

For Case II + Choice II(v2) : 
1 54φ∗ H  ∆2 (N = 0) = D4 + −Y(3 + Y)2 −Mc φL + 4Λc φ3L + H 2 Y(3 + Y) 3 2 3 54H Y (3 + Y) + H (−4Λc φ3L + Mc φL )(18 + 6Y(3 + Y))
 − H 2 Y(3 + Y)(9 + 2Y(3 + Y)) − (2φL (Mc − 4Λc φ2L )D1 + 18H 2 D3 ) . (8.162) For Case III : 
1 27φ∗ H  ∆2 (N = 0) = D4 + −Y(3 + Y)2 Γξ Θξ + H 2 φL Y(3 + Y) 3 2 3 27H φL Y (3 + Y) + H (−Γξ Θξ (18 + 6Y(3 + Y))
 − H 2 φL Y(3 + Y)(9 + 2Y(3 + Y)) + (Γξ Θξ D1 − 9H 2 φL D3 ) . (8.163) Finally in the exact de Sitter case the contribution from all the slow-roll roll corrections completely vanishes and in that case the solution of Eq (8.113) takes the following form for the all different cases considered here:

For Case I : ∆2 (N = 0) = E4 −

φ∗ 2 H Y(3 +

Y)

  4Λc φ3L + H 2 Y(3 + Y) +

 1  4ΛC φ3L E1 − 9H 2 E3 . 3 27H (8.164)

For Case II : ∆2 (N = 0) = E4 −

   φ∗ 1  3 2 3 2 −4Λ φ + H Y(3 + Y) − 4Λ φ E + 9H E . c C 1 3 L L H 2 Y(3 + Y) 27H 3 (8.165)

For Case II + Choice I(v1) : ∆2 (N = 0) = E4 −

   φ∗ 1  2 3 2 3 −4Λ φ + H Y(3 + Y) − 4Λ φ E + 9H E . c C 1 3 L L H 2 Y(3 + Y) 27H 3 (8.166)

For Case II + Choice I(v2) : ∆2 (N = 0) = E4 −

   φ∗ 1  3 2 3 2 4Λ φ + H Y(3 + Y) + 4Λ φ E − 9H E . 1 3 c C L L H 2 Y(3 + Y) 27H 3 (8.167)

– 106 –

For Case II + Choice II(v1) : ∆2 (N = 0) = E4 −

φ∗ 2 H Y(3 +

Y)

  Mc φL − 4Λc φ3L + H 2 Y(3 + Y) +

(8.168)

 1  2 2 )E − 18H E . 2φ (M − 4Λ φ 1 3 L c c L 54H 3

For Case II + Choice II(v2) : ∆2 (N = 0) = E4 −

φ∗ 2 H Y(3 +

Y)

  −Mc φL + 4Λc φ3L + H 2 Y(3 + Y) −

(8.169)

 1  2φL (Mc − 4Λc φ2L )E1 + 18H 2 E3 . 3 54H

For Case III :     φ∗ 1 2 2 Γ Θ + H Y(3 + Y) − Γ Θ E − 9H φ E . ξ ξ ξ ξ 1 L 3 H 2 Y(3 + Y)φL 27H 3 φL (8.170) In the present context all of the sets of scaled integration constants parameterizes different trajectories and for our computation we set: ∆2 (N = 0) = E4 −

ˆ k ) = φ∗ , φ(0, W

(8.171)

ˆ k is defined as the collection of all integration constants in a specific situation as where W defined as:    ˆ 1, C ˆ 2, C ˆ 3, C ˆ 4 ].  [C     ˆ k = [W ˆ 1, W ˆ 2, W ˆ 3, W ˆ 4 ] = [D ˆ 1, D ˆ 2, D ˆ 3, D ˆ 4 ]. W       ˆ 1, E ˆ 2, E ˆ 3, E ˆ 4 ].  [E

for quadratic in slow-roll for linear in slow-roll

(8.172)

for de Sitter.

ˆ k , we have Further inverting Eq (8.142), for a specified set of values of the constants W obtained the following simplified expression for δN as a implicit function of the inflaton ˆ k as: field φ, additional field Ψ and W ˆ k ) = N (φ + δφ, Ψ + δΨ, W ˆ k ) − N (φ, Ψ, 0) δN = δN (φ, Ψ, W 2 X 4 X X 1 n m n ˆ m = ∂φnα ∂W ˆ m {N (φα , 0)} δφα Wk . k n!m! α=1 k=0 n,m

– 107 –

(8.173)

Explicitly Eq (8.173) can be written in terms of the constants of integrations as:

δN =

 2  XX 1 n m ˆm  ∂φnα ∂Cˆ m {N (φα , 0)} δφnα C  1  1 n!m!   α=1 n,m    2 X  X 1 n m   ˆm  + ∂φnα ∂Cˆ m {N (φα , 0)} δφnα C  2  2 n!m!   α=1 n,m   2 X  X  1 n m  ˆm  ∂φnα ∂Cˆ m {N (φα , 0)} δφnα C +  3  3  n!m!  α=1 n,m    2 X  X  1 n m  ˆ m.  + ∂φnα ∂Cˆ m {N (φα , 0)} δφnα C  4  4 n!m!   α=1 n,m           2 X  X  1 n m   ˆm ∂φnα ∂Dˆ m {N (φα , 0)} δφnα D  1  1  n!m!  α=1 n,m    2 X  X  1 n m  ˆm  ∂φnα ∂Dˆ m {N (φα , 0)} δφnα D  2 + 2 n!m! α=1 n,m 2 X X  1 n m   ˆm ∂φnα ∂Dˆ m {N (φα , 0)} δφnα D +  3  3  n!m!  α=1 n,m    2 X  X  1 n m  ˆ m.  ∂φnα ∂Dˆ m {N (φα , 0)} δφnα D +  4  4 n!m!   α=1 n,m           2 X  X  1 n m   ˆm  ∂φnα ∂Eˆ m {N (φα , 0)} δφnα E 1   1 n!m!  α=1 n,m    2 X  X  1 n m  ˆm  + ∂φnα ∂Eˆ m {N (φα , 0)} δφnα E  2  2 n!m!   α=1 n,m   2 X  X  1 n m   ˆm  ∂φnα ∂Eˆ m {N (φα , 0)} δφnα E + 3   3 n!m!  α=1 n,m    2 X  X  1 n m  ˆ m.  + ∂ n ∂ m {N (φα , 0)} δφnα E  4  n!m! φα Eˆ 4

for quadratic in slow-roll

for linear in slow-roll

for de Sitter.

α=1 n,m

(8.174) For this computation we have introduced the shift of the inflaton field φ, additional field Ψ and the number of e-foldings N as: φ → φ + δφ, Ψ → Ψ + δΨ, N → N + δN ,

– 108 –

(8.175) (8.176) (8.177)

in both the sides of Eq (8.142), to compute the analytical expression for δN in a iterative way from our present setup. Additionally it is important to note that in this present context φ field and Ψ field are not independent. They are related via Eq (8.99), as we have already mentioned earlier. In the present setup, we have already obtained the second order perturbative solutions of the scalar inflaton field trajectories around the particular reference solution, φL = φ∗ eYHt = φ∗ e−YN , as we have already pointed earlier. Additionally important to note that if we neglect the sub dominant contribution of the form ∆1 ∝ eYHt , then the analysis only holds good only at thsufficiently late time epochs. This directly implies that in this computation if we use such assumption then we choose the initial time in such a way that it is very close to the final time for the number of e-folds N ≤ 1. To serve this purpose the simplest possibility is to choose the initial time epoch is infinitesimally close to the time scale at φ = φ∗ = φ(Ncmb ) = φcmb . For the sake of simplicity one can further assume that the final expression for curvature ˆ k at N = 0 for which the perturbation in δN formalism is independent of the coefficients W following constraints holds good perfectly: m ∂W ˆ k N = 0 ∀m = 1, ......., ∞.

(8.178)

Consequently we get the following simplified expression: 2 X X 1 n ∂φnα {N (φα , 0)} δφnα ζ = δN = n! α=1 n

= [N,φ δφ + N,Ψ δΨ] +

1 [N,φφ δφδφ + (N,φΨ + N,Ψφ ) δφδΨ + N,ΨΨ δΨδΨ] 2!

1 [N,φφφ δφδφδφ + (N,φΨΨ + N,ΨφΨ + N,ΨΨφ ) δφδΨδΨ 3! + (N,φφΨ + N,φΨφ + N,Ψφφ ) δφδφδΨ + N,ΨΨΨ δΨδΨδΨ] + · · ·   0 V (φ) = 2N,φ δφ + 2N,φφ − N,φ δφδφ V(φ)   02   00 00 4 V (φ) 5 V (φ) 1 V (φ) + N,φφφ − 2 N,φφ + − N,φ δφδφδφ + · · · , 3 V(φ) 3 V 2 (φ) 6 V(φ) (8.179) +

where the function V(φ) we have explicitly defined earlier for all the derived effective potentials. Here · · · corresponds to the higher order contributions, which are very very small compared to the leading order contributions appearing from cosmological perturbation theory for scalar fluctuations. Next we take the derivatives of both sides of Eq (8.142) and further set the following two constraints: N = 0, ˆ k = 0 ∀k, W

(8.180) (8.181)

at the final stage of the calculation. Our next task is to derive the analytical expression ˆ k , which are generated via quantum for inflaton fluctuation δφ∗ and the coefficients W

– 109 –

fluctuations on the flat slice of δφ. To implement this computational technique let us consider the evolution of fluctuation in the inflaton field δφ on super horizon scales. The field fluctuation or more precisely the shift in the inflaton field φ can be expressed as: δφ(N ) =

2 X

δφi (N ) = φ∗

i=1

2 X

ˆ i (N ), ∆

(8.182)

i=1

where the subscript ”1” and ”2” signify the linear and second order solution appearing from cosmological perturbation. Additionally, it is important to note that both the solutions ˆ 1 (N ) and ∆ ˆ 2 (N ), contain the growing and decaying mode characteristics. Further im∆ posing the appropriate boundary condition from the end of the non-attractor region, where the number of e-folds N = 0, we get the following analytical expression for the shift in the inflaton field from linear order and second order cosmological perturbation theory: ˆ 1 (N = 0) δφ1 (N = 0) = δφ1∗ = φ∗ ∆   φ∗ φ∗ ˆ ˆ C1 + Y(3 + Y)3 = φ∗ C2 − 3H Y(3 + Y)3
 + (3 + Y)H 9 − 6η + 6Y − 4ηH Y + 2Y 2 + 2H (18 + 6Y(3 + Y)) , and For Case I : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆  
φ∗ 27H 3 3 2 ˆ = φ∗ C4 + − Y(3 + Y) 4Λ φ + H Y(3 + Y) c L 27H 3 Y 2 (3 + Y)4
+ 2H 4Λc φ3L (54 + 20Y(3 + Y)) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2 + HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (18 + 6Y(3 + Y) i  ˆ 1 − 9H 2 C ˆ3 . − 2ηH (6 + 4Y))))} + 4Λc φ3L C For Case II : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆  
27H 3 3 2 ˆ 4 + φ∗ − Y(3 + Y) −4Λ φ + H Y(3 + Y) = φ∗ C c L 27H 3 Y 2 (3 + Y)4
+ 2H −4Λc φ3L (54 + 20Y(3 + Y)) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2

+ HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (2(ηH (6 + 2Y) − 3(3 + Y(3 + Y)))))  i ˆ 1 + 9H 2 C ˆ3 . − 4Λc φ3L C

– 110 –

For Case II + Choice I(v1) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ 
 φ∗ 27H ˆ = φ∗ C4 + − 2 Y(3 + Y)3 −4Λc φ3L + H 2 Y(3 + Y) 3 4 27H Y (3 + Y)
+ 2H −4Λc φ3L (54 + 20Y(3 + Y)) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2

+ HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (2(ηH (6 + 2Y) − 3(3 + Y(3 + Y)))) i  2ˆ 3 ˆ − 4Λc φL C1 + 9H C3 .

For Case II + Choice I(v2) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆  
φ 27H ∗ 3 3 2 ˆ4 + = φ∗ C − Y(3 + Y) 4Λ φ + H Y(3 + Y) c L 27H 3 Y 2 (3 + Y)4
+ 2H 4Λc φ3L (54 + 20Y(3 + Y) + H (3 + Y) H 2 Y(3 + Y) 9 − 6ηH + 6Y − 4YηH + 2Y 2

+HY(3 + Y)(2ηH − 2Y − 3)) + 4Λc φ3L (18 + 6Y(3 + Y) − 2ηH (6 + 4Y))  i 3 ˆ 2ˆ + 4Λc φL C1 − 9H C3 .

For Case II + Choice II(v1) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ 
 φ 54H ∗ 3 3 2 ˆ4 + = φ∗ C − Y(3 + Y) M φ − 4Λ φ + H Y(3 + Y) c L c L 54H 3 Y 2 (3 + Y)4 + H (3 + Y) H 2 Y(3 + Y)(9 − 6ηH + 6Y − 4ηH Y + 2Y 2 ) + 4Λc φ3L (2(ηH (6 + 4Y) − 3(3 + Y(3 + Y)))) + Mc φL (18 + 6Y(3 + Y) − 2ηH (6 + 4Y))) + 2H ((Mc φL − 4Λc φ3L )(54 + 20Y(3 + Y)) + H 2 Y(3 + Y)(18 + 6Y(3 + Y))}  i ˆ 1 − 18H 2 C ˆ3 . + 2φL (Mc − 4Λc φ2L )C

– 111 –

For Case II + Choice II(v2) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ 
 φ∗ 54H ˆ = φ∗ C4 + − 2 Y(3 + Y)3 −Mc φL + 4Λc φ3L + H 2 Y(3 + Y) 3 4 54H Y (3 + Y) + H (3 + Y) H 2 Y(3 + Y)(9 − 6ηH + 6Y − 4ηH Y + 2Y 2 ) − 4Λc φ3L (2(ηH (6 + 4Y) − 3(3 + Y(3 + Y)))) − Mc φL (18 + 6Y(3 + Y) − 2ηH (6 + 4Y))) + 2H ((4Λc φ3L − Mc φL )(54 + 20Y(3 + Y)) + H 2 Y(3 + Y)(18 + 6Y(3 + Y))}  i ˆ 1 + 18H 2 C ˆ3 . − 2φL (Mc − 4Λc φ2L )(1 + 3Ht)C

For Case III : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆   φ∗ 27H ˆ = φ∗ C4 + − Y(3 + Y)3 (Γξ Θξ + H 2 φL Y(3 + Y)) 27H 3 φL Y 2 (3 + Y)4 +H (3 + Y)(H 2 φL Y(3 + Y)(9 − 6ηH + 6Y − 4ηH Y + 2Y 2 + Γξ Θξ (18 + 6Y(3 + Y) − 2ηH (6 + 4Y)))
+ 2H Γξ Θξ (54 + 20Y(3 + Y)) + H 2 φL Y(3 + Y)(18 + 6Y(3 + Y))  i ˆ 1 − 9H 2 φL C ˆ3 , − Γξ Θξ C

where we have considered the contributions from the quadractic slow-roll correction terms. Next we have taken the contributions from the first order slow-roll contribution and for this we get:

ˆ 1 (N = 0) δφ1 (N = 0) = δφ1∗ = φ∗ ∆ ˆ2 − = φ∗ D

 φ∗ φ∗ ˆ D1 + −Y(3 + Y)2 2 3H Y(3 + Y)

+ H (−9 + Y(3 + Y) {−2 + H(3 + 2Y)t})] .

– 112 –

For Case I : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ 
 φ∗ 27H ˆ = φ∗ D4 + −Y(3 + Y)2 4Λc φ3L + H 2 Y(3 + Y) 3 2 3 27H Y (3 + Y) i
 ˆ 1 − 9H 2 D ˆ3 . +H −4Λc φ3L (18 + 6Y(3 + Y)) − H 2 Y(3 + Y)(9 + 2Y(3 + Y)) + 4Λc φ3L D For Case II : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ 
 φ∗ 27H 3 2 2 ˆ −4Λ φ + H Y(3 + Y) = φ∗ D4 + −Y(3 + Y) c L 27H 3 Y 2 (3 + Y)3 i
 3 2 3 ˆ 2ˆ +H 4Λc φL (18 + 6Y(3 + Y)) − H Y(3 + Y)(9 + 2Y(3 + Y)) − 4Λc φL D1 + 9H D3 . For Case II + Choice I(v1) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ 
 φ∗ 27H ˆ = φ∗ D4 + −Y(3 + Y)2 −4Λc φ3L + H 2 Y(3 + Y) 3 2 3 27H Y (3 + Y) i
 ˆ 1 + 9H 2 D ˆ3 . +H 4Λc φ3L (18 + 6Y(3 + Y)) − H 2 Y(3 + Y)(9 + 2Y(3 + Y)) − 4Λc φ3L D For Case II + Choice I(v2) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆  
27H 2 3 2 ˆ 4 + φ∗ = φ∗ D −Y(3 + Y) 4Λ φ + H Y(3 + Y) c L 27H 3 Y 2 (3 + Y)3 i
 3 2 3 ˆ 2ˆ +H −4Λc φL (18 + 6Y(3 + Y)) − H Y(3 + Y)(9 + 2Y(3 + Y)) + 4Λc φL D1 − 9H D3 . For Case II + Choice II(v1) :  
φ∗ 54H 2 3 2 ˆ −Y(3 + Y) M φ − 4Λ φ + H Y(3 + Y) ∆2 (N = 0) = φ∗ D4 + c L c L 54H 3 Y 2 (3 + Y)3 + H (4Λc φ3L − Mc φL )(18 + 6Y(3 + Y)) i
 ˆ 1 − 18H 2 D ˆ3 . − H 2 Y(3 + Y)(9 + 2Y(3 + Y))) + 2φL (Mc − 4Λc φ2L )D (8.183)

– 113 –

For Case II + Choice II(v2) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ 
 φ∗ 54H ˆ = φ∗ D4 + −Y(3 + Y)2 −Mc φL + 4Λc φ3L + H 2 Y(3 + Y) 3 2 3 54H Y (3 + Y) + H (−4Λc φ3L + Mc φL )(18 + 6Y(3 + Y)) i
 ˆ 1 + 18H 2 D ˆ3 . − H 2 Y(3 + Y)(9 + 2Y(3 + Y)) − 2φL (Mc − 4Λc φ2L )D For Case III : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆  
φ 27H ∗ 2 2 ˆ4 + = φ∗ D −Y(3 + Y) Γ Θ + H φ Y(3 + Y) ξ ξ L 27H 3 φL Y 2 (3 + Y)3 + H (−Γξ Θξ (18 + 6Y(3 + Y)) i
 2 ˆ ˆ − H φL Y(3 + Y)(9 + 2Y(3 + Y)) + Γξ Θξ D1 − 9H φL D3 . 2

Further in the exact de Sitter case if we neglect the contributions from all slow-roll corrections we get the following result: ˆ2 − ˆ 1 (N = 0) = φ∗ E δφ1 (N = 0) = δφ1∗ = φ∗ ∆

φ∗ ˆ E1 − φ∗ . 3H

(8.184)

For Case I : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ ˆ4 − = φ∗ E

i   φ∗ φ∗ h 3 2 3 ˆ 2ˆ 4Λ φ E − 9H E 4Λ φ + H Y(3 + Y) + c L C L 1 3 . H 2 Y(3 + Y) 27H 3

For Case II : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ ˆ4 − = φ∗ E

i   φ∗ φ∗ h 3 2 3 ˆ 2ˆ −4Λc φL + H Y(3 + Y) − 4ΛC φL E1 + 9H E3 . H 2 Y(3 + Y) 27H 3

For Case II + Choice I(v1) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ ˆ4 − = φ∗ E

i   φ∗ φ∗ h 3 2 3 ˆ 2ˆ −4Λ φ + H Y(3 + Y) − 4Λ φ E + 9H E c L C L 1 3 . H 2 Y(3 + Y) 27H 3

– 114 –

For Case II + Choice I(v2) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ ˆ4 − = φ∗ E

i   φ∗ φ∗ h 3 2 3 ˆ 2ˆ E 4Λ φ + H Y(3 + Y) + 4Λ φ E − 9H 3 . c L C L 1 H 2 Y(3 + Y) 27H 3

For Case II + Choice II(v1) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ ˆ4 − = φ∗ E

φ∗ 2 H Y(3 +

Y)

  Mc φL − 4Λc φ3L + H 2 Y(3 + Y) +

i φ∗ h 2 ˆ 2ˆ E . 2φ (M − 4Λ φ ) E − 18H 3 L c c 1 L 54H 3

For Case II + Choice II(v2) : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ ˆ4 − = φ∗ E

φ∗ 2 H Y(3 +

Y)

  −Mc φL + 4Λc φ3L + H 2 Y(3 + Y) −

i φ∗ h 2 ˆ 2ˆ 2φ (M − 4Λ φ ) E + 18H E 1 3 . L c c L 54H 3

For Case III : ˆ 2 (N = 0) δφ2 (N = 0) = δφ2∗ = φ∗ ∆ ˆ4 − = φ∗ E

h i   φ∗ φ∗ 2 2 ˆ ˆ Γ Θ E − 9H φ E Γ Θ + H Y(3 + Y) − ξ ξ ξ ξ 1 L 3 . H 2 Y(3 + Y)φL 27H 3 φL

Consequently shift in inflaton field at φ = φ∗ is given by: δφ∗ = δφ(0) =

2 X i=1

δφi∗ = φ∗

2 X

  ˆ i (0) = φ∗ ∆ ˆ 1 (0) + ∆ ˆ 2 (0) . ∆

(8.185)

i=1

Now in the present context as we have started our computation from the reference solution φ ∝ e−YN , then using this relationship one can write down the explicit expression for the number of e-folds in terms of the inflaton field value as:   1 φ∗ N (φ) = ln , (8.186) Y φ which is consistent with the boundary condition that at φ = φ∗ the number of e-folds is N = 0 in the present case.

– 115 –

Further using Eq (8.186) one can compute the expressions for the various derivatives of number of e-folds as: 1 , Yφ 1 = , Yφ2 2 = − 3. Yφ

N,φ = − N,φφ N,φφφ

(8.187) (8.188) (8.189)

Using these results at φ = φ∗ one can write down following expression for the curvature perturbation using δN formalism as:

ζ = δN 2 X X 1 n ∂φn∗α {N (φ∗α , 0)} δφn∗α = n! α=1 n = [N,φ δφ + N,Ψ δΨ]∗ +

1 [N,φφ δφδφ + (N,φΨ + N,Ψφ ) δφδΨ + N,ΨΨ δΨδΨ]∗ 2!

1 [N,φφφ δφδφδφ + (N,φΨΨ + N,ΨφΨ + N,ΨΨφ ) δφδΨδΨ 3! + (N,φφΨ + N,φΨφ + N,Ψφφ ) δφδφδΨ + N,ΨΨΨ δΨδΨδΨ]∗ + · · ·   0 V (φ) = 2 {N,φ }∗ δφ∗ + 2N,φφ − N,φ δφ∗ δφ∗ V(φ) ∗   02   0 00 4 V (φ) 5 V (φ) 1 V (φ) + N,φφφ − 2 N,φφ + − N,φ δφ∗ δφ∗ δφ∗ + · · · , 3 V(φ) 3 V 2 (φ) 6 V(φ) ∗ = A(φ∗ )δφ∗ + B(φ∗ )δφ∗ δφ∗ + C(φ∗ )δφ∗ δφ∗ δφ∗ + · · · , (8.190) +

where A(φ∗ ), B(φ∗ ) and C(φ∗ ) is defined as:

2 , A(φ∗ ) = − Yφ∗   0   2 V (φ) 1 B(φ∗ ) = + , Yφ2∗ V(φ) ∗ Yφ∗   00  2 4 V (φ) 1 C(φ∗ ) = − 3 − 2 Yφ∗ 3 V(φ) ∗ Yφ2∗   02   00   5 V (φ) 1 V (φ) 1 − − . 3 V 2 (φ) ∗ 6 V(φ) ∗ Yφ∗

– 116 –

(8.191) (8.192)

(8.193)

For the derived effective potentials B(φ∗ ) and C(φ∗ ) can be recast as:   3     Yφ2∗      φ4V   3 +  φ4∗ 1       Yφ2 φ4V   ∗ 1 − φ4  ∗   B(φ∗ ) =   2 2  1 2λmc φ∗     3−   m2c  Yφ2∗  2 − λφ2 )2 1 −  (m  2 2 c ∗  (mc −λφ ∗)      φ2   1 + ξ(3φ2∗ − φ2V ) − φV2   2  ∗  2+   φ2V   Yφ2∗  2 2 1 + ξ(φ∗ − φV ) + φ2

for Case I & II

for Case II+Choice I(v1& v2)

for Case II+Choice II(v1& v2)

for Case II+Choice III.

∗

(8.194) and   19 1   −  3   ∗  3 Yφ    2    φ4V φ4V      1  8 2 1 + 2 φ4∗ 5 1 + 3 φ4∗     −  − 3 +  2   φ4V  φ4V Yφ∗  3 3     1 − 4 1 −  4 φ φ∗ ∗       2 2λm2c φ2∗  c   8 2 1 − m2m  2 − (m2 −λφ2 )2 −λφ 1  c ∗  c ∗ 2   − 3 +  m   Yφ 3 c  ∗ 1 − m2 −λφ  2  c ∗    2  2  mc 2λm2c φ2∗   1 − −  m2c −λφ2∗ (m2c −λφ2∗ )2 5   +   2   2 3  c  1 − m2m  2 c −λφ∗    C(φ∗ ) ≈ 6m2c λφ∗ 8λ2 m2c φ3∗ +  1 (m2c −λφ2∗ )2 (m2c −λφ2∗ )3       +  m2c   6  1 − m2 −λφ2   c ∗      φ2V  2 2   2 1 + ξ(3φ − φ ) − 2  ∗ V φ∗ 1 8   − 3 +  2  φ  Yφ∗  3  1 + ξ(φ2∗ − φ2V ) + φV2  ∗   2    φ2V  2 2     5 1 + ξ(3φ∗ − φV ) − φ2∗  +     3 (1 + ξ(φ2 − φ2 ) + φ2V2 )2   ∗ V φ∗        φ2V 2   6ξφ + 2  2 ∗  φ∗ 1     −     6 1 + ξ(φ2 − φ2 ) + φ2V  2 ∗ V φ

for Case I & II

for Case II+Choice I(v1& v2)

for Case II+Choice II(v1&v2)

for Case II+Choice III.

∗

(8.195) Next we decompose the product of the fluctuation in the inflaton field δφ∗ δφ∗ and δφ∗ δφ∗ δφ∗ into two parts which comes from linear and second order cosmological perturbation in the

– 117 –

following way: δφ∗ δφ∗ = δφ(0)δφ(0) 2 X 2 X = δφi∗ δφj∗ i=1 j=1

= φ2∗ =

2 X 2 X

ˆ i (0)∆ ˆ j (0) ∆

i=1 j=1 2 δφ1∗ + δφ22∗

= φ2∗



+ 2δφ1∗ δφ2∗  ˆ 1 (0)∆ ˆ 2 (0) . ˆ 22 (0) + 2∆ ˆ 21 (0) + ∆ ∆

(8.196)

δφ∗ δφ∗ δφ∗ = δφ(0)δφ(0)δφ(0) 2 X 2 X 2 X = δφi∗ δφj∗ δφk∗ i=1 j=1 k=1 2 X 2 X 2 X

=

φ3∗

=

i=1 j=1 k=1 3 δφ1∗ + δφ32∗ +

= φ3∗



ˆ i (0)∆ ˆ j (0)∆ ˆ k (0) ∆

3δφ21∗ δφ2∗ + 3δφ1∗ δφ22∗  2 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ∆1 (0) + ∆2 (0) + 3∆1 (0)∆2 (0) + +3∆1 (0)∆ (0) . 2

(8.197)

an write down following expression for the curvature perturbation using δN formalism as: ζ = δN = A(φ∗ )δφ∗ + B(φ∗ )δφ∗ δφ∗ + C(φ∗ )δφ∗ δφ∗ δφ∗ + · · · = A(φ∗ )δφ(0) + B(φ∗ )δφ(0)δφ(0) + C(φ∗ )δφ(0)δφ(0)δφ(0) + · · · 2 2 X 2 2 X 2 X 2 X X X δφi∗ δφj∗ δφk∗ + · · · = A(φ∗ ) δφi∗ + B(φ∗ ) δφi∗ δφj∗ + C(φ∗ ) i=1

= φ∗ A(φ∗ )

2 X

i=1 j=1

ˆ i (0) + φ2 B(φ∗ ) ∆ ∗

i=1

= A(φ∗ ) (δφ1∗ + δφ2∗ ) + B(φ∗ )

2 X 2 X

i=1 j=1 k=1

ˆ i (0)∆ ˆ j (0) + φ3 C(φ∗ ) ∆ ∗

i=1 j=1

δφ21∗

2 X 2 X 2 X

ˆ i (0)∆ ˆ j (0)∆ ˆ k (0) + · · · ∆

i=1 j=1 k=1

δφ22∗




+ + 2δφ1∗ δφ2∗
+ C(φ∗ ) δφ31∗ + δφ32∗ + 3δφ21∗ δφ2∗ + 3δφ1∗ δφ22∗ + · · ·     ˆ 21 (0) + ∆ ˆ 22 (0) + 2∆ ˆ 1 (0)∆ ˆ 2 (0) ˆ 1 (0) + ∆ ˆ 2 (0) + φ2∗ B(φ∗ ) ∆ = φ∗ A(φ∗ ) ∆   3 3 3 2 2 ˆ ˆ ˆ ˆ ˆ ˆ + φ∗ C(φ∗ ) ∆1 (0) + ∆2 (0) + 3∆1 (0)∆2 (0) + 3∆1 (0)∆2 (0) + · · · ,

(8.198)

Further using local configuration in momentum space one can define the non-Gaussian amplitude associated with the three point function using δN formalism as [122]: fNlocL =

B(k1 , k2 , k3 ) 5 N,IJ N,I N,J 5 = , 6 [Pζ (k1 )P (k2 ) + Pζ (k2 )Pζ (k3 ) + Pζ (k3 )Pζ (k1 )] 6 (N,K N,K )2

– 118 –

(8.199)

Here B(k1 , k2 , k3 ) is the bispectrum and Pζ (k) is the power spectrum for scalar perturbations. Here I, J, K are the field configuration indices i.e. I, J, K = φ, Ψ. In terms of inflaton field Φ and additional field Ψ we get the following simplified expression for the non-Gaussian amplitude associated with the three point function:   5 N,φφ N,φ N,φ + N,ΨΨ N,Ψ N,Ψ + (N,φΨ + N,Ψφ ) N,φ N,Ψ loc . (8.200) fN L = 6 (N,φ N,φ + N,Ψ N,Ψ )2 ∗ Now as the Ψ field can be expressed in therms of φ field, using this crucial fact we get the following result of non-Gaussian amplitude in the attractor regime as:    0 2 1 V (φ) 1  5  1 + V 2 (φ) + V 4 (φ) N,φφ V 3 (φ)  − (8.201) fNlocL =   .  2 2 1 6 N,φ N,φ 1 1 + 1 + V 2 (φ) V 2 (φ) ∗

Further substituting the explicit form of the function V(φ) and N,φ , N,φφ for all derived effective potentials at φ = φ∗ we get: 5Y [G1 (φ∗ ) + G2 (φ∗ )φ∗ ] , 6 where the functions G1 (φ∗ ) and G2 (φ∗ ) are defined as:   12Mp2 36Mp4  1 + +  2 4 81φ∗ 6561φ∗   for     2 2 6M  p  1 +  2 81φ∗      12Mp2 36Mp4   1 + + 2 4  φ∗ φ∗   for 2   2  6Mp   1 +  φ2∗       4 2   1 + 12Mp 2 + 36Mp 4    φ4 φ4   φ2∗ 1− V4 φ4∗ 1− V4  φ φ∗  ∗  for  2     2   1 +  6Mp 2    4 φ  φ2∗ 1− V4 φ∗  G1 (φ∗ ) =    2 4   1 +  12Mp 2 +  36Mp 4    m2 m2  φ2∗ 1− 2 c 2 φ4∗ 1− 2 c 2  mc −λφ∗ mc −λφ∗    for  2     2  1 +  6Mp 2    2  m c  φ2∗  2 m2  c −λφ∗        12Mp2 36Mp4  1 +  2 + 4    2  φ φ2  2 2 V V 2 2 4 2 φ∗ 1+ξ(φ∗ −φV )+ 2 φ∗ 1+ξ(φ∗ −φV )+ 2   φ∗ φ∗   for  2     2 6Mp   1 +  2    φ2 2  V 2 2 fNlocL =

φ∗ 1+ξ(φ∗ −φV )+

(8.202)

Case I

Case II

Case II+Choice I(v1&v2)

Case II+Choice II(v1&v2)

Case II+Choice III.

φ2 ∗

(8.203)

– 119 –

and

G2 (φ∗ ) =

 6Mp2   3    81φ∗ 2    6M   1 + 81φp2  ∗    6Mp2   3    φ∗ 2    6M   1 + φ2p   ∗   4   2 1+3 φV  6M p 4  φ ∗    3  φ4  V 3  φ∗ 1− 4  φ∗         2  6Mp    1 + 2  φ4V 2   φ∗ 1− 

for Case I

for Case II

for Case II+Choice I(v1&v2)

φ4 ∗  m2 2λm2 φ2 1− 2 c 2 − 2 c 2∗ 2 mc −λφ∗ (mc −λφ∗ ) 3  2 mc φ3∗ 2 2 mc −λφ∗

 6Mp2                2   1 +  6Mp 2     m2 c  φ2∗  2 m2  c −λφ∗     φ2  2 2 2 6Mp 1+ξ(3φ∗ −φV )− V2   φ∗  3   2  φ  3 1+ξ(φ2 −φ2 )+ V φ  ∗ ∗ 2 V  φ∗         6Mp2   2   1 +   φ2  φ2∗ 1+ξ(φ2∗ −φ2V )+ V2

for Case II+Choice II(v1&v2)

for Case II+Choice III.

φ∗

(8.204) Here it is important to note that the exact momentum dependence will not be calculable using the semi classical techniques used in δN formalism in the attractor regime of cosmological perturbations. But to know the exact momentum dependence of the non-Gaussian amplitude obtained from the three point function of the scalar curvature fluctuation it is always useful to follow exact quantum mechanical techniques used in in-in formalism as discussed earlier part of this section. In case of in-in formalism we freeze the the value of the additional field Ψ at the Planck scale and perform the calculation in the non-attractor regime of perturbation theory. But to get the correct estimate one can claim that the results obtained using both of the techniques should match at the horizon crossing iff we freeze the value of the Ψ field at the Planck scale in δN formalism. This is also a strong information from the observational point of view, as Planck and the other future observation trying to probe the value of non-Gaussianity at this scale. In this work, we have done both the calculations for three point function for scalar curvature fluctuation by following semi classical and quantum mechanical techniques. In case of in-in formalism we have computed the results we use two physical shape configurations or templates- equilateral and squeezed to analyse the non-Gaussian amplitude obtained from the three point function for scalar curvature fluctuation by freezing the value of the additional Ψ field at the Planck scale. Now to implement the equality between two results at the horizon crossing we have to fix

– 120 –

the value of the additional field Ψ in the δN formalism also. After freezing the value of Ψ in the all derived effective potentials we get the following result for curvature perturbation in terms of δN at φ = φ∗ : ζ = δN = D(φ∗ )δφ∗ + E(φ∗ )δφ∗ δφ∗ + F(φ∗ )δφ∗ δφ∗ δφ∗ + · · · = D(φ∗ )δφ(0) + E(φ∗ )δφ(0)δφ(0) + F(φ∗ )δφ(0)δφ(0)δφ(0) + · · · 2 2 X 2 2 X 2 X 2 X X X = D(φ∗ ) δφi∗ + E(φ∗ ) δφi∗ δφj∗ + F(φ∗ ) δφi∗ δφj∗ δφk∗ + · · · i=1

= φ∗ D(φ∗ )

2 X

i=1 j=1

ˆ i (0) + φ2 E(φ∗ ) ∆ ∗

i=1

2 X 2 X

i=1 j=1 k=1

ˆ i (0)∆ ˆ j (0) + φ3 F(φ∗ ) ∆ ∗

i=1 j=1

= D(φ∗ ) (δφ1∗ + δφ2∗ ) + E(φ∗ )

δφ21∗

2 X 2 X 2 X

ˆ i (0)∆ ˆ j (0)∆ ˆ k (0) + · · · ∆

i=1 j=1 k=1

δφ22∗




+ + 2δφ1∗ δφ2∗
+ F(φ∗ ) δφ31∗ + δφ32∗ + 3δφ21∗ δφ2∗ + 3δφ1∗ δφ22∗ + · · ·     ˆ 1 (0) + ∆ ˆ 2 (0) + φ2∗ E(φ∗ ) ∆ ˆ 21 (0) + ∆ ˆ 22 (0) + 2∆ ˆ 1 (0)∆ ˆ 2 (0) = φ∗ D(φ∗ ) ∆   3 3 3 2 2 ˆ ˆ ˆ ˆ ˆ ˆ + φ∗ F(φ∗ ) ∆1 (0) + ∆2 (0) + 3∆1 (0)∆2 (0) + 3∆1 (0)∆2 (0) + · · · ,

(8.205)

where the new functions D(φ∗ ), E(φ∗ ) and F(φ∗ ) are defined as: 1 , Yφ∗ 1 1 , E(φ∗ ) = (N,φφ )∗ = 2 2Yφ2∗ 1 1 . F(φ∗ ) = (N,φφφ )∗ = − 6 3Yφ3∗ D(φ∗ ) = (N,φ )∗ = −

(8.206) (8.207) (8.208)

After freezing the value Ψ in the Planck scale in the non attractor regime of cosmological perturbation theory we get the following expression for the non-Gaussian amplitude from three point scalar curvature fluctuation as:   5 N,φφ 5 loc fN L = = Y. (8.209) 6 N,φ ∗ 6 Now further we use the general momentum dependent result at the horizon crossing and also use two different templates to equate with the results obtained from δN formalism and finally we get folowwing expression for the unknown factor Y as: " 3 X 1 ∗ ∗ Y = P3 2(2H − ηH ) ki3 2 i=1 ki3 i=1 !# 3 3 3 X X X 8 + ∗H − ki3 + ki kj2 + ki2 kj2 . (8.210) K i=1 i,j=1,i>j i,j=1,i6=j

– 121 –

or equivalently in terms of potential dependent slow-roll parameter we get: Y ≈

"

1 2

2(3∗W˜

P3

3 i=1 ki

−

∗ ηW ˜)

3 X

ki3

i=1

+

∗W˜

−

3 X i=1

ki3

+

3 X i,j=1,i6=j

ki kj2

8 + K

3 X

!# ki2 kj2

.

(8.211)

i,j=1,i>j

However it is crucial to note that, without freezing the value of the addition field Ψ in the Planck scale in the non attractor regime of cosmological perturbation theory one can perform the exact quantum mechanical in-in calculation where solution of the Ψ field is related to the inflaton field φ and finally match with the results obtained from the δN formalism. In this paper we have not computed this in case in in formalism and we also hope to generalize this methodology in the attarctor regime as well in near future. Next we use use the two physical templates for the shape configurations-equilateral and squeezed to determine the functional form of the unknown factor Y which is appearing in δN formalism. In this context we get: 1. Equilateral limit configuration:

Y =

 1 1 ∗ ∗ ∗ [23∗H − 6ηH ]≈ 29W˜ − 6ηW ˜ . 6 6

(8.212)

2. Squeezed limit configuration: "  3 #  2 kS 1 k S ∗ ∗ Y = 4(3∗H − ηH ) + 10∗H − (∗H + 2ηH ) 2 kL kL "  3 #  2 kS 1 k S ∗ ∗ ∗ ∗ − (2ηW ≈ 4(4∗W˜ − ηW . ˜) ˜ ) + 10W ˜ ˜ − W 2 kL kL

(8.213)

which are correct results of the unknown factor Y at the level of three point function computed from scalar curvature perturbation. 8.2 8.2.1

Four point function Using In-In formalism

Here we discuss about the constraint on the primordial four point scalar correlation function in the non attractor regime of soft inflation. In general one can write down the following expressions for the four point function of the scalar fluctuation as [148–153]: hζ(k1 )ζ(k2 )ζ(k3 )ζ(k4 )i = (2π)3 δ (3) (k1 + k2 + k3 + k4 )T (k1 , k2 , k3 , k4 ).

– 122 –

(8.214)

In our computation we choose Bunch-Davies vacuum state and for single field soft slow-roll inflation we get the following expression for the trispectrum: h H6 1 ˆ S (k1 , k2 , k3 , k4 ) + G ˆ S (k1 , k3 , k2 , k4 ) G 8Mp6 (∗H )2 (k1 k2 k3 k4 )3 ˆ S (k1 , k4 , k3 , k2 ) − W ˆ S (k1 , k2 , k3 , k4 ) +G ˆ S (k1 , k3 , k2 , k4 ) − W ˆ S (k1 , k4 , k3 , k2 ) −W n oi S S S ˆ ˆ ˆ − 2 R (k1 , k2 , k3 , k4 ) + R (k1 , k3 , k2 , k4 ) + R (k1 , k4 , k3 , k2 ) , h ˜ 3 (φcmb , Ψ) 1 W ˆ S (k1 , k2 , k3 , k4 ) + G ˆ S (k1 , k3 , k2 , k4 ) ≈ G 216Mp12 (∗W˜ )2 (k1 k2 k3 k4 )3 ˆ S (k1 , k4 , k3 , k2 ) − W ˆ S (k1 , k2 , k3 , k4 ) +G ˆ S (k1 , k3 , k2 , k4 ) − W ˆ S (k1 , k4 , k3 , k2 ) −W n oi ˆ S (k1 , k2 , k3 , k4 ) + R ˆ S (k1 , k4 , k3 , k2 ) , ˆ S (k1 , k3 , k2 , k4 ) + R −2 R

T (k1 , k2 , k3 , k4 ) =

(8.215)

ˆ S (k1 , k2 , k3 , k4 ), W ˆ S (k1 , k2 , k3 , k4 ) and where the momentum dependent functions G ˆ S (k1 , k2 , k3 , k4 ) are defined as: R ˜ ˜ ˆ S (k1 , k2 , k3 , k4 ) = S(k, k1 , k2 )S(k, k3 , k4 ) × G |k1 + k2 |3   [k1 .(k1 + k2 )] [k3 .(k3 + k4 )] k1 .k3 + |k + k2 |2  1  [k2 .(k1 + k2 )] [k4 .(k3 + k4 )] × k2 .k4 + |k1 + k2 |2   [k1 .(k1 + k2 )] [k4 .(k3 + k4 )] + k1 .k4 + |k + k2 |2  1  [k2 .(k1 + k2 )] [k3 .(k3 + k4 )] × k2 .k3 + |k1 + k2 |2   [k1 .(k1 + k2 )] [k2 .(k3 + k4 )] − k1 .k2 + |k + k2 |2  1  [k3 .(k3 − k4 )] [k4 .(k3 − k4 )] × k3 .k4 + |k1 + k2 |2 (8.216) with "

3 X 1 Y ˜ k1 , k2 ) = K − 1 ki kj − 2 ki S(k, K i>j K i=1

– 123 –

# , ˜ k=−(k 1 +k2 )

(8.217)

and ˆ S (k1 , k2 , k3 , k4 ) = 1 A1 (k1 , k2 , k3 , k4 ) + 1 A2 (k1 , k2 , k3 , k4 ) + 1 A3 (k1 , k2 , k3 , k4 ) R ˆ ˆ2 ˆ3 K K K 3 X 1 = A (k , k , k , k ), (8.218) ˆn n 1 2 3 4 K n=1

where ˆ = k1 + k2 + k3 + k4 = K

4 X

ki = K + k4 ,

(8.219)

i=1

and the momentum dependent functions A1 (k1 , k2 , k3 , k4 ), A2 (k1 , k2 , k3 , k4 ) and A3 (k1 , k2 , k3 , k4 ) are defined as:  (k3 .k4 )((k1 .k2 )(k12 + k22 ) + 2k12 k22 ) + (1, 2 ↔ 3, 4) A1 (k1 , k2 , k3 , k4 ) = 8|k1 + k2 |2 (k 2 k 2 (k2 .k3 ) + k12 k32 (k2 .k4 ) + k22 k42 (k1 .k3 ) + k12 k42 (k2 .k3 )) − 1 4 2|k1 + k2 |2 ((k1 .k2 )(k12 + k22 ) + 2k12 k22 )((k3 .k4 )(k32 + k42 ) + 2k32 k42 ) . (8.220) − 8|k1 + k2 |4 

[k3 k4 (k3 + k4 )((k1 .k2 )(k12 + k22 ) + 2k12 k22 )(k3 k4 + k3 .k4 ) + (3, 4 ↔ 1, 2)] 8|k1 + k2 |4  2 2 1 − k1 k4 (k2 .k3 )(k2 + k3 ) + k12 k32 (k2 .k4 )(k2 + k4 ) 2 2|k1 + k2 |  + k22 k42 (k1 .k3 )(k1 + k3 ) + k22 k32 (k1 .k4 )(k1 + k4 )  (k1 .k2 )  ((k1 + k2 )((k3 .k4 )(k32 + k42 ) + 2k32 k42 ) + 8|k1 + k2 |2 + k3 k4 (k3 + k4 )(k3 k4 + k3 .k4 ))] + (1, 2 ↔ 3, 4)} (8.221)

A2 (k1 , k2 , k3 , k4 ) = −

k1 k2 k3 k4 (k1 + k2 )(k3 + k4 )(k1 k2 + k1 .k2 )(k3 k4 + k3 .k4 ) 4|k1 + k2 |4 k1 k2 k3 k4 (k1 k4 (k2 .k3 ) + k1 k3 (k2 .k4 ) + k2 k4 (k1 , k3 ) + k2 k3 (k1 , k4 )) − |k1 + k2 |2 [k3 k4 (k3 + k4 )((k1 .k2 )(k12 + k22 ) + 2k12 k22 )(k3 k4 + k3 .k4 ) + (3, 4 ↔ 1, 2)] + 2|k1 + k2 |2 3k1 k2 k3 k4 (k1 k2 + k1 .k2 )(k3 k4 + k3 .k4 ) + . (8.222) 4|k1 + k2 |2

A3 (k1 , k2 , k3 , k4 ) = −

– 124 –

  [k1 .(k1 + k2 )] [k3 .(k3 + k4 )] × k1 .k3 + |k1 + k2 |2   [k2 .(k1 + k2 )] [k4 .(k3 + k4 )] k2 .k4 + |k1 + k2 |2   [k1 .(k1 + k2 )] [k4 .(k3 + k4 )] + k1 .k4 + × |k1 + k2 |2   [k2 .(k1 + k2 )] [k3 .(k3 + k4 )] k2 .k3 + |k1 + k2 |2   [k1 .(k1 + k2 )] [k2 .(k3 + k4 )] − k1 .k2 + × |k1 + k2 |2   [k3 .(k3 − k4 )] [k4 .(k3 − k4 )] k3 .k4 + |k1 + k2 |2 "( k1 k2 (k1 + k2 )2 ((k1 + k2 )2 − k32 − k42 − k3 k4 ) × ˆ − 2(k3 + k4 ))2 K ˆ 2 ((k1 + k2 )2 − |k1 + k2 |2 ) (K  k1 + k2 k1 + k2 k1 + k2 − 2 + × − 2 2 2k1 k2 k3 + k4 + 4k3 k4 − (k1 + k2 ) |k1 + k2 |2 − (k1 + k2 )2 ! ) 1 3 1 + (1, 2 ↔ 3, 4) − + + ˆ − 2(k1 + k2 ) K ˆ 2(k1 + k2 ) K

ˆ S (k1 , k3 , k2 , k4 ) = −2 W

 |k1 + k2 |3 (|k1 + k2 |2 − k12 − k22 − 4k1 k2 )(|k1 + k2 |2 − k32 − k42 − 4k3 k4 ) − 2(|k1 + k2 |2 − k12 − k22 − 2k1 k2 )(|k1 + k2 |2 − k32 − k42 − 2k3 k4 ) "( ˆ S (k1 , k3 , k2 , k4 ) 2|k1 + k2 |3 G k1 k2 (k1 + k2 )2 ((k1 + k2 )2 − k32 − k42 − k3 k4 ) =− ˜ k1 , k2 )S(k, ˜ k3 , k4 ) ˆ − 2(k3 + k4 ))2 K ˆ 2 ((k1 + k2 )2 − |k1 + k2 |2 ) S(k, (K  k1 + k2 k1 + k2 k1 + k2 × − − 2 + 2 2k1 k2 k3 + k4 + 4k3 k4 − (k1 + k2 )2 |k1 + k2 |2 − (k1 + k2 )2 ! ) 1 1 3 + − + + (1, 2 ↔ 3, 4) ˆ − 2(k1 + k2 ) K ˆ 2(k1 + k2 ) K  |k1 + k2 |3 (|k1 + k2 |2 − k12 − k22 − 4k1 k2 )(|k1 + k2 |2 − k32 − k42 − 4k3 k4 ) − 2(|k1 + k2 |2 − k12 − k22 − 2k1 k2 )(|k1 + k2 |2 − k32 − k42 − 2k3 k4 ) (8.223) Here it is important to mention that, our derived result is consists of three following parts: ˆ S , which appears 1. First of all, we have the contribution from contact interaction term R due to the longitudinal graviton S-channel propagator as given by: " 4 # Y ˆ S (k1 , k2 , k3 , k4 ). (8.224) RS (k1 , k2 , k3 , k4 ) = 16(2π)3 δ (3) (k1 + k2 + k3 + k4 ) φ(kI ) R I=1

ˆ S , which comes from the contri2. Next we have the contribution from the terms like W

– 125 –

bution which appears due to the transverse graviton propagator as given by: Z Z 00 0 0 00 3 f ˜ ij,kl (z1 , x1 ; z2 , x2 )δ kk δ ll T 00 00 (z2 , x2 ), W = dz1 d x1 dz2 d3 x2 Ti0 j 0 (z1 , x1 )δ i i δ j j G k l

(8.225)

˜ ij,kl (z1 , x1 ; z2 , x2 ) is given by: where the transverse graviton Green’s function G Z d3 k ik.(x1 −x2 ) ˜ Gij,kl (z1 , x1 ; z2 , x2 ) = e (2π)3 " # Z ∞  3 (pz1 )J 3 (pz2 )  J 1 2 dp2 × Peik Pejl + Peil Pejk − Peij Pekl . √2 2 + p2 ) 4 z z (k 1 2 0 Here Peij is the which is the transverse traceless projector onto the directions perpendicular to k as given by:   ki kj e (8.227) Pij = δij − 2 , k and J 3 (x) is the Bessel function with characteristic index 3/2, which can be expressed 2 in terms of the following simplified form: r   2 sin x − cos x J 3 (x) = 2 πx x r 2 (1 − ix) eix − (1 + ix) e−ix = . (8.228) πx 2ix Additionally in the present context the expression for stress tensor Tij (z, x) in terms of scalar field inflaton fluctuation δφ(z, x) is given by the following simplified expression:   Tij (z, x) = 2(∂i δφ)(∂j δφ) − δij (∂z δφ)2 + η kl (∂k δφ)(∂l δφ) . (8.229) Here it is important to mention that, two different insertions of the stress tensor corresponds to two different values of the radial variable z = (z1 , z2 ) which we finally integrate out. Finally, for S-channel contribution substituting for δφ in Fourier space we get the following expression for the transverse graviton propagator: " 4 # Y f S (k1 , k2 , k3 , k4 ) = 16(2π)3 δ (3) (k1 + k2 + k3 + k4 ) W φ(kI ) I=1

×

k1i k2j k3k k4l



 e e e e e e Pik Pjl + Pil Pjk − Pij Pkl Θ(k1 , k2 , k3 , k4 ),

where Θ(k1 , k2 , k3 , k4 ) is defined as: Z ∞ dp2 Θ(k1 , k2 , k3 , k4 ) = 2(p2 + Ks2 ) 0 Z ∞ 3 dz1 × (1 + k1 z1 )(1 + k2 z1 )z12 J 3 (pz1 ) e−(k1 +k2 )z1 2 2 z Z0 ∞ 1 3 dz2 × (1 + k3 z2 )(1 + k4 z2 )z22 J 3 (pz2 ) e−(k3 +k4 )z2 , 2 2 z2 0

– 126 –

(8.230)

(8.231)

(8.226)

(a) S channel diagram.

(b) T channel diagram.

(c) U channel diagram.

Figure 21. Representative S, T and U channel diagram for bulk interpretation of four point scalar correlation function in presence of graviton exchange contribution. In all the diagrams graviton is propagating on the bulk and the end point of scalars are attached with the boundary at z = 0.

– 127 –

where Ks is defined as the norm of the total momentum required for graviton exchange in S-channel: Ks = |k1 + k2 | = | − (k3 + k4 )| = |k3 + k4 |. (8.232) Further we use the following result for the integration: r 3 Z ∞ 3 2 p 2 (k12 + k22 + p2 + 4k1 k2 ) dz1 −(k1 +k2 )z1 2 3 (pz1 ) e = J (1 + k z )(1 + k z )z , 1 1 2 1 1 2 z12 π ((k1 + k2 )2 + p2 )2 0 r 3 Z ∞ 3 dz2 2 p 2 (k32 + k42 + p2 + 4k3 k4 ) −(k +k )z 2 3 4 2 3 (pz2 ) e (1 + k z )(1 + k z )z J = . 3 2 4 2 2 2 z22 π ((k3 + k4 )2 + p2 )2 0

(8.233) (8.234)

Further substituting these results in the integral Θ(k1 , k2 , k3 , k4 ) we get the following simplified expression: Z 2 ∞ (k 2 + k22 + p2 + 4k1 k2 )(k32 + k42 + p2 + 4k3 k4 ) Θ(k1 , k2 , k3 , k4 ) = dp p4 21 π 0 (p + Ks2 )((k1 + k2 )2 + p2 )2 ((k3 + k4 )2 + p2 )2 2k1 k2 (k1 + k2 )2 ((k1 + k2 )2 − k32 − k42 − 4k3 k4 ) =− ˆ − 2(k3 + k4 ))2 K ˆ 2 ((k1 + k2 )2 − Ks2 ) (K ×

1 k1 + k2 3 1 − − − ˆ ˆ 2(k1 + k2 ) K K − 2(k3 + k4 ) 2k1 k2

k1 + k2 k1 + k2 + 2 − 2 2 2 Ks − (k1 + k2 ) k3 + k4 + 4k3 k4 − (k1 + k2 )2 K 3 (K 2 − k 2 − k 2 − 4k1 k2 )(Ks2 − k32 − k42 − 4k3 k4 ) + s2 s 2 1 2 2 . (Ks − k1 − k2 − 2k1 k2 )2 (Ks2 − k32 − k42 − 2k3 k4 )2

 + (1, 2 ↔ 3, 4) (8.235)

Additionally here we have the following expression for the transverse projector along with appropriate index contraction in momentum direction:     (k1 .(k1 + k2 ))(k3 .(k3 + k4 )) i j k l e e e e e e k1 k2 k3 k4 Pik Pjl + Pil Pjk − Pij Pkl = k1 .k3 + |k1 + k2 |2   (k2 .(k1 + k2 ))(k4 .(k3 + k4 )) k2 .k4 + |k1 + k2 |2   (k1 .(k1 + k2 ))(k4 .(k3 + k4 )) + k1 .k4 + |k1 + k2 |2   (k2 .(k1 + k2 ))(k3 .(k3 + k4 )) k2 .k3 + |k1 + k2 |2   (k1 .(k1 + k2 ))(k2 .(k3 + k4 )) − k1 .k2 − |k1 + k2 |2   (k3 .(k1 + k2 ))(k4 .(k3 + k4 )) k3 .k4 − . (8.236) |k1 + k2 |2 Finally substituting all of these expressions in Eq (8.230) we get the following simplified expression for the S-channel contribution in transverse graviton propagator: " 4 # Y f S (k1 , k2 , k3 , k4 ) = 16(2π)3 δ (3) (k1 + k2 + k3 + k4 ) ˆ S (k1 , k2 , k3 , k4 ), (8.237) W φ(kI ) W I=1

– 128 –

ˆ S (k1 , k2 , k3 , k4 ) is defined as: where W   ˆ S (k1 , k2 , k3 , k4 ) = k1i k2j k3k k l Peik Pejl + Peil Pejk − Peij Pekl Θ(k1 , k2 , k3 , k4 ). W 4

(8.238)

Here it is important to note that, the contribution from the T and U -channel can be obtained by replacing the following momenta: T − channel : U − channel :

k2 ↔ k3 , k2 ↔ k4 .

(8.239) (8.240)

The representative S, T and U channel diagrams for bulk interpretation of the four point scalar correlation function in presence of graviton exchange is shown in shown in fig. (21(a)), fig. (21(b)) and fig. (21(c)). In these diagrams we have explicitly shown that, graviton is propagating on the bulk and the end point of scalars are attached with the boundary at z = 0. In or computation all the representative diagrams are important to explain the total four point scalar correlation function. ˆ S appears due to integrating out the metric 3. Additionally, the extra contributions G perturbation. In the present context, including the contribution from four point function one can parameterize non-Gaussianity phenomenologically via a non-linear correction to a Gaussian perturbation ζg in position space as:   3 9 loc 3 ζ(x) = ζg (x) + fNlocL ζg2 (x) − hζg2 (x)i + gN ζ (x) + · · · , 5 25 L g

(8.241)

where · · · represent higher order non-Gaussian contributions. In case local non-Gaussianity amplitude of the bispectrum from the three point function is defined as [137, 148]: # " X X 54 loc Pζ (ki )Pζ (kj )Pζ (kp ) , T (k1 , k2 , k3 , k4 ) = τNlocL Pζ (kij )Pζ (kj )Pζ (kp ) + gN L 25 i