COSMOS- \(e'\) -soft Higgsotic attractors

The European Physical Journal C, Jul 2017

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 \(R^2\) gravity, from which we have studied the attractor and non-attractor phases by exactly computing two point, three point and four point correlation functions from scalar fluctuations using the In-In (Schwinger–Keldysh) and the \(\delta \mathcal{N}\) formalisms. 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 the field excursion formula are explicitly presented for the attractor and non-attractor phase. Further, reheating constraints, scale dependent behavior 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.

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COSMOS- \(e'\) -soft Higgsotic attractors

Eur. Phys. J. C COSMOS-e -soft Higgsotic attractors Sayantan Choudhury 0 0 Department of Theoretical Physics, Tata Institute of Fundamental Research , Colaba, Mumbai 400005 , India 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 phases by exactly computing two point, three point and four point correlation functions from scalar fluctuations using the InIn (Schwinger-Keldysh) and the δN formalisms. 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 the field excursion formula are explicitly presented for the attractor and non-attractor phase. Further, reheating constraints, scale dependent behavior 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. - S. Choudhury: Presently working as a Visiting (Post-Doctoral) fellow at DTP, TIFR, Mumbai. Contents 10.2 Dynamical dilaton at late times . . . . . . . . . 10.3 Details of the δN formalism . . . . . . . . . . 10.3.1 Useful field derivatives of N . . . . . . 10.3.2 Second-order perturbative solution with various source . . . . . . . . . . . . . . 10.3.3 Expressions for perturbative solutions in final hypersurface . . . . . . . . . . . 10.3.4 Shift in the inflaton field due to δN . . . 10.3.5 Various useful constants for δN . . . . . 10.4 Momentum dependent functions in four point function . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction The inflationary paradigm is a theoretical proposal which attempts to solve various long-standing issues with standard Big Bang cosmology and has been studied earlier in various works [ 1–12 ]. But apart from the success of this theoretical framework it is important to note that no single model exists till now using which one can explain the complete evolution history of the universe and also one is unable to break the degeneracy between various cosmological parameters computed from various models of inflation [ 13–33 ]. It is important to note that we have the vacuum energy contribution generated by the trapped Higgs field in a metastable vacuum state which mimics the role of an effective cosmological constant in effective theory. At the later stages of the universe such a vacuum contribution dominates over other contents and correspondingly the universe expands in an exponential fashion. But using such metastable vacuum state it is not possible to explain the tunneling phenomenon and also impossible to explain the end of inflation. To serve both of the purposes the effective potential for inflation should have a flat structure. Due to such a specific structure the effective potential for inflation satisfies the 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 in existence in the cosmological literature where one has derived such a specific structure of inflation [ 14,34– 39 ]. For example, the Coleman–Weinberg effective potential serves this purpose [ 40,41 ]. Now if we consider the finite temperature contributions in the effective potential [ 42,43 ] 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 inflation 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. This class of models is a 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 [ 44–46 ]) and at late times these perturbations act as the seeds for the large scale structure formation, which we observe at the present epoch. Apart from this huge success of the inflationary paradigm in the slowly varying regime it is important to mention that these density fluctuations generated from various classes 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 the 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 to possibly satisfy these cosmological and particle physics constraints. But the prime theoretical challenge at this point is that for such a 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), φ = 0|φ|0 = 0, where |0 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 [ 44–46 ] computable from various inflationary observables from this setup. However, apart from the success of the 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 and allows huge fine-tuning in the theoretical setup. This is obviously a not recommendable prescription from a model builder’s point of view. Additionally, all these classes of models are not ruled out completely by the present observed data (Planck 2015 and other joint data sets [ 44–47 ]), as they are degenerate in terms of the determination of inflationary observables and associated cosmological parameters in precision cosmology. There are various ideas existing in the cosmological literature which can drive inflation. These are: • Category I: In this class of models, inflation is driven through a field theory which involves a very high energy physics phenomenon. Example: string theory and its supergravity extensions [ 13,15–17,19,22,23,25,48–82 ], various supersymmetric models [ 14,34–39 ], 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 [ 83–86 ]. Example: the Starobinsky inflationary framework, which is governed by the model [83] f (R) = a R + b R2, where the coefficients a and b are given by a = M 2p 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 we will explore the cosmological consequences from the scale free theory of gravity. 2. Introducing higher derivative terms of the form of Gauss–Bonnet gravity coupled with a scalar field in a non-minimal fashion, where the contribution in the effective action can be expressed as [ 87,88 ] (1.1) SG B = d4x √−g f (φ)[Rμναβ Rμναβ − 4Rμν Rμν + R2], where f (φ) is the inflaton dependent coupling which can be treated as the non-minimal 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 the Gauss–Bonnet term is a topological surface term. 3. Another possibility is to incorporate the effect of nonminimal coupling of the inflaton field and the gravity sector [ 89–91 ]. The simplest example is f (φ)R gravity theory. For Higgs inflation [ 89 ], f (φ) = (1 + ξ φ2), where ξ is the non-minimal coupling of the Higgs field. Here one can consider a more complicated possibility as well by considering a non-canonical interaction between inflaton and f (R) gravity by allowing an f (φ) f (R) term in the 4D effective action [ 92 ]. For the construction of the 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 [ 93–102 ]. For example one can consider the possibilities R f1( )R, Rμν f2( )Rμν , Rμναβ f3( )Rμναβ , R f4( )∇μ∇ν ∇α∇β Rμναβ , Rμναβ f5( )∇α∇β ∇ν ∇ρ ∇λ∇γ Rμρλγ , Rμναβ f6( )∇α ∇β ∇ν ∇μ∇λ∇γ ∇η∇ξ Rλγ ηξ , where is defined as = √1−g ∂μ[√−g gμν ∂ν ]; it is the d’Alembertian operator in 4D and the fi ( )∀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 the matter sector of the effective theory. One of the examples is to use Jordan–Brans–Dicke (JBD) gravity theory [ 103,104 ] along with extended inflationary models which includes non-canonical interactions. By adjusting the value of the Brans–Dicke 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 the soft inflationary paradigm in an Einstein frame, where a chaotic Higgsotic 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 α R2 gravity. Here it is important to mention that, in the case of a soft inflationary model, the dilaton exponential potential is multiplied by a 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 [ 105,106 ] (see Appendix 10.1 for more details). In this context also it is important to specify that one can treat the field dependent couplings in the simple effective potentials or maybe in generalized P(X, φ) functionals, entailing a decaying behavior with dilaton field value as it contains an overall exponential factor which is coming from the dilaton potential itself in an Einstein frame. This is a very interesting feature from the point of view of RG flow in QFT as the field dependent coupling in an Einstein frame captures the effect of field flow (energy flow). In this context instead of solving directly the RGE for the effective coupling, we solve the dynamical equations for the fields and the effective coupling for power-law and exponential attractors. Due to the similarities in the two techniques here one can arrive at the conclusion that in cosmology solving a dynamical attractor problem in the presence of effective coupling in an Einstein frame mimics the role of solving RGE in QFT. Thus due to the exponential suppression in the effective coupling in an Einstein frame it is naturally expected from the prescribed framework that for suitable choices of the model parameters soft cosmological constraints can be obtained [ 107,108 ]. As in this prescribed framework the dilaton exponential coupling plays a very significant role, one can ask the very crucial question of its theoretical origin. Obviously there are various sources in existence from which one can derive exponential effective couplings or more precisely the effective potentials for dilaton. These possibilities are: Scale free gravity + Higgso c scalar field (in Jordan frame) Two,three and four point (using δN) func on for infla on +Rehea ng constraints Consistent with observa on (Planck and other joint data) Consistent with rehea ng and late me accelera on Conformal Transforma on Conformal Transforma on Attractor phase Power law (stable) a ractor Einstein gravity+ Higgso c dilaton coupled two-field theory (in Einstein frame) Dynamical A ractor solu ons Exponen al (unstable) a ractor Stability from String Theory, non-minimal coupling and contribu on from mass Scale free gravity + Higgso c scalar field (in Jordan frame) Two, three and four point (using Schwinger Keldysh/In-In) func on for infla on +Rehea ng constraints Consistent with observa on (Planck and other joint data) Consistent with rehea ng and late me accelera on Non attractor phase Einstein gravity+ Higgso c dilaton coupled two-field theory (in Einstein frame) Assume dilaton is heavy field Freezing dilaton at UV cut-off of Effec ve theory Dynamical non a ractor solu on (a) Diagrammatic representation of attractor phase of soft (b) Diagrammatic representation of non-attractor phase of Higgsotic inflation. In this representative diagram we have soft Higgsotic inflation. In this representative diagram we shown the steps followed during the computation. have shown the steps followed during the computation. Dynamical dilaton Fixed dilaton Infla on from so Higgso c sector coupled with dilaton (in Einstein frame) Attractor phase Non attractor phase New consistency rela ons for NG+ PGW for two a ractors Old consistency rela on for NG+ New consistency rela on for PGW (c) Diagrammatic representation of attractor and non-attractor dynamical phase of soft Higgsotic inflation which is coupled with dilaton in Einstein frame. • Source I: One of the sources for dilaton exponential potential is string theory, appearing in the Category I. Specifically, superstring theory and low energy supergravity models are the theoretical possibilities in string theory [ 109–118 ] where dilaton exponential potential appears in the gravity part of the action in a Jordan frame and after a conformal transformation in the Einstein frame such dilaton effective potential is coupled with the matter sector. The most important example is the α-attractor which mimics a class of inflationary models in N = 1 supergravity in 4D. For details see Refs. [ 119–129 ]. • Source II: Another possible source of the dilaton exponential potential is coming from a modified gravity theory framework such as f ( R) gravity [ 83–86 ], f (φ) f ( R) gravity [ 89–92 ] and Jordan–Brans–Dicke theory [ 103, 104 ] in the Jordan frame, which appear in the Category II (1 and 3) and Category III. After transforming the theory in the Einstein frame via conformal transformation one can derive the dilaton exponential potential. In Fig 1a–c, we have shown the diagrammatic representation of attractor and non-attractor phases 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: • Transition from scale free gravity to scale dependent gravity have discussed and its impact on the solutions in the attractor and non-attractor regime of inflation have also discussed. • Explicit calculation of the δN formalism is presented by considering the effect up to second-order perturbation in the solution of the field equation in attractor regime. Additionally deviation in the consistency relation between the non-Gaussian amplitude for four point and three point scalar correlation function a.k.a. Suyama– Yamaguchi relation is presented to explicitly show the consequences from attractor and non-attractor phase. • Additionally, new sets of consistency relations are presented in attractor and non-attractor phases of inflation to explicitly show the deviation from the results obtained from a canonical single field slow-roll model. • Detailed numerical estimations are given for all the inflationary observables for attractor and non-attractor phases 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 in terms of S, T and U channel contribution for all the individual terms obtained from three and four point correlation function. • Scale dependent behaviors of the non-minimal coupling between inflaton field and additional dilaton field are given in an Einstein frame for power-law and exponential types of attractor. • Three possible theoretical proposals have presented to overcome the tachyonic instability [ 130–134 ] at the time of late time acceleration in a Jordan frame and due to this fact the structure of the effective potentials changes in an Einstein frame as well. These proposals are inspired by: – I. Non-BPS D-brane in superstring theory [23,135– 140]. – 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 α R2 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 α R2 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 α R2 gravity sector. For such canonical slow-roll models the effective sound speed cS = 1. But for completeness one can consider a most generalized version of P(X, φ) models, where X = − 21 gμν ∂μφ∂ν φ and the effective sound speed cS < 1 for such models. For example one can consider the following structure [ 56,105 ]: 1 P(X, φ) = − f (φ) 1 1 − 2X f (φ) + f (φ) − V (φ), (1.2) which is exactly similar to the 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 [ 56 ] f (φ) ≈ φg4 , which is known as the 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 the general class of P(X, φ) theories which follows the above structure. Similarly one can consider the following structure of P(X, φ) for tachyon and Gtachyon models given by [ 23,141 ] For Tachyon: P(X, φ) = −V (φ)√1 − 2X α , (1.3) For GTachyon: P(X, φ) = −V (φ)(1 − 2X α )q (1/2 < q < 2), (1.4) where α is the Regge slope. Here we consider the most simple canonical form, P(X, φ) = X − V (φ), where V (φ) is the effective potential for the monomial φ4 model considered here for our computation. 4. As a choice of the initial condition or precisely as the choice of vacuum state we restrict our analysis using Bunch–Davies vacuum. If we relax this assumption, then we can generalize the results for α vacua as well. 5. During our computation we have restricted ourselves up to the minimal interaction between the α R2 gravity and matter sector. Here one can consider the possibility of non-minimal interaction between α R2 gravity and matter sector. 6. During the implementation of the In-In formalism [ 2 ] to compute three and four point correlation function we have use the fact that the additional dilaton field is fixed at Planckian field value to get the non-attractor behavior of the present setup. One can relax this assumption and can redo the analysis of the In-In formalism to compute three and four point correlation function without freezing the dilaton field and also use the attractor behavior of the model to simplify the results. 7. During the computation of correlation functions using a semi classical method, via the δN formalism [ 23,142– 146 ], 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 an Einstein frame. For 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. 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 the presence of time dependent coupling f (φ (η)) to study the features of primordial magnetic field through inflationary magnetogenesis [ 147–149 ]. The plan of this paper is as follows: • In Sect. 2, we start our discussion with f (R) = α R2 gravity where a scalar field is minimally coupled with the gravity sector and contains only canonical kinetic term. Next in the matter sector we choose a very simple monomial model of potential, V (φ) = λ4 φ4, which can be treated as a Higgs like potential as at the scale of inflation, the contribution from the VEV of Higgs is almost negligible. • Further, in Sect. 3, we provide the field equations in a Jordan frame written in a spatially flat FLRW background. Next, we perform a conformal transformation in the metric to the Einstein frame and introduce a new dilaton field. Further, we derive the field equations in an Einstein frame and try to solve them for two dynamical attractor features: a power-law solution, and exponential solution. However, the second case give rise to tachyonic behavior which can be resolved by considering-I. non-BPS D-brane in superstring theory, II. via switching on the effect of quadratic term in the effective potential and III. by introducing a non-minimal coupling between matter and α R2 gravity sector. • Next, in Sect. 4, using two dynamical attractors, powerlaw and exponential solution, we study the cosmological constraints in the 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 Sect. 4.2. Further, we repeat the analysis for tensor modes and also comment on the future observables – the amplitude of the tensor fluctuations and tensor-to-scalar ratio in Sect. 4.3. Additionally, in Sect. 4.4, we study the constraint for the reheating temperature. Finally, in Sects. 5.1 and 5.2, we derive the expression for the inflaton and the non-minimal coupling at horizon crossing, during reheating and at the onset of inflation for the two above mentioned dynamical cosmological attractors. • Further, in Sect. 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 Sect. 7.1. To serve this purpose we have used the ADM formalism using which we compute two point functions and associated present observables using the Bunch–Davies initial condition for scalar fluctuations in Sects. 7.2.1 and 7.2.2. Further, in Sects. 7.3.1 and 7.3.2, we repeat the procedure for tensor fluctuations as well where we have computed two point function and the associated future observables. We also derive a few sets of consistency relations in this context which are different from the usual single field slow-roll models. Further, in Sect. 7.4, we derive the constraints on reheating temperature in terms of observables and the number of e-foldings. • Next, in Sects. 8.1.1 and 8.1.2, as a future probe, we compute the expression for three point function and the bispectrum of scalar fluctuations using the In-In formalism for the non-attractor case and the δN formalism for the attractor case. Further, we derive the result for a nonGaussian amplitude fNloLc for equilateral and squeezed limit triangular shape configuration. Also we give a bulk 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 in the Planck scale and redo the analysis of the δN formalism. Here we show that the expression for the three point non-Gaussian amplitude is slightly different as expected for the single field case. Further, in Sects. 8.1.1 and 8.1.2, we compare the results obtained from the In-In formalism and δN formalism for the non attractor phase, where the additional field is fixed in Planck scale. Finally, we give a theoretical bound on the scalar three point non-Gaussian amplitude. • Finally, in Sects. 8.2.1 and 8.2.2, as an additional future probe, we have also computed the expression for the four point function and the trispectrum of scalar fluctuations using the In-In formalism for the non-attractor case and δN formalism for the attractor case. Further, we derive the results for non-Gaussian amplitude gNloLc and τ NloLc for equilateral, counter-collinear or folded kite and squeezed limit shape configuration from the In-In formalism. Further we give a bulk 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 derive the expressions for the four point non-Gaussian amplitude gNloLc and τ NloLc. Next we have shown that the consistency relation connecting three and four point non-Gaussian amplitude aka Suyama–Yamaguchi relation is modified in the attractor phase and further given an estimate of the amount of deviation. Further, for the consistency check we freeze the additional field in Planck scale and redo the analysis of the δN formalism. Here we show that the four point non-Gaussian amplitude is slightly different as expected for the single field case. Finally, we give a theoretical bound on the scalar four point non-Gaussian amplitude. 2 Model building from scale free gravity To describe the theoretical setup let us start with the total action of f (R) gravity coupled minimally along with a scalar inflaton field φ: S = d4x √−g f (R) − (∂μφ)(∂ν φ) − V (φ) (2.1) gμν 2 where in general f (R) can be an arbitrary function of the Ricci scalar R. For example one can choose a generic form given by [ 150,151 ] mial powers of R. Here also one can treat the α Rn 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, which is known as scale free gravity in a Jordan frame: f (R) = α R2, where α is a dimensionless scale free coefficient. For this type of theory if we perform the conformal transformation from Jordan to Einstein frame then we will induce a constant term in the effective potential and this 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 a Jordan frame, which we identified to be the inflaton. After a conformal transformation in an 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 a Jordan frame. We will show that here the two fields, dilaton and inflaton form dynamical attractors using which one can very easily solve this two-field complicated model in the context of cosmology. Next we will discuss the structure of the inflational as appearing in Eq. (2.1). Generically in 4D effective theory the effective potential can be expressed as f (R) = an Rn, ∞ 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 = M 2p/2, an = 0∀n > 1, then one can get back the well-known Einstein–Hilbert action (GR) in Jordan frame: f (R) = M 2p R/2. In this particular case Jordan frame and Einstein frame are exactly the same because the conformal factor for the frame transformation is unity. This directly implies that no dilaton potential appears 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 are more significant in the above mentioned generic expansion of f (R) gravity. 2. If we set, a1 = a = M 2p/2, a2 = b = α, an = 0∀n > 2, then one can get back the specific structure of the very well-known Starobinsky model: f (R) = a R + b R2 = M 2p R/2 + α R2. Here one can treat the α R2 term as an additional quantum correction to the Einstein gravity. 3. One can also set a1 = a = M 2p/2, an = α∀n ≥ 2, then one can get back the specific structure f (R) = M 2p R/2 + α Rn, which describes the situation where the Einstein–Hilbert gravity action is modified by the mono(2.2) V (φ) = Vren(φ) Renormalizable part + ∞ δ=5 Jδ(g) φδ M δp−4 Non-renormalizable part = ∞ δ=0 Cδ(g) φδ M δp−4 where Jδ(g) and Cδ(g) are the Wilson coefficients in effective theory. Here g stands for the scale of theory and the dependences of the Wilson coefficients on the scale can be exactly computed for a full UV complete theory using renormalization group equations. In this paper a similar scale dependence on the couplings we will calculate using dynamical attractor method in an 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 V (φ) = Cδ(g) 4 δ=0 φδ M δp−4 . Next to get the Higgslike monomial structure of the potential we set C3(g) = 0, as in this paper our prime motivation is to look into only Higgsotic potentials. Consequently we get (2.3) (2.4) V (φ) = C0 + C2(g)M 2pφ2 + C4(g)φ4. To get the Higgsotic structure of the potential one should set C0(g) = λ4 v4, C2(g) = − λ2 v2, C4(g) = λ4 . Here v is the VEV of the field φ. Consequently, one can write the potential in the following simplified form: V (φ) = λ4 (φ2 − v2)2. Now we consider a situation where scale of inflation as well as the field value are very much 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 λ 4 V (φ) = 4 φ . Further varying Eq. (2.1) with respect to the metric and using Eqs. (2.2) and (2.8) the equation of motion (modified Einstein equation) for the α R2 scale free gravity can be written as G˜ μν := α[{Rμν + 2(gμν − ∇μ∇ν )} + Gμν ]R = Tμν wgαhβe∇reα∇thβe =D’Aglαeβm∇bαe∂rβtian= op√e1−ragt∂oαr( √is−dgegfiαnβe∂dβ )asand th=e energy-momentum stress tensor can be expressed as Tμν = − √−g 2 δ(√−gLM ) δgμν = ∂μφ∂ν φ − gμν 1 gαβ ∂αφ∂β φ + λ4 φ4 . 2 (2.5) (2.6) (2.7) (2.9) (2.10) (2.11) (2.12) Here it is important to note that the Einstein tensor is defined gμν as Gμν := Rμν − 2 R. Now after taking the trace of Eq. (2.9) we get R R = 6Tα , where the trace of the energy-momentum tensor is characterized by the symbol T = Tμμ. In this modified gravity picture we have ∇μG˜ μν = 4α[∇μ, ]R = 0 where we use ∇μ Rμν = g2μν ∇μ R, which directly follows from the Bianchi identity ∇μGμν = 0. Now varying Eq. (2.1) with respect to the field φ we get the following equation of motion in curved space-time: φ = −V (φ) = −λφ3 = −λφ3. 1 ⇒ √−g ∂α √−ggαβ ∂β φ Further assuming the flat (k = 0) FLRW background metric the Friedmann equations can be written from Eq. (2.9) as a 2 H 2 = a˙ 2H˙ + 3H 2 = 2 a¨ a˙ 2 a + a pφ where we have assumed the energy-momentum tensor can be described by a perfect fluid as Tνμ = diag(−ρφ , pφ , pφ , pφ ) where the energy density ρφ and the pressure density pφ can be expressed for the scalar field φ as ρφ = φ˙22 + λ4 φ4, pφ = φ˙22 − λ4 φ4. Similarly the field equation for the scalar field φ in the flat (k = 0) FLRW background can be recast as φ¨ + 3H φ˙ + λφ3 = 0. (2.8) In the flat (k = 0) FLRW background we have the following expressions: R = 6( H..˙. + 2H 2), R¨ = 6(H + 4H˙ 2 + 4H H¨ ). R˙ = 6(H¨ + 4H H˙ ), Substituting these results in Eqs. (2.12) and (2.13) the Friedmann equations can be recast in the Jordan frame as 2H (H¨ + 3H H˙ ) − H˙ 2 = 1ρ8φα , 9H˙ (H˙ + H 2) + 6H H¨ + H... = − 6pαφ . In the slow-roll regime (φ˙ 2/2 λ4 φ4) the energy density ρφ and the pressure density pφ can be approximated by ρφ ≈ λ φ4, pφ ≈ − 4 4 λ φ4. Consequently Eqs. (2.15), (2.17) and (2.18) can be recast as 3H φ˙ + λφ3 ≈ 0, V (φ) 2H (H¨ + 3H H˙ ) − H˙ 2 ≈ 18α , ... V (φ) 9H˙ (H˙ + H 2) + 6H H¨ + H ≈ − 6α , λ φ4. Further combining Eqs. (2.20) and where V (φ) =... 4 (2.21) we get H = 3H˙ (3H 2 − 4H˙ ). For further analysis one can also define the following sets of slow-roll parameters in a Jordan frame: H = − HH˙2 , δH = − HH¨3 = H˙H − 2 2H , ... γH = − HH4 = 3 H (3 + 4 H ) , ηH = − Hφ¨φ˙ . Further using these new sets of parameters in Eqs. (2.20) and (2.21) can be recast into the following simplified form: γH 21 2δH + 12 + 4 V (φ) λφ4 H ≈ − 18α H 4 = − 72α H 4 . (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) However, solving this two-field problem in the presence of scale free gravity is itself very complicated for the following reasons: • Complication I: First of all, for a given structure of inflationary potential in a Jordan frame (here it is the Higgsotic potential as mentioned earlier) it is impossible to solve directly the dynamical equations (2.20), (2.21) and (2.23) due its complicated coupled structural form. • Complication II: One can use various solution Ansatzes to get approximated numerical results, but this is also dependent on the structure of the inflaton potential in a Jordan frame and how one can able to implement initial condition (starting point) of inflation for arbitrary structure of the effective potential. • Complication III: In connection with the implementation of the initial condition and to check the sufficient condition for inflation in this complicated field theoretical setup one needs to define the expression for number of e-foldings in terms of effective potentials. But this cannot be very easy in the present context as the field equations are coupled. Due to these huge number of difficulties in a Jordan frame we transform the total action into the Einstein frame using a conformal transformation. After transforming the Jordan frame action into the Einstein frame in the present context we need the solve a two interacting field problem in the presence of Einstein gravity. There are several ways one can solve this problem. These possibilities are: • Solution I: the first solution to this problem is to follow the well-known approach to solving two-field models of inflation by following the method of curvature and isocurvature perturbation in the semiclassical δN formalism. For more accurate results one can also solve directly the Mukhanov–Sasaki equation for this two-field model and directly treat fluctuations quantum mechanically. Since this methodology has been discussed in various earlier works, we will not discuss this issue in this paper. See Refs. [ 152–156 ] fore more details. • Solution II: a second way of solving this problem is to use dynamical attractor mechanism in the present context where the two fields are connected through specific relations, which can be obtained by solving dynamical field equations in cosmology. This is equivalent to solving renormalization group equations in the context of quantum field theory as the dynamical attractor solution of two fields captures the effects of all the energy scale. In our computation we explore the possibility of two dynamical attractors: 1. Power-law attractor 2. Exponential attractor Here they have different cosmological consequences. But they originate from the Higgsotic structure of the effective potential which we will discuss in the next section in detail. • Solution III: a final possibility is to freeze the dilaton field in the Planck scale or in the vicinity, so that one can absorb it in the effective couplings in the Higgsotic theory. This is identified as the non-attractor phase in the context of cosmology. The physical justification for such possibilities can also be explained from the UV behavior of the 4D effective theory, which is known as the UV completion of the effective theory. According to this proposal we have two sectors in the theory: 1. Hidden sector: the hidden sector is made up of a heavy field (in our case dilaton) which lies around the UV cutoff of the effective theory, which is the Planck scale. We are not able to probe directly this sector. But we can visualize how its imprints on the low energy effective theory. 2. Visible sector: the visible sector is made up of a light field (in our case inflaton) which one can probe directly. For present discussion the visible sector is important to explain the cosmological evolution. Usually in such a prescription one integrates the heavy fields and finally gets an effective theory in the visible sector. Here we use the fact that such a procedure mimics the role of freezing the heavy dilaton field near the Planck scale. The only difference is that in the case of freezing the dilaton field we only concentrate on the Higgsotic potential. But the integration of the heavy field allows for all relevant and irrelevant operators. However, by applying a similar argument one can look into only the renormalizable Higgsotic part of the total effective potential. Additionally, it is important to note that at late times the dynamical picture is completely opposite where the inflaton field freezes in the vicinity of the Planck scale and the dynamical contribution for late time acceleration comes from the dilaton field. In a simpler way one can interpret this physical prescription as the competitive dynamical description of the two fields. During inflation the Higgsotic field wins the game and at late times the dilaton serves the same purpose. More precisely, within this prescription dynamic features transfer from dilaton to Higgsotic field (or any scalar inflaton) during inflation and at late times a completely opposite situation appears, where a similar transfer takes place from inflaton to dilaton field. In this paper we explore the possibility of Solution II and Solution III in detail in the next section. For completeness we briefly review also Solution I in the appendix. 3 Soft attractor: a two-field approach where after applying C.T. the total potential can be recast as In the present context let us introduce a scale dependent mode , which can be written in terms of a no scale dilaton mode as 2 = f (R)M −p2 = 2α R M −p2 = e 3 Mp which mimics the role of a Lagrange multiplier and arises in the Jordan frame without space-time derivatives. In terms of the newly introduced no scale dilaton mode the total action of the theory (see Eq. (2.1)) can be recast as S = − g2μν (∂μφ)(∂ν φ) − λ4 φ4 . To study the behavior of the proposed R2 theory of gravity here we introduce the following conformal transformation (C.T.) in the metric from Jordan frame to the Einstein frame: gμν −C−.→T. g˜μν = 2gμν , gμν −C−.→T. g˜μν = −2gμν , √−g −C−.→T. −g˜ = 4√−g, which satisfies the condition gμν gνβ = g˜μκ g˜κβ = δμβ. In the present context the conformal factor is given by = √ √2 = e 2√3 Mp . With this proposed C.T. in the metric the Ricci curvature scalar in the Jordan frame (R) is related to the Einstein frame (R˜ ) as R = 2[R˜ + 6 ln − 6g˜μν ∂˜μln ∂˜ν ln ] where ∂˜μ = ∂x∂˜μ and ln ≡ √1−g˜ ∂α( −g˜ g˜αβ ∂β ln ). After doing C.T. the total action can be recast in the Einstein frame as1: 1 Here we apply Gauss’ theorem to remove the following contribution in the total effective action: d4x −g˜ 23 Mp ˜ = = where V0 = M 4p/8α exactly mimics the role of the cosmological constant and the effective matter coupling (λ( )) in the potential sector is given by λ( ) = λ4 = λe− 2√√32 Mp . Now varying Eq. (3.7) with respect to the metric the field equations can be expressed as G˜μν := R˜μν − g˜2μν R˜ = T˜μν (φ, ) where the energy-momentum tensor T˜μν (φ, ) for the dilaton–inflaton coupled theory can be expressed as (3.8) (3.9) T˜μν (φ, ) = − 2 δ −g˜ −g˜L(φ, ) δg˜μν = ∂˜μφ∂˜ν φ + ∂˜μ ∂˜ν − g˜μν + 21 g˜αβ ∂˜α ∂˜β + W˜ (φ, ) . 2 ˜ 1 gαβ ∂˜αφ∂˜β φ Here for the matter part of the action the following property holds between the Einstein frame and Jordan frame energy 2 δ(√−g˜LM ) momentum tensor: T˜μν (φ, ) ⊃ T˜μν = − √−g˜ δg˜μν = Tμ2ν , which implies that using the perfect fluid assumption one can write T˜νμ = μdiag(−ρ˜φ , p˜φ , p˜φ , p˜φ ) = 14 diag(−ρφ , pφ , pφ , pφ ) = Tν4 . Assuming the flat (k = 0) FLRW background metric in an Einstein frame the Friedmann equations can be written from Eq. (3.9) as2: Additionally, the Hubble parameter in the Einstein frame (H˜ ) can be expressed as its Jordan frame (H ) counterpart 2 It is important to mention here that the time interval in an Einstein frame dt˜ is related to the time interval in a Jordan frame dt as dt˜ = dt. H˜ 2 = d H˜ dt˜ + H˜ 2 = d ln a 2 dt˜ d2a dt˜2 ρ ˜ , = 3M 2p = − (ρ˜ + 3 p˜) 6M 2p ρ˜ = p˜ = d dt˜ d dt˜ 2 2 + + dφ dt˜ dφ dt˜ 2 2 + W˜ (φ, ), − W˜ (φ, ). where the effective energy density (ρ˜) and the effective pressure ( p˜) can be written in an Einstein frame as (3.10) (3.11) (3.12) 1 = e− √6 Mp as H˜ = 1 H + 21 d lndt 2 H + √6˙Mp . Also the Klein–Gordon field equations for the inflaton field φ and the new field can be written in the flat (k = 0) FLRW background as d2φ dφ dt˜2 + 3H˜ dt˜ + ∂φ W˜ (φ, ) = 0 d2 d dt˜2 + 3H˜ dt˜ + ∂ W˜ (φ, ) = 0. Now in the slow-roll regime the field equations are approximated by dφ 3H˜ dt˜ + λ( )φ3 = 0 d λ( )φ4 3H˜ dt˜ − √6M p = 0, To study the behavior of the proposed model let us consider two cases, where the dynamical features are characterized by 1. Case I: power-law solution, 2. Case II: exponential solution. which we discuss in the next subsection. 3.1 Case I: Power-law solution We consider here large α, small V0(≈ 0) with λ > 0 with effective potential W˜ (φ, ) ≈ λ(4 ) φ4 = λ4 e− 2√√32 Mp φ4 (for Case I). Consequently the field equations can be recast as dφ 2√2 3H˜ dt˜ + λe− √3 Mp φ3 = 0, λ ˜ = 12M 2p d λφ4 3H˜ dt˜ − √6M p 2√2 H 2 e− √3 Mp φ4. 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 Eqs. (3.19)–(3.21) are given by3: 3 Throughout the paper the subscript ‘0’ is used to describe the inflationary epoch. Case I − , (a) Case I : Power-law behavior. (b) Case II : Tachyonic behaviour. Fig. 2 Behavior of the inflationary potential for a V0 ≈ 0 and λ > 0 (Case I) and b V0 = 0 and λ < 0 (Case II). In a the inflaton rolls down from a large field value and inflation ends at φ f ≈ 1.09 M p. On the other hand in b the inflaton field rolls down from a small field value = − 16αλ( )φ02 ⎣⎢ 1 − M 4p ≈ 2αλ( )φ04 ln 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 Fig. 2a, b we have shown the behavior of the inflationary potential for the two cases, 1. V0 ≈ 0 and λ > 0, 2. V0 = 0 and λ < 0. Figure 2a implies that the inflaton rolls down from a large field value and inflation ends at φ f ≈ 1.09 M p. Also the potential has a global minimum at φ = 0, around which field is start to oscillate and take part in reheating. On the other hand in Fig. 2b the inflaton field rolls down from a small field value and the inflation ends at the field value φ f = 2.88 α1/8 M p, where the lower bound on the parameter α is, α ≥ 2.51 × 107, which is consistent with Planck 2015 data [ 44–46 ]. 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 α R2 gravity theory. It is a general notion that the reheating phenomenon can only be explained if the effective potential has a local minimum and a remnant contribution (vacuum energy or equivalent to and the inflation ends at the field value φ f = 2.88 α1/8 M p, where the lower bound on the parameter α is α ≥ 2.51 × 107, which is consistent with Planck 2015 data [ 44–46 ] cosmological constant) in the total effective potential finally produce the observed energy density at the present epoch as given by4 ρnow ≈ 10−47 GeV4, which is necessarily required to explain the late time acceleration of the universe. Now here one can ask a very relevant question: if we include some additional features to the effective Higgsotic potential, which also can be treated as a massless tachyonic potential, then how one can interpret the justifiability as well as the behavior of effective field theory framework around the minimum of the potential which will significantly control the dynamical behavior 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 an Einstein frame here we discuss a few physical possibilities: • Choice I: The first possible solution of the mentioned problem is motivated from non-BPS D-brane in superstring theory. In this prescription the effective potential have a pair of global extrima at the field value, φextrema = φ = ±φV for the non-BPS D-brane within the framework of superstring theory [ 23, 135–140 ]. Additionally, it is important to note that here a one parameter (γ ) family of global extrima exists at the field value, φ = φV eiγ for the brane–antibrane system. Here φV is identified to be the field value where the reheating phenomenon 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 4 For Einstein gravity one can write the observed energy density at the present epoch in the following form: ρnow ≈ 3H02 M2p, where H0 is the Hubble parameter at the present epoch. appearing in the expression for effective potential in an Einstein frame. Here for the sake of simplicity we relax a little bit the constraints as appearing exactly in Case II. To explore the behavior of the derived effective potential here we have allowed both of the signatures of the coupling parameter λ. This directly implies the following constraint condition: 2√2 λ e− √3 Mp φV4 + − 4 2√2 λ e− √3 Mp φV4 + 4 p = 0 (for λ < 0), p = 0 (for λ > 0), (3.32) (3.33) (3.42) , (3.43) (3.44) (3.45) (3.47) (3.48) The solutions of Eqs. (3.37)–(3.42) are given by Choice I(v1) − , d 3H˜ dt˜ − λ(φ4 − φV4 ) e− √3 Mp = 0, (3.41) 2√2 √6M p H 2 ˜ where p is the above mentioned additional contribution and in the context of superstring theory this is given by p = √2(2π )− p gs−1 for non-BPS Dp-brane, 2(2π )− p gs−1 for non-BPS Dp−D¯ p brane pair, (3.34) with string coupling constant gs. This implies that the inflaton energy density vanishes at the minimum of the tachyon type of the derived effective potential and in this connection the remnant energy contribution is given by V0 = M 4p/8α, which serves the explicit role of cosmological constant in the context of late time acceleration of the universe. In this case considering the additional contribution as mentioned above the total effective potential can be modified as M 4p 2√2 4 λ e− √3 Mp (φ4 − φV ) v1: W˜ (φ, ) = 8α − 4 (for λ < 0), M 4p 2√2 4 λ e− √3 Mp (φ4 − φV ) v2: W˜ (φ, ) = 8α + 4 (for λ > 0). Here to avoid any confusion we have taken out the signature of the coupling λ outside in the expression for the effective potential for the λ < 0 case. In the present context the field equations can be expressed as dφ 2√2 For v1: 3H˜ dt˜ − λe− √3 Mp φ3 = 0, M 2p λ ˜ = 24α − 12M 2p 2√2 e− √3 Mp (φ4 − φV4 ). (3.35) (3.36) (3.37) (3.39) (3.40) dφ 2√2 For v2: 3H˜ dt˜ + λe− √3 Mp φ3 = 0, In Fig. 3a, b we have shown the variation of the potential with respect to the inflaton field for both cases. For Fig. 3a the inflaton can roll down in both ways. Firstly, it can roll down to a global minimum at the field value φV = 0 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 from higher to lower field value in an opposite fashion. (a) Choice I(v1) : Modified potential from superstring the- (b) Choice I(v2) : Modified potential from superstring theory with λ < 0. ory with λ > 0. Fig. 3 Behavior of the modified effective potential for case II with a Choice I(v1): V0 = 0, λ < 0, b Choice I(v2): V0 = 0, λ > 0, where Mp = 2.43 × 1018 GeV In that case the inflaton goes up to the zero energy level of the effective potential and cannot 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 the Einstein frame is around the field value, φV = 0.42 M p. Figure 3b 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. 3a. 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 cases the inflaton field settles down to a local minimum at, φmin = φV = 0.42 M p and within the vicinity of this point it will produce particles via reheating. In the two situations the lower bound on the parameter α is fixed at, α ≥ 2.51 × 107, which is perfectly consistent with Planck 2015 data [ 44–46 ]. • Choice II: It is possible to explain the reheating as well as the light 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 an Einstein frame can be written as M 4p v1: W˜ (φ, ) = 8α + m2c2 φ2 − λ4 φ4 e− 2√√32 Mp M 4p v2: W˜ (φ, ) = 8α − m2c2 φ2 − λ4 φ4 e− 2√√32 Mp (for mc2 > 0, λ < 0), (for mc2 < 0, λ > 0). (3.49) (3.50) 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. In this context during inflation the inflaton field satisfies the constraint field satisfies φ |λ2| |mc|. After inflation when reheating starts, the |λ2| |mc|. Finally at the field value φ = |λ2| |mc| the remnant energy V0 = M 4p/8α serves the purpose of explaining the late time acceleration of the universe. In the present context the field equations can be expressed as dφ 2√2 For v1: 3H˜ dt˜ + (mc2φ − λφ3)e− √3 Mp = 0, (3.51) 2√2 e− √3 Mp . dφ 2√2 For v2: 3H˜ dt˜ − (mc2φ − λφ3)e− √3 Mp = 0, (3.54) 2√2 2√2 e− √3 Mp = 0, The behavior of the effective potential in an Einstein frame is plotted in Fig. 4a, b, where the inflaton field is rolling down from a large field to lower value or the lower to larger field value and after inflation take part in (a) Choice I(v1) : Modified potential from superstring the- (b) Choice I(v2) : Modified potential from superstring theory with λ < 0. ory with λ > 0. where ξ represents the non-minimal coupling parameter and φV represents the VEV of the field φ in this context. After performing conformal transformation, the effective action in the Einstein frame can be written as where after applying C.T. the total modified effective action can be written as In the present context the field equations can be expressed as 2√2 e− √3 Mp = 0, The solutions of Eqs. (3.66)–(3.68) are given by: − 0 ≈ 2√√23Mp ln a0 a = 3M√2pα (t − t0) (φ2 − φ02) 1 + ξ2 φ2 + φ02 − 2φV2 + 2φV2 ln φφ0 , 1 + ξφV2 In Fig. 5, we have shown the behavior of the effective potential with respect to inflaton field in the presence of non-minimal coupling parameter, ξ = M −p2 and ξ = 10−8 M −p2 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 becomes steeper. For both 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 = M p. 4 Constraints on inflation with soft attractors Here we require the following constraints to study the inflationary paradigm in the attractor regime: 4.1 Number of e-foldings To get a sufficient amount of inflation from the proposed setup (for both Case I and Case II), necessarily + |N (φ0) − N (φ f )| ≈ ++ln + a f + + a0 ++ which is a necessary quantity to be able to solve the 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 Eqs. (3.24) and (3.31) the field value at the end of inflation can be explicitly computed for the above mentioned two cases as • For Case I the expression for the field associated with the end of inflation φ f is completely fixed by the value initial field value φ0. Here no information for the field dependent coupling λ(ψ f ) = λ( = f ) is required for this case as the expression for φ f is independent of the dilaton field dependent coupling. • For Case II the expression for the field associated with the end of inflation φ f is fixed by the value initial field value φ0 as well as by the field dependent coupling λ(ψ f ) = λ( f ). 4.2 Primordial density perturbation 4.2.1 Two point function The next observational constraint comes from the imprints of density perturbations through scalar fluctuations. Such fluctuations in CMB map directly implies that5: δρ ρ δρ ρ cr = AS ∼ 10−5 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 cases one can write σ δρ ρ t1 = σ δρ 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 + 3(1+w) , w = ρp . It is important to note that (t1, t2) 2 represent the times when the perturbation first left and reentered the horizon, respectively. At time t1, Eq. (3.12) 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 the radiation- and matterdominated epochs, respectively. For the potential dominated inflationary epoch, w ≈ −1 and consequently one can write the following constraint condition: δρ = φ˙ δφ˙ + ˙ δ ˙ − 3H˜ φ˙ δφ + ˙ δ ≈ −2H˜ φ˙ δφ + ˙ δ where we use the symbol as ˙ ≡ d/dt˜ and one can write down, δφ˙ ≈ H˜ δφ, δ ˙ ≈ H˜ δ , δφ ≈ H˜ , δ ≈ H˜ , and finally the fractional density contrast can be expressed as δρ ρ t2 = with the following constraint on the parameter C as given by, C ∼ O(1) 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 δρ Region I: |φ˙ | < | ˙ | ⇒ ρ ≈ δρ Region II: |φ˙ | > | ˙ | ⇒ ρ ≈ H˜ 2 W˜ h , | ˙ | ≈ 2√2M 2p H 2 ˜ |φ˙ | ≈ W˜ h3/2 M 3p(∂φ W˜ )h . δρ ρ t2 ≈ 1 1 − σ δρ ρ t1 . Further using Eq. (3.12) 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 5 Here one equivalent notation for the amplitude of the scalar perturbation used as √Pcmb = √P(Ncmb), which we have used in the non-attractor case. (4.6) (4.7) (4.8) (4.9) (4.3) (4.4) (4.5) Here one can interpret the results as • In 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( h). • In Region II, the amplitude of the density fluctuation at the horizon crossing is given by δρ ρ Region II = √ 2 W˜ h δρ ρ Region I . (4.10) This implies that contribution from the first slow-roll parameter, as given by W˜ = M22p ∂φW˜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( h). 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 δρ Case I: ρ ∼ δρ Case II: ρ ∼ ⎧ φ02 ⎪⎪⎪ 4M2p ⎨ ⎧ ⎪⎪⎪⎪⎪⎨ 8√1 α • In Region I and Region II of Case I, the amplitudes of the density fluctuation at the horizon crossing are related by δρ ρ δρ ρ ( h − 0) Region I . Region I 1/2 (4.13) δρ ρ δρ ρ ≈ This implies 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 ∼ √2M p ∼ O(M p) then by evaluating the amplitude of the density perturbation in Region I one can easily quantify the amplitude of the density perturbation in Region II. In this setup within the range 50 < N f / h < 70, we get ∼ δρ ρ which is consistent with Planck 2015 data. But if inflation starts at the following field value, φ0 = √2 M p, where the parameter ≷ 1 then one ca write the following relationship between the amplitude of the density perturbation in Region I and Region II as δρ ρ δρ ρ δρ ρ δρ ρ δρ ρ (4.11) (4.12) (4.16) (4.17) (4.18) (4.20) This implies that for ≷ 1 in Region II we get tightly constrained result for the amplitude for the density perturbation. • In Region I and Region II of Case II, the amplitude of the density fluctuation at the horizon crossing are related by Region II M 3p ≈ √8αλ( h)φ03 δρ ρ Region I . (4.19) 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( h) 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(M p) and we have an additional constraint: λ( h) ∼ √ 1 8α , (4.15) then by evaluating the amplitude of the density perturbation in the Region I one can easily quantify the amplitude of the density perturbation in Region II. Here one can also consider an equivalent constraint: φ0 ∼ 1 √8αλ( h) 1/3 M p. For both situations in the present setup within the range 50 < N f / h < 70, we get ∼ δρ ρ which is also consistent with Planck 2015 data. But if inflation starts at the field value φ0 = M p, where the parameter ≷ 1, and we define where the parameter ≷ 1, then one can write the following relationship between the amplitude of the density perturbation in Region I and Region II as δρ ρ . δρ ρ This implies that for ≷ 1 and ≷ 1 in Region II we get a tightly constrained result for the amplitude for the density perturbation. (4.23) (4.24) (4.25) (4.26) (4.27) Table 1 Inflationary observables and model constraints in the light of Planck 2015 data [ 44–46 ] for the dynamical attractors considered in Case I and Case II N f/h 50 60 70 AS (×10−9) δρ ρ = δρ ρ δρ ρ δρ ρ δρ ρ (4.21) In this context the scalar spectral tilt can be written at the horizon crossing as6 nS − 1 = d ln AS d f h ≈ ⎧ 3 ⎪⎨ (N f/h+1)2 3 ⎪⎩ N 2f/h ⎧ ⎪⎨ ≈ 6 − (N f/h+1)3 6 ⎩⎪ − N 3f/h for Case I, for Case II, for Case I, for Case II. (4.29) (4.30) (4.31) Finally combining Eqs. (4.29), (4.30) and (4.31) we get the following consistency relation for both Case I and Case II: (nS − 1)2 κS 2/3 βS = 3 = 3 − 6 . (4.32) This is obviously a new consistency relation for the present Higgsotic model of inflation and it is also consistent with Planck 2015 data [ 44–46 ]. 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 < N f / h < 70. In Fig. 6, we have plotted the running of the spectral tilt for scalar perturbation (κS = d2nS/d2 ln k) vs. spectral tilt for scalar perturbation (nS) in the light of Planck 2015 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 numbers of e-foldings N f / h = 70, N f / h = 60 and N f / h = 50, respectively. 6 Here we use a new symbol N f/h, which is defined as + N f/h = ++ln + aahf ++++ = |N (φh) − N (φ f )| ∼ 50−70, (4.28) Fig. 6 Plot for running of the running of spectral index κS = d2nS/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 the confidence contours obtained from various joint constraints [ 44–46 ] 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 2015 joint data sets [ 44–46 ]. It is clear from Fig. 6 that, for Case I we cover the range 0.59 × 10−3 < βS = ddlnnSk < 1.16 × 10−3 and −1.65 × 10−5 > κS = dd22lnnSk > −4.56 × 10−5 in the (κS, βS) 2D plane. Similarly for Case II we cover the dnS range 0.62 × 10−3 < βS = d ln k < 1.20 × 10−3 and −1.78×10−5 > κS = dd22lnnSk > −4.80×10−5 in the (κS, βS) 2D plane. 4.3 Primordial tensor modes and future observables In terms of the number of e-foldings (N ) the most useful parametrization of the primordial scalar and tensor power spectrum or equivalently the tensor-to-scalar ratio can be written near the horizon crossing Nh = N (φh) as 8 r (N ) = M 2p dφ dN 2 = r (Nh)e(N −Nh){Ah+Bh(N −Nh)} 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 r = r (Nh) ≈ 8M2p Vh Vh 2 = ⎧ 128M2p ⎪⎪⎨ φh2 ⎪ 512α2λ2( h)φh6 ⎪⎩ M6p (4.33) for Case I, for Case II, (4.34) and the symbols Ah, Bh and Ch are expressed in terms of the inflationary observables at horizon crossing as Ah = nT − nS+1, Bh = 21 (βT−βS). In the above parametrization Ah Bh i.e. βS − 2(nS − 1) βT − 2nT is always required for convergence of the Taylor expansion. Using this assumption the relationship between field excursion, φ = φh − φ f and tensor-to-scalar ratio r (Nh) can be computed as | φ| M p ≈ A2h 0 2π + r (Nh) e− 2Bh +erfi 8 Bh ++ Ah √2Bh − erfi √A2hBh − B8h N f / h +++++ . (4.35) Now the scale of inflation is connected with the tensor-toscalar ratio in the following fashion: V 1/4 h = 23 π 2 ASr ( fh) 1/4 M p ∼ 7.9 × 10−3 M p × r ( fh) 1/4 Substituting Eq. (4.36) in Eq. (4.35) we compute the relationship between field excursion and the scale of inflation as | φ| M p ≈ 0 Vh 6π M 4p AS Bh A2 e− 2Bhh ++erfi + + Ah √2Bh − erfi √A2hBh − B8h N f / h +++++ . (4.37) Also using Eq. (4.36) the tensor-to-scalar ratio can be written as r = r (Nh) = ⎪⎨ (2×λ1(0−h2)φMh4p)4 ⎧ Further using Eqs. (4.34) and (4.38) we get the following constraints from the primordial tensor perturbation: Table 5 Constraint on scalar four point non-Gaussian amplitude from equilateral, folded kite and squeezed configuration with assuming Suyama–Yamaguchi consistency relation Table 6 Constraint on scalar four point non-Gaussian amplitude from equilateral configuration without assuming Suyama–Yamaguchi consistency relation Scanning region I II III I II III I II III IV I + II + III + IV Scanning region I II III IV I + II + III + IV Scanning region IV I + II + III + IV Scanning region IV I + II + III + IV τNeqLuil In Table 5, we give the numerical estimates and constraints on the four point non-Gaussian amplitude from equilateral configuration with assuming Suyama–Yamaguchi consistency relation. Also in Table 6, we give the numerical estimates and constraints on the four point non-Gaussian amplitude from equilateral configuration without assuming Suyama–Yamaguchi consistency relation. Here all the obtained results are consistent with the two point and three point constraints as well as with the Planck 2015 data [ 44– 46 ]. 8.2.2 Using δN formalism In this section using the prescription of δN formalism in the attractor regime of cosmological perturbation we derive the expression for the non-Gaussian amplitudes associated with the four point function of scalar curvature fluctuation as gNeqLuil −0.004 < gNeqLuil < −0.023 Now we already know that in the attractor regime cosmological perturbation, solution for the additional field can be expressed in terms of the inflaton field φ and using this fact the expression for the non-Gaussian amplitudes associated with the four point function of scalar curvature fluctuation can be recast as where the new functions X1(φ), . . . , X6(φ) are defined as X1(φ) = f (φ) 1 + V22(φ) + V42(φ) + V61(φ) , X2(φ) = −3 f (φ) VV3((φφ)) + VV5((φφ)) , X3(φ) = f (φ) V 2(φ) V 2(φ) V6(φ) + V4(φ) , X4(φ) = f (φ) 1 + V23(φ) + V43(φ) + V61(φ) , X5(φ) = −3 f (φ) VV7((φφ)) + 2 VV5((φφ)) + VV3((φφ)) , X6(φ) = − f (φ) VV3((φφ)) − 2 VV42((φφ)) − 3 VV82((φφ)) − 5 V 2(φ) . V6(φ) −3 where f (φ) = 1 + V21(φ) . Further substituting the explicit form of the function V(φ) and N,φ , N,φφ , N,φφφ for all derived effective potentials at φ = φ∗ we get τ NloLc = Y2 X1(φ∗) + X2(φ∗)φ∗ + X3(φ∗)φ∗2 , 25 2 2X4(φ∗) + X5(φ∗)φ∗ + X6(φ∗)φ∗2 . gNloLc = 54 Y Now we comment on the consistency relation between the non-Gaussian parameters derived from four point and three (8.177) (8.178) (8.179) point scalar correlation function in the attractor regime of inflation. To establish this connection we start with Eqs. (8.64), (8.178) and (8.179) and finally get new set of consistency relations: 36 τ NloLc = 25 ( fNloLc)2 X1(φ∗) + X2(φ∗)φ∗ + X3(φ∗)φ∗2 , 10 gNloLc = 27 ( fNloLc)2 2X4(φ∗) + X5(φ∗)φ∗ + X6(φ∗)φ∗2 . gNloLc = 418265 τ NloLc 2222XX44((φφ∗∗)) ++ XX55((φφ∗∗))φφ∗∗ ++ XX66((φφ∗∗))φφ∗∗2233 . It is a very well-known fact that in the non-attractor regime, where the additional field is frozen in the Planck scale Suyama–Yamaguchi consistency relation [ 176–178 ] holds true, which states: 36 τ NloLc = 25 ( fNloLc)2. (8.183) Further using this results one can estimate the deviation in the Suyama–Yamaguchi consistency relation if we go from attractor regime to non-attractor regime of cosmological perturbation as | τ NloLc| = |[τ NloLc|non-attractor − τ NloLc|attractor] 36 = 25 ( fNloLc)2|non-attractor|{1 − Qcorr}|, where the correction factor Qcorr can be written as ( fNloLc)2|attractor Qcorr = ( fNloLc)2|non-attractor X1(φ∗) + X2(φ∗)φ∗ + X3(φ∗)φ∗2 . (8.180) (8.181) (8.182) (8.184) (8.185) Here we need to point out a few crucial issues: • First of all, to estimate the magnitude of the deviation factor Qcorr we need to concentrate on two physical situations, I. Super Planckian field regime and II. Sub Planckian field regime. • In the super Planckian field regime the deviation factor Qcorr can be expressed as ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎪⎪⎪⎪ ⎛ ⎪⎪ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎪⎪⎪⎪ ⎛ ⎪⎪ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ 1 − ⎪⎪⎪⎪ ⎛ ⎪⎪ ⎪⎪⎪⎪⎪⎪ ⎜⎝ 1 − ⎪⎪⎪ ⎪⎪⎪⎪ ⎛ ⎨ ⎜ 1 − ⎪⎪ ⎝ ⎪⎪⎪ ⎪⎪⎪⎪ ⎛ ⎪⎪ ⎪⎪⎪⎪⎪⎪ ⎜⎝ 1 − ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ − 18M2p 72M4p 3888M8p 81φ∗2 + 6561φ∗4 + 43046721φ∗8 + · · · 18M2p φ2 + ∗ 72M4p φ4 + ∗ 3888M8p φ8 ∗ 3888M8p mc2 φ∗8 1− mc2−λφ∗2 72M4p 4 4 + φ∗4 1− φφV4 ∗ 3888M8p φ∗8 1− φφV44 8 + · · · ⎠⎟ ∗ ⎞ 72M4p mc2 φ∗4 1− mc2−λφ∗2 ⎞ 4 8 + · · · ⎠⎟ 18M2p 72M4p 2 4 φ∗4 1+ξ(φ∗2−φV2 )+ φφV2 ∗ where the factor f is defined as ( fNloLc)2|attractor f = ( fNloLc)2|non-attractor . Now to give a proper estimate of the deviation in the magnitude of the amplitude of non-Gaussian parameter computed from four point function in terms of the three point nonGaussian amplitude for the time being we assume that the results obtained from the attractor and non-attractor formalism is almost at the same order of magnitude. In that case we (8.187) where the correction factor Jcorr 1 is highly suppressed in the super Planckian region of the perturbation theory, but those small corrections are important as precision measurement is concerned in the context of cosmology. In the case of our derived effective potentials we get the following approximate expressions for the correction factor: ⎞ 8 + · · · ⎟⎠ ⎞ for Case I for Case II for Case II + Choice I for Case II + Choice II for Case II + Choice III. for Case II + Choice III. (8.186) (8.189) Page 64 of 82 • In the sub Planckian field regime the deviation factor Qcorr can be expressed as 1 + 198683 Mφ∗66p + · · · for Case I for Case II for Case II + Choice I for Case II + Choice II for Case II + Choice III. f × So it is clear that | Jcorr| captures the effect of the deviation in the Suyama–Yamaguchi consistency relation which are very small and highly suppressed in the super Planckian regime of inflation. But as far as precision cosmology is concerned, this small effect is also very useful to discriminate between all derived effective models considered in this paper. If in the near future Planck or any other observational probe detects the signature of primordial non-Gaussianity with high statistical significance then one can also further comment on the significance of attractors and non-attractors in the context of early universe cosmology. Further using this results one can estimate the deviation in the Suyama–Yamaguchi consistency relation if we go from attractor regime to non-attractor regime of the cosmological perturbation as 36 | τ NloLc| = 25 ( fNloLc)2|non-attractor|{1 − f (1 + Ccorr)}| 36 ∼ 25 ( fNloLc)2|non-attractor|Ccorr|. (8.195) where the factor f is defined earlier, which is f ∼ O(1). Consequently the deviation factor can be recast as where the correction factor Ccorr 1 is suppressed in the sub Planckian region of the perturbation theory, but those small corrections are important as precision measurement is concerned in the context of cosmology. In the case of our derived effective potentials we get the following approximate expressions for the correction factor: (8.192) (8.194) Also the fractional change can be expressed as +++ τ NloLc +++ ++ (τ NloLc)non-attractor ++φ∗ Mp f (1 + Ccorr)| ∼ |Ccorr|. (8.196) So it is clear that |Ccorr| captures the effects of the deviation in the Suyama–Yamaguchi consistency relation which are very small and suppressed in the sub Planckian regime of inflation. • From the study of sub Planckian and super Planckian regime it is evident that when f ∼ O(1) i.e. the nonGaussian amplitude obtained from three point function in attractor and non-attractor regime for all the derived effective potentials are of the same order then the deviation from Suyama–Yamaguchi consistency relation is very small. The only difference is in the sub Planckian case this correction is greater than unity and on the other hand in the super Planckian case this correction factor is less than unity. But since we are interested in the precision cosmological measurement, such small but distinctive corrections will play a significant role in discriminating between the classes of effective models of inflation derived in this paper. • Finally, if we relax the assumption that the deviation factor, f = 1, then one can consider the following two situations1. First we consider, f 1. In this case in the super Planckian and sub Planckian regime we get the following results for the deviation in the Suyama– Yamaguchi consistency relation: 36 | τ NloLc|φ∗ Mp = 25 ( fNloLc)2|non-attractor| f (1 − Jcorr)|. 36 | τ NloLc|φ∗ Mp = 25 ( fNloLc)2|non-attractor| f (1 + Ccorr)|. Also the fractional change in the Suyama–Yamaguchi consistency relation can be expressed as +++ τ NloLc +++ + (τ NloLc)non-attractor ++φ∗ Mp + +++ τ NloLc +++ ++ (τ NloLc)non-attractor ++φ∗ Mp = | f (1 − Jcorr)|, = | f (1 + Ccorr)|. (8.197) (8.198) (8.199) In this specific situation the deviation factor is large and consequently one can achieve a maximum amount of violation in the Suyama–Yamaguchi consistency relation. Here the results of the super Planckian and sub Planckian field values differ due to the presence of the correction factors Jcorr and Ccorr. Here both Jcorr < 1 and Ccorr < 1, but for model discrimination such small corrects are significant as mentioned earlier. 2. Next we consider, f 1. In this case in the super Planckian and sub Planckian regime we get the following results for the deviation in the Suyama– Yamaguchi consistency relation: 36 | τ NloLc|φ∗ Mp = 25 ( fNloLc)2|non-attractor|1 − 36 | τ NloLc|φ∗ Mp = 25 ( fNloLc)2|non-attractor|1 − f |. f |. (8.200) (8.201) (8.202) (8.203) (8.204) Also the fractional change in the Suyama–Yamaguchi consistency relation can be expressed as +++ τ NloLc +++ + (τ NloLc)non-attractor ++φ∗ Mp + +++ τ NloLc +++ ++ (τ NloLc)non-attractor ++φ∗ Mp In this specific situation deviation factor is small and consequently one can achieve very small amount of violation in Suyama–Yamaguchi consistency relation. Here the results of the super Planckian and sub Planckian field value are almost the same as we have neglected the terms f Jcorr 1 and f Ccorr 1. Now to derive the results of non-Gaussian amplitudes in the non-attractor regime using the δN formalism we need to freeze the value of the additional field in the Planck scale. If we do this job then the expression for the four point nonGaussian amplitude computed from scalar fluctuation can be expressed as +++ τ NloLc +++ = ++++ 2257 W f − 1+++ . (8.207) ++ (τ NloLc)In-In ++ + Now if we claim that at the horizon crossing non-Gaussian amplitudes obtained from the δN and In-In formalism are of the same order then in that case we get W f ∼ O(1). Consequently the deviation in the Suyama–Yamaguchi consistency relation can be recast as 72 | τ NloLc| = |(τ NloLc)δN − (τ NloLc)In-In| ∼ 625 (( fNloLc)2)In-In. + τ NloLc + Consequently the fractional deviation is given by +++ (τ NloLc)In-In +++ ∼ 225 . 9 Conclusion | τ NloLc| = |(τ NloLc)δN − (τ NloLc)In-In| To summarize, in the present article, we have addressed the following points: • Firstly we have started 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. For simplicity we consider the case where the matter field contains only canonical kinetic term. To build effective potential from this toy setup of modified gravity in 4D we choose f (R) = α R2 gravity. • Next to start with in the matter sector we choose a very simple model of potential, V (φ) = λ4 φ4, where φ is a real scalar field and λ is a real parameter of the monomial model. This type of potential can be treated as a Higgs like potential as the structure of Higgs potential is given by V (H ) = λ4 (H † H − V 2), where λ is Yukawa coupling, H is the Higgs SU(2) doublet and 0|H |0 = V ∼ 125 GeV is the VEV of the Higgs field. Now one can write the Higgs SU(2) doublet as H † = (φ 0) and the corresponding Higgs potential can be recast as V (φ) = λ4 (φ2 −V2)2. Now at the scale of inflation, which is at O(1016 GeV), contribution from the VEV is almost negligible and consequently one can recast the Higgs potential in the mono λ φ4. The only difference is in the mial form, V (φ) ≈ 4 case of Higgs where λ is the Yukawa coupling and in the (8.206) (8.208) case of a general monomial model λ is a free parameter of the theory. Due to the similar structural form of the potential we call the general φ4 monomial model as Higgsotic potential. • Further, we provide the field equations in a spatially flat FLRW background, which are extremely complicated to solve for this setup. To simplify, next we perform a conformal transformation in the metric and write down the model action in the transformed Einstein frame. Next, we derive the field equations in a spatially flat FLRW background and try to solve them for two dynamical attractor features: I. a power-law solution and II. an exponential solution. However, the second case gives rise to tachyonic behavior which can be resolved by considering the non-BPS D-brane in superstring theory, considering the effect of mass like quadratic term in the effective potential and considering the effect of non-minimal coupling between f (R) = α R2 scale free gravity sector and the matter field sector. • Next, using two dynamical attractors, a power-law and an exponential solution, we have studied the cosmological constraints in the presence of two fields in an Einstein frame. We have studied the constraints from primordial density perturbation, by deriving the expressions for two point function and the present observables-amplitude of power spectrum for density perturbations, corresponding spectral tilt and associated running and running of the running for inflation. We have repeated the analysis for tensor modes and also comment on the future observables – the amplitude of the tensor fluctuations, associated tilt and running, and the tensor-to-scalar ratio. We also provide a modified formula for the field excursion in terms of the tensor-to-scalar ratio, scale of inflation and the number of e-foldings. Further, we have compared our model with Planck 2015 data and constrain the parameter α of the scale free gravity and non-minimal coupling parameter λ( h). Additionally, we have studied the constraint for the reheating temperature. Finally, we derive the expression for the inflaton and the coupling parameter at horizon crossing, during reheating and at the onset of inflation which are very useful to study the scale dependent behavior. Most importantly, in the present context one can interpret such scale dependence as an outcome of RG flow in the usual Quantum Field Theory. • Further, we have explored the cosmological solutions beyond attractor regime. We have shown that this possibility can be achieved if we freeze the field value of the dilaton field in Einstein frame. This possibility can be treated as a single field model where an additional field freezes at a certain field value, which we fix at the reduced Planck scale. To serve this purpose we have used the ADM formalism and computed the two point function and associated present inflationary observables using Bunch–Davies initial condition for scalar fluctuations. We have repeated the procedure for tensor fluctuations as well. In the non-attractor regime, we have also derived a modified version of the field excursion formula in terms of the tensor-to-scalar ratio, scale of inflation and the number of e-foldings. We have also derived few sets of consistency relations in this context which are different from the usual single field slow-roll models. For example, instead of getting r = −8nT here we get sidered the contribution from contact interaction term, scalar and graviton exchange. In the attractor phase following the prescription of the δN formalism we also derive the expressions for the four point non-Gaussian amplitude gNloLc and τ NloLc. Next we have shown that the consistency relation connecting three and four point nonGaussian amplitude aka Suyama–Yamaguchi relation is modified in the attractor phase and further given an estimate of the amount of deviation. Further, for the consistency check we freeze the dilaton field in the Planck scale and redo the analysis of the δN formalism. By doing this we have found that the expression for the four point nonGaussian amplitude is slightly different as expected for the single field case. Next we have also shown that the exact numerical deviation of the consistency relation is of the order of 2/25 by assuming non-Gaussian three point amplitude for attractor and non-attractor phase are of the same order of magnitude. Further, we compare the results obtained from the In-In formalism and δN formalism for the non-attractor phase, where the dilaton field is fixed in the Planck scale. Here, finally, we give a theoretical bound on the scalar four point non-Gaussian amplitude computed from equilateral, folded kite and squeezed limit configurations. The obtained results are consistent with the Planck 2015 data. The future prospects of our work are appended below: • 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. • To investigate the role of scale free theory of gravity, as an example we have used α R2 gravity. But the present analysis can be generalized to any class of f (R) gravity models and other class of higher derivative gravity models. • In the matter sector for completeness one can consider most generalized version of P(X, φ) models, where X = − 21 gμν ∂μφ∂ν φ. DBI is one of the examples of P(X, φ) model which can be implemented in the matter sector instead of simple canonical kinetic contribution. • In this work, we have not given any three point computation and found point scalar correlation function and representative non-Gaussian amplitudes using the In-In formalism in the attractor regime in the presence of both fields, φ and , for all classes of Higgsotic models. In near future we are planning to present the detailed calculation on this important issue. • Generation of primordial magnetic field through inflationary magnetogenesis is one of the important issues in the context of primordial cosmology, which we have not explored yet from our setup. One can consider such inter Page 68 of 82 actions by breaking conformal invariance of the U (1) gauge field in the presence of time dependent coupling f (φ (η)) to study the features of primordial magnetic field through inflationary magnetogenesis. We have also a future plan to address this issue. • In this work we have restricted our analysis within the class of Higgsotic models. For completeness in the future we will extend this idea to all class of potentials allowed by the presently available observed Planck data. We will also include the effects of various types of nonminimal and non-canonical interactions in the present setup. • In the same direction one can also carry forward the present analysis in the context of various types of higher derivative gravity setup and comment on the constraints on the primordial non-Gaussianity, reheating and generation of primordial magnetic field through inflationary magnetogenesis for completeness. Also one can consider the possibility of non-minimal interaction between α R2 gravity and matter sector. In future we will investigate the possibility of appearing new consistency relations in the presence of higher derivative gravity setup and will give proper estimate of the amount of violation from Suyama– Yamaguchi consistency relation. • During the computation of correlation functions using semi classical method, via the δN formalism, 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 an Einstein frame. For more completeness, one can relax these assumptions and redo the analysis by taking care of all such contributions. Additionally, we have a future plan to extend the semi classical computation of the δN formalism of cosmological perturbation theory in a more sophisticated way and will redo the analysis in the present context. • In this work, we also have not investigated the possibility of getting dark matter and dark energy constraints from the present up. Most importantly the present structure of interactions in the Einstein frame shows that the two fields, φ and , are coupled and due to this fact if we want to explain the possibility of dark matter and dark energy together; from this setup it is very clear that they are coupled. But this is not very clear at the level of analytics and detailed calculations. Here one can also investigate these possibilities from this setup. • In this work we have not investigated 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 specific numerical contribution of such terms and comment on the effects of such terms in precision cosmology measurement. • Here one can generalize the results for α vacua and study its cosmological consequences for all types of derived potential in the present context. • In the present context one can also study the quantum entanglement between the Bell pairs, which can be created through the Bell inequality violation in cosmology [ 179–181 ]. Acknowledgements SC would like to thank Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai and specially the Quantum Structure of the Spacetime Group for providing a Visiting (Post-Doctoral) Research Fellowship. The work of SC was supported in part by Infosys Endowment for the study of the Quantum Structure of Space Time. SC takes this opportunity to thank sincerely to Ashok Das, Sudhakar Panda, Shiraz Minwalla, Sandip P. Trivedi, Gautam Mandal and Varun Sahni for their constant support and inspiration. SC also thanks the organizers of Indian String Meet 2016 and Advanced String School 2017 for providing the local hospitality during the work. SC also thanks Institute of Physics, Bhubaneswar for providing the academic visit during the work. Last but not least, SC would like to acknowledge debt to the people of India for their generous and steady support for research in natural sciences, especially for theoretical high energy physics, string theory and cosmology. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. 10 Appendix 10.1 Effective Higgsotic models for generalized P ( X, φ) theory In this section, to give a broad overview of the effective Higgsotic models let us start with a general f ( R) theory in the gravity sector and generalized P ( X, φ) theory in the matter sector. The representative actions in a Jordan frame is given by S = d4 x √−g [ f ( R) + P ( X, φ)] , where P ( X, φ) is a arbitrary function of single scalar field φ and the kinetic term X = − 21 gμν ∂μφ∂ν φ. In general f ( R) is any arbitrary function of R. But for our purpose we choose f ( R) = α R2 to study the consequences from scale free gravity. From this representative action one can write down the field equations in a spatially flat FLRW background as H 2 = a 2 ˙ a Here the total potential can be recast as where V0 = M 4p/8α, exactly mimics the role of cosmological constant as mentioned earlier. In the case of Higgsotic model we can rewrite the total potential as (10.7) the specific form of P(X, φ) as stated in Eq. (10.6) after a conformal transformation we get (10.3) G(X˜ , φ, ) = 1 4 2H˙ + 3H 2 = 2 aa¨ + aa˙ 2 pφ where for generalized P(X, φ) theory pressure pφ and density ρφ can be written as pφ = P(X, φ), ρφ = 2X P,X (X, φ) − P(X, φ). (10.4) Here the effective speed of sound parameter cS is defined as cS = 0 P,X (X, φ) P,X (X, φ) + 2X P,X X (X, φ) . If we choose the following functional form of P(X, φ): as pointed out earlier, then we get the following simplified expression for pφ and density ρφ : Also one can consider any arbitrary slow-roll effective potential but for our purpose we choose monomial Higgsotic λ φ4 in the Jordan frame. model, V (φ) = 4 In the present context let us introduce a scale dependent mode , which can be written in terms of a no scale dilaton mode as = f (R)M −p2 = 2α R M −p2 = e 23 Mp = 2, which plays the role of a Lagrange multiplier and arises in the Jordan frame without space-time derivatives. Here is the conformal factor of the conformal transformation that we perform from Jordan frame to Einstein frame. In terms of the newly introduced no scale dilaton mode the total action of the theory (see Eq. (2.1)) can be recast as S = After doing C.T. the total action can be recast in the Einstein frame as M 2p 2 R˜ + G(X˜ , φ, ) (10.9) where after applying C.T. the functional G(X˜ , φ, ) is defined in an Einstein frame as G(X˜ , φ, ) = 1 4 P(X˜ , φ) − M8α4p e2 23 Mp . (10.10) Here X˜ is the kinetic term after a conformal transformation, which is defined as X˜ = − 21 g˜μν ∂˜μφ∂˜ν φ. Now in the case of The rest of the computation is exactly similar to what we have performed earlier, only the structure of the total effective potential changes. Here it is important to note that apart from f (R) gravity one can consider various other possibilities. To give a clear picture about various classes of two-field attractor models one can consider the following 4D effective action in an Einstein frame: where J (X, Y, φ, ) is the general functional of the two fields φ and : the following specific mathematical structure: c J (X, Y, φ, ) = e− M1 p X + Y − W (φ, ). (10.16) Here c1 and c2 characterize the effective coupling constant in 4D, which are different for various types of source theories. In the EFT setup these are identified as the Wilson coefficients. Additionally, it is important to note that the kinetic terms for (10.11) (10.12) (10.13) (10.14) gμν the φ and field are defined as X = − 2 ∂μφ∂ν φ and Y = gμν − 2 ∂μ ∂ν . Here W (φ, ) is the 4D effective potential, which is given by the following expression: This is a non-separable form of the two-field effective poten c tial where one can treat V (φ) as usual inflaton field and e− M2 p as the dilaton exponential coupling. This type of effective theory can be derived from the following class of models: 1. Type I: Consider an action in a Jordan frame where the scalar field is non-minimally coupled with the gravity sector as given by (10.17) S = d4x √−g 2 f1( )R − f2( )gμν ∂μ ∂ν − U ( ) + X − V (φ)] . (10.18) Here f1( ) is the non-minimal coupling and f2( ) is the non-canonical interaction. This type of theories include the following subclass of models: • Jordan Brans Dicke (JBD) theory: In this case we have , U ( ) = 0, Here ω is the JBD parameter and for power-law inflation ω > 1/2. • Induced gravity theory: In this case we have ω f1( ) = 16π , f2( ) = 16π M p . Here g1 and g2 are coupling constants. For power-law inflation g1 < 1/2. • Non-minimally coupled theory: In this case we have f1( ) = M22p − ξ2 2, f2( ) = 21 , U ( ) = 0, (10.20) (10.21) (10.22) c1 = c22 , c2 = =<>= 2(6ξ3MξM2p 2p+ 1) , (10.23) for ξ = 1/6 for ξ = 1/6. (10.24) After doing the conformal transformation in an Einstein frame one can derive the required form of the effective action from all these models. 2. Type II: Consider an action in a Jordan frame where the scalar field is minimally coupled with the f (R) gravity sector as given by S = d4x √−g [ f (R) + X − V (φ)] . (10.25) Here f (R) is an arbitrary functional of the Ricci scalar R. After doing the conformal transformation in an Einstein frame one can derive the required form of the effective action. 3. Type III: Consider a 4 + D dimensional Kaluza–Klein theory with an additional scalar field. This type of theories includes the following subclass of models: • Extra dimensional theory-I: In this case the inflaton is introduced in the 4D effective action in a Jordan frame: S = 1 2 R + 4 1 − D × gμν ∂μ ∂ν ; − U ( ) + X − V (φ) . (10.26) In this case we have c2 c1 = 2 , c2 = 8D D + 2 0 2 = M p 2 1 + D ln( ). (10.27) But from this type of model no power-law inflationary solutions are possible. • Extra dimensional theory-II: In this case the inflaton is introduced in the 4 + D dimensional action in a Jordan frame as given by S = d4+D x R + X − V (φ) . 1 2κ42+D (10.28) Here g4+D is the determinant of the 4 + D dimensional metric and κ42+D is the 4 + D dimensional gravitational coupling constant. In this case also we have c1 = 0, c2 = 2D D + 2 , From this type of model power-law inflationary solutions are possible for all extra D dimensions. 4. Type IV: Consider an action in a Jordan frame from superstring theory in 10 dimension with fixed Kalb– Ramond background. In this case the scalar field is non-minimally coupled with the gravity sector: S = d4x√−g e−2 R + 4gμν ∂μ ∂ν + X − V (φ) . In this case is known as the dilaton field. But from this type of model no power-law inflationary solutions are possible. Here additionally we have two classes of the solutions: Class I: After the completion of the phase of reheating, the total system enters the radiation dominated stage, at the beginning of which the total energy density is governed by Eq. (7.47). At that stage, the scalar inflaton fields have almost settled down in one of the potential valleys of the derived EFT potentials and get its VEV for the proposed model in R2 gravity setup in an Einstein frame. To make the computation simpler we also assume that at the level of perturbations the dilaton field is almost decoupled from the Standard Model fields and the only dynamical field present in the model at late times. Henceforth, we will treat as a dynamical field minimally coupled to the R2 gravity in a conformally transformed Einstein frame and also assume that the field is non-interacting with other matter degrees of freedom and radiation content of the universe at late times. During this epoch the total potential is characterized by the following expression: W˜ (φˆ , ) = V0 1 + , Dominant at late time = V0 + λˆ exp (10.33) where V0 is defined as V0 = M8α4p , and the VEV of the inflaton field φ is denoted by the symbol φˆ . Here one can set φˆ ∼ O(M p) for the proposed model at late time scale. Once the contribution of the inflaton scalar field φ gets its VEV the corresponding energy density ρm ≡ ρφ = Constant. Now in the present context to characterize the features of late time acceleration of the universe let us introduce equation of state parameter wX(= w ), which is defined as pX wX = ρX = d dt˜ d dt˜ 2 2 − W˜ (φˆ , ) + W˜ (φˆ , ) and the continuity equation in the present context can be written as dρX + 3H˜ (1 + wX)ρX = 0. (10.35) dt˜ For the qualitative analysis of the prescribed system in the Einstein frame and in order to compare with present day observations, we introduce the following sets of dimensionless density parameters and shifted equation of state parameter: ρX = 3H˜ 2 M 2p , m ≡ ρm φ = 3H˜ 2 M 2p X ≡ X ≡ m ≡ ρr r ≡ 3H˜ 2 M 2p , = 1 + w = 1 + wX, φ = 1 + wφ = 1 + wm . (10.34) (10.36) (10.37) In order to transform the cosmological equations into a simplified autonomous system, we define the following dimensionless auxiliary variables for the study of present dynamical system at late time scale: x ≡ √ 6H˜ M p W˙˜ (φˆ , ) , y ≡ √ 3H˜ M p ≡ −M p∂ ln W˜ (φˆ , ), Case I: δ φV4 1 − φ4 1 − (mc2m−c2λφ2) 2 2 φV2 1 + ξ2 (φ2 + φ0 − 2φV2 ) + ξ2 (φ2 − φ0 ) + φ2 V(φ) = − √ φ 6M p × for Case I for Case II for Case II + Choice I(v1&v2) for Case II + Choice II(v1&v2) for Case II + Choice III. dx dN dy dN d dN d r dN d m which can be recast in the autonomous form as W˜ (φˆ , )∂ W˜ (φˆ , ) (∂ W˜ (φˆ , ))2 r ) − r ), dN together with an additional constraint condition, X + r + m = x 2 + y2 + m + r = 1. Also using these dimensionless variables Eqs. (10.34) and (10.36) can be recast as (10.39) weff − 3r , X One can also define the total effective equation of state as peff pX + pm + pr weff ≡ ρeff = ρX + ρm + ρr p + pφ + pr = ρ + ρφ + ρr = x 2 − y2 + 3r . For an accelerated expansion effective equation of state satisfy the following constraint, weff < −1/3. Using this methodology mentioned in this section one can study the constraints on the model from late time acceleration which is beyond the scope of our discussion in this paper. 10.3 Details of the δN formalism 10.3.1 Useful field derivatives of N To simplify the calculation for δN let us consider all these possibilities to write down the infinitesimal change in field in terms of the inflaton field φ: (10.40) (10.41) ∂φ ∂ ∂φ ∂φ ∂ ∂φ ∂ ∂φ = 9φ = − √6M p φ δφ, Case II: δ = − √6M p Case II + Choice I(v1&v2): δφ, δ δ φ , = − √6M p Case II + Choice II(v1&v2): φ mc2 δ = − √6M p δφ 1 − (mc2 − λφ2) , Case II + Choice III: φ δφ 1 + ξ2 (φ2 + φ02 − 2φV2 ) = − √6M p (10.42) (10.43) (10.44) (10.45) (10.46) (10.47) (10.48) (10.49) (10.50) (10.51) This additionally implies that one can write down the following differential operator for the field: 1 ∂ = V(φ) ∂φ , ∂2 = 1 2 V (φ) V2(φ) ∂φ − V3(φ) ∂φ , ∂3 = V31(φ) ∂φ3 − 3 VV4((φφ)) ∂φ2 + 3 V 2(φ) ∂φ , V5(φ) V(1φ) ∂φ2 − VV2((φφ)) ∂φ , ∂ ∂φ = V(1φ) ∂φ2, V(1φ) ∂φ3 − 2 VV2((φφ)) ∂φ2 V2(φ) − 2 V 2(φ) V (φ) V3(φ) V(1φ) ∂φ3 − VV2((φφ)) ∂φ2 , ∂φ , ∂ ∂φ∂φ = V(1φ) ∂φ, 3 ∂φ∂ ∂ = V2(φ) ∂φ3 − 3VV3((φφ)) ∂φ2 + 3 VV42((φφ)) ∂φ , 1 ∂ ∂φ∂ = V21(φ) ∂φ3 − 2VV3((φφ)) ∂φ2 + 2 VV42((φφ)) ∂φ , (10.52) (10.53) +3 VV42((φφ)) N,φ , N, φ = V2(φ) ∂φ3 − 2VV3((φφ)) ∂φ2 + 2 VV42((φφ)) ∂φ N 1 1 = V2(φ) N,φφφ − 2VV3((φφ)) N,φφ +2 VV42((φφ)) N,φ , fiwehlCedroeφnsii.eseq.duee=finntl∂eyφdo.ansethcaenpawrrtiiatel derivative with respect to the N, φ = V211(φ) ∂φ3 − VV3((φφV)) (∂φφ2) N = V2(φ) N,φφφ − V3(φ) N,φφ . N, = V(1φ) ∂φN = V(φ) N,φ, 1 1 2 V (φ) N, = V2(φ) ∂φ − V3(φ)∂φ N 1 V (φ) = V2(φ) N,φφ − V3(φ) N,φ , 1 2 N,φ = V(φ) ∂φ − VV2((φφ)) ∂φ N (10.54) If we neglect the quadratic slow-roll corrections then the solutionofEq.(8.40)takesthefollowingformforalldifferent cases considered here: 10.3.2 Second-order perturbative solution with various source (10.60) (10.61) (10.62) 1 = V(φ) N,φφ − VV2((φφ)) N,φ , 1 2 1 N, φ = V(φ) ∂φ N = V(φ) N,φφ, N, = V3(φ) ∂φ3 − 3VV4((φφ)) ∂φ2 + 3VV52((φφ)) ∂φ N 1 (10.55) = V3(φ) N,φφφ − 3VV4((φφ)) N,φφ + 3VV52((φφ)) N,φ , 1 N,φφ = V(1φ) ∂φ3 − 2VV2((φφ)) ∂φ2 − VV2((φφ)) − 2VV32((φφ)) ∂φ N, 1 = V(φ) N,φφφ − 2VV2((φφ)) N,φφ − VV2((φφ)) − 2VV32((φφ)) N,φ , N,φ φ = V(1φ) ∂φ − VV2((φφ)) ∂φ2 N 3 1 = V(φ) N,φφφ − VV2((φφ)) N,φφ , 1 3 1 N, φφ = V(φ) ∂φ N = V(φ) N,φφφ, N,φ = V21(φ) ∂φ3 − 3VV3((φφ)) ∂φ2 + 3 VV42((φφ)) ∂φ N 1 = V2(φ) N,φφφ − 3VV3((φφ)) N,φφ (10.57) (10.58) (10.59) (10.56) +e−3Ht(4 cφL3(1 + 3Ht)D1 − 9H2D3) . (10.63) For Case I: 1 27φ∗HeHYt 2 = D4 + 27H3 Y2(3 + Y)3 {−Y(3 + Y)2 ×(4 cφL3 + H2Y(3 + Y)) + H(4 cφL3(−18 + Y(3 + Y)(−6 + Ht(3 + 2Y))) + H2Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))))} +9H2 cφL3t(φL + 4D2) For Case II: 1 27φ∗HeHYt 2 = D4 + 27H3 Y2(3 + Y)3 × 4−Y(3 + Y)2 −4 cφL3 + H2Y(3 + Y) + H −4 cφL3(−18 + Y(3 + Y)(−6 + Ht(3 + 2Y))) +H2Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))) 5 +9H2t β − cφL3 (φL + 4D2) −e−3Ht 4 cφL3(1 + 3Ht)D1 + 9H2D3 . (10.64) For Case II + Choice I(v1): 1 27φ∗HeHYt 2 = D4 + 27H3 Y2(3 + Y)3 4−Y(3 + Y)2 −4 cφL3 + H2Y(3 + Y) + H −4 cφL3(−18 + Y(3 + Y)(−6 + Ht(3 + 2Y))) +H2Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))) 5 For Case II + Choice II(v2): 1 2 = D4 + 54H 3 54φ∗ H eHYt Y2(3 + Y)3 :−Y(3 + Y)2 −McφL + 4 cφL3 + H 2Y(3 + Y) + H (−4 cφL3 + McφL)(18 − Y(3 + Y)(−6 + H t (3 + 2Y))) +H 2Y(3 + Y)(−9 + Y(3 + Y)(−2 + H t (3 + 2Y))) ; +9H 2t 2β + φL(−Mc(φL + 2D2) + cφL2 (φL + 4D2)) −e−3Ht 2φL(Mc − 4 cφL2 )(1 + 3H t)D1 + 18H 2D3 For Case III: 1 2 = D4 + 27H 3φL 27φ∗ H eHYt Y2(3 + Y)3 × :−Y(3 + Y)2 ξ ξ + H 2φLY(3 + Y) ξ ξ (−18 + Y(3 + Y)(−6 + H t (3 + 2Y))) + H +H 2φLY(3 + Y)(−9 + Y(3 + Y)(−2 + H t (3 + 2Y))) ; +9H 2t φL(β + ξ ) + ξ ξ D2 (10.69) (10.67) . . (10.68) +9H 2t β + cφV4 − cφL3 (φL + 4D2) −e−3Ht 4 cφL3 (1 + 3H t)D1 + 9H 2D3 . (10.65) 27φ∗ H eHYt Y2(3 + Y)3 × :−Y(3 + Y)2 4 cφL3 + H 2Y(3 + Y) + H 4 cφL3 (−18 + Y(3 + Y)(−6 + H t (3 + 2Y))) +H 2Y(3 + Y)(−9 + Y(3 + Y)(−2 + H t (3 + 2Y))) ; +9H 2t β − cφV4 + cφL3 (φL + 4D2) +e−3Ht 4 cφL3 (1 + 3H t)D1 − 9H 2D3 . (10.66) 54φ∗ H eHYt Y2(3 + Y)3 × :−Y(3 + Y)2 McφL − 4 cφL3 + H 2Y(3 + Y) + H (4 cφL3 − McφL)(18 − Y(3 + Y)(−6 + H t (3 + 2Y))) +H 2Y(3 + Y)(−9 + Y(3 + Y)(−2 + H t (3 + 2Y))) ; +9H 2t 2β + φL(Mc(φL + 2D2) − cφL2 (φL + 4D2)) +e−3Ht 2φL(Mc − 4 cφL2 )(1 + 3H t)D1 − 18H 2D3 × 4−Y(3 + Y)2 4 cφL3 + H 2Y(3 + Y) + H −4 cφL3 (18 + 6Y(3 + Y)) 10.3.3 Expressions for perturbative solutions in final hypersurface Neglecting the contribution from the quadratic slow-roll term and taking up to linear-order term in slow-roll we get the following result: 1 1(N = 0) = D2 − 3H D1 + (10.70) Similarly if we neglect the quadratic slow-roll corrections then the solution of 2(N = 0) takes the following form for all different cases considered here: For Case I: 1 2(N = 0) = D4 + 27H 3 27φ∗ H Y2(3 + Y)3 × 4−Y(3 + Y)2 4 cφL3 + H 2Y(3 + Y) + H −4 cφL3 (18 + 6Y(3 + Y)) −H 2Y(3 + Y)(9 + 2Y(3 + Y)) 5 + 4 cφL3 D1 − 9H 2D3 For Case II: 1 2(N = 0) = D4 + 27H 3 27φ∗ H Y2(3 + Y)3 × 4−Y(3 + Y)2 −4 cφL3 + H 2Y(3 + Y) + H 4 cφL3 (18 + 6Y(3 + Y)) −H 2Y(3 + Y)(9 + 2Y(3 + Y)) 5 − 4 cφL3 D1 + 9H 2D3 For Case II + Choice I(v1): 1 2(N = 0) = D4 + 27H 3 × 4−Y(3 + Y)2 −4 cφL3 + H 2Y(3 + Y) + H 4 cφL3 (18 + 6Y(3 + Y)) −H 2Y(3 + Y)(9 + 2Y(3 + Y)) 5 . − 4 cφL3 D1 + 9H 2D3 For Case II + Choice I(v2): 1 2(N = 0) = D4 + 27H 3 27φ∗ H Y2(3 + Y)3 27φ∗ H Y2(3 + Y)3 (10.71) (10.72) (10.73) −H2Y(3 + Y)(9 + 2Y(3 + Y)) 5 + 4 cφL3D1 − 9H2D3 . × 4−Y(3 + Y)2 McφL − 4 cφL3 + H2Y(3 + Y) + H (4 cφL3 − McφL)(18 + 6Y(3 + Y)) −H2Y(3 + Y)(9 + 2Y(3 + Y))) 5 (10.74) × 4−Y(3 + Y)2 −McφL + 4 cφL3 + H2Y(3 + Y) + H (−4 cφL3 + McφL)(18 + 6Y(3 + Y)) − 2φL(Mc − 4 cφL2)D1 + 18H2D3 . (10.76) Analytical expression for the shift in the inflaton field from linear-orderandsecond-ordercosmologicalperturbationtheory can be written up to considering the contributions from the first-order slow-roll contribution as δφ1(N = 0) = δφ1∗ = φ∗ ˆ 1(N = 0) = φ∗ Dˆ2 − 3φH∗ Dˆ 1 + Y(3φ+∗Y)2 −Y(3 + Y)2 + H (−9 + Y(3 + Y){−2 + H(3 + 2Y)t})]. (10.78) For Case I: δφ2(N = 0) = δφ2∗ = φ∗ ˆ 2(N = 0) φ∗ 27H = φ∗ Dˆ4 + 27H3 Y2(3 + Y)3 × 4−Y(3 + Y)2 4 cφL3 + H2Y(3 + Y) + H −4 cφL3(18 + 6Y(3 + Y)) −H2Y(3 + Y)(9 + 2Y(3 + Y)) 5 + 4 cφL3 Dˆ1 − 9H2 Dˆ3 . For Case II: + H 4 cφL3(18 + 6Y(3 + Y)) −H2Y(3 + Y)(9 + 2Y(3 + Y)) 5 − 4 cφL3 Dˆ1 + 9H2 Dˆ3 . × 4−Y(3 + Y)2 −4 cφL3 + H2Y(3 + Y) For Case II + Choice I(v1): δφ2(N = 0) = δφ2∗ = φ∗ ˆ 2(N = 0) φ∗ 27H = φ∗ Dˆ4 + 27H3 Y2(3 + Y)3 + H 4 cφL3(18 + 6Y(3 + Y)) −H2Y(3 + Y)(9 + 2Y(3 + Y)) 5 − 4 cφL3 Dˆ1 + 9H2 Dˆ3 . For Case II + Choice I(v2): δφ2(N = 0) = δφ2∗ = φ∗ ˆ 2(N = 0) φ∗ 27H = φ∗ Dˆ4 + 27H3 Y2(3 + Y)3 × 4−Y(3 + Y)2 4 cφL3 + H2Y(3 + Y) + H −4 cφL3(18 + 6Y(3 + Y)) + 4 cφL3 Dˆ1 − 9H2 Dˆ3 . (10.79) (10.80) (10.81) (10.82) For Case II + Choice II(v1): 2(N = 0) = φ∗Dˆ 4 + φ ∗ 54H 3 54H Y2(3 + Y) ξ ξ Dˆ1 − 9H 2φLDˆ 3 . 10.3.5 Various useful constants for δN For the derived effective potentials (φ ) and (φ ) can be recast as B ∗ C ∗ φ ∗ 54H 3 54H Y2(3 + Y) φ ∗ 27H 27H 3φL Y2(3 + Y) 4 3 −Y(3 + Y) −McφL + 4 cφL + H 2Y(3 + Y) 3 4 3 −Y(3 + Y) ξ ξ + H 2φLY(3 + Y) 4 3 −Y(3 + Y) McφL − 4 cφL + H 2Y(3 + Y) 3 for Case II + Choice I(v1&v2) for Case II + Choice II(v1&v2) for Case II + Choice III. (10.84) (10.85) (10.86) (10.87) for Case I & II for Case II + Choice I(v1&v2) for Case II + Choice II(v1&v2) for Case II + Choice III, 3 − 2 + 2λmc2φ∗2 mc2 (mc2−λφ2)2 1− (mc2−λφ∗2) ⎭ ∗ 2 1− mc2−λφ∗2 − (mc2−λφ∗2)2 2λmc2φ∗2 4 2 ⎫ φ 1+3 V4 ⎪ φ∗ ⎬ 4 2 φV 1− φ4 ⎪⎭ ∗ 1− mc2−λφ∗2 2λmc2φ∗2 2 2 ∗ mc2 1− mc2−λφ∗2 1 (mc2−λφ∗2)2 + (mc2−λφ∗2)3 6mc2λφ∗ 8λ2mc2φ∗3 ⎞ ⎫ mc2 1− mc2−λφ∗2 ⎟ ⎠ ⎪ ⎬ ⎪ ⎭ 2 2 φV 1+ξ(3φ∗2−φV2 )− φ2 φ2 6ξφ∗2+2 V φ2 ∗ 2 ⎟ φV ⎠ 1+ξ(φ∗2−φV2 )+ φ2 ⎞ ⎫ ⎪ ⎬ ⎪ ⎭ 3 ⎪ Yφ∗2 ⎪ ⎪ ⎪ ⎪ ⎪⎪⎪ φV4 ⎪⎪⎪⎪⎪⎪⎪ Y1φ∗2 13−+ φφφV4∗44 ⎪ ⎪ ⎪⎨ ∗ 1 ⎨ 1 ⎨ 8 3 + 3 + 1 ⎨ 8 3 + B(φ∗) = C(φ∗) ≈ Additionally the constants G1(φ∗) and G2(φ∗), as appearing in the expression for fNloLc, are defined as 6M2p −2 6M2p −2 4 2 ⎟ φV ⎠ φ∗2 1− φ4 6M2p mc2 12M2p 12M2p 36M4p 36M4p ⎞ −2 ⎛ 2 ⎟ ⎠ 6M2p 2 2 ⎟ φV ⎠ φ∗2 1+ξ(φ∗2−φV2 )+ φ2 + 81φ2 ∗ + φ2 ∗ 6M2p −1 6M2p 6M2p −1 6M2p φ3 ∗ G1(φ∗) = and G2(φ∗) = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎜ 1 + ⎪⎪ ⎝ ⎨ for Case I for Case II for Case II + Choice I(v1&v2) for Case II + Choice II(v1&v2) for Case II + Choice III 36M4p for Case I for Case II Choice I(v1&v2) + Choice II(v1&v2) + Choice III. + φ4 6M2p 1+3 V φ4 ∗ 4 3 φ V φ3 1− φ4 ∗ 6M2p 4 2 ⎟ 6M2p mc2 2 ⎟ 6M2p 2 2 ⎟ 6M2p 1− mc2−λφ2 − (mc2−λφ2)2 ∗ ∗ 2λmc2φ∗2 mc2 ∗ mc2−λφ∗2 3 φ φ3 1+ξ(φ∗2−φV2 )+ φ2 V ∗ ∗ 2 3 ∗ 10.4 Momentum dependent functions in four point function Momentum dependent functions Gˆ S(k1, k2, k3, k4), Wˆ S (k1, k2, k3, k4) and Rˆ S(k1, k2, k3, k4) as appearing in four point function are defined as S Gˆ (k1, k2, k3, k4) = S(k˜ , k1, k2)S(k˜ , k3, k4) |k1 + k2|3 × k2.k4 + × k2.k3 + × k3.k4 + [k2.(k1 + k2)] [k4.(k3 + k4)] [k2.(k1 + k2)] [k3.(k3 + k4)] [k3.(k3 − k4)] [k4.(k3 − k4)] |k1 + k2|2 |k1 + k2|2 |k1 + k2|2 k1.k3 + + k1.k4 + − k1.k2 + |k1 + k2|2 |k1 + k2|2 |k1 + k2|2 [k1.(k1 + k2)] [k4.(k3 + k4)] [k1.(k1 + k2)] [k2.(k3 + k4)] 12M2p 36M4p ⎞ −2 ⎛ 2 + 36M4p φ∗4 1− mc2−λφ∗2 12M2p 4 ⎟ ⎠ 2 2 + φV φ∗2 1+ξ(φ∗2−φV2 )+ φ2 2 4 ⎟ φV ⎠ φ∗4 1+ξ(φ∗2−φV2 )+ φ2 (10.88) (10.89) (10.90) with and where 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) + 2k12k22) 8|k1 + k2|2 A1(k1,k2,k3,k4) = +(1,2 ↔ 3,4) (10.94) A2(k1,k2,k3,k4) = − 2k3k4(k3 + k4)((k1.k2)(k12 + k22) + 2k12k22)(k3k4 + k3.k4) + (3,4 ↔ 1,2)3 8|k1 + k2|4 1 −2|k1 + k2|2 k12k42(k2.k3)(k2 + k3) + k12k32(k2.k4)(k2 + k4) + k22k42(k1.k3)(k1 + k3) + k22k32(k1.k4)(k1 + k4) (k1.k2) + 8|k1 + k2|2 ((k1 + k2)((k3.k4)(k32 + k42) + 2k32k42) + k3k4(k3 + k4)(k3k4 + k3.k4)) + (1,2 ↔ 3,4) , (10.95) 2|k1 + k2|2 3k1k2k3k4(k1k2 + k1.k2)(k3k4 + k3.k4) . 4|k1 + k2|2 2|k1 + k2|3Gˆ S(k1,k3,k2,k4) k1k2(k1 + k2)2((k1 + k2)2 − k32 − k42 − k3k4) (Kˆ − 2(k3 + k4))2Kˆ 2((k1 + k2)2 − |k1 + k2|2) k1 + k2 k1 + k2 k1 + k2 × − 2k1k2 − k32 + k42 + 4k3k4 − (k1 + k2)2 + |k1 + k2|2 − (k1 + k2)2 1 1 3 Kˆ − 2(k1 + k2) − Kˆ + 2(k1 + k2) + (1,2 ↔ 3,4) |k1 + k2|3(|k1 + k2|2 − k12 − k22 − 4k1k2)(|k1 + k2|2 − k32 − k42 − 4k3k4) . − 2(|k1 + k2|2 − k12 − k22 − 2k1k2)(|k1 + k2|2 − k32 − k42 − 2k3k4) (10.96) (10.97) 1. 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Sayantan Choudhury. COSMOS- \(e'\) -soft Higgsotic attractors, The European Physical Journal C, 2017, 469, DOI: 10.1140/epjc/s10052-017-5001-8