Reconstructing inflation and reheating in fϕTgravity Ram on Herrera1and Carlos R ıos1 2 1Instituto de F ısica Pontificia Universidad Cat olica de Valpara ıso

2025-04-29 0 0 2.31MB 18 页 10玖币

侵权投诉

Reconstructing inﬂation and reheating in f(ϕ)Tgravity

Ram´on Herrera1, ∗and Carlos R´ıos1, 2, †

1Instituto de F´ısica, Pontiﬁcia Universidad Cat´olica de Valpara´ıso,

Avenida Brasil 2950, Casilla 4059, Valpara´ıso, Chile.

2Departamento de Ense˜nanza de las Ciencias B´asicas,

Universidad Cat´olica del Norte, Larrondo 1281, Coquimbo, Chile.

In this work we study the reconstruction of an inﬂationary universe in the context of a theory

of gravity f(ϕ)T, in which Tcorresponds to the trace of energy momentum tensor. To realize this

reconstruction during the inﬂationary epoch, we consider as attractor the scalar spectral index ns

in terms of the number of e-folds N, in the framework of the slow-roll approximation. By assuming

a speciﬁc function f(ϕ) together with the simplest attractor ns(N), we ﬁnd diﬀerent expressions

for the reconstructed eﬀective potential V(ϕ). Additionally, we analyze the era of reheating occurs

after of the reconstruction obtained during the inﬂationary epoch. In this scenario we determine the

duration and temperature during the reheating epoch, in terms of the equation of state parameter

and of the observational parameters. In this context, the diﬀerent parameters associated to the

reconstructed model are restricted during the scenarios of inﬂation and reheating by considering the

recent astronomical observations.

I. INTRODUCTION

It is well known that during the evolution of the early universe, it exhibited a short period of rapid growth

called inﬂationary epoch or simply inﬂation [1–3]. In this context, the inﬂationary epoch provides solutions to long

standing cosmological problems associate to the hot big bang model. However, the inﬂationary stage not only resolves

the problematic of the standard hot model, but also explicates the large-scale structure (LSS) [4, 5], as well the

anisotropies observed in the cosmic microwave background (CMB) radiation of the early universe [6, 7].

In concern to the diﬀerent inﬂationary models that produce an adequate evolution during the early universe, we

can stand out those models that utilize modiﬁcations to the Einstein’s theory from a generalization in the Lagrangian

density of Einstein-Hilbert action. In this sense, we can mention the function f(R) in place of Ricci scalar Rin the

Einstein-Hilbert action corresponds to a simple generalization of this action [8], see also Refs.[9, 10]. A modiﬁcation to

the function f(R) in the action was developed in Ref.[11], where the authors replace the function f(R) by f(R, Lm),

in order to include a coupling between an arbitrary function of the Ricci scalar and the matter of the universe

characterized by the matter Lagrangian density Lm. In this context, diﬀerent analysis in astrophysical and cosmology

in relation to the non minimal coupling matter geometry coupling f(R, Lm) were developed in Refs.[12, 13], see

Ref.[14] for another formulations.

In fact, we have another extensions to the framework of standard General Relativity (GR) and in particular we

distinguish the f(R, Lm, T ) modiﬁed gravity or commonly called f(R, T ) gravity in which the quantity Tdenotes

the trace of the energy momentum tensor Tµν related to the matter i.e., the trace T=gµν Tµν [15], see also [16]. In

this framework, an arbitrary function of the scalar Ricci Ras well as of the trace of the tensor Tµν are considered

to describe the early and present universe. The motivation to introduce the trace of the energy momentum tensor

through the f(R, T ) gravity comes from to consider some exotic matters or quantum eﬀects to describe the universe.

In the literature, diﬀerent forms of the function f(R, T ) have been analyzed and in particular this function has been

decomposed as combinations of arbitrary functions related to the Ricci scalar R(R) and of the trace of the energy

momentum tensor T(T) such that; f(R, T ) = R(R)+T(T) as well the multiplication f(R, T ) = R(R)T(T). Another

proposed theory along the lines f(R, T ) is the energy momentum squared gravity proposed in Ref.[17], in which the

authors add to the standard Einstein-Hilbert action a quadratic term described by T2=Tµν Tµν in which as before

Trepresents to the trace of the energy momentum tensor. In this context another modiﬁed gravity analyzed in the

literature associated to trace Tcorresponds to add the f(ϕ)Tterm to the Einstein Hilbert action[18], where in this

situation f(ϕ) is an arbitrary function of the scalar ﬁeld (inﬂaton) coupled with the trace of the energy momentum

tensor, see also recently Ref.[19]. Here the authors studied the inﬂationary epoch (observational parameters) assuming

diﬀerent eﬀective potentials such as; chaotic inﬂation, natural inﬂation and the eﬀective potential of the Starobinsky

inﬂation obtained in the conformal frame. In this context, the idea of considering an extension of the modiﬁed gravity

∗ramon.herrera@pucv.cl

†carlos.rios@ucn.cl

arXiv:2210.10080v2 [gr-qc] 11 Oct 2023

from the term f(ϕ, T ) where the scalar ﬁeld ϕcouples to the trace of energy momentum Tbecomes interesting,

but also this extension oﬀers an alternative approach to resurrect some inﬂationary models that do not work in the

framework of the standard GR from the observational data such as; the chaotic model, natural inﬂation or other that

are strongly disfavored from observations. In this sense, the introduction of this extension of the modiﬁed gravity

transforms the expressions associated to the observational parameters which are sensitive to f(ϕ, T ) gravity.

Additionally, another models in which the trace Tplays an important role during inﬂation, correspond to the models

related with f(Q, T ) gravity, in with Qdenotes the non-metricity scalar Refs.[20, 21] and for another extensions in

which is considered the trace see e.g., Refs.[22, 23].

On the other hand, the concept of reconstruction associated to the physical variables that make up the background

dynamics of the diﬀerent models during the inﬂationary scenario, considering the parameterization of observational

quantities such as; the scalar spectrum, scalar spectral index and the tensor to scalar ratio, have been analyzed

by diﬀerent authors [24–27]. In particular, an interesting reconstruction mechanism to ﬁnd the physical quantities

during inﬂation under the slow roll stage, corresponds to the parameterization of the scalar spectral index ns(N) and

the tensor to scalar ratio r(N) (called attractors) in terms of the number of e-folds N. It is well known that the

parameterization for the scalar spectral index ns(N) as a function of the number of e-folds deﬁned as ns(N)=1−2/N

is well supported from observations for values of number N≃50-70 from the data taken by the Planck[28].

In the theoretical context of the general relativity (GR), various inﬂationary models can be reconstructed under a

single parameterization or attractor given by ns≃1−2/N assuming large Nsuch as; the hyperbolic tangent model

or T-model [29], E-model[30], R2or Starobinsky model[1] and the famous model of chaotic inﬂation[2]. However,

the methodology used for the reconstruction in the models of warm and Galileon inﬂation was required to introduce

two attractors ns(N) and r(N), in order to build the background variables [31, 32], see also Ref.[33]. In the same

way, it is possible to utilize the slow roll parameters ϵ(N) and η(N) as a function of the number of e-folds Nto

build the background variables and the observational parameters such as; the scalar spectrum index, power spectrum,

tensor to scalar ratio among other[34–36]. For example, in Ref.[34] was used some types of parameterization for the

slow roll parameter ϵ(N) in order to ﬁnd the eﬀective potential in terms of the inﬂaton ﬁeld. In the same way, in

Ref.[37] diﬀerent eﬀective potentials were reconstructed by considering as parameterization the slow roll parameters

as a function of the number of e-folds N, see also [38, 39].

In relation to the reheating of the universe, it is known that to recover the standard big-bang model, the early

universe has to be reheated after of inﬂationary epoch. During the process of reheating of the early universe, the

components of matter and radiation are generated generally through the decay of the scalar ﬁeld or another ﬁelds,

while the temperature of the universe increases in magnitude and then the universe connects with the radiation epoch

and then with the standard big-bang model[40]. However, there are various reheating models (mechanisms) in order

to increase the temperature during the early universe. Thus, we have the mechanism of reheating in which from

the perturbative decay of an oscillating inﬂaton ﬁeld at the end of inﬂationary epoch produces the reheating of the

universe[41], the mechanism associated to non-perturbative processes as parametric resonance decay[42], the reheat-

ing from tachyonic instability[43], instant preheating in which this mechanism takes place from a non-perturbative

processes and it occurs almost instantly[44], and also for non oscillating models or called the NO models, in which

the mechanism of reheating occurs from the another ﬁeld “curvaton” ﬁeld (decay) [45], see also Ref.[46].

During the reheating era we have the diﬀerent parameters associated to the reheating. Thus, we have that this

period can be characterized by the reheating temperature Treh, an eﬀective equation of state (EoS) wreh associated to

the matter content in this process and one parameter related with the reheating duration and characterized by number

of e-folds Nreh. In relation to the reheating temperature a lower limit is restricted by primordial nucleosynthesis (BBN)

in which the temperature during the primordial nucleosynthesis TBBN ∼10 MeV, see e.g., [47]. In the context of

the EoS parameter wreh, we can mention that diﬀerent numerical analysis were developed in order to characterize an

eﬀective EoS parameter from speciﬁc interactions between the inﬂaton ﬁeld and another matter ﬁelds, see e.g.,[48, 49].

Thus, we can consider that the EoS parameter wreh is a function of the cosmological time during the diﬀerent scenarios

of the reheating epoch. In this context, for example for the canonical reheating stage assuming a chaotic potential the

EoS parameter at the end of inﬂation takes the value wreh = 0, but from the numerical analysis the authors in Ref.[49]

showed that this parameter increases to values of wreh ∼0.3 for cosmological time t > 200/Mpl. In another scenarios

such as a massive ﬁeld the EoS parameter wreh increases from a negative value at the end of inﬂation wreh =−1/3[50]

to wreh = 0, see e.g., [51–53]. In this context, in a ﬁrst approximation we can assume that the EoS parameter wreh

during the stage reheating can be considered as a constant in time through all the reheating epoch [52].

The goal of this paper is to rebuild an inﬂationary model during the early universe from a modiﬁed f(ϕ)Tgravity.

In this sense, we shall analyze an interaction between the scalar ﬁeld ϕand the trace of energy-momentum tensor T

of the form f(ϕ)T, in order to reconstruct the inﬂationary stage assuming the parameterization of the scalar spectral

index nsas a function of the number of e-folds Ni.e., ns=ns(N). In this framework, we study how the background

dynamics in which there is an interaction between the ﬁeld ϕand the trace Tmodiﬁes the reconstruction of the

eﬀective potential in terms of the scalar ﬁeld assuming as attractor for large Nthe scalar spectral index ns(N). Thus,

from a general procedure, we will reconstruct the eﬀective potential from an attractor associated to the index ns(N)

(for large-N).

To reconstruct analytically the eﬀective potential in terms of the scalar ﬁeld, we will analyze a speciﬁc example for

the scalar spectral index parameterized in terms of N. In this form, we will assume the simplest attractor given by

ns(N) = 1 −2/N for large-N. In this framework, we will rebuild the eﬀective potential V(ϕ) and we will also obtain

the diﬀerent constraints on the parameters associated to the reconstruction (integration constants).

Additionally, we will study the reheating epoch from the reconstruction of the background variables obtained

during the inﬂationary stage. In this context, we will determine the reheating parameters such as; the duration of

the reheating from the number of e- folds, the temperature and the EoS during the reheating of the universe and how

using the cosmological parameters from Planck data (1σbound on ns) these reheating parameters are constrained.

The outline of the article is as follows: the Sect. II we give a brief description of the modify gravity from an

interaction between a coupling function that depends of the scalar ﬁeld f(ϕ) and the trace of energy-momentum

tensor T. Here we analyze the background equations under the slow-roll approximation and then we review the

cosmological perturbations in this modify gravity. In Sect. III we obtain, under a general formalism, explicit relations

for the eﬀective scalar potential in terms of the number of e−folds Nto consider the reconstruction from the scalar

spectral index ns(N). In the Sect. IV, we apply the reconstruction methodology in order to ﬁnd the scalar potential

V(N) analytically. Besides, we assume a speciﬁc case in which we consider the simplest attractor for the spectral

index ns(N)=1−2/N for large Ntogether with a determined coupling function f(ϕ), in order to rebuild the

eﬀective potential as a function of the scalar ﬁeld. In Sect. V, we study the reheating scenario for our model using the

reconstructed potential obtained from the simplest attractor ns(N). Here, we determine the reheating temperature

and the number of e-folds during the reheating era. Finally in Sect. VI we give our conclusions. We chose units so

that c=ℏ= 1.

II. THE f(ϕ)TMODIFIED GRAVITY AND THE INFLATIONARY PHASE

In this section, we shall consider the modiﬁed gravity from an interaction between the scalar ﬁeld and the trace of

the energy-momentum tensor through the term f(ϕ)T. In this context, we start by writing down the action for this

modiﬁed gravity as[18]

S=ZR

2κ+f(ϕ)T+Lm√−g d4x, (1)

where Rdenotes the scalar Ricci, gis the determinant of the metric gµν and the constant κ= 8π G =M−2

pl with

Mpl the Planck mass. The positive quantity f(ϕ) is an arbitrary function (dimensionless) associated to scalar ﬁeld or

inﬂaton ﬁeld ϕ, the expression Lmdenotes the matter Lagrangian density and T=gµν Tµν corresponds to the trace

of energy momentum tensor Tµν associated to Lm. In particular for the speciﬁc case in which the function f(ϕ)→0,

the action given by Eq.(1) reduces to standard General Relativity (GR).

It is well known that from the variation of the action (1) with respect to the metric, it gives rise to the Einstein’s

equation; Gµν =Rµν −(1/2)gµν R=κ˜

Tµν , where the tensor Gµν denotes the Einstein tensor and we have deﬁned

the tensor energy momentum ˜

Tµν associated to the matter Lagrangian together with the coupling between the trace

Tand the arbitrary function f(ϕ) such that

Tµν =−2

√−g

∂(√−g˜

Lm)

∂gµν ,(2)

where we have deﬁned ˜

Lmas a combination of the matter Lagrangian density and the additional term f(ϕ)Tsuch

that

Lm=f(ϕ)T+Lm.(3)

In this form, we can write the tensor ˜

Tµν given by Eq.(2) as

Tµν =Tµν −2f(ϕ)Tµν −1

2gµν T+ Πµν ,where Tµν =−2

√−g

∂(√−gLm)

∂gµν ,(4)

corresponds to the energy momentum tensor related to the matter Lagrangian Lmand the quantity Πµν is deﬁned as

Πµν =gαβ ∂Tαβ

∂gµν =−2Tµν +gµν Lm−2gαβ δ2Lm

δgµν δgαβ .(5)

In the following we will assume a single scalar ﬁeld ϕfor the matter, in order to analyze the reconstruction of our

inﬂationary model in the framework of f(ϕ)Tgravity. In this sense, we consider that the matter Lagrangian density

Lmcan be written as

Lm=−1

2gµν ∂µϕ∂νϕ−V(ϕ),(6)

where V(ϕ) corresponds to the eﬀective potential associated to scalar ﬁeld. By assuming an inﬂaton ﬁeld homogeneous

i.e., ϕ=ϕ(t) and considering a perfect ﬂuid of the form ˜

Tν

µ=diag(−˜ρ, ˜p, ˜p, ˜p), in which ˜ρand ˜pdenote the eﬀective

energy density and pressure, then we can identify these densities as

˜ρ=1

2˙

ϕ2[1 + 2f(ϕ)] + [1 + 4f(ϕ)] V, (7)

and

˜p=1

2˙

ϕ2[1 + 2f(ϕ)] −[1 + 4f(ϕ)] V, (8)

where the dots mean derivatives with respect to the cosmological time and we have assumed the metric signature

(−,+,+,+).

In order to ﬁnd the equation of motion of the scalar ﬁeld we can utilize the continuity equation ˙

˜ρ+ 3H(˜ρ+ ˜p)=0

and then we obtain the modiﬁed Klein-Gordon equation given by

[1 + 2f(ϕ)] ( ¨

ϕ+ 3H˙

ϕ) + f′(ϕ)˙

ϕ2+ [1 + 4f(ϕ)] V′+ 4f′(ϕ)V= 0,(9)

where the notation V′corresponds to V′=∂V

∂ϕ ,f′=∂f/∂ϕ,V′′ =∂2V/∂ϕ2, etc.

Also, the Friedmann equation can be written as

H2=κ

3˜ρ=κ

31

2˙

ϕ2[1 + 2f(ϕ)] + [1 + 4f(ϕ)] V.(10)

Following Ref.[18], we can consider the slow roll approximation in which ˙

ϕ2≪V,¨

ϕ≪H˙

ϕand f′(ϕ)˙

ϕ2≪H˙

ϕ,

respectively. Under this approximation the scalar ﬁeld and Friedmann equations are reduce to

3H˙

ϕ[1 + 2f(ϕ)] + [1 + 4f(ϕ)] V′+ 4f′(ϕ)V= 3H˙

ϕ F1+F2V′+F′

2V≃0,(11)

and

H2≃κ

3[1 + 4f(ϕ)] V=κ

3F2V, (12)

where we have deﬁned, the quantities F1and F2in terms of the scalar ﬁeld as

F1(ϕ) = F1= 1 + 2f(ϕ) and F2(ϕ) = F2= 2F1−1 = 1 + 4f(ϕ).(13)

On the other hand, in order to give a measure of the expansion during inﬂation, we can introduce the number of

e-folds Ndeﬁned from the relation ∆N=N−Nend =Rtend

tH dt =Rϕend

ϕ(H/ ˙

ϕ)dϕ, with which using the slow roll

approximation yields

∆N= ln[a(tend)/a(t)] ≃κZϕ

ϕend (1 + 2f(ϕ))(1 + 4f(ϕ))V

(1 + 4f(ϕ))V′+ 4f′(ϕ)Vdϕ, (14)

where Ncorresponds to the value of the number of e-folds at the cosmological time“t ” during inﬂation and Nend is

the e-folds at the end of inﬂation.

Introducing the dimensionless slow roll parameters, we have [18]

ϵV=1

2κ(1 + 2f(ϕ)) V′

V+4f′(ϕ)

(1 + 4f(ϕ))2

,(15)

and

ηV=1

κ(1 + 2f(ϕ)) V′′

V+f′(ϕ)(7 + 12f(ϕ))

(1 + 2f(ϕ))(1 + 4f(ϕ))

V′

V−4f′(ϕ)2

(1 + 2f(ϕ))(1 + 4f(ϕ)) +4f′′(ϕ)

(1 + 4f(ϕ)),(16)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Reconstructinginflationandreheatinginf(ϕ)TgravityRam´onHerrera1,∗andCarlosR´ıos1,2,†1InstitutodeF´ısica,PontificiaUniversidadCat´olicadeValpara´ıso,AvenidaBrasil2950,Casilla4059,Valpara´ıso,Chile.2DepartamentodeEnse˜nanzadelasCienciasB´asicas,UniversidadCat´olicadelNorte,Larrondo1281,Coquimbo,Chile....

展开>> 收起<<

Reconstructing inflation and reheating in fϕTgravity Ram on Herrera1and Carlos R ıos1 2 1Instituto de F ısica Pontificia Universidad Cat olica de Valpara ıso.pdf

共18页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Reconstructing inflation and reheating in fϕTgravity Ram on Herrera1and Carlos R ıos1 2 1Instituto de F ısica Pontificia Universidad Cat olica de Valpara ıso

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: