Issues in Palatini R2ination Bounds on the Reheating Temperature A. B. Lahanasa aNational and Kapodistrian University of Athens

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Issues in Palatini R2inﬂation : Bounds on the Reheating Temperature

A. B. Lahanasa∗

aNational and Kapodistrian University of Athens,

Department of Physics,

Nuclear and Particle Physics Section,

GR–157 71 Athens, Greece

We consider R2-inﬂation in Palatini gravity, in the presence of scalar ﬁelds coupled to gravity.

These theories, in the Einstein frame, and for one scalar ﬁeld h, share common features with

K- inﬂation models. We apply this formalism for the study of single-ﬁeld inﬂationary models,

whose potentials are monomials, V∼hn, with na positive even integer. We also study the Higgs

model non-minimally coupled to gravity. With R2-terms coupled to gravity as ∼αR2, with α

constant, the instantaneous reheating temperature Tins, is bounded by Tins ≤0.290 mP lanck/ α1/4,

with the upper bound being saturated for large α. For such large αneed go beyond slow-roll to

calculate reliably the cosmological parameters, among these the end of inﬂation through which Tins

is determined. In fact, as inﬂaton rolls towards the end of inﬂation point, the quartic in the velocity

terms, unavoidable in Palatini gravity, play a signiﬁcant role and can not be ignored. The values of

α, and other parameters, are constrained by cosmological data, setting bounds on the inﬂationary

scale Ms∼1/√αand the reheating temperature of the Universe.

Keywords: Modiﬁed Theories of Gravity, Inﬂationary Universe

PACS: 04.50.Kd, 98.80.Cq

I. INTRODUCTION

The Palatini formulation of General Relativity (GR), or ﬁrst-order formalism, is an alternative to the

well-known metric formulation, or second-order formalism. In the latter the space time connection is

determined by the metric while in the Palatini approach the connection Γµ

λσ is treated as an independent

variable [1–9]. It is through the equations of motion that Γµ

λσ receive the well known form of the

Christoﬀel symbols, describing thus a metric connection. Within the context of GR the two formulations

are equivalent. However in the presence of ﬁelds that are coupled in a non-minimal manner to gravity

this no longer holds [1–3], and the two formulations describe diﬀerent physical theories.

Encompassing the popular inﬂation models into Palatini Gravity, in an eﬀort to describe the cosmo-

logical evolution of the Universe, leads to diﬀerent cosmological predictions, from the metric formulation,

due to the fact that the dynamics of the two approaches diﬀer. A notable example is the Starobinsky

model, for instance, where except the graviton there exists an additional propagating scalar degree of

freedom, the scalaron, whose mass is related to the coupling of the R2term. In the Einstein frame this

emerges as a dynamical scalar ﬁeld, the inﬂaton, moving under the inﬂuence of the celebrated Starobinsky

potential, [10–12]. Within the framework of the Palatini Gravity, in any f(R) theory [3], there are no

extra propagating degrees of freedom, that can play the role of the inﬂaton, and hence the inﬂaton has

to be put in by hand as an additional scalar degree of freedom.

∗Electronic address: alahanas@phys.uoa.gr

arXiv:2210.00837v3 [gr-qc] 15 Dec 2022

The diﬀerences between metric and Palatini formulation in the cosmological predictions, as far as

inﬂation is concerned, arise from the non-minimal couplings of the scalars, that take-up the role of the

inﬂaton. These couplings are diﬀerent in the two approaches. This has been ﬁrst pointed out in [13] and

has attracted the interest of many authors since, [14–52], with still continuing activity, [53–83].

Measurements of the cosmological parameters, by various collaborations, has tighten the allowed win-

dow of these observables which in turn constrain, or even exclude, particular inﬂationary models, [84–87].

In particular, the spectral index nsand the bounds on the tensor-to-scalar ratio rimpose severe restric-

tions and not all models can be compatible with the observational data 1. The precise measurements of

the primordial scalar perturbations, and of the associated power spectrum amplitude As, imply constrains

for the scale of inﬂation in models encompassed in the framework of the metric or Palatini formulation,

which are more stringent in the case of Palatini Gravity as has been shown in [51].

In this work we shall consider R2theories, in the framework of the Palatini Gravity, and study the

cosmological predictions of some popular models existing in literature, with emphasis on the maximal

reheating temperature, or instantaneous reheating temperature. We will show that there are strict

theoretical bounds on it which are saturated when the couplings α, associated with the R2-term, is large.

To this goal need go beyond slow-roll approximation, to extract reliable predictions, since quartic in the

velocity terms of the inﬂaton play a crucial role. Assuming instantaneous reheating the cosmological data

impose upper bounds on α, or same, lower bound on the inﬂationary scale, which also hold for lower

reheating temperatures.

This paper is organized as follows :

In section II, we present the salient features and give the general setup of f(R) - Palatini Gravity 2

, in the presence of an arbitrary number of scalar ﬁelds, coupled to Palatini Gravity in a non-minimal

manner, in general. Although this is not new, as this eﬀort has been undertaken by other authors, as well,

we think that the general, and model-independent, expressions we arrive at, are worth being discussed.

We focus on R2theories for which the passage to the Einstein frame is analytically implemented. These

theories have a gravity sector, speciﬁed by two arbitrary functions, sourcing, in general, non-minimal

couplings of the scalars involved in Palatini Gravity, and a third function which is the scalar potential.

In the Einstein frame, and when a single ﬁeld is present, these models have much in common with the

K- inﬂation models [89].

In section III, we discuss the arising background equations of motions and discuss the slow-roll mecha-

nism, paying special attention to end of inﬂation and its validity within the slow-roll scheme. We ﬁnd that

in some cases need go beyond slow-roll to determine the end of inﬂation which controls the instantaneous

reheating temperature and the cosmological parameters.

In section IV we discuss various aspects of the inﬂationary evolution of these models, in the general

case, and extract useful conclusions, which hold even when the evolution of inﬂaton, as it approaches the

minimum of the scalar potential, deviates signiﬁcantly from slow-roll.

Section V deals with the instantaneous reheating temperature and its bounds set on it which are

dictated by the pertinent backrground equations. Strict upper bounds are derived which are saturated

when the parameter α, deﬁning the coupling of the R2-terms to gravity, is large. These could not have

been predicted within the slow-roll scheme. Moreover, assuming that reheating is instantaneous, we

explore the bounds set by the cosmological observables, on the parameters of particular inﬂation models,

namely the class of models in which the scalar ﬁeld h, is characterized by monomial potentials ∼hn,

with na positive even integer, and the Higgs model. The power spectrum amplitude Asresults to ﬁne

tuning of the parameters of the potential, while the spectral index nsand the tensor to scalar ratio r, set

bounds on α, and therefore bounds on the inﬂation scale and the instantaneous reheating temperature,

Tins. The latter can be as large as ∼1015 GeV , the larger values attained for the smaller allowed value

of the parameter α.

In sections VI, we end up with our conclusions.

1In this work, standard assumptions are made for neutrino masses and their eﬀective number. Relaxing these it induces

substantial shifts in ns[88].

2Throughout this paper the Ricci scalar will be denoted by R.

II. THE MODEL

In this section we shall outline the general setup, and follow the methodology and notation used in

[51]. More details, if needed, can be found in this reference. The starting point is an action involving

scalar ﬁelds hJwhich are coupled to Palatini gravity in the following manner,

S=Zd4x√−gf(R, h) + 1

2GIJ (h)∂hI∂hJ−V(h).(1)

In it Ris the scalar curvature, in the Palatini formalism, and f(R, h) an arbitrary function of the scalars

hJand R. Following standard procedure we write this action in the following manner, introducing an

auxiliary ﬁeld Φ,

S=Zd4x√−gf(Φ, h) + f0(Φ, h) (R − Φ) + 1

2GIJ (h)∂hI∂hJ−V(h).(2)

In this f0(Φ, h) denotes the derivative with respect Φ. This action can be written as follows, in Jordan

frame,

S=Zd4x√−gψR+1

2GIJ (h)∂hI∂hJ−ψΦ + f(Φ, h)−V(h),(3)

where ψin deﬁned by,

ψ=∂f(Φ, h)

∂Φ,with inverse Φ = Φ(ψ, h).(4)

One can go to the Einstein frame by performing a Weyl transformation of the metric

gµν = ¯gµν /2ψ(5)

and that done the theory receives the following form,

S=Zd4x√−¯gR

2+1

4ψGIJ (h)∂hI∂hJ−1

4ψ2(ψΦ−f(Φ, h) + V(h)) .(6)

We can further eliminate the ﬁeld ψ, using its equation of motion,

ψ(∂h)2=ψΦ−2f(Φ, h)+2V(h),(7)

where, in order to speed up notation, we have denoted GIJ (h)∂hI∂hJ= (∂h)2. Note that (7) is not

solvable, in general, however in R2-theories this is feasible.

In the following we shall focus on such theories, with a single ﬁeld hpresent, with f(h, R) quadratic

in the curvature, having therefore the form

f(R, h) = g(h)

2R+R2

12M2(h).(8)

Since a single scalar ﬁeld is assumed its kinetic term can be always brought to the form (∂h)2/2, that is

in the action (1) the ﬁeld can be taken canonically normalized. Therefore in this theory there are three

arbitrary functions, namely g(h), M2(h), V (h), and any choice of them speciﬁes a particular model. We

have set the reduced Planck mass mP lanck ≡mP= (8πGN)−1/2dimensionless and equal to unity and thus

all quantities in (8) are dimensionless. When we reinstate dimensions the functions g, V have dimensions

mass2, mass4, respectively, while M2is dimensionless. Note that a non-trivial ﬁeld dependence of the

functions g(h), and/or M2(h), is a manifestation of non-minimal coupling of the scalar hto Palatini

Gravity. We recall that in Palatini formalism there is no a scalaron ﬁeld, associated with an additional

propagating degree of freedom, which in the Einstein frame of the metric formulation plays the role of

the inﬂaton.

With the function f(R, h), as given by (8), we get from Eq. (4),

ψ=g(h)

2+Φ

6M2(h),(9)

and (7) is solved for ψin a trivial manner, yielding

ψ=4V+ 3M2g2

2(∂h)2+ 6M2g.(10)

In this way ψ, an hence Φ, from (9), are expressed in terms of h, (∂h)2. Plugging ψ, Φ into (6) we get, in

a straightforward manner

S=Zd4x√−gR

2+K(h)

2(∂h)2+L(h)

4(∂h)4−Uef f (h).(11)

In this action we have suppressed the bar in the scalar curvature and also in √−g, and in order to simplify

notation we have denoted ∂µh∂µhby (∂h)2and (∂µh∂µh)2by (∂h)4. Note the appearance of quartic

terms (∂h)4in the action. As for the functions K, L, Ueff , appearing in (11), they are analytically given

L(h) = (3M2g2+ 4V)−1, K(h) = 3M2gL , Ueff = 3M2V L . (12)

Observe that since terms up to R2have been considered, in f(R, h) , higher than (∂h)4terms do not

appear in the action (11).

The above Lagrangean may feature, under conditions, K - inﬂation models [89], which involve a single

ﬁeld, described by an action whose general form is

S=Z√−gR

2+p(h, X)d4x . (13)

where X≡(1/2)∂µh∂µh. The cosmological perturbations of such models were considered in [90] and the

importance of a time-dependent speed of sound csin K - inﬂation models was emphasized in [91] and

cosmological constraints were derived, where improved expressions for the density perturbations power

spectra were used. Speciﬁc models were also considered in [92]. See also [93–98], for more recent works

on these models, in various contexts.

In a ﬂat Robertson-Walker metric, where the background ﬁeld his only time dependent, the energy

density and pressure are given by

ρ(h, X) = K(h)X+ 3 L(h)X2+Uef f (h), p(h, X) = K(h)X+L(h)X2−Ueff (h),(14)

with Xbeing, in this case, half of the velocity squared, X=˙

h2/2.

We shall assume that the function L(h) is always positive to avoid phantoms, which may lead to an

equation of state with w < −1. This may occur when L < 0 and Xbecomes suﬃciently large. However,

there is no restriction on the sign of K(h) which may be negative in some regions of the ﬁeld space,

signaling that the kinetic term has the wrong sign in those regions. Obviously the sign of K(h) should

be positive at the minimum of the potential. Options where Kis negative in some regions, although

interesting, will not be pursued in this work. Besides, we shall assume that the potential is positive

Ueff (h)≥0 and bears a Minkowski vacuum. This ensures that the energy density is positive deﬁnite

even when the velocity is vanishing. The location of the Minkowski vacuum can be taken to be at h= 0,

without loss of generality, by merely shifting appropriately the ﬁeld h. Then having a positive deﬁnite

potential which vanishes at h= 0 entails Ueff (0) = 0 and also U0

eff (0) = 0. When inﬂation models are

considered, the inﬂaton will roll down towards this minimum signaling the end of inﬂation and beginning

of Universe thermalization.

Concerning the potential Ueff , appearing in the Lagrangian (11) in the Einstein frame, using Eq. (12)

it is trivially shown that it can be cast in the following form 3,

Ueff =1

43M2−1

R=3M2

4(1 −gK) where R=L

K2.(15)

The two forms of the potential above are equivalent, if the relation K= 3M2gL of Eq. (12) is used.

The quantity Rappearing in this equation may play an important role, as we shall see, in inﬂationary

evolution. From (15) we see that positivity of Ueff ≥0 entails to having R−1≤3M2. In terms of the

potential V(h) appearing in the action (1) this simply reads V≥0, as can be seen from the last of Eqs.

(12) . Dealing with positive deﬁnite potentials, an upper bound is then established,

Ueff ≤3M2

4.(16)

as is evident from (15). Although not necessary, this upper bound can be easily saturated, for large h,

by choosing appropriately the functions involved. Actually the asymptotic behavior of these functions,

for large h, control the behavior of the potential in this regime. Choosing for instance the function Rto

increase, as hbecomes large, then saturation of the above bound is easily obtained If, moreover, we opt

that the function M2approaches or even be a constant, for large h-values, while the function Rincreases

in this regime, then the potential reaches a plateau which may yield enough inﬂation. This is a rather

plausible scenario, which may drive successful inﬂation, and is obtainable under rather mild assumptions.

However, other less obvious choices may be available.

The models studied in this work can be, in general, classiﬁed in three main categories :

•Models with g, M2=constants, named M1 for short for future reference.

These are dubbed minimally coupled models. In this case the constant gcan be taken equal

to unity without loss of generality. This is implemented by rescaling the metric as gµν →g−1

0gµν ,

where g0=g, accompanied by a redeﬁnition of the ﬁeld h→g1/2

0hin the action (1), with f(R, h)

as given by (8) , before going to Einstein frame.

•Models with M2=constant and ga function of h. These we name M2 for short.

In this class of models gcan be a function of the ﬁeld h,g(h). In this case we can take g(0) = 1

by rescaling the metric and the ﬁeld h, in the way described previously for the M1 models, with g0

identiﬁed with g(0) . In both cases, M1 or M2, it is tacitly assumed that g0>0.

•Models with both g, M2functions of h. These we shall name M3.

The models M1, M2 cover a broad range of interesting models, studied in the past in various contexts,

and shall concern us most. Models belonging to the class M3have been studied in [61].

As we discussed, we shall be interested in models with positive semi-deﬁnite potential having a single

Minkowski vacuum at a point, which without loss of generality we can take it to be located at h= 0.

Then, besides Ueff >0, we must have dUeff

dh >0 for h > 0, with the sign of the derivative reversed when

h < 0. These imply restrictions for the functions describing the aforementioned models. For instance,

for the minimally coupled models this entails dK

dh <0 (>0) for h > 0 (h < 0), using Eq. (15) and the

fact that g > 0. The above requirements are rather mild and can be easily satisﬁed. Therefore many

options are available for scalar potentials bearing the characteristics demanded for successful inﬂation to

be possibly implemented. This will be exempliﬁed in speciﬁc models, to be discussed later.

Concluding this section, we presented a general, and model independent, framework of R2- theories,

in the Palatini formulation of Gravity, which may be useful for the study of inﬂation. In the Einstein

frame these theories may be considered as generalizations of K-inﬂation models. This formalism will be

implemented, for the study of particular inﬂationary models.

3The quantity Rshould not be confused with the Ricci scalar R.

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IssuesinPalatiniR2ination:BoundsontheReheatingTemperatureA.B.LahanasaaNationalandKapodistrianUniversityofAthens,DepartmentofPhysics,NuclearandParticlePhysicsSection,GR{15771Athens,GreeceWeconsiderR2-inationinPalatinigravity,inthepresenceofscalareldscoupledtogravity.Thesetheories,intheEinsteinframe...

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Issues in Palatini R2ination Bounds on the Reheating Temperature A. B. Lahanasa aNational and Kapodistrian University of Athens

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