Eects of a Pre-inationary de Sitter Bounce on the Primordial Gravitational Waves infRGravity Theories V.K. Oikonomou1

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Eﬀects of a Pre-inﬂationary de Sitter Bounce on the Primordial Gravitational Waves

in f(R)Gravity Theories

V.K. Oikonomou,1∗

1)Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece

In this work we examine the eﬀects of a pre-inﬂationary de Sitter bounce on the energy spectrum of

the primordial gravitational waves. Speciﬁcally we assume that the Universe is described by several

evolution patches, starting with a de Sitter pre-inﬂationary bounce which is followed by an quasi-

de Sitter slow-roll inﬂationary era, followed by a constant equation of state parameter abnormal

reheating era, which is followed by the radiation and matter domination eras and the late-time

acceleration eras. The bounce and the inﬂationary era can be realized by vacuum f(R) gravity

and the abnormal reheating and the late-time acceleration eras by the synergy of f(R) gravity

and the prefect matter ﬂuids present. Using well-known reconstruction techniques we ﬁnd which

f(R) gravity can realize each evolution patch, except from the matter and radiation domination

eras which are realized by the corresponding matter ﬂuids. Accordingly, we calculate the damping

factor of the primordial de Sitter bounce, and as we show, the signal can be detected by only one

gravitational wave future experiment, in contrast to the case in which the bounce is absent. We

discuss in detail the consequences of our results and the future perspectives.

PACS numbers: 04.50.Kd, 95.36.+x, 98.80.-k, 98.80.Cq,11.25.-w

I. INTRODUCTION

The two acceleration eras of our Universe, including the reheating era, are undoubtedly the most mysterious eras of

all the evolutions eras we assume that our Universe experienced. The late-time acceleration era though is conﬁrmed

and the diﬃculty lies to pinpointing the physical theory and mechanism which controls it. However, the other two

eras are speculated to have occurred in the primordial epoch of the Universe, and to date no ﬁrm evidence is provided

that these eras have actually occurred. With regard to the inﬂationary era [1–4], the occurrence of this era could be

veriﬁed by the direct detection of the B-modes in the Cosmic Microwave Background (CMB) temperature ﬂuctuations.

This for example can be veriﬁed in the stage 4 CMB experiments [5, 6] in some years from now, or alternatively, the

primordial stochastic tensor modes can be directly detected in future gravitational waves experiments [7–14], see also

Ref. [15] for some up to date information on cosmological studies of the LISA mission.

Now the plot may thicken with the existence or not of the inﬂationary era. The results of the future CMB and

gravitational waves experiments will play a crucial role, in both cases of detection or not of a signal. In the unlikely

event of non-observation of a signal, the scientists will be confronted with the diﬃcult task to explain why no signal

is detected. Is this non-detection because inﬂation did not occur, or simply because inﬂation is described by a theory

which yields a negative tensor spectral index and a non-detectable by the current experiments signal, while it also

yields a standard General Relativistic (GR) reheating era? On the antipode of this, there lies the detection of a

signal. Many questions can be asked then, how strong is the signal, is it detectable by several experiments in various

frequency ranges, or by some of the experiments? With regard to how strong a signal can be, this is very important.

The observation of a signal in some but not all the detectors may signify some physical process which causes damping

of the signal, such as supersymmetry breaking after or during reheating. This will also determine the era for which

the physics change occurred in the Universe. However, with regard to how strong the signal is, many things can be

said, and many questions can be asked. The answers to these questions may vary and strongly depend on the ﬁnal

form of the signal. Thus the strength and form of the signal may determine whether this signal is obtained by a

theory with positive tensor spectral index, or by a standard inﬂationary theory with an abnormal reheating era. In

the literature, many aspects on primordial gravitational waves are studied [16–75], and with regard to the abnormal

reheating perspective and eﬀects on the energy spectrum of the primordial gravitational waves, this aspect has recently

been studied in Refs. [69, 71, 75] and it was shown that the gravitational waves energy spectrum of standard f(R)

gravity inﬂation can be enhanced signiﬁcantly by the presence of an f(R) gravity generated reheating era. This is in

contrast to a GR compatible reheating era of course and the ﬂatness, form and strength of the detected signal may

reveal many properties regarding the underlying theory. Regarding the strength, this may vary and one mechanism

∗v.k.oikonomou1979@gmail.com,voikonomou@auth.gr

arXiv:2210.02861v1 [gr-qc] 6 Oct 2022

of enhancement or damping may be the presence of some peculiar pre-inﬂationary era, see for example Ref. [65] for

the eﬀects of a primordial bounce on the energy spectrum of the primordial gravitational waves. Speciﬁcally, in Ref.

[65] the eﬀects of a primordial pre-inﬂationary bounce on the energy spectrum of the inﬂationary gravitational waves

were considered. In this work we shall also consider the eﬀects of a pre-inﬂationary de Sitter bounce on the energy

spectrum of the primordial gravitational waves, in the context of f(R) gravity. In standard string theory scenarios,

pre-inﬂationary epochs may actually lead to an overall ampliﬁcation of the gravitational wave energy spectrum [76],

see also Refs. [77–83]. This was also the case in Ref. [65], however in this work we shall demonstrate that it is

possible a primordial bounce to lead to an overall damping of the gravitational waves energy spectrum, which is

quite signiﬁcant. In general, bouncing cosmology [84–89] is a possible alternative to the inﬂationary scenario, so in

this work we combine the presence of a pre-inﬂationary bounce with a standard post-bounce slow-roll inﬂationary

era. Our assumption is that the Universe’s dynamics is controlled by an f(R) gravity during the pre-inﬂationary and

inﬂationary era, and post-inﬂationary the evolution is controlled by the synergy of f(R) gravity in the presence of

matter and radiation perfect ﬂuids. As we will show, the predicted energy density of the primordial gravitational

waves is damped due to the primordial de Sitter bounce, and the strength of the eﬀect mainly depends on the duration

of the de Sitter bounce after during the initial expanding phase of the de Sitter bounce. The pre-inﬂationary bounce

is followed by a slow-roll quasi-de Sitter phase described by vacuum R2gravity, which may or may not be followed

by an f(R) gravity controlled reheating era. In all the cases, the late-time era can be described in a viable way by

some appropriate f(R) gravity. We will compute all the f(R) gravities which can realize the diﬀerent patches of the

Universe’s evolution and accordingly, we shall directly determine the energy spectrum of the primordial gravitational

waves for the resulting theories.

This paper is organized as follows: In section II we present in brief our proposal for the primordial era of our

Universe. We describe in detail the three evolution patches of our Universe primordially, which consist of a pre-

inﬂationary de Sitter bounce, followed by a quasi-de Sitter era, followed by a geometrically generated reheating era.

In section III, we discuss how the diﬀerent patches of our Universe’s evolution can be generated by f(R) gravity, and

we also discuss the qualitative eﬀects of the geometrically realized reheating era on the inﬂationary era. In section IV

we study several theoretical scenarios and their predictions for the energy spectrum of the primordial gravitational

waves. The conclusions follow in the end of the article.

II. PRE-INFLATIONARY DE SITTER BOUNCE AND PRIMORDIAL EVOLUTION

Let us discuss the scenario we propose in this work, which is based on the fact that the Universe pre-inﬂationary was

experiencing a de Sitter bounce, which is followed by a quasi-de Sitter era. Accordingly, after the quasi-de Sitter era

we will assume that the Universe enters a reheating era with constant equation of state (EoS) parameter w, followed

by the standard patches of evolution, namely a canonical reheating era with EoS parameter w= 1/3 and ﬁnally the

matter and dark energy era. Giving the Universe’s evolution in distinct patches is the best we can do as cosmologists,

since it is not possible to ﬁnd the exact scale factor which describes the Universe, this extends beyond the reach of the

human mind. Hence, assuming several evolutionary patches for the Universe is the best that we can do, and in fact,

some of these patches may be directly determined, as it happens with the dark energy era and also may happen with

the inﬂationary and post-inﬂationary era, via the future stage 4 CMB experiments and the future gravitational waves

experiments. However, pre-inﬂationary evolution patches are quite hard to be probed, and our proposal in this paper

is that these pre-inﬂationary eras may have a direct eﬀect on the energy spectrum of the primordial gravitational

waves, causing a signiﬁcant damping of the spectrum. In this line of research, let us quote the scale factor for the

pre-inﬂationary, inﬂationary and the ﬁrst moments of the post-inﬂationary epoch, which is,

a(t) = abcosh(jt)e−t/ti+aieH0t−Hit2+aw

3(w+1) ,(1)

and let us explain the diﬀerent patches for the above evolution. The ﬁrst term describes the de Sitter bounce [90],

which is followed by the quasi-de Sitter inﬂationary epoch described by the second term, followed by the constant

EoS parameter wreheating epoch, described by the third term. Now, ab,aiand awdenote the size of the Universe at

the beginning of the de Sitter bounce, at the beginning of the inﬂationary era and at the beginning of the reheating

era with constant EoS parameter w. The parameters j,H0and Hihave mass dimensions eV, eV and eV2, while the

time instances tiand t0are characteristic and denote the time that the bounce ends and the time instance that the

inﬂationary era ends. So for cosmic times tti, the exponential term is practically equal to unity, while for times

t≥tithe exponential term causes a damping of the ﬁrst term, thus the other two start to dominate. In the left and

right upper plots of Fig. 1 we plot the scale factor (1) vs the cosmic time (blue curve) and the de Sitter bounce scale

factor described by the ﬁrst term (red curve). Also in the bottom plot we present the Hubble radius RH=1

a(t)H(t)

FIG. 1. Upper plots: The scale factor (1) (blue curve) and the de Sitter bounce scale factor (red curve) vs the cosmic time.

Bottom plot: the Hubble radius RH=1

a(t)H(t)as a function of the cosmic time for the scale factor (1) (blue curve) and for the

de Sitter bounce (red curve).

as a function of the cosmic time for the scale factor (1) (blue curve) and for the de Sitter bounce (red curve). As it

can be seen, the scale factor (1) is described by a bounce pre-inﬂationary, in which case the scale factor decreases,

and the Hubble radius increases, after that the Universe experiences a short period of acceleration, in which case the

Hubble radius decreases, and this short acceleration period is followed by a deceleration period, in which case the

Hubble radius starts to decrease again. The blue curve has exactly the behavior described by the scale factor (1), so

the Universe starts with a per-inﬂationary bounce, followed by a short period of inﬂation, followed by a a power-law

evolution with constant EoS parameter. For the plots we assumed that w= 0 and this is also what we will assume

for the rest of the article. Hence basically, the reheating era is abnormal and has an EoS parameter w= 0, diﬀerent

from w= 1/3 which describes an ordinary reheating era.

III. INFLATION AND POST-INFLATION EVOLUTION WITH f(R)GRAVITY

Let us now proceed in the modiﬁed gravity description of the cosmological evolution we presented in the previous

section. Our basic assumption is that f(R) gravity controls the whole evolution, from the pre-inﬂationary era to the

late-time era. It is f(R) gravity which realizes the various evolutionary patches we described in the previous section.

Schematically, the f(R) gravity which realizes the evolution patches which we described in the previous section will

have the following form,

f(R) = 









FB(R)R≥RB,

R+R2

6M2R∼RI,

Fw(R)R∼RP I RI,

FDE (R)R∼R0RP I ,

with RIstands for the curvature scale of inﬂation, at the ﬁrst horizon crossing, RP I stands for the post-inﬂationary

curvature scale during the abnormal reheating era, and RBis the curvature scale near the bouncing point, when the

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EectsofaPre-inationarydeSitterBounceonthePrimordialGravitationalWavesinf(R)GravityTheoriesV.K.Oikonomou,11)DepartmentofPhysics,AristotleUniversityofThessaloniki,Thessaloniki54124,GreeceInthisworkweexaminetheeectsofapre-inationarydeSitterbounceontheenergyspectrumoftheprimordialgravitationalwaves.S...

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Eects of a Pre-inationary de Sitter Bounce on the Primordial Gravitational Waves infRGravity Theories V.K. Oikonomou1

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