pedro antonio pinto frazao- stochastic gravitational waves from a new type of modied chaplygin gas

8/3/2019 Pedro Antonio Pinto Frazao- Stochastic Gravitational Waves from a new type of modi ed Chaplygin Gas

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Stochastic Gravitational Waves from a new type ofmodified Chaplygin Gas

Pedro Antonio Pinto Frazao

Dissertacao para a obtencao de Grau de Mestre em

Engenharia Fsica Tecnologica

Juri

Presidente: Prof. Joao Seixas

Orientador: Prof. Alfredo Barbosa Henriques

Co-orientadora: Dra. Mariam Bouhmadi-Lopez

Vogais: Prof. Jose Pedro Mimoso

Novembro 2009


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Resumo

As ondas gravitacionais de origem cosmologica sao produzidas devido a flutuacoes quanticas do vacuodurante a expansao do universo, em particular nas primeiras fases de formacao, e, por este motivo,poderao vir a ser uma fonte importante de informacao acerca do universo primordial.

Neste trabalho investigamos as possveis assinaturas no espectro de energia do fundo estocastico deondas gravitacionais durante a transicao do regime inflacionario para a fase de domnio de radiacaodo Universo, assumindo uma nova unificacao para o universo primordial. Tal unificacao e no fundosemelhante a introduzida por Kamenshchik et al. [1] (ver tambem Refs. [2, 3]), para unificar materia eenergia escuras.

Tal objectivo e alcancado introduzindo um novo modelo cosmologico que corresponde a uma modi-ficacao na equacao de estado, ou, equivalentemente, na densidade de energia, conduzindo a um novo tipode gas de Chaplygin generalizado [4]. Neste modelo existe uma transicao suave entre um universo do tipode Sitter, e uma epoca de domnio de radiacao, onde o conteudo de materia e modelado por um fludoexotico, ou, num formalismo equivalente, por um campo escalar.

Desta forma, para realizar um estudo sobre o espectro de ondas gravitacionais, utilizamos o metododos coeficientes de Bogoliubov derivado por Parker [5], e usado pela primeira vez por Allen [6], paracalcular o respectivo espectro de energia, tal como seria medido na actualidade, mostrando que, para a

gama de altas frequencias, o espectro depende fortemente de um parametro do nosso modelo.Por outro lado, usando o numero de e-folds, o espectro de potencia e ndice espectral das perturbacoes

escalares, podemos efectuar um constrangimento ao modelo cosmologico, usando medicoes da RadiacaoCosmica de Fundo de Microondas, provenientes do WMAP [7].

Palavras-chave: Chaplygin Gas, Inflacao, Ondas Gravitacionais Cosmologicas, e Coeficientesde Bogoliubov.

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Abstract

The gravitational waves of cosmological origin, due to quantum fluctuations of the vacuum, areproduced during the expansion of the universe, in particular from the early phases, and, therefore, theycan give us important information about the very early universe.

In this work, we investigate the possible imprints in the power spectrum of the stochastic backgroundof the gravitational waves, during the transition from the inflationary regime to the radiation dominatedera of the Universe, assuming a new unified early scenario. Such unification is similar in spirit to thatintroduced by Kamenshchik et al. [1] (see also Refs. [2, 3]), to unify the dark sectors of the universe, i.e.the dark matter and the dark energy.

This unification is accomplished through the introduction of a new cosmological model. This modelcorresponds to a new type of modified generalised Chaplygin gas [4], because it requires a modificationin the equation of state, or, equivalently, in the energy density. The transition, described by this model,from an early de Sitter-like stage to a radiation dominated era of the Universe, is very smooth. Finally,the matter content is modelled by this exotic background fluid, with an underlying scalar field description.

Then, in order to study the gravitational waves, we use the method of the Bogoliubov coefficientsintroduced by Parker [5], and first used by Allen [6], to calculate its power spectrum as it would bemeasured today. We show that, at the high frequencies range, the spectrum depends strongly on a

parameter of our model. On the other hand, using the number of e-folds, the power spectrum and spectralindex of the scalar perturbations, we constrain this new cosmological model using CMBR measurementscoming from WMAP [7].

Keywords: Chaplygin Gas, Inflation, Cosmological Gravitational Waves and Bogoliubov coef-

ficients.

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Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

I Introduction 1

1 Introduction 3

1.1 Elements of FRW Cosmology and Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Generalised Chaplygin Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Inflationary Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

II Cosmological Model 9

2 The Model 112.1 Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Cosmological Evolution - Cosmic Time t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Cosmological Evolution - Conformal Time . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Inflationary Dynamics 15

3.1 Inflaton Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Number of e-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Slow-roll Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Observational Constraints 19

4.1 Power Spectrum of Density Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Spectral Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Comoving Wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4 nS r Parameter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

III Gravitational Waves 23

5 Gravitational Waves 25

5.1 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Cosmological Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3.1 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3.2 Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3.3 - Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3.4 CMBR/LSS Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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IV Conclusions 35

6 Conclusions 37

A Numerical Accuracy 39

A.1 Gravitational Energy Spectrum for = 1.03 with CMBR/LSS Constraints . . . . . . . . 40

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List of Figures

1.1 Current limits (solid lines) and projected sensitivities (dashed lines) to a stochastic gravitational-wave background versus the gravitational-wave frequency. The dotted curves are scale-invariant spectra. (From Ref. [27]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Plot of the potential (3.5), as a function of the scalar , for = 1.04 . . . . . . . . . . . 153.2 Evolution of and for the range of the -parameter of Table (3.1). The bluer lines

correspond to values that are closer to =

1, and there is a soft black line that represents

the unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Evolution of and for the range of the -parameter of Table (3.2). The bluer lines

correspond to values that are closer to = 1, and there is a soft black line that representsthe unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Evolution of(, ), for the range of the -parameter of Table (3.2), in a log-log plot. . . 18

3.5 Evolution of|(, )| for the range of the -parameter of Table (3.2), in a log-log plot. . . 18

4.1 Constraints on the nS r parameter space from CMBR/LSS measurements, for severalmodels, with the tensor-to-scalar ratio plotted on a log scale, from Ref. [38]. . . . . . . . . 19

4.2 Variation of the inflationary scale with the number of e-folds in the range 47 < Nc < 62.The limits correspond to Nc = 47 in blue, and Nc = 62 in purple. The dashed red linecorresponds to a second method for getting the value c through the mode function, see

Section (4.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Constraints on the nS r parameter space for two values of the number of e-folds: Nc = 47

(blue), and Nc = 62 (purple). The lines in black are for a variation with a constant -parameter, parameterized by the number of e-folds, 47 Nc 62. . . . . . . . . . . . . . 22

4.4 The slow-roll parameter evaluated at the moment the mode kc exits the horizon, for twovalues of the number of e-folds, Nc = 47 (blue), and Nc = 62 (purple), and for the methodof using the mode function to calculate c (dashed red). . . . . . . . . . . . . . . . . . . 22

5.1 Potential a/a for all the cosmological evolution (blue), and the Hubble co-moving wavenum-ber k2H (purple). Dashed gray lines refer to the method of integration: from a startinginflationary phase, ai, to aint, where it ends the GCG period and begins CDM, until af,the end of the integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 Evolution ||2, proportional to the number of gravitons created, in terms of the scale factor,and a zoom for the end of integration of large modes. These results are for = 1.03 andfor a energy scale of 1.4 1016 GeV, given by Eq. (4.3) and using the mode function tocalculate c. The second figure stands for a zoom between aeq1 and aeq2. . . . . . . . . . 29

5.3 Spectrum for the energy scales E = 0.11016 (red), 0.51016 (orange) and 1.51016 GeV(yellow), with = 1.4 . The spectrum suffers a upward shift due to an increase in theenergy scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.4 The potential a/a for the energy scales E = 1.5 1016, 0.5 1016 and 0.1 1016 GeV,in blue, with = 1.4 . The potential suffers a reduction with a decrease in the energyscale. Dashed gray lines refer to the method of integration. It is also shown the variationin the comoving wavenumber k2H, in purple. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.5 Gravitational-wave spectra for the values =

1.03,

1.04,

1.05,

1.10,

1.40, fol-

lowed by = 2.00, and 3.00. The value of moves away from 1 as we move to redderlines. The spectrum suffers an increase in the high frequency region with an increase inthe value of ||. All the curves are calculated for an energy scale of 1 .5 1016 GeV. . . . 31

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5.6 The potential a/a for = 1.4, 1.3, 1.2, and 1.1 , with the energy scale of 1.0 1016 GeV. The potential suffers a clear decrease in the maximum with a decrease in the||-value. It is also shown the variation in the comoving wavenumber k2H, in purple. . . . 31

5.7 Gravitational-wave spectra for the values = 1.05 (red), 1.04 (orange), and 1.03 (yellow) , with Nc = 47 (solid) and Nc = 62 (dashed). The spectrum suffers adownward shift as moves away from

1, and, simultaneously, an increase in the high

frequency region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.8 Gravitational-wave spectra for the values = 1.05 (red), 1.04 (orange), and

1.03 (yellow) . The spectrum suffers a downward shift as moves away from 1, and, atthe same time, an increase in the high frequency region. . . . . . . . . . . . . . . . . . . . 33

A.1 Gravitational-wave energy spectrum for = 1.03, with an inflationary scale given byCMBR/LSS constraints; and the condition ||2 ||2 evaluated for each point of thespectrum. To each point in the spectrum it corresponds a ||2 evolution, with the respectivecolor, in Figure (A.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.2 Evolution of||2 for each mode calculated in the spectrum, from an initial moment whenthe mode exits the horizon, until it reenters again. . . . . . . . . . . . . . . . . . . . . . . 40

A.3 Relative error in the derivative of |(a)|2, with respect to the scale factor, for each modecalculated in the spectrum of Fig. (A.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A.4 Relative error in the derivative of X(a), with respect to the scale factor, for each modecalculated in the spectrum of Fig. (A.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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Part I

Introduction

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Chapter 1

Introduction

1.1 Elements of FRW Cosmology and Inflation

Cosmology is in a new era, in which, for the first time, it is becoming possible to make detailed quantitative

tests of models of the early universe, through cosmological observations. This observational data comesfrom experiments such as COBE and WMAP for the Cosmic Microwave Background Radiation (CMBR),and, in the future, LIGO and LISA, attempting to directly detect the Gravitational Waves (GW). Suchobservations are at the present the most plausible route towards the possibility of testing some of thespeculative ideas which, in the recent years, have generated much interest, and also to learn more aboutthe aspects of physics at extremely high energies.

One of the most important paradigms in the early universe is the so-called inflationary era, whichassumes a period of accelerated expansion in the universes distant past. Inflation was introduced by A.Guth [8] and A. Starobinsky [9], as a possible solution to a large number of problems, directly relatedwith the foundations of the big-bang model, such as the horizon, flatness, and the relic density problem.The inflationary paradigm provides an excellent way to solve these problems, and it has also the usefulproperty that it generates spectra of both density perturbations and gravitational waves. For example,quantum fluctuations during the inflationary epoch become macroscopic density fluctuations, which leave

distinct imprints in CMBR, and are seeds for Large-Scale Structure (LSS).The standard cosmology lies in the cosmological principle of a homogeneous and isotropic universe at

large scales. Then, in an appropriate coordinate system, the Friedmann-Robertson-Walker (FRW) metricbecomes

ds2 = dt2 + a2(t)

dr2

1 Kr2 + r2

d2 + sin2 d2

, (1.1)

where a(t) is the scale factor with respect to the cosmic time t, and the constant K is the spatialcurvature, leading to a closed, flat, and hyperbolic spatial geometry, for positive, zero, and negativevalues, respectively. Substituting the FRW metric in the well known Einsteins equations

G

R

1

2g R =

8

m2P

T

g , (1.2)

where mp = 1/

G = 1.22 1019 GeV, in natural units, one gets the Friedmann equations:

H2 =8

3m2P K

a2, (1.3)

a

a= 4

3m2P( + 3p) , (1.4)

where a dot denotes the derivative with respect to t, H = a/a is the Hubble parameter. These relationslead to the equation for the conservation of energy,

+ 3H( +p) = 0 . (1.5)

Defining the density parameter as the ratio of the energy density to the critical density,

c

, with c 3H2m2P

8, (1.6)

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we can rewrite the Friedmann equation (1.3) in the form

1 = Ka2H2

. (1.7)

Let us now assume that the equation of state, for the matter in the universe, has the form pi = wii,where w

iis a constant, and i refers to a particular type of matter. This includes the main types of matter

that have an important role in cosmology, namely:

radiation, wr = 1/3; non relativistic matter, wm 0; cosmological constant, w = 1.Then, for a flat universe (K = 0), it is easy to combine the Friedmann equation (1.3) with the

energy conservation (1.5), in order to determine the cosmological evolution. For the energy density weget i a3(1+i), which, for radiation and non relativistic matter, leads to power-law expansions:a(t) t2/3 and a(t) t1/2, respectively. In the last case, for the cosmological constant, w = 1, theevolution is an exponential one: a(t)

eH t .

Therefore, one can then rewrite the first Friedmann equation (1.3) as a sum over the different typesof matter in the universe 1

H

H0

2=

i

i0

a

a0

3(1+wi), (1.8)

which, for the three mentioned types of matter, becomes

H

H0

2= r0

a

a0

4+ m0

a

a0

3+ . (1.9)

This form will be used in this work to describe the late time evolution of the universe, and corresponds

to the well known CDM model.The so-called concordance model, or CDM (-Cold Dark Matter), is the simplest model whichadequately fits the observational data coming from CMBR/LSS observations. It also agrees with theaccelerated expansion of the universe in the supernovae observations.

In this model, the acceleration of the Universe is described by a cosmological constant , correspondingto dark energy, and the cold dark matter (CDM), which stands for a non-relativistic type of dark matterat the epoch of radiation-matter equality. Finally, to include the inflationary era, this model is usuallysupplemented by some inflationary scenario: a new modified generalised Chaplygin gas [4] in our work.This model also assumes a nearly scale-invariant spectrum of primordial perturbations and a flat universe.

However, on the other hand, the CDM model has the well known cosmological constant problems:the magnitude problem and the coincidence problem. The first one points out that the observed value ofthe cosmological constant is extremely small to be attributed to the vacuum energy of matter fields; andthe second one the question about why, since there is just a very short period of time in the evolution

of the universe in which the energy density of the cosmological constant is comparable with the energydensity of matter, is this happening just at the present time, such that we can observe it.

1.2 Generalised Chaplygin Gas

As we saw in the last section, there is a large evidence that the Universe, at the present time, is dominatedby a small energy density component, with a negative pressure, the so-called dark energy, that leads to theobserved accelerated expansion, behaving like a cosmological constant. While the most evident candidatefor such component is the vacuum energy, another possible alternative is quintessence [ 10].

However, instead of a cosmological constant, or the dynamics of a scalar field rolling down an under-lying potential, the evidence for a dark energy component can also be explained by a perfect fluid with a

modified equation of state [1], avoiding the fine-tuning problems associated with CDM and quintessencemodels.

1The subscript 0 refers to quantities evaluated at the present time.

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Such modification consists in the introduction of an exotic background fluid, the Chaplygin gas,described by the equation of state [1],

p = A

, (1.10)

where A is a positive constant, and = 1. Inserting this equation of state into the equation of the energyconservation (1.5), we get an energy density that scales with the scale factor in the form [1],

=

A +

B

a6, (1.11)

where B is also a positive constant. Therefore, this energy density interpolates between a dust dominatedphase, where Ba3, at small scale factors, and a de Sitter phase at large scale factors, where p .Subsequently, this model admits a generalisation which can be regarded as a unification of dark matterand dark energy.

If we consider the constant a free parameter [1, 2, 3], with values lying in the range 0 < 1, weget an interpolation between a universe dominated by dust and a De Sitter stage via a phase describedby another equation of state, p = , with

= 1, behaving like soft matter. Then, from the equation

of the energy conservation (1.5) and the generalised equation of state Eq. (1.10), with in the range0 < 1, we get the generalised version of Eq. (1.11), the so-called generalised Chaplygin gas (GCG):

=

A +

B

a3(1+)

11+

. (1.12)

In the last years, the GCG has received a large attention, and has been constrained with various cosmo-logical observations [11, 12, 13, 14, 15]. It seems to describe the effective behaviour of dark energy, morethan that of dark matter, and it accounts also the advantage that some GCG models can be excellentframeworks to analyse dark energy related singularities [17, 18, 19, 20, 21, 16].

In this work we propose a phenomenological model for the inflationary epoch and the subsequentradiation dominated era of the universe, through a unification similar to the one of the Chaplygin gas,

but with respect to the first periods of the universe.

1.3 Inflationary Dynamics

In a large range of inflationary models, the implicit dynamics is simply that of a single scalar field the inflaton rolling down in some underlying potential. In the following, we will see the equationsgoverning the dynamics of the scalar field, , whose potential energy, V(), can lead to the acceleratedexpansion of the universe.

Neglecting spatial gradients, the energy density, , and the pressure, p, for a homogeneous singlescalar field are given by

=1

2 2

+ V() , (1.13)

p =1

22 V() . (1.14)

Substituting Eqs. (1.13) and (1.14) into Eqs. (1.3) and (1.5), we obtain,

H2 =8

3m2P

1

22 + V()

, (1.15)

+ 3H + V() = 0 , (1.16)

where V dV /d .These last two equations do not always give an accelerated expansion of the inflationary era, but

only in specific ranges, like in the so-called slow-roll regime. The condition for inflation, + 3p < 0,requires 2 < V() or, classically, that the potential energy dominates over the kinetic energy. Hence,one requires a sufficiently flat potential for the inflaton in order to lead to sufficient inflation. Then,

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the slow-roll conditions are given by 2/2 V() and || 3H||, and Eqs. (1.15) and (1.16) areapproximated by

H2 83m2P

V() , (1.17)

3H

V() . (1.18)

The so-called slow-roll parameters are defined by [22]

m2P

16

VV

2, m

2P

8

VV

, 2 m4P

642VV

V2. (1.19)

Therefore, the above slow-roll approximations are valid when 1 and || 1 for a long period of time.and also |2| 1, and the inflationary phase ends when and || approach one.

A very useful quantity, to describe the amount of inflation, is the number of e-folds of expansionbefore the end of inflation, defined by

N ln aa

= t

t

Hdt, (1.20)

from a given moment t, and a corresponding scale factor a, to the end of inflation at t with a = a. Or,equivalently, in terms of a scalar field, using Eqs. (1.17) and (1.18), from a given value that increasesuntil the end of inflation at = :

N 8m2P

V

Vd . (1.21)

By definition, the number of e-folds is zero at the end of inflation.The power spectrum, to be evaluated at the moment when a given scale exists the horizon, is deter-

mined by the k-space weighted contribution of modes with given wavenumber. To leading order in theslow-roll approximation, the amplitudes of the power spectra for scalar perturbations, density perturba-tions, and tensorial perturbations, i.e. gravitational waves, can be written as [23, 24],

Ps(k) 1283m6P

V3

V2

k=aH

, (1.22)

Pt(k) 1283

V

m4P

k=aH

, (1.23)

which are a function of the comoving wavenumber k. The scalar field potential, V, and its derivative, V,must be evaluated when a relevant scale in consideration exits the horizon during inflation. The powerspectra can also be expanded in power laws,

Ps(k)

Ps(kc)

k

kc

1ns+(s/2)ln(k/kc), (1.24)

Pt(k) Pt(kc)

k

kc

nt+(t/2)ln(k/kc), (1.25)

where kc is a pivot wavenumber at which the spectral parameters, the spectral index and its running [22],

ns(k) 1 6 + 2 , (1.26)s(k) 16 242 2 , (1.27)

for the scalar perturbations, and,

nt(k) 2 , (1.28)t(k)

4

82 , (1.29)

for the tensorial perturbations, are to be evaluated. In Section 4.1, we will evaluate Ps, ns, and s at thedistance scales of the CMBR and LSS, where they are measured or constrained.

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To a first approximation, the power spectra are power laws with power-law indices ns and nt, althoughthese indices may run slightly with k, with a running parameterized by s and t [25]. Finally, usingthe slow-roll parameters (1.19) in the expressions (1.22) and (1.23) for the scalar and the tensorial powerspectra, one gets the tensor-to-scalar ratio,

r

Pt(k)

Ps(k)

= 16 . (1.30)

1.4 Gravitational Waves

The gravitational waves of cosmological origin, due to quantum fluctuations of the vacuum, are producedduring the expansion of the universe, in particular from the early phases, and, therefore, can give usimportant information about the very early universe. Despite the fact that the gravitational waves ofcosmological origin have not yet been directly detected, they have received increased attention and therehas been an important effort of research, for their study in the last years.

In this work, we investigate the possible imprints in the power spectrum of the stochastic backgroundof gravitational waves, for a new model that describes the transition from the inflationary to the radiationdominated eras. This transition is far from being well know, and the signatures associated with it, inthe spectrum, could be a smoking gun leading to the inflationary model behind the early stages of

the cosmological evolution. This would give us a new insight into the physics underlying the very earlyuniverse.

Figure 1.1: Current limits (solid lines) and projected sensitivities (dashed lines) to a stochasticgravitational-wave background versus the gravitational-wave frequency. The dotted curves are scale-invariant spectra. (From Ref. [27])

Then, we integrate numerically the differential equations, derived by Parker [5], for the time depen-dent continuous Bogoliubov coefficients, that allow to model all the transitions between distinct periodsof the cosmological evolution as continuous. Hence, this method has advantages over the sudden transi-

tion approximation, because, associated with it, there is always an overproduction of gravitons of largefrequencies, requiring an explicit cut-off for frequencies above the rate of expansion of the universe [6],which appears in a natural way in the method of the Bogoliubov coefficients [26].

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This last advantage is an important feature because, as we will see, the high frequency range of ourmodel depends strongly on a parameter of our model, and it is important to have this range withoutany numerical artifact. This is also a very useful method to calculate the full spectrum, from the lowfrequencies, corresponding to the present cosmological horizon, to the large ones, directly related withthe inflation to radiation transition.

Although the present sensitivities of the gravitational-wave detectors, for the relative energy spec-

trum of gravitational waves, do not include the high frequency range, as be can seen in the Fig. 1.1, therehas been a considerable effort to improve these sensitivities like, for instance, in the NASAs Big BangObservatory (BBO) [28], the japanese Deci-hertz Interferometer Gravitational Wave Observatory (DE-CIGO) [29], the Laser Interferometer Gravitational-Wave Observatory (LIGO) [30], and, in the future,the space-based Laser Interferometer Space Antenna (LISA) [31].

1.5 Outline

This work is organized as follows. After this last review, Part I, of the basic and relevant issues relatedwith the principal topics for this work, we present a new cosmological model in Part II, and derive thesubsequent cosmological evolution by an analytical integration of the Friedmann equation. Then wecalculate some inflationary parameters, from the equivalent scalar field potential, and we constrain our

model using data coming from CMBR/LSS.In Part III, we present the method of the Bogoliubov coefficients and the cosmological evolution from

the inflationary era to the present time, and compare the results of our numerical simulations for thegravitational wave spectra resulting from different combinations of the parameters of our model. Finally,in Part IV, we summarize and make some concluding remarks.

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Part II

Cosmological Model

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Chapter 2

The Model

2.1 Energy Density

In this work, it is proposed a new cosmological model to describe the transition between the inflationary

regime and the radiation dominated era of the early universe, leading to a unification of these two stagesof the early evolution, and a transition which evolves in a smooth and instantaneous way. In this model,the matter content is modelled by a new modified generalised Chaplygin gas, with an underlying scalarfield description. Such unification is similar to that introduced by Kamenshchik et al. [1], but, in this lastcase, to unify the dark sectors of the universe, i.e. the dark matter and the dark energy.

With the above taken into consideration, the new cosmological model, that describes such transition,corresponds to a simple perfect fluid with an energy density evolving as

=

A +

B

a4(1+)

11+

, (2.1)

where A and B are constants, necessarily positive, and is a free parameter of the model, that mustsatisfy the condition 1 + < 0 in order to obtain an interpolation, between a positive cosmological

constant at small scale factors, and a radiation fluid at large scale factors1

:

A 11+ i , a a, (2.2)

B1

1+

a4 r , a a, (2.3)

where i is the energy density of the inflationary regime, r is the energy density during the radiation-dominated era, and a is the value of the scale factor at the end of inflation, which it is determined bythe condition for the end of inflation, i.e. + 3p = 0.

From these two limits of the energy density, we can determine the values of the constants A and B:

i = A1

1+ A = 1+i , (2.4)

r = r0a0

a4

=

B1

1+

a4 B = 0 a 40 1+ . (2.5)Therefore, the energy density (2.1) can be written in the more intuitive form =

1+i +

1+r

11+ ,

showing the interpolation between the de Sitter and radiation stages.Then, an early universe, dominated by this fluid, would experience a primordial inflationary era

through a de Sitter expansion, because it automatically leads to an asymptotic phase where the equation

of state is dominated by a cosmological constant 83m2P

A1

1+ . Finally, the universe exits this accelerated

expansion at a = a, entering in a radiation dominated epoch, with an almost instantaneous transitionbetween the two stages.

Using this energy density (2.1) in the energy conservation equation (1.5), we can derive the equationof state satisfied by its pressure, p, and energy density, ,

p =1

3 4

3

A

. (2.6)

1The subscript stands for the value of the quantities at the end of the inflationary period.

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Although this equation of state corresponds to a mixture of a radiation fluid and a generalised Chaplygingas [2], it is only the total energy density which is conserved. This type of equation of state has beenpreviously introduced in the context of dark energy models [32, 33, 34, 35], as far as we know initially inRef. [32]. However, it has never been analysed in the context of the early-time evolution of the universe.Notice that, for 0 < < 1, a fluid with the energy density (2.1) interpolates between a radiation fluid atsmall scale factors and a cosmological constant at large scale factor [34].

On the other hand, for < 1, as we saw in the limits (2.2) and (2.3), a fluid with the energy density(2.1) interpolates between a cosmological constant at small scale factors and a radiation fluid at largescale factors. This latter behaviour is the one we are looking for in the present work, as it may correspondto a viable and a simple phenomenological model for unifying the inflationary primordial era and theradiation epoch of the universe.

2.2 Cosmological Evolution - Cosmic Time t

The evolution of the universe is described by the time variation of the scale factor, that can be determinedusing the energy density (2.1) in the Friedmann equation (1.3). Initially, we have a constant Hubbleparameter, since, by the limit (2.2),

H2

8

3m2P A

11+

, (2.7)

and the initial evolution of the universe is described by a de Sitter expansion. The universe expands inan accelerated way, a > 0, as long as, by Eq. (1.4), we have + 3p 0, where the equality stands for theend of the inflationary period, from which we can determine the corresponding value of the scale factor,

a =

B

A

14(1+)

. (2.8)

Thereafter, the scale factor keeps increasing, leading to a radiation dominated universe, with the Fried-mann equation (1.3) given by

H2

8

3m2

P

B1

1+

a4

, (2.9)

using the limit (2.3). These two limits have a well known solution that was presented in the Section (1.1),with power-law solutions.

The general case for the transition (2.1), that we are studying, is more difficult, and there is not anexact solution. However, we can get an implicit relation between the cosmic time and the scale factor.More precisely, using the energy density (2.1) in the Friedman equation (1.3) we can do an analyticalintegration 2 [36, 37]

2

8

3m2PA

12(1+) (t t) = (y + 1)rF

1, r; 1 r; 1

1 + y

F(r, r; 1 r; 1) . (2.10)

In the previous expression t is the cosmic time, t a constant, F(a,b;c;y) corresponds to a hypergeometric

function 3 and

y =

a

a

4(1+), r = 1

2(1 + )> 0 . (2.11)

The time t t < 0 measures the cosmic time elapsed since the universe has a radius a < a until it exitsthe inflationary era when a = a at t = t. Similarly, t t > 0 measures the cosmic time elapsed sincethe universe has a radius a = a until it reaches a radius a larger than a. In this model, the universe hasan infinite past; in fact the cosmic time reaches infinite negative values when a 0 or y 1, becausethe hypergeometric function, F

1, r; 1 r; 1

1+y

, diverges at y = 0. On the other hand, the cosmic

time gets very large values when a 1; i.e. y 1, and the universe becomes radiation dominated.2The Friedmann equation (1.3), for introduced in Eq. (2.1), can be integrated in two steps by (i) performing it for

1 + > 0 [36] and (ii) performing an analytical continuation of the hypergeometric function to values of such that

1 + < 0 [37].3A hypergeometric series F(b, c; d; x), also called a hypergeometric function, converges at any value x such that |x| 1,whenever b + c d < 0. However, if 0 b + c d < 1 the series does not converge at x = 1. In addition, if b + c d 1,the hypergeometric function blows up at |x| = 1 [36].

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Unfortunately, Eq. (2.10) is quite involved and we cannot obtain an explicit expression of the scale factora as a function of the cosmic time t.

By choosing

2

8

3m2PA

12(1+) t = F (r,r; 1 r; 1) , (2.12)

i.e. by performing a shift on the cosmic time, Eq. (2.10) simplifies and reads

(y + 1)rF

1, r; 1 r; 1

1 + y

= 2

8

3m2PA

12(1+) t . (2.13)

Of course, this redefinition of the cosmic time does not modify the fact that in this model the universehas an infinite past, where it is asymptotically de Sitter, and an infinite future, where it is radiation-dominated.

2.3 Cosmological Evolution - Conformal Time

As we will see, in order to obtain the gravitational wave power spectrum, it is convenient to introducethe conformal time (see Eq. (5.15)), , and see if the factor a/a, where the prime denots a derivativewith respect to conformal time, has a particular simple form. Using the relation dt = a d we can rewriteEq. (2.10) in terms of :

aA1

2(1+)

8

3m2P( ) = F

r,r

2; 1 r

2; 1

y r2 (1 + y)r F

r, 1; 1 r

2;

y

1 + y

. (2.14)

At the conformal time , where the scale factor has the value a = a, the universe exits the inflationaryera. This can be easily checked by noticing that [37]

Fr,r

2; 1 r

2; 1

= 2r F

r, 1; 1 r

2;

1

2

. (2.15)

We recall that r, defined in Eq. (2.11), is positive. Therefore, the hypergeometric series in the previousequality are well defined (see footnote 3). The primordial universe starts its de Sitter expansion at

where a 1. On the other hand, the universe gets radiation-dominated at where1 a. We note that [37]

Fr, 1; 1 r

2; 1

= 1 . (2.16)Like in the previous section, by choosing

aA1

2(1+)

8

3m2P = F

r,r

2; 1 r

2; 1

, (2.17)

the relation between the conformal time, , and the scale factor given in Eq. (2.14) can be simplified to

aA1

2(1+)

8

3m2P = y r2 (1 + y)r F

r, 1; 1 r

2;

y

1 + y

. (2.18)

Again, we notice that this redefinition of the conformal time does not modify the fact that in this modelthe universe has an infinite past (in terms of the conformal time), where it is asymptotically de Sitter,and an infinite future (in terms again of the conformal time), where it is radiation-dominated.

In Section 5.1 it will become clear that, to determine the gravitational wave power spectrum, we haveto solve a set of differential equations that is equivalent to the second order differential equation (5.15),

X +

k2 a

a

X = 0 , (2.19)

in terms of the conformal time. As this equation closely resembles a Shrodinger equation equation withthe potential barrier V a/a, leading to graviton production. Then, in the following, we will refer tothe factor a/a as the potential.

For a de Sitter expansion, or a radiation-filled universe, this equation has an exact solution, but, on the

other hand, the intermediate evolution, described by the model ( 2.1), does not. Hence, the cosmologicalevolution of the scale factor in terms of conformal time is implicit given in Eq. (2.18). Therefore, tocalculate the spectrum, it will be necessary a numerical integration of the Eq. ( 2.19).

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Chapter 3

Inflationary Dynamics

3.1 Inflaton Field

In order to describe the inflationary dynamics of the model presented in the previous section, it is quite

helpful to introduce an equivalent description in terms of a scalar field, , rolling down the underlyingpotential, V(), as we can see in Fig. (3.1). The respective energy density and pressure are given by(1.13),

=2

2 a2+ V() , (3.1)

p =2

2 a2 V() , (3.2)

where the prime stands for derivative with respect to conformal time.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

mP

V

V0

Figure 3.1: Plot of the potential (3.5), as a function of the scalar , for =

1.04 .

In a FRW universe, the scalar field, , is equivalent to the modified Chaplygin gas with the energydensity, , and pressure, p, introduced in Eqs. (2.1) and (2.6), as long as = and p = p. Therefore,the scalar field, , can be determined analytically in terms of the scale factor, cancelling V() in theequations (3.1) and (3.2), and, with a change of variable, integrating from a given point to the end ofinflation:

(a) = 18|1 + |arcsinh

B

Aa2(1+)

mP . (3.3)

For simplicity, we will restrict to the solution (3.3) with the plus sign (+), as our results do not dependon which of the two signs we choose.

In a similar way, adding Eq. (3.1) with Eq. (3.2), the scalar field potential becomes

V() =1

2( p) , (3.4)

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and, using the energy density (2.1) and pressure (2.6), with Eq. (3.3) solved for a = a(), we get thepotential in terms of the scalar field,

V() =V03

cosh

21+

8|1 + | /mP

+ 2 cosh

21+

8|1 + | /mP

, (3.5)

where V0 = A1

1+ .

The universe starts its inflationary phase in a de Sitter state where the scalar field is sitting at thetop of the potential; i.e. at = 0 (see Fig. (3.1)). Then it starts rolling down the potential until it exitsthe inflationary era when (a) = :

=1

8|1 + | ln(1 +

2) mP . (3.6)

Finally, the radiation dominated phase starts for large values of the scalar field, i.e. .The derivatives of the scalar field potential (3.5), with respect to the scalar field are long expressions

because V() does not have a simple form,

V =V0

8

3mPcosh

`8|1 + |/mP

21+

h1 4 + cosh `28|1 + |/mPi tanh `8|1 + |/mP , (3.7)

V

=V016

3m2Pcosh `8|1 + |/mP

21+

n

cosh`

2

8|1 + |/mP

+ h

1 + 4 2(1 + 3) sech`8|1 + |/mP2io , (3.8)V =V0

(8)3/2

3m3P

cosh`

8|1 + |/mP

21+

h`

3 + 4`

1 + 2(1 + 5)

+ 2`

2 + 2 43 cosh `28|1 + |/mP+ cosh `48|1 + |/mPi sech`8|1 + |/mP2 tanh `8|1 + |/mP . (3.9)

This expressions will be used in Section 3.3 to calculate the slow-roll parameters.

3.2 Number of e-folds

The number of e-folds of expansion, from the moment that a given mode k = aH exits the horizon duringthe inflationary era, at = c, until the end of inflation, can be given by ( 1.21). Using = c, we get:

Nc 8m2P

c

V

Vd . (3.10)

Substituting the scalar field potential (3.5) on the previous expression, we obtain

Nc =1

2(1 + )(2 1) ln8

pedro antonio pinto frazao- stochastic gravitational waves from a new type of modied chaplygin gas

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