Bielefeld University

Faculty of physics

Bachelor thesis

The number of e-foldings of the radiationdominated epoch and the effect of cosmic

transitions

Author:Pascal Krelingpkreling@physik.uni-bielefeld.deBielefeld UniversityWertherstraße 442, 33619 BielefeldGermany

1st Supervisor:Prof. Dr. Dominik J. Schwarz

dschwarz@physik.uni-bielefeld.deBielefeld University

2nd Supervisor:

Prof. Dr. Dietrich Bödekerbodeker@physik.uni-bielefeld.de

Bielefeld University

May 10, 2017

Contents

1. Introduction 1

2. The early universe 12.1. Friedmann-Lemaître-Robertson-Walker model . . . . . . . . . . . . . . . . . . . 1

2.1.1. FLRW line element and metric tensor . . . . . . . . . . . . . . . . . . . . 22.1.2. Friedmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2. Standard model of cosmology: The ΛCDM-model . . . . . . . . . . . . . . . . . 32.3. Inflationary model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3. Effective degrees of freedom 73.1. Neglecting interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2. Hadron resonance gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4. Speed of sound 124.1. Thermodynamic derivation for the speed of sound . . . . . . . . . . . . . . . . . 134.2. Speed of sound during transitions in the universe . . . . . . . . . . . . . . . . . . 15

5. Number of e-foldings of the radiation epoch 16

6. Conclusion 18

References 20

A. Notation 21

B. Program code for numerical integration 22

1. Introduction

The number of e-foldings is an indicator of the duration of the inflationary epoch in the earlyuniverse. The concept of e-foldings can also be used in other epochs of the universe. Wediscuss the number of e-foldings of the radiation dominated epoch of the universe under dif-ferent circumstances. Our starting point is the case in which we assume our universe to be anideal gas or fluid of massless and interactionless particles. Subsequently, we consider particlemasses and take strong interactions, especially the QCD transition, into account. The stronginteraction leads to a modification of the evolution of the universe and affects the total numberof e-foldings of the radiation dominated epoch.

The cosmic expansion is considered to be very slow so that we always stay in the adiabaticcase and we neglect chemical potentials.

This thesis is structured as follows. We first discuss some theoretical basics in chapter 2 todescribe our universe in terms of mathematical equations and relations. Also we derive a setof equations to describe the evolution of the universe and briefly take a look at the standardmodel of cosmology. In chapter 3 we investigate the effective degrees of freedom in theuniverse as a function of temperature for interacting and non-interacting particles. Includingparticle interaction leads to modifications to the speed of sound. In chapter 4 we deduce thedefinition of the speed of sound for non entropy conserving cases and phase transitions. Inchapter 5 we use that the number of e-foldings and the speed of sound are connected. Wecalculate the modification to the number of e-foldings due to the modifications of the speed ofsound based on results from [12].

Notation is explained in the appendix A.

2. The early universe

Cosmology studies the origin, evolution and eventual fate of the universe. For any cosmolog-ical conclusion we need astronomical observations and for this observations the Copernicanprinciple is very important. The principle says ’we are a typical observer’ so anybody fromanother point of view in the universe should see the same type of structures. This principle wasextended to the Cosmological principle which reads as ’viewed on a sufficiently large scale,the properties of the universe are the same for all observers. Embedded in the cosmologicalprinciple is that the universe is homogeneous and spatial isotropic at large scales. As gravityhas the most impact on the large scale in the universe we consider some basics of general rel-ativity. The cosmological principle together with general relativity leads to a conclusion thatour universe is non-static and expanding today, which is in good accordance with astronomicalobservations. So a theory is needed to describe an expanding space-time.

2.1. Friedmann-Lemaître-Robertson-Walker model

Due to the fact that we want to describe the universe with general relativity we need a theoryaccording with the Einstein field equations. A complete solution to these equations and con-taining a homogeneous and spatially isotropic universe is the Friedmann-Lemaître-Robertson-

1

Walker (FLRW) model (1920’s). The main parts of this model are the Friedmann equationsand the line element with the embedded metric.

2.1.1. FLRW line element and metric tensor

The line element is defined as,

ds2 = −gµν(x)dxµdxν. (2.1)

Here gµν(x) is the metric tensor, which is a function of the four-dimensional space-time coor-dinates. The spatial part of the metric tensor for an expanding space and therefore the spatialline element dl is [1],

dl2 = a2

dx idx i +(x idx i)2

1− K x i x i

. (2.2)

The factor a is known as the scale factor and K is the curvature of the space with,

K =

+1 spherical

−1 hyperspherical

0 Euclidean

. (2.3)

Now we can extend this spatial line element to a the line element of the space-time by addingthe time coordinate and let a become a function of time,

ds2 = −dt2 + a2(t)

dx idx i +(x idx i)2

1− K x i x i

. (2.4)

Because we have an isotropic space, it is wise to use spherical coordinates. In this case theline element reads,

ds2 = −dt2 + a2(t)

dr2

1− Kr2+ r2(dθ 2 + sin2 θdφ2)

. (2.5)

With spherical coordinates the metric tensor becomes diagonal,

gr r =a2(t)

1− Kr2, gθθ = a2(t)r2, gφφ = a2(t)r2 sin2 θ , gt t = −1. (2.6)

2.1.2. Friedmann equation

Now that we have found the metric tensor of an expanding universe, we want to describe theexpansion itself. The cosmological expansion is determined by the Einstein equation,

Rµν −12

gµνR+Λgµν = 8πGTµν, (2.7)

2

where Rµν is the Ricci tensor, R is the scalar curvature, Λ is the cosmological constant, G isNewton’s gravitational constant and Tµν is the energy-momentum tensor. The Ricci tensor isdefined as,

Rµν = ∂λΓλµν− ∂µΓ λνλ + Γ

λµνΓσλσ− Γ λ

µσΓσλν

, (2.8)

where Γ is the Christoffel symbol with,

Γµ

νλ=

12

gλσ(∂νgλσ + ∂λgνσ − ∂σgνλ). (2.9)

The Ricci scalar is the contracted Ricci tensor,

R= Rµµ. (2.10)

We also need the energy-momentum tensor [2],

Tµν = pgµν + (ε + p)uµuν, (2.11)

with uµ the fluid four-velocity, ε the energy density and p the pressure of the fluid. Calculatingall non-zero Christoffel symbols and taking equation (2.11) into account the tt-component ofthe Einstein equation leads to the Friedmann equation,

H2 +ka2=

8πG3ε +Λ

3. (2.12)

In this equation H is the Hubble rate and is defined as,

H =aa

. (2.13)

The spatial components of the Einstein equation provide the second Friedmann equation,

−

2H + 3H2 +ka2

= 8πG −Λ. (2.14)

Instead of equation (2.14) often the energy-momentum conservation is used, which is thecombination of equation (2.12) and equation (2.14) and has the form,

ε + 3H(ε + p) = 0. (2.15)

The set of equations (2.12), (2.13) and (2.15), together with the equation of state p = p(ε),form a closed set of equations to describe the evolution of the universe.

2.2. Standard model of cosmology: The ΛCDM-model

With this set of equations we can study different cosmological models by inserting an energydensity and a pressure. This has been done in the past and most of the models do not fitto observations already made. The best fitting model is the ΛCDM-model. This model is a

3

cosmological model which contains cold dark matter (CDM) besides baryonic matter, darkenergy described by the cosmological constant Λ and starts with a big bang. Also the modelmay have non-vanishing spatial curvature. This components of the universe have to be takeninto account in the energy density in the Friedmann equation (2.12) which leads to,

H2 =a2

a2=

8πG3(ε + εrad + εΛ + εcurv), (2.16)

where εM , εrad and εΛ are energy densities of non-relativistic matter, relativistic matter (radi-ation) and dark energy. The energy density of curvature εcurv is defined as,

8πG3εcurv = −

ka2

. (2.17)

We can now introduce a critical energy density εc by[3],

εc =3

8πGH2

0 . (2.18)

With this critical energy density we can define dimensionless parameters referring to thepresent universe,

ΩM =εM ,0

εc, Ωrad =

εrad,0

εc, ΩΛ =

εΛm0

εc, Ωcurv =

εcurv,0

εc. (2.19)

It follows from equations (2.16) and (2.18) that,∑

i

Ωi = ΩM +Ωrad +ΩΛ +Ωcurv = 1. (2.20)

Therefore the parameters Ωi are equal to relative contributions of different sorts of energy andalso of spatial curvature. Important to mention is that the matter density parameter contain thebaryonic matter as well as the dark matter,

ΩM = ΩB +ΩDM . (2.21)

By using different observational methods we can determine the value of the density param-eters and get [3],

ΩM ≈ 0.27, ΩΛ ≈ 0.73, Ωcurv ≈ 0, (2.22)

with an accuracy of about 5% [3] and ΩM is consisting of ΩB = 0.046 and ΩDM = 0.23. Inter-esting to mention is that the spatial curvature of our universe is very small if not exactly zero.This fact will lead to a problem which will be investigated later on. Spatially flat cosmologicalmodels with non-relativistic cold dark matter and dark energy with parameters close to thosementioned before are called flat ΛCDM models.

4

2.3. Inflationary model

The ΛCDM model as described before has some weaknesses. As already mentioned ouruniverse is spatial flat. To reach this flatness today after the big bang the initial conditionshave to be very fine tuned which is very unlikely. Another problem is that the fluctuations inthe Cosmic Microwave Background (CMB) which is a relic of the big bang are very small.These small fluctuations indicate that the different regions in the CMB have been physicallyconnected in the early universe. But the size of the apparently connected areas are to large tobe described by the ΛCDM model which is called the horizon problem.

To solve these and other problems not mentioned here the ΛCDM model was extended byan inflationary epoch in the early universe which was invented in the 1980’s. The inflationaryepoch is distinguished by a simple definition,

inflation⇔ a > 0. (2.23)

For constant acceleration, this definition is equivalent to an exponentially expanding universein a very short period of time. Considering this rapid expansion we can solve the horizonproblem because of the rapid expansion our present observable universe can be originated bya tiny region that was physically connected during inflation. To solve the flatness problem wecan follow a similar idea. The present observational universe is spatially flat as mentionedbefore which is alright if we acknowledge that the observational universe is just a small partof the whole universe and locally flat. To arrive at the point that the universe is spatial flata certain amount of inflation must have happened during inflation. The amount of inflationhappened is quantified by the ratio of the scale factor at the end of inflation to its value atsome initial time. Typically this quantity is a large number so that the logarithm is taken andthe result is called the number of e-foldings N ,

N(t)≡ ln

a(tend)a(t)

. (2.24)

To solve the flatness problem, roughly 50 to 60 e-foldings are needed. However, the conceptof e-foldings can be applied to any time interval in the evolution of the universe and is notbound to the inflation epoch.

During the inflationary epoch the universe is dominated by a scalar field φ with a certainpotential. This potential can be chosen in some different ways and so there are different infla-tionary models. But all of these different models share that the potential energy of the scalardrops because of the expansion of the universe. At some point it starts oscillating which iscalled reheating. Due to the fast expansion during inflation quantum fluctuations in the earlyuniverse are blow up and lose their physical connection so that they do not annihilate eachother. These fluctuations together with the reheating therefore form later in the evolution theuniverse radiation and matter. Also the fluctuations have a density perturbation with an ampli-tude AS and a spectral index nS, which differ for different inflation models. These parametersdescribe the physics of inflation have a measurable value today. The density perturbationspectral index is connected to the number of e-foldings[4],

ns ≈ 1−2N

. (2.25)

5

ln a = N

ln λphys.

a0

inflation RD MD ΛD

1/H

Figure 1: Kinematics of scales. During Inflation and Λ domination (ΛD) the Hubble scale is constant,during radiation domination (RD) it is proportional to a2 and during matter domination (MD)proportional to a3/2. The line at a0 describes the scale factor today. The dotted line stand forthe wavelength of a physical object, for example quantum fluctuations which grow propor-tional to a.

So by measuring the parameter nS can be related to the number of e-foldings which have to beoccurred from reheating up to today.

With the inflationary epoch added to the ΛCDM model let us have a look at the history ofour universe and especially at the different epochs, which can be seen in figure 1. After theBig Bang occurred the universe was dominated by inflation and the Hubble scale 1/H wasconstant as a function of time. This epoch ended with the reheating where the energy of theinflation field was partially transformed into radiation. After the reheating the universe wasradiation dominated. In this epoch some cosmic transitions like the electroweak or the QCDtransition happened and the scale factor a∝ t2. Also the Big Bang nucleosynthesis happenedin this era when light nuclei were formed. The universe therefore is expanding the whole timeand is cooling down. At some point the universe is so large that the radiation and matter in theuniverse reach equality of their respective energy densities and after this equality the universeis matter dominated. The scale factor during the matter dominated epoch is a ∝ t3/2. Thematter dominated case holds up to the younger past when the universe starts acceleratingagain. The era today is called Λ domination which is the currently best fitting explanation tothe observed accelerated expansion and the scale factor becomes constant again.

6

3. Effective degrees of freedom

Before radiation and matter equality the universe is dominated by radiation. A reasonableapproximation for the equation of state is [5],

p =13ε. (3.1)

To calculate the energy density and to see how the energy density and the spin degrees offreedom are connected we will first look at the situation in which we neglect the effect ofinteractions on the equation of state.

3.1. Neglecting interactions

Let us first have a look at the non-interacting case. We assume the universe is a relativisticideal gas or fluid. To calculate its energy density we use methods of statistical mechanicswhich is similar to a calculation by Weinberg [5]. Assuming an equal number of particlesand antiparticles the number density (as all chemical potentials are assumed to vanish) of aspecies i of fermions or bosons in a finite volume of phase space with momentum between pand p+ dp is,

gi

(2π)3f±(p, T ), (3.2)

with

f± =1

eE/T ± 1, (3.3)

the distribution function of either fermions (with a plus sign) or bosons (with a minus sign)and gi are the spin degrees of freedom of the particle species i. The energy density of a speciesof particles is,

εi(T ) =

∫

d3p(2π)3

E(p)gi f±(p, T ). (3.4)

Because we consider a relativistic gas or fluid E(p) is given by the relativistic dispersionrelation E(p) =

Æ

p2 +m2i . Using now spherical coordinates in momentum space we achieve,

εi(T ) = gi

∫

dp2π2

p2q

p2 +m2i f±(p). (3.5)

If we consider the ultra-relativistic limit (m= 0, p = E), the integral becomes,

εi =gi T

4

2π2

∞∫

0

dxx3

ex − 1= gi T

4 12π2

ζ(4)Γ (4) = giπ2

30T 4, (3.6)

7

m/MeV F/B gi

γ 0 B 2ν 0 F 6u 2.4 F 12d 4.8 F 12s 104 F 12c 1270 F 12b 4200 F 12t 177000 F 12g 0 B 16e 0.5 F 4µ 106 F 4τ 1700 F 4W 80300 B 6Z 91100 B 3H 125300 B 1

Table 1: Used particles for the calculation of the degrees of freedom as a function of temperature duringthe evolution of the universe.Pions are used below, gluons and Quarks above the temperatureof the QCD-transition at 155 MeV. [8]

with the substitution x = ET , ζ the Zeta-function and Γ the Gamma-function. This result holds

for bosons. For fermions the distribution function comes with a plus sign and the result ismodified by a factor of 7/8.

The general case for any temperature is now rewriting equation (3.5) with the substitutionp→ E and normalize the equation by the result of equation (3.6). This rewriting leads to theconnection between the energy density and the effective degrees of freedom,

εi(T )π2

30 T 4= gi

15π4

∞∫

mi/T

dE

√

√

E2 −mi

T

2f±(E)E

2 = gi,e f f (T ). (3.7)

With this definition of the effective degrees of freedom we can compute the effective degreesof freedom as a function of temperature. The overall degrees of freedom are the sum over eachindividual degrees of freedom of each type of particle,

ge f f =∑

i

gi,e f f = 2+78

6+ ge f f ,e− + ge f f ,µ + ... . (3.8)

Here the neutrinos are assumed to be massless. The particles, masses and degrees of freedomrelevant for the expansion of the universe are shown in table 1. Now we plot the effectivedegrees of freedom as given in equation (3.8) for all particles species shown in table 1 inclusivethe pions from table 2 which leads to the graphic in figure 2. The thresholds for each crossingare defined by a temperature which is equal to a third of the mass of the new particle type. This

8

0.01 1 100 104

10

20

50

100

T/MeV

gϵ

γ,ν

e

μ

π

u,d,g,s

c τb W Z

H

t

Figure 2: Effective degrees of freedom as a function of temperature. The red dashed line is the QCDcrossing at T = 155 MeV. The blue lines are Heaviside functions for each particle type andthe black line is the continuous.

assumption can be made because the average kinetic energy of a particle in a gas is E ∼ 3kB T .So the gas is relativistic as long as the temperature is a third of the mass.

So far we assumed that the particles are not interacting which is relevant because the pionsare left out before the QCD crossing and those particles did not arrive spontaneously duringthe evolution of the universe. So one should have a look at interactions.

3.2. Hadron resonance gas

Taking interactions into account we have to modify the relation between the energy densityand the degrees of freedom (eq. (3.6)). With a perturbed calculation which holds if T TQC D

we can find that the energy density changes [9],

εi(T ) = giπ2

30T 4(1+ c1 g2

c + c2 g2c ln(gc) + ...+ cn−1 g5

c ln(gc) + cn g6c ), (3.9)

where gc is a coupling constant for the interactions between the particles in the universe. Ingeneral the coupling constant is a function of the temperature (running coupling), i.e. thetemperature dependence of the energy density and so the effective degrees of freedom aremodified.

To have a look at the impact of interactions on the degrees of freedom the integral for thelower section before 155 MeV was computed up to 550 MeV. Also there were new particles

9

particles with m< 3 TQC D particles with m< 6 TQC D particles with m< 9 TQC D

m/MeV F/B gi m/MeV F/B gi m/MeV F/B gi

π0 135 B 1 η 547.9 B 1 η′ 957.8 B 1π± 139 B 2 f0 600 B 1 f0 990 B 1

ρ 775 B 9 a0 980 B 4ω 782 B 3 φ 1019 B 3K± 493 B 2 h1 1170 B 3K0 497 B 2 b1 1229 B 12K∗ 893 B 12 a1 1230 B 12

f2 1275 B 5f1 1281 B 3η 1295 B 1π 1300 B 4a2 1318 B 20f0 1350 B 1π1 1354 B 12K1 1272 B 4p 938.27 F 4n 939.57 F 4∆ 1232 F 32Λ0 1115 F 4Σ± 1192 F 8Σ0 1193 F 4Ξ± 1314 F 4Ξ0 1321 F 4

Table 2: Used particles for the calculation of the degrees of freedom as a function of temperature duringthe evolution of the universe. For the case m < 3 TQC D no new particles were added becausethere are no particles with masses in this range except the already used ones as in Fig. 2. [8]

with higher masses added which are compound particles made of quarks and gluons for exam-ple the ∆ or proton and neutron. So the strong interaction is taken into account because it hasthe strongest coupling constant and the most impact at the perturbed calculation made in eq.(3.9). In the sector below the QCD transition temperature the degrees of freedom are given bythe bound states of quarks and gluons for example in hadrons. To see now the impact on theoverall degrees of freedom we look at the heavier particles as excited states of lighter particleswhich is called hadron resonance gas. These added particles are separated by the temperatureof QCD transition (TQC D) so that in the first step particles with mass up to 3 times TQC D wereadded, in the second step particles with mass up to 6 times TQC D, third up to 9 times TQC D andfourth up to 12 times TQC D. The tables 2 and 3 show the added particles. For the new particlesnew integrals were added like in equation (3.7), calculated up to 550 MeV and the sum ofall these integrals leads to the graphic shown in Fig. 3. One can see that the more particles

10

particles with m< 12 TQC D

m/MeV F/B gi m/MeV F/B gi m/MeV F/B gi

η 1405 B 1 Λ 1520 F 8 K∗±,02 1770 B 20

f1 1420 B 3 Λ 1600 F 4 K∗±,03 1780 B 28

ω 1420 B 3 Σ±,0 1385 F 24 K±,02 1820 B 20

ρ±,0 1450 B 9 Ξ±,0 1530 F 16 p 1650 F 4a±,0

0 1450 B 3 ∆ 1620 F 16 n 1650 F 4η 1475 B 1 ∆ 1700 F 32 p 1675 F 12f0 1500 B 1 Λ 1670 F 4 n 1675 F 12f2’ 1525 B 5 Λ 1690 F 8 p 1680 F 12

K±,01 1400 B 12 π±,0

1 1600 B 9 n 1680 F 12K∗±,0 1410 B 12 η2 1645 B 5 p 1700 F 8K∗±,0 1430 B 4 ω 1650 B 3 n 1700 F 8K∗±,0

2 1430 B 20 ω3 1670 B 7 o 1710 F 4p 1440 F 4 π±,0

2 1670 B 15 n 1710 F 4n 1440 F 4 φ 1680 B 3 p 1720 F 8p 1520 F 8 ρ±,0

3 1690 B 21 n 1720 F 8n 1520 F 8 ρ±,0 1700 B 9 Λ 1800 F 4p 1535 F 4 f0 1710 B 1 Λ 1810 F 4n 1535 F 4 π±,0 1800 B 3 Λ 1820 F 12∆ 1600 F 32 φ3 1850 B 7 Λ 1830 F 12Λ 1405 F 4 K∗±,0 1680 B 12 Σ±,0 1660 F 12Σ 1670 F 24 Σ 1750 F 12 Σ±,0 1775 F 36Ξ±,0 1820 F 16 Ω± 1672 F 8

Table 3: Used particles for the calculation of the degrees of freedom as a function of temperature. [8]

are added the more degrees of freedom one obtains. Also the new graph with masses up to 3times TQC D is not shown because there are no new particles to add to the old graph. But theinteresting point of this plot is now the crossing at the QCD temperature where we can seean increase of the degrees of freedom from the graph with only pions up to the graph withparticles with mass up to 12 times TQC D. The exact numbers are gc,0 = 16.869, gc,6 = 21.707,gc,9 = 24.995 and gc,12 = 28.521 at TQC D.

We see that including interactions the effective degrees of freedom change. Thus the energydensity and so the equation of state is modified as well as the quantity ε − 3p which is thetrace of the energy-momentum tensor and is often called the interaction measure.

In general the equation of state is often defined by a dimensionless number w which is,

w=pε

. (3.10)

So the interaction measure is (1− 3w)ε and would be zero in the prefect radiation dominatedcase but differs now from zero because of the consideration of interactions.

11

0.01 1 100 104

10

20

50

100

T/MeV

gϵ

particles of mass up to 9 × QCD transition

particles of mass up to 6 × QCD transition

particles of mass up to 12 x QCD transition

only pions

Figure 3: Effective degrees of freedom as a function of temperature including a hadron resonance gas.Black line and dashed orange line as in figure 2. The blue line contains particles with massup to 6 times TQC D, the red line contains all particles with mass up to 9 times TQC D and theorange line all particles up to 12 times TQC D.

4. Speed of sound

As we have seen that interactions change the degrees of freedom and so the circumstances ofthe expansion of the universe, we should now further investigate the impact at the equationof state how we have to modify the assumptions made so far. The equation of state for theradiation dominated case is,

p =13ε. (4.1)

In general the equation of state for any case in an adiabatic expansion is,

δp =

∂ p∂ ε

sδε +

∂ p∂ ε

ε

δs

= c2s δε +

∂ p∂ ε

ε

δs, (4.2)

where c2s is the speed of sound in the universe and s = S/NB is the specific entropy per particle

number NB. In the case with constant entropy (isentropic) the δs = 0 and the second summandvanishes. For example we know that c2

s =13 for the radiation dominated case and for the matter

dominated case this would be c2s = 0. This definition therefore assumes an adiabatic or even

12

isentropic expansion of an ideal and relativistic gas or fluid which is a good and commonmodel for the universe in cosmology .

4.1. Thermodynamic derivation for the speed of sound

The assumption of an isentropic expanding universe is not always the right choice because wealready used that there are interactions in the universe to change the degrees of freedom andtherefore entropy. To pursued with the interacting case we want to have a look at the speed ofsound in a more general case.

This has already been done in a publication [11] by Weinberg (1971). To start of he firsttakes a look at the energy moment tensor of a fluid similar to equation (2.11) and the particlecurrent which are,

Tαβ = pηαβ + (ε + p)uαuβ , (4.3)Nα = nuα, (4.4)

with ηαβ as the metric tensor, n the number density and uα the normalized velocity four-vectorwith,

uαuα = −1.

Also important are the equations of motion of a fluid which are contained in the conservationlaws which are,

∂ Tαβ

∂ xβ= 0, (4.5)

∂ Nα

∂ xα= 0. (4.6)

By the use of equation (4.4) equation (4.6) becomes,

∂ uβ

∂ xβ= −

uβ

n∂ n∂ xβ

. (4.7)

Another equation which is needed, is the second law of thermodynamics which gives thevariation in the specific entropy s, already defined earlier, as,

Tds = dε

n

+ pd

1n

=1n

dp−ε + p

ndn

, (4.8)

where T is the temperature. In the case of adiabatic motion the second law of thermodynamicsbecomes with usage of equation (4.7),

uβ∂ T∂ xβ

=

∂ ε

∂ T

−1

n

n

∂ ε

∂ n

T− ε − p

∂ uβ

∂ xβ, (4.9)

13

where the index denotes which parameter is constant for the derivative. Also Weinberg men-tions, because ds must be a perfect differential the second law of thermodynamics (equation(4.8)) can be written as,

∂

∂ T

1nT

∂ ε

∂ n

T−ε + p

n

n=

∂

∂ n

1nT

∂ ε

∂ T

n

T, (4.10)

or more simply,

T

∂ p∂ T

n= ε + p− n

∂ ε

∂ n

T. (4.11)

Now we can rewrite equation (4.7) by the use of equation (4.11) to,

uβ∂ T∂ xβ

= −T (∂ p/∂ T )n(∂ ε/∂ T )n

∂ uβ

∂ xβ. (4.12)

The next step done is to use comoving coordinates such that,

ui = 0, ut = 1, ∂ u0/∂ xα = 0. (4.13)

In this case equation (4.12) becomes,

∂ T∂ t= −

T (∂ p/∂ T )n(∂ ε/∂ T )n

∇ · u. (4.14)

This is the evolution of temperature in a fluid such that we can identify the speed of sound as,

c2s =(∂ p/∂ T )n(∂ ε/∂ T )n

. (4.15)

We see that the speed of sound has a temperature dependence if the pressure and or energydensity depend on the temperature.

Later on in the publication Weinberg gives a more explicit expression for the speed of soundwithout an exact thermodynamic derivation,

c2s =

T (∂ p/∂ T )2n(ε + p(∂ ε/∂ T )n

+

nε + p

∂ p∂ n

T. (4.16)

In the limit of small chemical potentials (nµ ε+p) which holds for the radiation dominatedepoch, we can send n → 0. This limit together with the identities ε + p = TS and S =(∂ p/∂ T )n which can be derived by the total differential of the inner energy [10], is,

c2s =

T (∂ p/∂ T )2n(ε + p)(∂ ε/∂ T )n

+

nε + p

∂ p∂ n

T=(∂ p/∂ T )n(∂ ε/∂ T )n

. (4.17)

This equation is the same result as in equation (4.15). For the case of radiation domination andto describe the behavior of the speed of sound at the QCD transition we have to take equation(4.15) for the next calculations.

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4.2. Speed of sound during transitions in the universe

To calculate now the exact speed of sound during a transition we need the energy density andthe pressure if we want to use equation (4.15). But these two parameters are not known exactlyfor a transition. So we will use some data [12]which has been calculated via perturbed thermalfield theory completed by fits to Lattice QCD by Mikko Laine and York Schröder. Thereforewe have a look at the effective degrees of freedom which were calculated in chapter 3 andto those calculated by Laine and Schröder. The data for the speed of sound which was also

*

** * ******* * * * *****

****************************************************************

*

*

*

*********************************************************************************************************************************************************************

*****************************************************************

******************

*************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************

0.1 10 1000 105

10

20

50

100

T/MeV

gϵ

particles of mass up to 9 × QCD transition

particles of mass up to 6 × QCD transition

particles of mass up to 12 × QCD transition

only pions

data from M. Laine

Figure 4: Effective degrees of freedom as a function of temperature. The red, blue, orange and blacklines are the same as in Fig. (3). The purple dots show the calculated d.o.f.s at a certaintemperature by Mikko Laine [12].

made by Laine and Schröder (compare Fig. 5) indicates that at temperatures around the QCDtransition the speed of sound squared drops noticeable below the value of 1/3.

15

0.1 10 1000 105

0.20

0.25

0.30

T/MeV

cs2

Figure 5: Speed of sound squared as a function of temperature taken from [12]. The data points takenfrom [12] are connected via a spline interpolation. We see that in the region of the QCDtransition temperature the speed of sound drops noticeable.

5. Number of e-foldings of the radiation epoch

Now that we have seen transitions in the universe do have an impact at the degrees of freedomand the speed of sound, we want to see how these transitions or especially the QCD transitionimpacts the number of e-foldings. Therefore we need the evolution of temperature in anexpanding volume. This evolution is described by the equation [11],

dTdt= −3c2

s HT. (5.1)

We can solve this differential equation with the definition for the number of e-foldings (N =lnaend/lna). This method leads to an expression for the e-foldings,

dN =1

3c2s

dlnT. (5.2)

For the non-interacting radiation dominated case N(T ) ∝ ln(T ). Taking interactions andparticle masses into account, case the speed of sound squared becomes a function of temper-ature. Because we are interested in the effect of mass thresholds and interactions we couldplot both cases and subtract these two. Instead we change the differential equation for thee-foldings (eq. (5.2)) by subtracting the case c2

s = 1/3 from the case c2s = c2

s (T ). This method

16

1 10 100 1000 104 105

0.0

0.2

0.4

0.6

0.8

T/MeV

ΔN(T)

Figure 6: Modification of the number of e-foldings during the radiation dominated epoch due to thedeviations of the speed of sound based on the data set from [12].

minimizes the numerical error and leads to an expression for the modification of the numberof e-foldings,

d∆N =

13c2

s (T )− 1

dlnT. (5.3)

This differential equation can be solved by integrating,

∆N(T ) =

T∫

Tβ

dlnT ′

13c2

s (T )− 1

. (5.4)

Computing now this integral by using the data points from [12]we obtain the∆N as a functionof temperature. Because the data contains not a continuous line but points we have to do theintegral numerically with trapezoidal numerical integration. This numerical integration is doneby a program written in C which can be find in the appendix B. We use Tbeta = 1 MeV becauseat this temperature the calculated c2

s is roughly 1/3 (compare Fig. 5). Interpolating this pointslead to a continuous function of the change of e-foldings as a function of temperature whichcan be seen in Fig. 6. We see from the plot of ∆N(T ) that the number of e-foldings changesby roughly 0.7 e-folds due to the QCD transition. Taking this change into account we cannow calculate the number of e-foldings as function of temperature before the reference point

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at 1 MeV,

N(T ) = N1 MeV + ln

T1 MeV

+∆N(T ). (5.5)

The N1 MeV is the number of e-foldings between the time when the universe had a temperatureof 1 MeV and today. This number of e-foldings is,

N1 MeV = ln

1 MeVTν,0

= ln

1 MeV

411

1/3Tγ,0

!

= 22.51, (5.6)

with Tν,0 is the neutrino temperature today and Tγ,0 is the measured CMB temperature todaywhich is T0 = 2.726±0.001K [11]. The ratio between the CMB and the neutrino temperature

is the factor of

411

13 [13]. With equations (5.5) and (5.6) we can calculate the number of e-

foldings from a given temperature up to today. For example, the number of e-foldings areN(155 MeV) = 28.01 at the QCD transition , N(200 GeV) = 35.42 at the electroweaktransition and N(1 TeV) = 37.06 which is the last data point from [12].

6. Conclusion

ln a = N

ln λphys.

a0

inflation RD MD ΛDNinf ΔN

1/H

Figure 7: Kinematics of scales. Similar to figure 1 except that the course during the RD epoch ischanged due to the impact of QCD transition. The ∆N is the offset between the to pointswhere the inflation epoch is reheating and proceeding into the radiation dominated epoch.

Let us now recall all the results of this thesis. The main idea was to see how interactions duringthe QCD transition in the evolution of the universe change the speed of sound and therefore

18

the number of e-foldings of the radiation dominated epoch. To calculate this modification weused a data set calculated with perturbed thermal field theory completed by fits to Lattice QCDby Mikko Laine and York Schröder [12]. The good fit of our results of the effective degreesof freedom to the data by Laine and Schröder indicates that the hadron resonance gas givesuseful results not only below but also around 155 MeV.

The result for the modification to the number of e-foldings due to the QCD transition isroughly 0.7 (compare figure 6). That means the radiation dominated epoch was a little bitlonger than expected in the non-interacting case which is sketched in figure 7. Therefore theduration of inflation was shorter by the same amount of e-foldings. This shorter durationhas a direct impact on the various inflation models because if we compare the modificationof e-foldings with the number of e-foldings predicted by a certain model the cosmologicalparameters AS and nS change. These two parameters depend both on the number of e-foldingsand are measured. The relation between the density perturbation spectral index is given inequation (2.25). The measured values for the parameters AS and nS are [4],

ln(1010AS) = 3.089± 0.036, (6.1)nS = 0.9655± 0.0062. (6.2)

Taking the modification to the number of e-foldings into account the parameter nS is modifiedby 0.0005. This value lies within the error of the measurement and is therefore not resolv-able. But as the measure methods are improving with time the error becomes smaller and byconsidering not only the QCD transition it is possible that the value of the modification of thenumber of e-foldings due to the transitions becomes resolvable. If the modification becomesmeasurable we have a criterion to prove different inflation models to be accurate.

19

References

[1] S. Weinberg, Cosmology (Oxford Univ. Press, Oxford, 2008), pp. 2-5[2] D. S. Gorbunov, V. A. Rubakov, Hot Big Bang Theory (World Scientific Publishing, Sin-gapore, 2011), p. 47[3] D. S. Gorbunov, V. A. Rubakov, Hot Big Bang Theory (World Scientific Publishing, Sin-gapore, 2011), pp. 61-63[4] J. Aumont, G. P. Efstathiou, A. Lewis, A. Renzi, D. Scott, et al. (Planck collaboration),Planck 2015 results. XIII. Cosmological parameters, (2016), A& A 594, A13[5] S. Weinberg, Cosmology (Oxford Univ. Press, Oxford, 2008), pp. 149-151[6] Gel’fan, I. M.; Shilov, G. E., Generalized functions 1-5, Academic Press, Vol. 1 §II.2.5,1966-1968[7] D. H. Rischke, Fluid dynamics for relativistic nuclear collisions (1998) [nucl-th/9809044][8]K. A. Olive, et al. (Particle Data Group), Particle Data Booklet Chin. Phys. C, 38, 090001,2014.[9] K. Kajantie, M. Laine, K. Rummukainen, Y. Schröder, The pressure of hot QCD up to gˆ6ln(1/g), Phys. Rev. D 67 (2003) 105008 [hep-ph/0211321][10] L. D. Landau, E. M. Lifschitz, Statistische Physik/[1], Lehrbuch der theoretischen Physik(1975), p. 67[11] S. Weinberg, Entropy Generation and the Survival of Protogalaxies in an ExpandingUniverse, Astrophysical Journal, vol. 168, pp.175-194,1971[12] M. Laine, Y. Schröder, Quark mass thresholds in QCD thermodynamics, Phys.Rev. D73:085009, 2006[13] D. J. Fixsen, The temperature of the Cosmic Microwave Background, Astrophys. J. 707,916 (2009)[14] S. Weinberg, Cosmology (Oxford Univ. Press, Oxford, 2008), p. 154

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A. Notation

Latin indices i, j, k, etc. on gerneally run over the three spatial coordinate labels.

Greek indices µ,ν, etc. generally run over the four space-time coordinate labels 0, 1, 2, 3with x0 as the time coordinate.

Einstein summation convention is used, repeated inidces are generally summed.

The flat space-time metric ηµν is diagonal with η00 = −1 and η11 = η22 = η33 = 1.

A dot over any quantity denotes the time-derivative of that quantity respect to the general time.

A prime at any quantity denotes the time-derivative of that quantity with respect to its propertime.

21

B. Program code for numerical integration

# i n c l u d e < s t d i o . h># i n c l u d e <math . h># i n c l u d e < s t d l i b . h>

c o n s t i n t nmax = 1158 ;

i n t main ( vo id ) do ub l e x [ nmax ] ;do ub l e y [ nmax ] ;c h a r o u t p u t F i l e n a m e [ 1 1 5 8 ] ;c h a r i n p u t F i l e n a m e [ 1 1 5 8 ] ;

p r i n t f ( " I n p u t f i l e n a m e ? : " ) ;s c a n f ("% s " , i n p u t F i l e n a m e ) ;p r i n t f ("(% s ) \ n " , i n p u t F i l e n a m e ) ;

p r i n t f ( " Outpu t f i l e n a m e ? : " ) ;s c a n f ("% s " , o u t p u t F i l e n a m e ) ;p r i n t f ("(% s ) \ n " , o u t p u t F i l e n a m e ) ;

FILE ∗ fp ;FILE ∗ o u t p u t F i l e ;

fp = fopen ( i n p u t F i l e n a m e , " r " ) ;

i f ( fp == NULL) p r i n t f ( " D a t e i ko nn t e n i c h t g e o e f f n e t werden . \ n " ) ;

e l s e p r i n t f ( " D a t e i ko nn t e g e o e f f n e t werden . \ n " ) ;

i n t i = 0 ;do ub l e a , b ;

do i f ( i > nmax )

p r i n t f ( " r e a d i n e r r o r : Number o f i n p u t l i n e s >= %d ! \ n " , nmax ) ;e x i t ( 2 ) ;

i f ( f s c a n f ( fp , "% l f %l f " , &a , &b ) == 2) x [ i ]= a ; y [ i ]= b ; i ++;

22

w h i l e ( ! f e o f ( fp ) ) ;

o u t p u t F i l e = fopen ( o u t p u t F i l e n a m e , "w " ) ;

i n t j ;do ub l e A=0;f o r ( j =0 ; j <=1157; j ++)

A =A + 0 . 5 ∗ ( x [ j +1]−x [ j ] ) ∗ ( y [ j ]+ y [ j + 1 ] ) ; / / t r a p e z e i n t e g r a t i o np r i n t f ( " Zahl %.10E \ n " , A ) ;

f p r i n t f ( o u t p u t F i l e , "%.10E %.10E \ n " , x [ j ] , A ) ;

p r i n t f ( " I n t e g r a t i o n a b g e s c h l o s s e n ! \ n " ) ;

f c l o s e ( o u t p u t F i l e ) ;

f c l o s e ( fp ) ;

23

Eigenständigkeitserklärung

Hiermit bestätige ich, dass ich die vorliegende Arbeit selbständig verfasst und keine anderenals die angegebenen Hilfsmittel benutzt habe. Die Stellen der Arbeit, die dem Wortlaut oderdem Sinn nach anderen Werken (dazu zählen auch Internetquellen) entnommen sind, wurdenunter Angabe der Quelle kenntlich gemacht.

Datum, Ort Unterschrift

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