The Early Universe Before First Light

Abstract

In the present chapter we explore the Universe at times before photon decoupling which corresponds to ages smaller than ${\simeq }3\times 10^5\,$y after the big bang, when the Universe was smaller than today by a factor ${\simeq }1100$ or more. During that epoch most species first were in thermodynamic equilibrium and subsequently fell out of equilibrium when their interaction rates became smaller than th expansion rate. Thus we start with a recapitulation of equilibrium thermodynamics. The next three sections then discuss the three most important events in which species decoupled from the rest of the plasma: The formation of the cosmic microwave background which marks photon decoupling, the freeze-out of weakly interacting neutrinos and dark matter, and big bang nucleosynthesis which determines the primordial nuclear abundances. We then describe how the expansion of the Universe can also lead to a change of the equilibrium state, typically in phase transitions which in turn are often associated with the breaking of symmetries of interactions. Such phase transitions can leave primordial relics such as magnetic fields and small excesses of lepton and baryon number as we see them today which is developed next. The following two sections concern the earliest phase of the Universe before an equilibrium state was even established. There we focus on the most widely discussed scenario, namely cosmic inflation in which an epoch of quasi-exponential expansion is followed by a reheating phase in which the first hot equilibrium state would be created. Finally, we close with some more speculative ideas on connections between cosmology, thermodynamics, primordial initial conditions and the arrow of time .

Problems

4.1

Redshifted Thermal Distributions

Derive the energy spectrum of fermions and bosons that results when a thermal spectrum at temperature T is redshifted by a factor $1+z$. Show that in case of massless particles the new energy spectrum is again a thermal distribution. Which temperature does the redshifted energy spectrum correspond to?

4.2

Speed of Sound in Thermal Distributions

Show that the speed of sound of a relativistic ideal fluid is given by $c_s=1/\sqrt{3}$. What is the speed of sound in the non-relativistic limit?

4.3

Electron-Positron Annihilation

(a) Consider the annihilation of electron-positron pairs at a temperature $T\sim 1\,$MeV: Determine the effective number of relativistic degrees of freedom $g_r$ from Eq. (4.21) of the electromagnetic plasma before and after annihilation. By which factor does the photon temperature increase after the end of pair annihilation if annihilation is approximated as instantaneous and the entropy density $s\propto g_rT^3$ is roughly conserved? Since neutrinos are already decoupled when the pairs annihilate, this factor is also equal to the ratio between the temperatures of the neutrinos and the photons. Which temperature $T_\nu$ should the cosmological relic neutrino background , therefore, have today?

(b) Using the result above and Eqs. (4.21) and (4.22) show that for $T\lesssim 1\,$MeV and as long as the neutrinos are still relativistic the total relativistic energy density is given by $\varOmega _r\simeq 1.68\varOmega _\mathrm{CMB}$. Compute the effective number of degrees of freedom $g_r$ at that epoch.

4.4

Thermodynamics of Non-Relativistic Particles

(a) Derive Eq. (4.6),

$$\begin{aligned} n_\mathrm{eq}(T)\simeq & {} \frac{gm^2T}{2\pi ^2}\,K_2\left( \frac{m}{T}\right) \,, \nonumber \\ \rho _\mathrm{eq}(T)\simeq & {} -T^2\frac{\partial n_\mathrm{eq}(T)}{\partial T}\,,\nonumber \end{aligned}$$

for the number and energy densities of a non-relativistic particle species from the definition Eq. (4.7) of the modified Bessel functions of the second kind.

(b) Derive the non-relativistic limits Eq. (4.9) by taking the limes of the modified Bessel function of the second kind, Eq. (4.7), for $x\gg 1$.

4.5

Ionization Fraction and Saha Equation

(a) Derive Eq. (4.30) from (4.29) using the general expression for the non-relativistic densities at a given temperature T. Equation (4.30) is known as the Saha equation.

(b) Derive Eq. (4.31) for the ionization fraction in thermal equilibrium from Eq. (4.29).

4.6

Approximate Cold Dark Matter Abundance for Thermal Freeze-Out

Derive the scaling between the cold dark matter relic abundance and the annihilation cross section in Eq. (4.55). More concretely, show that

$$\begin{aligned} \varOmega _Xh^2\sim \frac{64\sqrt{5}\pi ^{5/2}}{3}\frac{g_f^{1/2}}{H_0^2\langle \sigma _{X{\bar{X}}}v\rangle } \left( \frac{T_0}{M_{\mathrm{Pl}}}\right) ^3\,, \end{aligned}$$

(4.334)

where $g_f\equiv g_r(T_f)$ is the effective number of relativistic degrees of freedom at dark matter freeze-out at $T_f\simeq m_X/20$.

4.7

Boltzmann Equation and Dark Matter Freeze-Out

Derive Eqs. (4.60) and (4.63) from the Boltzmann equation Eq. (4.54) by using the expressions for the entropy density and Hubble rate .

4.8

Effective Potentials for Phase Transitions

Fill in the details of the calculations with the effective potential given by Eq. (4.85). Hint: When calculating quantities at the extrema $\phi _{1,2}$, it can be useful to eliminate quadratic terms by using

$$\lambda (T)\phi ^2_{1,2}=3ET\phi _{1,2}+2D(T_0^2-T^2)\,.$$

4.9

Some Properties of Inflation and the Inflaton

(a) Using the general definition Eq. (2.322) show that the energy-momentum tensor for a complex scalar field in a general curved space–time is given by Eq. (4.246). Hints: Use the fact that the Lagrange density Eq. (2.328) does not depend on derivatives of the metric and the relation $\delta g^{1/2}=-g^{1/2}g_{\mu \nu }\delta g^{\mu \nu }/2$ which follows from Eq. (2.177). Show that the definition Eq. (2.72) would give the same result. Verify that energy density and pressure of a real scalar field in a potential $V(\phi )$ are given by Eq. (4.247).

(b) Show that the slow roll parameter $\epsilon _V$ defined in Eq. (4.252) can also be expressed as

$$\begin{aligned} \epsilon _V=-\frac{d\ln H}{d\ln a}=-\frac{1}{H}\frac{dH}{dN}=-\frac{\dot{H}}{H^2}\,, \end{aligned}$$

(4.335)

where the number of e-folds N has been defined in Eq. (4.249).

(c) Derive the second expression in Eq. (4.259) by using Eq. (4.252).

(d) Show that in the slow roll approximation the slow roll parameters can be approximated by Eq. (4.261).

4.10

Scalar Field Induced Fluctuations

(a) Show that the canonical quantization condition Eq. (2.78) applied to the scalar field expansion Eq. (4.273) into comoving momentum modes and the canonically conjugated momentum Eq. (4.276) leads to the normalization condition for the eigen-mode functions $u_\mathbf{k}$ given by Eq. (4.277) if one assumes the standard commutation relations Eq. (1.62) for the annihilation and creation operators $a_\mathbf{k}$ and $a^\dagger _\mathbf{k}$.

(b) Using the scalar field expansion Eq. (4.273) and the standard commutation relations Eq. (1.62) show that the fluctuations $\langle \phi ^2\rangle$ can be written in the form Eq. (4.283).

(c) Show that in the slow roll approximation the scale dependence of the power spectrum of the tensor perturbations is given by $P_T(k)\propto k^{n_T}$ with $n_T\simeq -2\epsilon _V$ for $\epsilon _V\ll 1$, see Eq. (4.287). Hint: Use Eq. (4.335) in (4.286).

(d) Derive Eq. (4.288) for the intrinsic spatial curvature of the three-dimensional spatial metric $ds^2=a^2(t)\left[ 1-2\psi (\mathbf{r})\right] d\mathbf{r}^2=[g_{ij}+a^2(t)h_{ij}(\mathbf{r})]dx^idx^j$ by using Eq. (2.336) with $\Box \equiv g^{ij}\partial _i\partial _j=\varDelta /a^2(t)$ to compute $^{(3)}R\equiv R_i^i$.

4.11

de Sitter space as a Closed Friedmann Universe

(a) Show that the de Sitter hyperboloid Eq. (4.253) can be parametrized as a closed Friedmann Universe as

$$\begin{aligned} z_0= & {} H^{-1}\sinh Ht\,,\quad z_4=H^{-1}\cosh Ht\,,\nonumber \\ z_1= & {} H^{-1}\cosh Ht\cos \chi \sin \theta \cos \phi \,,\quad z_2=H^{-1}\cosh Ht\cos \chi \sin \theta \sin \phi \,,\nonumber \\ z_3= & {} H^{-1}\cosh Ht\cos \chi \cos \theta \,,\\ -\infty< & {} t<\infty \,,\quad 0\le \chi \le 2\pi \,,\quad 0\le \theta \le \pi \,,\quad 0\le \phi \le 2\pi \,,\nonumber \end{aligned}$$

(4.336)

which induces the metric

$$\begin{aligned} ds^2=dt^2-H^{-2}\cosh ^2Ht\left[ d\chi ^2+sin^2\chi (d\theta ^2+\sin ^2\theta d\phi ^2)\right] \,\, . \end{aligned}$$

(4.337)

Note that for $t<0$ this describes a contracting phase.

(b) Show that the parametrization Eq. (4.336) covers the whole de Sitter hyperboloid which is, therefore, said to be geodesically complete . This is in contrast to the parametrizations Eqs. (4.254) and (4.256) which only cover half of de Sitter space.

4.12

First Order Differential Equation for Inflaton Field Evolution

(a) Show that the ansatz

$$\begin{aligned} \dot{\phi }=\frac{\partial W}{\partial \phi }\,,\quad H=-4\pi G_\mathrm{N}W \end{aligned}$$

(4.338)

solves the equation of motion Eq. (4.244) for a real homogeneous inflaton exactly, provided that the function $W(\phi )$ satisfies

$$\begin{aligned} V(\phi )=6\pi G_\mathrm{N}W^2-\frac{1}{2}\left( \frac{\partial W}{\partial \phi }\right) ^2\, \end{aligned}$$

(4.339)

in terms of the inflaton potential $V(\phi )$.

(b) Show that this allows to write the equation of motion for $\phi$ as

$$\begin{aligned} \frac{d\phi }{dN}=-\frac{1}{4\pi G_\mathrm{N}W}\frac{\partial W}{\partial \phi }= \pm \left( \frac{\epsilon _V}{4\pi G_\mathrm{N}}\right) ^{1/2}\equiv \beta (\phi )\,. \end{aligned}$$

(4.340)

This form is equivalent to the renormalization group equations for the coupling constants of gauge fields , see Eq. (2.357) in Sect. 2.10.1. Fixed points are given by $\beta (\phi )=0$ and slow roll inflation corresponds to $\beta (\phi )\ll M_\mathrm{Pl}$. Note that a fixed point corresponds to the de Sitter limit .

4.13

Preheating

(a) Show that Eq. (4.319) is an approximate solution of the equation of motion of the inflaton field Eq. (4.244) for the potential Eq. (4.318) provided that $H\ll m$ and $g^2\chi ^2\ll Hm$. What is the corresponding condition on the values of the inflaton field $\phi$?