LOG#108. Basic Cosmology (III).

Universe_History2

The current Universe has evolved since its early phase of thermal equilibrium until the present state. The departure from thermal equilibrium in the early Universe made a fossil record we can observe at current time!

There are some easy rules for thermal equilibrium. The easiest one, is that coming from the “interaction rate” \Gamma_{int}. It can be expressed in the following way:

\Gamma_{int}>\mbox{Expansion rate H}

and then, at a given temperature T, we get

\Gamma_{int}(T)=n(T)\langle \sigma \vert v \vert \sigma \rangle^T

and where H\approx \dfrac{T^2}{M_p}

Remark: If \Gamma_{int}=aT^n \forall n>2, then

N_{int}=\int_t^\infty T_{int}(t')dt'=\dfrac{\Gamma (H)\vert_t}{n-2}<1

and it implies that a particle interacts less than once after the time \Gamma =H.

Moreover, we can understand roughly the so-called decoupling era:

1st. Any interaction mediated by a massless gauge boson provides

\sigma\sim \dfrac{\alpha^2_X}{s} with s\sim E^2 and \alpha_X=\dfrac{g^2_X}{4\pi}

and this implies that

\Gamma \sim n\langle \sigma v\rangle\sim T^3\dfrac{\alpha_X^2}{T^2}=\alpha^2_XT

and

\dfrac{\Gamma}{H}\sim \alpha_X^2\dfrac{M_p}{T}

so the equilibrium temperature is found whenever T\leq \alpha_X^2M_p!

2nd. Interactions mediated by any massive gauge boson provides

\sigma\sim G_X^2s with G_X\sim \dfrac{\alpha_X}{m_X^2}

and this implies that

\Gamma \sim T^3G_X^2T^2=G_X^2T^5

and

\dfrac{\Gamma}{H}\sim G_X^2M_pT^3

and then

\left( G_X^2M_p\right)^{-1/3}\leq T\leq m_X\longrightarrow \mbox{Equilibrium temperature (E.T.)}

Moreover,

T\leq \left( G_X^2M_p\right)^{-1/3}\sim \left(\dfrac{m_X}{100\mbox{GeV}}\right)^{4/3}\mbox{MeV}\longrightarrow \mbox{Freeze out}

As a consequence, we can realize that the out-of-equilibrium phenomena in the early and current Universe are very important processes! In particular:

1) They provide the formation of light elements during the Big Bang Nucleosynthesis (BBN), also known as primordial nucleosynthesis, i.e., the formation of the first light elements after the Big Bang (circa 300000 years after the Universe “birth”).

2) They provide the path of recombination of electrons and protons into hydrogen atoms.

3) They imply the  (likely) production of dark matter (or equivalently the presence of some kind of “modified gravity” or/and modified newtonian dynamics).

Boltzmann’s equation for annihilation of particles in equilibrium

There is a beautiful equation that condenses the previous physical process of equilibrium at a given temperature and the particle production it yields. Conceptually speaking, we have

\begin{pmatrix}\mbox{Boltzmann}\\ \mbox{Equation}\end{pmatrix}:

\begin{pmatrix}\mbox{Rate of change}\\ \mbox{in the abundance}\end{pmatrix}=\begin{pmatrix}\mbox{Rate of}\\ \mbox{particle production}\end{pmatrix}-\begin{pmatrix}\mbox{Rate of}\\ \mbox{particle erasing/annihilation}\end{pmatrix}

Consider a process like

\mbox{particle type 1}+\mbox{particle type 2}\leftrightarrow \mbox{particle type 3}+\mbox{particle type 4}

and where the particle 1 is the one we are interested in. Then, we deduce that

\underbrace{\dfrac{1}{a^3}\dfrac{d(n_1a^3)}{dt}}_\text{change in comoving volume}=\underbrace{\int\dfrac{d^3p_1}{(2\pi)^32E_1}\int\dfrac{d^3p_2}{(2\pi)^32E_2}\int\dfrac{d^3p_3}{(2\pi)^32E_3}\int\dfrac{d^3p_4}{(2\pi)^32E_4}}_\text{phase space invariant}\times A

where A is certain complicated facter involving “delta functions” of the energies and momenta of the particles 1,2,3,4 and an additional term depending on the statistics of the particle. Explicitly, it takes the form

A=\left[(2\pi)^4\delta^3(p_1+p_2-p_3-p_4)\delta (E_1+E_2-E_3-E_4)\vert M\vert^2\right]\times S

with S=\left[ f_3f_4(1\pm f_1)(1\pm f_2)-f_1f_2(1\pm f_3)(1\pm f_4)\right]

and where

f_i=\dfrac{1}{e^{(E_i-\mu_i (t))/T}\pm 1}

is the Fermi-Dirac (FD, -)/Bose-Einstein (BE,+) distribution. In fact, the above FD/BE factors provide the so-called Pauli blocking/”Bose-Einstein” enhancement effects for the particle production in the processes 3+4\rightarrow 1+2 and 1+2\rightarrow 3+4. Indeed, particle physics enter into the game here (see above formulae again) and we assume

M(1+2\rightarrow 3+4)=M(3+4\rightarrow 1+2)

Do you recognize the principle of detailed balance in this equation?

We can simplify the assumptions a little bit:

1st. The kinetic equilibrium is taken to be a rapid elastic scattering and we input the FD/BE statistics without loss of generality.

2nd. The annihilation in thermal equilibrium will be calculated from the sum of the chemical potential in any balanced equation.

3rd. Low temperature approximation. Suppose that

T<< (E-\mu)

then we obtain the Maxwell-Boltzmann approximation to the FD/BE statistics

f\approx e^{-(E-\mu)/T} and 1+f\approx 1, so, since E_1+E_2=E_3+E_4, we get

f_3f_4(1\pm f_1)(1\pm f_2)-f_1f_2(1\pm f_3)(1\pm f_4)\approx e^{-(E_1+E_2)/T}\left( e^{\frac{(\mu_3+\mu_4)}{T}}-e^{\frac{(\mu_1+\mu_2)}{T}}\right)-\star

What is \star? After a change of variable

\mu_i (t)\longrightarrow n_i(t)=g_ie^{\mu_i/T}\int \dfrac{d^3p}{(2\pi)^3}e^{-E_i/T}

\star is the “equilibrium number density” deducen from the expression

n_i^{0}\equiv g_i\int {d^3p}{(2\pi)^3}e^{-E_i/T}=\begin{cases}g_i\left(\dfrac{m_iT}{2\pi}\right)^{3/2}e^{-m_i/T},\;\; \mbox{if}\;\; m_i>>T\\ g_i\dfrac{T^3}{\pi^2},\;\;\mbox{if}\;\; m_i<<T\end{cases}

and it yields

n_i=e^{\mu_i/T}n_\gamma^{0}

and then we finally get that

\star equals e^{-(E_1+E_2)/T}\left[\dfrac{n_3n_4}{n_3^0n_4^0}-\dfrac{n_1n_2}{n_1^0n_2^0}\right]

Now, we can define the thermally averaged cross section

\langle \sigma v\rangle\equiv \dfrac{1}{n_1^0n_2^0}\int \dfrac{d^3p_1}{(2\pi)^32E_1}\cdots \dfrac{d^3p_4}{(2\pi)^32E_4}e^{-(E_1+E_2)/T}(2\pi)^4\delta^3 (p_1+p_2-p_3-p_4)\times

\times \delta (E_1+E_2-E_3-E_4)\vert M\vert^2

The Boltzmann equation becomes with these conventions

\dfrac{1}{a^3}\dfrac{d(n_1a^3)}{dt}=n_1^0n_2^0\langle \sigma v\rangle \left(\dfrac{n_3n_4}{n_3^0n_4^0}-\dfrac{n_1n_2}{n_1^0n_2^0}\right)

Remark (I): LHS is similar to \dfrac{n_1}{t}\sim n_1H and the RHS is similar to n_1n_1\langle \sigma v\rangle

Remark (II): If the reaction rate is n_1\langle \sigma v\rangle >> H, then it provides the chemical equilibrium condition well known in the nuclear statistical equilibrium as the Saha equation, i.e.,

\dfrac{n_3n_4}{n_3^0n_4^0}-\dfrac{n_1n_2}{n_1^0n_2^0}\approx 0

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LOG#106. Basic Cosmology (I).

cosmologyLettersCMB_Timeline300_no_WMAP

The next thread is devoted to Cosmology. I will intend to be clear and simple about equations and principles of current Cosmology with a General Relativity background.

First of all…I will review the basic concepts of natural units I am going to use here. We will be using the following natural units:

\hbar=c=k_B=1

We will take the Planck mass to be given by

M_P=\sqrt{8\pi G_N}\approx 1\mbox{.}2\cdot 10^{19}GeV

The solar mass is M_\odot=2\cdot 10^{30}kg and the parsec is given by the value

1pc=3\mbox{.}26lyr=3\mbox{.}1\cdot 10^{16}m

Well, current Cosmology is based on General Relativity. Even if I have not reviewed this theory with detail in this blog, the nice thing is that most of Cosmology can be learned with only a very little knowledge of this fenomenal theory. The most important ideas are: metric field, geodesics, Einstein equations and no much more…

In fact, newtonian gravity is a good approximation in some particular cases! And we do know that even in this pre-relativistic theory

\mbox{Gravitational force}=\mbox{Matter/Mass density}

via the Poisson’s equation

\nabla^2\phi =4\pi G_N\rho

This idea, due to the equivalence principle, is generalized a little bit in the general relativistic framework

\mbox{Spacetime geometry}=\mbox{Matter content/Energy-momentum}

The spacetime geometry is determined by the metric tensor g_{\mu\nu}(x). The matter content is given by the stress-energy-momentum tensor T_{\mu\nu}. As we know one of these two elements, we can know, via Eisntein’s field equations the another. That is, given a metric tensor, we can tell how energy-momentum “moves” in space-time. Given the energy-momentum tensor, we can know what is the metric tensor in spacetime and we can guess how the spacetime bends… This is the origin of the famous motto: “Spacetime says matter how to move, energy-momentum says spacetime how to curve”! Remember that we have “deduced” the Einstein’s field equations in the previous post. Without a cosmological constant term, we get

G_{\mu\nu}=\kappa^2T_{\mu\nu}=8\pi G_NT_{\mu\nu}

Given a spacetime metric g_{\mu\nu}, we can calculate the (affine/Levi-Civita) connection

\Gamma^\sigma_{\;\;\mu\nu}=\dfrac{1}{2}g^{\sigma\rho}\left(\partial_\mu g_{\nu\rho}+\partial_\nu g_{\rho\mu}-\partial_\rho g_{\mu\nu}\right)

The Riemann tensor that measures the spacetime curvature is provided by the equation

R^\rho_{\;\; \sigma\mu\nu}=\partial_\mu \Gamma^\rho_{\;\;\mu\sigma}-\partial_\mu \Gamma^\rho_{\;\; \mu \sigma}+\Gamma^\rho_{\;\;\mu\lambda}\Gamma^\lambda_{\;\;\nu\sigma}-\Gamma^\rho_{\;\;\nu\lambda}\Gamma^\lambda_{\;\;\mu\sigma}

The Ricci tensor is defined to be the following “trace” of the Riemann tensor

R_{\mu\nu}=R^\lambda_{\;\;\mu\lambda \nu}

The Einstein tensor is related to the above tensors in the well-known manner

G_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}Rg_{\mu\nu}

The Einstein’s equations can be derived from the Einstein-Hilbert action we learned in the previous post, using the action principle and the integral

\boxed{S_{EH}=\int d^4x \sqrt{-g} \left(\kappa^{-2}R+\mathcal{L}_M\right)}

The geodesic equation is the path of a freely falling particle. It gives a “condensation” of the Einstein’s equivalence principle too and it is also a generalization of Newton’s law of “no force”. That is, the geodesic equation is the feynmanity

\boxed{\dfrac{d^2 x^\mu}{d\tau^2}+\Gamma^\mu _{\rho\sigma}\dfrac{dx^\rho}{d\tau}\dfrac{dx^\sigma}{d\tau}=0}

Finally, an important concept in General Relativity is that of isometry. The symmetry of the “spacetime manifold” is provided by a Killing vector that preserves transformations (isometries) of that manifold. Mathematically speaking, the Killing vector fields satisfy certain equation called the Killing equation

\boxed{\xi_{\mu ; \nu}+\xi_{\nu ; \mu}=0}

Maximally symmetric spaces have n(n+1)/2 Killing vectors in n-dimensional (nD) spacetime. There are 3 main classes or types of 2D maximally symmetric that can be generalized to higher dimensions:

1. The euclidean plane E^2.

2. The pseudo-sphere H^2. This is a certain “hyperbolic” space.

3. The spehre S^2. This is a certain “elliptic” space.

The Friedmann-Robertson-Walker Cosmology

Current cosmological models are based in General Relativity AND  a simplification of the possible metrics due to the so-called Copernican (or cosmological) principle: the Universe is pretty much the same “everywhere” you are in the whole Universe! Remarkbly, the old “perfect” Copernican (cosmological) principle that states that the Universe is the same “everywhere” and “every time” is wrong. Phenomenologically, we have found that the Universe has evolved and it evolves, so the Universe was “different” when it was “young”. Therefore, the perfect cosmological principle is flawed. In fact, this experimental fact allows us to neglect some old theories like the “stationary state” and many other “crazy theories”.

What are the observational facts to keep the Copernican principle? It seems that:

1st. The distribution of matter (mainly galaxies, clusters,…) and radiation (the cosmic microwave background/CMB) in the observable Universe is homogenous and isotropic.

2nd. The Universe is NOT static. From Hubble’s pioneer works/observations, we do know that galaxies are receeding from us!

Therefore, these observations imply that our “local” Hubble volume during the Hubble time is similar to some spacetime with homogenous and isotropic spatial sections, i.e., it is a spacetime manifold M=\mathbb{R}\times \Sigma. Here, \mathbb{R} denotes the time “slice” and \Sigma represents a 3D maximally symmetric space.

The geometry of a locally isotropic and homogeneous Universe is represented by the so-called Friedmann-Robertson-Walker metric

\boxed{ds^2_{FRW}=-dt^2+a^2(t)\left[\dfrac{dr^2}{1-kr^2}+r^2\left(d\theta^2+\sin\theta^2d\phi^2\right)\right]}

Here, a(t) is the called the scale factor.  The parameter k determines the geometry type (plane, hyperbolic or elliptical/spherical):

1) If k=0, then the Universe is “flat”. The manifold is E^3.

2) If k=-1, then the Universe is “open”/hyperbolic. The manifold would be H^3.

3) If k=+1, then the Universe is “closed”/spherical or elliptical. The manifold is then S^3.

Remark: The ansatz of local homogeneity and istoropy only implies that the spatial metric is locally one of the above three spaces, i.e., E^3,H^3,S^3. It could be possible that these 3 spaces had different global (likely topological) properties beyond these two properties.

Kinematical features of a FRW Universe

The first property we are interested in Cosmology/Astrophysics is “distance”. Measuring distance in a expanding Universe like a FRW metric is “tricky”. There are several notions of “useful” distances. They can be measured by different methods/approaches and they provide something called sometimes “the cosmologidal distance ladder”:

1st. Comoving distance. It is a measure in which the distance is “taken” by a fixed coordinate.

2nd. Physical distance. It is essentially the comoving distance times the scale factor.

3rd. Luminosity distance. It uses the light emitted by some object to calculate its distance (provided the speed of light is taken constant, i.e., special relativity holds and we have a constant speed of light)

4th. Angular diameter distance. Another measure of distance using the notion of parallax and the “extension” of the physical object we measure somehow.

There is an important (complementary) idea in FRW Cosmology: the particle horizon. Consider a light-like particle with ds^2=0. Then,

dt=a(t)\dfrac{1}{\sqrt{1-kr^2}}

or

\dfrac{dr}{\sqrt{1-kr^2}}=\dfrac{dt}{a(t)}

The total comoving distance that light have traveled since a time t=0 is equal to

\boxed{\eta=\int_0^{r_H}\dfrac{dr}{\sqrt{1-kr^2}}=\int_0^t\dfrac{dt'}{a(t')}}

It shows that NO information could have propagated further and thus, there is a “comoving horizon” with every light-like particle! Here, this time is generally used as a “conformal time” as a convenient tiem variable for the particle. The physical distance to the particle horizon can be calculated

\boxed{d_H(t)=\int_0^{r_H}\sqrt{g_{rr}}dr=a(t)\int_0^t\dfrac{dt'}{a(t')}=a(t)\eta}

There are some important kinematical equations to be known

A) For the geodesic equation, the free falling particle, we have

\Gamma^0_{ij}=\dfrac{\dot{a}}{a}\overline{g}_{ij}

\Gamma^i_{0j}=\Gamma^i_{j0}=\dfrac{\dot{a}}{a}\delta_{ij}

\Gamma^i_{jk}=\overline{\Gamma}^i_{jk}

for the FRW metric and, moreover, the energy-momentum vector P^\mu=(E,\mathbf{p}) is defined by the usual invariant equation

P^\mu=\dfrac{dx^\mu}{d\lambda}

This definition defines, in fact, the proper “time” \lambda implicitely, since

\dfrac{d}{d\lambda}=\dfrac{dx^0}{d\lambda}\dfrac{d}{dx^0}=E\dfrac{d}{dt}

and the 0th component of the geodesic equation becomes

E\dfrac{dE}{dt}=-\Gamma^0_{ij}p^ip^j=-\delta_{ij}a\dot{a}p^ip^j

g_{\mu\nu}p^\mu p^\nu=-E^2+a^2\delta_{ij}p^ip^j=-m^2

EdE=a^2\vert \mathbf{p}\vert d\vert \mathbf{p}\vert

a^2 p\dfrac{dp}{dt}=-a\dot{a} p^2

\dfrac{1}{\vert \mathbf{p}\vert }\dfrac{d\vert \mathbf{p}\vert}{dt}+\dfrac{\dot{a}}{a}=0

Therefore we have deduced that \vert \mathbf{p}\vert \propto a^{-1}. This is, in fact, the socalled “redshift”.  The cosmological  redshift parameter is more generally defined through the equation

\boxed{\dfrac{a(t_0)}{a(t)}=1+z=\dfrac{\lambda_0}{\lambda}}

B) The Hubble’s law.

The luminosity distance measures the flux of light from a distant object of known luminosity (if it is not expanding). The flux and luminosity distance are bound into a single equation

\boxed{F=\dfrac{L}{4\pi d^2_L}}

If we use the comoving distance between a distant emitter and us, we get

\chi (a)=\int_t^{t_0}\dfrac{dt'}{a(t')}=\int_a^1\dfrac{da'}{a'^2 H(a')}

for a expanding Universe! That is, we have used the fact that luminosity itself goes through a comoving spherical shell of radius \chi (a). Moreover, it shows that

F=\dfrac{L (\chi)}{4 \pi \chi (a)^2 a_0^2}=\dfrac{L}{4\pi (\chi (a)/a)^2}

The luminosity distance in the expanding shell is

d_L=\dfrac{\chi (a)}{a}=\left(\dfrac{L}{4\pi F}\right)^{1/2}

and this is what we MEASURE in Astrophysics/Cosmology. Knowing a(t), we can express the luminosity distance in terms of the redshift. Taylor expansion provides something like this:

H_0d_L=z+\dfrac{1}{2}(1-q_0)z^2+\ldots

where higher order terms are sometimes referred as “statefinder parameters/variables”. In particular, we have

\boxed{H_0=\dfrac{\dot{a}_0}{a_0}}

and

\boxed{q_0=-\dfrac{a_0\ddot{a}_0}{\dot{a}_0^2}}

C) Angular diameter distance.

If we know that some object has a known length l, and it gives some angular “aperture” or separation \theta, the angular diameter distance is given by

\boxed{d_A=\dfrac{l}{\theta}}

The comoving size is defined as l/a, and the coming distance is again \chi (a). For “flat” space, we obtain that

\theta=\dfrac{l/a}{\chi (a)}

that is

d_A=a\chi (a)=\dfrac{\chi}{1+z}

In the case of “curved” spaces, we get

d_A=\dfrac{a}{H_0\sqrt{\vert \omega_k\vert}}\cdot\begin{cases}\sinh \left( \sqrt{\Omega_k}H_0\chi\right),\Omega_k>0\\ \sin \left( \sqrt{-\Omega_k}H_0\chi\right),\Omega_k<0\end{cases}

FRW dynamics

Gravity in General Relativity, a misnomer for the (locally) relativistic theory of gravitation, is described by a metric field, i.e., by a second range tensor (covariant tensor if we are purist with the nature of components). The metric field is related to the matter-energy-momentum content through the Einstein’s equations

G_{\mu\nu}=-\kappa^2 T_{\mu\ nu}

The left-handed side can be calculated for a FRW Universe as follows

R_{00}=-3\dfrac{\ddot{a}}{a}

R_{ij}=(a\ddot{a}+2\dot{a}^2+2k)\overline{g}_{ij}

R=6\left(\dfrac{\ddot{a}}{a}+\dfrac{\dot{a}^2}{a^2}+\dfrac{k}{a^2}\right)

The right-handed side is the energy-momentum of the Universe. In order to be fully consistent with the symmetries of the metric, the energy-momentum tensor MUST be diagonal and T_{11}=T_{22}=T_{33}=T. In fact, this type of tensor describes a perfect fluid with

T_{\mu\nu}=(\rho+p)U_\mu U_\nu+pg_{\mu\nu}

Here, \rho, p are functions of t (cosmological time) only. They are “state variables” somehow. Moreover, we have

U_\mu =(1,0,0,0)

for the fluid at rest in the comoving frame. The Friedmann equations are indeed the EFE for a FRW metric Universe

3\left(\dfrac{\dot{a}^2}{a^2}+\dfrac{k}{a^2}\right)=\kappa^2\rho for the 00th compoent as “constraint equation.

2\dfrac{\ddot{a}}{a}+\dfrac{\dot{a}^2}{a^2}+\dfrac{k}{a^2}=-\kappa^2p for the iith components.

Moreover, we also have

G_{\mu\nu}^{;\nu}=T_{\mu\nu}^{;\nu}=0

and this conservation law implies that

\dot{\rho}+3\dfrac{\dot{a}}{a}(\rho+p)=0

Therefore, we have got two independent equations for three unknowns (a, \rho, p). We need an additional equation. In fact, the equation of state for p=p(\rho) provides such an additional equation. It gives the “dynamics of matter”!

In summary, the basic equations for Cosmology in a FRW metric, via EFE, are the Friedmann’s equations (they are secretly the EFE for the FRW metric) supplemented with the energy-momentum conservations law and the equation of state for the pressure p=p(\rho):

1) \boxed{\dfrac{\dot{a}^2}{a^2}+\dfrac{k^2}{a^2}=\dfrac{\kappa^2}{3}\rho}

2) \boxed{\dot{\rho}+3\dfrac{\dot{a}}{a}(\rho+p)=0}

3) \boxed{p=p(\rho)}

There are many kinds of “matter-energy” content of our interest in Cosmology. Some of them can be described by a simple equation of state:

\boxed{p=\omega \rho}

Energy-momentum conservation implies that \rho\propto a^{-3(\omega +1)}. 3 special cases are used often:

1st. Radiation (relativistic “matter”). \omega=1/3 and thus, p=1/3\rho and \rho\propto a^{-4}

2nd. Dust (non-relativistic matter). \omega=0. Then, p=0 and \rho\propto a^{-3}

3rd. Vacuum energy (cosmological constant). \omega=-1. Then, p=-\rho and \rho=\mbox{constant}

Remark (I): Particle physics enters Cosmology here! Matter dynamics or matter fields ARE the matter content of the Universe.

Remark (II): Existence of a Big Bang (and a spacetime singularity). Using the Friedmann’s equation

\dfrac{\ddot{a}}{a}=-\dfrac{\kappa^2}{6}(\rho+3p)

if we have that (\rho+3p)>0, the so-called weak energy condition, then a=0 should have been reached at some finite time in the past! That is the “Big Bang” and EFE are “singular” there. There is no scape in the framework of GR. Thus, we need a quantum theory of gravity to solve this problem OR give up the FRW metric at the very early Universe by some other type of metric or structure.

Particles and the chemical equilibrium of the early Universe

Today, we have DIRECT evidence for the existence of a “thermal” equilibrium in the early Universe: the cosmic microwave background (CMB). The CMB is an isotropic, accurate and non-homogeneous (over certain scales) blackbody spectrum about T\approx 3K!

Then, we know that the early Universe was filled with a hot dieal gas in thermal equilibrium (a temperature T_e can be defined there) such as the energy density and pressure can be written in terms of this temperature. This temperature generates a distribution f(\mathbf{x},\mathbf{p}). The number of phase space elements in d^3xd^3p is

d^3xd^3p=\dfrac{d^3\mathbf{x}d^3\mathbf{p}}{(2\pi\hbar)^3}

and where the RHS is due to the uncertainty principle. Using homogeneity, we get that, indeed, f(x,p)=f(p), and where we can write the volume d^3x=dV. The energy density and the pressure are given by (natural units are used)

\rho_i=g_i\int \dfrac{d^3p}{(2\pi)^3}f_i(p)E(p)

p_i=g_i\int \dfrac{d^3p}{(2\pi)^3}f_i (p)\dfrac{p^2}{3E(p)}

When we are in the thermal equilibrium at temperature T, we have the Bose-Einstein/Fermi-Dirac distribution

f(p)=\dfrac{1}{e^{(E-\mu)/T}\pm 1}

and where the + is for the Fermi-Dirac distribution (particles) and the - is for the Bose-Einstein distribution (particles). The number density, the energy density and the pressure are the following integrals

\boxed{\mbox{Number density}=n=\dfrac{N}{V}=\dfrac{g}{2\pi^2}\int_m^\infty \dfrac{(E^2-m^2)^{1/2}}{e^{(E-\mu)/T}\pm 1}dE}

\boxed{\mbox{Density energy}=\rho=\dfrac{E}{V}=\dfrac{g}{2\pi^2}\int_m^\infty \dfrac{(E^2-m^2)^{1/2}E^2}{e^{(E-\mu)/T}\pm 1}dE}

\boxed{\mbox{Pressure}=p=\dfrac{g}{6\pi^2}\int_m^\infty \dfrac{(E^2-m^2)^{3/2}}{e^{(E-\mu)/T}\pm 1}dE}

And now, we find some special cases of matter-energy for the above variables:

1st. Relativistic, non-degenerate matter (e.g. the known neutrino species). It means that T>>m and T>>\mu. Thus,

n=\left(\dfrac{3}{4}\right)\dfrac{\zeta (3)}{\pi^2}gT^3

\rho=\left(\dfrac{7}{8}\right)\dfrac{\pi^2}{30}gT^4

p=\dfrac{1}{3}\rho

2nd. Non-relativistic matter with m>>T only. Then,

n=g\left(\dfrac{mT}{2\pi}\right)^{3/2}e^{-(m-\mu)/T}

\rho= mn+\dfrac{3}{2}p, and p=nT<<\rho

The total energy density is a very important quantity. In the thermal equilibrium, the energy density of non-relativistic species is exponentially smaller (suppressed) than that of the relativistic particles! In fact,

\rho_R=\dfrac{\pi^2}{30}g_\star T^4 for radiation with p_R=\dfrac{1}{3}\rho_R

and the effective degrees of freedom are

\displaystyle{\boxed{g_\star=\sum_{bosons}g_b+\dfrac{7}{8}\sum_{fermions}g_f}}

Remark: The factor 7/8 in the DOF and the variables above is due to the relation between the Bose-Einstein and the Fermi-Dirac integral in d=3 space dimensions. In general d, the factor would be

(1-\dfrac{1}{2^d})=\dfrac{2^d-1}{2^d}

Entropy conservation and the early Universe

The entropy in a comoving volume IS a conserved quantity IN THE THERMAL EQUILIBRIUM. Therefore, we have that

\dfrac{\partial p_i}{\partial T}=g_i\int \dfrac{d^3p}{(2\pi)^3}\dfrac{df}{dT}\dfrac{p^2}{3E(p)}=g_i\int \dfrac{4\pi pE dE}{(2\pi)^3}\dfrac{df}{dE}\left(-\dfrac{E}{T}\right)\dfrac{p^2}{3E}

and then

\dfrac{\partial p_i}{\partial T}=\dfrac{g_i}{2\pi^2}\int \left(-\dfrac{d}{dE}\left(f\dfrac{p^3E}{3T}\right)+f\dfrac{d}{dE}\left(\dfrac{p^3E}{3T}\right)\right)dE

or

\dfrac{\partial p_i}{\partial T}=\dfrac{1}{T}(\rho_i+p_i)

Now, since

\dfrac{\partial \rho}{\partial t}+3\dfrac{\dot{a}}{a}(\rho+p)=0

then

\dfrac{\partial}{\partial t}\left(a^3(\rho+p)\right)-a^3\dfrac{\partial p}{\partial t}=0

\dfrac{1}{a^3}\dfrac{\partial (a^3(\rho +p))}{\partial t}-\dfrac{\partial \rho}{\partial t}=0

if we multiply by T and use the chain rule for \rho, we obtain

\dfrac{1}{a^3}\dfrac{\partial}{\partial t}\left(\dfrac{a^3(\rho+p)}{T}\right)=0

but it means that a^3s=\mbox{constant}, where s is the entropy density defined by

\boxed{s\equiv \dfrac{\rho+p}{T}}

Well, the fact is that we know that the entropy or more precisely the entropy density is the early Universe is dominated by relativistic particles ( this is “common knowledge” in the Stantard Cosmological Model, also called \Lambda CDM). Thus,

\boxed{s=\dfrac{2\pi^2}{45}g_\star T^3}

It implies the evolution of temperature with the redshift in the following way:

T\propto g_\star^{-1/3}a^{-1}

Indeed, since we have that n\propto a^{-3}, s\propto a^{-3}, the yield variable

Y_i\equiv \dfrac{n_i}{s}

is a convenient quantity that represents the “abundance” of decoupled particles.

See you in my next cosmological post!