Time Evolution of CMB

Problem 1

Show that in the expanding Universe the quantity $aT$ is an approximate invariant.

solution

Expanding of Universe according to the Friedmann equations is adiabatic ($S=const$). Entropy of the Universe is mainly determined by the photon component with $S\propto VT^3$. Using $V\propto a^3$, one obtains $aT=const$.

Problem 2

Show that the electromagnetic radiation frequency decreases with expansion of Universe as $\omega(t)\propto a(t)^{-1}$.

solution

Problem 3

Show that if the radiation spectrum was equilibrium at some initial moment, then it will remain equilibrium during the following expansion.

solution

Equilibrium character of the spectrum is equivalent to the following Planck form of the distribution function \[N(\omega)=\frac{1}{e^{\frac{\hbar\omega}{kT}}-1}=\frac{1}{e^{\frac{2\pi\hbar c}{kTa(t)\lambda_0}}-1},\] where $a(t)$ is scale factor.
Taking into account that $aT=const$, one can see that the Planck form of the distribution does not change with time.

Problem 4

Find the CMB temperature one second after the Big Bang.

solution

Problem 5

Show that creation of the relic radiation (the photon decoupling) took place in the matter-dominated epoch.

solution

The energy density of non-relativistic matter became equal to that of radiation at time moment (see): $$ t_{md} = \frac{2} {3H_0}\frac{\Omega _{r0}^{3/2}} {\Omega _{m0}^2} \approx 50\ 000\ \mbox{years.} $$ The primordial hydrogen was $50\%$ ionized at temperature ${T_{rec}} \approx 3\ 400K$ in the age $t_{rec}\simeq 300\ 000\mbox{years}.$ Thus at $t=t_{rec}$ the mater dominated already long time.

Problem 6

What color had the sky at the recombination epoch?

solution

The newborn relic radiation had the black body spectrum with temperature $3\ 400$ K, which is very close to the radiation of the incandescent lamp-white color with orange admixture.

Problem 7

When the night sky started to look black?

solution

It happened when the CMB temperature became of order $270$ K, which corresponds to the scale factor value $a\sim0.01$. In the matter dominated Universe it happened in the age $t\approx14\cdot10^6$ years.

Problem 8

Estimate the moment of time when the CMB energy density was comparable to that in the microwave oven.

solution

Typical microwave oven has power of order $10^3\mbox{W}$ and volume of order $10$~liters, which with characteristic time of operation $10^3\mbox{s}$ provides the energy density $10^8\mbox{J/m}^3$. According to Stephan-Boltzmann law: \[\rho _{_{CMB}} = \alpha T_{CMB}^4,\; \alpha = \frac{\pi^2}{15}\frac{(kT)^4}{(\hbar c)^3},\] such density corresponds to CMB temperature $T = 6\cdot10^5\mbox{ K}$, which took place at the radiation dominated epoch with the scale factor value $a = 2.725/T\simeq4.5\cdot10^{-6}$, when the Universe had age $t = t_0a^2 = 9\cdot10^6 \mbox{ s}$, or just three months and a half.

Problem 9

Estimate the moment of time when the CMB wavelength will be comparable to that in the microwave oven, which is $\lambda=12.6\ cm$.

solution

Presently the CMB wavelength equals to $\lambda_0\simeq1.9 \mbox{mm}$. Using the invariant $a/\lambda = const$, and explicit dependence $a(t) = (t/t_0)^{2/3}$ for the matter-dominated epoch, one obtains $t = t_0(\lambda/\lambda_0)^{3/2}\simeq2.4\cdot10^{20}\mbox{s}\simeq 8\cdot10^{12}$ years. Taking into account the accelerated expansion of Universe it will happen much earlier though.

Problem 10

When the relic radiation obtained formal right to be called CMB? And for what period of time?

solution

Microwave range includes wavelengths from 1mm 1m. Maximum intensity of the Planck spectrum falls on the wavelength \[\lambda_0\approx\frac{2\pi\hbar}{5kT}.\] Therefore microwaves correspond to the temperature range $0.003\,K<T<3\,K$, which takes place for the scale factor values in the interval $0.9<a<900$. For matter-dominated Universe it corresponds to the time interval between $12\cdot10^9$ years and $378\cdot10^{12}$ years.
Therefore the relic radiation can formally b called the CMB already for $2\cdot10^9$ years and will loose it after almost $400\cdot10^{12}$ years.

Problem 11

Calculate the presently observed density of photons for the CMB and express it in Planck units.

solution

Concentration of the photons equals to the integral of the photon distribution function (the Planck distribution), divided by the energy of a single photon. It results in the following: $${n_{_{CMB}} } = \frac{c^3} {\pi ^2}\int\limits_0^\infty \frac{\omega ^2d\omega } {e^{\hbar \omega /kT} - 1} = \frac{1} {\pi ^2}\left( \frac{kT} {\hbar c} \right)^3\int\limits_0^\infty \frac{x^2dx} {{e^x} - 1} = \frac{2\zeta \left( 3 \right)} {\pi ^2 l_{Pl}^3}\left( \frac{kT} {M_{Pl}c^2} \right)^3 \approx 410\;\mbox{cm}^{-3}. $$

Problem 12

Find the ratio of CMB photons' energy density to that of the neutrino background.

solution

$$ \frac{\rho_\nu}{\rho _{_{CMB}}} = 3 \cdot \frac{7}{8} \cdot \left( \frac{4}{11} \right)^{4/3} \simeq 0.68. $$

Problem 13

Determine the average energy of a CMB photon at present time.

solution

Average the CMB photon energy with the Planck distribution to obtain: $$ \left\langle E \right\rangle _\omega = \frac{1} {\pi ^2c^3}\int\limits_0^\infty \hbar \omega \frac{\omega ^2d\omega } {e^{\hbar \omega /kT} - 1} = \frac{\hbar c} {\pi ^2}\left( \frac{kT} {\hbar c} \right)^4\int\limits_0^\infty \frac{x^3dx} {{e^x} - 1} = \frac{\pi ^2} {15}\frac{(kT)^4} {(\hbar c)^3} = 0.398 \mbox{eV/cm}^3. $$ In problem there was obtained that ${n_{_{CMB}} } \approx 410\:\mbox{cm}^{ - 3}.$ So that the average energy of a single photon evidently equals to $\varepsilon_{_{CMB}}\simeq 9.7\cdot 10^{-4}\mbox{eV}$.

Problem 14

Why, when calculating the energy density of electromagnetic radiation in the Universe, we can restrict ourself to the CMB photons?

solution

Integral density of the CMB equals to ${\left\langle E \right\rangle _\omega } \approx 0.398\mbox{MeV/m}^3$, and current energy density of radiation from galaxies is roughly $ nL \approx 2 \cdot 10^8L_{\odot} \mbox{Mpc}^{-3} \approx 2.6 \cdot {10^{ - 33}}\mbox{W/m}^{ 3}. $ Assume that the latter luminosity was constant during the whole period of the Universe lifetime, then $ \rho _{st} \sim nLt_0 \approx 1.1 \cdot 10^{ - 15}\mbox{J/m}^{3}\approx 0.007\mbox{MeV/m}^3.$ Even such overestimated value shows that $ \rho _{CMB} \approx 60\rho _{st},$ so that CMB energy density is many times higher than integral density of radiation from other sources (such as stars, galaxies, radiogalaxies, quasars), calculated with its probable evolution in the past.

Problem 15

The relation $\rho_\gamma\propto a^{-4}$ assumes conservation of photon's number. Strictly speaking, this assumption is inaccurate. The Sun, for example, emits of the order of $10^{45}$ photons per second. Estimate the accuracy of this assumption regarding the photon's number conservation.

solution

Problem 16

Can hydrogen burning in the thermonuclear reactions provide the observed energy density of the relic radiation?

solution

CMB energy density is $\left\langle E \right\rangle _\omega \approx 0.398 \mbox{ eV/cm}^3$, and that of baryon matter is $\simeq 160\mbox{ eV/cm}^3.$ Therefore burning of a small fraction of hydrogen could provide sufficient amount of energy to fill the space with the radiation of temperature $\sim 3 \:K.$ However the burning of nuclear fuel in stars is known to give the radiation spectrum which is very distinct from the observed $3K$ equilibrium radiation, say nothing about the discrete angular distribution instead the uniform one. If the burning took place in the distant past then the high-temperature radiation emitted by the sources could in principle transform into the low-temperature one, but due to the cosmological expansion its energy density would significantly decreases, so that the required energy outcome in the past would go beyond any reasonable limit.

Problem 17

Find the ratio of relic radiation energy density in the epoch of last scattering to the present one.

solution

Temperature of the CMB photons on the last scattering surface was $T_{dec}\simeq 3400K,$ so \[ \frac{\rho_{ls}}{\rho_0}=\left(\frac{T_{dec}}{T_0}\right)^4\simeq 3\cdot 10^{12}.\]

Problem 18

Find the ratio of average number densities of photons to baryons in the Universe.

solution

Concentrations of photons and baryons depend on the scale factor in the same way: $n\sim a^{-3}$, so their ratio is invariant and according to present observational data it equals to $n_{_{CMB}}/n_{_B}\simeq 1.7\cdot 10^9.$

Problem 19

Explain qualitatively why the temperature of photons at the surface of last scattering (0.3 eV) is considerably less than the ionization energy of the hydrogen atom (13.6 eV).

solution

One would naively expect that relative concentration of neutral hydrogen becomes significant when the Universe temperature falls below than $13.6$ eV which is the ionization energy of the hydrogen atom. However it happens at much lower temperatures in fact, because, firstly, the photon number density considerably exceeds that of baryon (their ratio equals to $n_{_{CMB}}/n_{_B} \sim 10^{9} $, see problem, and secondly, the Planck distribution has high-energy tails. it can be shown that the numbers of neutral and ionized hydrogen atoms become equal at temperature $T \approx 0.3\mbox{eV}\left( {3400K} \right),$ which is called the recombination temperature.

Problem 20

Estimate the moment of the beginning of recombination-transition from ionized plasma to gas of neutral atoms.

solution

The recombination took place at the matter dominated epoch: $a(t=t_{rec})=(t_{rec}/t_0)^{2/3}=2.725/T_{rec}\simeq 8\cdot 10^{-4}\Rightarrow t_{rec}\simeq 2\cdot 10^{-5},\;t_0=320\ \mbox{ kyr.}$

Problem 21

Determine the moment of time when the mean free path of photons became of the same order as the current observable size of Universe).

solution

Problem 22

How will the results of problem and problem change if one takes into account the possibility of creation of neutral hydrogen in excited states?

solution

Problem 23

Why is the cosmic neutrino background (CNB) temperature at present lower than the one for CMB?

solution