Difference between revisions of "Time Evolution of CMB"

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 $aT=const$.

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.

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.

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.

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.

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.

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.

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.

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: $${n_{_{CMB}} } = \frac{c^3} {\pi ^2}\int\limits_0^\infty \frac{\omega ^2d\omega } {e^{\hbar \omega /kT} - 1} = \frac{1} {\pi ^2}\left( \frac{kT} {\hbar c} \right)^3\int\limits_0^\infty \frac{x^2dx} {{e^x} - 1} = \frac{2\zeta \left( 3 \right)} {\pi ^2 l_{Pl}^3}\left( \frac{kT} {M_{Pl}c^2} \right)^3 \approx 410\;\mbox{\it cm}^{-3}. $$

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

Average the CMB photon energy with the Planck distribution to obtain: $$ \left\langle E \right\rangle _\omega = \frac{1} {\pi ^2c^3}\int\limits_0^\infty \hbar \omega \frac{\omega ^2d\omega } {e^{\hbar \omega /kT} - 1} = \frac{\hbar c} {\pi ^2}\left( \frac{kT} {\hbar c} \right)^4\int\limits_0^\infty \frac{x^3dx} {{e^x} - 1} = \frac{\pi ^2} {15}\frac{(kT)^4} {(\hbar c)^3} = 0.398 \mbox{\it eV/cm}^3. $$ 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 $\varepsilon_{_{CMB}}\simeq 9.7\cdot 10^{-4}\mbox{eV}$.

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{\it Mpc}^{-3} \approx 2.6 \cdot {10^{ - 33}}\mbox{\it 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{\it 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.

CMB energy density is $\left\langle E \right\rangle _\omega \approx 0.398 \mbox{\it eV/cm}^3$, and that of baryon matter is $\simeq 160\mbox{\it eV/cm}^3.$ 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.

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

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 $n_{_{CMB}}/n_{_B}\simeq 1.7\cdot 10^9.$

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.

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