Difference between revisions of "Time Evolution of CMB"

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[[Category:Cosmic Microwave Background (CMB)|2]]
 
[[Category:Cosmic Microwave Background (CMB)|2]]
  
 
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=== Problem 1 ===
+
=== Problem 1: temperature ===
 
Show that in the expanding Universe the quantity $aT$ is an approximate invariant.
 
Show that in the expanding Universe the quantity $aT$ is an approximate invariant.
 
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=== Problem 1 ===
+
=== Problem 2: redshift ===
Show that the electromagnetic radiation frequency decreases with expansion of Universe
+
Show that the electromagnetic radiation frequency decreases with expansion of the Universe
 
as $\omega(t)\propto a(t)^{-1}$.
 
as $\omega(t)\propto a(t)^{-1}$.
<div class="NavFrame collapsed">
+
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=== Problem 1 ===
+
 
 +
=== Problem 3: Planck spectrum ===
 
Show that if the radiation spectrum was equilibrium at some initial moment, then it will remain equilibrium during the following expansion.
 
Show that if the radiation spectrum was equilibrium at some initial moment, then it will remain equilibrium during the following expansion.
 
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=== Problem 1 ===
+
=== Problem 4: one second after Big Bang ===
 
Find the CMB temperature one second after the Big Bang.
 
Find the CMB temperature one second after the Big Bang.
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<div id="cmb7"></div>
 
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=== Problem 1 ===
+
=== Problem 5: decoupling time ===
 
Show that creation of the relic radiation (the photon decoupling) took place in the matter-dominated epoch.
 
Show that creation of the relic radiation (the photon decoupling) took place in the matter-dominated epoch.
 
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=== Problem 1 ===
+
=== Problem 6: color of the sky ===
What color had the sky at the recombination epoch?
+
What was the color of the sky at the recombination epoch?
 
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   </div>
 
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<div id="cmb_tmp_2"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
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=== Problem 1 ===
+
 
 +
=== Problem 7: when the night sky appeared ===
 
When the night sky started to look black?
 
When the night sky started to look black?
 
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   </div>
 
</div></div>
 
</div></div>
 
  
  
 
<div id="cmb4"></div>
 
<div id="cmb4"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 8: CMB vs microwave ===
 
Estimate the moment of time when the CMB energy density was comparable to that in the microwave oven.
 
Estimate the moment of time when the CMB energy density was comparable to that in the microwave oven.
 
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     <p style="text-align: left;">Typical microwave oven has power of order $10^3\mbox{\it W}$
+
     <p style="text-align: left;">Typical microwave oven has power of order $10^3\mbox{W}$
and volume of order $10$~liters, which with characteristic time of operation $10^3\mbox{\it s}$
+
and volume of order $10$~liters, which with characteristic time of operation $10^3\mbox{s}$
provides the energy density $10^8\mbox{\it J/m}^3$. According to Stephan-Boltzmann law:
+
provides the energy density $10^8\mbox{J/m}^3$. According to Stephan-Boltzmann law:
 
\[\rho _{_{CMB}} = \alpha T_{CMB}^4,\; \alpha =
 
\[\rho _{_{CMB}} = \alpha T_{CMB}^4,\; \alpha =
\frac{\pi^2}{15}\frac{(kT)^4}{(\hbar c)^3},\] such density corresponds to CMB temperature $T = 6\cdot10^5\mbox{\it K}$, 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 $t = t_0a^2 = 9\cdot10^6 \mbox{\it s}$, or just three months and a half.</p>
+
\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.</p>
 
   </div>
 
   </div>
 
</div></div>
 
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<div id="cmb5"></div>
 
<div id="cmb5"></div>
 
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=== Problem 1 ===
+
=== Problem 9: wavelengths ===
 
Estimate the moment of time when the CMB wavelength will be comparable to that in the microwave oven, which is $\lambda=12.6\ cm$.
 
Estimate the moment of time when the CMB wavelength will be comparable to that in the microwave oven, which is $\lambda=12.6\ cm$.
 
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<div id="cmb_tmp_3"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 10: micro or not ===
 
When the relic radiation obtained formal right to be called CMB? And for what period of time?
 
When the relic radiation obtained formal right to be called CMB? And for what period of time?
 
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<div id="cmb10"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 11: photon number density ===
 
Calculate the presently observed density of photons for the CMB and express it in Planck units.
 
Calculate the presently observed density of photons for the CMB and express it in Planck units.
 
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\right)^3\int\limits_0^\infty  \frac{x^2dx} {{e^x} - 1}  =
 
\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}
 
\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}.
+
\right)^3 \approx 410\;\mbox{cm}^{-3}.
 
$$</p>
 
$$</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
 
  
  
 
<div id="cmb11"></div>
 
<div id="cmb11"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 12: $\gamma$ and $\nu$ backgrounds ===
 
Find the ratio of CMB photons' energy density to that of the neutrino background.
 
Find the ratio of CMB photons' energy density to that of the neutrino background.
 
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<div id="cmb12"></div>
 
<div id="cmb12"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 13: energy of a photon ===
 
Determine the average energy of a CMB photon at present time.
 
Determine the average energy of a CMB photon at present time.
 
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{\pi ^2}\left( \frac{kT} {\hbar c} \right)^4\int\limits_0^\infty
 
{\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
 
\frac{x^3dx} {{e^x} - 1}  = \frac{\pi ^2} {15}\frac{(kT)^4} {(\hbar
c)^3} = 0.398 \mbox{\it eV/cm}^3.
+
c)^3} = 0.398 \mbox{eV/cm}^3.
 
$$
 
$$
  
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   </div>
 
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<div id="cmb13"></div>
 
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=== Problem 1 ===
+
=== Problem 14: contribution to radiation's energy ===
Why, when calculating the energy density of electromagnetic radiation in the Universe, we can restrict ourself to the CMB photons?
+
Why, when calculating the energy density of electromagnetic radiation in the Universe, we can restrict ourselves to the CMB photons?
 
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     <p style="text-align: left;">Integral density of the CMB equals to ${\left\langle E
 
     <p style="text-align: left;">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
 
\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^{ -
+
10^8L_{\odot} \mbox{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
+
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{\it J/m}^{3}\approx
+
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.</p>
 
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.</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
 
  
  
 
<div id="cmb44"></div>
 
<div id="cmb44"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 15: conservation of photons' number ===
 
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.
 
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.
 
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<div id="cmb14"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 16: thermonuclear sources ===
 
Can hydrogen burning in the thermonuclear reactions provide the observed energy density of the relic radiation?
 
Can hydrogen burning in the thermonuclear reactions provide the observed energy density of the relic radiation?
 
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     <p style="text-align: left;">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.</p>
+
     <p style="text-align: left;">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.</p>
 
   </div>
 
   </div>
 
</div></div>
 
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<div id="cmb15"></div>
 
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=== Problem 1 ===
+
=== Problem 17: energy density evolution ===
 
Find the ratio of relic radiation energy density in the epoch of last scattering to the present one.
 
Find the ratio of relic radiation energy density in the epoch of last scattering to the present one.
 
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<div id="cmb16"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 18: photons and baryons ===
 
Find the ratio of average number densities of photons to baryons in the Universe.
 
Find the ratio of average number densities of photons to baryons in the Universe.
 
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<div id="cmb17"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 19: temperature at last scattering ===
 
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).
 
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).
 
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     <p style="text-align: left;">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 \ref{cmb16}), 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.</p>
+
     <p style="text-align: left;">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 [[#cmb16|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.</p>
 
   </div>
 
   </div>
 
</div></div>
 
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<div id="cmb20"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
 
Estimate the moment of the beginning of recombination-transition from ionized plasma to gas of neutral atoms.
+
=== Problem 20: recombination ===
 +
Estimate the moment of the beginning of recombination: transition from ionized plasma to gas of neutral atoms.
 
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     <p style="text-align: left;">The recombination took place at the matter dominated epoch:
 
     <p style="text-align: left;">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
 
$a(t=t_{rec})=(t_{rec}/t_0)^{2/3}=2.725/T_{rec}\simeq 8\cdot
10^{-4}\Rightarrow t_{rec}\simeq 2\cdot 10^{-5},\;t_0=320\ \mbox{\it kyr.}$</p>
+
10^{-4}\Rightarrow t_{rec}\simeq 2\cdot 10^{-5},\;t_0=320\ \mbox{ kyr.}$</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
 
  
  
 
<div id="cmb21"></div>
 
<div id="cmb21"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 21: mean free path ===
 
Determine the moment of time when the mean free path of photons became of the same order as the current observable size of Universe).
 
Determine the moment of time when the mean free path of photons became of the same order as the current observable size of Universe).
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+
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+
  
  
 
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<div id="cmb22"></div>
 
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<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 22: excited states ===
How will the results of [[#cmb20|problem]] and [[#cmb21|problem]] change if one takes into account the possibility of creation of neutral hydrogen in excited states?
+
How will the [[#cmb20|moment of recombination]] and the time when [[#cmb21|mean free path of photons becomes comparable with the size of observable Universe]] change if one takes into account the possibility of creation of neutral hydrogen in excited states?
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   </div>
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+
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+
  
  
 
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<div id="cmb35"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
+
=== Problem 23: what about neutrino? ===
 
Why is the cosmic neutrino background (CNB) temperature at present lower than the one for CMB?
 
Why is the cosmic neutrino background (CNB) temperature at present lower than the one for CMB?
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</div></div>
+
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Latest revision as of 20:19, 19 November 2012


Problem 1: temperature

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


Problem 2: redshift

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


Problem 3: Planck spectrum

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


Problem 4: one second after Big Bang

Find the CMB temperature one second after the Big Bang.


Problem 5: decoupling time

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


Problem 6: color of the sky

What was the color of the sky at the recombination epoch?


Problem 7: when the night sky appeared

When the night sky started to look black?


Problem 8: CMB vs microwave

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


Problem 9: wavelengths

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


Problem 10: micro or not

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


Problem 11: photon number density

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


Problem 12: $\gamma$ and $\nu$ backgrounds

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


Problem 13: energy of a photon

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


Problem 14: contribution to radiation's energy

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


Problem 15: conservation of photons' number

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.


Problem 16: thermonuclear sources

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


Problem 17: energy density evolution

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


Problem 18: photons and baryons

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


Problem 19: temperature at last scattering

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).


Problem 20: recombination

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


Problem 21: mean free path

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


Problem 22: excited states

How will the moment of recombination and the time when mean free path of photons becomes comparable with the size of observable Universe change if one takes into account the possibility of creation of neutral hydrogen in excited states?


Problem 23: what about neutrino?

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