Time Evolution of CMB - Revision history

Igor at 20:19, 19 November 2012

2012-11-19T20:19:12Z

Igor: /* Problem 6: color of the sky */

2012-11-19T20:16:19Z

‎Problem 6: color of the sky

Igor: /* Problem 2: redshift */

2012-10-15T11:40:39Z

‎Problem 2: redshift

Igor at 14:14, 14 October 2012

2012-10-14T14:14:43Z

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Cosmo All at 20:35, 1 October 2012

2012-10-01T20:35:31Z

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Cosmo All at 20:27, 1 October 2012

2012-10-01T20:27:43Z

Cosmo All: /* Problem 1 */

2012-10-01T20:26:55Z

‎Problem 1

Cosmo All: /* Problem 1 */

2012-10-01T20:25:02Z

‎Problem 1

Cosmo All: /* Problem 1 */

2012-10-01T20:24:18Z

‎Problem 1

Cosmo All at 20:23, 1 October 2012

2012-10-01T20:23:36Z

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@@ Line 21: / Line 21: @@
 Show that the electromagnetic radiation frequency decreases with expansion of the Universe
 as $\omega(t)\propto a(t)^{-1}$.
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 === Problem 4: one second after Big Bang ===
 Find the CMB temperature one second after the Big Bang.
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 === 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).
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 === Problem 22: 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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 === Problem 23: what about neutrino? ===
 Why is the cosmic neutrino background (CNB) temperature at present lower than the one for CMB?
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@@ Line 80: / Line 80: @@
 === 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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 === Problem 7: when the night sky appeared ===
 When the night sky started to look black?

@@ Line 19: / Line 19: @@
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 === 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 $\omega(t)\propto a(t)^{-1}$.
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@@ Line 31: / Line 31: @@
 <div id="cmb34"></div>
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 === 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.

@@ Line 168: / Line 168: @@
 \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}.
+\right)^3 \approx 410\;\mbox{cm}^{-3}.
 $$</p>
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@@ Line 206: / Line 206: @@
 {\pi ^2}\left( \frac{kT} {\hbar c} \right)^4\int\limits_0^\infty
 \frac{x^3dx} {{e^x} - 1}  = \frac{\pi ^2} {15}\frac{(kT)^4} {(\hbar
-c)^3} = 0.398 \mbox{\it eV/cm}^3.
+c)^3} = 0.398 \mbox{eV/cm}^3.
 $$
@@ Line 225: / Line 225: @@
      <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
-^8L_{\odot} \mbox{\it Mpc}^{-3} \approx 2.6 \cdot {10^{ -
+^8L_{\odot} \mbox{Mpc}^{-3} \approx 2.6 \cdot {10^{ -
-}}\mbox{\it W/m}^{ 3}. $ Assume that the latter luminosity was constant during the whole period of the Universe lifetime, then $ \rho _{st} \sim
+}}\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
 .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>
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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>
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@@ Line 313: / Line 313: @@
      <p style="text-align: left;">The recombination took place at the matter dominated epoch:
 $a(t=t_{rec})=(t_{rec}/t_0)^{2/3}=2.725/T_{rec}\simeq 8\cdot
-^{-4}\Rightarrow t_{rec}\simeq 2\cdot 10^{-5},\;t_0=320\ \mbox{\it kyr.}$</p>
+^{-4}\Rightarrow t_{rec}\simeq 2\cdot 10^{-5},\;t_0=320\ \mbox{ kyr.}$</p>
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@@ Line 297: / Line 297: @@
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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>
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@@ Line 305: / Line 305: @@
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 === Problem 1 ===
 Estimate the moment of the beginning of recombination-transition from ionized plasma to gas of neutral atoms.

@@ Line 115: / Line 115: @@
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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:
 \[\rho _{_{CMB}} = \alpha T_{CMB}^4,\; \alpha =
-\frac{\pi^2}{15}\frac{(kT)^4}{(\hbar c)^3},\] 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>
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@@ Line 127: / Line 127: @@
 <div id="cmb5"></div>
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 === Problem 1 ===
 Estimate the moment of time when the CMB wavelength will be comparable to that in the microwave oven, which is $\lambda=12.6\ cm$.

@@ Line 71: / Line 71: @@
 $$
 t_{md} = \frac{2} {3H_0}\frac{\Omega _{r0}^{3/2}} {\Omega _{m0}^2}
-\approx 50\ 000\ \mbox{\it years.}
+\approx 50\ 000\ \mbox{years.}
 $$
 The primordial hydrogen was $50\%$ ionized at temperature ${T_{rec}}  \approx 3\ 400K$ in the age $t_{rec}\simeq
-\ 000\mbox{\it years}.$ Thus at $t=t_{rec}$ the mater dominated already long time.</p>
+\ 000\mbox{years}.$ Thus at $t=t_{rec}$ the mater dominated already long time.</p>
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@@ Line 83: / Line 83: @@
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 === Problem 1 ===
 What color had the sky at the recombination epoch?