View source for Connection between Temperature and Redshift

[[Category:Thermodynamics of Universe|4]]

__NOTOC__


<div id="therm23"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
Find the dependence of radiation temperature on the redshift.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">$$
T \propto a^{ - 1};\quad T = T_0(1 + z).
$$
</p>
  </div>
</div></div>



<div id="therm24"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 2 ===
Find the dependence of free non-relativistic gas temperature on the redshift.
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">$$
T \propto a^{ - 2};\quad T = T_0(1 + z)^2.
$$
</p>
  </div>
</div></div>



<div id="therm25"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 3 ===
Estimate the time moment when the recombination started, i.e. when ionized plasma transited to the gas of neutral atoms, and determine the corresponding redshift value. The recombination temperature equals to $T_{rec} \approx 0.3\,eV.$
<div class="NavFrame collapsed">
  <div class="NavHead">solution</div>
  <div style="width:100%;" class="NavContent">
    <p style="text-align: left;">$$
z_{rec} = \frac{T_{rec}}{T_0} - 1.
$$
As $1\mbox{\it eV} \approx 11600\,K$, then
$$
z_{rec} \approx \frac{0.3}{2.73}11600 \approx 1300.
$$
For a rough estimate assume that the transition $T_{rec} \to T_0$ occurs mainly in the matter-dominated epoch to obtain
$$
t_{rec} = t_0\left( \frac{a}{a_0} \right)^{3/2} =
t_0\left(\frac{T_0}{T_{rec}} \right)^{3/2}\approx
300\,000\,years.
$$
</p>
  </div>
</div></div>