<p>A more intuitive way to understand the comparison between the reaction rate <span class="math-container">$\Gamma$</span> and the Hubble rate <span class="math-container">$H$</span> is to take the reciprocal. <span class="math-container">$\Gamma^{-1}$</span> is the time between reactions. <span class="math-container">$H^{-1}$</span> is approximately the age of the Universe. If <span class="math-container">$\Gamma\gg H$</span>, then <span class="math-container">$\Gamma^{-1}\ll H^{-1}$</span>, i.e. the time between reactions is much shorter than the age of the Universe. If on the other hand <span class="math-container">$\Gamma\ll H$</span>, then <span class="math-container">$\Gamma^{-1}\gg H^{-1}$</span>, i.e. the time between reactions is much longer than the age of the universe. In the latter case, clearly equilibrium cannot be maintained.</p>

CC BY-SA 4.0
2023-05-09T20:52:58.767Z
2023-05-09T20:52:58.767Z
Sten
<p>For a spatially flat Universe (described by FLRW metric), which is consistent with current observation, the scale factor itself is not so relevant. One can always redefine/scale it so that it is e.g. unity at some given time. Thus, if we want an energy scale of physical relevance, we need to eliminate this arbitrary constant. Also, the scale factor has dimension of length (it becomes apparent for an open or closed Universe: the scale factor is the radius of spatial curvature). Comparing two numbers with different units doesn't make sense.</p>
<p>You can find the derivation of these statements in most theoretical (particle) cosmology books, e.g. Introduction to the Theory of the Early Universe by Gorbunov and Rubakov and The Early Universe by Kolb and Turner.</p>

CC BY-SA 4.0
2023-05-10T11:03:34.220Z
The scale factor $a$ is indeed dimensionless and normalized to unity at the present epoch.
pela
CC BY-SA 4.0
2023-05-10T07:32:23.903Z
2023-05-10T07:32:23.903Z
Chen
<p>I've come across sentences like</p>
<blockquote>
<p>...the reaction
rates (the number of reactions per particle per unit time) must
be larger than the cosmic expansion rate <span class="math-container">$H(t)$</span> in order for the
particles to maintain equilibrium.</p>
</blockquote>
<p>The quote is from the book &quot;Extragalactic Astronomy and Cosmology&quot; by Peter Schneider, page 195.
Why does the reaction rate have to be greater than <span class="math-container">$H(t)$</span> and not <span class="math-container">$\dot a$</span>? <span class="math-container">$\dot a$</span> feels like it should be the cosmic expansion rate.</p>

<cosmology><particles>
Hubble expansion rate and reaction rates
Arnab Chowdhury
# Hubble expansion rate and reaction rates

(Asked by: username_0 on 2023-05-09)

I've come across sentences like

> ...the reaction
> rates (the number of reactions per particle per unit time) must
> be larger than the cosmic expansion rate $H(t)$ in order for the
> particles to maintain equilibrium.

The quote is from the book "Extragalactic Astronomy and Cosmology" by Peter Schneider, page 195.
Why does the reaction rate have to be greater than $H(t)$ and not $\dot a$? $\dot a$ feels like it should be the cosmic expansion rate.

**Tags:** cosmology, particles

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## 2 answers

### Answer by username_1 on 2023-05-09. Score: 9

A more intuitive way to understand the comparison between the reaction rate $\Gamma$ and the Hubble rate $H$ is to take the reciprocal. $\Gamma^{-1}$ is the time between reactions. $H^{-1}$ is approximately the age of the Universe. If $\Gamma\gg H$, then $\Gamma^{-1}\ll H^{-1}$, i.e. the time between reactions is much shorter than the age of the Universe. If on the other hand $\Gamma\ll H$, then $\Gamma^{-1}\gg H^{-1}$, i.e. the time between reactions is much longer than the age of the universe. In the latter case, clearly equilibrium cannot be maintained.


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### Answer by username_2 on 2023-05-10. Score: 4

For a spatially flat Universe (described by FLRW metric), which is consistent with current observation, the scale factor itself is not so relevant. One can always redefine/scale it so that it is e.g. unity at some given time. Thus, if we want an energy scale of physical relevance, we need to eliminate this arbitrary constant. Also, the scale factor has dimension of length (it becomes apparent for an open or closed Universe: the scale factor is the radius of spatial curvature). Comparing two numbers with different units doesn't make sense.

You can find the derivation of these statements in most theoretical (particle) cosmology books, e.g. Introduction to the Theory of the Early Universe by Gorbunov and Rubakov and The Early Universe by Kolb and Turner.

> **Comments:**
> **username_3**: The scale factor $a$ is indeed dimensionless and normalized to unity at the present epoch.  

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