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.


2 Answers 2


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.


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.

  • 1
    $\begingroup$ The scale factor $a$ is indeed dimensionless and normalized to unity at the present epoch. $\endgroup$
    – pela
    May 10, 2023 at 11:03

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