# How to comoving volumes depend on the evolution of the Universe?

I'm reading a paper which states that

Neutron star binary merger rate at redshift $z$ per unit observer time interval per unit volume is $\dot{n}_{m} = \dot{n}_{0} (1+z)^{2} (1+z)^{\beta}$, where $\dot{n}_{0}$ is the local neutron star binary merger rate per unit volume, $(1+z)^2$ accounts for the shrinking of volumes with redshift (assuming constant comoving volume density of the merger rate) and time dilation, $\beta$ describes evolutionary effects.

I don't understand where they get the $\beta$ part from, could anyone shead some light on this?

Edit: The above is from this paper, and the paragraph is the one at the top of page 3. The author uses $\beta=0$ and $\beta =1$ in the plots.

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Can you post a link to the paper? –  John Rennie Feb 7 '13 at 14:54
@JohnRennie Thanks for the interest. I have made an edit that includes a link to the paper. –  user12345 Feb 7 '13 at 16:11

Have a look at this paper. The authors are concerned with modelling gamma ray bursts from neutron star binaries, but the same argument will apply to gravitational waves emitted from the mergers. The point is that the rate of neutron star formation has varied since the big bang, and the population at any particular red shift depends on the neutron star binary formation rate in preceding times. The paper goes into some detail about fitting the NS binary population and explains why the $(1 + z)^\beta$ fit is used. As FrankH says, it's phenomenological rather than derived from any fundamental principles.

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Ah, it's 'evolution of neutron stars' not 'evolution of the Universe'. –  user12345 Feb 7 '13 at 17:46

In astrophysics processes that are not well understood are often modeled as power laws and that is what I believe the authors are doing here. First of all $1/(1+z)\$ is the scale factor $a(t)\$ from the FRW metric that describes the evolution of the universe. The case $\beta = 0$ is assuming there is no evolution Of the neutron star population as the universe expands and the case $\beta = 1$ is assuming that the evolution is linear in $a(t)$. I don't think there is any real significance beyond this simple kind of model.

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I see what you mean, but according to this* webpage, we have the $(1+z)^{2}$ but I don't understand why $0<\beta <1$. Why those numbers? What does a $\beta=1$ Universe have in it? [This is why reading papers is hard - because authors just assume you know everything and give the result.] *ned.ipac.caltech.edu/level5/Hogg/Hogg9.html –  user12345 Feb 7 '13 at 16:55
@user16307 I don't think there is any significance to the $0 < \beta < 1$ range - it is just a model they are trying... –  FrankH Feb 7 '13 at 20:16
Just one correction: $a = 1/(1+z)$. –  Chris White Feb 8 '13 at 3:43
@ChrisWhite Thanks, made correction! –  FrankH Feb 8 '13 at 6:47