<p>So I wanted know the order of magnitude of this phase <a href="http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1203.1.K.pdf" rel="nofollow noreferrer">space invariant</a> (equation <span class="math-container">$3.31a$</span> )<span class="math-container">$R$</span>  <a href="https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" rel="nofollow noreferrer">in our universe</a> (FLRW metric):</p>
<p><span class="math-container">$$R = \int_{p} \frac{\mathcal{N}(x,p,t)}{\mathcal{E}} dp_x dp_y dp_z$$</span></p>
<p>where <span class="math-container">$\mathcal{N}(x,p,t)$</span> is the phase space distribution of matter in our universe, <span class="math-container">$\mathcal{E}= p_0$</span> is the energy and <span class="math-container">$p_x$</span>, <span class="math-container">$p_y$</span> and <span class="math-container">$p_z$</span> are the spatial momentum.</p>
<p>One can assume low energy and temperature limit (also assume the formation of the molecules has taken place)? What is it's observed value?</p>

CC BY-SA 4.0
2022-06-06T22:53:40.780Z
Is $R \sim 10^7$ or $R\sim 10^6$ in SI units?
More Anonymous
CC BY-SA 4.0
2022-06-10T20:09:28.880Z
What is $R$ physically?
Daddy Kropotkin
CC BY-SA 4.0
2022-06-10T20:14:48.893Z
@DaddyKropotkin "This is the sum, over all particles
in a unit 3-volume, of the inverse energy. "
More Anonymous
CC BY-SA 4.0
2022-06-11T12:46:30.490Z
Might be better to ask on physics SE
Daddy Kropotkin
<general-relativity>
Order of magnitude of this phase space invariant?
More Anonymous
# Order of magnitude of this phase space invariant?

(Asked by: username_0 on 2022-06-06)

So I wanted know the order of magnitude of this phase [space invariant](http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1203.1.K.pdf) (equation $3.31a$ )$R$ [in our universe](https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric) (FLRW metric):

$$R = \int\_{p} \frac{\mathcal{N}(x,p,t)}{\mathcal{E}} dp\_x dp\_y dp\_z$$

where $\mathcal{N}(x,p,t)$ is the phase space distribution of matter in our universe, $\mathcal{E}= p\_0$ is the energy and $p\_x$, $p\_y$ and $p\_z$ are the spatial momentum.

One can assume low energy and temperature limit (also assume the formation of the molecules has taken place)? What is it's observed value?

**Tags:** general-relativity
> **Comments:**
> **username_0**: Is $R \sim 10^7$ or $R\sim 10^6$ in SI units?  
> **username_1**: What is $R$ physically?  
> **username_0**: @DaddyKropotkin "This is the sum, over all particles in a unit 3-volume, of the inverse energy. "  
> **username_1**: Might be better to ask on physics SE  

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## 0 answers

