<p>Matter is indeed distributed in knots (clusters), filaments, and sheets (e.g. <a href="https://ui.adsabs.harvard.edu/abs/1996Natur.380..603B/abstract" rel="nofollow noreferrer">Bond et al. 1996</a>). This is predicted from theories and numerical simulations of structure formation under gravitational collapse, and is directly observable in large galaxy surveys by eye, such as <a href="http://www.2dfgrs.net/" rel="nofollow noreferrer">The 2dF Galaxy Redshift Survey</a>:</p>

<p><a href="https://i.sstatic.net/UxMUp.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UxMUp.png" alt="2dfgrs"></a></p>

<h2>Two-point correlation function</h2>

<p>To quantify this clustering, usually a <em>correlation function</em> is used, either in 2D (<span class="math-container">$x$</span> and <em>y</em> position on the sky), or 3D (if redshifts are available for the galaxies, giving the <span class="math-container">$z$</span> dimension). The (two-point) correlation function <span class="math-container">$\xi(r)$</span> gives the excess probability of finding two galaxies at a distance <span class="math-container">$r$</span> from each other, relative to a random (Poisson) distribution.
If <span class="math-container">$\bar{n}$</span> is the mean number density of galaxies, then the probability <span class="math-container">$dP$</span> of finding a pair of galaxies at a separation <span class="math-container">$r_{12}$</span> in volume elements <span class="math-container">$dV_1$</span> and <span class="math-container">$dV_2$</span> is
<span class="math-container">$$
dP = \bar{n}^2 \big(1 + \xi(r_{12})\big) dV_1 dV_2.
$$</span>
The power spectrum <span class="math-container">$P(k)$</span> of the galaxies is related to the two-point correlation function by
<span class="math-container">$$
\xi(r) = \frac{1}{2\pi^2} \int dk\,k^2\,P(k)\,\frac{\sin kr}{kr}
$$</span>
Observationally it is found that the two-point correlation function is given by a power-law:
<span class="math-container">$$
\xi(r) = \left( \frac{r}{r_0} \right)^{-\gamma},
$$</span>
where <span class="math-container">$r_0$</span> is the correlation length (of the order <span class="math-container">$5\,h^{-1}\mathrm{Mpc}$</span>) and <span class="math-container">$\gamma=1.7$</span>–<span class="math-container">$1.8$</span>.</p>

<p>Note that the value of <span class="math-container">$r_0$</span> depends on galaxy type; brighter and redder galaxies have a longer correlation length, i.e. they are more strongly clustered (e.g. <a href="https://ui.adsabs.harvard.edu/abs/2008ApJ...687..919W" rel="nofollow noreferrer">Wang et al. 2008</a>).</p>

<h2>Excursion set theory</h2>

<p>Using a formalism know as <em>excursion set theory</em>, the mass fraction in knots, filaments, sheets, respectively, can be predicted analytically. The idea of this theory is to model the collapse of structure in an expanding Universe as a stochastic process, where an overdensity fluctuates and forms a structure if it crosses a given threshold. See e.g. <a href="https://ui.adsabs.harvard.edu/abs/2006ApJ...645..783S" rel="nofollow noreferrer">Shen et al. (2006)</a>.</p>

<h2>Dynamical classification</h2>

<p><a href="https://ui.adsabs.harvard.edu/abs/2009MNRAS.396.1815F" rel="nofollow noreferrer">Forero-Romero et al. (2009)</a> proposed a dynamical classification of the cosmic web, based on evalution of the <em>deformation tensor</em>
<span class="math-container">$$
T_{\alpha\beta} = \frac{\partial^2 \phi}{\partial r_\alpha \partial r_\beta}
$$</span>
of the gravitational potential <span class="math-container">$\phi$</span>, counting the number of eigenvalues above a certain threshold at each grid point in a cosmological N-body simulation, where the case of zero, one, two or three such eigenvalues corresponds to void, sheet, filament or a knot grid point.</p>

<h2>Other methods</h2>

<p>I quick <a href="https://www.google.dk/#q=quantifying+the+cosmic+web&amp;start=10" rel="nofollow noreferrer">googling</a> leads me to other methods I don't really know anything about, e.g. "the Shapefinder diagonistic" (<a href="https://ui.adsabs.harvard.edu/abs/2016IAUS..308..250S/abstract" rel="nofollow noreferrer">Prakash 2016</a>), and large-scale halo ellipticity–ellipticity and ellipticity–direction correlations (<a href="https://ui.adsabs.harvard.edu/abs/2008MNRAS.389.1266L" rel="nofollow noreferrer">Lee et al. 2008</a>).</p>

CC BY-SA 4.0
2016-11-10T19:16:06.897Z
2020-05-06T10:51:10.010Z
2020-05-06T10:51:10.010Z
pela
<p>I occasionally hear claims along the line of ' the mass of the universe is not uniformly distributed but actually is concentrated along a '  line, or some other lower  dimensional manifold &amp; I would like like to know what statistical techniques, if any,  are used to make support these claims.  I'm not interested  in whether the claims have held, just in the what was used for the original analysis.  I have data  whose dimension I would like to estimate, and am familiar with correlation dimension.</p>

CC BY-SA 3.0
2016-11-10T13:03:41.397Z
Recommend closing as too broad
Carl Witthoft
CC BY-SA 4.0
2020-05-06T10:45:12.840Z
So, did I answer your question, Mike?
pela
<mass>
statistical techniques for estimating distribution of mass
mike
# statistical techniques for estimating distribution of mass

(Asked by: username_0 on 2016-11-10)

I occasionally hear claims along the line of ' the mass of the universe is not uniformly distributed but actually is concentrated along a ' line, or some other lower dimensional manifold & I would like like to know what statistical techniques, if any, are used to make support these claims. I'm not interested in whether the claims have held, just in the what was used for the original analysis. I have data whose dimension I would like to estimate, and am familiar with correlation dimension.

**Tags:** mass

> **Comments:**
> **username_1**: Recommend closing as too broad  
> **username_2**: So, did I answer your question, Mike?  

---

## 1 answers

### Answer by username_2 on 2016-11-10. Score: 3

Matter is indeed distributed in knots (clusters), filaments, and sheets (e.g. [Bond et al. 1996](https://ui.adsabs.harvard.edu/abs/1996Natur.380..603B/abstract)). This is predicted from theories and numerical simulations of structure formation under gravitational collapse, and is directly observable in large galaxy surveys by eye, such as [The 2dF Galaxy Redshift Survey](http://www.2dfgrs.net/):

[![2dfgrs](https://i.sstatic.net/UxMUp.png)](https://i.sstatic.net/UxMUp.png)

## Two-point correlation function

To quantify this clustering, usually a *correlation function* is used, either in 2D ($x$ and *y* position on the sky), or 3D (if redshifts are available for the galaxies, giving the $z$ dimension). The (two-point) correlation function $\xi(r)$ gives the excess probability of finding two galaxies at a distance $r$ from each other, relative to a random (Poisson) distribution.
If $\bar{n}$ is the mean number density of galaxies, then the probability $dP$ of finding a pair of galaxies at a separation $r\_{12}$ in volume elements $dV\_1$ and $dV\_2$ is
$$
dP = \bar{n}^2 \big(1 + \xi(r\_{12})\big) dV\_1 dV\_2.
$$
The power spectrum $P(k)$ of the galaxies is related to the two-point correlation function by
$$
\xi(r) = \frac{1}{2\pi^2} \int dk\,k^2\,P(k)\,\frac{\sin kr}{kr}
$$
Observationally it is found that the two-point correlation function is given by a power-law:
$$
\xi(r) = \left( \frac{r}{r\_0} \right)^{-\gamma},
$$
where $r\_0$ is the correlation length (of the order $5\,h^{-1}\mathrm{Mpc}$) and $\gamma=1.7$–$1.8$.

Note that the value of $r\_0$ depends on galaxy type; brighter and redder galaxies have a longer correlation length, i.e. they are more strongly clustered (e.g. [Wang et al. 2008](https://ui.adsabs.harvard.edu/abs/2008ApJ...687..919W)).

## Excursion set theory

Using a formalism know as *excursion set theory*, the mass fraction in knots, filaments, sheets, respectively, can be predicted analytically. The idea of this theory is to model the collapse of structure in an expanding Universe as a stochastic process, where an overdensity fluctuates and forms a structure if it crosses a given threshold. See e.g. [Shen et al. (2006)](https://ui.adsabs.harvard.edu/abs/2006ApJ...645..783S).

## Dynamical classification

[Forero-Romero et al. (2009)](https://ui.adsabs.harvard.edu/abs/2009MNRAS.396.1815F) proposed a dynamical classification of the cosmic web, based on evalution of the *deformation tensor*
$$
T\_{\alpha\beta} = \frac{\partial^2 \phi}{\partial r\_\alpha \partial r\_\beta}
$$
of the gravitational potential $\phi$, counting the number of eigenvalues above a certain threshold at each grid point in a cosmological N-body simulation, where the case of zero, one, two or three such eigenvalues corresponds to void, sheet, filament or a knot grid point.

## Other methods

I quick [googling](https://www.google.dk/#q=quantifying+the+cosmic+web&start=10) leads me to other methods I don't really know anything about, e.g. "the Shapefinder diagonistic" ([Prakash 2016](https://ui.adsabs.harvard.edu/abs/2016IAUS..308..250S/abstract)), and large-scale halo ellipticity–ellipticity and ellipticity–direction correlations ([Lee et al. 2008](https://ui.adsabs.harvard.edu/abs/2008MNRAS.389.1266L)).


---

