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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.

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  • $\begingroup$ Recommend closing as too broad $\endgroup$ – Carl Witthoft Nov 10 '16 at 13:03
  • $\begingroup$ So, did I answer your question, Mike? $\endgroup$ – pela May 6 '20 at 10:45
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Matter is indeed distributed in knots (clusters), filaments, and sheets (e.g. Bond et al. 1996). 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:

2dfgrs

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).

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).

Dynamical classification

Forero-Romero et al. (2009) 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 leads me to other methods I don't really know anything about, e.g. "the Shapefinder diagonistic" (Prakash 2016), and large-scale halo ellipticity–ellipticity and ellipticity–direction correlations (Lee et al. 2008).

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