Density fluctuations of the present Universe with Large scale structures

Question

Disclaimer The question below is based on a vague knowledge, and hence, statements can be potentially wrong or misleading.

An important quantity in Cosmology is the density fluctuation of matter quantified in terms of the ratio $$\frac{\delta\rho(\textbf{x})}{\bar{\rho}}=\frac{\rho(\textbf{x})-\bar{\rho}}{\bar\rho}\tag{1}$$ where $\bar{\rho}$ is the average density. If I understand it correct, the information of CMB anisotropies enables us to predict the density fluctuations at the time of recombination.

Question $1$ Is it meaningful to talk about density fluctuations $\frac{\delta\rho(\textbf{x})}{\bar{\rho}}$ for the present universe?

Question $2$ If yes, how can it be measured and what is its present value, and can it be related to its primordial value?

Answer 1

The term that you are referring to is called density contrast.

$$\delta = \frac{\delta\rho(\textbf{x})}{\bar{\rho}}$$ Usually its denoted as $\delta$. However some books uses different notations such as in Longair its denoted as $\Delta$.

Now for galaxies, the present $\delta$ can be found by

$$\delta_{galaxy} = \frac{\rho_{galaxy}}{\rho_{cric}}$$ where $\rho_{crit} = 10^{-26}~~kg/m^3$$(Equivalent to the critical energy density)

For a galaxy which has an average density, this is equal to $\delta_{galaxy} \approx 10^{6}$

For a cluster $\delta_{cluster} \approx 10^3$

Since the $\rho$ (average matter density) changes as $(1+z)^3$ for $z \approx 100$ the $\delta_{galaxy} \approx 1$.

For more information, Longair, Galaxy Formation, 2nd Ed, (Chp 11. Page 312)