cosmic density perturbations grow with time ?

[Widdekind]

For sake of simplicity, assume a flat cosmology. Then, from the Friedmann equations:

 

[math]H^2 = \frac{8 \pi G \rho}{3}[/math]

 

[math]\dot{\rho} = - 3 H \rho = - \sqrt{24 \pi G} \; \rho^{3/2}[/math]

 

If, at one moment of time, two separate regions of the universe have slightly different densities; then they will have slightly different scale factors, and expansion rates, according to the above equations. Seemingly, the density difference would evolve as:

 

[math]\frac{\partial}{\partial t} \left( \rho - \bar{\rho} \right) = - \sqrt{24 \pi G} \; \left( \rho^{3/2} - \bar{\rho}^{3/2}\right)[/math]

 

[math]\approx - \sqrt{24 \pi G} \; \left( \bar{\rho}^{3/2} \left( 1 + \frac{3}{2} \frac{\delta \rho}{\bar{\rho}} \right) - \bar{\rho}^{3/2}\right)[/math]

 

[math]= - \frac{9}{2} \sqrt{ \frac{8 \pi G \bar{\rho}}{3} } \; \delta \rho[/math]

 

[math]= - \frac{9 H_0}{2} \sqrt{\Omega_0} \; \alpha^{-3/2} \delta \rho[/math]

 

[math]= - \frac{3}{t_0 } \alpha^{-3/2} \delta \rho[/math]

 

[math]\boxed{ \frac{\partial (\delta \rho) }{\partial \tau} \approx - 3 \frac{\delta \rho}{\tau} }[/math]

 

wherein the scale factor and time have been normalized:

 

[math]\alpha \equiv \frac{a(t)}{a_0}[/math]

 

[math]\tau \equiv \frac{t}{t_0} = \frac{3 t}{2 H_0^{-1}} [/math]

 

So, seemingly, density perturbations (in a flat cosmology) decay away as [math]\delta \rho \propto t^{-3}[/math]. According to the Friedmann equations, denser regions expand faster, et vice versa. So, expanding (flat) space-time tends to "smooth" and "unwrinkle" itself as it stretches. Regions initially denser expand faster, and "catch up"; regions initially diffuser expand slower, and "drop back". So, why do textbooks state that density perturbations grow with time? Instead, i understand, from the above, that all density perturbations derive from the flow of matter through space-time, i.e. from "peculiar velocities" as seen in galaxies residing in cosmic large-scale Structures -- for matter fixed within the fabric of space-time, the expansion of the universe tends to smooth such density differences.

Edited December 11, 2012 by Widdekind