Introduction to Cosmology Ryden · Problem 11.6

Structure Formation: Gravitational Instability

Physics — Astrophysics · gaussian-density-field-mass-fluctuations-power-spectrum

Problem

Suppose that the density fluctuations

\delta(\vec{r})

in the early universe constitute a Gaussian field with a power spectrum

P(k)

that equals zero above some maximum wavenumber

k_{\rm max}

. This maximum wavenumber corresponds to a minimum length scale

r_{\rm min} = 2\pi/k_{\rm max}

and a minimum mass scale

M_{\rm min} = (4\pi/3)\rho_m r_{\rm min}^3

. Show that for

M < M_{\rm min}

, the mean square mass fluctuation,

\langle(\delta M/M)^2\rangle

, is equal to

\sigma_\delta^2

for the density field.