# I Small scale homogeneity

#### Apashanka

From the continuity equation $\frac{\partial \rho}{\partial t}+\rho (\nabla • u)=0$ where $\rho$ is the mass density and is homogeneous and $u$ is the velocity of expansion or contraction.
For an expanding volume this becomes $\nabla•u=\frac{\dot v(t)}{v(t)}=\Theta$ which gives the rate of expansion ,$v(t)$ is the volume.
Now if we consider our volume $V(t)$ to be consist of small patches of volume $v(t)$ where $\rho$ is homogeneous in $v(t)$ ,local expansion rate of volume $V(t)$ is therefore $\Theta$ ,now if the variance of this $\Theta$ is taken over $V(t)$ it will be $\sigma_\Theta=[\bar{\Theta^2}-(\bar \Theta)^2]$.
Now my question is it small scale homogeneity is sufficient for this non-zero variance of volume expansion rate??

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