- #1

ergospherical

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- Homework Statement
- Show linearisation of

##\frac{\partial \rho}{\partial t} + 3H \rho + \frac{1}{a} \nabla \cdot (\rho \mathbf{v}) = 0##

is

##\frac{\partial \delta}{\partial t} + \frac{1}{a} \nabla \cdot (\delta \mathbf{v}) = 0##

where

##\delta \equiv \delta \rho / \bar{\rho}##, ##\rho = \bar{\rho} + \epsilon \delta \rho##, ##\mathbf{v} = \epsilon \delta \mathbf{v}##, and ##\epsilon \ll 1##.

- Relevant Equations
- N/A

After expanding to first order in ##\epsilon## and subtracting off the unperturbed equation, I get\begin{align*}

\frac{\partial \delta \rho}{\partial t} + 3H \delta \rho + \frac{\bar{\rho}}{a} \nabla \cdot \delta \mathbf{v}=0

\end{align*}I'm not sure how to deal with the ##3H \delta \rho## term. Where does ##H## enter? (##H = \dot{a}/a## is the Hubble parameter).

\frac{\partial \delta \rho}{\partial t} + 3H \delta \rho + \frac{\bar{\rho}}{a} \nabla \cdot \delta \mathbf{v}=0

\end{align*}I'm not sure how to deal with the ##3H \delta \rho## term. Where does ##H## enter? (##H = \dot{a}/a## is the Hubble parameter).