Linearisation of continuity equation (cosmology)

In summary, when expanding to first order in ##\epsilon## and subtracting off the unperturbed equation, the resulting equation becomes $$ \frac{\partial \delta \rho}{\partial t} + 3H \delta \rho + \frac{\bar{\rho}}{a} \nabla \cdot \delta \mathbf{v}=0 $$ The term ##3H \delta \rho## causes confusion, but it can be better understood by using the fact that ##H = \dot{a}/a##, where ##\dot{a}## is the Hubble parameter. To solve for this term, the 0'th order equation $$ \partial_t \bar\rho ~+~
  • #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).
 
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  • #2
The 0'th order eqn is $$ \partial_t \bar\rho ~+~ 3 H \bar\rho ~=~ 0 ~.$$ You must use this in the 1st order eqn.

Additional hint: before writing out the 1st order eqn, compute ##\,\partial_t \left( \frac{\delta\rho}{\bar\rho} \right)## carefully, separately, using the 0'th order eqn.

[Question for other HW helpers: does the above give away too much of the solution in one go? I'm never really sure where the balance lies.]
 
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Related to Linearisation of continuity equation (cosmology)

What is the continuity equation in cosmology?

The continuity equation in cosmology describes the conservation of energy density in an expanding universe. It relates the rate of change of the energy density of a fluid (such as matter, radiation, or dark energy) to the expansion rate of the universe. Mathematically, it is often written as: \[ \frac{d\rho}{dt} + 3H(\rho + p) = 0 \]where \(\rho\) is the energy density, \(H\) is the Hubble parameter, and \(p\) is the pressure.

Why is linearisation of the continuity equation important in cosmology?

Linearisation of the continuity equation is important because it simplifies the complex, non-linear equations governing the dynamics of the universe into a form that is easier to solve and analyze. This is particularly useful for studying small perturbations in the cosmic fluid, which can help us understand the formation of large-scale structures like galaxies and clusters of galaxies.

How is the continuity equation linearised in cosmology?

To linearise the continuity equation, cosmologists typically assume small perturbations around a homogeneous and isotropic background. The energy density \(\rho\) and pressure \(p\) are expressed as the sum of a background component and a small perturbation: \(\rho = \rho_0 + \delta\rho\) and \(p = p_0 + \delta p\). The continuity equation is then expanded to first order in the perturbations, resulting in a linearised form that is easier to handle mathematically.

What are the applications of the linearised continuity equation in cosmology?

The linearised continuity equation is used in various applications, including the study of cosmic microwave background (CMB) anisotropies, the growth of cosmic structures, and the evolution of density perturbations in the early universe. It is a crucial tool for understanding how small initial fluctuations grew into the large-scale structures we observe today.

What are the limitations of using a linearised continuity equation in cosmology?

One of the main limitations of using a linearised continuity equation is that it is only valid for small perturbations. When perturbations become large, non-linear effects become significant, and the linear approximation breaks down. In such cases, more complex, fully non-linear treatments are required to accurately describe the dynamics. Additionally, the linearised approach may not capture all the relevant physical processes, such as interactions between different components of the cosmic fluid.

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