A Newtonian theory of cosmological perturbations

[spaghetti3451]

In the Newtonian theory of cosmological perturbations, a density excess $\delta\rho$ in a localised region of spacetime leads to the equation of motion $\ddot{\delta\rho} \sim G\delta\rho$. I can see that this follows directly from Newton's gravitational law.

Why does this equation lead to an exponential instability of flat spacetime to the development of fluctuations?

 

[mfb]

Just solve the differential equation?

 

[spaghetti3451]

Well, I get $\delta\rho = A\ e^{\sqrt{G}x} + B\ e^{-\sqrt{G}x}$.

I can see that this is the sum of an exponential growth and an exponential decay of the initial density fluctuation in a localised region of spacetime.

Does the exponential instability of flat space-time to the development of fluctuations refer to exponential growth or exponential decay?

 

[mfb]

Does the exponential instability of flat space-time to the development of fluctuations refer to exponential growth or exponential decay?

It is called instability...

 

[spaghetti3451]

Thanks!

I was also wondering why it is inconsistent in General Relativity to consider density fluctuations in a non-expanding background.

Why can't Einstein's field equations be used to analyse density fluctuations (in the early universe) in a non-expanding universe?

 

[Chalnoth]

Why can't Einstein's field equations be used to analyse density fluctuations (in the early universe) in a non-expanding universe?

I'm pretty sure it's just because it's not stable. There is no static solution to the Einstein field equations that both contains matter and is stable.

 

[spaghetti3451]

Got it!

Thanks!