A Newtonian theory of cosmological perturbations

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?
 
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Just solve the differential equation?
 
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?
 
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Does the exponential instability of flat space-time to the development of fluctuations refer to exponential growth or exponential decay?
It is called instability...
 
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

Science Advisor
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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.
 
Got it!

Thanks!
 

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