# 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

Mentor
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

Mentor
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!

"Newtonian theory of cosmological perturbations"

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