# A Newtonian theory of cosmological perturbations

1. Sep 18, 2016

### 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?

2. Sep 18, 2016

### Staff: Mentor

Just solve the differential equation?

3. Sep 18, 2016

### 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?

4. Sep 18, 2016

### Staff: Mentor

It is called instability...

5. Sep 18, 2016

### 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?

6. Sep 18, 2016

### Chalnoth

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.

7. Sep 19, 2016

Got it!

Thanks!