What Is the Rate of Continuous Creation Needed to Maintain Universal Density?

[epsilon]

Question:
Estimate the rate of continuous creation required to keep the density of the universe constant at 10^-26kg/m³. Express your answer in protons/year/km³.

Attempt:
Assuming a spherical matter-dominated Friedmann universe, we know from solving the fluid equation that ρ ∝ 1/a³ (where ρ is the energy density and a is the scale factor).

I believe the result of the continuous creation is that ∂ρ/∂r = 0 - hence the change in density due to expansion is canceled out by the change in density due to the continuous creation:
∂ρ_expansion/∂r = -∂ρ_creation/∂r
Thus I am expecting that the continuous creation's energy density will vary as ρ_creation ∝ a³.

However I do not know how to progress this any further, such that any advice, suggestions or solutions would be greatly appreciated! Note: I do not know what the correct answer is!

 

[Simon Bridge]

You need to know how the scale parameter $a$ changes with time.

 

[epsilon]

And so this is the Hubble parameter then?

H = 1/a × da/dt

 

[Simon Bridge]

That seems a reasonable place to go... a lot depends on your coursework to date.

 

[epsilon]

OK thank you! :-)