My lecturer has written the following:(adsbygoogle = window.adsbygoogle || []).push({});

Given [itex] \frac{4 \pi \rho a^3 c^2 }{3}[/itex] and [itex] V = \frac{4 \pi a^3}{3} [/itex], then substituting [itex]\frac{dE}{dt} = -p\frac{dV}{dt}[/itex] we have:

[itex]\frac{d(\rho a^3)}{da} = -\frac{3pa^2}{c^2} \leq 0 [/itex] (1)

Ok that part is fine - substitute and use the chain rule:

So, if we assume [itex]p \geq 0 [/itex] we then see that:

[itex]d(\rho a^3)} \geq 0 [/itex] if [itex] da < 0[/itex] or [itex]d(\rho a^3)} \leq 0 [/itex] if [itex] da > 0[/itex]

I understand how he has got this inequality, simply working from the above inequality, however I'm a little unsure of what da actually represents here, it then goes on to say:

From the first of the inequality, we see that as a decreases, [itex]\rho a^3[/itex] increases. In particular, [itex]\rho a^2[/itex] increases at least as fast as 1/a.

That is the bit I'm really struggling to get.

By the way, the he says that a is the scale factor and is a function of t only. He says it gives the "universal expansion rate" - I don't know if he means that it is the universal expansion rate, or just that you kind find the universal expansion rate from it. Either way I fail to see how it's a universal expansionratesince it is just a time dependent function and not a derivative itself.

So the way I interpret it as follows:

d(a(t)) = da/dt x dt - the total differential, qualitatively - to me this means the amount a changes when you make an infinitesimal change in t. So for da < 0, this means that as time goes on, the scale factor a is decreasing - the universal expansion is slowing?

So when he says "when a is decreasing" - he is using the first inequality - because da < 0 is the same as saying a is decreasing. So why does that mean [itex]\rho a^3[/itex] increases?

Any help is much appreciated guys, thanks.

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# Cosmology inequality that I'm struggling to make sense of

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