Is there an alternate way to describe the expansion of the universe?

[properman]

As the title suggests, I am wondering if there is an alternate way to describe the expansion of the universe outside of the Friedmann Equation. Specifically, can the expansion of the universe be modeled on time alone?

 

[BillSaltLake]

The expansion factor "a" can be written as a function of time (although not in closed form). After the period of inflation, a~t^1/2 until about 10,000 yr. Then a~t^2/3. Now a is increasing a little faster, and since around t = 1 billion yr, a has increased about 1.2x what it would have been if it were at the t^2/3 rate alone. At present a~t¹, or more precisely, (1/a)(da/dt)~1/t now. (It was [1/2]/t and [2/3]/t at the earlier eras.)

 

[Chalnoth]

As the title suggests, I am wondering if there is an alternate way to describe the expansion of the universe outside of the Friedmann Equation. Specifically, can the expansion of the universe be modeled on time alone?

What do you mean, modeled on time alone?

 

[properman]

The expansion factor "a" can be written as a function of time (although not in closed form). After the period of inflation, a~t^1/2 until about 10,000 yr. Then a~t^2/3. Now a is increasing a little faster, and since around t = 1 billion yr, a has increased about 1.2x what it would have been if it were at the t^2/3 rate alone. At present a~t¹, or more precisely, (1/a)(da/dt)~1/t now. (It was [1/2]/t and [2/3]/t at the earlier eras.)

Oh well. Thank you anyway.

What do you mean, modeled on time alone?

I meant was looking for a single variable function of expansion over time, ie. a(t). But apparently from the post above, that does not exist as a smooth function.

 

[Chalnoth]

Oh well. Thank you anyway.
I meant was looking for a single variable function of expansion over time, ie. a(t). But apparently from the post above, that does not exist as a smooth function.

Sure, a(t) is a smooth function. It is continuous because a discontinuity in a(t) would mean a discontinuity in distances, which would mean infinite velocities. That isn't happening.

Its first derivative is continuous because its first derivative is a function of the matter density, and conservation laws ensure that the matter density can't be discontinuous in time.

Even its second derivative is continuous because that depends upon the matter density and the pressure, and the pressure also doesn't change in a discontinuous fashion (though pressure can come pretty close to changing in a discontinuous fashion, it still doesn't change instantly).

The Friedmann equations tell us this, by the way, because they are differential equations in a(t) as a function of the matter content. They tell us not just about the expansion, but about how the expansion changes depending upon the matter density.

 

[properman]

Right. Didn't really think that through... So then would there be such a function that is not the piece-wise function mentioned above? Would it be to solution to one of the Friedmann equations?

Thanks for putting up with my random questions, I just was stricken by an idea earlier, and in order for it to work I would need a function for the expansion of the universe with regard to time.

 

[Chalnoth]

Right. Didn't really think that through... So then would there be such a function that is not the piece-wise function mentioned above? Would it be to solution to one of the Friedmann equations?

Thanks for putting up with my random questions, I just was stricken by an idea earlier, and in order for it to work I would need a function for the expansion of the universe with regard to time.

I don't understand what you're asking here.

 

[properman]

The expansion factor "a" can be written as a function of time (although not in closed form). After the period of inflation, a~t^1/2 until about 10,000 yr. Then a~t^2/3. Now a is increasing a little faster, and since around t = 1 billion yr, a has increased about 1.2x what it would have been if it were at the t^2/3 rate alone. At present a~t¹, or more precisely, (1/a)(da/dt)~1/t now. (It was [1/2]/t and [2/3]/t at the earlier eras.)

I don't understand what you're asking here.

Those above equations require the use of different equations for different times in the expansion of the universe. What I am looking for is one equation that can model the expansion for the entirety of the expansion.

 

[Chalnoth]

Those above equations require the use of different equations for different times in the expansion of the universe. What I am looking for is one equation that can model the expansion for the entirety of the expansion.

Oh, well, the matter content of the universe changes with time, and the matter content determines how the expansion changes over time. So if you want to have one equation that models the whole expansion, it's going to have to take into account the matter content and how it's changed.

 

[properman]

Oh, well, the matter content of the universe changes with time, and the matter content determines how the expansion changes over time. So if you want to have one equation that models the whole expansion, it's going to have to take into account the matter content and how it's changed.

Ok, so you are telling me that in order to get a perfect model it would need to be a differential equation, which of course is what the Friedmann equations are. well there goes that idea.

 

[Chalnoth]

Ok, so you are telling me that in order to get a perfect model it would need to be a differential equation, which of course is what the Friedmann equations are. well there goes that idea.

Pretty much, yes :)

Basically, there are three main ways to extend the Friedmann equations:
1. Propose a different theory of gravity. The Friedmann equations are based upon General Relativity. Quantum gravity may potentially modify these equations, particularly at very high densities.
2. Propose some new sort of matter density. This obviously isn't really an extension of the Friedmann equations, but can be an extension of the standard cosmological model (which is basically normal matter + cold dark matter + cosmological constant).
3. Take into account the fact that the universe isn't perfectly smooth. The Friedmann equations assume this, but obviously it isn't actually true. There are numerous approximations that can be used to estimate the effects of the fact that the universe isn't perfectly smooth.

 

[properman]

Alright, thank you very much for your help.

 

[BillSaltLake]

Before the recent acceleration, there is a closed-form solution for t as a function of a, but not for a(t). The solution for recent acceleration is model-dependent, and for ΛCDM model, a is ~proportional to [sinh(bt)]^2/3, where b is a constant. (The sinh form ignores very early time when energy dominated.) This assumes flat space.