Time dependence of the Ratio Hubble length to scale factor

[timmdeeg]

Let's assume a universe like ours which after inflation expands decelerated and accelerated thereafter.
How will the ratio Hubble length $1/H$ to scalefactor $a$ evolve over time? And how could one calculate the time dependence of this ratio.
Any help appreciated.

 

[Jorrie]

Let's assume a universe like ours which after inflation expands decelerated and accelerated thereafter.
How will the ratio Hubble length $1/H$ to scalefactor $a$ evolve over time? And how could one calculate the time dependence of this ratio.

The standard Friedmann equation for H(a) gives this relationship directly and it looks like this:

.

To find the relationship against time, you will have to integrate H(a) over time.

If you are prepared to ignore radiation density, you can use Marcus' "Hypersine model" to simplfy calculations.

 

[Chalnoth]

Let's assume a universe like ours which after inflation expands decelerated and accelerated thereafter.
How will the ratio Hubble length $1/H$ to scalefactor $a$ evolve over time? And how could one calculate the time dependence of this ratio.
Any help appreciated.

What is the point of this ratio?

 

[timmdeeg]

The standard Friedmann equation for H(a) gives this relationship directly and it looks like this:
View attachment 91691.

To find the relationship against time, you will have to integrate H(a) over time.

If you are prepared to ignore radiation density, you can use Marcus' "Hypersine model" to simplfy calculations.

Thanks. I will try to learn to use your calculator.

The ratio I'm asking for is $1/Ha$. Replacing $H$ by $(da/dt)/a$ yields $1/(da/dt)$. Therefore $1/Ha$ should increase as long as the universe expands decelerated and decrease during accelerated expansion then. Kindly correct.

 

[timmdeeg]

What is the point of this ratio?

During Inflation the scalefactor grows exponentially and thus much faster than the Hubble length, being almost constant, which solves the horizon problem. I'm interested to learn how that evolves in case the universe expands accelerated (but not exponentially).

 

[Jorrie]

During Inflation the scalefactor grows exponentially and thus much faster than the Hubble length, being almost constant, which solves the horizon problem. I'm interested to learn how that evolves in case the universe expands accelerated (but not exponentially).

Yes, I think this would be the case. The LightCone calculator actually provides the inverse of the function that you are looking for; it is labeled V_gen, representing the recession speed history of a 'generic galaxy' that presently happens to be on our Hubble sphere, where the recession speed equals c.

The red curve V_gen/c is the product of the gold curve (H/H₀) and the blue curve a(t). The minimum point is at about 7.5 Gy, where the expansion goes from decelerating to accelerating. As I said before: "If you are prepared to ignore radiation density, you can use Marcus' "Hypersine model" to simplfy calculations."

 

[Chalnoth]

During Inflation the scalefactor grows exponentially and thus much faster than the Hubble length, being almost constant, which solves the horizon problem. I'm interested to learn how that evolves in case the universe expands accelerated (but not exponentially).

Well, what growth rate are you thinking of if not exponential?

If the current accelerated expansion is described by a cosmological constant, then it will asymptotically approach exponential expansion.

 

[timmdeeg]

Yes, I think this would be the case. The LightCone calculator actually provides the inverse of the function that you are looking for; it is labeled V_gen, representing the recession speed history of a 'generic galaxy' that presently happens to be on our Hubble sphere, where the recession speed equals c.

The red curve V_gen/c is the product of the gold curve (H/H₀) and the blue curve a(t). The minimum point is at about 7.5 Gy, where the expansion goes from decelerating to accelerating. As I said before: "If you are prepared to ignore radiation density, you can use Marcus' "Hypersine model" to simplfy calculations."

It's impressing that the Cosmological Calculator can deal with products of curves. Thanks for the advise to use Marcus' "Hypersine model".

The inflection point (hopefully this is the right expression) of the blue curve seems to coincide with the minimum of the red curve, as it should.

Thanks for helping.

 

[timmdeeg]

Well, what growth rate are you thinking of if not exponential?

Sorry, it was misleading to mention inflation. If the scale factor grows faster than the Hubble length, then far away galaxies will become invisible (causally disconnected) over time. I wasn't really sure regarding the criterion for that. But that has been clarified.

 

[Jorrie]

It's impressing that the Cosmological Calculator can deal with products of curves.

The LightCone calculator does not generally handle products of curves, it is just that some are built in, because they are very useful. There is limit on how many is possible in such a calculator.

What I do if I need more such functionality is to use the tabular "Office" output with an increased number of steps and then paste it into a spreadsheet or similar. Further manipulation and/or and graphing is more feasible there.

 

[timmdeeg]

Just probing the time dependence of $aH$ another way: At $t=0$, the very beginning of decelerated expansion (after inflation has ended) some comoving observers shall have distances to each other such that their Hubble spheres just don't overlap. Then these spheres should start to overlap increasingly until deceleration switches to acceleration. From this time on the overlapping areas should shrink to zero a certain time $t$ later. If correct, how would one calculate this time $t$?
Later said observers are causally disconnected forever.
Its my best guess, thanks for commenting.

 

[Jorrie]

Then these spheres should start to overlap increasingly until deceleration switches to acceleration. From this time on the overlapping areas should shrink to zero a certain time tt later. If correct, how would one calculate this time tt?

It is not very clear what time you are referring to, but it is possible to calculate the Hubble radius and the cosmological event horizon distance. LightCone7 does both and they compare as follows:

The model used is in the link at the bottom of the LightCone page. Maybe this will help you decide on which integration function you need.

 

[timmdeeg]

It is not very clear what time you are referring to, but it is possible to calculate the Hubble radius and the cosmological event horizon distance.

Its interesting, but I was looking for the comparison of the Hubble radius (blue curve in 12#) with a (blue curve in #6). Is that possible? Otherwise I can put the 2 diagrams together.

 

[Jorrie]

Yes, it is possible, but with the standard Lightcone 7, the two vertical scales are quite different and the comparison does not mean much. However Lightcone 7z (both links in my sig) gives compatible values on the normalized 'zeit scale'. Give it a try.

 

[timmdeeg]

Yes, it is possible, but with the standard Lightcone 7, the two vertical scales are quite different and the comparison does not mean much. However Lightcone 7z (both links in my sig) gives compatible values on the normalized 'zeit scale'. Give it a try.

I will. Jorrie, you have been very helpful, thanks.