Cosmology: Estimating CMBR Horizon Distances & Inflation Size

[RingNebula57]

Hello everyone!
I have found a pretty interesting problem on the internet about cosmology. I'm new into cosmology and I don't know exactly how to solve it... That's why I need a little help. I wrote under the problem text how I would do it.Measurement of the cosmic microwave background radiation (CMB) shows that its temperature is practically the same at every point in the sky to a very high degree of accuracy. Let us assume that light emitted at the moment of recombination (T r ≈3000 K, t r ≈ 300000 years) is only reaching us now. Scale factor S is defined as such So= S (t=to) = 1 and S t= S (t<to) < 1
Note that the radiation dominated period was between the time when the inflation stopped(t=10^-32 seconds) and the time when the recombination took place, while the matter dominated period started at the recombination time. During the radiation dominated period S is proportional to t ^ 1/2, while during the matter dominated period S is proportional to t ^ 2/3 .

a. Estimate the horizon distances when recombination took place. Assume thattemperature T is proportional to 1/ S , where S is a scale factor of the size of the Universe

..Note: Horizon distance in degrees is defined as maximum separation between thetwo points in CMBR imprint such that the points could “see” each other at the timewhen the CMBR was emitted

b.Consider two points in CMBR imprint which are currently observed at a separationangle α = 5 °
. Could the two points communicate with each other using photon?(Answer with “YES” or “NO” and give the reason mathematically)

c.Estimate the size of our Universe at the end of inflation period.

At a. I assume that by knowing the temperatures we could find the scale factor and then related to the radius of universe from the formula R = R o * S(t), where R is the radius at time t and R o is current size

At b. I would calculate the rate at which the radius of the universe is increasing and then comparing it to the speed of a photon (c)

At c. I would calculate this rate in the same way as b and then integrate on the given time interval.

Thank you!

 

[PeterDonis]

RingNebula57, I have moved this post to the advanced physics homework forum, since you are basically asking for help with the solution of a set problem. To increase your chances of getting useful feedback, you should post the equations you would use for your proposed solution methods, and your attempt at using those methods to solve the problem.

 

[RingNebula57]

a. The scale factor now is by definition 1, and by knowing that it is proportional to the inverse of the time observation we can determine de scale factor at recombination :

S = To / T ; where To= 2,73k and T = 3000k
I don't really know how to solve after this, I thought of integrating hubbles law .

b. whitout integration , just calculating dR/dt from Hubble's law, where we coult asume that H=2/3 * t^-1

c. I would integrate like in a. from the recombination time to the current age of the universe:

dR/R = H dt ==> ln(R/R') = 2/3 * ln ( t/t') where t= 300000 yr , t'=1,5 bil yr

 

[PeterDonis]

S = To / T ; where To= 2,73k and T = 3000k

Yes, this is fine. But the question asks about "horizon distance", so you also need an equation for horizon distance. Can you find one?

whitout integration , just calculating dR/dt from Hubble's law, where we coult asume that H=2/3 * t^-1

How would this tell you whether two points can "communicate", i.e., can send light signals to each other? Remember that that's what "horizon distance" means: it's the maximum distance apart that two points can be, at a given time, and still send light signals to each other. So this question is basically asking how far apart two points, that appear 5 degrees apart in the sky now, were at the time the CMBR was emitted.

I would integrate like in a. from the recombination time to the current age of the universe

The recombination time is not the same as the time of the end of inflation. You first need to find the time of the end of inflation (which should be available in various references on inflationary cosmology); then the question is asking what the scale factor is at that time (at least, that's how I'm reading it).

 

[RingNebula57]

well , at a) after figuring out the scale factor I would find the rate at which the universe is expanding but I don't know how because if I were to find it from Hubble's law , I wouldn't know the Hubble constant because it varies in time. But if I knew it , I would do so: v=H(t) * R ; R is the universe size at recombination time given by R=Ro * S

the speed of light c = 2*v*cos(90-alpha/2) , and form here , knowing v , i would calculate alpha

at b) I would do the same thing as at a) but backwards: if 2*v* cos( 90-alpha/2) >c ( they can't see each other)
if 2*v * cos(90-alpha/2) < c ( they can see each other)

And the v is calculated the same as it is in a) but now we know the Hubble's constant = 72km/s/Mpc, I don't know how we would determine it if we don't know the size of the universe... v=Ho * Ro
And at c) I was wrong when I wrote recombination time. So:

S = To/T ; S(i)/S=(t(i)/t)^(1/2) , where t(i)=10^-32 sec, t=300000 years, and from here we found S(i)= scale factor an end of inflation, but how would we determine the size of the universe?

 

[PeterDonis]

after figuring out the scale factor I would find the rate at which the universe is expanding

How would that help you find the horizon distance?

i would calculate alpha

Part a of the question is not asking for an angle, it's asking for a distance. Do you have a formula for horizon distance?

Even for part b, the diagram you drew is not what the angle alpha means. The universe is not expanding from a center, so alpha is not the angle between two velocity vectors. Alpha is the angle between two CMB light rays reaching us now; it has nothing to do with velocities of anything.

at b) I would do the same thing as at a) but backwards

You can't. See above. First, you need to calculate the horizon distance at the time of the CMB emission, which means you need a formula for horizon distance. Then, you need to calculate, for two points in the sky that appear to us an angle alpha apart now, how far apart (distance, not angle) they were at the time of the CMB emission. Then you can compare that distance to the horizon distance at the time of the CMB emission.

S = To/T ; S(i)/S=(t(i)/t)^(1/2) , where t(i)=10^-32 sec, t=300000 years,

What does 10^-32 sec have to do with anything? The question is not asking anything about the time of inflation.

 

[PeterDonis]

I wouldn't know the Hubble constant because it varies in time.

Yes; but if you know the dynamics of the universe, you know how it varies in time. It looks like the question wants you to assume that the universe is matter dominated from the time of CMB emission to the present (which is not actually true, the universe has been dark energy dominated for the past few billion years, but it works for purposes of this problem). That implies a certain relationship between the scale factor and time.

 

[RingNebula57]

Ok, the Hubble constant can be determined using Friedmann s equation or using the given proportionality, but I don't fully understand the concept of horizon distance... could you expkain it in more detail please? Maybe with a drawing.
In my opinion, the horizon distance is the apparent radius of the universe . The formula I would use is d=c/H , H is the Hubble constant.

 

[PeterDonis]

I don't fully understand the concept of horizon distance... could you expkain it in more detail please?

The concept is related to the "horizon problem" in cosmology, which is what it looks like the problem you posted is getting at. See here for an overview:

https://en.wikipedia.org/wiki/Horizon_problem

Also see further comments below.

In my opinion, the horizon distance is the apparent radius of the universe .

If you mean the radius of the observable universe, that's not the same as the horizon distance, at least not the "horizon" that is referred to in the horizon problem in cosmology. There are actually several different things that can be referred to as "horizons" in cosmology; see this article:

https://en.wikipedia.org/wiki/Cosmological_horizon

To briefly summarize what the article is saying, as it relates to this discussion:

(1) The "particle horizon" is the boundary of the observable universe. Note that, strictly speaking, this boundary is not a "distance", it's our past light cone; i.e., it's the set of all light rays from distant objects in the universe that are just reaching us at this instant. But we can convert it to a distance by calculating how far away an object whose light rays are just reaching us at this instant would be "now" (i.e., at this cosmological time), assuming it is "commoving", i.e., at rest in standard FRW coordinates.

(2) The "Hubble horizon" is the distance given by $d = H c$, the formula you gave. Note that this is not the distance to the particle horizon or the cosmological horizon (see below). That is because the expansion of the universe is not linear.

(The article talks about other kinds of horizons as well, but those aren't relevant for this discussion.)

The "horizon distance" that is relevant for the horizon problem in cosmology is related to the particle horizon distance. It should be evident from the definition above that, while we usually talk about our particle horizon distance, we could equally well apply the concept to any "spatial point" in the universe. The distance to a given point's particle horizon is the same, regardless of which point we choose, because the universe is homogeneous; so we can just calculate a "particle horizon distance" for any instant of cosmological time, without having to refer to any specific spatial point. Two points separated by this distance will "just" have been able to send light signals to each other since the time of the Big Bang.

Basically, part a of the question you posted is asking what the particle horizon distance, calculated in this way, is at the time of the CMB emission. Part b then wants you to calculate the following: take two points in the universe that were separated by the particle horizon distance at the time of CMB emission. Assume that CMB light rays from both points are just reaching Earth today. What will be the separation angle that we see between those two light rays? Will that angle be greater or less than 5 degrees?