How Large Is a CMB Hot Spot Today Compared to the Virgo Supercluster?

[damasgate]

CMB hot spots have a physical size that corresponds to the size of the horizon at age
300; 000 yrs. Assume one such hot spot region has been expanding together with the
universe, how big in physical size (express in unit of light-year) has it become today?
For this exercise, use a model of the universe that is matter-dominated and assume

Omega = 1, so a proportional to t^(2/3) For comparison, the Virgo super-cluster currently has a size of 100 million light years, and it is marginally expanding with the Hubble
ow. There
are no structure larger than a super-cluster in our universe today.

What I know and tried:

-I know that a matter dominated univese is 30% of the density of the univese.
-It look like "the physical size" is measured in (light years) somehow which also confuses me

I think I just need a jumpstart explanation to get this but I just don't know how to start

 

[cepheid]

CMB hot spots have a physical size that corresponds to the size of the horizon at age
300; 000 yrs.

What you have to do is figure out the physical horizon scale at t = 300,000 years. Do you know how to do this? Are you familiar with the Friedmann equation and the Robertson-Walker metric?

The horizon scale is the largest distance over which light can have traveled since the beginning of the universe. In other words, it is the farthest possible distance over which information can have propagated. An observer cannot yet have been influenced by events occurring outside of his or her horizon (therefore he cannot have any knowledge of them).

For a universe that has always been matter-dominated, it turns out that the physical horizon scale is on the order of ct, where t is the age of the universe (just as one would expect naively). But it's actually a bit larger than that due to the expansion, and you may be expected to compute it more exactly than that using the Friedmann equation.

-I know that a matter dominated univese is 30% of the density of the univese.

I'm not sure what you mean here. What "matter-dominated" means is that the dynamics of the expansion of the universe are dominated by the density of (non-relativistic) matter. That's because the density of this matter is much larger than the density of radiation and relativistic particles, which contribute negligibly.

-It look like "the physical size" is measured in (light years) somehow which also confuses me

Why does it confuse you that the size of some region is measured in light years?