Event Horizon of Universe

[Cheetox]

Homework Statement

Compute the horizon of the universe as a function of $$\Omega$$_m in a flat universe with both matter and a cosmological constant but no radiation.

Homework Equations

Event horizon distance
r = a(t)c $$\int_0^tcdt'/a(t')$$

The Attempt at a Solution

No idea how I'm going to be able to transform that integral into something I can do,

Using the FRW equation you can get H² = $$\Lambda/ 3($$$$\Omega$$_m -1)

possibly using H = (da/dt)/a thus dt = da/aH ?

and then if you do substitute for an integral in a would the limits be 0 to 1 as we set a = 1 at present or 0 to a?

 

[Jhero]

Thank you for your post. Calculating the horizon of the universe as a function of \Omegam can seem daunting at first, but with some careful consideration and the use of the Friedmann equations, it is possible to arrive at a solution.

First, let's define some terms:
- r is the event horizon distance, which is the maximum distance that light can travel since the beginning of the universe.
- t is the age of the universe.
- a(t) is the scale factor, which describes the expansion of the universe.
- c is the speed of light.
- \Omegam is the density parameter for matter.
- \Lambda is the cosmological constant.

To start, we can use the Friedmann equation for a flat universe with both matter and a cosmological constant, given by:

H2 = (8πG/3)ρ - (kc2)/a2 + (Λc2)/3

Where H is the Hubble parameter, G is the gravitational constant, ρ is the density of matter, k is the curvature of the universe (which is 0 for a flat universe), and Λ is the cosmological constant.

Since we are considering a universe with no radiation, we can set the term (kc2)/a2 to 0. This leaves us with the following equation:

H2 = (8πG/3)ρ + (Λc2)/3

We can rearrange this equation to solve for the density of matter, ρ:

ρ = (3H2)/(8πG) - (Λc2)/8πG

Now, we can substitute this value of ρ into the event horizon distance equation:

r = a(t)c \int_0^tcdt'/a(t')

We can also substitute for the Hubble parameter using the relation H = (da/dt)/a, giving us:

r = c \int_0^tcdt'/a(t')

We can further simplify this equation by substituting for a(t) using the Friedmann equation we found earlier:

a(t) = (Λ/3)ρ/(H2)

Substituting this into the equation for r, we get:

r = c \int_0^tcdt'/[(Λ/3)ρ/(H2)]

We can now substitute in the value for ρ that we found earlier, giving us:

r