Find how long a beam of light emitted at a certain time takes to reach

[xdrgnh]

Homework Statement

In the model of “Empty Universe” consider a galaxy X, which starts emitting light at time t = tH/1000 (14 My). If, at that time, the proper distance between “X” and our galaxy was 10 Mly at what time the first light from “X” is expected to arrive to our galaxy?

I posted this in the intro physics section because I suspect the math to do this is elementary. However the question is from my level 4000 Cosmology class.

Homework Equations

V=HD
Lp=Cth*ln(z+1)/z

Lp is proper length at time emitted and Cth is Hubble length.

The Attempt at a Solution

I'm pretty sure I just need to find how long it takes for the light beam to reach Earth from galaxy X. To do that I just integrate the Hubbles law equation and get this exponential for D which is De^(Ht) then set it equal to x=ct where c is speed of light. However I'm being given equations for other things and I don;t know how they are relevant to this problem.

 

[mark726]

First, let's define some terms to make sure we're on the same page. The Hubble length, denoted by Cth, is defined as the distance at which the recession velocity of a galaxy due to the expansion of the universe is equal to the speed of light. This is given by Cth = c/H, where c is the speed of light and H is the Hubble constant. The proper distance, denoted by D, is the distance between two objects at a given time, taking into account the expansion of the universe.

Now, in the "Empty Universe" model, the proper distance between two objects is given by D = De^(Ht), where De is the proper distance at time t=0 and H is the Hubble constant.

In this problem, we are given that the proper distance between galaxy X and our galaxy at the time of emission (t=tH/1000) is 10 Mly. We can use this information to find the proper distance at time t=0, which is given by De = D/e^(HtH/1000) = 10/e^(HtH/1000) Mly.

Next, we need to find the time it takes for the light from galaxy X to reach our galaxy. This can be done by using the equation for proper length at time t emitted, which is given by Lp = Cth*ln(z+1)/z, where z is the redshift. Since the light from galaxy X is emitted at t=tH/1000, we can set t=tH/1000 in this equation. This gives us Lp = Cth*ln(z+1)/z = Cth*ln(1+z)/z. We know that the redshift, z, is given by z = (a(tH)/a(t emitted)) - 1, where a(t) is the scale factor of the universe at time t. Since we are in an "Empty Universe" model, the scale factor at any time is given by a(t) = e^(Ht). Plugging this into our equation for z, we get z = (e^(HtH)/e^(HtH/1000)) - 1 = e^(HtH/1000) - 1.

Now, we can plug this value of z into our equation for Lp to get Lp = Cth*ln(e