[Astrophysics] CBR and angular size

[Tloh]

Question:

Assuming a flat universe, find the angular size of the largest causally connected region of the CBR. Hint: We see this region as it existed at the time of decoupling, when the CBR photons were set free.

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Alright, flat universe, k = 0, tdec --> z~= 1100

Theta = (H0 * D / 2c) * ((1+z)^1.5) / ((1+z)^0.5)-1)

H0 = Hubble constant
D = Linear diameter
c = constant speed of light

I am stuck here. I can solve the z portion of the equation, but I don't know how to find the value of D, the causally connected section of the CBR. My guess is that I'm supposed to use the size of the universe at the tdec point (time just before the photons were set free). Any ideas?

-Thomas

 

[anathema]

Hello Thomas,

To find the angular size of the largest causally connected region of the CBR, we first need to understand what causally connected means in this context. In a flat universe, the causally connected region refers to the maximum distance that light (or any other information) can travel from one point to another since the beginning of the universe. This is limited by the speed of light, which is represented by the constant "c" in your equation.

Since we are looking at the CBR at the time of decoupling, we can assume that the photons have been traveling since the beginning of the universe and have reached the maximum distance they can travel. This means that the size of the causally connected region at the time of decoupling is the same as the size of the entire observable universe.

To find the size of the universe at the time of decoupling, we can use the equation for the comoving distance (D) at a given redshift (z): D = c/H0 * integral from 0 to z of (1/sqrt(Ωm(1+z)^3 + Ωk(1+z)^2 + ΩΛ)) dz.

Here, Ωm, Ωk, and ΩΛ represent the density parameters for matter, curvature, and dark energy, respectively. For a flat universe, Ωm = 1 and Ωk = ΩΛ = 0. Plugging these values into the equation and solving for D, we get D = (c/H0) * (2/3) * (1/sqrt(1+z)).

Now, we can plug this value of D into your original equation to find the angular size of the causally connected region of the CBR at the time of decoupling. Keep in mind that the value of z for this calculation will be around 1100, as you mentioned in your post.

I hope this helps! Let me know if you have any further questions.

 

[thepatientmental]

Great question, Thomas! To find the angular size of the largest causally connected region of the CBR, we need to first understand what causally connected means in this context. In astrophysics, causally connected refers to regions of space that have had enough time to communicate with each other since the beginning of the universe. This means that the photons in these regions have had enough time to travel and interact with each other, creating the CBR that we see today.

At the time of decoupling, when the CBR photons were set free, the universe was about 379,000 years old. This means that the largest causally connected region of the CBR would be the size of the observable universe at that time, which is about 42 billion light years in diameter.

To calculate the angular size, we can use the formula you provided, with D being the diameter of the observable universe at tdec. This gives us an angular size of about 1.7 degrees. This means that the largest causally connected region of the CBR would appear to us as a circle with a diameter of about 1.7 degrees on the sky.

I hope this helps! Keep exploring the wonders of astrophysics.