Question on Cosmology Solution for Tired Light Hypothesis

[TRB8985]

Homework Statement

A hypothesis once used to explain the Hubble relation is the "tired light hypothesis". The tired light hypothesis states that the universe is not expanding, but that photons simply lose energy as they move through space (by some unexplained means), with the energy loss per unit distance being given by the law below.

Show that this hypothesis gives a distance relation that is linear in the limit of z << 1 (where k is a constant). What must the value of k be in order to yield a Hubble constant of 68 (km/s)/Mpc?

Relevant Equations

dE/dr = -kE

Good evening, I have a question on a cosmology problem I have solved from Barbara Ryden’s Introduction to Cosmology 2nd Edition. I believe I have answered the question correctly, resulting in the following linear redshift relation when using separation by variables and some algebra manipulation:

z ≈ kr

Which is just the Hubble law, with k having the following value:

k ≈ 2.3E-4 Mpc^-1

Here’s my question: Is there something in particular about this value that is ridiculous and warrants dismissing the hypothesis? I know that photons don't lose energy as they traverse through the universe, so perhaps the absurdity lie in the value of redshift a photon attains every Mpc? I'm not entirely sure.

Thank you for taking the time to read my question, and I appreciate any insights you might have.

 

[Nate810]

The value of k that you have derived is consistent with the accepted value of the Hubble constant, which is approximately 2.3 x 10^-18 s^-1. This is the rate of expansion of the universe, and it is not at all ridiculous. In fact, this value is one of the most important constants in cosmology and is used to measure distances and ages in the universe. What you have done is correctly derived the linear redshift relation, which relates the recessional velocity of an object to its distance from us (z ≈ kr). This is a very useful equation in cosmology and is used to calculate distances and ages of galaxies and other objects in the universe. So, there is nothing ridiculous about the value you have derived and it is consistent with the accepted value of the Hubble constant.