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Thanks Is it a problem with the induction proof? I'm not sure how to start.

Hello. This is one of my coursework questions I was wondering if I could get some insight here.. here is the question: The size of the Universe if conveniently parameterized by a scale factor, a(t), which simply describes how big the Universe is at other times relative to its present size (ie. at the present we say that a is 1, and at some time in the past when the Universe was half as big as it was today, then a was 0.5). A matter-dominated Universe grows with time as [math]a \propto t^{\frac{2}{3}} [/math]. Assuming the Universe is 13.5 billion years old at present, how old is the Universe at redshifts, z, of z = 0.5 ... etc, z= 100? Assume that we presently live in a matter-dominated Universe, and that the Universe is matter-dominated out to redshifts of at least 100. The formula for redshift relative to scale factor is [math] 1 + z = \frac{a_{now}}{a_{then}} [/math] Then, since [math]a \propto t^{\frac{2}{3}}[/math] then [math] 1 + z = t^{\frac{2}{3}}[/math] Then I plug in z and solve for t, then divide the current age by t?

a is the scale factor for the size of the universe where the a = 1 at the present time.. and 0.5 sometime in the past when it was half the size, etc.. and z is the redshift.

Hello. This is question for my course work, I was wondering if I could get some insight, here is the question: Assume that the vast majority of the photons in the present Universe are cosmic microwave radiation photons that are a relic of the big bang. For simplicity, also assume that all the photons have the energy corresponding to the wavelength of the peak of a 2.73K black-body radiation curve. At Approximately what redshift will the energy density in radiation be equal to the energy density in matter? (hint: work out the energy density in photons at the present time. Then work it out for baryons, assuming a proton for a typical baryon. Remember how the two quantities scale with redshift to work out when the energy density is the same.) [math] \rho_M \propto a^{-3} [/math] [math] \rho_\gamma \propto a^{-4} [/math] [math] T \propto a^{-1} [/math] [math] 1 + z = \frac{v}{v_0} = \frac{\lambda_0}{\lambda} = \frac{a(t_0)}{a(t)} [/math] How do I calculate the energy density of photons and protons at the present time? Do I use E = mc^2?