Introduction to Cosmology Ryden · Problem 6.8

Measuring Cosmological Parameters

Physics — Astrophysics · olbers-paradox-flux-intensity

Problem

A spatially flat universe contains a single component with equation-of-state parameter ww. In this universe, standard candles of luminosity LL are distributed homogeneously in space. The number density of the standard candles is n0n_0 at t=t0t = t_0, and the standard candles are neither created nor destroyed. Show that the observed flux from a single standard candle at redshift zz is f(z)=L(1+3w)216π (c/H0)21(1+z)2[1−(1+z)−(1+3w)/2]−2,f(z) = \frac{L(1+3w)^2}{16\pi\,(c/H_0)^2}\frac{1}{(1+z)^2}\left[1-(1+z)^{-(1+3w)/2}\right]^{-2}, when w≠−1/3w \neq -1/3. What is the corresponding relation when w=−1/3w = -1/3? Show that the observed intensity (that is, the power per unit area per steradian of sky) from standard candles with redshifts in the range z→z+dzz \to z + dz is dJ(z)=n0L (c/H0)4π(1+z)−(7+3w)/2 dz.dJ(z) = \frac{n_0 L\,(c/H_0)}{4\pi}(1+z)^{-(7+3w)/2}\,dz. What will be the total intensity JJ of all standard candles integrated over all redshifts? Explain why the night sky is of finite brightness even in universes with w≤−1/3w \leq -1/3, which have an infinite horizon distance.

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