Throughout this question, assume that the universe the universe only contains matter and a possible cosmological constant. use the definition of the density parameter to show that at any epoch we can write the matter density as constant-ω0h0 given that in any realistic cosmology s2mat ~ 1 at early times, compute the hubble parameter at decoupling (zdec 1000) as a fraction of its present value. as the only important characteristic scale in the young universe, the hubble length ch gives the characteristic scale of the first peak in the microwave background. in a spatially-flat cosmology with a cosmological constant, the present physical distance to an object with z »> 1 is given approximately by 0 [if you wish, you can verify this approximation by numerical integration along the lines of problem a2.4.] by considering the angular diameter distance, demonstrate that the angle subtended by the hubble length at decoupling is approximately inde- pendent of s2o in spatially-flat cosmologies, and compute its value in degrees. this demonstrates that the peak position is nearly independent of 2a for spatially-flat geometries, the approximations being the formula for aoto above and the assump- tion that the universe is perfectly matter dominated at last-scattering.