Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 10E (Cosmology)

For neighbouring comoving observers separated by a small physical distance $d\ell$, the recession velocity is $H(t)d\ell$, where $H=\dot{a}/a$ is the Hubble parameter. During the photon's travel time $dt=d\ell /c$, the local Doppler effect gives $d\lambda /\lambda =Hdt$. Integrating along the ray gives the cosmological wavelength-redshift relation:The same ratio stretches the arrival interval of successive wave crests and of successive emitted pulses: $d{t}_0=(1+z)d{t}_e$. Each photon loses a factor $1+z$ in energy because $E=hc/\lambda$. Thus energy arriving through the sphere per unit observer time has one factor from cosmological redshift and one from time dilation:This is the total power crossing the sphere; division by its area would give the observed flux.