Friedmann cosmology and redshift

Use the FLRW line element $ds^2=c^2dt^2-a(t)^2[d\chi^2/(1-K\chi^2)+\chi^2d\Omega^2]$. Here a is dimensionless, chi has length units, and constant K has inverse-length-squared units. Normalize $a(t_0)=1$ at the chosen present reference time. Ideal comoving observers have fixed spatial coordinates and proper time t. Exact homogeneity and isotropy are modeling assumptions for the background, not claims that galaxies and empty intergalactic regions have identical local density.

For neighboring comoving points on a fixed-time slice, proper separation scales with a. Gravitationally bound or electromagnetically bound systems do not automatically follow this relation; their internal dynamics determine their size. The Hubble parameter $H=\dot a/a$ describes the instantaneous fractional expansion rate. It can change with cosmic time, so extrapolating a present value as a constant over the entire cosmic history is generally incorrect.

Local covariant stress conservation gives $\dot\varepsilon+3H(\varepsilon+p)=0$ for a homogeneous perfect fluid. It is the expanding-volume energy balance: pressure performs work as the comoving volume changes. For constant equation-of-state parameter $w=p/\varepsilon$, integrating gives $\varepsilon\propto a^{-3(1+w)}$. This applies to a separately conserved component with constant w; interacting components or changing equations of state require their own evolution equations.

Cold matter has negligible pressure, so its density falls as inverse volume, $a^{-3}$. Radiation additionally loses energy per photon as wavelengths stretch, giving $a^{-4}$. A cosmological-constant component has $w=-1$ and constant energy density. Photon redshift in an expanding geometry is not evidence for a missing local conservation law: a general FLRW background lacks the global time-translation symmetry that would supply an ordinary conserved total photon energy.

With our Riemann and signature conventions write the cosmological field equation as $G_{\mu\nu}-\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}/c^4$, defining positive Lambda to contribute positive vacuum expansion. The FLRW time-time equation is $H^2=8\pi G\varepsilon/(3c^2)-Kc^2/a^2+\Lambda c^2/3$. Here epsilon excludes any vacuum energy already counted through Lambda; counting both would double the same contribution.

For flat geometry with only pressureless matter and no Lambda, the expanding solution has $a\propto t^{2/3}$ after choosing the big-bang time origin. A radiation-only counterpart has $a\propto t^{1/2}$. These are idealized eras, not complete descriptions of the observed universe. Determining cosmic distances requires integrating the expansion history; redshift alone does not supply a unique proper distance or a universally valid recession speed through the special-relativistic Doppler formula.