Tolman surface brightness test

Remarks

Proof idea: the luminosity distance $d_L$ and the angular diameter distance $d_A$ satisfy Etherington's reciprocity relation $d_L = (1+z)^2 d_A$. The received flux is $L/(4\pi d_L^2)$ and the solid angle of a patch of area $A$ is $A/d_A^2$, so the surface brightness scales as $(d_A/d_L)^2 = (1+z)^{-4}$. One factor of $1+z$ comes from the energy loss of each photon. One factor comes from the lower arrival rate of photons. Two factors come from the apparent enlargement of the source.

Example: a galaxy at $z = 1$ appears $2^4 = 16$ times dimmer per unit solid angle than an identical galaxy nearby, which is $3$ magnitudes per square arcsecond.

In a static universe where redshift comes from a "tired light" mechanism, only the energy loss acts and the dimming is $(1+z)^{-1}$. Richard Tolman proposed comparing the two exponents as a test of the expansion. In practice the test is difficult because galaxies were intrinsically brighter in the past, so the evolution of their luminosity must be modelled.

Going deeper

For a student

The result is purely kinematic: it uses only the metric and photon conservation, not the Friedmann equations. It is equivalent to the reciprocity relation $d_L = (1+z)^2 d_A$, so a measured violation would point to photon loss or to physics beyond a metric theory. For a surface brightness per unit frequency, the exponent depends on the spectral shape of the source. Observers correct for this shift of the band by the K-correction.