Conservation of the phase-space density of photons plus Lorentz invariance requires that the cosmological luminosity distance be larger than the angular diameter distance by a factor of ( 1 + z ) ^ { 2 } , where z is the redshift .
Because this is a fundamental symmetry , this prediction—known sometimes as the “ Etherington relation ” or the “ Tolman test ” —is independent of the world model , or even the assumptions of homogeneity and isotropy .
It depends , however , on Lorentz invariance and transparency .
Transparency can be affected by intergalactic dust or interactions between photons and the dark sector .
Baryon acoustic feature ( BAF ) and type Ia supernovae ( SNeIa ) measures of the expansion history are differently sensitive to the angular diameter and luminosity distances and can therefore be used in conjunction to limit cosmic transparency .
At the present day , the comparison only limits the change \Delta \tau in the optical depth from redshift 0.20 to 0.35 at visible wavelengths to \Delta \tau < 0.13 at 95 % confidence .
In a model with a constant comoving number density n of scatterers of constant proper cross-section \sigma , this limit implies n \sigma < 2 \times 10 ^ { -4 } h~ { } { \mathrm { Mpc } } ^ { -1 } .
These limits depend weakly on cosmological world model .
Assuming a concordance world model , the best-fit value of \Delta \tau to current data is negative at the 2 \sigma level .
This could signal interesting new physics or could be the result of unidentified systematics in the BAF/SNeIa measurements .
Within the next few years , the limits on transparency could extend to redshifts z \approx 2.5 and improve to n \sigma < 1.1 \times 10 ^ { -5 } h~ { } { \mathrm { Mpc } } ^ { -1 } .
Cosmic variance will eventually limit the sensitivity of any test using the BAF at the n \sigma \sim 4 \times 10 ^ { -7 } h~ { } { \mathrm { Mpc } } ^ { -1 } level .
Comparison with other measures of the transparency is provided ; no other measure in the visible is as free of astrophysical assumptions .