Cosmological parallax–distance formula

Abstract

The standard cosmological parallax–distance formula, as found in the literature, including text-books and reference books on cosmology, requires a correction. This correction stems from the fact that in the standard text-book derivation it has been ignored that any chosen baseline in a gravitationally bound system does not partake in the cosmological expansion. Though the correction is available in the literature for some time, the text-books still continue to use the older, incorrect formula, and its full implications are not yet fully realized. Apart from providing an alternate correct, closed-form expression that is more suitable and convenient for computations for certain limiting cases of FRW ($\varLambda=0$) world models, we also demonstrate how one can compute parallax distance for the currently favored flat-space accelerating-universe ($\varLambda>0$, $k=0$) cosmologies. Further, we show that the correction in parallax distance at large redshifts could amount to a factor of three or even more. Moreover, even in an infinite universe the parallax distance does not increase indefinitely with redshift and that even the farthest possible observable point may have a finite parallax angle, a factor that needs to be carefully taken into account when using distant objects as the background field against which the parallax of a foreground object is to be measured. Some other complications that could arise in parallax measurements of a distant source, like that due to the deflection of incoming light by the gravitation field of the Sun and other planetary bodies in the solar system, are pointed out.