The Instantaneous Redshift Difference of Gravitationally Lensed Images: Theory and Observational Prospects

Abstract

Due to the expansion of our universe, the redshift of distant objects changes with time. Although the amplitude of this redshift drift is small, it will be measurable with decade-long campaigns by the next generation of telescopes. Here we present an alternative view of the redshift drift which captures the expansion of the universe in single-epoch observations of the multiple images of gravitationally lensed sources. Considering a sufficiently massive lens, with an associated time delay of order decades, simultaneous photons arriving at a detector would have been emitted decades earlier in one image compared to another, leading to an instantaneous redshift difference between the images. We also investigated the peculiar velocity which may influence the redshift difference in observation. While still requiring the observational power of the next generation of telescopes and instruments, the advantage of such a single-epoch detection over other redshift drift measurements is that it will be less susceptible to systematic effects that result from requiring instrument stability over decade-long campaigns.

Redshift drift and strong gravitational lensing

“…After the draft of this paper had been written, I became aware of the papers by Wucknitz et al (2021) and Wang et al (2022), which have much overlap with the present paper. I have left the text essentially the same as in the original draft, amending it to include additional points discussed by Wucknitz et al (2021) and Wang et al (2022) which are relevant to this paper (and of course citing them in the relevant passages).…”

“…For the typical velocity dispersion of a rich cluster (1000-1500 km s −1 ), the typical time delay (which of course depends on the exact configuration) is in the range of several hundred to a few thousand years. Loeb (1998), however, pointed out that, in practice, that might not be possible, because in general a gravitational lens will have a velocity component perpendicular to the observer, which will lead to a difference in redshift between the images of the lensed source (Birkinshaw & Gull 1983) which in general is larger than that due to redshift drift; for a velocity of 300 km s −1 and a separation of 20 arcsec, the difference in redshift is about 1 × 10 −7 , corresponding to about 30 m s −1 (Wang, Bolejko & Lewis 2022) (see also Wucknitz, Spitler & Pen 2021), about an order of magnitude larger than the difference due to redshift drift (Wang et al 2022). Peculiar velocities and peculiar accelerations due to cosmological perturbations, i.e.…”

“…Subtracting this difference from the observed difference then gives the difference due to the redshift drift. 5 Wucknitz et al (2021) discuss a similar scheme involving a quadruply imaged source rather than an Einstein ring; Wang et al (2022) point out the advantage of having more than one lensed source to correct for the same effect.…”

“…Another approach is to make use of the fact that there can be more than one source which is multiply imaged. Wang et al (2022) have shown that if one has ten or more sources then it is possible to remove the signal due to the transverse velocity (see their fig. 4), concluding that 'the effects of peculiar velocity are either negligible or can be subtracted with sufficient precision'.…”

“…Of course, the method can work only if there is sufficiently high spectral resolution (which is within the reach of instruments planned or being built now), but also sufficiently sharp spectral features. Wang et al (2022) give an example of a cluster lens (velocity dispersion 1500 km/s, z l = 0.4, z s = 4 and eccentricity parameter ε = 0.6) with ∆z = 2.67 × 10 −8 , corresponding to a wavelength difference of ∆λ = 4.45 × 10 −5 Å. Thus, one might need an AGN (for the sharp emission lines) for the source, but in that case the precision is within the reach of next-generation optical telescopes (and similar precision should be possible in the radio (Lu et al 2022).…”