Cosmology with extragalactic proper motions: harmonic formalism, estimators, and forecasts

Abstract

We conduct a thorough study into the feasibility of measuring large-scale correlated proper motions of galaxies with astrometric surveys. We introduce a harmonic formalism for analysing proper motions and their correlation functions on the sphere based on spin-weighted spherical harmonics, and study the statistics of the transverse velocity field induced by large-scale structure. We use a likelihood formalism to derive optimal estimators for the secular parallax due to the Solar System's motion relative to distant objects, and compute the variance and bias due to peculiar velocities and relativistic aberration. We use a simulated catalogue of galaxy proper motions with radial distributions and noise properties similar to those expected from Gaia to forecast the detectability of the proper motion dipole, whose amplitude may be considered a proxy for the Hubble constant. We find cosmic variance to be the limiting source of noise for this measurement, forecasting a detectability of 1-2σ on a single component of the local velocity, increasing to 2-4σ (equivalent to a 25%-50% measurement of the Hubble constant) if the CMB dipole is included as prior information. We conduct a thorough study into the radial dependence of the signal-to-noise, finding that most of the information comes from galaxies closer than a few hundred Mpc. We forecast that the amplitude of peculiar transverse velocities can potentially be measured with 10σ significance; such a measurement would offer a unique probe of cosmic flows and a valuable test of the cosmological model. 1 http://sci.esa.int/gaia/ 2 http://ngvla.nrao.edu/ like', i.e. ellipticals with large bulge-to-disk components. Additionally, a large population of quasars will be observed to pin down the celestial reference frame (see, e.g. Gaia Collaboration et al. 2018).There are several potential uses of such a dataset for cosmology (see Darling et al. 2018 for a review). Firstly, since the Solar System moves relative to distant objects, there is a 'secular parallax' (SP) proper motion in the opposite direction to our local velocity. This proper motion has an amplitude of roughly 80 r/1 Mpc µas yr −1 , where r is the comoving distance to the object, and has a dipolar dependence on angle, anti-aligned with the velocity vector of the Solar System with respect to the CMB rest frame 3 . With spectroscopic redshifts as a proxy for distance, the SP proper motion may 3 We will often express quantities in the CMB rest frame, the frame in which the CMB dipole vanishes (Planck Collaboration et al. 2014). We assume that the matter rest frame (the frame in which the dipole anisotropy of peculiar velocities vanishes) and the CMB rest frame coincide, i.e. we do not consider the 'tilted universe' scenario of Turner

Secular Extragalactic Parallax: Measurement Methods and Predictions for Gaia

“…The transverse peculiar velocity dipole causes proper motions of similar amplitude to secular parallax for galaxies within ∼ 100 Mpc. Hall (2019) therefore predict transverse peculiar velocities dominate the error on secular parallax measurements for local galaxies. The predicted peculiar velocity dipole decreases in power as a function of distance, which causes the observed peculiar proper motion dipole amplitude to decrease faster than 1/D.…”

“…Peculiar velocities induce distance-dependent proper motions at all angular scales, including a dominant dipole for local galaxies. Hall (2019) presented predictions for the LSS transverse velocity power spectrum and its correlation to the secular parallax dipole. The transverse peculiar velocity dipole causes proper motions of similar amplitude to secular parallax for galaxies within ∼ 100 Mpc.…”

“…Darling &Hall 2019) proper motions may probe these peculiar transverse velocities and therefore the matter power spectrum. Hall (2019) derive the peculiar velocity power spectrum for low redshift based on the matter power spectrum computed with the CAMB cosmology code. The peculiar velocity field is expected to be nearly curl-free and corresponds to an E-mode transverse velocity power spectrum where the dominant mode is distance dependent.…”

“…Secular aberration drift is observed as a distance-independent dipole with amplitude of ∼ 5 µas yr −1 (Titov & Lambert 2013;Truebenbach & Darling 2017), which should be separable from a distance dependent dipole. LSS, however, induces distance-dependent proper motions at all angular scales with amplitudes comparable to secular parallax (Hall 2019). For individual galaxies, separating the peculiar and secular parallax components of the proper motion is not possible without independent estimates of the peculiar velocity and distance of the galaxy, though the inferred secular parallax distance would not provide an independent distance estimate in this case.…”

“…For individual galaxies, separating the peculiar and secular parallax components of the proper motion is not possible without independent estimates of the peculiar velocity and distance of the galaxy, though the inferred secular parallax distance would not provide an independent distance estimate in this case. However, a statistical detection of secular parallax may be possible for large sample of galaxy proper motions, which may be used to calibrate existing extragalactic distance measures and to constrain the Hubble parameter (Ding & Croft 2009;Hall 2019).…”