Kinematic cosmic dipole from a large sample of strong lenses

Abstract

Measurements of the kinematic cosmic dipole continue to show an intriguing tension between the value inferred from the CMB and that obtained from high-redshift source number counts. While the measured dipole direction appears consistent, the amplitude, set by the observer's peculiar velocity $v_{o}$, remains in significant disagreement. In this paper, we propose using strong gravitational lenses with well-measured Einstein radii to estimate the kinematic cosmic dipole, through the relativistic distortion of the Einstein angle induced by the observer's motion. We show that this effect could be detected solely from measurements of the Einstein radius in wide, high-resolution imaging surveys such as Euclid. However, the precision achievable using Einstein-radius measurements alone, without redshift or lens-galaxy mass information, appears insufficient to discriminate between the CMB value of $v_{o}$ and that derived from source number counts at high statistical significance. Nevertheless, we demonstrate that including a large sample of lenses with available kinematic information, either via the Fundamental Plane relation or, ideally, through spectroscopic velocity-dispersion measurements, drastically reduces the noise and substantially improves the constraining power of this method. We show that, for a realistic sample of strong lenses detected by Euclid and complemented with spectroscopic velocity dispersion measurements from 4MOST or DESI, it is possible to discriminate between the CMB- and source-number-counts-inferred values at the $\sim 4σ$ level using a new, fully independent method. We further demonstrate that this technique is only weakly sensitive to strong-lensing selection effects, with selection biases and threshold effects estimated to be well below the 1% level.

Figures (5)

• Figure 1: Left panel: Coordinate system used in this work. We consider two observers, located at the origin $\mathcal{O}$. The first observer is stationary in this frame and measures comoving quantities, which are written without primes, e.g. $\theta_{\rm cm}$. Because of the symmetry around the $\boldsymbol{e}_z$ axis, the system is fully captured by lenses which lie in the $\boldsymbol{e}_y-\boldsymbol{e}_z$ plane. The stationary observer sees a perfect circular Einstein ring with Einstein angle $\theta_E$ in the direction $\boldsymbol{\hat{n}}$. They associate a coordinate system orthogonal to $\boldsymbol{\hat{n}}$ on the sky $\{\boldsymbol{\hat{e}}_1,\boldsymbol{\hat{e}}_2\}$, depicted in the middle panel. This coordinate system is such that the $\boldsymbol{\hat{e}}_1$ vector is orthogonal both to $\boldsymbol{\hat{n}}$ and $\boldsymbol{v}_o$ (here it is aligned with $\boldsymbol{e}_x$), while $\boldsymbol{\hat{e}}_2$ lives in the $\boldsymbol{e}_y-\boldsymbol{e}_z$ plane. The second observer moves with peculiar velocity $\boldsymbol{v}_o$ aligned with $\boldsymbol{\hat{e}}_z$ and observes the Einstein ring in the $\boldsymbol{\hat{n}}'$ direction, which forms an angle $\theta'_{\rm cm}$ with the $\boldsymbol{e}_z$ axis. For this moving observer, the Einstein ring appears squashed along the $\boldsymbol{\hat{e}}'_2$ direction, but unchanged along the $\boldsymbol{\hat{e}}'_1$ direction. It appears approximately as an ellipse in the coordinate system defined by $\{\boldsymbol{\hat{e}}_1', \boldsymbol{\hat{e}}_2'\}$, as can be seen in the right panel. The Einstein angle as a function of the observed boosted azimuthal angle $\delta'$ can be computed from the components $\theta'_{E 1}(\delta')$ and $\theta'_{E 2}(\delta')$. The amplitude of this deformation is proportional to $\cos(\theta_{\rm cm})v_o/c$.
• Figure 2: Properties of the simulated Euclid lens sample considered in this work.
• Figure 3: Posterior probability distributions of the kinematic dipole amplitude and direction inferred from samples of $100\,000$ (blue contours) and $400\,000$ (orange contours) strong lenses with Euclid-quality imaging. The black lines indicate the true value of $v_{o} = 369.82$${\, \mathrm{km}\, \mathrm{s}^{-1}}$, pointing toward Galactic coordinates $(l_o,b_o) = (264.021^\circ,\; 48.253^\circ)$. The contours enclose 39.3% and 86.5% of the posterior probability.
• Figure 4: Posterior probability distributions of the kinematic dipole amplitude and direction inferred from samples of $100\,000$ strong lenses with Euclid-quality imaging (blue contours), in combination with spectroscopic stellar velocity dispersion measurements (red contours), or with FP estimates of the velocity dispersion using spectroscopic redshifts (green contours) or photometric redshifts (orange contours). The black lines indicate the true value of $v_{o} = 369.82$${\, \mathrm{km}\, \mathrm{s}^{-1}}$, pointing towards Galactic coordinates $(l_o,b_o) = (264.021^\circ,\; 48.253^\circ)$. The contours enclose 39.3% and 86.5% of the posterior probability.
• Figure 5: Posterior probability distributions of the kinematic dipole amplitude and direction inferred from samples of $100\,000$ strong lenses with Euclid-quality imaging and different type of ancillary kinematic information, listed in Table \ref{['tab:Euclid_scenario']}. The black lines indicate the true value of $v_{o} = 1109.42$${\, \mathrm{km}\, \mathrm{s}^{-1}}$, pointing towards Galactic coordinates $(l_o,b_o) = (264.021^\circ,\; 48.253^\circ)$. The contours enclose 39.3% and 86.5% of the posterior probability.