On distance and velocity estimation in cosmology

TL;DR

The paper separates two Malmquist biases arising when reconstructing peculiar velocity fields from distance indicators: distance Malmquist bias ($dMB$) for individual distances and velocity Malmquist bias ($vMB$) that corrupts continuous velocity fields when using inferred positions. Using the Tully–Fisher relation as a concrete example, it shows that unbiased per-object velocities do not guarantee unbiased velocity fields and that even bias-corrected distances leave $vMB$ intact; the authors advocate placing galaxies at redshift coordinates ($s=cz/H_0$) for robust large-scale reconstructions, with a practical residual bias only on small scales set by $\sigma_v/H_0$. A modified Wiener filter is developed to marginalize over $P(r|d)$, but while it removes systematic bias, it reduces amplitude, and machine learning approaches converge to the Wiener filter in the Gaussian limit, suggesting limited gains. Overall, redshift-space placement emerges as the most reliable strategy for velocity-field reconstruction in typical surveys, with implications for velocity–gravity comparisons and growth-rate measurements, and the FP framework is discussed as an analogous distance-indicator context.

Abstract

Scatter in distance indicators introduces two conceptually distinct systematic biases when reconstructing peculiar velocity fields from redshifts and distances. The first is distance Malmquist bias (dMB) that affects individual distance estimates and can in principle be approximately corrected. The second is velocity Malmquist bias (vMB) that arises when constructing continuous velocity fields from scattered distance measurements: random scatter places galaxies at noisy spatial positions, introducing spurious velocity gradients that persist even when distances are corrected for dMB. Considering the Tully-Fisher relation as a concrete example, both inverse and forward formulations yield unbiased individual peculiar velocities for galaxies with the same true distance (the forward relation requires a selection-dependent correction), but neither eliminates vMB when galaxies are placed at their inferred distances. We develop a modified Wiener filter that properly encodes correlations between directly observed distance $d$ and true distance $r$ through the conditional probability $P(r|d)$, accounting for the distribution of true distances sampled by galaxies at observed distance $d$. Nonetheless, this modified filter yields suppressed amplitude estimates. Since machine learning autoencoders converge to the Wiener filter for Gaussian fields, they are unlikely to significantly improve velocity field estimation. We therefore argue that optimal reconstruction places galaxies at their observed redshifts rather than inferred distances; an approach effective when distance errors exceed $σ_v/H_0$, a condition satisfied for most galaxies in typical surveys beyond the nearby volume.

Figures (6)

• Figure 1: Normalized density distributions along a line of sight through the overdensity center at $r_0 = 80$ Mpc. The solid line shows the true distance distribution $p(r)$, while the dashed line shows the observed (iTF) distance distribution $p(d)$ from the toy model catalog. The decline at large distances reflects the imposed magnitude limit.
• Figure 2: Upper panel: true distances $r$ versus observed iTF distances $d$. Lower panel: true distances $r$ versus $\bar{r}$. Solid lines: regression of $r$ on the distance estimator. Dashed lines: regression of the estimator on $r$.
• Figure 3: Various line-of-sight velocity estimators as functions of different distance estimators for the toy model magnitude-limited catalog. Each panel shows binned mean velocities in distance binns. Top-left: velocities vs true distance $r$, including $V^{\rm obs}$ vs redshift-space distance $s$ (green). Top-right: velocities vs observed iTF distance $d$. Bottom-left: velocities vs posterior mean distance $\bar{r}$, with the corresponding velocity estimator $\bar{V} = s - \bar{r}$. Bottom-right: velocities vs sampled distance $\hat{r}$ from $P(r|d)$, with $\hat{V} = s - \hat{r}$.
• Figure 4: Spatial derivatives of the velocity field, $\partial V_z / \partial z$ in the true, observed, and redshift--space representations (top and bottom--left) All fields were smoothed with a Gaussian kernel of width $20~h^{-1}\,\mathrm{Mpc}$. Mock distance errors with a Gaussian scatter of $10~h^{-1}\,\mathrm{Mpc}$ were applied to generate the observed and redshift--space quantities.
• Figure 5: Probability density functions of $\partial_z V$ for the four three--dimensional fields shown in the previous figure. Values of the standard deviations, $\sigma$, are listed in the figure. The comparison shows the enhancement of the fluctuations in $V^{\mathrm{obs}}(d)$ (orange) and the suppression in the Wiener--filtered field $V^{\mathrm{WF}}$ (green), while the redshift--space field $V^{\mathrm{obs}}(s)$ (red) is close to the true field $V(r)$ (blue).