Line-of-Sight Peculiar Velocities

• Line-of-sight peculiar velocities are measurements of a galaxy’s motion along the observer’s sightline, isolating deviations from the Hubble expansion.
• They are inferred using redshift and distance indicators such as the Fundamental Plane, Tully-Fisher relation, and Type Ia supernovae, with Bayesian corrections for biases.
• Analysis via spherical harmonic decomposition and power spectrum estimation allows constraints on cosmological parameters and tests of gravity.

Line-of-sight (LOS) peculiar velocities quantify the component of a galaxy or matter tracer's velocity along the observer’s direct line of sight relative to the unperturbed Hubble expansion. These velocities encode deviations from pure Hubble flow caused by gravitational interactions with large-scale structure, the growth of density perturbations, and astrophysical processes. Measurement and analysis of these velocities are fundamental for probing the growth of structure, constraining cosmological parameters, and understanding the interplay between matter density, dark energy, and gravity on cosmological scales.

1. Theoretical Framework and Mathematical Formalism

The LOS peculiar velocity $v_{\rm pec, LOS}$ is defined as the scalar projection of the full peculiar velocity vector, $\mathbf{v}$, onto the observer’s line of sight. In linear theory and in the presence of a large-scale gravitational potential, peculiar velocities are related to the underlying matter density fluctuations and potential gradients by

$$\mathbf{v}(t, \mathbf{r}) = -\frac{2 a \dot{D}(t)}{3 \Omega_0 H_0^2} \nabla \Phi_0(\mathbf{r}),$$

where $D(t)$ is the growth factor, $a$ the scale factor, $\Omega_0$ the present matter density parameter, and $\Phi_0$ is the (initial) gravitational potential (Nusser et al., 2012). The key observable in redshift surveys is the dimensionless redshift deviation field:

$$\Theta \equiv \frac{z - z_{\rm cos}}{1 + z_{\rm cos}} = \frac{V_{\rm LOS}}{c} - \frac{\Phi}{c^2} - \frac{2}{c^2} \int_{t(r)}^{t_0} dt\, \frac{\partial \Phi(r(t), t)}{\partial t},$$

where $z$ is the observed redshift, $z_{\rm cos}$ is the cosmological (unperturbed) redshift, $V_{\rm LOS}$ is the LOS peculiar velocity, and the remaining terms account for gravitational and integrated Sachs-Wolfe effects.

Spherical harmonic analysis is typically used to decompose the $z - z_{\rm cos}$ field over wide-area surveys, producing angular power spectra $C_\ell$ that quantify the amplitude of peculiar velocity fluctuations as a function of angular scale.

2. Measurement Strategies and Data Sources

LOS peculiar velocities are traditionally inferred via a combination of redshift and independent distance indicators, such as the Fundamental Plane (FP) for early-type galaxies (Springob et al., 2014), the Tully-Fisher relation for spirals, or Type Ia supernovae. The general approach involves computing the difference between the redshift-derived distance (assuming pure Hubble flow) and the “standard candle/ruler” distance to each object:

$\Delta d = \log \left( D_z / D_H \right),$

where $D_z$ is the inferred comoving distance from redshift, and $D_H$ is the true distance from the distance indicator. Bayesian techniques are used to model the probability distributions and to correct for selection effects and Malmquist bias.

Large photometric surveys (e.g. Euclid-like) enable the use of apparent magnitudes as noisy distance proxies. By exploiting the mean $z_{\rm cos}(m)$ relation derived from ensembles of galaxies, the LOS peculiar velocity field can be characterized even in the presence of large individual errors, provided the sample is sufficiently large (Nusser et al., 2012). The error on $z_{\rm cos}(m)$ is typically $\sigma_z \sim 0.3$ for Euclid.

In simulations, LOS velocities can be directly extracted and used to model the kinematics of galaxies subject to spiral, bar, or environmental perturbations (Grand et al., 2015).

Additional techniques involve:

• Measuring the kinematic Sunyaev–Zel'dovich (kSZ) effect in galaxy clusters to estimate bulk LOS velocities (Du et al., 21 Feb 2025).
• Using redshift-space distortions (RSD) in galaxy clustering to extract information about the LOS peculiar velocity field via multipole expansions (Turner et al., 2022).
• Neural network and machine learning estimators trained on mock catalogs to reconstruct LOS velocities from environmental features (Chen et al., 2023).

3. Power Spectrum and Statistical Characterization

The angular power spectrum $C_\ell$ of the LOS peculiar velocity field encodes cosmological information about the amplitude and scale-dependence of density fluctuations and gravitational potentials. In the method of (Nusser et al., 2012), one computes:

$$C_\ell = \langle |f_{\ell m}|^2 \rangle_m - \frac{\sigma_f^2}{\bar{n}},$$

where $f_{\ell m}$ are spherical harmonic coefficients of the LOS deviation field, $m$ indexes the $2\ell+1$ modes at each $\ell$, and $\sigma_f^2$ is the field variance.

Key regimes:

• On large angular scales ($\ell \lesssim 15$), the power spectrum is dominated by LOS velocity (Doppler) terms.
• On intermediate scales ($15 \lesssim \ell \lesssim 60$), gravitational lensing magnification effects become significant.

High-precision surveys ($\sim 10^9$ galaxies at $z \lesssim 2$) like Euclid enable measurement of the combined velocity+lensing power spectrum at $z\sim 1$ to better than 5% accuracy.

Analysis of configuration-space and Fourier-space correlation functions, including both auto- and cross-correlations (e.g., between density and velocity fields), further constrains cosmological parameters. These statistics are sensitive to the growth rate $f\sigma_8$ and the redshift distortion parameter $\beta$ (Turner et al., 2022).

4. Systematic Effects and Mitigation

Several sources of systematic contamination and bias must be managed:

• Environmental Effects: The luminosity function may vary with local density; empirical corrections assuming $\Delta m \simeq 0.2 \delta$ can be applied (Nusser et al., 2012).
• Malmquist Bias: Non-uniform spatial distribution of tracers and magnitude limits introduce biases. Forward-modeling approaches marginalizing over true distances and incorporating external density estimates via $n(r) \propto r^2 [1 + \hat{\delta}_g(r)]$ correct for inhomogeneous Malmquist bias (Boruah et al., 2021).
• Gravitational Lensing Magnification: Lensing shifts apparent magnitudes and thus inferred distances, affecting the $z_{\rm cos}(m)$ relation. Lensing terms can dominate intermediate angular scales and must be included in modeling.
• Non-Gaussianity of Errors: Gaussianization techniques (e.g., Box–Cox transformation) are used to render velocity error distributions suitable for standard maximum-likelihood (ML) estimators, improving bulk flow measurements and reducing bias (Qin, 2021).
• Wide-Angle Effects and Observer Geometry: Large-area surveys whose lines of sight vary significantly across the footprint require full 3D treatment of velocity correlation functions to account for multipole mixing and break-down of the plane-parallel approximation (Castorina et al., 2019).
• Intrinsic Astrophysical Processes: Intra-cluster gas dynamics, AGN feedback, and star formation induce local deviations in cluster velocities inferred from kSZ measurements (Du et al., 21 Feb 2025).

Optimal data analysis strategies often involve weighting schemes (e.g., to minimize shot noise), explicit modeling and removal of contaminants, and multipole expansions explicitly accounting for survey geometry and wide-angle corrections.

5. Applications and Cosmological Implications

LOS peculiar velocities are central to several key cosmological and astrophysical applications:

• Constraints on Growth of Structure: Direct estimation of the velocity power spectrum provides stringent tests of structure formation and ΛCDM consistency, independently of galaxy bias (Nusser et al., 2012, Turner et al., 2022).
• Tests of Gravity: The ability to separately measure the velocity and gravitational lensing power spectra enables constraints on modifications to General Relativity on large scales, particularly via gravitational slip parameters.
• Hubble Constant Measurements and Tension: Proper correction for LOS peculiar velocities in SNIa and time-delay lensing measurements is required to avoid biases in $H_0$. Such corrections are shown to have $\lesssim 0.1\%$ impact on the global $H_0$ in large samples but can be at the percent level for individual systems depending on LOS alignment and observer velocity (Sedgwick et al., 2019, Dalang et al., 2023).
• Cluster Mass Estimation: Peculiar velocity “waves” in the Hubble diagram around massive clusters can be quantitatively related to cluster mass through analytic (Gaussian-plus-continuum) fits and machine learning models, offering an independent mass calibration (Sorce et al., 2023).
• Void Cosmology: LOS peculiar velocities introduce shape distortions in cosmic voids but affect cosmological measurements based on void properties only at the $\sim$10–20% level, a bias that can be controlled through cuts on void size and density contrast (Pisani et al., 2015).
• Epoch of Reionization 21-cm mapping: Accurate light-cone simulations require pixel-by-pixel LOS integration of the 21-cm intensity, properly accounting for peculiar velocity gradients to predict the observed power spectrum and temperature distribution (Chapman et al., 2019).

6. Relativistic and Curvature Effects

Recent advances demonstrate that the relativistic evolution of peculiar velocities predicts a stronger linear growth with scale factor than Newtonian treatments. Specifically, on superhorizon scales the peculiar velocity grows as $v \propto a^2$ (relativistic), versus $v \propto a^{1/2}$ (Newtonian), due to the additional gravitational effect of the energy flux in general relativity (Tsaprazi et al., 2019). This faster growth holds across Friedmann backgrounds with curvature, with the effect being most pronounced in closed (overdense) regions and slower in underdense (open) regions (Miliou et al., 29 Apr 2024). These insights have direct implications for interpreting observed bulk flows and adjusting models of large-scale cosmic motion.

7. Outlook and Future Survey Prospects

Next-generation photometric redshift and CMB surveys (e.g., Euclid, LSST, DESI, Simons Observatory, CMB-S4) will deliver unprecedented numbers of galaxy redshifts, peculiar velocity tracers, and kSZ measurements. High signal-to-noise measurement of the LOS velocity angular power spectrum and the integrated gravitational lensing signal will refine constraints on the nature of dark energy, test gravity, and enable bias-independent probes of structure growth.

Machine learning, Bayesian forward modeling, and high-resolution simulations will enable more precise velocity reconstructions and cosmological parameter extraction. Complementary kinematic probes (transverse velocities, moving lens effect, 21-cm tomography) will further enhance the three-dimensional mapping of cosmic flows and provide new cross-checks for model consistency and departures from ΛCDM.

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In conclusion, line-of-sight peculiar velocities are a vital and richly informative observable in cosmology, encoding the dynamical response of matter to gravitational perturbations. Continued advances in measurement, statistical treatment, and theoretical modeling promise further insight into cosmic structure, the nature of gravity, and the fundamental parameters of our Universe.