DESI Peculiar Velocity Tracers

• DESI Peculiar Velocity Tracers are galaxies with redshift-independent distance measures from the Fundamental Plane and Tully–Fisher relations, enabling direct peculiar velocity estimation.
• They facilitate precision reconstruction of the local cosmic velocity field, constraining key cosmological parameters like the Hubble constant and growth rate of structure.
• The DESI Data Release 1 improves sample size, calibration uniformity, and cross-survey integration, boosting analyses such as kSZ measurements and velocity reconstructions.

Peculiar velocity tracers in the Dark Energy Spectroscopic Instrument (DESI) survey are galaxies for which redshift-independent distances and hence line-of-sight peculiar velocities can be measured directly, primarily via the Fundamental Plane (FP) relation for early-type galaxies and the Tully–Fisher (TF) relation for spiral galaxies. These tracers allow reconstruction of the local cosmic velocity field, enabling precision constraints on the cosmic growth rate, the Hubble constant, and deviations from General Relativity on cosmologically relevant scales. The DESI Data Release 1 (DR1) marks a substantial advance in sample size, calibration uniformity, and precision over all previous PV surveys.

1. Galaxy Selection and Distance Indicator Methodology

DESI PV tracers comprise two principal classes: early-type galaxies used for FP distances, and late-type (disk) galaxies used for TF distances (Douglass et al., 2 Dec 2025, Ross et al., 2 Dec 2025, Saulder et al., 2023).

Early-type/Fundamental Plane (FP) galaxies are selected using:

• Photometric morphological criteria: Sersic index $n_s>2.5$ or de Vaucouleurs (DEV) light profile, axis ratio $b/a>0.3$
• Red-sequence color boundaries: $(g-r)>0.68$, $(g-r)>1.3(r-z)-0.05$, $(g-r)<2.0(r-z)-0.15$
• Apparent magnitude: $10 < m_r < 18$
• Redshift: $0.0033CMB frame)
• Velocity dispersion: $50<\sigma<420$ km/s measured via pPXF on DESI spectra

The FP relation is fit as

$$\log_{10} R_e = a\,\log_{10} \sigma_0 + b\,\log_{10} I_e + c$$

with $R_e$ the effective (half-light) radius, $\sigma_0$ central stellar velocity dispersion, and $I_e$ mean surface brightness. FP offsets provide log-distance ratios and thus peculiar velocities (Ross et al., 2 Dec 2025).

Late-type/Tully–Fisher (TF) galaxies are selected with:

• Sersic index $n_s<2$ (disk dominated)
• Inclination $i>20^\circ$ ($b/a<0.94$)
• $D_{25}>20''$ (isophotal diameter)
• Apparent magnitude and redshift: $0.03$r<18$
• Multi-fiber placement: one fiber at center; two off-center at $0.4 R_{26}$

The TF calibration uses

$M_r = a\,\log_{10}(V/ V_0) + b$

with rotational velocity $V$ measured by redshift difference between center and off-center fibers. The DESI DR1 calibration is $a = -7.22\pm0.10$ in $r$-band, with intrinsic scatter $\sigma=0.466$ mag (Douglass et al., 2 Dec 2025).

Sample sizes in DR1 after quality cuts are 98,292 FP galaxies and 10,262 TF galaxies, with PV-quality clusters for cosmological analyses containing 73,822 FP and 6,807 TF galaxies (Ross et al., 2 Dec 2025, Douglass et al., 2 Dec 2025).

2. Distance, Peculiar Velocity, and Catalog Construction

Distances are determined photometrically, using either the FP or TF relations, and then compared to observed redshifts to derive peculiar velocities. For both indicators, key steps include:

• Correction for Milky Way dust, internal extinction, photometric zero-points, and k-corrections.
• Calculation of a log-distance ratio $\eta$ that serves as a Gaussian variable for likelihoods.
• Peculiar velocity estimator for FP:

$$\hat v(\eta) = c\,\ln 10\,[(1+z_\mathrm{CMB})\,c/(H(z_\mathrm{CMB})\,d(z_\mathrm{CMB})) -1 ]^{-1} \eta$$

• For TF, peculiar velocity:

$$v_\mathrm{pec} = cz_\mathrm{CMB} - H_0 D\,,\quad D = 10^{(\mu_\mathrm{TF} - 25)/5}\,\mathrm{Mpc}$$

with $H_0$ adopting fiducial cosmology $\Omega_m = 0.3151$.

Key catalog columns include galaxy ID, celestial coordinates, redshift, photometric and spectroscopic parameters, TF/FP distance modulus, log-distance ratio $\eta$, and derived peculiar velocity with propagated uncertainties. Typical uncertainties: FP $\sigma_\eta\sim0.12$ dex ($\sim$26% distance), TF $\sigma_\eta\sim0.11$, $\sigma_{v,\rm pec} \sim 200$ km/s (for main TF sample) (Ross et al., 2 Dec 2025, Douglass et al., 2 Dec 2025).

3. Calibration, Systematic Error Control, and Zero-Point Anchoring

Absolute distance calibration is achieved via hierarchical cross-matching of PV galaxies to external calibrators:

• SH0ES/Pantheon+ SNe Ia ($\sim100$ hosts at $z<0.1$) provide a primary ladder for zero-point fixes.
• Group catalogs (e.g. Lim et al. 2017 2dFGRS/6dFGS/2MRS/SDSS) link PV galaxies to SNe, boosting calibrator statistics ($\sim571$ group-based links in DR1).
• Alternative calibrators include SBF distances, masers, and Coma cluster measurements, although overlap is less complete (Carr et al., 2 Dec 2025).

The Hubble constant is determined by simultaneous zero-point fitting across FP, TF, and SNe ladders, yielding $$H_0 = 73.7 \pm 0.06\,(\mathrm{stat.}) \pm 1.1\,(\mathrm{syst.})$$ km s$^{-1}$ Mpc$^{-1}$. The systematic floor is set by SN calibration; DESI-specific systematics (grouping, catalog choice) contribute $\sim0.3$ km s$^{-1}$ Mpc$^{-1}$ (Carr et al., 2 Dec 2025).

Systematic control includes:

• Internal and external photometry and velocity-dispersion validation (DECaLS vs BASS/MzLS; pPXF vs SDSS Portsmouth).
• Sky-pattern checks for systematics in velocity-dispersion and photometric calibration.
• Selection/Malmquist corrections and group-richness bias fitting for FP.
• Propagation of all error contributions, including intrinsic scatter, into $\eta$ and $v_\mathrm{pec}$ uncertainties.

4. Cosmological Application: Growth Rate, Bulk Flow, and Gravity Tests

DESI PV tracers are central to direct measurements of the cosmic velocity field, growth rate of structure ($f\sigma_8$), and bulk flows in the local Universe.

• Joint analyzes of FP, TF, and galaxy density fields (via DESI Bright Galaxy Survey) enable robust two-point correlation function and power spectrum fits to the growth rate. DR1 measurements yield (Turner et al., 2 Dec 2025, Lai et al., 2 Dec 2025):
  • $f\sigma_8(z=0.07) = 0.4497 \pm 0.0548$ (12.2% precision consensus)
  • This is mutually consistent across maximum-likelihood fields, correlation function, and power spectrum analyses.
  • The gravitational growth index constraint, $\gamma_L = 0.580\pm0.11$, is consistent with the General Relativity prediction $\gamma\approx0.55$.
• Cosmological flow reconstruction: bulk velocity amplitudes $\langle |V_\mathrm{bulk}| \rangle = 292 \pm 34$ km/s at $R=150\,h^{-1}$ Mpc, dynamical homogeneity scale not reached within $200-300\,h^{-1}$ Mpc (Courtois et al., 3 Feb 2025).
• Large-scale velocity field enables direct tests of deviations from GR via the velocity-density relation on scales of $20-150\,h^{-1}$ Mpc.

An integrated PV catalog with calibration to external reference frames is critical for $H_0$ inferences, providing an independent cross-check on local universe expansion and the ongoing "Hubble tension" debate (Carr et al., 2 Dec 2025).

5. Cross-Survey Synergy and Statistical Combination

DESI PV tracers have been integrated into larger cosmography efforts:

• The CF4++ compendium merges WALLABY, FAST, and DESI PV data for a more comprehensive velocity field reconstruction at $z<0.1$. DESI provides $\sim4,191$ high-quality unique PV tracers with recalibrated distances on a common $H_0$ zero-point (Courtois et al., 3 Feb 2025).
• Statistical combination employs inverse-variance weighting and Hamiltonian Monte Carlo (HMC) reconstruction of density and velocity fields, with each survey’s error model explicitly included.

DESI’s uniform selection and extended NGC+SDSS coverage significantly improve northern sky velocity uniformity, reduce uncertainties in the $100-200\,h^{-1}$ Mpc shell by 20% compared to CF4 alone, and enhance detection of otherwise obscured superclusters (e.g., Vela) (Courtois et al., 3 Feb 2025).

6. Advanced Applications: kSZ Measurements and Velocity Reconstruction

DESI Luminous Red Galaxies (LRGs), though not classical redshift-independent PV tracers, are instrumental in velocity-field studies via the kinematic Sunyaev–Zel'dovich (kSZ) effect. By stacking CMB measurements at cluster positions, DESI LRGs enable the highest significance pairwise kSZ measurement to date (SNR=9.3), allowing direct estimation of cluster-scale peculiar velocities (Gong et al., 28 Nov 2025). Machine-learning models, trained on simulations, further refine optical depth and velocity inference for individual clusters (e.g., $N\sim450,000$ clusters for which peculiar velocities are estimated).

Velocity reconstruction from galaxy density fields (continuity equation, Zeldovich approximation) provides a complementary indirect route to velocity field inference from DESI spectroscopic and photometric samples, with $r_\parallel\approx0.64$ (cross-correlation coefficient between reconstructed and true velocities for DESI Y1 LRGs) (Hadzhiyska et al., 2023).

7. Future Prospects and Limitations

DESI DR2 and subsequent data releases are projected to expand the PV sample well beyond $10^5$ galaxies, directly overlap with Cepheid and TRGB-calibrated galaxies, and push precision errors on $H_0$ toward the sub-percent level, independent of SNe Ia (Carr et al., 2 Dec 2025). Forecasts predict $\sim4\%$ errors on $f\sigma_8$ for $z<0.15$ with the full PV+BGS sample, a factor of $\sim2.5$ improvement over redshift-only constraints (Saulder et al., 2023). Limitations include:

• Intrinsic scatter in FP and TF relations (per-tracer $\sigma_\eta\sim0.1$–$0.12$).
• Reliance on external zero-point calibrators for absolute cosmology.
• Selection incompleteness at low $z$ for group catalogs.

Integration with other velocity probes (SNe Ia, surface brightness fluctuations, gravitational wave standard sirens) and cosmic variance–canceling joint analysis strategies remains under active development.

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References:

(Douglass et al., 2 Dec 2025, Ross et al., 2 Dec 2025, Carr et al., 2 Dec 2025, Turner et al., 2 Dec 2025, Lai et al., 2 Dec 2025, Saulder et al., 2023, Courtois et al., 3 Feb 2025, Gong et al., 28 Nov 2025, Hadzhiyska et al., 2023)