4.1 Velocity dispersion cuts

Once velocity dispersion measurements had been obtained, three additional cuts were applied to our sample. The first was a minimum velocity dispersion measurement of $\sigma=50\,\textrm{km}\,\textrm{s}^{-1}$ . This cut removes galaxies from our sample for which we were unable to successfully measure velocity dispersions. Additionally, galaxies with intrinsic velocity dispersions under or close to the resolution limit of the spectrograph can result in artificially inflated spectral velocity dispersions. Such measurements are unreliable, and so these galaxies were removed from our sample by setting the minimum required velocity dispersion slightly above the DESI blue camera resolution limit. The second cut we applied to our sample is a maximum velocity dispersion measurement of $\sigma = 420\,\textrm{km}\,\textrm{s}^{-1}$ . This cut removes galaxies that had much higher velocity dispersion measurements than expected, usually the result of a poor or unsuccessful measurement. Finally, we require that the velocity dispersion error (from pPXF), $\delta\sigma \gt0$ , as a zero-error measurement indicates that something failed during the fitting process.

4.2 Primary observation identification

As our sample consists of a compilation of all DR1 spectra that pass our selection cuts, many of the galaxies have been observed multiple times. The majority of these repeat observations are coadded by the DESI data reduction pipeline (Guy et al. Reference Guy2023). However, there are some repeat observations that are not merged, falling into 3 categories:

1. 1. the same TARGETID but different PROGRAM and/or SURVEY,

2. 2. overlapping target location but different TARGETID,

3. 3. and offset targets on the same galaxy.

Galaxies in the first category are usually observed in both dark and bright time. However, there are several galaxies that have been observed as ‘backup’ targets (during poor observing conditions) before later being re-observed, as well as galaxies that have repeat observations across multiple survey stages – sv1, sv2, sv3, and/or main, where sv are survey validation observations that were mostly released in the Early Data Release, EDR. In all situations, the spectra for each galaxy are coadded and saved separately for each TARGETID, PROGRAM, SURVEY combination.

The second category occurs when a galaxy has been targeted separately by different surveys, usually BGS and peculiar velocity targets, with slightly different fibre positions. In this situation, it is possible that different surveys are trying to target different regions of the galaxy (e.g. the galaxy centre as well as a star forming region or orbiting dwarf galaxy). Due to their very small offsets of less than the fibre aperture radius of $\theta_{\rm ap}=0.75$ arcsec, there is a high percentage of overlap between repeat observations, resulting in both observations observing the same area on the sky.

The third category usually occurs intentionally in large, nearby galaxies. In the peculiar velocity target selection, some of the FP galaxies were specifically targeted multiple times using different fibre offsets to assist in understanding systematic effects associated with non-central fibre placement (‘extended’ FP targets, Saulder et al. Reference Saulder2023). Additionally, other surveys have targeted different points of interest such as star-forming regions, supernovae, etc. within nearby galaxies, some of which may have made it into our sample. This third category of repeat observations are usually low priority and are only observed when there are no other targets available within the fibre patrol radius.

Figure 3 shows the distribution of the separations between primary and duplicate fibre locations. Almost all duplicate observations have significant overlap with the chosen ‘primary’ observation for the galaxy, with separations much less than the fibre radius (red line). As such, the majority of repeat observations fall into either the first or second categories. This happens when a galaxy has been observed with slightly different fibre positions across different SURVEY/PROGRAM/TARGETID combinations, either due to repeatability of the fibre positioning or slightly different targeting coordinates. In our sample, only one of the 10 492 repeatedly observed galaxies falls into category 3; the primary and secondary observations for this galaxy are offset by $0.78$ arcsec which corresponds to an overlap of 40% of the primary fibre-area. The reason so few category 3 repeats exist in the DR1 sample is likely due to the low-priority of extended PV targets, which are yet to be observed, and the light-profile and colour cuts that remove star-forming, clumpy, and disrupted galaxies targeted by other projects. As such, contamination from significantly offset observations is not an issue for the DR1 FP catalogue but will become increasingly important to identify and analyse such galaxies in future data releases as the survey nears completion.

Figure 3.

Distribution of on-sky separation (in arcsec) between duplicate and primaryVdisp fibre locations for repeat observations of the same galaxy. The red dashed line is the on-sky fibre radius ( $\theta_{\text{ap}} = 0.75$ arcsec) for DESI fibres. All but one duplicate observation overlaps with the primary observation.

As all but one of our repeat observations are have very high overlap, we treat them as if they are the measuring the same region of the galaxy. We chose the observation with the best relative velocity dispersion error as the primary observation, labelling them as such using the primaryVdisp flag. We use the repeated observations to perform internal consistency checks on our velocity dispersion measurements (Section 4.3), but only the primary observations are used when doing external consistency checks (Section 4.4), when fitting the Fundamental Plane parameters (Section 5.2.1), and during zero pointing (Section 5.2.4). However, all observations are included in the final catalogue with independently calculated log-distance and peculiar velocity measurements to allow for easy crossmatching to other catalogues.

4.3 Internal consistency

As the Fundamental Plane calibration is performed unanchored, it is not crucial that any of the 3 FP parameters is consistent with other surveys, as any offsets will be corrected during zero-pointing. More complex deviations, such as tilts, bends or redshift-dependent and coordinate-dependent effects are not removed during this process and need to be identified. As such, consistency of measurements across subsamples is essential when they represent the same population of galaxies. The most obvious way to select such a subsample is by observing conditions (bright vs dark time) and by position on the sky. Additionally, we check that we are accurately estimating the velocity dispersion errors by comparing primary and secondary observations as described in Section 4.2.

In Figure 4, we check for deviations in velocity dispersion measurements across a variety of different sky positions. We do this in HEALPix bins of width $\sim$ $3.7^\circ$ (Górski et al. Reference Górski2005). Masking out bins with 20 observations, we calculate the mean velocity dispersion in each bin and colour-code by the percent offset from the global mean. In the DR1 data, are no noticeable patterns in the figure based off sky-position.

Figure 4.

Variation of velocity dispersion measurements across the sky for cells with more than 20 observed galaxies. Each cell is colour coded based of the mean velocity dispersion of the cell which is then normalised to the global mean velocity dispersion. The cell standard deviation is 5% across the $1\,156$ cells with observations. With the maximum cell offset at 16% and no correlations with sky location there is no evidence of bias.

Next, we analyse dependencies between bright and dark time observations. In Figure 5 we look at the relationship between the signal to noise ratio (SNR) and the relative velocity dispersion errors for both subsamples. From the plot we can see that the bright-time observations (orange) are consistent with the relation (black line) found by Said et al. (Reference Said2024). When looking at the dark-time observations (purple) the measurements have higher SNR and smaller relative velocity dispersion errors when compared to bright observations with a similar SNR.

Figure 5.

Relative velocity dispersion error vs SNR for both bright (orange) and dark time (purple) observations. The bright time observations are consistent with the early data release (EDR) relation observed in Said et al. (Reference Said2024), over-plotted in black with the dark time observations on average obtaining tighter constraints on the velocity dispersions measurements compared to equivalent SNR bright time observations.

In Figure 6, we directly compare the velocity dispersion measurements made in bright and dark time using the $7\,570$ galaxies with observations in both programmes. We find that there is an offset between the bright and dark samples with the dark samples measuring larger velocity dispersions. When weighting by the combined errors, this offset is $\bar{\varepsilon} = -0.22$ (i.e. 22% of the contoured error). We correct for this offset by applying a $+2.3\pm0.1\,\textrm{km}\,\textrm{s}^{-1}$ correction to all bright-time velocity dispersion measurements, calculated by taking the variance-weighted mean of the offset, which brings them into better consistency with the dark-time measurements. The source of this offset will be further investigated in the DR2 analysis.

Figure 6.

Pairwise, galaxy by galaxy comparison between velocity dispersion measurements obtained from bright and dark time spectra. Left panel: One-to-one comparison of pPXF velocity dispersion measurements with best-fit linear relation (red) and one-to-one line (black). Middle panel: Difference between the measurements as a function of the mean velocity dispersion. Over-plotted in red are the sample means in bins. Right panel: Distribution of the velocity dispersion measurement pairwise relative errors, with best-fit Gaussian (red) and unit Gaussian distribution (black). There is an offset of $\bar{\varepsilon} = -0.22$ indicating that dark time velocity dispersion measurements tend to be larger then bright time after accounting for errors. To correct for this we apply a $+2.3\pm0.1\,\textrm{km}\,\textrm{s}^{-1}$ correction to all bright-time velocity dispersion measurements.

After applying this correction, we compare the the primary and secondary velocity dispersion measurements to check the velocity dispersion error calculations. Figure 7 shows that the primary and secondary velocity dispersion measurements are consistent ( $\bar{\varepsilon}= 0.02$ ) but the errors are underestimated slightly ( $\sigma_\varepsilon =1.08$ ). The slightly higher primary measurements indicated by $\bar{\varepsilon}\gt0$ are due to our primaryVdisp selection criterion being the smallest relative error; if two observations have the same measurement error, we take the largest measurement. Since this offset is small, it should not cause any significant biases for the DR1 sample, but we will revisit using an alternative criteria for primaryVdisp in the DR2 analysis. Finally, we correct for our underestimated errors by manually inflating our velocity dispersion errors across our entire sample by a factor of 1.13 so that $\sigma_\varepsilon =1.0$ .

Figure 7.

As in Figure 6 but between velocity dispersion measurements obtained from ‘primary’ and ‘secondary’ spectra after applying the bright-dark correction. This sample consists of all repeated observations in our Fundamental Plane sample. The best fit Gaussian distribution (red line, right panel) is given by $\bar{\varepsilon}= 0.02$ $\sigma_\varepsilon =1.08$ , indicating good agreement between primary and secondary measurement but an underestimation of the velocity dispersion errors which we account for by scaling the velocity dispersion errors of our full sample by 1.13.

4.4 External consistency

To validate the consistency of our velocity dispersion measurements, we compared velocity dispersions measured using DESI spectra to those measured in SDSS DR16 for galaxies present in both samples. This was done by crossmatching our entire Fundamental Plane sample to SDSS DR16 using the external match files (survey-dr9-<region>-specObj-dr16.fits) provided by the Legacy Surveys as well as the north/south region of each galaxy and the BRICKID and BRICK_OBJID parameters. From this crossmatch we obtain SDSS object IDs as well as pipeline velocity dispersion measurements. We then further crossmatched using the internal SDSS IDs to the Portsmouth group catalogue (Thomas et al. Reference Thomas2013) which provides pPXF velocity dispersion measurements. This process leaves us with a crossmatched sample of $37\,038$ unique galaxies with both DESI (this work) and SDSS (Portsmouth catalogue) velocity dispersion measurements. Before completing any analysis, we apply the SDSS velocity dispersion cuts $70 \,\textrm{km}\,\textrm{s}^{-1} \lt\sigma_{\rm vdisp}\lt420 \,\textrm{km}\,\textrm{s}^{-1}$ (Howlett et al. Reference Howlett2022) on both the DESI and SDSS measurements, further restricting our sample to $35\,067$ galaxies with valid velocity dispersion measurements.

Figure 8.

As in Figure 7 but between DESI measurements (this work) and those from the Portsmouth catalogue (SDSS, Thomas et al. Reference Thomas2013). All DESI measurements are post corrections (as discussed in Section 4.3) and in the case of multiple observations, we use the ‘primary’ observation. Overall, there is good agreement between the two catalogues with the overall offset within 5% of errors. A small discrepancy ( $\lesssim$ 5%) emerges at high velocity dispersion ( $\sigma\gt250\,\textrm{km}\,\textrm{s}^{-1}$ ) as data becomes sparse, but this has minimal impact on the overall sample agreement.

The Portsmouth velocity dispersion measurements were obtained using the pPXF method, the MILES stellar library (Sánchez-Blázquez et al. Reference Sánchez-Blázquez2006) and SDSS spectra. The method is the same as used in this work. The pPXF fitting code has been extensively tested using simulations (Cappellari & Emsellem Reference Cappellari and Emsellem2004; Cappellari Reference Cappellari2017) and compared against measurements obtained using other methods in the SAURON (Cappellari & Emsellem Reference Cappellari and Emsellem2004) and LEGA-C (Cappellari Reference Cappellari2023) surveys where the method has been shown to consistently reproduce galaxy velocity dispersions. As the DESI and SDSS instrumentation and analysis pipelines are otherwise independent, we treat the respective velocity dispersions as being fully independent observations.

In Figure 8, we compare the velocity dispersion measurements obtained by this work and Thomas et al. (Reference Thomas2013). Both sets of measurements are in good agreement, with a preference for slightly smaller velocity dispersions measured by DESI but within 5% of the contoured errors. When analysing the differences in measurement bins (middle panel), the DESI and Thomas et al. measurements are very consistent until $\sigma \approx 250 \,\textrm{km}\,\textrm{s}^{-1}$ , at which point observations begin to become more sparse and the DESI measurements tend to be slightly larger than those of Thomas et al. This is a shift in the opposite direction to what is seen in the sample as a whole. The effect of these high velocity dispersion measurements on the mean offset is minimal, with the disagreement only increasing to within 7% of the contoured error bars when removing all galaxies with either velocity dispersion measurement greater than $250\,\textrm{km}\,\textrm{s}^{-1}$ . Both of these comparisons are much more consistent then what was observed during EDR (Said et al. Reference Said2024) where consistency with SDSS was only within 18% of the contoured errors.

When looking at the error measurements in Figure 8 (right panel) we obtain $\sigma_{\epsilon} = 1.1$ indicating a slight underestimate in one of or both of the DESI and SDSS errors. Unfortunately, with our current analysis it is impossible to determine which of the two samples (or both) are underestimating errors. It is important to note that, although this issue is not completely resolved, the combined error underestimate has been reduced significantly from the $\sigma_{\epsilon} = 1.17$ that was measured by Said et al. (Reference Said2024) in DESI EDR.

5.1 Parameters

We compute an angular effective radius from the half-light radius $r^{\text{morph}}$ and axial ratio $b/a$ using the formula:

(7) \begin{equation} \theta_e = r^{\text{morph}}\sqrt{b/a}.\end{equation}

From this we can calculate an observed physical effective radius, $r_{z}$ , by using the observed CMB-frame redshifts as a proxy to calculate comoving distances, $d(z_{\rm cmb})$ , assuming no peculiar velocities. This gives us the first distance-dependent Fundamental Plane parameter in units of $h^{-1}\textrm{kpc}$ :

(8) \begin{equation} r_z = \log\!(\theta_e) + \log\!(d(z_{\text{cmb}})) - \log\!(1+z_{\text{helio}}) \\+\log\!\left(\frac{1\,000\pi}{180 \times 3\,600}\right)\!.\end{equation}

We note that this observed physical radius, $r_{z}$ , will differ from the true effective radius of the galaxy due to the presence of peculiar velocities, which is what later enables us to extract the peculiar velocity of each galaxy given a calibrated FP relation. More concretely, given the fitted Fundamental Plane, we can define

(9) \begin{equation} r_{z} - as - bi - c = \log\!(d(z_{\text{cmb}})) - log(d(\bar{z})) \equiv \eta,\end{equation}

where $d(\bar{z})$ is the true comoving distance to the galaxy (with cosmological redshift $\bar{z}$ ) and $\eta$ is the so-called log-distance ratio.

The second Fundamental Plane parameter, s, is calculated from the velocity dispersions described in Section 4. However, these measurements are obtained by running pPXF on spectra measured through a fixed diametre fibre. To be used as a distance-independent parameter in the Fundamental Plane, we need to convert this measurement into the intrinsic central velocity dispersion for each galaxy by centralising the observed measurement. This is done because comparably sized galaxies at the same distance will have similar intrinsic velocity dispersions. However, when one of these galaxies is further away, it will appear smaller in the sky, and thus a larger proportion of the galaxy will be covered by the fibre, resulting in a decreased velocity dispersion measurement. Similarly, different sized galaxies at the same distance will have different proportions of their light captured by the fibre, artificially decreasing the measured velocity dispersion of the smaller galaxy. By centralising the velocity dispersion measurements we remove the effect of the angular size of each galaxy, making our measurements consistent across our sample.

To obtain the centralised velocity dispersion measurements, we apply the following aperture correction:

(10) \begin{equation} s = \log\!(\sigma) + \alpha_{\rm ap}\big[\log\!(\theta_e) - \log\!(8\theta_{\rm ap})\big], \end{equation}

where s is the centralised velocity dispersion in $\,\textrm{km}\,\textrm{s}^{-1}$ , $\sigma$ is the measured velocity dispersion obtained from the pPXF fit described in Section 4, $\theta_e$ is the angular effective radius, $\theta_{\rm ap} = 0.75$ is the aperture radius (in arcsec) of the DESI fibres (DESI Collaboration et al. 2022), and $\alpha_{\rm ap}$ is an aperture correction parameter.

A variety of aperture correction parameters $\alpha_{\rm ap}$ have been reported in the literature, ranging from $\alpha_{\rm ap} = -0.066\pm 0.035$ (Cappellari et al. Reference Cappellari2006) to $\alpha_{\rm ap} = -0.033\pm 0.003$ (de Graaff et al. 2021) with varying sample selection criteria. The choice of this parameter makes little difference to the individual log-distance measurements, causing shifts on the order of $\Delta \eta \approx 0.005$ , well within the typical Fundamental Plane-derived error of $\sigma_\eta \approx 0.12$ dex. However, as is discussed in Section 6.1.1, across the entire sample the choice of aperture correction can introduce a redshift-dependent tilt. For our baseline analysis, we used $\alpha_{\rm ap} = -0.06 \pm 0.03$ which flattens this tilt while encapsulating the range of literature values measured from early-type galaxy samples.

The last distance-independent Fundamental Plane parameter is the effective surface brightness. This parameter is measured from the apparent magnitude and angular effective radius of each galaxy

(11) \begin{align} i = 0.4(M^r_\odot - m_r^{\text{morph}} - Qz_{\text{cmb}} + k_r)- \log\!(2\pi\theta_e^2) &\notag\\ + 4\log\!(1+z_{\text{helio}}) + 2\log\!(64800/\pi )& \end{align}

where $M^r_\odot= 4.65$ mag is the r-band absolute solar magnitude, $m_r^{\text{morph}} = \log\!(f_r/A_r)$ is the galactic-extinction-corrected, r-band apparent magnitude for each galaxy, $k_r$ is the r-band k-correctionFootnote ^a and $2\log\!(64\,800/\pi)$ is a unit conversion term. We use the evolution correction parameter of $Q=1.1\pm0.4$ , which is calculated by taking the mean of the values found in the literature (Bernardi et al. Reference Bernardi2003; Blanton et al. Reference Blanton2003; Loveday et al. Reference Loveday2012; Loveday Reference Loveday2014; McNaught-Roberts et al. Reference McNaught-Roberts2014; Loveday et al. Reference Loveday2015) with error equal to the standard deviation of these values. The effects of varying this correction are explored Section 6.1.1.

5.2 Fitting methodology

To obtain log-distance ratios and peculiar velocities for our Fundamental Plane sample, we fit a 3D Gaussian Fundamental Plane model using a maximum likelihood method. This method was initially formulated by Saglia et al. (Reference Saglia2001) and Colless et al. (Reference Colless2001) for the EFAR Fundamental Plane analysis but has been further refined in subsequent Fundamental Plane studies (Magoulas et al. Reference Magoulas2012; Springob et al. Reference Springob2014; Said et al. Reference Said, Colless, Magoulas, Lucey and Hudson2020; Howlett et al. Reference Howlett2022; Said et al. Reference Said2024). The method is a three-step process that first fits a Fundamental Plane model assuming no peculiar velocities, then uses the obtained Fundamental Plane parameters to fit the most likely log-distance ratio to each galaxy individually. After the fitting process is complete, the sample is anchored to a subsample of galaxies with known distances, returning any (if present) radial bulk-flow motion to the sample.Footnote ^b

The likelihood of our Fundamental Plane model for N galaxies is:

(12) \begin{align} \mathcal{L} = \prod^N_{n=1} &\Bigg( \frac{1}{(2\pi)^{3/2} \lvert\textbf{C}_n\rvert^{1/2}f_n} \nonumber\\ &\;\;\; \times\exp\left[-\frac{1}{2}(\boldsymbol{x}_n -\bar{\boldsymbol{x}})\textbf{C}_n^{-1}(\boldsymbol{x}_n - \bar{\boldsymbol{x}})^T\right] \Bigg)^{1/S_n} ,\end{align}

where $\boldsymbol{x}_n = \{r_n,s_n,i_n\}$ are the Fundamental Plane parameters (as described in Section 5.1) for each observed galaxy n, $S_n$ is an inverse weighting based off the commonly used $1/V_{\text{max}}$ weighting (Schmidt Reference Schmidt1968) and accounts for galaxies missing from our sample due to the redshift, magnitude and velocity dispersion cuts through an up-weighting of the galaxies within the sample. Here $\bar{\boldsymbol{x}} = \{\bar{r}, \bar{s}, \bar{i}\}$ are the mean values of the Fundamental Plane model and $\mathbf{C}_n$ is the covariance matrix describing both the intrinsic scatter and measurement uncertainty in the Fundamental Plane. This can be determined from the measured r, s, and i uncertainties, two Fundamental Plane coefficients a and b, as well as three orthogonal intrinsic scatter parameters $\sigma_1$ , $\sigma_2$ , and $\sigma_3$ . The $f_n$ function is a normalisation term that corrects for the ‘selection bias’ of each individual galaxy and has been modified from Howlett et al. (Reference Howlett2022) to include an upper velocity dispersion cut. It can be found below by substituting in the minimum and maximum velocity dispersion and magnitude cuts in log-space (i.e. for this work $s_{\text{min}} = \log\!(50)$ , $s_{\text{max}} = \log\!(420)$ , $m_{\text{min}} = 10$ , and $m_{\text{max}} = 18$ ):

(13) \begin{align} f_n &= \frac{1}{(2\pi)^{3/2}\lvert\boldsymbol{C}_n\rvert^{1/2}}\int_{-\infty}^{\infty}\int_{r_{\text{min}}-i/2}^{r_{\text{max}}-i/2}\int_{s_{\text{min}}}^{s_{\text{max}}}\text{d}s\,\text{d}r\,\text{d}t \nonumber\\[5pt] &\quad \times \exp\left[-\frac{1}{2}(\boldsymbol{x}_{\boldsymbol{n}}-\bar{\boldsymbol{x}})\boldsymbol{C}_n^{-1}(\boldsymbol{x}_{\boldsymbol{n}}-\bar{\boldsymbol{x}})^T\right]\end{align}

where,

(14) \begin{multline} r_{\text{min(max)}} = \frac{1}{5}\big[10 + M_\odot^r + 5\log\!(1+z_{\text{helio}}) - Qz_{\text{cmb}}\\-2.5\log\!(2\pi) + k_r + 5\log\!(d(\bar{z}))-m_{\text{min(max)}}\big] \end{multline}

During this calculation, $f_n$ is calculated individually for each galaxy at each proposed cosmological redshift $\bar{z}$ and comoving distance $d(\bar{z})$ . As a result, each galaxy has many $f_n$ values that vary as the galaxy’s proposed distance increases. A full derivation of equations (12), (13) and (14) along with contained terms, can be found in Howlett et al. (Reference Howlett2022). The aforementioned paper describes a method to reduce the $f_n$ integral to a sum of elementary functions for fast computation. Using these equations, we are left with an 8-parameter model ( $a,\, b,\, \bar{r},\, \bar{s},\, \bar{i},\, \sigma_1,\, \sigma_2,\, \sigma_3$ ) which we fit to our data to obtain the Fundamental Plane relation for our dataset.

5.2.1 Fitting the Fundamental Plane

To fit the Fundamental Plane model we first remove all repeat observations from our sample by limiting our sample to primaryVdisp = True, reducing our sample to $98\,292$ unique galaxies. We then use the likelihood function in equation (12) to fit the Fundamental Plane. The maximisation of this likelihood is done using the SciPy differential evolution optimisation algorithm (Storn Rainer Reference Storn Rainer1997; Virtanen et al. Reference Virtanen2020) with a tolerance of $10^{-6}$ and a max iteration count of $10\,000$ . Uniform priors were set on each of the 8 free parameters with limits as shown in Table 2.

Table 2.

Uniform priors for each of the 8 fitted parameters of the Fundamental Plane model, where $\mathcal{U}(a,b)$ denotes a uniform distribution between a and b.

Table 2. Long description

A table with two columns and eight rows. The first column is labeled ’Parameter’ and the second column is labeled ’Prior’. The table lists the following parameters and their corresponding prior distributions: Row 1: Parameter alpha, Prior uniform distribution between 1.0 and 1.8. Row 2: Parameter b, Prior uniform distribution between -1.5 and -0.5. Row 3: Parameter r bar, Prior uniform distribution between -0.5 and 0.5. Row 4: Parameter s bar, Prior uniform distribution between 2.0 and 2.4. Row 5: Parameter s hat, Prior uniform distribution between 2.4 and 3.0. Row 6: Parameter sigma 1, Prior uniform distribution between 0.01 and 0.12. Row 7: Parameter sigma 2, Prior uniform distribution between 0.05 and 0.5. Row 8: Parameter sigma 3, Prior uniform distribution between 0.1 and 0.3.

Following the fit, we compute the $\chi^2$ difference between each data point and the best fitting Fundamental Plane model, from which we calculate the p-value for whether or not each galaxy is part of the distribution. Galaxies that have $p\lt 0.01$ are identified as outliers and removed from our sample. The entire fitting process is then repeated until the total number of outliers remains constant, taking the final iteration as our fitted model. This procedure converged after 6 iterations, removing $1\,534$ outlier galaxies from our sample, leaving us with $96\,758$ calibration galaxies which are flagged as FPcalibrator in our final PV catalogue.

5.2.2 Fitting the log-distance ratios

As described in Section 5.1 the $r_z$ values used as inputs to the Fundamental Plane model represent the distance-dependent parameter of the Fundamental Plane. When computing these values we use redshift as a proxy for distance. However, the true distance is required in order to determine galaxy peculiar velocities. Using the Fundamental Plane parameters calculated in the previous step, we can estimate the true intrinsic size of the galaxy $r_t$ . The difference between the measured and estimated sizes is equal to the log-distance ratio;

(15) \begin{equation} r_z-r_t = \eta \equiv \log\!(d(z_{\rm cmb})/d(z_t)).\end{equation}

This estimate of the true distance of the galaxy allows the peculiar velocity to be determined.

We fit the log-distance ratios by generating an array of 1001 uniformly distributed candidate log-distance ratios in the range $[\!-1.5,1.5]$ for each galaxy. We compute the likelihood for each log-distance and target combination using equation (12), fixing $N = 1$ , $S_n = 1$ and $\bar{\boldsymbol{x}}$ and $\textbf{C}_n$ to the best-fitting parameters from the previous step. This likelihood is combined with a flat prior on the log-distance ratio to obtain a posterior PDF for each galaxy, which is then normalised. From this PDF we produce summary statistics assuming a skew-normal distribution, recording the mean (logdist $= \langle\eta_n\rangle$ ), standard deviation (logdist_err $= \sigma_{\eta_n}$ ), and skew (logdist_alpha $ = \gamma_{\eta_n}$ ) as the log-distance location, scale, and shape parameters for each galaxy.

We compute log-distance ratios for all 108,810 targets that pass our initial selection criteria including non-primary observations (primaryVdisp = False) and Fundamental Plane outliers. These additional observations are useful for crossmatches to other DESI catalogues and for internal consistency checks. However, when doing population and cosmological analyses, only primaryVdisp = True targets should be used.

5.2.3 Correction for group richness

When carrying out preliminary analyses of the clustering of the log-distance ratios in the FP catalogue (see companion papers Turner et al. Reference Turner2025; Qin et al. Reference Qin2025; Lai et al. Reference Lai2025), an excess of clustering was identified in the velocity correlation function/power spectrum on small scales. We hypothesised this was due to a correlation between the richness of a group in which a FP galaxy resides and the measured log-distance ratio. A trend of this nature was found in Howlett et al. (Reference Howlett2022) using data from the SDSS. A correction was carried out in that work by fitting separate Fundamental Planes in bins of group richness, where it was found that the bias in the log-distance ratios could be removed by allowing the mean surface brightness $\bar{i}$ of the FP fit to vary between richness bins. The resultant fits exhibited a relatively smooth preference for lower mean surface brightnesses in richer groups, although at the time it was not clear whether this was due to intrinsic, physical, correlations between group environment and location of a galaxy on the FP, or data systematics such as difficulties de-blending photometry in dense clusters.

Unfortunately, there is not yet a group catalogue available for the DESI DR1 data, making testing and correcting for a similar systematic in this work more challenging. In lieu of this correction, a crude grouping algorithm was used to explore the relationship between the average log-distance ratio as a function of the number of other nearby FP measurements. Firstly, all pairs of galaxies in the full FP catalogue with a perpendicular and parallel separation (relative to the line of sight) of $r_{\perp}\lt1.75h^{-1}\mathrm{Mpc}$ and $r_{||}\lt8.5h^{-1}\mathrm{Mpc}$ , were identified.Footnote ^c Then, these pairs were converted into $12\,736$ distinct groups by identifying repeated indices (i.e. if a pair [A, B] and a pair [B, C] exist, these were grouped into a triplet [A, B, C] and so on). Accounting for only galaxies with primaryVdisp = True, $67\,055$ of the FP galaxies were found to reside in groups of at least two members. The remaining $31\,238$ were designated as field galaxies.

Using this grouping, a clear difference between the average log-distances of galaxies in groups and in the field, increasing with group size, was identified. This is shown in Figure 9. The direction of this trend – larger groups having, on average, more negative log-distance ratios – matches the trend seen in SDSS data (Howlett et al. Reference Howlett2022), although its amplitude in DESI is roughly half that found in the SDSS. The fact that a similar trend is found in both independent datasets is intriguing and points perhaps more towards a physical origin for the correlation. On the other hand, the difference in amplitude works against this hypothesis. It should be noted, however, that a weaker signal may also arise from our ad-hoc method of identifying groups.

Figure 9.

Average log-distance ratio as a function of group-size in the FP catalogue before (solid points) and after (open points) correction for this bias. The solid line indicates the piecewise fit used for the correction. The dotted horizontal line is the average log-distance ratio for field galaxies (i.e. those not in any groups).

In any case, the log-distance ratios in the FP catalogue are corrected for this trend by fitting a piece-wise linear $+$ constant function to the bias as a function of group-size relative to the field-average. Our best-fitting function is given by

(16) \begin{equation}\langle \eta_{\mathrm{group}} \rangle - \langle \eta_{\mathrm{field}} \rangle = \begin{cases} -0.0145\log_{10}(N_\mathrm{group}) & N_\mathrm{group} \lt 71 \\ -0.0268 & \mathrm{otherwise} \end{cases},\end{equation}

and gives a $\chi^{2}/\mathrm{dof} = 16.3/16$ . This correction was applied to each individual galaxy based its group occupancy. Log-distance ratios both with and without this correction are included in the final FP catalogue, but the group-corrected distances were used for subsequent zero-point calibration (Carr et al. Reference Carr2025) and clustering measurements (Turner et al. Reference Turner2025; Qin et al. Reference Qin2025; Lai et al. Reference Lai2025).

5.2.4 Zero-point calibration and peculiar velocities

Initially, the zero-point of the Fundamental Plane relation is unanchored to independent distance measurements, but set under the assumption of zero bulk radial flow. In other words, on average, log-distance ratios are 0 with no redshift evolution. The physical zero-point is set as the last step of fitting the Fundamental Plane since it is a simple offset $\Delta\eta$ to shift the Hubble Diagram to that of any given independent distance indicators (the ‘calibrators’).

The calibration of the zero-point for the DESI DR1 PV sample is done in the companion paper by Carr et al. (Reference Carr2025). There, we perform a joint fit of FP and TF data and independent supernovae Ia distances from the Pantheon+SH0ES sample (Riess et al. Reference Riess2022; Brout et al. Reference Brout2022). To do this, we make use of galaxy groups by equating the distance to any group member to all members of the group, then correct for any sample-wide average offset between DESI DR1 distances to those groups and the independently measured distances (the ‘zero-point’ offset). This step is essential to improve the statistical uncertainty introduced by minimal direct overlap of SNe Ia in DR1 FP and TF galaxies. Once the zero-point of the FP and TF relations have been tied to independent distances, we can measure the expansion rate, $H_0$ , directly from the DR1 distance-redshift relation. The method of finding the galaxy groups is described in Carr et al. (Reference Carr2025) along with the FP+TF global zero-pointing process and $H_0$ constraints.

Log-distance ratios are a measure of the difference between observed and true galaxy sizes, which we attribute to peculiar velocities; peculiar velocities alter observed redshifts such that the distance we infer differs from the true distance. Estimating the true sizes/distances from the FP relation introduces significant scatter that is not related to peculiar velocities, but in principle $\eta$ is a direct inference of PV. The same is true for TF measurements, except in that case we measure distance moduli $\mu$ and compare to a fiducial model such that the PV measure is $\Delta\mu=-5\eta$ . To convert from $\eta$ to peculiar velocities $\hat{v}$ , we use the Carreres et al. (Reference Carreres2023) estimator

(17) \begin{equation} \hat{v}(\eta) = c\ln{10}\left(\frac{(1+z_{\text{cmb}})c}{H(z_{\text{cmb}})d(z_{\text{cmb}})}-1\right)^{-1}\eta,\end{equation}

which reduces to other common estimators used in peculiar velocity studies under different approximations.Footnote ^d

Carr et al. (Reference Carr2025), uses the same fiducial cosmology as this work (flat $\Lambda$ CDM model with $H_0 = 100$ $\,\textrm{km}\,\textrm{s}^{-1}\, \textrm{Mpc}^{-1}$ and $\Omega_m = 0.3151$ ) during their zero-pointing and H0-fitting processes which is later updated to the best-fit value $H_0 =73.7$ $\,\textrm{km}\,\textrm{s}^{-1}\, \textrm{Mpc}^{-1}$ for computing peculiar velocities. This means that when applying the $\Delta\eta $ zero-point shift to the DESI DR1 FP data we need to take into consideration the difference between the fiducial and fitted models. We do so by taking the fiducial zero-point $\eta_\mathrm{zp} = -0.139 \pm 0.007$ (Table 2, Carr et al. Reference Carr2025) and subtracting the log ratio of the $H_0$ values to obtain the final shift $\Delta\eta =-0.136 -\log\left(H_0^\mathrm{fit}/H_0^\mathrm{fid}\right) = -0.006 \pm 0.007$ which is applied to our data.

Figure 10.

The projected Fundamental Plane for the DR1 FPcalibration sample. The figure shows the measured effective radii ( $r_z$ ) as a function of the predicted effective radii ( $r_t = as+bi+c$ ). The solid black line represents the one-to-one relation and the fiducial fitted Fundamental Plane parameters are superimposed.

6.1 Internal validation

The Fundamental Plane is a population-calibrated relation, meaning it is heavily dependent on the properties of the sample to which it is fit and can vary significantly between different populations of galaxies. In this work, we have attempted to isolate our sample to luminous red early-type galaxies using strict selection criteria as described in Sections 2 and 4.1. Some of these cuts, while important, are known to introduce biases, such as Malmquist bias from the magnitude and redshift cuts. These biases have been accounted for to the best of our ability in the likelihood function $f_n$ and $S_n$ terms (Equation 12). However, it is important that we test our methodology for additional sources of biases by exploring our fiducial analysis choices for how they affect the recovered Fundamental Plane relation and log-distance ratios.

Table 3.

Fundamental Plane parameters for the DESI DR1 sample and variations. In each row we make a single change to our selection criteria, method, or parameter choice, with each change described in the left column. The uncertainties on the fiducial Fundamental Plane parameters are derived from the scatter seen in mock realisations of our data (Bautista et al., in prep), and values that lie more than $3\sigma$ from our fiducial parameters have been bolded.

Table 3. Long description

The table presents Fundamental Plane parameters for the DESI DR1 sample and variations. It has 25 rows and 10 columns. The columns are labeled as follows: N gal, a, b, i, s, i, σ1, σ2, σ3, and σ2 int. Each row represents a different variation in selection criteria, method, or parameter choice. Row 1: Fiducial, 96758, 1.17, -0.803, 0.257, 2.132, 2.458, 0.0582, 0.456, 0.227, 0.097. Row 2: No sigma clipping, 98292, 1.23, -0.787, 0.256, 2.124, 2.443, 0.0630, 0.475, 0.248, 0.106. Row 3: σ vdisp < 360 km s^-1, 96745, 1.18, -0.803, 0.256, 2.131, 2.457, 0.0581, 0.457, 0.224, 0.097. Row 4: σ vdisp < 500 km s^-1, 96770, 1.18, -0.803, 0.256, 2.131, 2.457, 0.0581, 0.456, 0.224, 0.097. Row 5: σ vdisp > 60 km s^-1, 95153, 1.17, -0.803, 0.267, 2.145, 2.464, 0.0581, 0.459, 0.206, 0.096. Row 6: σ vdisp > 46 km s^-1, 97199, 1.19, -0.803, 0.251, 2.125, 2.454, 0.0581, 0.456, 0.233, 0.098. Row 7: m r < 17 mag, 69348, 1.17, -0.805, 0.246, 2.113, 2.440, 0.0598, 0.453, 0.236, 0.097. Row 8: m r > 14 mag, 94250, 1.18, -0.801, 0.250, 2.128, 2.460, 0.0581, 0.455, 0.223, 0.096. Row 9: Hα EW < 10 Å, 88733, 1.19, -0.798, 0.256, 2.146, 2.471, 0.0566, 0.452, 0.224, 0.095. Row 10: Hα EW < 1Å, 66240, 1.22, -0.783, 0.228, 2.156, 2.507, 0.0533, 0.439, 0.227, 0.090. Row 11: Δχ2 > 50, 96767, 1.18, -0.803, 0.256, 2.131, 2.457, 0.0581, 0.457, 0.224, 0.097. Row 12: no Δχ2 cut, 96770, 1.18, -0.803, 0.256, 2.131, 2.457, 0.0581, 0.456, 0.224, 0.097. Row 13: n s > 4, 59281, 1.19, -0.799, 0.336, 2.177, 2.400, 0.0584, 0.446, 0.204, 0.090. Row 14: B/A > 0.5, 76276, 1.19, -0.823, 0.285, 2.130, 2.411, 0.0608, 0.440, 0.229, 0.094. Row 15: Q = 1.5, 96771, 1.17, -0.802, 0.256, 2.131, 2.446, 0.0582, 0.457, 0.223, 0.097. Row 16: Q = 0.7, 96772, 1.19, -0.802, 0.256, 2.131, 2.468, 0.0585, 0.456, 0.224, 0.097. Row 17: Q = 0.85 (SDSS value), 96770, 1.19, -0.803, 0.256, 2.131, 2.464, 0.0585, 0.456, 0.224, 0.097. Row 18: α = -0.09, 96720, 1.21, -0.827, 0.256, 2.145, 2.457, 0.0595, 0.456, 0.222, 0.099. Row 19: α = -0.03, 96817, 1.15, -0.780, 0.256, 2.116, 2.455, 0.0588, 0.457, 0.228, 0.095. Row 20: α = -0.04 (SDSS value), 96797, 1.16, -0.788, 0.256, 2.121, 2.456, 0.0589, 0.457, 0.226, 0.095.

We looked at two primary factors when deciding on our final fiducial selection criteria: (1) minimising any tilt in the relationship between redshift and log-distance, (2) the trade-off between minimising the intrinsic scatter and maximising the sample size. The first of these criteria arises from our underlying assumption that the recovered peculiar velocities should not depend strongly on redshift. We evaluate this by plotting the mean log-distances, with uncertainties in redshift bins. The second criterion is more subjective and as such, unless there was a clear and significant improvement in the dispersion $\sigma_r^2$ , we maintained the selection criteria used in target selection (Saulder et al. Reference Saulder2023) and the DESI EDR Fundamental Plane (Said et al. Reference Said2024).

In addition, we wanted to identify possible weaknesses in our selection criteria for further analysis in DR2. We tested a variety of samples that fell within the targeting criteria, including modifications to velocity dispersion and $\Delta \chi^2_z$ cuts, tighter constraints on magnitude, Sérsic index and $b/a$ ratio as well as introducing a cut on the H $\alpha$ equivalent width. We also tested removal of the sigma-clipping step in our method as well as using a variety of evolution correction and aperture correction values from the literature to compute our Fundamental Plane input parameters. The Fundamental Plane parameters for each of these fits, including the resulting sample size and r-direction intrinsic scatter, $\sigma_r^2$ , are presented in Table 3.

Variations to the $b/a$ ratio, upper and lower velocity dispersion cuts and $\Delta\chi^2$ cuts had no or minimal effect on the resulting log-distance ratios and less than 5% variation in $\sigma_r^2$ . Decreasing the maximum magnitude cut shifted the mean log-distance higher but did so in a very uniform way across the sample, as such this shift would be removed during the zero-point process, resulting in equivalent distances and peculiar velocities in the fiducial sample. Additionally, while decreasing the magnitude limit slightly improves the surface brightness error, it significantly decreases the total sample size and increases the intrinsic $\sigma_r^2$ scatter, having the opposite effect to our goal.

If we instead tighten our lower magnitude cut to $m_r \gt 14$ mag, we observe that the tilt in the log-distance redshift relation flattens out earlier than the fiducial model at $z\sim0.025$ compared to $z\sim0.04$ . This cut also results in a minimal decrease in the overall sample size (loss of only $\sim$ $2\,500$ galaxies) which is ideal at first glance. However, these removed galaxies lie almost entirely at low redshifts with the cut removing $\sim$ 70% of the galaxies with $z\lt 0.02$ and $\sim$ 25% of galaxies with $z\lt 0.04$ . This is the primary redshift domain used for our zero-point calibration and as a result we reduce our total number of calibrators by 35%. Additionally, as FP uncertainties scale with redshift, this is also the domain on which we have the tightest constraint on galaxy distances and PVs. As such, we do not want to remove such a large percentage of these galaxies for a minimal decrease in the intrinsic scatter.

Increasing the Sérsic index cut or implementing a hydrogen alpha equivalent width (H $\alpha$ EW) cut both slightly reduce the $\sigma_r^2$ scatter but significantly reduce the sample size and exacerbate the redshift-log-distance tilt. It is important to note that the Sérsic index cut, and the H $\alpha$ EW cut have the greatest improvement on the $\sigma_r^2$ scatter out of the tested samples. As these parameters are both indicators of star formation, it indicates that there is likely some small residual spiral contamination in our sample. We have visually identified some of these contaminants and will investigate further ways to clean the sample in the DR2 analysis based on comprehensive visual inspection of the accumulated data.

6.1.1 Evolution and aperture corrections

Two parameters that were found to effect the log-distance redshift tilt are the two correction parameters; the aperture correction parameter, $\alpha$ , and the evolution correction parameter, Q. To reiterate, the aperture correction parameter enters into the Fundamental Plane as a redshift-independent correction to the s input parameter (Equation 10). It corrects for the fact that the proportion of the galaxy being captured by the spectrograph varies with the on-sky radius of the galaxy (in arcsec). This has the strongest effect of our two ‘corrections’, with more negative aperture correction indices rotating the log-distance-redshift plane clockwise around $z\sim0.06$ . The evolution correction parameter – a redshift-dependent term that corrects for galaxies dimming slightly as they evolve and is incorporated into the calculation of the Fundamental Plane input parameter i – has a smaller effect, although similar in nature to the aperture correction, rotating the mean log-distance ratios clockwise around $z\sim0.07$ for larger values of Q. During our analysis, we attempted to fit for Q, both as a Fundamental Plane parameter and separately following the method outlined by Saulder et al. (Reference Saulder, Mieske, Zeilinger and Chilingarian2013). Unfortunately, we were unable to converge on a value using these methods due to degeneracies with other parameters and analysis choices. Instead, we incorporated the wide spread of literature values for both the aperture correction and evolution correction terms into our fiducial analysis. Due to the similar effect these parameters had on the measured log-distances, we chose our fiducial parameters as $\alpha_\mathrm{fid} = -0.06\pm 0.03$ , $Q_\mathrm{fid} = 1.1\pm0.04$ , the pair that best flattened the log-distance-redshift tilt while staying within the published literature measurements.

In the literature, Q is often included in works that fit a linearly evolving luminosity function to galaxy populations spanning a wide redshift range. We identified five papers that measured the r-band luminosity function for early-type galaxies and fit a single population Q correction. They measure values as follows: $Q_{Bl03} = 1.62 \pm 0.3$ (Blanton et al. Reference Blanton2003), $Q_{L12} = 1.7\pm 0.2$ (Loveday et al. Reference Loveday2012), $Q_{L14} = 0.79$ (Loveday Reference Loveday2014), $Q_{MN14} = 0.80\pm0.26$ (McNaught-Roberts et al. Reference McNaught-Roberts2014), and $Q_{L15} = 0.73 \pm 0.1$ (Loveday et al. Reference Loveday2015). We also include the Bernardi et al. (Reference Bernardi2003) value $Q_{Be03} = 0.85$ that was measured using a modified Fundamental Plane relation. This is the value that was used by both the SDSS and DESI EDR Fundamental Plane surveys (Howlett et al. Reference Howlett2022; Said et al. Reference Said2024). Our fiducial evolution correction value comes from the mean of the previously listed literature values, taking the standard deviation as error, giving us $Q_\mathrm{fid} = 1.1\pm 0.4$ . In Figure 13, we plot the log-distance ratios resulting from the Fundamental Plane fits for each of the aforementioned Q values and their respective uncertainties, while fixing the aperture correction at the fiducial value. In the solid black line, we show the fiducial mean log-distance ratios with error bars representing the $1\sigma$ measurement uncertainties within each of the redshift bins, $\text{std}(\eta)/\sqrt{N_{\rm bin}}$ . Over-plotted in grey we show the range of the mean log-distances spanned by varying $Q_\mathrm{fid}$ by $1\sigma$ between $Q=0.7$ and $Q=1.5$ . This demonstrates that our choice of fiducial evolution correction and adopting/propagating through a suitable uncertainty on Q spans the results we would have obtained using any of the aforementioned literature values.

Figure 13.

Mean log-distance ratio in redshift bins for Fundamental Plane fits using the Fiducial aperture correction $\alpha_\mathrm{fid} = -0.06\pm 0.03$ while varying the evolution correction parameter. In the solid black we show our fiducial sample with the shaded region indicating the spread between $1\sigma$ variations of our fiducial evolution correction. For comparison the Fundamental Plane realisations using the literature evolution correction values are plotted in the coloured dashed lines. From this plot we can see that the chosen fiducial model is close to flat between $0.04\lt z\lt 0.09$ , bringing it into agreement with the assumption of redshift-independent peculiar velocities.

Figure 14.

As in Figure 13, but with fixed evolution correction $Q_\mathrm{fid} = 1.1\pm 0.4$ and varying aperture correction values from prior works. The Jorgensen et al. (J04, orange), Falcón-Barroso et al. (FB17, green), Mehlert et al. (M03, red) and Cappellari et al. (C06, purple) values are all measured using samples containing only early-type galaxies, while the de Graaff et al. value (G21, Blue) was measured from a mixed sample. As with Figure 13, the chosen fiducial aperture correction model is consistent with the assumption of redshift-independent peculiar velocities and has sufficiently large error to encompass variation arising from the different correction values.

For the aperture correction, we found the following values in the literature: $\alpha_{G21} = -0.033 \pm 0.003$ (de Graaff et al. 2021), $\alpha_{J95}=-0.04$ (Jorgensen et al. Reference Jorgensen, Franx and Kjaergaard1995),Footnote ^f $\alpha_{FB17} = -0.055\pm 0.02$ (Falcón-Barroso et al. Reference Falcón-Barroso2017), $\alpha_{M03} = -0.06$ (Mehlert et al. Reference Mehlert, Thomas, Saglia, Bender and Wegner2003), and $\alpha_{C06} = -0.066 \pm 0.035$ (Cappellari et al. Reference Cappellari2006). For each measurement, we fit the Fundamental Plane and plot the resulting log-distances in Figure 14. All the measurements we tested, except for the de Graaff et al. (2021) measurement, $\alpha_{G21}$ , which was taken on a combined sample, were measured using samples that contained only early-type galaxies. Zhu et al. (Reference Zhu2023) also provide a range of aperture corrections that depend on Sérsic index, absolute r-band magnitude and $g-i$ colour, but this work was not used in our analysis as their range of binned measurements was found to have poor overlap with our DR1 sample. For our fiducial value, we took $\alpha_\mathrm{fid} = -0.06 \pm 0.03$ as when combined with our choice of evolution correction $Q_\mathrm{fid}= 1.1\pm 0.4$ the log-distance-redshift tilt is close to flat through the body of our sample $0.04\lt z\lt 0.09$ (black line, Figure 13), and the error bar on this correction ( $\pm 0.03$ ) spans the range of measurements found in the literature.

To summarise our internal consistency checks, the log-distance ratio measurements resulting from our Fundamental Plane fit are not heavily impacted by our choice of selection criteria. The selection cuts that have the largest reduction in intrinsic scatter are the Sérsic index and the hydrogen alpha equivalent width (H $\alpha$ EW) cuts. Both of these parameters are used to identify late-type star-forming galaxies which indicates that there is some small spiral contamination in our sample. Due to the large reduction in sample size and relatively little gain for the cost, neither of these cuts has been implemented in the DR1 analysis, but we plan to further investigate this contamination and possible solutions in DR2. The two parameters that have the biggest effect on the log-distance ratio measurements are the evolution correction and aperture correction parameters. Varying either of these corrections within $1\sigma$ of the fiducial model has minimal effect on the recovered log-distance ratios of individual galaxies; the evolution correction giving $\eta\pm 0.002$ dex and the aperture correction giving $\eta \pm 0.005$ (typical log-distance ratio errors are $0.12$ dex). It should be noted that this is a redshift-dependent effect that more heavily impacts nearby galaxies, with $\Delta \eta \approx 0.05$ for the closest galaxies in our sample. As such, fixing these two parameters to a particular choice can affect results – such as the bulk flow – derived from a Fundamental Plane catalogue, highlighting the importance of marginalising over our uncertainty in these parameters as we have done in this work.

6.2 Comparison to CosmicFlows-4

It is important to compare our final distance measurements to external datasets in order to verify our results are free from bias. In this work we compare our results to the complete CosmicFlows-4 catalogue (CF4) as it is the most comprehensive distance catalogue currently available. We crossmatch to this catalogue by taking all matches within 5^″ of each primaryVdisp = True DR1 FP galaxy. This results in a total of $13\,112$ galaxies with measurements in both catalogues.

Our work presents distances in the form of log-distance ratios, $\eta$ , which are cosmology independent. This differs from CF4 who report the distance modulus. By choosing a cosmological model it is possible to convert from log-distance ratio to distance modulus using

(18) \begin{equation} \mu = 5 \log\!(d_{L}( z_{\rm cmb} )) - 5\eta + 25 ,\end{equation}

where $d_{L}( z_{\rm cmb})$ is the luminosity distance of the galaxy in Mpc. For the DESI results we use the group and zero-point corrected log-distance ratios, Flat $\Lambda$ CDM with $\Omega_m = 0.3151$ and $H_0 = 73.7$ $\,\textrm{km}\,\textrm{s}^{-1}\, \textrm{Mpc}^{-1}$ as fit by Carr et al. (Reference Carr2025).

We present the comparison between DESI DR1 FP and CF4 in Figure 15. For this comparison we use the weighted average distance modulus across all tracers. We compute a best fit relation $\mu_{\rm DESI} = 0.987^{+0.005}_{-0.005}\mu_{CF4} + 0.48^{+0.20}_{-0.18}$ which indicates good agreement between the samples. In orange we highlight the 279 CF4 galaxies that have distance measurements but not via the Fundamental Plane. These galaxies account for the majority of the outliers in the sample – given the almost constant offset between the orange and blue points in Figure 15 this could be indicative of either an internal zero-point difference between FP and non-FP measurements in the CF4 data, but it could also be that these galaxies are contaminants in our sample that should not be on the FP and so have slightly biased distances. It should also be noted that the CF4 and DESI measurements diverge for galaxies $\mu\lesssim33$ mag, which corresponds to redshift $z\sim 0.01$ . These galaxies are removed from our clustering sample before computing any cosmology fits (see section 7.1). The reason for these small number of disagreements are currently unknown but will be investigated further in DR2.

Figure 15.

Comparison of distance modulus as measured by CosmicFlows-4 (weighted average across tracers, Tully et al. Reference Tully2023) and DESI (this work). Overall, we find decent agreement between the two samples. Of the $13\,112$ crossmatches, 279 galaxies did not have a FP measurement in CF4, these have been highlighted in orange and account for the majority of outliers.

7.1 Clustering catalogues

From the full FP catalogue, we construct a subset of galaxy peculiar velocities for the purposes of cosmological measurements, in particular, clustering measurements of positions and velocities for constraining the growth rate of structure as in Qin et al. (Reference Qin2025), Lai et al. (Reference Lai2025), Turner et al. (Reference Turner2025). The clustering catalogue is designed to retain only the most robust peculiar velocity measurements and provide additional systematics weights to account for the effects of redshift completeness, fibre assignment and density variations in the imaging used for target selection.

The FP clustering catalogues are produced by first cutting out outlying galaxies with FPCalibrator = True, as well as very low redshift galaxies by setting $z_{\min}=0.01$ . We then crossmatch the remaining FP galaxies to the BGS large-scale structure (LSS) catalogues presented in Ross et al. (Reference Ross2025), based on their targetid. This keeps only those objects within the well-defined angular mask of the LSS catalogues and provides the previously mentioned systematics weights for each FP galaxy.

The final FP clustering catalogue contains 73 822 peculiar velocity measurements out of the original set of 98 292 non-duplicate measurements. Two catalogues containing sets of matching random non-clustered data points are also produced, with $20\times$ and $200\times$ the number of real galaxies. These random catalogues have the same sky, redshift, and peculiar velocity error distributions. From the random catalogues, we compute a gridded number density of BGS redshifts and FP-based peculiar velocities from which we extrapolate the number density at the location of each real FP galaxy in the clustering catalogue. These are used, for example, for measurements of the density and velocity correlation functions in Turner et al. (Reference Turner2025). The clustering catalogue and associated random catalogues will be released alongside the full FP sample.