2.1 Description of parameters

The relevant parameters for our study are the redshifts and peculiar velocities. The heliocentric redshift ( ${z_{\text{hel}}}$ ) is the ‘observed’ redshift.Footnote b We convert from ${z_{\text{hel}}}$ to the CMB frame redshift ( $z_{\text{CMB}}$ ) using the standard formulae in Section 5 and emphasise that we do not approximate these transformations. CMB-frame redshift refers to the redshift after we correct for only our own peculiar velocity, that is we correct for the Planck-observed CMB dipole. The peculiar velocity ( ${v_{\text{p}}}$ ) and corresponding peculiar redshift ( ${z_{\text{p}}}$ ) refer to the motion of the distant galaxy that is in addition to the Hubble flow. The final Hubble-diagram redshift that is used for cosmology ( $z_{\text{HD}}$ ) is the final stage, after we have corrected $z_{\text{CMB}}$ for the peculiar velocity of the distant galaxy. The standard formulae are also given in Section 5, and the derivation of the peculiar velocities themselves is described in Section 6. Each of these parameters have uncertainties represented by $\unicode{x03C3}$ . When the parameters come from the host galaxy group (from Peterson et al. Reference Peterson2021) instead of the individual host or SN, the symbol is preceded by ‘Group’.

2.2 Corrections and additions to previous data

Pantheon+ carries over the same SNe from Pantheon and includes many more SNe from FSS, DES, LOSS, SOUSA and CNIa0.02, as defined in Scolnic et al. (Reference Scolnic2021). Therefore, redshift and bookkeeping mistakes are carried over from Pantheon which were in turn carried over from their original sources, mostly from older SN compilations. After examining each of the samples listed in Table 1 we found and fixed various errors, and added improvements as follows:

• GOODS+PANS and SCP had Right Ascension (RA) and Declination (Dec) listed as zeros. As a result, the heliocentric corrections had been made to the incorrect part of the sky. We assign coordinates from the original datasets (Riess et al. Reference Riess2004; Riess et al. Reference Riess2007; Suzuki et al. Reference Suzuki2012) and recompute $z_{\text{CMB}}$ .

• We provide updates to DES redshifts from their 3-yr values (Smith et al. Reference Smith2020a) to their (previously unpublished) values that will be used in DES 5-yr cosmology. This includes reassigning uncertainty based on whether the redshift comes from the host or SN spectrum: 5 ${\times 10^{-4}}$ and 5 ${\times 10^{-3}}$ respectively.

• All SCP and GOODS+PANS SNe had redshift uncertainties set to 1 ${\times 10^{-3}}$ regardless of redshift source, so we reassigned the uncertainties the same way as with DES.

• Six FSS SNe had coordinates that disagreed with both the NED entry and FSS-assigned-host by up to tens of degrees. We therefore correct the SN coordinates to the NED coordinates, as listed in Table 3.

• One CfA4 SN had its location mistaken for a SN discovered around the same time. The record for 2008cm had the coordinates of SNF 20080514-002, but since the redshift was in agreement with the host of 2008cm, we update the SN coordinates using NED (Table 3).

• We update the coordinates of CfA1 SN 1996C to those of SIMBAD because the NED coordinates are incorrect (Table 3).

• Where we have identified the host of a SN, host coordinates are provided separately to SN coordinates. There was previously no record of SN hosts in Pantheon or some source catalogues. Heliocentric corrections are performed using host coordinates where possible.

• Where an International Astronomical Union name (IAUC) for a SN exists, we record it alongside the internal ID (SNID). The IAUC links SNe that are common across samples that use different internal names. However, we recommend that in future, a dictionary of all names for a SN be implemented since SNe without an IAUC must be matched via position instead.

• In collaboration with Peterson et al. (Reference Peterson2021), we include group-centre coordinates and group-average heliocentric redshifts, from which we derive group $z_{\text{CMB}}$ , group $z_{\text{HD}}$ and group ${v_{\text{p}}}{}$ (see Table 2).

Table 2.

Master table of updated redshifts. Symbols are defined in Section 2.1. The full machine readable table is available online from VizieR and at https://github.com/PantheonPlusSH0ES/DataRelease. The online versions of this table include columns for peculiar velocity uncertainty and binary classifications for if a SN has an associated host, if the redshift is from a host and if the supernova has group values. All peculiar velocities have an uncertainty of 250 $\mathrm{km\,s}^{-1}$ , and all ${z_{\text{hel}}}$ share the same uncertainty as their corresponding $z_{\text{CMB}}$ . Blank entries for group information mean the SN has no associated group, and blank IAUC entries mean there is no IAU name for the SN.

As a result of these changes, all SNe now have the same information: both the SNID and IAUC where applicable, both SN and host co-ordinates where applicable, redshifts in the heliocentric and CMB frame, and finally our updated peculiar velocities.

Table 3.

Corrections to SN positions.

4.1 Typical uncertainties

More problematic than reporting only class-based uncertainties is the non-reporting of uncertainties. Some SN data releases did not include uncertainties (e.g. CSP, CfA), in which case the only way to obtain an uncertainty estimate is to instead use an original measurement of the redshift. Our aim is for every redshift to have an uncertainty estimate, so we must estimate uncertainties where none are reported. Despite identifying the best primary source according to the hierarchy above, a small number of sources did not provide uncertainties. With no information about the spectra themselves, we opt to set missing uncertainties to a typical value, according to whether they are host or SN redshifts.

We examine the redshift uncertainties of five surveys with redshifts in Pantheon+ or that provide their own galaxy redshift uncertainty statistics: the Two-degree Field Galaxy Redshift Survey (2dFGRS; Colless et al. Reference Colless2001), the WiggleZ Dark Energy Survey (Drinkwater et al. Reference Drinkwater2018), SDSS, DES/OzDES (Lidman et al. Reference Lidman2020) and the Digital Archive of H i 21 Centimeter Line Spectra from Springob et al. (Reference Springob, Haynes, Giovanelli and Kent2005). However, we defer the discussion of SDSS-measured redshifts to Section 4.2 as SDSS redshift uncertainties are consistently smaller than other optical estimates.

While each method of redshifting naturally provides a way of estimating uncertainty, the more common way to estimate uncertainty is by comparing multiple measurements of the same object. Each survey found:

• 2dFGRS: Over $0<z<0.3$ , the RMS of multiple measurements was $\unicode{x03C3}_z=2.8{\times 10^{-4}}$ . The authors found a slight upwards trend with redshift.

• WiggleZ: Over $0<z<1.3$ , the standard deviation of multiple measurements ranged from $\unicode{x03C3}_z=1.7{\times 10^{-4}}$ for the highest quality spectra to $\unicode{x03C3}_z=2.7{\times 10^{-4}}$ for the lower quality but successfully redshifted spectra. The authors found no trend with redshift.

• OzDES: The collaboration opted to estimate uncertainties for classes of objects, assigning general SN host galaxies uncertainties of $\unicode{x03C3}_z=1.5{\times 10^{-4}}$ , which the authors state is a lower bound.Footnote d

• Springob et al. (Reference Springob, Haynes, Giovanelli and Kent2005) 21 cm Archive: Compared with optical, redshifts determined from the 21 cm line offer impressive precision. After reanalysing nearly 9000 nearby ( $-0.005<z<0.08$ ) H i galaxies, the mean and median redshift uncertainty was 1.7 ${\times 10^{-5}}$ and 1.2 ${\times 10^{-5}}$ respectively.

For the Pantheon+ low-z sample, which has a mix of optical and radio redshifts from the literature, we find a median redshift uncertainty of 9 ${\times 10^{-5}}$ , which is consistent with the above measurements. We also include in this test the 30 PS1MD and SNLS SNe whose host galaxies we find third party redshifts for, extending to $z\sim0.50$ . We show in Figure 4 the distribution of the redshift uncertainties of this modified low-z sample. Since there remain 30 host galaxies that lack redshift uncertainties, all at low-z, we opt to assign the median uncertainty calculated from the rest of our low-z sample (see Table A3).

Figure 4.

Comparison of SDSS DR13 redshift uncertainties and all other low-z SN uncertainties. SDSS DR13 has a tight distribution peaking around 1 ${\times 10^{-5}}$ , while low-z is much broader due to its heterogeneous redshift sub-samples. Redshifts larger than 0.0006 have been collected into a single bin, as the maximum is 0.002 (Perlmutter et al. Reference Perlmutter1999). The spike in low-z at $\unicode{x03C3}_z=0.00015$ is due to 6dF, while the SDSS spike at $\unicode{x03C3}_z=0.0005$ corresponds to the uncertainty typically set for a redshift from emission/absorption lines in a SN spectrum.

Until now, we have discussed only host galaxy redshifts because these have been studied in detail. Estimating redshifts from SN spectra has had less attention. Like host galaxy redshifts, there are SN redshifts without uncertainty estimates. In these cases, we rely on a less concrete estimation of $\unicode{x03C3}_z=0.005$ , based on the population of SN redshifts to date and expert opinion. We reiterate that this is a somewhat conservative but fair estimation. As an example, as given by SNID (Blondin & Tonry Reference Blondin and Tonry2007), the mean redshift uncertainty of the 51 supernovae with Type Ia SNID classifications from the CfA Supernova Group websiteFootnote e is $\unicode{x03C3}_z=4.1{\times 10^{-3}}$ with a standard deviation of 1.7 ${\times 10^{-3}}$ .

See Table A3 for a list of all SNe previously missing uncertainties.

4.2 Underestimated SDSS uncertainties

We noted in Section 3 that when multiple redshift measurements were available in SDSS, the dispersion in redshift values is usually higher than expected from their typical uncertainties of 1 ${\times 10^{-5}}$ . This comparison indicates that typical SDSS redshift uncertainties are somewhat optimistic. We quantify this comparison by calculating both average uncertainty, $\overline{\unicode{x03C3}_z}$ , and dispersion in z, $\unicode{x03C3}(z)$ , for all multiply-measured objects in SDSS DR13. Only objects with two or more reliable redshifts are used.

The average uncertainty versus z dispersion for all multiply-measured objects is shown in Figure 5. We perform a linear fit to the objects with five or more measurements weighted by number of redshift measurements. Calculating standard deviation from few points is systematically biased low (grey points), so we exclude objects with two–four measurements. We also exclude dispersions above 0.01 since these dispersions are caused by multiple confident but disparate redshifts. Thus the fit was performed on 1000 of 31000 objects.

Figure 5.

Standard deviation of multiple SDSS DR13 reliable redshift measurements of the same object against the average of their quoted uncertainties. We expect a slope of 1.0 (red dashed line) if the uncertainties are appropriate. The solid blue line is a linear fit to all data with five or more measurements and a dispersion of $\unicode{x03C3}(z)<0.01$ , which shows that, consistently, $\unicode{x03C3}(z)>\overline{\unicode{x03C3}_z}$ . Every point with $\unicode{x03C3}(z)\gtrsim0.01$ is a catastrophic redshift failure caused by at least two distinct confident redshifts.

The uncertainty of 9 ${\times 10^{-5}}$ that we apply to sources with no provided uncertainty reflects the mean dispersion of 9.5 ${\times 10^{-5}}$ we calculate for repeated non-outlying SDSS measurements. Since we see a gradient of almost unity and a positive offset of roughly 3 ${\times 10^{-5}}$ between dispersion and estimated uncertainty, increasing the SDSS uncertainties by this amount would be reasonable, and we test the impact of this increase in our cosmological analysis in Section 7.Footnote f

5.1 Heliocentric corrections

Most publicly available heliocentric corrections, including those currently in NED and Pantheon used a low-redshift approximation (although the approximation in NED will be corrected; see Carr & Davis Reference Carr and Davis2021). When performing the heliocentric correction, the low-redshift approximation assumes the observed redshift ${z_{\text{hel}}}$ is an additive combination of $z_{\text{CMB}}$ and the redshift due to our Sun’s peculiar motion, $z_{\text{Sun}}$ ,

(1) \begin{equation}{z_{\text{hel}}}={z_{\text{CMB}}^{\mathrm{+}}}+z_{\text{Sun}}.\end{equation}

However, the correct way to combine redshifts is to multiplicatively combine factors of $(1+z)$ , so

(2) \begin{equation}(1+{z_{\text{hel}}})=\left(1+{z_{\text{CMB}}^{\mathrm{\times}}}\right)(1+z_{\text{Sun}}).\end{equation}

This gives the correct CMB-frame redshift, which is

(3) \begin{equation}{z_{\text{CMB}}^{\mathrm{\times}}} = \frac{1+{z_{\text{hel}}}}{1+z_{\text{Sun}}}-1.\end{equation}

The difference between using the additive approximation and the correct multiplicative equation is exactly $z_{\text{CMB}}z_{\text{Sun}}$ . Since $z_{\text{Sun}}$ is our own motion with respect to the CMB (our velocity $v_{\text{CMB}}$ in the direction of the CMB dipole), it is of order ${10^{-3}}$ . Therefore, at low-z, the difference between ${z_{\text{CMB}}^{\mathrm{\times}}}$ and ${z_{\text{CMB}}^{\mathrm{+}}}$ appears almost negligible; however, by $z\sim 1$ , the error is on the order of ${10^{-3}}$ , which is an order of magnitude larger than most reported statistical uncertainties in redshifts.

We have ensured that all sub-samples in the new Pantheon+ sample consistently use the multiplicative correction.

5.1.2 Calculating $z_{\text{Sun}}$

The projection of the Sun’s peculiar velocity along the line of sight to an object is

(4) \begin{equation} v_{\text{Sun}} = \boldsymbol{v}_{\text{Sun}}\cdot\hat{\boldsymbol{n}}_{\text{obj}} = {v_{\text{Sun}}^{\mathrm{max}}}\cos\alpha, \end{equation}

where $\hat{\boldsymbol{n}}_{\text{obj}}$ is the object’s position vector, and $\alpha$ is the angle separating the dipole direction and the object.

Since the Sun’s velocity is small (order of ${10^{2}}\,\mathrm{km\,s}^{-1}$ ) compared to c, the low-z approximation $z_{\text{Sun}} \approx -v_{\text{Sun}}/c$ is adequate, but we use the full special relativistic calculation

(5) \begin{equation} z_{\text{Sun}} = \sqrt{\frac{1+({-}v_{\text{Sun}})/c}{1-({-}v_{\text{Sun}})/c}}-1,\end{equation}

because there is negligible computational advantage to the approximation. The minus signs before $v_{\text{Sun}}$ have been left explicit to emphasise that at zero angular separation ( $\alpha=0$ in Equation (4)) the object should appear slightly blueshifted due the our velocity directly towards it.

5.2 Peculiar velocity corrections

The peculiar redshifts arising due to the peculiar velocities of the supernova host galaxies also need to be treated multiplicatively. Equation (2) becomes

(6) \begin{equation}(1+{z_{\text{hel}}})=(1+z_{\text{HD}})(1+z_{\text{Sun}})(1+{z_{\text{p}}}{}).\end{equation}

Here we use the ‘Hubble diagram redshift’ $z_{\text{HD}}$ , which is the cosmological redshift we are interested in. This differs in our nomenclature from the CMB-frame redshift $z_{\text{CMB}}$ , because $z_{\text{CMB}}$ takes into account our motion but not the peculiar velocity of the source. Combined with Equation (2) we see

(7) \begin{equation}z_{\text{HD}}=\frac{1+z_{\text{CMB}}}{1+{z_{\text{p}}}{}} -1.\end{equation}

Thus the Hubble diagram redshift requires knowledge of the SN host’s peculiar redshift ${z_{\text{p}}}$ , so we turn to how we derive peculiar velocities.

6.1 Estimating peculiar velocities

The most precise way to estimate peculiar velocities is to measure the density field (e.g. through a redshift survey) and use that to predict the expected peculiar velocity field.Footnote g This is known as velocity field reconstruction. Importantly, this method does not use supernova distances, and therefore does not introduce correlations between the peculiar-velocity-corrected SN redshift and its measured distance. The reconstruction does require an assumed cosmological model, but the cosmological dependence is weak. We quantify the impact of these peculiar velocity corrections on cosmological parameters in Section 7.

In the linear regime, peculiar velocity is related to the gravitational acceleration via:

(8) \begin{equation} v(r) = \frac{f}{4\pi}\int \mathrm{d}^3\boldsymbol{r}^\prime \unicode{x03B4}(\boldsymbol{r}^\prime) \frac{\boldsymbol{r}^\prime-\boldsymbol{r}}{|\boldsymbol{r}^\prime-\boldsymbol{r}|^3},\end{equation}

where f is the growth rate of the cosmic structure and $\unicode{x03B4}(\boldsymbol{r})$ is the density contrast. This equation has two limitations:

• Galaxy surveys do not measure the total matter density, so it is assumed that the observed galaxy density ( $\unicode{x03B4}_g$ ) linearly traces the total density, $\unicode{x03B4} = \unicode{x03B4}_g/b$ . Here b is the linear biasing parameter, which is different for different types of galaxies, and therefore has to be measured or marginalised over.

• The region over which we have a sufficient number density of measured galaxies to do reconstruction is smaller than the region for which we need to estimate peculiar velocities. This has been addressed by estimating the ‘external velocity’, ${\boldsymbol{V}_{\text{ext}}}$ , which arises due to structures outside the survey volume, and estimating how that would theoretically decay with distance (Section 6.3).

We use the velocity field reconstruction created by Carrick et al. (Reference Carrick, Turnbull, Lavaux and Hudson2015), which uses data from the 2M $++$ compilation from Lavaux & Hudson (Reference Lavaux and Hudson2011). 2M $++$ includes data from 2MRS (Huchra et al. Reference Huchra, Fairall and Woudt2005), 6dFGS (Jones et al. Reference Jones2009), and SDSS (Abazajian et al. Reference Abazajian2009) and extends to a radius of $r_{\text{max}}=200\;{h}^{-1}\mathrm{Mpc}$ . One slice ( $SGZ=0$ ) of the reconstructed density field is shown in Figure 6 in the supergalactic plane ( $SGX-SGY$ ). Figure 7 shows the 2M $++$ velocity field in redshift-space on a regular spatial grid.

Figure 6.

Slice of the 2M $++$ reconstructed density field plotted in the supergalactic plane ( $SGZ=0$ ). White areas are regions for which data is missing, therefore the density reconstruction is uncertain.

Figure 7.

2M $++$ velocity field plotted on a regular grid of redshift-space positions. Note the white regions in Figure 6 have been interpolated over to make a complete velocity field out to 200 ${h}^{-1}\mathrm{Mpc}$ , which means those regions of the velocity field will have higher uncertainty than other regions.

A key model parameter to evaluate is $\unicode{x03B2}\equiv f/b$ . The rate of growth is often parameterised by $f={\Omega_{\text{m}}}^\gamma$ , where $\gamma=0.55$ in the standard cosmological model, $\Lambda$ CDM. Importantly, however, $\unicode{x03B2}$ is determined empirically rather than computed from the $\Lambda$ CDM model.

Both $\unicode{x03B2}$ and ${\boldsymbol{V}_{\text{ext}}}$ are derived from a combination of density field reconstruction and observation. The reconstruction process delivers a normalised peculiar velocity field, $v_{\rm p,recon.}(\boldsymbol{r})$ , which gives the directions and relative magnitudes of the peculiar velocities as a function of position. Predictions from this peculiar velocity field are compared with galaxies that have peculiar velocities derived from distance measures such as the Tully-Fisher relation or Fundamental Plane relation. The calibrated peculiar velocity field is

(9) \begin{equation}v_{\rm p}(\boldsymbol{r}) = \unicode{x03B2} v_{\rm p,recon.}(\boldsymbol{r}) + V_{\text{ext}}(\boldsymbol{r}).\end{equation}

The parameter $\unicode{x03B2}$ thus acts as a scaling of the normalised velocity field (subject to the sample of observed ${v_{\text{p}}}$ ), and ${\boldsymbol{V}_{\text{ext}}}$ is the residual mean velocity.

Carrick et al. (Reference Carrick, Turnbull, Lavaux and Hudson2015) measured $\unicode{x03B2}=0.431\pm0.021$ and an external velocity of $|{\boldsymbol{V}_{\text{ext}}}{}| = 159\pm23\,\mathrm{km\,s}^{-1}$ in the direction of $(l, b) = (304{^{\circ}}\pm11{^{\circ}}, 6{^{\circ}}\pm13{^{\circ}})$ . While we use the velocity field measured by Carrick et al. (Reference Carrick, Turnbull, Lavaux and Hudson2015), we use an updated value of $\unicode{x03B2}=0.314^{+0.031}_{-0.047}$ derived in Said et al. (Reference Said, Colless, Magoulas, Lucey and Hudson2020), which gives a better fit when comparing SDSS Fundamental Plane peculiar velocities to the predicted peculiar velocity field. We confirm that this lower $\unicode{x03B2}$ value results in a lower scatter in the supernova Hubble diagram, see Peterson et al. (Reference Peterson2021).

Carrick et al. (Reference Carrick, Turnbull, Lavaux and Hudson2015) estimated the peculiar velocity uncertainty to be 250 $\mathrm{km\,s}^{-1}$ for the galaxies (the particle velocity field) and 150 $\mathrm{km\,s}^{-1}$ for galaxy groups (the haloes). In other words, the uncertainty on an individual galaxy’s peculiar velocity is higher than the uncertainty on peculiar velocity of the group in which it resides. We note, however, that some regions of the reconstruction are less certain than others because of incomplete sampling. Unfortunately, sampling is not accounted for in current models, so we adopt the value of $\unicode{x03C3}_{v_{\rm p}}=250$ $\mathrm{km\,s}^{-1}$ and leave a more precise estimate of the peculiar velocity uncertainty for future work.

6.2 Real vs redshift-space

Carrick et al. (Reference Carrick, Turnbull, Lavaux and Hudson2015) provide the velocity field in ‘real-space’, so the position and distance of a galaxy can be used to draw the peculiar velocity directly from the modelled velocity field. However, in supernova cosmology, the distance measured via the distance modulus should be independent of the redshift. This distance should not be used to predict the peculiar velocity to correct the redshift. Converting the observed redshift to distance (by assuming a cosmology) to estimate the peculiar velocity is valid, but less precise than using the redshift. We therefore convert the reconstructed velocity field of Carrick et al. (Reference Carrick, Turnbull, Lavaux and Hudson2015) to redshift-space. While we assume a cosmology for this conversion, any reasonable choice has a negligible effect on the velocity field. Thus we query the peculiar velocity model using the coordinates of each host galaxy or SN (RA, Dec, $z_{\text{CMB}}$ ), and Equation (9).

The conversion to redshift-space takes two steps. First, for each real-space grid point i we convert the real-space position, $\boldsymbol{r}_i$ , into redshift position $\boldsymbol{z}_i$ using the predicted peculiar velocity at that grid point, $\boldsymbol{v}_{\text{p},i}$ . Second, we use inverse distance weighting to interpolate and adjust the irregularly-spaced grid in redshift-space to a regular grid. This process is described in more detail in Appendix B.

An alternative method is to integrate along the line of sight over real-space. This technique is used by the online toolFootnote h associated with Carrick et al. (Reference Carrick, Turnbull, Lavaux and Hudson2015), which was previously used to estimate the Pantheon peculiar velocities. Both methods agree very well, within the uncertainty, as seen in Figure 8. The mean difference is only 5 km s^–1 (50 times smaller than the individual uncertainty).

Figure 8.

The predicted peculiar velocity for a sample of 851 host galaxies using two different approaches. Each ${v_{\text{p}}}$ has an uncertainty of 250 $\mathrm{km\,s}^{-1}$ . We use the 2M++ velocity field converted from real-space to redshift-space whereas the method associated with Carrick et al. (Reference Carrick, Turnbull, Lavaux and Hudson2015) (previously used for Pantheon), integrates over real-space along the line of sight for each host galaxy. There are only negligible systematic differences between the results of these techniques, and all scatter is within 1 $\unicode{x03C3}$ .

6.3 Velocities beyond $r_{\rm max}$

It is difficult to properly account for velocities outside $r_{\rm max}$ because we do not have an adequate measurement of the density field to predict individual velocities precisely. However, we expect velocities to continue to behave largely according to the bulk flow trend beyond $r_{\rm max}$ as a consequence of $\Lambda$ CDM large scale structure. In standard $\Lambda$ CDM a theoretical bulk flow magnitude of ${\sim}20\,\mathrm{km\,s}^{-1}$ is expected even for a sphere with radius $z\sim1$ (grey dashed lines in Figure 9). Accordingly, peculiar velocities of galaxies outside $r_{\rm max}$ should not be set to zero.

Figure 9.

Top: Pantheon (original) peculiar velocities converted to redshift. The expected decay of peculiar velocity amplitude according to $\Lambda$ CDM is over-plotted (grey dashed), outside the limit of the reconstruction at 200 ${h}^{-1}\mathrm{Mpc}$ ( $z\approx0.067$ ). The spurious increase in peculiar velocities as redshift increases is driven by the erroneous use of the low-redshift approximation (Equation (1)). The error term is plotted (red dashed) only for the four SNLS fields. Bottom: Pantheon+ (this work) peculiar velocities, converted to redshift, which are now well-behaved beyond low-redshift.

To ensure a smooth transition across $r_{\rm max}$ we have chosen to model the bulk flow as a decaying function consistent with $\Lambda$ CDM expectations, and in the direction of the bulk flow of the 200 ${h}^{-1}\mathrm{Mpc}$ sphere. While there is a $\Lambda$ CDM model dependence, the impact of this high-z correction on cosmological inferences is small both because the corrections are small (at most ${\sim}5 \times 10^{-4}$ when in the direction of the bulk flow), and because at high-z these peculiar redshifts represent a small fraction of the total redshift.

We note that there is a slight difference between ${\boldsymbol{V}_{\text{ext}}}$ and the bulk flow of the 200 ${h}^{-1}\mathrm{Mpc}$ sphere. The bulk flow of the sphere is the average of the internal velocities (which is small but non-zero), plus the external velocity. We calculate the bulk flow at the 2M $++$ maximum radius of 200 ${h}^{-1}\mathrm{Mpc}$ to be $182\pm23\,\mathrm{km\,s}^{-1}$ in the direction of $(l, b) = (302{^{\circ}}\pm10{^{\circ}}, 2{^{\circ}}\pm9{^{\circ}})$ . At this large radius, the bulk flow is dominated by the external bulk flow ( $V_{\text{ext}} \approx 170$ km s^–1: Said et al. Reference Said, Colless, Magoulas, Lucey and Hudson2020; Boruah, Hudson & Lavaux Reference Boruah, Hudson and Lavaux2020) plus a small contribution from the mean internal velocities.

Contrary to common expectations, the bulk flow should not necessarily converge on the direction of the CMB dipole. The observed CMB dipole is particular to our own motion, and is removed with the correction from the heliocentric to the CMB frame. Dramatic changes to the magnitude and direction of bulk flow direction become unlikely as we average over spheres that approach the scale of homogeneity. Therefore, we fix the direction of the decaying bulk flow in the direction of the 200 ${h}^{-1}\mathrm{Mpc}$ sphere’s bulk flow.

The Pantheon sample peculiar velocities outside the velocity reconstruction suffered three main issues which can be seen in Figure 9. The most apparent issue is the increasing ${v_{\text{p}}}$ with redshift. We show that this artefact is caused by the low-z approximation of the heliocentric correction by plotting the error term (i.e. the difference between Equations (1) and (2)) for the four SNLS fields, as these stretch to high-z. This particular error also occurs for PS1MD and SDSS but is less visible since these surveys do not extend as far in redshift. Also visible in Figure 9 are the HST SNe with ${v_{\text{p}}}{}=0$ , and the somewhat random scatter around $z\approx0.15$ which may have been due to assigning 2M++ peculiar velocities outside the 2M++ sphere. We show in the bottom panel of Figure 9 that these issues are now resolved.

7.1 Dependence of distance modulus on redshift change

We analyse each variation independently, meaning that for each: each SN light curve is fit using the SALT2 model Guy et al. (Reference Guy2010) as derived in Brout et al. (Reference Brout2021) using SNANA (Kessler et al. Reference Kessler2009), biases are corrected following the BEAMS with Bias Correction (BBC) framework established in Kessler & Scolnic (Reference Kessler and Scolnic2017), and distance moduli $\unicode{x03BC}$ are determined, all within the Pippin framework (Hinton & Brout Reference Hinton and Brout2020). The details of this process are given in Brout et al. (Reference Brout2022). The resulting distance moduli changes can be seen in Figure 10.

The total number of supernovae in each variation changes slightly due to various reasons. When we restrict the redshift range for fitting $H_0$ and w, some borderline-redshift supernovae are shifted in and out of the sample due to redshift changes. However, shifting redshifts also affects the light curve fit parameters. The quality cuts we apply are to these fit parameters (among others) and also their errors. A redshift shift can therefore also shift a supernova in or out of the sample. For example, a light curve parameter may pass the cut with the shifted redshift, or the fit may be poorer at the shifted redshift.

We expect $\unicode{x03BC}$ to change when a supernova redshift changes because the duration of a light curve (stretch) is affected by time-dilation and its colour is affected by K-corrections, both of which are dependent on the measured heliocentric redshift (but not the peculiar velocity correction); the theoretical basis for these variations is explained in Huterer et al. (Reference Huterer, Kim, Krauss and Broderick2004). However, the distance modulus of a supernova may also change between sample variations without changing the redshift. The procedure for deriving distance moduli for a supernova sample (e.g. using BBC; Kessler & Scolnic Reference Kessler and Scolnic2017) determines the peak magnitude ( $m_B$ ) from the light curve, the global parameters $\alpha$ and $\unicode{x03B2}$ that adjust the stretch ( $x_1$ ) and colour (c) of the supernovae, and a correction term for selection biases ( $\unicode{x03B4}\unicode{x03BC}_{\text{bias}}$ ), according to a modified version of the Tripp relation (Tripp Reference Tripp1998), following Kessler et al. (Reference Kessler2019),Footnote i

(10) \begin{equation}\unicode{x03BC} = m_B-M+\alpha x_1 - \unicode{x03B2} c -\unicode{x03B4}\unicode{x03BC}_{\text{bias}}.\end{equation}

Therefore the distance modulus may change even if the redshift is not altered since the calibration of $\alpha$ , $\unicode{x03B2}$ and absolute magnitude (M) depend on the sample as a whole. This explains the slight changes in distance moduli for variation 6 (the host-z sample), in which we do not alter any redshifts.

The dependence of the change in a supernova’s $\unicode{x03BC}$ between variations on the change in z is demonstrated in Figure 11, which shows that the two are strongly correlated (see also Huterer et al. Reference Huterer, Kim, Krauss and Broderick2004). This correlation results in a cancellation that reduces the impact of redshift uncertainties, particularly at mid-range redshifts (around $z\sim0.5$ ). This can be seen in the solid lines in Figure 11, which show the slope (d $\unicode{x03BC}/\textit{d}z$ ) of the Hubble diagram at different redshifts. At mid-range redshifts the slope is the same as the degeneracy direction between redshift change and distance modulus change. Thus uncertainties in redshift essentially cancel out at these redshifts, as they cause points to be shifted along the magnitude-redshift relation instead of deviating from it. This may be particularly helpful for supernova cosmology using photometric redshifts whose uncertainties are larger than spectroscopic redshifts (Chen et al. Reference Chen2022).

Figure 11.

Change in distance modulus versus change in redshift (circles), and $\unicode{x03BC}(z)$ derivatives d $\unicode{x03BC}$ /dz for given redshifts (black lines). The largest changes occurred due to amending ${z_{\text{hel}}}$ . The black lines represent the purely theoretical $\Delta\unicode{x03BC}$ that would occur on the Hubble diagram given the $\Delta z$ on the horizontal axis (in other words, the slope of the Hubble diagram at those redshifts). The positive trend demonstrates that a change in redshift is partially cancelled the corresponding change in $\unicode{x03BC}$ (see discussion in Section 7.1). This trend is seen in each variation.

For a simplified assessment of how redshift changes impact cosmology, it is natural to simply use the published $\unicode{x03BC}$ values, and shift the redshifts (as was done in Davis et al. Reference Davis, Hinton, Howlett and Calcino2019; Steinhardt et al. Reference Steinhardt, Sneppen and Sen2020). For the low-z sample, we show using the large, transparent symbols in Figure 12 that this gives a reasonable approximation to the full analysis that uses recalculated $\unicode{x03BC}$ values (smaller symbols). The main difference of not recalculating $\unicode{x03BC}$ is less cancellation of the effect of systematic redshift changes. At intermediate redshifts one might expect that neglecting the cancellation in d $\unicode{x03BC}$ /dz may overestimate the deviation due to redshift shifts. Interestingly, we find that when fitting for w without re-deriving $\unicode{x03BC}$ the change in w often becomes slightly larger (e.g. variations 0–2) but in some cases (e.g. variations 8–9, shifting the redshifts by 1 $\unicode{x03C3}_z$ ) we find the shift in w goes in the opposite direction (likely due to $\unicode{x03C3}_z$ increasing with z). Overall we find that doing approximate cosmology fits by changing z without changing $\unicode{x03BC}$ gives reasonable results, but for precision cosmology one should refit $\unicode{x03BC}$ whenever ${z_{\text{hel}}}$ changes.

Figure 12.

The impact on inferences of $H_0$ and w due to changes in the redshifts. The regular symbols are the nominal results while the faint, larger symbols come from not recalculating $\unicode{x03BC}$ after changing redshift so that the partial cancellations seen in Figure 11 are not present. The descriptions on the left hand side and variation numbers on the right hand side both correspond to the descriptions in Table 4. The $H_0$ fit uses the redshift range $0.0233<z_{\text{HD}}<0.15$ , while the w fit uses the range $z_{\text{HD}}>0.01$ , where the maximum redshift of the sample is 2.26. The uncertainties in both $H_0$ and w only represent the statistical uncertainty from the SN magnitude uncertainties—they do not contain any uncertainty from distance ladder calibration nor intrinsic dispersion in SN magnitudes. All variations are small, showing that the cosmology results are robust to small changes in redshift and there is no indication of systematic redshift errors biasing previous results.

7.2 Fitting $H_0$

We first fit $H_0$ using the method in Riess et al. (Reference Riess2016), that compares the distance modulus data to a low-redshift approximation of the recession velocity.Footnote j The only free parameter in this fit is $H_0$ .

For this fit we focus on only the low-redshift regime of $0.0233<z_{\text{HD}}<0.15$ , which is the standard range used by previous supernova cosmology analyses such as Riess et al. (Reference Riess2016); Riess et al. (Reference Riess2018). In the Final dataset this redshift range contains 512 SNe of which only 17 lack host galaxy redshifts.

When calculating the uncertainties we only consider the statistical uncertainties in the distance moduli of the supernovae, not the uncertainties in the absolute magnitude calibration. Thus the uncertainties in $H_0$ in Figure 12 reflect the size of shifts due only to redshift/sample variations and not the size of uncertainties in the $H_0$ measurements (for example the current uncertainty on $H_0$ from SN cosmology is about 5 times larger).

The results are shown in Table 4 and Figure 12, where we see that the redshift improvements we have made have a small impact on the value of $H_0$ relative to the uncertainty from the SH0ES $H_0$ measurement with uncertainty of $1.0\, \mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ (Riess et al. Reference Riess2022). We define the difference in cosmological parameters for each variation to be the variation minus the Final value, that is $\Delta H_0 = H_0^n - H_0^{\text{Final}}$ for variation n. Applying all updates to the original redshifts results in $\Delta H_0 = -0.12\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ . The largest redshift variations, that is shifting all redshifts by their $1\unicode{x03C3}$ uncertainties (variations 8 and 9) result in $|\Delta H_0|\leq0.2\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ .

7.3 Fitting wCDM

We also evaluate the impact of the redshift updates on the best fit parameters in the flat-wCDM model. This model has two free parameters: the matter density ${\Omega_{\text{m}}}$ , and the equation of state of dark energy w, but we implement a Planck-like prior on the matter density of ${\Omega_{\text{m}}}=0.311\pm0.010$ so we can isolate the impact on w. We marginalise over absolute magnitude and $H_0$ , and we fit over the redshift range $z_{\text{HD}}>0.01$ . To estimate the changes in w we use the fast minimisation routine wFit in SNANA (Kessler et al. Reference Kessler2009) which outputs marginalised cosmology parameters w and $\Omega_M$ ; the complete fit with a thorough covariance analysis can be found in Brout et al. (Reference Brout2022).

The results are shown in Table 4 and Figure 12; changes in w are smaller than the statistical uncertainty of $\lesssim$ 0.03 given in the full Pantheon+ w measurement from Brout et al. (Reference Brout2022) and the uncertainty of 0.04 from Pantheon (Scolnic et al. Reference Scolnic2018). Applying all updates to the original redshifts results in $\Delta w = 0.003$ . The largest redshift variations, that is shifting all redshifts by their $1\unicode{x03C3}$ uncertainties (variations 8 and 9) result in $|\Delta w|\leq0.015$ .

7.4 SN-z vs host-z

Using the original Pantheon sample, Steinhardt et al. (Reference Steinhardt, Sneppen and Sen2020) find a statistically significant shift in the cosmological parameters derived for the subset of SNe that have redshifts measured from the supernova (SN-z) and those that have host-galaxy redshifts (host-z). Here we revisit this analysis with our updated data.

There are several important differences between Steinhardt et al. (Reference Steinhardt, Sneppen and Sen2020) and our analysis: (1) Steinhardt et al. (Reference Steinhardt, Sneppen and Sen2020) fit all wCDM parameters simultaneously, (2) our definition of the SN-z sample is stricter than their not-host-z, in that we allow host emission-lines in SN-dominated spectra to be assigned to the host-z sample, and (3) our allocation of SN-z particularly for PS1MD and SDSS SNe differs from theirs. We expect the SDSS classifications to differ because, as we addressed in Section 3, we updated 81 SN-zs to host-zs (and further, applied the $+2.2{\times 10^{-3}}$ systematic offset to the remaining SN-zs as determined in Sako et al. Reference Sako2018). However, the reason for the PS1MD classification differences are unclear; our classifications come directly from reported redshift uncertainties (SN-z have uncertainties of 0.01 and host-z 0.001).

We find that restricting the data to only those SNe with host-z (variation 6 in Table 4 and Figure 12) gives $\Delta H_0=0.05\pm0.20\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ (i.e. $0.3\unicode{x03C3}$ ) relative to the nominal Final dataset. By contrast, when we restrict the data to the SN-z sample (of which only 17 are in the redshift range of the $H_0$ fit), we find $\Delta H_0=-2.8\pm1.3\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ .

These results are broadly consistent with Steinhardt et al. (Reference Steinhardt, Sneppen and Sen2020), who found that $\Delta H_0= 0.45\pm 0.25\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ for the host-z sample and $\Delta H_0=-0.96\pm0.50\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ for the SN-z sample. The significance of the shift in the host-z sample is lower in our case, which is likely due to the greater proportion of host-z redshifts in our sample and the corrections we have implemented to the SN-z based on the systematic offset correction determined by Sako et al. (Reference Sako2018). On the other hand, the $H_0$ shift we find in the SN-z sample is larger than Steinhardt et al. (Reference Steinhardt, Sneppen and Sen2020), but our results are of similar significance (approximately $2\unicode{x03C3}$ in each case) since we have fewer SNe in the SN-z sample.

Increasing the redshift range over which we fit for $H_0$ adds some model dependence, but allows us to include more of the SN-z sample. When we do so the deviation from the nominal dataset vanishes, with the results from SN-z alone agreeing with the result from host-z alone (within $1\unicode{x03C3}$ ) for all $z_{\rm max}\gtrsim 0.25$ . Figure 13 shows the impact of including or excluding SN-z from our fit for $H_0$ , as a function of redshift range used in the fit. The impact is small, with $|\Delta H_0|\lesssim 0.05\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ .

Figure 13.

The dependence of $H_0$ on the redshift-range we fit over. The horizontal axis shows the maximum redshift used in the fit, keeping the minimum redshift as $z_{\text{HD}}=0.0233$ . Red shows the Final sample, and blue shows the host-z sample in order to demonstrate the impact of excluding SNe that only have SN-z. As redshift increases the cosmological model dependence becomes increasingly important so the grey shading shows the range of $H_0$ values from fitting with $0.25<{\Omega_{\text{m}}}<0.35$ in the flat- $\Lambda$ CDM model ( ${\Omega_{\text{m}}}=0.35$ being the lower edge). At the nominal upper redshift limit of $z_{\rm max}=0.15$ the statistical variance of the sample (black error bar) remains larger than the uncertainty due to even this quite wide range of ${\Omega_{\text{m}}}$ .

As expected, Figure 13 shows that as we increase the maximum redshift in our $H_0$ fit the cosmological model-dependence becomes increasingly apparent. The $v_{\rm approx}(z,q_0,j_0)$ equation in Section 7.2 is a good approximation to the full equation $v(z)=c\int_0^z\frac{dz}{E(z)}$ where $E(z)\equiv H(z)/H_0$ , as long as ${\Omega_{\text{m}}}\sim0.3$ for the flat- $\Lambda$ CDM model. Over the nominal redshift range of $0.0233<z_{\text{HD}}<0.15$ (black point) a shift of $\Delta{\Omega_{\text{m}}}\pm0.05$ results in $\Delta H_0\mp0.2\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ , showing the cosmological model-dependence is still sub-dominant to the sampling uncertainties on $H_0$ (which is the purpose of using the restricted redshift range).

For the equation of state of dark energy, the host-z only case (6) shows a shift of $\Delta w = -0.022\pm0.048$ . This is the largest impact on w of any variation, albeit still insignificant ( $0.6\unicode{x03C3}$ ). Using only SN-z gives $\Delta w=0.25\pm0.16$ . The reason for this shift can be seen in Figure 10, where the SN-z sample shows a systematic positive offset in $\Delta \unicode{x03BC}$ at low redshift, but a negative offset at high redshift. Any shift with a redshift dependence will have an impact on w whereas a constant shift would mainly affect $H_0$ .

Given these results, in Section 8 we discuss whether SN-z samples should be included in cosmological fits.

7.5 Redshift systematics and uncertainty values

As expected, some of the largest changes in $H_0$ and w occur when we add a systematic shift to all redshifts in the sample (variations 8–11). However, even when we shift the redshifts systematically by their uncertainties (variations 8–9) the impact is only $|\Delta H_0|\sim 0.2\pm0.2\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ and $|\Delta w|\sim0.015\pm0.045$ , smaller than the sample uncertainties in each case.

Changing the size of the uncertainties on the redshifts has a negligible impact on the cosmological parameters (variations 12–13).