Indefinitely Flat Circular Velocities and the Baryonic Tully–Fisher Relation from Weak Lensing

We use a new deprojection formula to infer the gravitational potential around isolated galaxies from weak gravitational lensing. The results imply circular velocity curves that remain flat for hundreds of kiloparsecs, greatly extending the classic result from 21 cm observations. Indeed, there is no clear hint of a decline out to 1 Mpc, well beyond the expected virial radii of dark matter halos. Binning the data by mass reveals a correlation with the flat circular speed that closely agrees with the baryonic Tully–Fisher relation known from kinematic data. These results apply to both early- and late-type galaxies, indicating a common universal behavior.


INTRODUCTION
The rotation curves of spiral galaxies become approximately flat at large radii (Rubin et al. 1978) and remain so well beyond the extent of the observed luminous mass (Bosma 1981).A constant rotation speed implies an enclosed mass that increases linearly without bound, a behavior that should not persist indefinitely.A longstanding question is just how far it does persist.
Rotation curves are commonly inferred from radio interferometry of the 21 cm line of atomic hydrogen.This technique allows probing many tens of kpc (Lelli et al. 2016a) and sometimes up to 100 kpc (Noordermeer et al. 2005;Lelli et al. 2010), without revealing any credible indication of a Keplerian decline (de Blok et al. 2008;Lelli 2022).Rotation curves have distinctive shapes that correlate with surface brightness (Lelli et al. 2013(Lelli et al. , 2016c) ) and are not perfectly flat (Casertano & van Gorkom 1991): it is common to see a gradual decline in massive galaxies, but the gradient is not Keplerian and the rotation curve tends to flatten out at the largest probed radii (Noordermeer et al. 2005;Di Teodoro et al. 2023).
Weak gravitational lensing offers another probe (e.g., Hudson et al. 1998;Kleinheinrich et al. 2006;Brimioulle et al. 2013;Milgrom 2013;Wang et al. 2016) that extends to much larger radii.Indeed, recent lensing observations imply rotation curves that remain remarkably flat to a few hundred kpc (Brouwer et al. 2021).Here we apply a new technique (Mistele et al. 2024) to further extend these results.
Galaxies follow the Baryonic Tully-Fisher relation (BTFR), which links the baryonic mass (stars plus gas) to the mean rotation speed along the flat part of the rotation curve (McGaugh et al. 2000;Lelli et al. 2016b;Schombert et al. 2020).The BTFR generalizes the original Tully-Fisher relation (Tully & Fisher 1977), which relates luminosity and line-width as proxies for stellar mass and rotation speed (Verheijen 2001;Ponomareva et al. 2017Ponomareva et al. , 2018;;Lelli et al. 2019).
Here, we derive circular velocity curves and the corresponding BTFR from weak lensing data.We measure circular velocities out to ∼ 1 Mpc, exploiting the KiDS DR4 weak-lensing data (Kuijken et al. 2019;Giblin et al. 2021;Bilicki et al. 2021), that was previously analyzed in Brouwer et al. (2021), but using a new robust deprojection method from Mistele et al. (2024).

DATA
Inspired by Brouwer et al. (2021), we analyze a sample of isolated galaxies from the KiDS survey (Kuijken et al. 2019).For distance dependent quantities, we adopt a Hubble constant H 0 = 73 km s −1 Mpc −1 .This choice is made for consistency with previous kinematic work (Lelli et al. 2016a) and H 0 measured with the BTFR (Schombert et al. 2020).We further assume a flat FLRW cosmology with Ω m = 0.2793 (Hinshaw et al. 2013) for consistency with Brouwer et al. (2021).Where useful, we employ the notation h 70 ≡ H 0 /(70 km s −1 Mpc −1 ).
We follow the procedure of Mistele et al. (2024) to select lens and source galaxy samples.Specifically, we use source galaxies from the KiDS-1000 SOM-gold catalog (Kuijken et al. 2019;Wright et al. 2020;Giblin et al. 2021;Hildebrandt et al. 2021) and lens galaxies from the KiDS-bright sample (Bilicki et al. 2021).We split the lens sample into four baryonic mass bins.To have a good compromise between bin width and number of galaxies in each bin, we use bin edges log 10 M b /M ⊙ = [9.0, 10.5, 10.8, 11.1, 11.5].Unlike Mistele et al. (2024) we do not impose an explicit cutoff on stellar mass.
We calculate stellar and baryonic masses as in Mistele et al. (2024) where we reanalyzed the KiDS data using the stellar population synthesis model of Schombert & McGaugh (2014), which was previously used for the kinematic BTFR in Lelli et al. (2019).Compared to Brouwer et al. (2021), we find excellent agreement for late-type galaxies (LTGs) but slightly larger stellar masses for early-type galaxies (ETGs).Consequently, we adopt the stellar masses of Brouwer et al. (2021) for LTGs and correct the stellar masses of ETGs by a factor of 1.4.Following Mistele et al. (2024) and Brouwer et al. (2021) we define LTGs and ETGs by the color split u − r ≷ 2.5.We correct all stellar masses to account for our choice of H 0 .
We use scaling relations to account for the baryonic mass in gas.For ETGs, we add a hot gas component according to the scaling relation which accounts for the X-ray-emitting coronae of massive ETGs (Chae et al. 2021).
Star-forming LTGs have a non-negligible interstellar medium of atomic and molecular gas.Thus, for LTGs we add a cold gas component according to the scaling relation where (3) The first term in Equation (2) represents atomic gas according to the scaling relation from Lelli et al. (2016a) with M * = 0.5L [3.6] .The second term takes into account molecular gas.Equation (3) accounts for the variation of the hydrogen fraction X as metallicity varies with stellar mass (McGaugh et al. 2020).
Since we are interested in the gravitational potential of isolated galaxies, we require that lens galaxies have no neighbor with a fraction f ⋆ > 0.1 of their stellar mass within a 3D distance R isol = 4 Mpc/h 70 .1As shown by Mistele et al. (2024), the weaker isolation criterion R isol = 3 Mpc/h 70 used by Brouwer et al. (2021) is not sufficient for ETGs, likely because these galaxies are more clustered than LTGs (Dressler 1980).
As discussed in Mistele et al. (2024) and Brouwer et al. (2021), the isolation criterion relies on the KiDS photometric redshifts, which have significant uncertainties.Using ΛCDM simulations, Brouwer et al. (2021) estimate that their weaker isolation criterion is reliable out to about 300 kpc.Empirically, we find that our overall results do not change out to ∼ 1 Mpc for any sensible choice of f ⋆ and R isol , so our results may be reliable out to radii ∼3 times larger than what Brouwer et al. (2021) estimate.In the following, we consider radii out to 1 Mpc but, if in doubt, 300 kpc is a conservative lower bound.In addition, to estimate the impact of redshift uncertainties, we compared to a sample of lenses with spectroscopic redshifts from GAMA III (Driver et al. 2022;Bellstedt et al. 2020;Taylor et al. 2011) and found consistent results (see Section 4.1).
The azimuthally averaged tangential shear that we use (Mistele et al. 2024) is a measure of the cumulative projected mass along the line-of-sight (e.g.Kaiser & Squires 1993), so an isolation criterion based on a cylinder along the line-of-sight instead of a 3D sphere may, in principle, be preferable.However, as discussed in Mistele et al. (2024), this may leave too few lenses to obtain a useful signal.This will be investigated in detail in future work.
After applying our isolation criterion, the four mass bins contain [24297, 17183, 22083, 11570] LTG lenses and [1795, 5439, 14220, 32440] ETG lenses.The lowest mass bin for ETGs has too few lenses to give any useful signal, so we do not consider it further.The isolation criterion leaves a fraction of 8%, 18%, 30%, and 42% of LTG lenses and 9%, 14%, and 22% of ETG lenses, respectively, in each mass bin.Not split by mass and type, the isolated fraction is 16%.

Circular velocities
The weak-lensing signal of any individual lens is indeed weak, so we stack all lenses in a given mass bin.In particular, we derive stacked circular velocities using the method from Mistele et al. (2024) to infer stacked radial accelerations, g stacked obs (R), in 15 logarithmic bins between 0.3 Mpc/h 70 and 3 Mpc/h 70 .These stacked radial accelerations are weighted averages of the radial accelerations g obs,l (R) of each lens l, with weights wl (R) and the normalization factor N −1 (R) = l wl (R).Following Mistele et al. (2024), we choose the weights wl (R) to be the inverse square of the statistical uncertainty of g obs,l (R), i.e. wl (R) = σ −2 g obs,l (R).The g obs,l (R) of each lens l is calculated from the excess surface density (ESD) profile ∆Σ l (R) of that lens using the deprojection formula (Mistele et al. 2024), where M (R) is the deprojected dynamical mass enclosed within a spherical radius R.This deprojection formula assumes spherical symmetry which is a reasonable approximation at the large radii we consider in this work.
The integral in Equation ( 5) is evaluated as in Mistele et al. (2024).In particular, the systematic uncertainties on g obs (R) are calculated considering different choices in how to interpolate between the discrete ∆Σ(R) data points and how to extrapolate ∆Σ(R) beyond the last data point at R = R max (see Mistele et al. 2024 for technical details).These systematic uncertainties become important only close to R max .In practice, this happens around 1 Mpc (see Section 4.1).For the statistical uncertainties, we adopt an improved procedure described in Appendix B.
The stacked radial accelerations g stacked obs (R) can be converted to stacked circular velocities using These V c values are not linear averages of the circular velocities of the individual stacked lens galaxies, i.e., they are not ⟨V c ⟩. Instead, they are ⟨V 2 c ⟩.

Averaged quantities for BTFR
In Section 4.1, we show that the circular-velocity curves from weak lensing are approximately flat at large radii.Thus, we can construct a BTFR, taking the flat circular velocity V flat to be a weighted average of V c (R) at different radii R. We choose the weight of each V c (R) to be proportional to the inverse square of its statistical uncertainty.We average over the radii 50 kpc < R < 1000 kpc, where the lower cutoff is chosen such that the baryonic mass of the lenses can effectively be treated as a point mass and the upper cutoff is chosen to avoid systematic uncertainties (see Section 3.1 and Figure 2) and because the isolation criterion is probably not reliable beyond 1 Mpc.We consider the effect of other choices in Section 4.3.
To construct the BTFR we need both V flat and M b .Our V flat is based on stacked velocities ⟨V 2 c ⟩ inferred from weak lensing.Thus, to test the BTFR using weaklensing data, we need a suitably stacked M b to compare to.Since the relation et al. 2000) concerns individual galaxies, the appropriate quantity to compare the stacked V c to is ⟨ √ M b ⟩ and not, for example, ⟨M b ⟩ 1/4 .This averaging procedure has the important property that, if the relation V c ∝ M 1/4 b holds for individual galaxies, it also holds for the stacked quantities.For V flat , one additionally needs to take into account the averaging over radial bins described above, see Appendix A for details. 2  For our lens sample, these properly-averaged baryonic masses M b,eff are smaller than those obtained from a naive unweighted linear average, but the difference plays only a minor role.Indeed, the properly and naively averaged masses differ by < 4%, except in the lowest mass bin where M b,eff is about 20% smaller.

Circular velocities
Figure 1 and Table 1 show the circular velocities inferred from weak lensing.We split the data by mass, doing so for the whole sample and for LTGs and ETGs separately.These circular velocity curves are approximately flat out to ∼ 1 Mpc, with no clear indication of a decline.This remarkable behavior persists in every mass bin for both ETGs and LTGs.
Since our isolation criterion is stricter than that of Brouwer et al. (2021), these results may be reliable all the way out to ∼ 1 Mpc (see Section 2).Even conservatively, our results are reliable out to at least R = 300 kpc, where the weaker isolation criterion of Brouwer et al. (2021) was shown to be reliable using ΛCDM simulations.
As a cross-check, Figure 1 shows the circular velocity curve of a sample of GAMA III lenses with spectroscopic redshifts.We analyse these in the same way as the KiDS lenses.The spectroscopic GAMA sample is much smaller (13, 957, or 9%, isolated lenses) than the KiDS sample with photometric redshifts, so we show only a single wide mass bin for GAMA and not split by type.We find the same behavior as for KiDS.GAMA shows a slightly stronger but not very significant decline beyond ∼ 500 kpc.
Figure 2 shows how the weak-lensing data compares to typical rotation curves from gas kinematics.3For illustration, we choose galaxies from the SPARC database (Lelli et al. 2016a) with comparable M b and V flat to Figure 1.Circular velocities implied by weak lensing for four baryonic mass bins (most to least massive from the top row to the bottom) for the whole sample (left column), for LTGs (middle column), and for ETGs (right column).The lowest ETG mass bin is not shown because it contains too few lenses.Instead we show results for lenses with spectroscopic redshifts from GAMA, without splitting by mass or type due to the small sample size (gray and white symbols).For comparison, we also show results for KiDS without splitting by mass or type (small yellow symbols).Open symbols at small radii indicate where lenses are not yet effective point masses.Light-colored symbols at large radii indicate data points that may still be reliable but where the isolation criterion is less certain.The error bars show the statistical errors.Horizontal lines and the corresponding shaded regions indicate the inferred V flat values and uncertainties that we use for the BTFR.The extent of the horizontal lines indicates the radial range we consider when calculating V flat .
Weak lensing data extend the circular velocity curves from gas kinematics by more than one order of magnitude in radius.Rotation curves remain flat to ∼ 1 Mpc.
Beyond 1 Mpc, the circular velocities in some mass bins possibly decline, but there is no clear departure from flatness, let alone any indication of a Keplerian decline.At these extreme radii, systematic uncertainties on V c become significant and the continued isolation of the lenses becomes dubious.The circular velocities from weak lensing (circles) compared with those from gas kinematics (diamonds).The individual galaxies illustrated here have among the most extended 21 cm rotation curves in their mass bins; the lensing data continue to much larger radii still.The error bars show the statistical error, while the gray band indicates the systematic uncertainty from converting ESD profiles to radial accelerations (Section 3.1).Symbol colors are as in Figure 1.Open symbols at large radii indicate where lenses are not sufficiently isolated.The solid green lines indicate the circular velocities of NFW halos and baryonic point masses appropriate for each mass bin.Green crosses indicate each NFW halo's virial radius.The light green band adds a qualitative estimate of a two-halo term contribution to the NFW halo, which may become important at large radii in case our isolation criterion is imperfect there.

Comparison with ΛCDM expectations
R)+GM b,eff /R with the averaged baryonic mass M b,eff (Section 3.2).For simplicity, we assume an NFW halo (Navarro et al. 1996).The halo parameters are determined using the WMAP5 massconcentration relation from Macciò et al. (2008) and the stellar mass-halo mass relation from Kravtsov et al. (2018), assuming the average stellar mass in each bin.The Kravtsov et al. (2018) relation is the most appro-priate because, unlike other common relations, it does not overshoot the circular velocities at small radii in the high-mass bins (Di Cintio & Lelli 2016;Li et al. 2022b).
The circular velocity curves of NFW halos do not remain flat indefinitely.This is in tension with the lensing-inferred circular velocities that remain flat out to  1) and the systematic errors from converting ESD profiles to radial accelerations (see Section 3.1).Covariance matrices are available on reasonable request to the authors.Measurements are omitted for radial bins with negative stacked radial acceleration.Table 1 is published in its entirety in the machine-readable format.A portion is shown here for guidance regarding its form and content.
∼ 1 Mpc.In the higher-mass bins, there is no strong discrepancy if we conservatively consider only R ≲ 300 kpc where there is no doubt about the isolation criterion.In the lower-mass bins, however, there is a clear discrepancy already at R ≲ 300 kpc because of the smaller DM halos.The lensing data now probe to the virial radius and beyond with no indication of the expected downturn in rotation speed.
Although we here assume a specific DM halo profile, our results apply more generally in the context of ΛCDM cosmology because our lensing data mainly probe the outer slopes of DM halos which are predicted to approximately follow ρ DM ∝ r −3 due to the hierarchical process of structure formation.The gravitational effect of baryons is expected to lead to halo contraction, which makes the circular velocities decline faster than for a fiducial NFW profile (Li et al. 2022b,a).Other baryonic processes, such as stellar and black-hole feedback, play a negligible role for the outer halo slope.Thus, assuming different reasonable DM halo profiles, such as those from hydrodynamic simulations of galaxy formation (see Li et al. 2020), will not affect our conclusions.Similarly, different stellar mass-halo mass or mass-concentration relations do not change our conclusions because these mainly change the overall normalization of V c but leave its shape at large radii unchanged.
Our isolation criterion may preferentially select void galaxies that, in principle, may be hosted by systematically different halos than the average galaxy.In practice, however, variations in halo concentration with cosmic environment are predicted to be small (Hellwing et al. 2021).If anything, they lead to slightly larger concen-trations for void galaxies in the relevant range of halo masses (> 10 11 M ⊙ ), which would make the circular velocities decline even earlier than shown in Figure 2.
No galaxy is completely isolated.To illustrate the possible effect of neighboring DM halos, the light green band in Figure 2 shows a qualitative estimate (see Appendix C) of the so-called two-halo term.The approach we employ should not be considered the immutable prediction of ΛCDM, but it does successfully reproduce the lensing signal at large radii around isolated lenses in ΛCDM simulations (Appendix C).

BTFR
Figure 3 and Table 2 show the BTFR from weak lensing data and compare it with binned kinematic data from Lelli et al. (2019).For visualization purposes, the mass bins for the kinematic data are chosen such that the data points do not overlap with those from weak lensing.The kinematic V flat values are averaged in each mass bin, with weights proportional to their inverse squared uncertainties.As with the weak-lensing data, we use properly averaged baryonic masses (see Section 3.2).
We consider V flat to be the average of the data points over two ranges of radii weighted by their statistical uncertainty (see Section 3.2).Our results are most robust over the radial range 50 kpc < R < 300 kpc, so Figure 3 shows the result of averaging over these data.We also show the result from averaging over 50 kpc < R < 1000 kpc.These give similar results; we have checked other reasonable choices of radii to include in the computation of V flat and have verified that dif- ferences between them are generally smaller than the statistical uncertainties.
Figure 3 shows good agreement between the BTFR from lensing data and that known from kinematics.The lensing data are consistent with being an extension of the flat rotation curves that are observed kinematically.This extension persists indefinitely in radius at the amplitude indicated by the baryonic mass.
When considering the entire lensing sample, there is no hint of an offset between the kinematic and lensing BTFR.When dividing by galaxy type, there is the suggestion of a small offset between ETGs and LTGs, but it is not statistically significant.Based on the random uncertainties in V flat alone, the difference corresponds to 0.54σ and 1.86σ when V flat is defined using radii up to 300 kpc and 1000 kpc, respectively.The lat- Note-The V flat values and the corresponding properly-averaged baryonic masses used for the BTFR (see Figure 3).The V flat errors listed are the statistical errors.The uncertainty on the mean M b is statistically negligible for the lensing data, but we adopt 0.1 dex as a systematic uncertainty in M * /L, see Figure 3.
ter is also subject to type-dependent differences in the isolation criterion, which is more robust for LTGs than for ETGs (Mistele et al. 2024).The net effect of this systematic difference would be a slight overestimate of V flat for ETGs when including larger radii.

DISCUSSION
The circular velocity curves from weak lensing observations remain flat for hundreds of kpc, possibly up to 1 Mpc, and imply a weak-lensing BTFR that is fully consistent with the kinematic BTFR.These results apply to both LTGs and ETGs separately.Evidently, the asymptotic flatness of rotation curves and the BTFR are independent of galaxy morphology (disks or spheroids) and evolutionary history (star-forming or passive).Galaxies seem inevitably to lie on the BTFR given the availability of an adequate tracer to measure V c out to large radii.Similar results, indeed, were found for ETGs that occasionally possess an outer extended HI disk (den Heijer et al. 2015;Shelest & Lelli 2020).
The mere existence of the BTFR (McGaugh et al. 2000) already suggests that this relation is independent of the diverse evolutionary histories of galaxies because the relative contribution of gas mass and stellar mass can greatly vary across the galaxy population (e.g., Lelli 2022).In general, the gas mass contribution becomes important below log 10 M b /M ⊙ = 10 ( McGaugh et al. 2000), so it should not matter much for our lens sample that contains mostly galaxies with log 10 M b /M ⊙ > 10 ( Mistele et al. 2024).Indeed, we have verified that rerunning our analysis using stellar masses instead of baryonic masses has only a small effect on the resulting Tully-Fisher relation, with the data points in Figure 3 moving towards slightly smaller masses.This is expected and implies that the details of our gas mass estimates from Equations ( 1) and ( 2) are relatively unimportant to our general results.
Given that kinematic measurements of V flat rely on data from relatively small radii where the disk geometry matters, rotation curves may have not yet reached their truly asymptotic value.Consequently, one may expect that lensing data gives slightly smaller V flat values compared to the kinematic BTFR.Indeed, a razor-thin disk galaxy with scale length 5 kpc that obeys the radial acceleration relation (Lelli et al. 2017;Brouwer et al. 2021;Mistele et al. 2024) has an asymptotic V flat that is about 5% smaller than the rotation curve velocity at 25 kpc (McGaugh & de Blok 1998).There are hints of just such an offset in the higher LTG mass bins, but in general the uncertainties in our weak-lensing analysis do not allow for such a small offset to be reliably observed.
Our results are difficult to understand in ΛCDM because the data should have reached the asymptotic V c ∼ log(r)/r decline of the DM halos.On the other hand, a universal BTFR and indefinitely flat rotation curves for isolated galaxies were predicted by Modified Newtonian Dynamics (MOND, Milgrom 1983a,b,c).Indeed, the lensing data now probe far into the deep MOND regime of extremely low accelerations without showing any deviation from the prediction that the asymptotic V flat is determined by M b as V flat ∝ M 1/4 b .Perhaps with further improvement in the data it might become possible to perceive a decline at large radii due to the socalled external field effect (Bekenstein & Milgrom 1984;Chae et al. 2020Chae et al. , 2021)).Given our strict isolation criterion and the stacking required to obtain the lensing signal, this external field effect may be undetectable in the present data.

CONCLUSION
We have derived circular velocities for isolated galaxies from weak gravitational lensing data.The circular velocity curves are consistent with being flat out to hundreds of kpc, perhaps even 1 Mpc, with no sign of having reached the edge of the DM halo.Using these circular velocities, we have constructed the BTFR implied by weak lensing, finding good agreement with previous kinematic determinations of the BTFR.These results hold for both LTGs and ETGs separately, suggesting a common universal behavior.
TM thanks Amel Durakovic for helpful discussions.This work was supported by the DFG (German Research Foundation) -514562826.Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017, 177.A-3018 and 179.A-2004, and on data products produced by the KiDS consortium.The KiDS production team acknowledges support from: Deutsche Forschungsgemeinschaft, ERC, NOVA and NWO-M grants; Target; the University of Padova, and the University Federico II (Naples).GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope.The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey.Complementary imaging of the GAMA regions is being obtained by a number of independent survey programmes including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio coverage.GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions.The GAMA website is https://www.gama-survey.org/ .2016) (see their Figure 4).We ignore data points with negative ∆Σ in the lowest mass bin.Right: The circular velocities implied by these ESD profiles according to our deprojection formula Equation ( 5).Colored bands indicate the systematic uncertainty from converting ESD profiles to accelerations (Section 3.1).Solid gray lines indicate the circular velocity of an NFW halo and a baryonic point mass as in Figure 2. Light gray bands additionally take into account contributions due to the local environment around not perfectly isolated lenses according to Equation (C11).In particular, the gray bands show the ±0.1 dex range around be values of 0.3, 0.57, 0.95, 1.25, respectively, for the four stellar mass bins.
where J 2 denotes the second Bessel function of the first kind, D l is the angular diameter distance of the lens, b e is the bias, z is the redshift of the lens, ρm,0 is the mean matter density at redshift z = 0, and P m (k ℓ ; z) is the linear matter power spectrum at k ℓ = ℓ/[(1 + z)D l ] (see below for why we use the linear power spectrum).This leads to an additional contribution g obs,e to the acceleration g obs that we infer when using Equation ( 5) The second equality follows by exchanging the order of the θ and ℓ integrals, substituting x ≡ (ℓR)/(D l sin θ) at fixed ℓ in the inner θ integral, and analytically evaluating the resulting x integral using Mathematica (Wolfram Research, Inc. 2024).We evaluate the remaining ℓ integral numerically, assuming z = 0.2 for a typical lens galaxy.We calculate the linear matter power spectrum using CAMB (Lewis & Bridle 2002).
One limitation is that Equation (C10) does not know that our lens sample consists of isolated galaxies.For simplicity, we here assume that this can be taken into account by choosing an appropriate normalization b e and using the linear, rather than non-linear, matter power spectrum P m .Indeed, using the linear power spectrum qualitatively mimics the effect of our isolation criterion of removing structure on relatively small spatial scales.To find reasonable values for b e we match our semi-analytical estimate to the lensing signal of isolated galaxies obtained from ΛCDM simulations by Wang et al. (2016).Since Wang et al. (2016) impose an isolation criterion different from ours (see below), this amounts to making the additional assumption that, at least qualitatively, the b e values we obtain in this way apply to our isolation criterion as well.
We use stacked ESD profiles for four stellar mass bins from the G11-P' model from Wang et al. (2016).These cover a similar range in V c as our KiDS data.Figure 4, left, shows these ESD profiles and Figure 4, right, shows the corresponding V c inferred using our deprojection formula Equation ( 5).The inferred circular velocities V c show an upwards trend at large radii.This is due to the two-halo term which leads to ∆Σ falling off slower than 1/R which in turn means our deprojection formula Equation (5) infers rising circular velocities.We note that, where the two-halo term becomes important, these inferred circular velocities no longer correspond to actual circular orbits of bound objects.They are simply what our method infers when applied to non-isolated lenses.
This behavior can, at least qualitatively, be reproduced by considering the environment-induced lensing-inferred acceleration g obs,e from Equation (C11) in addition to an NFW halo and a baryonic point mass.This is illustrated in Figure 4, right, where we show a ±0.1 dex band around b e values of 0.3, 0.57, 0.95, and 1.25, respectively, for the four stellar mass bins of Wang et al. (2016).These values are relatively small (Tinker et al. 2010) as might be expected for a sample of isolated galaxies.We also see a clear overall trend with lower masses requiring lower values of b e .As a rough approximation, we show these same four bands of b e values in Figure 2 in the main text.
The NFW halo plus baryonic point mass models shown in Figure 4 are calculated as in Section 4.1 but with log 10 M * /M ⊙ values 10.2, 10.5, 10.85, and 11.1, respectively, for the four stellar mass bins.Baryonic masses are calculated using these stellar masses assuming the cold gas mass estimate from Equation (2) for the three lower mass bins and the hot gas mass estimate from Equation (1) for the highest mass bin.These values were chosen to fit the overall normalization of the inferred circular velocities and may not be correct in detail.For our purposes, the important point is the tail where the two-halo term dominates.The details of the NFW halos and baryonic point masses are unimportant.
Nevertheless, one may notice from Figure 4, right, that our NFW plus baryonic point mass models do not match the shape of the circular velocities inferred from the Wang et al. (2016) ESD profiles, not even at small radii where the two-halo term should be unimportant.We speculate that this may be due the particular isolation criterion imposed by Wang et al. (2016).Indeed, one difference compared to our isolation criterion (Section 2) is that Wang et al. (2016) only exclude lenses with neighbors that are brighter than the lens itself, i.e. f * = 1.0 using our notation from Section 2 and assuming a constant M * /L.This allows for significant non-isolation effects already at small radii and may be why NFW plus baryonic point mass models are not a good match.In any case, as mentioned above, the details at small radii do not matter for our purposes, which is merely to illustrate the effects of the two-halo term.

Figure 2
Figure 2 also shows the rotation curve implied by a dark matter (DM) halo and a baryonic point mass,

Figure 2 .
Figure2.The circular velocities from weak lensing (circles) compared with those from gas kinematics (diamonds).The individual galaxies illustrated here have among the most extended 21 cm rotation curves in their mass bins; the lensing data continue to much larger radii still.The error bars show the statistical error, while the gray band indicates the systematic uncertainty from converting ESD profiles to radial accelerations (Section 3.1).Symbol colors are as in Figure1.Open symbols at large radii indicate where lenses are not sufficiently isolated.The solid green lines indicate the circular velocities of NFW halos and baryonic point masses appropriate for each mass bin.Green crosses indicate each NFW halo's virial radius.The light green band adds a qualitative estimate of a two-halo term contribution to the NFW halo, which may become important at large radii in case our isolation criterion is imperfect there.

Figure 3 .
Figure 3.The baryonic Tully-Fisher relation implied by weak lensing for the entire sample (yellow symbols, left column) and for ETGs and LTGs separately (red and blue symbols, right column).The V flat values are weighted averages of the Vc values shown in Figure 1 for 50 kpc < R < 300 kpc (first row) and 50 kpc < R < 1000 kpc (second row).Vertical error bars represent a 0.1 dex systematic uncertainty on M * /L.For comparison, we also show the best fit to the kinematic data from Lelli et al. (2019) (solid gray line) and the corresponding binned kinematic data (white diamonds).

Figure 4 .
Figure 4. Left: Stacked ESD profiles for four stellar mass bins from the G-11P' ΛCDM simulation from Wang et al. (2016) (see their Figure4).We ignore data points with negative ∆Σ in the lowest mass bin.Right: The circular velocities implied by these ESD profiles according to our deprojection formula Equation (5).Colored bands indicate the systematic uncertainty from converting ESD profiles to accelerations (Section 3.1).Solid gray lines indicate the circular velocity of an NFW halo and a baryonic point mass as in Figure2.Light gray bands additionally take into account contributions due to the local environment around not perfectly isolated lenses according to Equation (C11).In particular, the gray bands show the ±0.1 dex range around be values of 0.3, 0.57, 0.95, 1.25, respectively, for the four stellar mass bins.

Table 1 .
Circular Velocities from Lensing Note-The circular velocities Vc(R) inferred from weak lensing for four baryonic mass bins with bin edges log 10 M b /M⊙ = [9.0,10.5, 10.8, 11.1, 11.5].We separately list results for ETGs, LTGs, and the entire sample.The lowest mass bin for ETGs is not shown because it contains too few lenses to obtain a useful signal.The listed errors on Vc are the statistical errors (see the colored error bars in Figure