Superfluid dark matter in tension with weak gravitational lensing data

Superfluid dark matter (SFDM) is a model that promises to reproduce the successes of both particle dark matter on cosmological scales and those of Modified Newtonian Dynamics (MOND) on galactic scales. SFDM reproduces MOND only up to a certain distance from the galactic center, and only for kinematic observables: it does not affect trajectories of light. We test whether this is consistent with a recent analysis of weak gravitational lensing that has probed accelerations around galaxies to unprecedentedly large radii. This analysis found the data to be close to the prediction of MOND, suggesting they might be difficult to fit with SFDM. To investigate this matter, we solved the equations of motion of the model and compared the result to observational data. Our results show that the SFDM model is incompatible with the weak-lensing observations, at least in its current form.

SFDM postulates a new type of particle that condenses to a superfluid around galaxies.Inside this superfluid core of galaxies, the phonons of the superfluid mediate a force that, in a certain limit, resembles that of MOND.At larger radii, the condensate breaks down and one matches the superfluid's density profile to that of a standard NFW halo at a matching radius r NFW [6].
Models like SFDM reproduce MOND only near the centers of galaxies, typically in the range where rotation curves are measured.At larger radii, they deviate from MOND.This is a desired feature for galaxy clusters where observations require accelerations larger than what MOND predicts [10,11].This deviation, however, can cause problems [12] when one also tries to fit recent weak-lensing data which has shown MOND-like behavior around galaxies up to ∼ 1 Mpc [13].
SFDM in particular might conflict with this data for two reasons.First, the MOND phenomena in SFDM are related to the condensed phase of a superfluid.But at radii of about ∼ 1 Mpc, this condensed phase is expected to break down [5,6,14].Second, even in the condensed phase, the MOND-like force of SFDM does not affect photons -if it did, this would cause conflicts with the observations from GW170817 which saw the gravitational wave signal arrive at almost the same time as the electromagnetic signal [15][16][17].For this reason, gravitational lensing in SFDM works as in General Relativity (GR), just with the superfluid's mass as an additional source.This means that even in the condensed phase, predictions for gravitational lensing in SFDM will differ from those of MOND.
More concretely, the acceleration inferred from kinematic observables includes both the acceleration due to the phonon force, a θ , and the standard Newtonian gravitational acceleration a N .Here, a N includes both the Newtonian acceleration due to the baryons, a b , and that due to the superfluid's mass, a SF .That is, a N = a b + a SF .In contrast, the acceleration inferred from lensing observations does not include the phonon contribution a θ .That is, (1.1) Below we discuss whether or not SFDM can simultaneously explain kinematic and lensing observations, with a focus on the most recent weak lensing data [13].2 The superfluid dark matter model In the superfluid phase, the SFDM model is described by two fields, namely the Newtonian gravitational field ϕ N and the phonon field θ.Instead of the Newtonian gravitational potential ϕ N , it will be convenient to use the quantities Here, µ nr is the superfluid's non-relativistic chemical potential, m is the mass of the particles that make up the superfluid, α is a dimensionless model parameter, and a b is the Newtonian gravitational acceleration due to the baryons.In spherical symmetry, a b = GM b (r)/r 2 with the enclosed baryonic mass M b and the Newtonian gravitational constant G.
We assume spherical symmetry and a point-particle baryonic density, ρ b = M b δ(⃗ x).This is justified because the details of the baryonic mass distribution are not important at the large radii probed by weak lensing.
In spherical symmetry, the equation of motion for the phonon field θ reduces to an algebraic equation.In particular, we have for the acceleration due to the phonon field a θ [18,19], where a 0 and β are further model parameters that we discuss below and x β is determined by a cubic equation, Thus, a θ can be calculated as an algebraic function of a b and ε * .The quantity ε * can be calculated from μ and a b , where μ is determined by the usual Poisson equation, but with the superfluid's energy density ρ SF as an additional source, The superfluid density can be written as [19] ρ SF = 2 3 Solving the Poisson equation requires a choice of boundary condition for μ.Physically, this corresponds to a choice of chemical potential for the superfluid [18,20].Thus, unlike in MOND, the solutions are not determined by the baryonic mass distribution alone.They additionally depend on a choice of boundary condition ultimately set by galaxy formation.Since galaxy formation in SFDM is not yet well-understood, we here treat this boundary condition as a free parameter for each galaxy.See appendix A for how we obtain solutions numerically.
The total acceleration felt by matter is that due to the Newtonian gravitational potential, a N , plus that due to the phonon field, a θ , where we split a N into a part due to the baryonic mass, a b , and a part due to the superfluid, a SF .In the so-called MOND limit this total acceleration a tot is close to the acceleration in MOND with an acceleration scale a 0 [5,18], Specifically, this corresponds to MOND with an interpolation function ν(y) = 1 + 1/ √ y.
In MOND, this interpolation function is not sufficient to match observations in general.But here we are mostly interested in the large radii probed by weak lensing where all interpolation functions give a tot ≈ √ a 0 a b .Thus, for simplicity, we assume this interpolation function for

MOND.
Unless otherwise stated, we use the fiducial numerical parameter values proposed by the original authors of the model [6], namely m 2 /α = 0.18 eV 2 , β = 2, and a 0 = 0.87 • 10 −10 m/s 2 .This fiducial value of a 0 is a bit lower than the best-fit value in MOND, namely a MOND 0 ≈ 1.2 • 10 −10 m/s 2 [21].The reason for choosing this smaller value is to compensate for possible contributions from a SF .We argue in appendix C that changing the numerical parameter values does not qualitatively affect our conclusions.
For concreteness, when comparing to MOND, we always use the larger value a MOND 0 for MOND, not the SFDM value a 0 , (2.9) As already mentioned above, the total acceleration a tot = a θ + a N is relevant only for kinematic observables like galactic rotation curves.The phonon field does not contribute to gravitational lensing because light does not directly couple to the new field.As a result, lensing works as in ΛCDM just with the superfluid mass instead of the cold dark matter mass [17].That is, if one uses the same formalism as in ΛCDM to infer an acceleration around galaxies from lensing, this inferred acceleration corresponds to a N in SFDM.This is for the boundary condition for which a tot stays within a fraction δ = 0.15 from MOND up to the maximum possible radius r max (vertical dotted line).Dashed (dash-dotted) black lines show solutions with this optimal boundary condition multiplied by 1.5 and 1.2 (0.8 and 0.5).Also shown are the different components that make up a tot , namely the baryonic Newtonian acceleration plus the acceleration due to the phonon field, a b + a θ , (solid red line) and the superfluid's Newtonian acceleration a SF (solid green line).For the non-optimal boundary conditions, a b + a θ and a SF are shown as grey lines.All solutions are shown up to the radius where the corresponding superfluid density ρ SF drops to zero.

Deviations from MOND for kinematic observables
We first consider kinematic observables, like rotation curves, on their own.Since kinematic observations are well-explained by MOND, we first quantify the deviation of SFDM from MOND and identify the solutions with minimal deviations.We then compare these optimal, and close-to-optimal, solutions to observations.By solutions we here mean solutions of the SFDM equations of motion.Optimal and non-optimal solutions differ from each other in the boundary condition that one must choose when solving these equations.Physically, this means the superfluids of different solutions have different chemical potentials, see section 2.
The total acceleration felt by matter in SFDM follows the MOND prediction only at sufficiently small galactocentric radii.At larger radii, it deviates from MOND.Where exactly these deviations set in and how large they are depends on the model parameters, the boundary condition, and the baryonic mass of the galaxy.
Specifically, consider a galaxy with a fixed baryonic mass M b and suppose we allow the acceleration in SFDM to deviate from that in MOND by at most a fraction δ.Then, there is an optimal boundary condition for which this condition is fulfilled up to the maximum possible radius r max .
We illustrate this in Fig. 1 for one example galaxy with M b = 2 • 10 10 M ⊙ .For the Here, we show the matching radius r NFW that gives the largest dark matter acceleration a DM outside the superfluid core, but other choices are also possible (see for example Fig. 5).
optimal boundary condition (solid blue line), the total kinematically-inferred acceleration in SFDM stays within 15% from MOND up to a maximum radius r max of about 70 kpc.For all other boundary conditions (dashed and dash-dotted black lines), either deviations from MOND become larger than 15% before 70 kpc or the condensate phase ends before 70 kpc.
While this example illustrates a typical case, the precise numerical values will be different for different galaxies.
That the condensate phase ends means that the condensate density drops to zero, ρ SF = 0. Mathematically, the solutions can be continued beyond this point but the physical meaning of this regime is unclear.Most likely, the negative density indicates an instability, so we truncate all solutions at ρ SF = 0.
In SFDM, it is usually assumed [5,6] that the condensate phase transitions into a noncondensed cold dark matter phase already before ρ SF = 0.In particular, the idea is that the superfluid density is matched to an NFW halo at a radius r NFW .This is illustrated for one example galaxy in Fig. 2.
Here, we treat the matching radius r NFW as a free parameter for each galaxy since the transition from the condensed to the not-condensed phase is still poorly understood [18].To keep things simple, we use a 1/r 3 NFW tail instead of a full NFW profile.We do not expect this to qualitatively change the results [17].
In principle, how long the kinematically-inferred acceleration stays close to MOND could depend on where and how the superfluid core is matched to the NFW tail.However, we consider only the condensate without any NFW matching for our definition of the maximum radius r max and the corresponding optimal boundary condition.That is, Fig. 1 does illustrate the actual definitions we use although it does not involve any matching to an NFW halo.There are two reasons for using this definition.
First, the condensate core is what matters most for comparing SFDM to MOND.This is because the relation of SFDM to MOND comes from the phonon force which acts only inside the condensate.Outside, the total acceleration will in general be different from MOND.
Of course, in principle, SFDM can match the MOND prediction even outside the condensate.But this is possible only by carefully adjusting a boundary condition separately for each galaxy.This is similar to the situation in ΛCDM.One of the main motivations behind SFDM is precisely to reproduce MOND without such tuning.Thus, SFDM loses one of its main advantages if we allow to match MOND by tuning the boundary condition.
Second, the maximum radius and the optimal boundary condition often don't change when taking into account a transition to an NFW halo.The reason is that the phonon force does not act outside the superfluid core so that, at the transition radius r NFW , the kinematically-inferred acceleration drops from a θ + a N to just a N .Often, a θ is the most important contribution, see for example Fig. 1.When it drops out, the acceleration typically immediately deviates from MOND by more than a fraction δ at the transition radius r = r NFW .The optimal transition radius is then just where ρ SF drops to zero.Thus, the optimal boundary condition is the same with and without taking into account a transition to an NFW tail.
In MOND, there is a one-to-one relation between the observed acceleration a obs and the baryonic Newtonian acceleration a b .In particular, MOND predicts a obs = a b at large accelerations, a b ≫ a 0 , and a obs = √ a 0 a b at small accelerations, a b ≪ a 0 (Eq.(2.9)).Kinematic observations confirm this relation, see for example the Radial Acceleration Relation (RAR) [21].
In SFDM, there is no longer such a one-to-one relation.This is illustrated in Fig. 3 which shows the relation between a obs and a b for galaxies with various baryonic masses and boundary conditions.We see that there is a strong trend with baryonic mass: Smaller galaxies more easily follow the MOND prediction than larger galaxies.This was first noticed in Ref. [19, appendix A.1.2]and is theoretically explained in appendix B. This trend may make SFDM less attractive compared to MOND, but is by itself not enough to rule out SFDM using kinematic data.For a quantitative discussion of fitting kinematic data with SFDM, we refer the reader to Ref. [19].

Deviations from MOND for weak lensing
As already mentioned above, photons are not coupled to the superfluid's phonons in order to avoid constraints from GW170817.Thus, lensing works as in ΛCDM just with the superfluid's mass as an additional source.In particular, the acceleration inferred from lensing is not a N +a θ but just a N .
The consequences for the RAR are illustrated in Fig. 4. In contrast to the kinematic RAR shown in Fig. 3, the lensing RAR almost never follows the MOND prediction.Even when carefully adjusting the boundary condition, the lensing RAR of SFDM follows the MOND prediction only in a very small spatial region, not over an extended range of radii.This is in tension with lensing observations [13] that agree well with the MOND prediction.
Both Fig. 4 and Fig. 3 show only the superfluid core around a galaxy.But as discussed above, this superfluid core ends at a finite radius r NFW .At this radius, the superfluid density is matched to that of a standard NFW halo and the phonon force is no longer active, not even for kinematic observables.We show the lensing RAR including such an NFW tail for various The total acceleration a obs that would be inferred from kinematic observations for numerical solutions in SFDM as a function of a b for various baryonic masses and boundary conditions.Solid lines show, for each M b , the optimal boundary condition for δ = 0.8.That is, the boundary condition for which the kinematically-inferred a obs stays closer than a fraction δ to MOND up to the maximum possible radius.Solutions with this boundary condition multiplied by 0.5 and 2.0 are shown as dash-dotted and dashed lines, respectively.These solutions are valid inside the superfluid cores of galaxies.We do not continue them with an NFW tail (see Fig. 5 for that).For a given M b , solutions are cut off where the first of the solutions reaches zero superfluid density.Also shown are the observed kinematic RAR [21] and the observed lensing RAR [13].The lensing data does not include the hot gas estimate used in some places in the original weak lensing analysis [13].We correct the observed weak-lensing RAR to be consistent with the M/L * scale of the observed kinematic RAR (McGaugh 2022, priv.comm.).Data points below a b = 10 −13 m/s 2 have a lighter color since the isolation criterion assumed in the weak lensing analysis [13] is less reliable there.The error bars indicate the statistical uncertainties and the gray band indicates the 0.1 dex systematic uncertainty from the excess surface density to RAR conversion [13].
choices of r NFW in Fig. 5.We see that introducing an NFW tail allows the SFDM lensing RAR to more easily follow the MOND prediction.That is, outside the superfluid core, the shape of the lensing-inferred a obs is much closer to the MOND prediction.Of course, asymptotically, the MOND acceleration scales as 1/r while that of the NFW tail scales as ln(r)/r 2 which is very different.Still, as Fig. 5 demonstrates, an NFW tail can reasonably reproduce the shape of the MOND acceleration over a non-negligible range of radii.
However, a big problem remains: For the normalization -not just the shape -to come out correctly, one has to carefully choose the boundary condition and matching radius r NFW separately for each galaxy.For this SFDM has to rely on galaxy formation to pick the correct values.But the goal of SFDM was precisely to be able to reproduce the MOND prediction without having to rely on galaxy formation.So this is not satisfactory.
So far, we have considered the lensing or kinematic signal of a single galaxy in SFDM.The observed weak-lensing RAR [13], however, is available only in an averaged sense for a large number of stacked galaxies.Thus, accelerations larger than MOND from one galaxy  As Fig. 4, but for a fixed boundary condition corresponding to δ = 0.8 and with the superfluid density matched to an 1/r 3 NFW tail at various radii r NFW .Solid lines show the lensing-inferred a obs inside the superfluid core, at r < r NFW .Dash-dotted lines show a obs outside the superfluid, at r > r NFW .For each M b , we show a obs for the r NFW that gives the maximum possible lensing-inferred a obs outside the superfluid.We also show NFW tails with 0.35, 0.5, and 0.7 times this optimal r NFW .Outside the superfluid, at r > r NFW , the kinematic and the lensing a obs coincide since the phonon force does not act there [5].may cancel with accelerations smaller than MOND from another galaxy, giving a stacked lensing RAR that is more MOND-like than the lensing RAR of any individual galaxy.This gives a bit more freedom in the boundary conditions and NFW matching radii that galaxy formation needs to pick for SFDM to reproduce the observed weak-lensing RAR.However, it's still far from clear why galaxy formation would pick just the right boundary conditions and matching radii to produce the desired cancellations.
Another problem for SFDM is that observations continue without gap from the kinematic to the lensing RAR (see the data points in, for example, Fig. 3).In SFDM, as illustrated by Fig. 3, Fig. 4, and Fig. 5, the lensing RAR can be close to the MOND prediction only when the kinematic RAR is not and vice versa.
In principle, with special boundary conditions and special galaxy samples, it could be that lensing observations never see the relatively small galactocentric radii of galaxies where the lensing-inferred a obs is much smaller than in MOND while kinematic observations never reach sufficiently large radii where the kinematically-inferred a obs is significantly larger than in MOND.In this case, SFDM might match both kinematic and lensing observations.But this seems highly unlikely.
In some cases, we can make an even stronger statement about the tension between the lensing and kinematic RAR in SFDM.To see this, suppose we require the kinematic RAR to follow the MOND prediction within the superfluid core (where the phonon force is active).This gives an upper bound on the lensing-inferred a obs outside the superfluid core (see appendix D).In some cases, this upper bound is smaller than what lensing observations imply.In these cases, no amount of tuning the boundary condition and NFW matching radius can make the lensing RAR in SFDM agree with observations.
As an example, consider a fixed baryonic mass M b .Consider then the optimal solution that, kinematically, stays within a fraction δ = 0.8 of the MOND prediction up to the maximum possible radius r max .Given this kinematically optimal solution, there is an optimal matching radius r NFW which produces the largest value for the lensing-inferred a obs , see appendix D. This optimal matching radius r NFW and the corresponding maximal lensinginferred a obs are illustrated in Fig 5. We see that even for this optimal matching radius r NFW , the maximal lensing-inferred a obs only barely reaches the MOND prediction.Repeating the same procedure with a kinematically-optimal solution for a smaller allowed deviation from MOND, i.e. a smaller δ, the lensing-inferred a obs often stays significantly below the MOND prediction, no matter which matching radius r NFW is chosen.

Quantitative analysis
Finally, we illustrate some of the qualitative points we made above by a simple quantitative analysis.For each galaxy, we minimize the reduced χ 2 as a function of two parameters, namely the boundary condition and the NFW matching radius.We have where N = 15 is the number of weak-lensing data points, N dof = 2 is the number of fit parameters, i runs over the N data points, a obs,lensing,i is the i-th weak-lensing-inferred acceleration with statistical uncertainty σ i , and a obs,SFDM,i is the i-th a obs value calculated in SFDM.We minimize this reduced χ 2 using the Julia package 'Optim.jl'[22].
In Fig. 6 we show solutions for the "closest-match" parameters for three galaxies with log 10 M b /M ⊙ values of 8.5, 10.0, and 11.5.We call these "closest-match" rather than "best-Figure 6.As Fig. 5 but for the "closest-match" boundary condition and NFW matching radius for each M b .In addition, we show the kinematically-inferred acceleration for each M b (dotted lines).We use the term "closest match" rather than "best fit" for the solutions with minimum reduced χ 2 to indicate that none of these solutions match all of the data.This is explained in more detail in the main text.As also explained in the main text, the closest-match solutions for log 10 M b /M ⊙ = 8.5 and 10.0 represent the "universal" best way to fit the lensing data in SFDM.Galaxies with larger masses, such as log 10 M b /M ⊙ = 11.5, cannot reach this "universal" closest match.
fit" because none of the solutions match all of the data; these are the least bad results we can obtain.The corresponding minimized χ 2 red values are 15.0, 14.9, and 28.7.These are worse than what we find in MOND with no free parameters, N dof = 0, namely χ 2 red = 6.5.However, one should not read too much into the precise numerical values here.Formally, all of these χ 2 red values indicate bad fits, since they are all much larger than 1.But that does not mean that the corresponding models are all ruled out.Indeed, the systematic uncertainties -such as those in the stellar masses and in converting the excess surface density profiles to accelerations -are larger than the statistical uncertainties that enter the χ 2 calculation [13].The situation is similar for kinematic observations, see for example [23].
It is more informative to look at the qualitative features of the closest-match solutions shown in Fig. 6.For example, we see that the closest-match solutions for log 10 M b /M ⊙ = 8.5 and 10.0 are very similar.Indeed, all closest-match solutions would ideally follow exactly this "universal" closest match to the lensing data.That is, all SFDM closest-match solutions would ideally match to an NFW tail at about a b ∼ 10 −12 m/s 2 -slightly after the first lensing data point -with an a obs slightly larger than the data points.This is the "universal" best way to fit the lensing data in SFDM because the superfluid itself does not match the lensing data at all (Fig. 4) and an NFW tail does not have the right scaling (namely 1/r 3 compared to the 1/r 2 scaling required by the lensing data).Thus, the closest-match solutions for log 10 M b /M ⊙ = 8.5 and 10.0 are almost identical because they both follow this universal closest match.However, this is not possible for log 10 M b /M ⊙ = 11.5 because of the large baryonic mass.
As can be guessed from Fig. 4, for such large masses, the a obs in SFDM either significantly overshoots the observed a obs at a b ∼ 10 −12 m/s 2 (for relatively large boundary conditions) or the superfluid ends well before the weak-lensing data starts (for relatively small boundary conditions).In both cases one cannot match to an NFW tail at a b ∼ 10 −12 m/s 2 with an a obs slightly larger than that of the lensing observations.For smaller baryonic masses, this can always be achieved by adjusting the boundary condition.More generally, Fig. 6 shows that the closest-match solutions all transition to an NFW tail already at relatively large values of a b .So the best way to fit the lensing data with SFDM seems to be to just fit an NFW halo to the data.As mentioned above, NFW halos can -at best -explain the weak-lensing RAR when we allow to tune the halo parameters.For SFDM, this amounts to giving up one of its main motivations.Namely the idea that SFDM, like MOND, can reproduce the shape and normalization of the RAR without having to tune any boundary conditions or similar parameters.
Of course, in principle, it is possible that galaxy formation provides a mechanism that selects the needed boundary conditions and NFW matching radii with high probability.However, so far, we have no reason to believe that this is the case, at least not any more or less so than in standard CDM models which also cannot reproduce the observed RAR without galaxy formation picking just the right halo parameters.Thus, SFDM cannot explain the weak-lensing data without losing its main advantage over standard ΛCDM models.
Fig. 6 also shows that fitting the weak-lensing data sometimes implies a bad fit of the kinematic data.An example is the closest match for log 10 M b /M ⊙ = 8.5 which significantly undershoots the kinematic data.This is because, for such small masses, reaching the "universal" closest match of the lensing data means deviating a lot from the boundary conditions that are optimal for reproducing MOND for kinematic observables.This can be seen by comparing Fig. 6 and Fig. 3.

Discussion and conclusion
Our analysis used the fiducial numerical parameters proposed by the original authors of SFDM [6].As we argue in appendix C, different choices for these parameters can change the numerical details but cannot avoid the general problem that the model has with gravitational lensing.
Similarly, an improved SFDM model called two-field SFDM was proposed [18].For this model, the precise numerical details will again be different.But we expect the same basic problems as in the original SFDM model: In two-field SFDM, as in the original SFDM model, a MOND-like phonon force is an emergent phenomenon of a superfluid condensate around galaxies.To avoid constraints from GW170817, this phonon force does not couple to photons.As a result, the lensing RAR can follow the MOND prediction only when carefully adjusting the boundary condition and NFW matching radius separately for each galaxy.And even then, unlike in observations, there is no continuity between the kinematic and the lensing RAR.
In conclusion, we have explored whether or not superfluid dark matter can explain the recent data from weak gravitational lensing.We have found this to be not possible.The major problem is that the observational data continue smoothly from those obtained by kinematic measurements -from baryonic matter -to those obtained from lensing -from photons.Such a continuation is unexpected in superfluid dark matter because only the baryons couple to the new field, whereas the photons do not.

B Mass-dependence of deviations from MOND
In Fig. 3, we saw that, for kinematic observables, smaller galaxies follow the MOND prediction more easily than larger galaxies.To see why, first note that deviations from MOND become relevant when ε * violates the condition ε * ≪ 1 (see Eq. (2.7)).Consider then the rough analytical lower bound on ε * derived in earlier work [19], where ε * min is the value of ε * that gives ρ SF = 0 (see Eq. (2.5), ε * min ≈ −0.3 for the fiducial parameter value β = 2).Eq. (B.1) suggests that ε * becomes larger than 1, at the latest, at a universal radius that is independent of baryonic mass M b .That is, Eq. (B.1) suggests that deviations from MOND set in, at the latest, at this universal radius.This, in turn, suggests that the maximum radius r max is roughly independent of M b .This maximum radius corresponds to a minimum acceleration a bmin = GM b /r 2 max which is then proportional to M b .This qualitatively fits with the mass-dependence of deviations from MOND in Fig. 3.

C Dependence on parameter values
The phenomenology of SFDM around galaxies depends on the values of the model parameters a 0 , β, and m 2 /α [19].In the main text, we assumed the numerical parameter values proposed by the original authors [6].Here, we discuss the effects of changing these values.
Consider first a 0 .In the MOND limit ε * ≪ 1, the kinematically-inferred acceleration a obs has the form a b + √ a 0 a b .This roughly reproduces MOND as long as a 0 has roughly the value it usually has in MOND, i.e. a 0 ≈ 10 −10 m/s 2 .Thus, we cannot change this parameter much without giving up one of the main ideas behind SFDM, namely to reproduce MOND in this regime.Thus, for any reasonable value of a 0 , our conclusions remain the same.
The most important parameter for our purposes is m 2 /α.It multiplies both the superfluid energy density ρ SF as well as the quantity ε * that controls whether or not the kinematically-inferred a obs is in the MOND limit.With the original fiducial parameter values [6] and at large a b , the SFDM prediction for the lensing-inferred a obs is much smaller than observations (see section 4).To counter this, we could choose a larger value of m 2 /α which gives a larger superfluid mass and thus a larger lensing-inferred a obs .However, this also makes ε * larger.As a result, the kinematically-observed a obs starts to deviate from the MOND prediction already at larger a b .
That is, by adjusting the parameter m 2 /α we can change where the problems discussed in section 4 occur.But we cannot avoid these fundamental problems of SFDM.This is illustrated in Fig. 7, Fig. 8, and Fig. 9.These are the same as Fig. 3, Fig. 5, and Fig. 6 but with m 2 /α increased by a factor 10. We see that there is still a tension between the kinematic and the lensing RAR.And reproducing a MOND-like lensing RAR is, at best, possible when tuning the boundary condition and NFW matching radius.This leaves β.This parameter controls the precise shape of ρ SF and the phonon force a θ outside the MOND limit, i.e. for ε * ≳ 1 (see for example Appendix A of [19]).To avoid instabilities and to ensure a positive superfluid density, β must be between 3/2 and 3 [5].In Fig. 10, Fig. 11, and Fig. 12, we show the effect of using β = 1.55 instead of the original fiducial value β = 2.0 [5].We see that there are minor quantitative changes but our qualitative conclusions remain unchanged.6 but with m 2 /α increased by a factor 10. The reduced χ 2 values for the closest-match solutions shown here are 13.6, 43.8, and 66.9 for log 10 M b /M ⊙ = 8.5, 10.0, and 11.5, respectively.Note that the "universal" closest match described in Sec.4.1 is the same as before, but now it can be reached only for the lowest baryonic mass shown here, log 10 M b /M ⊙ = 8.5.Due to the increased m 2 /α, the other masses are too large to reach it.This is why all three closest-match solutions shown here are different while in Fig. 6 two of them are almost identical.Even if changing β did lead to significant changes, it would be best to not rely on them too much.This is because both the value of β and the form of the finite-temperature corrections that β is supposed to represent are completely ad-hoc [18].So any significant changes related to β might easily turn out to be unphysical.D Upper bound on lensing a obs as a function of r NFW Suppose we are given a solution of the equations of motion inside the superfluid core of a galaxy.This does not fix the acceleration outside the superfluid since this depends on where exactly we match the superfluid density to an NFW 1/r 3 tail.
There is an optimal r NFW that maximizes the acceleration outside the superfluid, as we will now show.Using the condition that the superfluid density ρ SF matches the NFW tail's density at r = r NFW , we have outside the superfluid core for the acceleration a DM due to dark matter, where we used M ′ SF (r) = 4πr 2 ρ SF (r).Note that this is independent of r.Thus, there is a single r NFW that maximizes the acceleration everywhere outside the superfluid.
In earlier work [19], we showed that -for a given solution inside the superfluid -the virial mass M 200 has a maximum as a function of r NFW .It turns out the r NFW that maximizes M 200 discussed there is the same r NFW that maximizes a DM discussed above.To see this, In extreme cases, it can happen that the virial radius r 200 is inside the superfluid core.In this case, r 200 and a DM are not maximized by the same a DM .For definiteness, above we numerically maximize a DM and not r 200 .Concretely, we use the Julia package 'Optim.jl'[22] to find which r NFW maximizes a DM for r NFW in the interval [0.001 kpc, r 0 ] where r 0 is the radius where the superfluid density drops to zero.

Figure 1 .
Figure1.The total kinematically-inferred acceleration a tot = a b +a SF +a θ inside the superfluid core (solid blue line) relative to the acceleration in MOND for an example galaxy with M b = 2 • 10 10 M ⊙ .This is for the boundary condition for which a tot stays within a fraction δ = 0.15 from MOND up to the maximum possible radius r max (vertical dotted line).Dashed (dash-dotted) black lines show solutions with this optimal boundary condition multiplied by 1.5 and 1.2 (0.8 and 0.5).Also shown are the different components that make up a tot , namely the baryonic Newtonian acceleration plus the acceleration due to the phonon field, a b + a θ , (solid red line) and the superfluid's Newtonian acceleration a SF (solid green line).For the non-optimal boundary conditions, a b + a θ and a SF are shown as grey lines.All solutions are shown up to the radius where the corresponding superfluid density ρ SF drops to zero.

Figure 2 .
Figure2.The superfluid density ρ SF (solid blue line) for the solution with δ = 0.15 for an example galaxy with M b = 2 • 10 10 M ⊙ .The dashed red line illustrates one possibility of matching an NFW tail, ρ NFW ∝ 1/r 3 , to the superfluid density ρ SF .Here, we show the matching radius r NFW that gives the largest dark matter acceleration a DM outside the superfluid core, but other choices are also possible (see for example Fig.5).

Figure 4 .
Figure 4.As Fig.3but for the lensing-inferred a obs instead of the kinematically-inferred a obs .These differ in SFDM because the phonon acceleration a θ contributes only to kinematic observables.

Figure 5 .
Figure 5.As Fig.4, but for a fixed boundary condition corresponding to δ = 0.8 and with the superfluid density matched to an 1/r 3 NFW tail at various radii r NFW .Solid lines show the lensing-inferred a obs inside the superfluid core, at r < r NFW .Dash-dotted lines show a obs outside the superfluid, at r > r NFW .For each M b , we show a obs for the r NFW that gives the maximum possible lensing-inferred a obs outside the superfluid.We also show NFW tails with 0.35, 0.5, and 0.7 times this optimal r NFW .Outside the superfluid, at r > r NFW , the kinematic and the lensing a obs coincide since the phonon force does not act there[5].

Figure 7 .
Figure 7. Same as Fig. 3 but with m 2 /α increased by a factor 10.

Figure 8 .
Figure 8. Same as Fig. 5 but with m 2 /α increased by a factor 10.

Figure 9 .
Figure 9. Same as Fig.6but with m 2 /α increased by a factor 10. The reduced χ 2 values for the closest-match solutions shown here are 13.6, 43.8, and 66.9 for log 10 M b /M ⊙ = 8.5, 10.0, and 11.5, respectively.Note that the "universal" closest match described in Sec.4.1 is the same as before, but now it can be reached only for the lowest baryonic mass shown here, log 10 M b /M ⊙ = 8.5.Due to the increased m 2 /α, the other masses are too large to reach it.This is why all three closest-match solutions shown here are different while in Fig.6two of them are almost identical.