Could Quasar Lensing Time Delays Hint to a Core Component in Halos, Instead of H0 Tension?

The time delay measured between the images of gravitationally lensed quasars probes a combination of the angular diameter distance to the source-lens system and the mass density profile of the lens. Observational campaigns to measure such systems have reported a determination of the Hubble parameter H0 that shows significant tension with independent determination based on the cosmic microwave background (CMB) and large-scale structure (LSS). We show that lens mass models that exhibit a cored component, coexisting with a cusp, probe a degenerate direction in the lens model parameter space, being an approximate mass sheet transformation. This family of lens models has not been considered by the cosmographic analyses. Once added to the model, the cosmographic error budget should become dependent on stellar kinematics uncertainties. We propose that a core component coexisting with a cusp could bring the lensing measurements of H0 into accordance with the CMB/LSS value.


Introduction and Main Result
There appears to be a tension between measurements of the Hubble parameter H 0 based on the classic cosmic distance ladder method and measurements obtained through a fit of the standard ΛCDM model to the cosmic microwave background (CMB) or large-scale structure (LSS) data (for a summary, see Verde et al. 2019 (Planck Collaboration et al. 2018), or with H 0 measurements obtained from galaxy clustering and galaxy lensing data (Abbott et al. 2018;D'Amico et al. 2019;Ivanov et al. 2019;Tröster et al. 2019) that are independent of but agree with the CMB result. The tension is not so strong in the analysis of a supernova sample calibrated with the tip of the red giant branch, which gives =   H 69.8 0.8 stat 1.7 sys 0 ( ) ( ) km s −1 Mpc −1 (Freedman et al. 2019). On the other hand, a third, alternative method to constrain H 0 independently of both the distance ladder and cosmological perturbation theory, is provided by measurements of time delays in strongly lensed systems (Refsdal 1964;Kochanek 2002Kochanek , 2006. The H0LiCOW collaboration (Suyu et al. 2017) used the time delays between multiple images of strongly lensed galaxies hosting a quasar to obtain = -+ H 73.3 0 1.8 1.7 km s −1 Mpc −1 (Suyu et al. 2017;Birrer et al. 2019;Bonvin et al. 2019;Chen et al. 2019;Wong et al. 2019), achieving 2.5% precision with a central value in agreement with SH0ES. Combining the SH0ES and H0LiCOW measurements, the result is in more than 5σ tension with CMB/LSS data.
Given the importance of these results to cosmology, it is worth investigating them from every angle. In this Letter we focus on the lensing time delay ("cosmography") measurements. It is well known that lensing analyses are subject to systematic uncertainty associated with the choice of the family of models used to reconstruct the lens potential (Falco et al. 1985;Schneider & Sluse 2013Xu et al. 2016;Unruh et al. 2017;Tagore et al. 2018;Sonnenfeld 2018;Gomer & Williams 2019;Kochanek 2019). The H0LiCOW collaboration is, of course, well aware of this problem, and had taken measures to mitigate it by considering different families of lens models. Nevertheless, H0LiCOW systems probe the baryonic-dominated inner part of the lens, where theoretical understanding of the mass profile is limited to challenging hydrodynamical simulations. Moreover, given that we do not know what makes up the dark matter, its distribution on galactic scales could exhibit unexpected features. With these issues in mind, the question we address in this Letter is: could a feature in the mass density profile of H0LiCOW lenses bring the cosmographic result for H 0 to agree with the CMB/LSS value? Another way to phrase this question could be to accept, tentatively and for the purpose of the exercise, the CMB/LSS value of H 0 ; and then ask, given this hypothesis, what would the cosmography data teach us about the inner structure of galaxies.
We believe that we have found an interesting answer to this question. To summarize, we find that if one took the simple power-law (PL) density models, shown by H0LiCOW to provide a good fit to the lensing data, and then added a core component in addition to the PL, then: (i) the lensing reconstruction problem should be equally well solved by the PL+core models as it is for the pure PL; and (ii) the addition of comparable cores to all H0LiCOW lenses would systematically shift the inferred cosmographic value of H 0 downward in all systems, in accordance with the fact that all of the systems analyzed by the standard H0LiCOW analysis pipeline consistently hint to high H 0 .
The cores we need are a moderate deformation of the nominal profile: at the Einstein radius (translating to a few kiloparsecs for H0LiCOW systems, where the lenses are massive elliptical galaxies), the core component need only make up 10% or less of the total enclosed mass of the lens.
Outside of the Einstein radius the relative core contribution could become larger, potentially reaching as much as (1) of the mass and opening a possible way to constrain our solution with detailed kinematic modeling. However, current lensing data do not constrain very well the outer extent of the core.
Single-source lensing data cannot distinguish a pure PL profile from PL+core, because moving along the PL+core family of models (as we shall define in Section 2) is an approximate mass sheet transformation (MST; Falco et al. 1985). Therefore, PL +core models probe a flat direction in the likelihood for H 0 . Stellar kinematics could, in principle, break the mass sheet degeneracy (MSD; Romanowsky & Kochanek 1999;Treu & Koopmans 2002;Jee et al. 2015Jee et al. , 2016Shajib et al. 2018). However, as mentioned above, to solve the H 0 tension we need an effect of no more than 10% in enclosed mass within q E . Constraining this with stellar kinematics would not be trivial and would suffer from systematic uncertainties related to, e.g., the velocity anisotropy modeling. Furthermore, kinematics modeling uncertainties would come to dominate the determination of H 0 , which should be revised (Kochanek 2019). Perhaps another potential way to resolve the MSD would be to have multiple sources lensed by the same object, as is usually the case in lensing by galaxy clusters (Grillo et al. 2018(Grillo et al. , 2020. It is worth pointing out that we are not aware of the presence of PL+core profiles in simulations. In this sense, introducing them is an ad hoc solution of the (cosmographic contribution to) the H 0 tension. But we are also not aware of observational data that excludes such profiles. If one accepts PL+core profiles, then the question arises what is the core component made of. Since the inner part of the lenses is baryon-dominated, we do not know at this point if the core is some baryonic structure, or dark matter. It is exciting to speculate that the H 0 tension could actually hint to new constraints on the nature of the dark matter, perhaps along the lines of models such as in Schive et al. (2014) or Spergel & Steinhardt (2000).
This Letter is organized as follows. In Section 2 we show that adding an inner core component, on top of a (rescaled) cusp component, is an approximate MST. We give some simple examples, estimate the MSD-breaking effects, and introduce λPL models as a family of models that is expected to probe a flat direction in cosmographic measurements of H 0 . In Section 3 we consider as input the CMB/LSS measurement of H 0 and use it to estimate the required morphology of H0LiCOW lenses. In Section 4 we discuss our findings, and the possibility that the (cosmographic part of the) H 0 tension might actually hint to a core component coexisting with a central cusp in galaxies. In the Appendix we collect some formulae for profiles that could serve as λPL models.

Adding a Core to a Cusp Is an MST
Lensing analyses (Suyu et al. 2017;Birrer et al. 2019;Bonvin et al. 2019;Chen et al. 2019;Wong et al. 2019) take as input a brightness map defined on the image plane, spanned by coordinates q, and constrain the deflection angle a q ( ) given by where b parameterizes positions on the source plane. The deflection angle is obtained by integration over the convergence, Here q S( ) is the projected surface mass density of the lens and D D D , , s l ls are the angular diameter distances from the source to the observer, from the lens to the observer, and from the source to the lens.
Given multiple images of a quasar contained in the host galaxy, one constructs the time delay Dt ij between quasar images q i and q j , If one has a model of q k( ), then it can be used to calculate t D ij in Equation (6). Given a measurement of Dt ij , one can extract The MSD comes from the fact that if the lensing reconstruction problem (Equation (2)) is solved by a model for q k( ), along with a model for the source position b, then the reconstruction problem is also solved equally well by the alternative MST model 3 1 0 ( ) leaving q unchanged. While the lensing image-plane geometry is invariant under the MST, the time delay is not invariant and it is easy to verify that t l t . This means that if we measure H 0 from some model κ, then the MST model k l would give The actual reconstruction problem of H0LiCOW deals with an extended source model, given by a map of the brightness b I s ( ) on multiple source-plane pixels. Under an MST, the distortion matrix , ( ) ( ), which means that the magnification m = A 1 det becomes m m l = l 2 . Importantly, the relative magnification between images is unchanged. For H0LiCOW systems the precise intrinsic luminosity of the source is unknown, so absolute magnification cannot be measured. Thus extended source information does not mitigate the MSD.
The "mass sheet" in MSD refers to the l -1 term in Equation (9), which is a constant convergence term and thus it acts as a q-independent mass sheet. H0LiCOW took careful measures to account for the MSD due to external convergence k ext (see dedicated discussions in Suyu et al. 2017;Birrer et al. 2019;Bonvin et al. 2019;Chen et al. 2019;Wong et al. 2019). This was done by using numerical simulations to estimate the cumulative contributions of mass along the line of sight in the field of the lens systems. However, a cored density profile (3D density ρ∼ const) extending over a finite radius R c , and dropping quickly afterward, can also give κ that is constant Thus, if the images in the lensing data only extend over angles q q < c , then the addition of a cored density component, with k c that is constant inside q c , is (i) an approximate MST, if it is done alongside a rescaling of the previous κ model, and (ii) is not equivalent to an external convergence term. 5 To make things more concrete we define the λPL family of profiles: Here, we take k PL to represent the elliptic PL profile as used by H0LiCOW to successfully model the lensing data in their systems. 6 The q k c ( ) term is chosen to satisfy q k » 1 c ( ) for q q < c and to fall faster than k PL at q q > c . We do not need to assume that q k c ( ) is isotropic, but in what follows for simplicity we will.
As a first example, consider the 3D cored density profile r = S + Equation (11) gives an approximate MSD inside of q q < c . We can estimate the corrections to the MSD by comparing the Einstein angle q E for k PL and the Einstein angle q l E for k l in Equation (11). For simplicity, in this exercise we take k PL to be isotropic and given by q k = . From the form of δ we can infer the parametric dependence of the breaking of the MSD. The corrections to the image-plane geometry enter at order q q c 2 2 , and if l » 1 (that is, if we only add a small core) are further suppressed by a factor l -1 . Note that for real systems H0LiCOW find g » 2 so g -» 1 1 1 ( ) (see Table 1). More generally, if in Equation (11) we use a core component that can be expanded as k q q = + + a 1 ... at q q < c , then the leading-order image-plane corrections to the MSD at q q < c scale as l q q -1 c 2 2 ( ) . This scaling remains true also when the baseline term k PL (or whatever other baseline model is considered, e.g., a composite stellar cusp+NFW model) is anisotropic.
As another example, consider the 3D cored Navarro-Frenk-White (NFW) density profile, which contains one extra parameter R c , defining the core, in addition to the usual NFW density r 0 and scale radius R s . The convergence k cNFW can be computed analytically even though is not particularly illuminating (in the Appendix we collect some formulae for profiles that could serve as the core component in λPL models). With proper normalization such that k = 0 1 cNFW ( ) it has the correct characteristics to function as k c in Equation (11). We show k cNFW by the dashed black line in Figure 1 To illustrate the MSD, in Figure 2 we calculate the lensing geometry and time delays for a toy model of a quasar sitting in an extended host galaxy. To make things simple we replace the extended host by a circle on the source plane, centered on the quasar. We first do the lensing exercise for a pure PL model with slope n=1.95 and ellipticity parameter q=0.8, similar to typical H0LiCOW systems. The source-plane host "galaxy" is shown by the red circle in the top panel (source plane). The "quasar" is denoted by the magenta cross. The lensed images are shown by red lines in the bottom panel (image plane). (It is difficult to see these lines because they lie underneath the green lines of the λPL model, as explained below.) We calculate the dimensionless time delays t D ij at the quasar image positions and show them next to the bottom panel (magenta, labeled PL). The convergence for this PL model (along the q x axis) is shown by the red line in Figure 1. We have chosen the PL normalization such that q » 1 E . Next, we consider a λPL model with l = 0.75. The convergence for this λPL model is shown by the blue line in Figure 1. The source-plane host model as given by the MSD is shown by the green circle in the top panel of Figure 2. The quasar is shown by the blue cross. The images are shown by the green line and blue crosses in the bottom panel. As expected, they sit almost on top of the pure PL. The time Note. Values for H 0 (in km s −1 Mpc −1 ) are from the PL fit ( Figure 6 in Millon et al. 2019). Approximate values for the PL index γ, the Einstein radius q E , and the NFW scale q s were read from PL and composite NFW+stellar fits reported by papers in the last column.
delays for the λPL model images are shown next to the bottom panel (blue, labeled λPL). As expected the λPL time delays satisfy t l t

If We Assume H 0 from CMB/LSS, What Do We Learn about H0LiCOW Lenses?
If one used the toy example of Figure 2 to measure H 0 , and if, assuming the pure PL model, one found, for example, = H 74 0 km s −1 Mpc −1 , then we expect that the λPL model with l = 0.75 would give acceptable likelihood with » H 56 0 km s −1 Mpc −1 . Our choice of λ in this example is, of course, an exaggeration. To solve the H 0 tension we only need l » 0.9. In Table 1 we collect some key numbers for six H0LiCOW systems. Taking » H 67 0 km s −1 Mpc −1 to represent the CMB/LSS measurement, we show in the third column the value of λ that is required to bring the cosmographic H 0 from each system down to the CMB/LSS value.
Noting that H0LiCOW found adequate fits to the lensing reconstruction with the PL model, and given an estimate of λ for each system from Table 1, we can use Equation (11) with some models for k c to investigate the implied physical shape of the lens galaxies. In Figure 3  if we project along the q x direction, or -gq 1 2 if we project along q y , without adjusting k c . Typical H0LiCOW lenses haveq 0.8 and g~2.
Considering Figure 3, it is important to note that the particular shape of the profiles at angles q q > E is probably poorly constrained by the lensing data. Despite the fact that H0LiCOW utilizes extended host information, in practice the host image pixels do not exceed q q  1.5 E or so. As we have seen, the impact of the edge of the core component, which breaks the MSD, enters the image plane geometry at order l q q -1 c 2 2 ( ) . The λ values needed to sort out the H 0 tension imply l -» 1 0 . 1or so for most systems; even for the most deviant system (PG1115) we have l -» 1 0 . 2. This means that q q  3 c E would be enough to bring the MSD-breaking deformation down to the 1% level for most systems. Moreover, in a real analysis, some of this deformation would probably be absorbed by the fitting for the source-plane host parameters. A full-fledged analysis such as H0LiCOW, fitting λPL models to the real data, would be needed to truly quantify the level of the degeneracy. At this point, however, we emphasize that the shape of the profiles in Figure 3 at q q > E comes from our particular choice of k c in this example, and is not necessitated by the data.
Finally, let us make a preliminary comparison with constraints from kinematics. Cappellari et al. (2015) presented an analysis of stellar kinematics in early-type galaxies with stellar masses in the range M M log 10 * ( )  ∼10.2-11.7. These systems may be reasonable analog systems to H0LiCOW lenses. According to Cappellari et al. (2015), the total density profiles of all of the analyzed galaxies are consistent within the modeling uncertainties with simple PL all the way from r R 0.1 e out to r∼4R e , where R e is the half-light radius. This range of kinematics coverage is interesting because it overlaps with and extends the range covered by the lensing analyses, which typically probe  r R e . In Figure 4 we compare the 3D density of a λPL model with the profiles found in the galaxy kinematics analysis of Cappellari et al. (2015). The kinematics constraint, shown for the example of the system NGC4649 (see Figure4(d) in Cappellari et al. 2015), is given by the shaded band that envelopes a collection of 100 profiles obtained by randomly selecting model parameters from the posterior distribution of the fit. The λPL models for l = 0.9 and l = 0.75 are shown by solid and dashed lines, respectively. In the left panel we show the 3D equivalent of the cNFW model considered in Figures 1 and 2 The comparison of λPL models to the results from Cappellari et al. (2015) should be regarded with caution, as the family of dark matter density profiles considered in Cappellari et al. (2015) was restricted to a generalized NFW form that does not overlap with the λPL shape. With that in mind, we take Figure 4 to suggest that, currently, constraints from kinematics most likely cannot exclude λPL with l~0.9, which is the range of λ that would be implied from cosmography if one calibrated H 0 from CMB/LSS data. That said, PL-core combinations with, e.g., l = 0.75 could perhaps be constrained by data, motivating a dedicated kinematics analysis specifically designed to test λPL profiles.

Discussion and Summary
Lensing data alone cannot resolve the MSD. Therefore, we think that the likelihood function in the cosmographic Figure 1. Convergence for a λPL model, with l = 0.75 (blue) and l = 0.9 (green). The l = 1 pure PL case is shown as the red line, and the cNFW profile is shown as the black dashed line. A value of l » 0.9 would bring the H0LiCOW determination of H 0 down to the CMB/LSS value. Note that for H0LiCOW lenses, both lensing and kinematics data reach outward only slightly beyond q E , and never constrain angles around the value of q c chosen in this example. measurement of H 0 would have a very flat (albeit not completely flat) direction, corresponding to the effective MST λ parameter of λPL models. The H0LiCOW collaboration could thoroughly test this hypothesis on the real data. The λPL models do not require more fitting parameters than, for example, the composite stars+NFW models considered already by H0LiCOW.
Stellar velocity data do resolve the MSD and could constrain λPL models. Probably too large variations over the pure PL, in terms of the total enclosed mass, are not allowed by the stellar velocity data. However, we doubt that stellar velocity data can test an λPL model with l » 0.9. Either way, if the systematic uncertainty comes to be dominated by the stellar velocity modeling, then the significance of the H 0 tension would need to be revised. In this respect, we agree with the recent discussion of Kochanek (2019).
It would be very interesting if indeed H0LiCOW has detected a core component in the lens galaxies. We are not aware of such cores in N-body or hydrodynamical simulations. Perhaps they could arise if, for example, the dark matter sector contains a component of ultralight (Schive et al. 2014) or selfinteracting (Spergel & Steinhardt 2000) dark matter. On the other hand, since typical H0LiCOW lenses have q q s E  (inferred in their composite stars+NFW models), it is clear that the lensing data probe the inner part of the lens halo where baryons either dominate the dynamics or at least make a large impact on it, making the simulations challenging. From this point of view, a detection of λPL profiles with l » 0.9 could be turned into a goal to explain. Before we venture to more speculations, though, it would be reassuring to see a dedicated lensing/time delay analysis that tests λPL models on the real data.