Halo concentration strengthens dark matter constraints in galaxy-galaxy strong lensing analyses

Our lensing systems feature:

• •

  a power-law mass profile to represent the main lens (Tessore et al., 2016), characterized by the following free parameters: $(x_{\rm l},y_{\rm l})$, the centre of the mass distribution; $\epsilon_{\rm l}$, the Einstein radius; $\beta_{\rm l}$, the slope of the mass profile; $(e_{1,l},e_{2,l})$, the two independent components of the profile’s ellipticity; $(\gamma_{1},\gamma_{2})$, the two independent components of the external shear.

• •

  a parametric source with a Sersic profile: projected centre, $(x_{\rm s},y_{\rm s})$; effective radius, $r_{\rm eff}$; ellipticity, $(e_{1,s},e_{2,s})$; Sersic index, $n_{\rm s}$; and total flux, $I_{\rm s}$.

These two components define our macromodel: $\theta_{\rm m}$ is therefore a 15-dimensional vector.

The perturbing haloes are modelled with spherically symmetric Navarro-Frenk-White mass profiles (Navarro, Frenk, & White, 1997), introducing the following additional 5 parameters: projected centre, $(x_{\rm h},y_{\rm h})$; redshift, $z_{\rm h}$; mass, $M_{\rm h}$; and concentration $c_{\rm h}$. Throughout the paper we take halo masses, $M_{\rm h}$, to be the virial mass, $M_{200}$, i.e. the mass contained within a sphere of density 200 times the critical density. We use the open source software PyAutoLens⁵⁵5PyAutoLens is open-source and available from https://github.com/Jammy2211/PyAutoLens (Nightingale & Dye, 2015; Nightingale, Dye, & Massey, 2018; Nightingale et al., 2021) to generate all of our mock data and for all our lensing modelling.

Within this framework, we choose two different lensing configurations for our exploration. Both have their source at $z_{\rm s}=1$ and lens at $z_{\rm l}=0.5$, but one is in a quad-configuration while the other features two asymmetric arcs. Values for the ground truth macromodel parameters, $\theta_{\rm m}$, are recorded in Table 1; Fig. 1 illustrates their geometry, displaying the corresponding model fluxes, $\bf{f}(\theta_{\rm m})$. The exact values are of no importance here: we simply choose sets of parameters that are qualitatively in line with what is found in lensing studies of real systems, and we use a pair of different configurations to ensure that the trends we identify are not a peculiarity of the specific system we happen to adopt. Throughout this work, we use pixel size and point-spread function (PSF) width typical for HST data, fixing both quantities at $0.05^{\prime\prime}$.

The sensitivity function of a lens scales with the quality of the available data, quantified here by the noise map, $\bf n$, and the associated maximum value of the signal-to-noise, $\textbf{n}\propto 1/SN_{\rm max}$. Notice, however, that while the noise map itself is known, the actual noise realization of the observed data is not. We are therefore interested in assessing how different noise realizations affect the value of the log-likelihood change.

Let us assume that the data are characterized by a noise realization, $\bf r$, so that $\bf d=\langle d\rangle+r$, where $\langle\cdot\rangle$ denotes an average over noise realizations. In the case of mock data, $\langle\bf d\rangle$ is the input model flux corresponding to the ground truth model parameters, which we indicate with $(\theta_{\rm m},\zeta_{\rm h})$: $\langle{\bf d}\rangle=\textbf{f}(\theta_{\rm m},\zeta_{\rm h})$. The log-likelihood gain is as in equation (2), with terms in the same order:

where we have implicitly assumed a sum over image pixels. The fluxes, $\textbf{f}(\hat{\theta}_{\rm m},\hat{\zeta}_{\rm h})$, and $\textbf{f}(\bar{\theta}_{\rm m})$, correspond to the model that best fit the noise-corrupted data, $\bf d$, with and without an extra halo. It is convenient to consider instead the model fluxes that provide the best fit to the noise-free data, $\langle\bf d\rangle$, which we refer to as $\textbf{f}_{\rm h}$ and $\textbf{f}_{\rm m}$. These models do not achieve the maximum log-likelihood values we require in equation (5). For example, for the model including a halo mass component,

where the difference, $l_{\rm h}^{\rm bf}$, is a function of the noise realization and has a positive value. Similarly,

Furthermore, in order to highlight the dependence on the noise realization, $\bf r$, we can recast the model fluxes, $\textbf{f}_{\rm h}$ and $\textbf{f}_{\rm m}$, in terms of their associated residuals, $\textbf{f}_{\rm h}=\langle\textbf{d}\rangle+\delta_{\rm h}$, and $\textbf{f}_{\rm m}=\langle\textbf{d}\rangle+\delta_{\rm m}$, which gives:

and

Equation (5) is therefore the difference between the right-hand sides of the two equations above. We are interested in the mean and the standard deviation of this difference across noise realizations.

Let us first consider the two shifts, $l^{\rm bf}_{\rm h}$, and $l_{\rm m}^{\rm bf}$. These are nonzero, but they are not the leading terms of equation (5) that we seek. We can show this by estimating their average magnitude across varying noise realizations. This can be calculated analytically under the assumption that the likelihood surface is Gaussian. If so, we can see that

which is valid for both $l^{\rm bf}_{\rm h}$ and $l_{\rm m}^{\rm bf}$, and in which $k$ is the number of independent parameters in the likelihood. In our case, the model which includes a halo mass component features 5 additional free parameters, so that

This is considerably smaller than the log-likelihood differences we are after and we therefore ignore these terms from now on.

By expanding the chi-square terms in Eqns. (8) and (9), we finally obtain the leading terms we are interested in:

Here, the first term quantifies the inability of a model that does not include a perturbing halo mass component to describe the perturbed data. This is what sensitivity mapping is after, and is independent of the noise realization. The second term introduces scatter in the measurement of the log-likelihood gain as a consequence of varying noise realizations, $\bf r$. The case we are interested in is the one in which a model featuring a halo mass component provides a satisfactory fit to the data, $\delta_{\rm h}/\textbf{n}\ll 1$. This is also the case of mock data in which the model used to generate data is the same used to fit it: $\langle\bf d\rangle=\bf f_{\rm h}$. In this case,

where we have used $\delta\equiv\delta_{\rm m}$ for compactness. By definition, the noise realization, $\bf r$, is a random variable with zero mean; furthermore, by construction, the residuals, $\delta$, are not correlated with $\bf r$. As a result, the second term in equation (13) averages to zero:

We can estimate the magnitude of the scatter introduced by the same term by taking the ratio, $\textbf{r}/\textbf{n}$, to be a set of independent normal random variables with unit variance, which results in a standard deviation of

We test this scaling in Appendix B, where Fig. 14 shows experiments that highlight the scaling of equation (15).

In conclusion, from equation (14) we deduce that the mean log-likelihood increase scales with the square of the maximum signal-to-noise ratio, $SN_{\rm max}$, and we note that the Bayesian evidence will also feature in the same scaling. From eqns. (12) and (13) we see that, in real data, a scatter of the order of $\sqrt{2\Delta\mathscr{L}}$ should be expected. In fact, the same scatter should be expected when mapping the sensitivity function using mock data that include a random noise realization. This implies that multiple noise realizations should be used and results averaged. However, the analysis above also shows that this can be avoided by using noise-free mock data (i.e. $\textbf{r}=0$), while at the same time using the appropriate noise map, $\bf n$, featuring the same maximum signal-to-noise as in the real data. This is the strategy we adopt in this work.

Having chosen our macromodels, $\theta_{\rm m}$, we can introduce intervening LOS haloes with input parameters, ${\zeta}_{\rm h}$, and simulate the resulting model fluxes, $\bf{f}(\theta_{\rm m},{\zeta}_{\rm h})=\bf f$. As outlined in Section 2, each determination of the likelihood gain, $\Delta\mathscr{L}({\zeta}_{\rm h},\bf{r})$, requires two non-linear searches. However, in our case, $\bf r$=0, so that we have, $(\hat{\theta}_{\rm m},\hat{\zeta}_{\rm h})=(\theta_{\rm m},\zeta_{\rm h})$, or equivalently, $\mathscr{L}(\hat{\theta}_{\rm m},\hat{\zeta}_{\rm h})=0$, by construction, and therefore,

Thus, for each set of halo parameters, $\zeta_{\rm h}$, we only require one non-linear search in order to determine the best fitting parameters, $\bar{\theta}_{\rm m}$, of the model that does not include a halo mass component. This is also the fit with fewer free parameters – and therefore both the fastest to run, and the least likely to get stuck in local minima during a likelihood optimization.

A common assumption in sensitivity mapping is that the perturber’s redshift can be recast in terms of its effective mass. Li et al. (2017) noticed that the mass and redshift of a perturber are highly degenerate, and used this to introduce the idea of rescaling a perturber’s mass as a function of redshift. Despali et al. (2018) investigated this equivalence further and provided a universal scaling between redshift and effective mass, which is obtained by requiring that the map of deflection angles be minimally changed. In practice, at any fixed projected location, the perturbing characteristics of a halo of mass, $M_{\rm h}$, at a redshift, $z_{\rm h}$, have been equated to those of a halo located at the redshift of the lens and having an ‘effective’ mass of $\log M_{\rm sh}=\log M_{\rm h}+\delta M(z_{\rm h})$. The mass shift, $\delta M(z_{\rm h})$, is clearly zero at $z_{\rm h}=z_{\rm l}$, and is found to be monotonically increasing with redshift, so that detecting a halo of fixed mass becomes more challenging with increasing redshift, and is easiest for close-by perturbers.

Figs 2 and 3 show that our calculations do not support this working hypothesis in previous work. When fitting image fluxes (rather than deflection angles), we find that haloes are harder to detect when they are behind or in front of the main lens. The details depend on the precise lensing configuration, mass and concentration of the perturbing halo, as well as on the precise projected location. However, a decrease in $\Delta\mathscr{L}$ when the halo redshift deviates from the lens redshift is a universal qualitative feature in our maps displayed by both our adopted lensing configurations and across halo masses. This means that the effect of foreground haloes of fixed mass is more easily ‘reabsorbed’ by suitable macromodels when these perturbers are at low redshifts. In other words, degeneracies in the lens modelling make the detection of perturbers of the same mass increasingly more difficult with decreasing redshift.

Fig. 4 summarizes the redshift dependence of the log-likelihood increase, showing the ratio between the log-likelihood increase for a perturber at some redshift, $\Delta\mathscr{L}(z_{\rm h},M_{\rm h},c_{\rm h})$, divided by that at the redshift of the lens, $\Delta\mathscr{L}(z_{\rm h}=z_{\rm l},M_{\rm h},c_{\rm h})$. This ratio is then averaged over the perturbers’ projected locations, $(x,y)$. The top row refers to our ‘quad’ configuration, the bottom one to our ‘arcs’. The right column is for a perturber of mass, $M_{\rm h}=10^{10}$ M_⊙, the left column to one of mass, $M_{\rm h}=10^{9.5}$ M_⊙. Profiles of different colours refer to different values of the perturbers’ concentrations. As described, for most haloes, the log-likelihood increase decreases for redshifts that are both higher and lower than the lens’ redshift, $z_{\rm l}$. The size of this decrease depends systematically and monotonically on halo concentration, with the less-concentrated haloes displaying sharper falloffs.

Depending on the lensing configuration, for haloes on the mass-concentration relation, $\Delta\mathscr{L}(z_{\rm h})$ decreases by a factor between 1.15 to 1.6 between $z_{\rm h}=z_{\rm l}$ and $z_{\rm h}=0.2$, and then drops more sharply towards lower redshift. For the highest halo concentrations, we see, instead, a mild apparent increase in detectability at low redshift. However, analysis of the top rows of Figs. 2 and 3 shows that this increase is due to the fact that Fig. 4 displays averages over projected locations. At the highest concentrations, the projected area in which perturbers result in ‘intermediate’ log-likelihood values, $50\lessapprox\Delta\mathscr{L}\lessapprox 100$ (corresponding to orange hues in Figs. 2 and 3) increases at the lowest redshifts. In the same regions, log-likelihood values are lower at $z_{\rm h}=z_{\rm l}$, which drives the mild increase apparent in the average quantities shown in Fig. 4. On the other hand, even at the highest concentrations, it remains true that the peak values of the log-likelihood gain, $\Delta\mathscr{L}$, decrease with decreasing redshift. In any case, this mild increase is limited to extremely concentrated haloes, and, therefore, is not a representative behaviour.

The qualitative contradiction between the predictions using deflection angle maps and our results implies that previous estimates of the number of detectable haloes obtained using the relation between mass and redshift proposed in Despali et al. (2018) are likely to overestimate the number of low redshift haloes. We will return to this point in Section 5.2.

Most previous studies have fixed the halo concentration to the mean for their mass for the adopted mass-concentration relation. However, Figs. 2 and 3 clearly show that concentration makes a significant difference to halo detectability. From top to bottom, the values of the log-likelihood gain decrease monotonically: the perturbations of less concentrated haloes are more easily reabsorbed by changes of the macromodel parameters. In turn, at fixed mass, more concentrated haloes are more easily detected.

Fig. 5 provides a summary of the dependence of the log-likelihood increase, $\Delta\mathscr{L}$, on halo concentration. This shows the ratio between the log-likelihood increase, $\Delta\mathscr{L}(z_{\rm h},M_{\rm h},c_{\rm h})$, for a perturber of any concentration, $c$, divided by that for one on the mass-concentration relation, $\Delta\mathscr{L}(z_{\rm h},M_{\rm h},c_{\rm h}=c(M_{\rm h},z_{\rm h}))$. We reiterate that $\delta\log c=0$ in our mapping of the sensitivity function corresponds to the mass-concentration relation measured by Ludlow et al. (2016). Similarly to Fig. 4, these ratios are then averaged over projected locations, $(x,y)$. The top row refers to our ‘quad’ configuration, the bottom to one to our ‘arcs’. The right column is for a perturber of mass, $M_{\rm h}=10^{10}$ M_⊙, the left column for one of mass, $M_{\rm h}=10^{9.5}$ M_⊙. Profiles of different colours refer to different values of the perturbers’ redshift. It is clear that $\Delta\mathscr{L}$ increases monotonically with concentration in all cases. The scalings appear qualitatively similar in all four panels, although, as for the dependence on redshift, quantitative details are still dependent on the lensing configuration and other halo parameters. In particular, we record a significant secondary dependence on redshift: the detectability of haloes at the lowest redshifts is most strongly boosted by concentration. The magnitude of this boost then decreases for redshifts approaching the redshift of the lens, where it has a minimum, to then increase again towards higher redshifts.

Notably, we find the dependence on concentration to be essentially exponential, al least when averaged over projected locations:

The top-right panel of Fig. 5 displays guiding lines for the exponent $\alpha=\{0.18,0.28,0.4\}$. The log-likelihood increase grows roughly by a factor between 1.5 to 2.5 for each additional $+1\sigma$ deviation from the mass-concentration relation. We will analyse the consequences of this on the expected number of haloes in Section 5.3.

Although our objective is not to provide absolute figures for the number of detectable haloes, Fig. 6 shows the cumulative distributions of detectable LOS haloes, $N_{\rm d}$, we obtain for our two lens configurations. Both panels are for a CDM universe and are calculated from our full maps of the log-likelihood increase, that is including both the full redshift dependence and the scatter in the mass concentration relation. We have used $SN_{\rm max}=50$. We stress that these figures cannot be directly applied to real data analysed with different techniques and featuring different lensing configurations. For definitiveness, we include all haloes with projected coordinates in a $4.5^{\prime\prime}\times 4.5^{\prime\prime}$ area at $z_{\rm h}\leq z_{\rm l}$, decreasing to $2.1^{\prime\prime}\times 2.1^{\prime\prime}$ at $z_{\rm h}=z_{\rm s}=1$, as displayed in Figs. 2 and 3.

Our two lensing configurations provide substantially different numbers of detectable haloes: a quad configuration appears less prone to modelling degeneracies and therefore more promising for the detection of perturbers. The number of expected detections is also a strong function of the imposed detection threshold. In our quad lens configuration, thresholds of $\Delta\mathscr{L}_{\rm th}=\{$10, 20, 35, 50$\}$ yield $N_{\rm d}=\{$0.2, 0.07, 0.02, 0.01$\}$ detections with $M_{\rm h}<10^{10}$ M_⊙ per lens, respectively.

The analysis of Section 3.2 also allows us to address systematically the dependence of the total number of detected haloes with the quality of the data, which in this work we have characterized by the value of $SN_{\rm max}$. Equation 14 allows us to equate changes in the value of $SN_{\rm max}$ with changes in the value of the log-likelihood threshold required for detection, $\Delta\mathscr{L}_{\rm th}$:

for any factor $\alpha$. We use this equivalence to focus on how the number of expected detections for a CDM universe would change for higher or lower values of the signal-to-noise, i.e. for longer or shorter exposure times.

Fig. 7 shows the number of expected detections of haloes of mass, $M_{\rm h}<10^{10}$ M_⊙ (left panel) and $M_{\rm h}<10^{9.5}$ M_⊙ (right panel) for varying values of $SN_{\rm max}$, normalized by the number of detections predicted for $SN_{\rm max}=50$. The figure displays the case of our ‘quad’ configuration; results for our ‘arcs’ configuration are similar, albeit with a more marked dependence on the $SN$ itself. Lines of different colour refer to different values of the log-likelihood threshold required for detectability. As expected, the number of detectable haloes is a rapidly increasing function of the signal-to-noise ratio. We find that, for a likelihood threshold of $\Delta\mathscr{L}_{\rm th}=20$, an increase in signal-to-noise ratio from 50 to 60 corresponds to a doubling in the number of expected detections $N_{\rm d}(M_{\rm h}<10^{10}$ M${}_{\rm\odot})$. The same increase in data quality makes the expected detections $N_{\rm d}(M_{\rm h}<10^{9.5}$ M${}_{\rm\odot})$ grow by a factor of three. As shown in the same figure, these factors are even larger if higher values of the log-likelihood ratio are required for detectability. For a value of $\Delta\mathscr{L}_{\rm th}=50$, analogous to what has been used in most previous studies, we find the corresponding figures to be 2.5 and 5 for haloes of $M_{\rm h}<10^{10}$ M_⊙ and $M_{\rm h}<10^{9.5}$ M_⊙ respectively. It should be noted that an increase in the maximum $SN$ ratio from 50 to 60 corresponds to an increase in the exposure time of a factor of $\approx 1.44$, which is therefore smaller than the corresponding gain in the number of detectable haloes.

We now examine the consequences of dropping the simplifying assumption of a tight relationship between halo mass and redshift. We isolate this effect by considering a population of CDM haloes assumed to lie on the Ludlow et al. (2016) mass-concentration relation, i.e. with reference to the previous Section, here we ignore the scatter in halo concentration. (We will consider this shortly.) Fig. 8 shows the redshift distribution of the population of detected haloes of mass, $M_{\rm h}<10^{10}$ M_⊙, we obtain when: for our two lensing configurations

• •

  using the mass shift proposed by Despali et al. (2018) and described in Section. 4.1, shown by a dashed line⁶⁶6We use the same mass shift, $\delta M(z)$, for all projected coordinates, $(x,y)$.;

• •

  using the full redshift dependence of our log-likelihood maps, shown by a solid line.

As expected, the two curves match at $z_{\rm h}=z_{\rm l}$, but we predict significantly fewer detections for foreground haloes, a reflection of the dependence on redshift of the log-likelihood increase described in Sect. 4.1. We also find that collapsing the redshift axis leads to an overestimate of detectable haloes also at $z_{\rm h}>z_{\rm l}$, though by a smaller factor.

The magnitude of the global overestimate varies with the lensing configuration. For definitiveness, we use a threshold of $\Delta\mathscr{L}_{\rm th}=20$ in Fig. 8. For the arcs configuration, the overestimate is a factor 1.95; for the quad configuration it is a factor of 1.63. We stress that these figures should not be used to ‘correct’ previous measurements of the number of detectable haloes and are meant only as an estimate of the magnitude of the effect.

We now consider the effect of accounting for scatter in the mass-concentration relation. We again focus on a population of CDM haloes, and compare the case in which all haloes are assumed to lie exactly on the median mass-concentration relation, to the case in which a lognormal scatter is included. For definitiveness, in both cases we use the full redshift dependence of the log-likelihood increase, $\Delta\mathscr{L}$, and, for simplicity, we restrict attention to our quad configuration. Results are analogous for our ‘arcs’ lensing morphology.

Fig. 9 shows the distribution of detectable haloes in the space of halo mass and shift relative to the median halo concentration, for the case in which the scatter in concentration is accounted for. The 2 dimensional histogram in the top panel is for an assumed threshold, $\Delta\mathscr{L}_{\rm th}=35$. The vertical dashed line shows the median mass-concentration relation, $\delta\log c=0$. It is clear that, thanks to the dependence on concentration of the log-likelihood increase described in Section  4.2, most detectable haloes are high-concentration haloes. This is quantified in the bottom panel of the figure, which collapses the mass axis to show the distribution of detected haloes over concentration shifts. For reference, the dashed line shows the Gaussian distribution of all cosmological haloes. Coloured lines show the distribution of the population of detectable haloes for different values of the log-likelihood threshold, $\Delta\mathscr{L}_{\rm th}$.

Haloes with high concentration achieve $\Delta\mathscr{L}>\Delta\mathscr{L}_{\rm th}$ more easily, so that higher thresholds for detectability correspond to increasingly concentrated populations of detectable haloes. For $\Delta\mathscr{L}_{\rm th}=50$ and our quad configuration, we find $\langle\delta\log c/\sigma_{\log c}\rangle=1.25$ when including all haloes of $M_{\rm h}<10^{10}$ M_⊙. The average concentration shift increases further for decreasing perturber masses, as shown by the coloured lines in the top panel. These represent the ‘mass-concentration relation of detectable haloes’. Lower mass haloes require stronger concentration boosts to achieve detectability, so that for $\Delta\mathscr{L}_{\rm th}=50$, $\langle\delta\log c/\sigma{\log c}\rangle=2.2$ for haloes of $M_{\rm h}\approx 10^{9.2}$ M_⊙.

The result is that the scatter in the mass-concentration relation boosts the number of expected detections. At low halo masses, for any fixed halo mass, there is a fraction of haloes with high enough concentration that become detectable if the scatter in the mass-concentration relation is accounted for, and which is lost when $c=c(M,z)$. This is quantified in Fig. 10, which shows the ratio between the cumulative number of detectable haloes, $N_{c}(<M_{\rm h})$, predicted when accounting for the scatter in the mass-concentration relation, to the corresponding number obtained when $c=c(M,z)$ for all haloes. Lines of different colours refer to different values of the detection threshold, $\Delta\mathscr{L}_{\rm th}$. For all thresholds, the ratios are a strong function of halo mass. The onset of the sharp rise identifies the halo mass that is not detectable at that threshold if $c=c(M,z)$, but for which detections are possible because of concentration effects. For $\Delta\mathscr{L}_{\rm th}=50$, even including all haloes with $M_{\rm h}<10^{10}$ M_⊙, concentration effects boost the number of detectable haloes by a factor of 2.75 for our ‘quad’ configuration. Since our ‘arcs’ configuration leads to lower values of the log-likelihood increase and fewer detections, the corresponding boost is a factor $\approx 26$, exemplifying, on the one hand, the importance of concentration effects, and, on the other, the need for estimates tailored to the specific lensing configuration.

Fig. 11 displays the same boost factor as a function of the number of expected detections, $N_{c}$ (which include the effect of concentration). Different values of the expected number of detections correspond to different values of the data $SN$ (or equivalently, different values of the log-likelihood threshold for detection). Once again we see that the exact figures depend on the lens configuration; for example, here the arcs configuration appears more sensitive to concentration effects than the quad.

In most previous estimates of the dependence of the number of detectable haloes on the properties of the DM model, particularly the mass of a WDM particle (or, equivalently, the cutoff mass in the mass function) it was assumed that all haloes lie exactly on the median mass-concentration relation of CDM haloes. Not only is this a poor approximation but it left the differences in the halo abundances alone as the statistic to differentiate amongst different models. Our results show that concentration is a crucial ingredient for halo detectability. Warmer WDM models make low-mass haloes that are progressively less concentrated, as quantified by equation 21. This makes the concentration parameter potentially helpful in boosting the spread among the expected numbers of detections in different DM models.

We test this in Fig. 12, in which we show, side by side, the cumulative number of expected detections, $N_{\rm d}(<M_{\rm h})$, for a range of WDM models, parameterized by their cutoff mass, $M_{\rm cut}$. The panel on the right displays results for the case in which concentration effects are neglected whereas the panel on the left includes both: (i) the scatter in the mass-concentration relation and (ii) the dependence of the median concentration on the DM model. For definitiveness, we show results pertaining to our ‘quad’ configuration, using a threshold for detection of $\Delta\mathscr{L}_{\rm th}=30$.

It is clear that concentration effects significantly enhance the dependence of the expected number of detections on the DM model. Reduced concentration values act together with reduced cosmological abundances to determine the expected number of detections. Once again, while the qualitative trend is clear and the magnitude of the effect significant, precise values are dependent on the lensing configuration and on the value of the threshold chosen for detection. We attempt to quantify how much concentration effects can actually sharpen substructure lensing constraints in Section 6.1.

In order to quantify the extent to which concentration effects can sharpen future substructure lensing constraints, we assume we have a set of strong lenses such that the total expected number of detections in CDM is

We ignore the contribution of satellite haloes, and assume that the figure above only includes haloes along the LOS, on which we have focussed in this work. While we are not able to tailor our analysis quantitatively to any specific set of observed strong lenses, this figure is representative of what is achievable with current HST data (Vegetti et al., 2018; Ritondale et al., 2019), and therefore provides a useful reference point. We use our maps of log-likelihood increase, $\Delta\mathscr{L}$, to calculate the number of expected detections in the same set of lenses for WDM models with different cutoff masses, $M_{\rm cut}$. We do so separately for our ‘quad’ and ‘arcs’ lensing configurations, requiring that the number of lenses in the two separate sets be such as to satisfy Equation 23 separately⁷⁷7 We require a detection threshold, $\Delta\mathscr{L}_{\rm th}=30$, for our quad configuration and $\Delta\mathscr{L}_{\rm th}=20$ for our arcs configuration, which, as we have shown, leads to systematically fewer detections per lens.. Furthermore, we set up predictions for both, the case in which concentration effects are accounted for and the case in which they are ignored. Then, we compare the inferences on the DM model that would result from actually detecting $i=(1,2,3)$ individual haloes, in the two different configurations. These are displayed in Fig. 13, which shows the likelihood ratio,

where $P(\cdot|m)$ is the Poisson probability distribution with mean, $m$, and $i$ is the number of actual detections. Inference resulting from predictions that ignore concentration effects are shown with dashed lines, while the likelihood ratio obtained when accounting for concentration effects is shown with solid lines. The vertical lines indicate the limits on the WDM cutoff mass corresponding to a likelihood ratio of $R=0.05$. The right and left columns correspond, respectively, to the two lensing configurations. Details are only marginally different and the magnitude of the effect is very similar in the two cases: at fixed number of expected haloes in a CDM universe, concentration effects make constraints on the WDM cutoff mass significantly more stringent. The suppression in the concentration of WDM haloes enhances the effect of lower cosmological halo abundances, allowing constraints that are about one order of magnitude more stringent in $M_{\rm cut}$.

Our results bring renewed confidence to the field of halo detection with strong lensing data, and boost confidence that meaningful constraints can be obtained from analysis of current optical data. A number of previous works have contributed to the realization that it is extremely challenging to use current optical HST data to obtain constraints on the cutoff of the DM halo mass function that are competitive with those obtained from the satellites of the Milky Way or measurements of the Lyman-$\alpha$ forest (see Enzi et al., 2021, and references therein). This is because, if halo abundances alone are used to differentiate between CDM and WDM, in order to be able to probe a WDM model with a cutoff mass, $M_{\rm cut}$ (the mass below which the abundance of haloes declines sharply), it is necessary to be sensitive to perturbers of that halo mass and below. However, evidence is mounting that detecting haloes of mass $M_{\rm h}\approx 10^{8.5}$ M_⊙ is extremely challenging with current lensing data, and therefore that it would be very difficult to place competitive constraints.

Concentration effects change this picture completely. For example, the limits displayed in Fig. 13 stem from detections of haloes of mass $M_{\rm h}>10^{9.5}$ M_⊙ – which is realistic with current HST data – but they can rule out values of the cutoff mass scale, $M_{\rm cut}\gtrsim 10^{8}$ M_⊙. This is a direct reflection of the effects of halo concentration, which, in contrast to halo abundances, first affects haloes of masses significantly above the cutoff mass itself. For this reason, concentration effects allow substructure lensing studies to probe WDM models with cutoff masses at least one order of magnitude below the lowest sensitivity mass scale. This implies that substructure lensing is, in fact, a much more sensitive probe of the identity of the DM than had been previously recognized.