Detecting the hidden population of low-mass haloes in strong lenses

TL;DR

This work demonstrates that a population of low-mass subhaloes, too small to be detected individually, can leave a measurable imprint on strongly lensed images when abundant, as in CDM. By combining a CNN-based substructure detector with sensitivity maps and forward modelling, the authors quantify how many such undetectable subhaloes would be inferred as detectable, and they observe a significant excess in detections in CDM that wanes with warmer DM (higher $M_{\mathrm{hm}}$). To account for this, they introduce a Gaussian-like pseudo-mass function characterized by $(n_{\mathrm{pop}}, m_{\mathrm{pop}}, \sigma_{\mathrm{pop}})$ and fit it to the data, finding the detectable population peaks around $M_{\mathrm{pop}}\sim 10^{9.8}\,M_\odot$, about two orders of magnitude below the individual-detection limit. They show that including this population improves dark matter constraints, potentially distinguishing DM models down to $M_{\mathrm{hm}} \sim 10^{7}\,M_\odot$, but they also reveal a strong degeneracy with angular structure in the lens galaxy, which can suppress the signal; elliptical multipole modelling may mitigate this. The results highlight the value of forward modelling and large surveys (Euclid, Roman, SKA) to exploit the collective signal of low-mass haloes in strong lensing for robust DM inferences.

Abstract

A generic prediction of particle dark matter theories is that a large population of dark matter substructures should reside inside the host haloes of galaxies. In gravitational imaging, strong gravitational lens observations are used to detect individual objects from this population, if they are large enough to perturb the strongly lensed images. We show here that low-mass haloes, below the individually detectable mass limit, have a detectable effect on the lensed images when in large numbers, which is the case in cold dark matter (CDM). We find that, in CDM, this population causes an excess of 40 per cent in the number of detected subhaloes for HST-like strong lens observations. We propose a pseudo-mass function to describe this population, and fit for its parameters from the detection data. We find that it mostly consists of objects two orders of magnitude in mass below the detection limit of individual objects. We show that including this modification, so that the effect of the population is correctly predicted, can improve the available constraints on dark matter from strong lens observations. We repeat our experiments using models that contain varying amounts of angular structure in the lens galaxy. We find that these multipole perturbations are degenerate with the population signal. This further highlights the need for better understanding of the angular mass structure of lens galaxies, so that the maximum information can be extracted from strong lens observations for dark matter inference.

Figures (7)

• Figure 1: Subhalo mass function for the lenses used in this work, with different values of $M_\mathrm{hm}$ indicated by colour. The lines give the median value and the shaded areas show the $1\sigma$ range over the $100$ lenses. Upper frame: the number of subhaloes per mass bin with bins of size 0.1 dex in $M_\mathrm{max}$. Lower frame: The cumulative distribution going from left to right, i.e., the total number of subhaloes in a lens with mass below the $M_\mathrm{max}$$x$-axis value. For example, in CDM, a typical lens used in this work has between $300$ and $1100$ subhaloes within $2\theta_\mathrm{E}$ in the chosen mass range, depending on the enclosed mass of the lens, $M_{2\theta_\mathrm{E}}$.
• Figure 2: Cumulative fraction of sensitivities for pixels inside $2\theta_\mathrm{E}$ for the $100$ mock lenses used in this work, for different multipole models.
• Figure 3: Method to test the detectability of populations of individually undetectable substructures. A subhalo mass function, effectively the dark matter model, is chosen first, parametrised by $M_\mathrm{hm}$ and $f_\mathrm{sub}$. From this populations of subhaloes are drawn to produce images. The images are run through a subhalo detector CNN, which is the same used to create the sensitivity map for each lens. The sensitivity map and the subhalo mass function give $\mu_\mathrm{sub}$, the mean number of detectable objects, and in turn $p_\mathrm{det}$, the probability that the CNN makes a detection. The actual number of detections made by the CNN is $f_\mathrm{det}$. Comparing these quantities is the basis for the results in this paper.
• Figure 4: Frequency of detections predicted by the sensitivity map, and achieved by the neural network as a function of the cutoff mass $m_\mathrm{cutoff}$. For each realisation of each lens, $m_\mathrm{cutoff}$ is drawn randomly and binned to produce this figure. The solid lines show the mean frequency per $m_\mathrm{cutoff}$ bin and the shaded area the standard deviation within each bin. The standard deviation is large relative to the number of detections because the number of detections in general is small.
• Figure 5: Properties of the subhaloes in the detectable population for different values of $M_\mathrm{hm}$. Upper frame: the modified mass function in \ref{['eq:mupop-integral']} using the best fit values for our HST strong lenses. Lower frame: the ratio of the individual masses of the detectable population objects to the pixel sensitivities in our data, randomly drawn.