Dark matter mass function from lensing

Constraining sterile neutrino cosmologies with strong gravitational lensing observations at redshift 0.2.

Abstract

We use the observed amount of subhaloes and line-of-sight dark-matter haloes in a sample of eleven gravitational lens systems from the Sloan Lens ACS Survey to constrain the free-streaming properties of the dark matter particles. In particular, we combine the detection of a small-mass dark-matter halo by Vegetti et al. (2010) with the non-detections by Vegetti et al. (2014) and compare the derived subhalo and halo mass functions with expectations from Cold Dark Matter (CDM) and sterile neutrino models. We constrain the half-mode mass, i.e. the mass scale at which the linear matter power-spectrum is reduced by 50 percent relatively to the CDM model, to be (equivalent thermal relic mass keV) at the 2- level. This excludes sterile neutrino models with neutrino masses of keV and lepton asymmetry and all values of for keV. Our constraints are weaker than currently provided by the number of Milky Way satellites, observations of the 3.5 keV X-ray line and the Lyman- forest. However, they are more robust than the former as they are less affected by baryonic processes. Moreover, unlike the latter they are not affected by assumptions on the thermal histories for the inter-galactic medium. Under the assumption of a CDM dominated universe, the expected number of detectable line-of-sight haloes is , in agreement with the single detection reported for this sample.

keywords:
cosmology: dark matter – galaxies: haloes – galaxies: structure – gravitational lensing: strong
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1 Introduction

The widespread presence of dark matter and its dominant contribution to the matter content of the Universe is supported by a significant number of astrophysical observations. However, the nature of dark matter is to this day still an unsolved problem. In the standard CDM cosmological model, dark matter is assumed to be cold, i.e. to have had negligible thermal velocity at early times, and possible candidates include the neutralino and the axions (e.g. Bottino et al. 2003; Ringwald 2016). While this model is very successful at reproducing large scale observations, some discrepancies arise at galactic and sub-galactic scales (e.g. Moore 1994; Klypin et al. 1999; Kuzio de Naray et al. 2008; de Blok 2010; Walker & Peñarrubia 2011; Amorisco & Evans 2012; Boylan-Kolchin et al. 2012).

In relation to these issues, alternative models have been considered (e.g. Lovell et al. 2014; Vogelsberger et al. 2016; Robles et al. 2017; Iršič et al. 2017b), among which are the so-called Warm Dark Matter models, where the corresponding particles have non-negligible thermal velocities. In particular, sterile neutrinos with masses of a few keV have been shown to be an interesting candidate for warm dark matter particles (e.g Boyarsky et al. 2012, and references therein). Moreover, the three sterile neutrino model, also known as the Neutrino Minimal Standard Model (MSM), can also explain baryogenesis processes and neutrino oscillations (Boyarsky et al. 2012, and references therein). Recently, sterile neutrinos have gained renewed attention following the apparent detections of a 3.5 keV X-ray line in several galaxy clusters as well as M31 and at the centre of the Milky Way (Boyarsky et al. 2014; Bulbul et al. 2014; Boyarsky et al. 2015), for a review see e.g. Iakubovskyi (2014). However, sterile neutrino decay as a source of the line remains contentious (e.g. Anderson et al. 2015; Gu et al. 2015; Jeltema & Profumo 2016). Recent results from the Hitomi collaboration (Aharonian et al. 2017) have rule out atomic transition as the origin of the 3.5 keV line, but a dark matter interpretation is still possible due to the different sensitivity of the Hitomi instrument to narrow (atomic) and broad lines. The 11- detection of the 3.5 keV signal by NuStar (Neronov et al. 2016) from a quiescent region of the sky and subsequent confirmation of this result with Chandra (Cappelluti et al. 2017) suggest that the line is not a statistical fluctuation nor an artefact of the XMM-Newton’s instrumentation. It is therefore important to find complimentary probes of these models as well as supporting future X-ray observatory missions such as XARM and ATHENA that will provide still stronger X-ray constraints.

Due to their appreciable thermal velocities at early times, warm dark matter particles free-stream out of density perturbations and are responsible therefore for a cutoff in the linear power-spectrum and by extension in the formation of structures at small scales. The exact scale at which the suppression happens is strongly related to the dark matter particle production mechanism, for example for 7 keV sterile neutrinos with a lepton asymmetries (define in Section 3) much smaller or larger than this happens at the scale of progressively larger dwarf galaxies (Lovell et al. 2016), while in a CDM model the suppression happens at much smaller scales. Quantifying the amount of small mass structures is therefore a key approach to distinguish among different dark matter models. For example, Kennedy et al. (2014) and Lovell et al. (2016) have used semi-analytic models to compare the abundance of luminous Milky Way satellites predicted by sterile neutrino dark matter models with the observed number of dwarf spheroidals, and derived constraints on the sterile neutrino mass and the lepton asymmetry . However, since luminous satellites are a biased tracer of the underlying mass distribution, the interpretation of these results is complicated by assumptions made on baryonic processes. Moreover, the not so precisely known mass of the Milky Way is also a source of degeneracy in the number of expected satellites.

Recently, measurements of the Lyman- forest from high redshift QSO spectra have argued that all values of for sterile neutrino mass  keV are ruled out. However, these results are highly sensitive to assumptions made on the thermal histories for the inter-galactic medium (Baur et al. 2017). It has been demonstrated by Garzilli et al. (2017) that if one does not make any assumption about the (experimentally unknown) state of the intergalactic medium at , these bounds get greatly relaxed. As a result, the Lyman- bounds are less stringent and in particular the 7 keV sterile neutrino model, consistent with the 3.5 keV line detection, are within the allowed range (Baur et al. 2017).

Strong gravitational lensing, being sensitive to gravity, provides a more direct method to quantify the dark matter distribution at sub-galactic scales and in a way which is less affected by baryonic processes (however see Despali & Vegetti 2017). The idea to constrain the amount of small mass structure in gravitational lens galaxies via their influence on the relative fluxes of multiply-imaged quasars was first proposed by Mao & Schneider (1998). Subsequently, Dalal & Kochanek (2002) were the first to use a relatively small sample of seven gravitationally lensed quasars to derive a statistical constraint on the projected mass fraction in substructure that is larger than, but consistent with, predictions from CDM numerical simulations. At the same time, analyses of CDM numerical simulations (Xu et al. 2009, 2015) have shown that the predicted substructure population is not sufficient to reproduce the level of flux-ratio anomalies currently observed, and that complex baryonic structures in the lens galaxies are a likely explanation (see also McKean et al. 2007; Gilman et al. 2017a; Hsueh et al. 2016; Hsueh et al. 2017a; Hsueh et al. 2017b). In the future, narrow-line observations for the large number of quadruply-imaged quasars, that are expected to be discovered by large scale surveys, will allow one to investigate these issues further and potentially provide new interesting constraints on the properties of dark matter (Nierenberg et al. 2017; Gilman et al. 2017b).

In the meantime, Koopmans (2005) and Vegetti & Koopmans (2009) have introduced the gravitational imaging technique that uses magnified arcs and Einstein rings to detect and measure the mass of individual subhaloes. Unlike other methods this technique treats substructures not as analytical mass clumps but as pixellated linear potential corrections to the main lensing potential. As such, it does not require any prior assumption on the number of substructures nor on their density profile and redshift. Moreover, it can easily distinguish a substructure from a smooth, but complex, mass distribution (Barnabè et al. 2009; Vegetti et al. 2014) and is therefore less prone to false detections. However, unlike studies of flux-ratio anomalies it relies on increasingly high resolution data for the detection of the smaller mass haloes. Using the gravitational imaging technique Vegetti et al. (2010) and Vegetti et al. (2012) have reported the detection of two small mass substructures, while Vegetti et al. (2014) have used a sample of eleven lenses to derive statistical constraints on the CDM substructure mass function, which are consistent with CDM predictions.

At the other end of the spectrum, several methods have been developed to detect mass substructures not individually, but via their collective gravitational effect (Fadely & Keeton 2012; Hezaveh et al. 2016; Cyr-Racine et al. 2016; Birrer et al. 2017; Daylan et al. 2017; Chatterjee & Koopmans 2018). Understanding the biases introduced by complex mass distributions of the main lensing potential, as highlighted by Hsueh et al. (2016); Hsueh et al. (2017a); Hsueh et al. (2017b), Gilman et al. (2017a) and Vegetti et al. (2014), will be key for a practical application of these techniques.

Most if not all analyses so far have considered subhaloes within lens galaxies as the only source of perturbation to the lensed images. However, as demonstrated by Metcalf (2005) and Despali et al. (2017), the contribution from line-of-sight haloes can be significant and even dominant, depending on the lens and source redshifts. Moreover, line-of-sight haloes are less affected by feedback and accretion processes and can therefore be used to get tighter and cleaner constraints on the properties of dark matter.

In this paper we reanalyse the sample by Vegetti et al. (2014), to include the contribution of both subhaloes and line-of-sight haloes to the total number of detectable objects. In particular, we combine the non-detection by Vegetti et al. (2014) and the detection by Vegetti et al. (2010) to derive statistical constraints on the halo and subhalo mass functions. As the key parameters that are set by the dark matter model are common to both mass functions, the inclusion of line-of-sight haloes represent a significant increase in the constraining power. We then compare our findings with predictions from sterile neutrino models. This paper is organised as follows. The data is introduced in Section 2. In Section 3 we present the sterile neutrino dark matter model under consideration. In Sections 4 and 5 we provide all the definitions of mass for subhaloes and line-of-sight haloes as well as the mass function expressions adopted for both populations. In Sections 6 and 7 we introduce the complete likelihood function, the model parameters and the corresponding priors. Finally, our results are presented and discussed in Sections 8 and 9, respectively.

2 Data

The analysis presented in this paper is based on a sample of eleven gravitational lens systems taken from the SLACS survey (Bolton et al. 2006; Auger et al. 2010, and references therein). A summary of the systems with the lens and source redshift can be found in Table 1. This sample has been previously modelled by Vegetti et al. (2010) and Vegetti et al. (2014) with the Bayesian grid-based adaptive code by Vegetti & Koopmans (2009). To summarise, Vegetti et al. (2010) have reported the detection of a dark-matter dominated perturber which was then interpreted as a substructure with a total mass of (under the assumption of a Pseudo-Jaffe profile located on the plane of the host lens, hereafter PJ, and corresponding roughly to ) in the lens system SDSSJ0946+1006, while Vegetti et al. (2014) have shown that no substructure is required for all the other lenses in the sample. For each lens, they have also derived the so called sensitivity function, that is the smallest detectable substructure mass at each relevant position on the image plane. Here, we make use of their results as an input to our analysis and refer to their papers for more details on the data and how these results were obtained. In particular, we relax the assumption that all perturbers are substructures and allow for the possibility of a contribution from line-of-sight haloes, for the statistical interpretation of both detections and non-detections.

Name (SDSS)

0.280 0.982

0.322 0.581

0.222 0.609

0.240 0.470

0.237 0.531

0.191 0.696

0.215 0.481

0.285 0.575

0.208 0.524

0.137 0.713

0.228 0.463
Table 1: The list of the gravitational lens systems considered in this paper, with the lens and source redshifts.

3 Dark matter model

The dark matter particle considered in this paper is the resonantly produced sterile neutrino (Shi & Fuller 1999; Dolgov & Hansen 2002; Asaka & Shaposhnikov 2005; Laine & Shaposhnikov 2008; Boyarsky et al. 2009). In this section we present a summary of the model and its application within astronomy. For an in-depth description of both, the interested reader is referred to Lovell et al. (2016).

The sterile neutrino is produced at high energies, around the gluon-hadron transition ( MeV). It originates from the oscillation of active neutrinos, and the probability of this oscillation is enhanced in the presence of a lepton asymmetry, i.e. an excess of leptons over anti-leptons. It is parametrized using the parameter: , where is the number density of leptons, the number density of anti-leptons, and the entropy density. The ability to enhance the production of sterile neutrinos via lepton asymmetry reduces the required mixing angle between sterile neutrinos and active neutrinos in order to obtain the measured universal dark matter density, and thus evades bounds on sterile neutrino parameters derived from X-ray observations.

A second important consequence of lepton asymmetry-induced production is the effect on the sterile neutrino velocities. This is because the enhancement in the production rate due to the lepton asymmetry scales with the momentum of the neutrinos.

Small asymmetries (, depending on the sterile neutrino mass) lead to excess production of low momentum sterile neutrinos, and are therefore ‘cooler’ distributions than the no asymmetry case. At higher asymmetries, the enhancement in production is stronger, and extends to high momentum sterile neutrinos. For the maximal lepton asymmetry () all momenta receive an equal boost compared to the zero asymmetry scenario and the resulting momentum distribution is therefore almost identical to that of no lepton asymmetry. Given that the mass of the sterile neutrino also plays a role in the momentum distribution, with less massive particles exhibiting higher velocities, the free-streaming scale, and thus the halo mass function, is specified by two parameters: sterile neutrino mass , and lepton asymmetry .

The momentum distribution functions are calculated using the methods of Laine & Shaposhnikov (2008), and are used as inputs for Boltzmann solver codes to calculate the linear matter power spectrum, . In our case, we use matter power spectra that have been computed using a modified version of camb (Lewis et al. 2000), and examples of these are displayed in Fig. 4 of Lovell et al. (2016). These curves exhibit an array of cutoff slopes and positions. For this study we characterise each curve by the transfer function of the sterile neutrino linear matter power-spectrum compared to its CDM counterpart, which is given by , and specifically the wavenumber at which the transfer function has the value 0.5. This is known as the half-mode wavenumber, . Its influence on the halo mass function is parametrized through a dependent parameter, the half-mode mass , which takes the following form:

(1)

where is the mean matter density of the Universe. In practice, is a function of both and , thus placing constraints on the former will lead to limits on the sterile neutrino parameters. In some cases does not fully encapsulate the fine details of the matter power spectra, such as the shallower slopes of some models: we comment below on how a more accurate parametrization of these curves would likely change the results.

4 Mass definition

We assume the true mass of line-of-sight haloes and substructures, , to be the virial mass of a Navarro et al. (1997, hereafter NFW) profile with a Duffy et al. (2008) concentration-mass relation. While this is a good description for the former it is only an approximation for the latter. By fitting the deflection angle of simulated subhaloes with NFW profiles, Despali et al. (2017) have shown that at fixed virial mass these have a larger concentration that is mildly dependent on the subhalo distance from the host centre. However, they have also shown that assuming a constant Duffy et al. (2008) mass-concentration relation plays a secondary effect and leads to an error on the mass which is as low as 5 per cent for subhaloes with masses of and as large as 20 percent for masses of ; the error on the expected number of substructure is of the order of 10 per cent. We also assume that the concentration does not change with the dark matter model. As shown by Ludlow et al. (2016) the concentration of WDM haloes differs from the CDM case only at low masses, where the number of structures is also strongly suppressed. Again, Despali et al. (2017) have shown that this assumption is of secondary importance. We refer to their paper for a more detailed discussion on the matter.

We assume that the measured mass of perturbers i.e. the mass derived from the gravitational lens modelling of the data, and the lowest detectable mass are PJ total masses located on the plane of the host lens. We then use the mass-redshift relation derived by Despali et al. (2017) to statistically relate the true and measured masses to each other as described in Section 6.2. This follows from the fact that while the detections were made in a pixellated model independent way, they have been then characterised in terms of PJ substructures. The sensitivity function used in this paper and derived by Vegetti et al. (2014) has also been obtained under the assumption of PJ substructures.

5 Dark matter mass function

While Vegetti et al. (2014) have assumed a CDM model and have focused only on substructures, here, we allow for a more general dark-matter model that includes the effect of particle free-streaming and also includes the contribution from small-mass dark matter haloes located along the line-of-sight. Following Lovell et al. (2014) and Schneider et al. (2012) we parametrize the (sub)structure and the halo mass function as follows

(2)

where the second factor expresses the effect of particle free streaming. A slightly different parametrization was found to be a better fit for the subhalo mass function by Lovell et al. (2014):

(3)

In this paper, we make use of both, with the first one leading to a larger number of WDM subhaloes and hence being more conservative. We assume the CDM mass function for substructures to be

(4)

For the line-of-sight halo CDM mass function we adopt the expression by Sheth & Tormen (1999), with the best fit parameters optimised for the Planck cosmology calculated by Despali et al. (2016). As discussed in the previous section for both populations the mass function is expressed in terms of a NFW virial mass of fixed concentration-mass relation. Scatter in this relation is taken into account for in a statistical sense as described in Section 6.2.

6 Likelihood function

In this section, we derive an expression for the likelihood of detecting perturbers (substructures plus line-of-sight haloes) with observed masses in the mass range at the projected positions range , with , and with no detection in all other mass and position ranges. Assuming a Poisson distribution we follow Marshall et al. (1983) and write the log-likelihood for a single lens galaxy as

(5)

Here, are the model free parameters that define the (sub)structure mass function, and are explicitly introduced in Section 7. and are the expected number of substructures and line-of-sight haloes, respectively, in the mass range and projected position range . A derivation of them is given in Section 6.1. All masses in the above equations are intended as measured ones and are defined according to Section 4. These are integrated between the lowest detectable mass at each projected position, , and . Both limits are intended as PJ total masses. It should be stressed that here is the position on the plane of the main lens where the perturber is detected, for substructures this corresponds with the projected position of the perturber, within a relatively small error. For line-of-sight haloes, due to the recursive nature of the lens equation in the multiple lens plane configuration, and are related to each other via the deflection angle of the main lens at .

6.1 Sensitivity function

For each of the considered lenses, the substructure sensitivity function, that is the lowest detectable mass as a function of position on the image plane , was derived by Vegetti et al. (2014). Briefly, this was calculated by identifying the smallest PJ total mass on the plane of the host lens responsible for a change of the Bayesian evidence by , relatively to a model with no substructure. Under the assumption of Gaussian noise this corresponds to a 10- detection. As a reference, the detections reported by Vegetti et al. (2010) and Vegetti et al. (2012), were respectively at the 16- and 12- limit. Lowering the threshold for this sample would give a higher statistical weight to the non-detections, but it would not increase the number of detections. As demonstrated by Vegetti et al. (2014) and Despali et al. (2017), assuming a constant sensitivity across the image plane can have a significant impact on the expected number of detectable objects and therefore on the inferred mass function parameters. Therefore, the sensitivity function was derived for each pixel on the image plane with a signal-to-noise ratio (SNR) larger than three.

6.2 Expectation values

We now provide explicit expressions for the expectation values of substructures and line-of-sight haloes. In the following and are respectively the true and measured masses of perturbers as defined in Section 4, and and are respectively the true and measured projected positions at which they are detected, relative to the centre of the main lens galaxy. Below, the integration limits are intended between and for the true NFW virial mass and within the SNR region for the true projected position. For the redshift of line-of-sight haloes we integrate between the observer and the source, but exclude the region within the virial radius of the main lens, i.e. .

The substructure expectation value is given by

(6)

where

(7)

is such that is the probability of finding one substructure in the mass range .

Introducing the projected dark matter mass of the main lens within the region of interest and the corresponding projected dark matter mass fraction in substructure we can express as follows

(8)

It should be stressed that in this paper is defined as the projected dark matter fraction in substructure with masses between and and within the considered region. This definition differs from the one by Vegetti et al. (2014), as it uses a different definition of substructure mass, as well as a different parametrization of the substructure mass function.

Similarly, the expectation value for line-of-sight haloes is given by

(9)

where is expressed as

(10)

is the probability of finding one line-of-sight halo in the mass range and in the redshift range , and is related to the halo mass function as follows

(11)

In both expressions (8) and (9), is a vector which is equal to one for detectable perturbers and zero otherwise, so that expresses the effect of the sensitivity function and is defined as

(12)

As show by Despali et al. (2017), the measurement error on the perturber positions is relatively small (i.e. within 2.5 times the PSF FWHM), hence for simplicity we assume and following the results by Xu et al. (2015) and Despali & Vegetti (2017) we assume a uniform probability for .

By comparing the lensing effect of PJ perturbers at the redshift of the main lens with those of NFW line-of-sight haloes and subhaloes, Despali et al. (2017) have derived a mass-redshift relation that allows one to map one population into the other; following their results and referring to the mass-redshift relation as we define as follows

(13)

Essentially, for a line-of-sight halo of NFW virial mass located at redshift , returns the PJ total mass located on the plane of the main lens with the most similar gravitational lensing effects. Here, we do not use the mean relation reported by Despali et al. (2017), but we derive a relation, the main parameters of which depend on the main lens model, for each of the lenses in our sample. The intrinsic scatter of the mass-redshift relation is also not the same as the one reported by Despali et al. (2017), but it is a sum in quadrature of the error on the measured mass and the uncertainty related to the scatter/different choice of the concentration-mass relation. The scatter does not account, therefore, for the measurement error on the main lens parameters. We have found this to be smaller than the scatter due to the perturber redshift degeneracy and mass density profile (Despali et al. 2017) once incorrect modelling of the main deflector has been ruled out or addressed.

7 Prior and posterior probabilities

The target parameters of the model , include the slope of the substructure mass function (equation 4), the mean projected mass fraction in substructures with virial masses between and , (equation 8), the half-mode mass and the slope (equations 2 and 4 ). Prior probabilities on these parameters are chosen as follows:

  1. is drawn from a Normal prior density distribution centred on and with a standard deviation of 0.2. This is in agreement with dark-matter-only and hydrodynamical numerical simulations (Despali & Vegetti 2017);

  2. for the normalization of the substructure mass function, we assume a uniform prior density distribution proportional to between 0 and 0.2;

  3. we assume to be normally distributed around (Lovell et al. 2017) with a standard deviation of 0.1;

  4. we adopt a prior distribution proportional to between and for the half-mode mass.

We derive the posterior probability for the mass function parameters from the likelihood function (equation 5) assuming the detection of perturbers from one lens galaxy to another to be independent events. We explore the posterior parameter space within the prior volume using MultiNest (Feroz & Hobson 2008), results are presented in the following section.

Figure 1: Posterior probability distribution for all model parameters, derived by taking into account the contribution from both substructures and line-of-sight haloes. The substructure mass function is parametrized according to equ. (2) and (3), for the black and red curves respectively. Contours corresponds to the 1- and 2- levels.
Figure 2: Left: half-mode mass versus lepton asymmetry for different values of the neutrino mass (coloured lines), 95 and 68 percent upper limit on the half-mode mass for our two parametrizations (equations 2 and 3) of the subhalo mass function (dashed and dashed-dotted black lines). Right: 95 percent exclusion region in the plane. The green shaded region is excluded from a non-observations of X-ray decay from M31 (Watson et al. 2012; Horiuchi et al. 2014). The purple and pink regions are excluded by the observed number of Milky Way satellites for two different feedback models from Lovell et al. (2017). The grey shaded region is in strong tension with Lyman- flux observations as described in the text. The yellow shaded region is excluded by the number of observed and non-observed mass perturbers in the sample of gravitational lens systems considered in this paper. We mark the position of the sterile neutrino model that explains the 3.55 keV line with an error bar.

8 Results

Our constraints on the model parameters are summarised in Table 2, where we report the maximum a posteriori (MAP) values, the 68 and the 95 percent confidence levels. For the substructure mass fraction and the half-mode mass, we only report the 68 and 95 percent upper limits, being their posterior distribution skewed towards high values. Posterior probability distributions for all parameters are shown in Fig. 1.

First of all, we note that the MAP value for the substructure fraction derived here is larger than the value reported by Vegetti et al. (2014) for the same sample of lenses. This difference can be attributed to a different definition of the substructure mass and mass limits as well as a different shape of the substructure mass function, which introduces a degeneracy between and .

We find a MAP value for the half-mode mass of . At the 95 percent confidence level (CL), we constrain the half-mode mass to be . This limit is independent of the subhalo mass function parametrizations, expressed by equations (2) and (3). Sterile neutrinos are a two parameter dark matter model, where combinations of neutrino masses and lepton asymmetry in the early Universe determine the particle momentum distribution and its colder or warmer behaviour. In the left-hand panel of Fig. 2, we plot the theoretical half-mode mass for different values of the neutrino mass and lepton asymmetry. Our 2- upper limit excludes sterile neutrino masses of keV and and models with keV for all values of . We have also derived a relationship between the mass of thermal relic particle and the half-mode mass using the results of Viel et al. (2005), which leads to a lower limit of keV at the 2- level.

We have tested the effect of our prior choices on the inferred upper limit of the half-mode mass: for example, a prior distribution uniform in logarithmic space between and leads to a posterior distribution where larger values of are statistically down-weighted, relatively to the results presented in Fig. 1. However, this does not significantly affect our constraints on the 2- upper limit, with .

We find that our single detection is responsible for setting the upper limit on the half-mode mass, as the dark matter has to be cool enough to lead to a non-negligible detection probability for such a massive object. For example, excluding this detection leads to a higher upper 2- limit of . Instead, if we had made a second equally massive detection we would have inferred a lower 2- upper limit of . This indicates, as expected, that a significantly larger number of detections is required to shift the upper constraints towards cooler model. On the other hand the non-detections provide constraints on the half-mode mass lower limit and a significant number of these, especially at the low-mass end, would be required to exclude CDM.

Taking into account only the contribution from the substructures we obtain instead at the 95 percent CL. This demonstrates how the contribution from the line-of-sight provides some extra constraining power on the properties of dark matter. However, for the sample of lenses considered in this paper, the relative redshift between the lens galaxies () and the background sources () is such that the size of the cosmological volume and the number of line-of-sight haloes are relatively small: for a CDM cosmology, the number density of detectable line-of-sight haloes is expected to be about twice that of subhaloes (Despali et al. 2017). Higher redshift lenses and sources are key to significantly increase the contribution from line-of-sight haloes.

In the right-hand panel of Fig. 2 we compare our constraints with those derived from the observed satellites in the Milky Way (Lovell et al. 2016), X-ray decay searches from M31 (Watson et al. 2012; Horiuchi et al. 2014) and from Lyman- forest constraints. The latter measure the one-dimensional (1D) matter power spectrum of Lyman- flux in QSO spectra. Comparing the limits from these studies with our results is complicated because their constraints are calculated using thermal relic matter power spectra, and a proper analysis requires simulations of structure formation to model the non-linear evolution of the power spectrum such as the flow from large scales to small scales. Another uncertainty is the thermal history around , where Iršič et al. (2017a) find their preferred power law prior requires  keV at the 95 per cent CL, whereas a freer prior on the thermal history relaxes the bound to  keV at 95 per cent CL.

We therefore take the following approach. We draw an exclusion region based on all sterile neutrino models that have a 1D power spectrum with less power at any point in the wavenumber range than the 3.5 keV thermal relic, where is the range of wavenumbers used in the analysis of Iršič et al. (2017a). When combined with the X-ray limit, this limit rules out all but a sliver of parameter space, which lies in the range  keV, ; the less conservative  keV limit instead rules out all  keV. Finally, we note that our method may rule out models in which the power transfer from large scales to small scales is stronger than for the thermal relic, therefore dedicated simulations of these sterile neutrino models will be required to confirm or correct this simple model (see also Baur et al. 2016).

Our 95 percent CL exclusion regions are significantly smaller than those derived from both the satellite counts and the Lyman- forest, and would potentially be weaker still if the shallower slopes of sterile neutrino power spectra were taken into account fully. However, they allow us to exclude models with neutrino masses smaller than those excluded by the X-ray line observations and are more robust than those from the Milky Way satellite counts, as they are less affected by feedback processes. They also have the potential to measure free-streaming length should enough lenses of sufficient quality be analysed.

In the near future, we will focus on samples of strong gravitational lens systems with higher sensitivities and with higher lens/source redshifts. This will allow us to improve our results thanks to the larger number of expected substructures and line-of-sight haloes (Despali et al. 2017).

Finally, we compute the number of line-of-sight haloes that we should have been able to detect with our sample in a CDM Universe. Using equation (9) we obtain an expectation value , in agreement with the single detection reported for this sample. Depending on the implementation of feedback processes the number of detectable CDM substructure should be about half the number of line-of-sight haloes for the considered combination of lens and source redshifts (Despali et al. 2017).

Parameter MAP
1.88 -0.17|+0.16 -0.33|+0.33
-1.27 -0.12|+0.07 -0.19|+0.14
0.017 0.094 0.18
9.69 10.75 10.96
Table 2: For each parameter we report the maximum a posteriori value as well as the 68 and 95 percent CL. For the substructure mass fraction and the half-mode mass, we report the 68 and 95 percent upper limits.

9 Conclusions

Sterile neutrinos with masses of a few keV have been shown to be a viable candidate for dark matter particles. Depending on the level of lepton asymmetry in the early Universe, these can behave as warmer or cooler models and introduce a cutoff in the primordial linear matter power spectrum. As a consequence, structure formation within sterile neutrino cosmologies is suppressed at a scale that is a function of the neutrino mass and the lepton asymmetry. In particular, 7 keV sterile neutrinos, have been shown to be a possible explanation for the apparent 3.5 keV X-ray line detection in several galaxy clusters as well as M31 and at the centre of the Milky Way. Recently, Lovell et al. (2016) have used semi-analytical models with different sterile neutrino models to quantify the predicted number of luminous satellites in the Milky Way halo, and concluded that the 7 keV sterile neutrinos are in good agreement with current observations. However, constraints on dark matter from the Milky Way satellites are limited by our knowledge of the Milky Way halo mass and the details of feedback processes.

Strong gravitational lensing allows one to gain new insight into the properties of dark matter by quantifying the number of substructure in gravitational lens galaxies and small-mass haloes along their line-of-sight, in a way that is less affected by galaxy formation models. In this paper, we have used the non-detections of mass perturbers obtained by Vegetti et al. (2014) in combination with the detection in the lens system SDSS J0946+1006 by Vegetti et al. (2010) to derive constraints on the two-parameter model of sterile neutrinos. At the 2 level we have excluded models with keV and values of lepton asymmetry and all values of for keV.

These constraints are currently less stringent than those provided by other, more established methods. We have made explicit comparisons to the Milky Way satellite counts (Polisensky & Ricotti 2011; Lovell et al. 2014; Kennedy et al. 2014; Lovell et al. 2016; Cherry & Horiuchi 2017), which almost always prefer a thermal relic mass  keV and, in the case of Lovell et al. (2016), set constraints on sterile neutrinos of mass  keV when applying their fiducial galaxy formation model; but their models with weaker reionisation feedback can however evade the constraints. There are also uncertainties in the abundance and spatial distribution of the observed Milky Way satellites, in the halo mass and, the issue that this analysis can only be performed for the Milky Way and Andromeda systems.

Still stronger constraints are obtained from the analysis of the matter power spectrum of perturbations in the Lyman- forest (Schneider 2016). In particular, high redshift QSO spectra from XQ-100, HIRES and MIKE have been argued to rule out all values of for sterile neutrino mass  keV; however, thermal histories for the inter-galactic medium return weaker limits (Baur et al. 2017). X-ray constraints instead set an upper limit on the sterile neutrino mass and a corresponding lower limit on . The strongest limits, particularly for the 3.55 keV line, have arguably been obtained using deep XMM-Newton observations of the Draco dwarf spheroidal galaxy, but the long integration times required for a detection, coupled to uncertainties in how best to conduct those observations, reduce the possibility that further studies can improve the situation in the new future (c.f. Jeltema & Profumo 2016; Ruchayskiy et al. 2016). Compared to all of the above methods, lensing studies benefit from the fact that they are sensitive to the total perturber mass and they are less affected by baryonic processes and should be therefore more robust.

In agreement with Despali et al. (2017) we have found that including the contribution of line-of-sight haloes can provide a boost to the constraining power of gravitational lensing. However, due to the low redshift of the current sources and lenses, this effect is relatively small for the gravitational lens systems in our sample. In the near future, we will provide more stringent constraints on sterile neutrino models and the properties of dark matter in general, by using higher resolution data and/or higher redshift samples. Finally, we have computed the expected number of detectable line-of-sight haloes in our sample for a CDM model and found it to be consistent with the single detection reported by Vegetti et al. (2014).

Acknowledgements

The authors are grateful to A. Boyarsky, O. Ruchayskiy and S. D. M. White for useful comments and discussions. M. R. L. is supported by a COFUND/Durham Junior Research Fellowship under EU grant 609412. M. R. L. acknowledges support by a Grant of Excellence from the Icelandic Research Fund (grant number 173929051).

Footnotes

  1. pubyear: 2015
  2. pagerange: Constraining sterile neutrino cosmologies with strong gravitational lensing observations at redshift 0.2.Constraining sterile neutrino cosmologies with strong gravitational lensing observations at redshift 0.2.

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