# Projection effects in the strong lensing study of subhaloes

Ran Li1, Carlos S. Frenk2, Shaun Cole2, Qiao Wang1, Liang Gao1
1Key laboratory for Computational Astrophysics, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, China
2Institute for Computational Cosmology, Department of Physics, University of
Durham, South Road, Durham, DH1 3LE
<EMAIL_ADDRESS>
###### Abstract

The defining characteristic of the cold dark matter (CDM) hypothesis is the
presence of a very large number of low-mass haloes, too small to have made a
visible galaxy. Other hypotheses for the nature of the dark matter, such as
warm dark matter (WDM), predict a much smaller number of such low-mass haloes.
Strong lensing systems offer the possibility of detecting small-mass haloes
through the distortions they induce in the lensed image. Here we show that the
main contribution to the image distortions comes from haloes along the line of
sight rather than subhaloes in the lens as has normally been assumed so far.
These interlopers enhance the differences between the predictions of CDM and
WDM models. We derive the total perturber mass function, including both
subhaloes and interlopers, and show that measurements of approximately 20
strong lens systems with a detection limit of $M_{\rm low}=10^{7}h^{-1}{\rm
M_{\odot}}$ would distinguish (at 3$\sigma$) between CDM and a WDM model
consisting of 7 keV sterile neutrinos such as those required to explain the
recently detected 3.5 keV X-ray emission line from the centres of galaxies and
clusters.

##  1 Introduction

Under the cold dark matter (CDM) hypothesis, the power spectrum of linear
density perturbations has power on all scales down to a very small cutoff
which depends on the nature of the cold particles but is typically of order
the Earth mass (Green et al., 2005). As a result, the mass function of CDM
haloes increases roughly as a power law to low masses (Springel et al., 2008b;
Diemand et al., 2008) and the defining characteristic of a CDM universe is the
existence of a very large number of low-mass haloes. Most of these are too
small for gas to have cooled in them to form visible galaxies (e.g Efstathiou,
1992; Benson et al., 2002; Sawala et al., 2016b).

Alternative candidates for the dark matter such as sterile neutrinos behave as
warm dark matter (WDM). Their free streaming in the early universe erases
perturbations much larger than the Earth mass, typically on the scale of dwarf
galaxies. As a result, these models predict far fewer small-mass haloes than
CDM and none at all below the corresponding cutoff mass in the power spectrum
which also depends on the properties of the particles. (e.g. Avila-Reese et
al., 2003; Lovell et al., 2012, 2016; Kang et al., 2013; Schneider et al.,
2012; Bose et al., 2016). A particularly topical candidate of this kind is a
sterile neutrino of mass of 7 kev whose decay could explain the 3.5 keV line
recently detected from the centres of galaxies and clusters (Bulbul et al.,
2014; Boyarsky et al., 2014). In the “coldest” example of a 7 kev sterile
neutrino, the cutoff occurs at a mass of a few times $10^{8}$ $h^{-1}{\rm
M_{\odot}}$ (Bose et al., 2016).

CDM and viable WDM models predict similar numbers of faint dwarf galaxies such
as those observed as satellites around the Milky Way (Kennedy et al., 2014;
Lovell et al., 2015). Although the recent discovery of new satellites may rule
out some currently acceptable WDM models Bose et al. (2016), a definitive test
of WDM and indeed of CDM, requires searching for the even smaller haloes which
failed to make a galaxy and thus remain dark. Analyses of strong lensing
systems offers the possibility of achieving exactly this: Koopmans (2005) and
Vegetti & Koopmans (2009a) showed that small haloes projected onto an Einstein
ring or giant arc cause a potentially detectable distortion of the image and
Vegetti & Koopmans (2009b) showed that a Bayesian analysis of sufficiently
deep photometric data can be used to constrain the subhalo mass function
(SHMF) (see also Vegetti et al., 2012, 2014; Hezaveh et al., 2016).

The technique proposed by Vegetti & Koopmans (2009a) has already returned the
detection of a halo of mass $1.9\pm 10^{8}$ $h^{-1}{\rm M_{\odot}}$ in the
Einstein ring of JVAS B1938+666 (Vegetti et al., 2012). These authors claim
that with imaging data of similar quality the detection sensitivity can reach
$2\times 10^{7}$ $h^{-1}{\rm M_{\odot}}$. In a recent paper (Li et al., 2016),
we showed that observations of approximately 100 strong lens systems with a
detection limit of $M_{\rm low}=10^{7}h^{-1}{\rm M_{\odot}}$ could, in
principle, distinguish CDM for even the coldest 7 kev sterile neutrino dark
matter model. Of course, failure to detect haloes of such low mass would
conclusively rule out CDM altogether.

A common assumption made in studies of strong lensing is that the haloes that
perturb the image lie at the same redshift as the main lens, i.e. that they
are subhaloes of the lens. However, it is possible that the large number of
haloes along the line of sight to a lens could be the dominant source of
distortion of the lensed image. Many previous studies have shown that the line
of sight haloes can play an important role in the modeling of lensed quasar
systems(e.g. Chen et al., 2003; Xu et al., 2009; Metcalf, 2005; Wambsganss et
al., 2005). In this paper, we calculate the contribution of these
“interlopers” and investigate how they affect the prospects of distinguishing
different dark matter candidates.

Since the majority of the distortions are produced by dark haloes and
subhaloes, at first sight baryon effects may seem to be irrelevant. However,
this is not quite true: the visible galaxy at the centre of the lens can, in
principle, destroy dark subhaloes by dynamical effects such as tidal
stripping. In this paper we neglect baryon effects but we investigate those in
a companion paper using the APOSTLE hydrodynamic simulations (Sawala et al.,
2016a).

The paper is organized as follows. In Section 2 we calculate the number
density of line of sight haloes both in CDM and WDM models using the
respective halo mass functions. In Section 3 we estimate the effect of
individual interlopers and derive the effective perturber mass functions. In
Section 4 we illustrate the constraining power of halo/subhalo detection from
multiple lens systems when including interlopers. Our conclusions are
summarized in Section 5.

##  2 Number density of haloes along the line of sight

Let $\theta_{E}$ be the Einstein radius of a lens. The light rays that cross
an annulus of thickness of $2\delta\theta$ around the lens form a light cone
whose volume is given by:

$V=\int_{0}^{z_{s}}\pi\left[R(z,\theta_{E}+\delta\theta)^{2}-R(z,\theta_{E}-\delta\theta)^{2}\right]\frac{d\chi(z)}{dz}dz\,,$
(1)

where $\chi(z)$ is the comoving distance from the observer at redshift 0 to
redshift $z$, and $R(z,\theta)$ is the transverse distance corresponding to
angle $\theta$ at redshift $z$. When $z<z_{l}$, $R(z,\theta)$ is simply
$D(0,z)\theta$, where $D(0,z)$ is the comoving distance from the observer to
redshift $z$. When $z>z_{l}$,
$R(z,\theta)=D(0,z)\theta-\hat{\alpha}D(z_{l},z)$ (see the sketch in Fig. 1),
where $\hat{\alpha}$ is the deflection angle of the lens. Thus, $R(z,\theta)$
may be written as:

$\displaystyle R(z,\theta)=\begin{cases}\theta D(0,z),&z<z_{l}\cr\theta
D(0,z)-\hat{\alpha}D(z_{l},z),&z>z_{l}\end{cases}$ (2)

For a singular isothermal sphere (SIS) lens,
$\hat{\alpha}=\theta_{E}D(0,z_{s})/D(z_{l},z_{s})$ is a constant. The total
number of haloes in the light cone with mass in the range, $[M_{1},M_{2}]$, is
given by:

$N_{\rm
los}(\theta_{E},\delta\theta)=\int_{0}^{z_{s}}n(M_{1},M_{2},z)\frac{dV}{dz}dz\,,$
(3)

where

$n(M_{1}<m<M_{2},z)=\int_{M_{1}}^{M_{2}}\frac{dn(m,z)}{dm}dm\,,$ (4)

where $\frac{dn(m,z)}{dm}$ is the halo mass function at redshift $z$.

In Fig. 2 we compare the projected number density of interlopers and lens
subhaloes in the Einstein ring region for a lens in a CDM halo of mass
$10^{13}$ $h^{-1}{\rm M_{\odot}}$ at $z_{l}=0.2$. To calculate the number
density of interlopers we use the formula for the halo mass function proposed
by (Sheth & Tormen, 1999). We use the projected number density of subhaloes
derived by Xu et al. (2015) from the Phoenix and Aquarius N-body simulations
(Gao et al., 2011; Springel et al., 2008a). The projected number density of
line of sight dark matter haloes is larger than that of subhaloes associated
with the lens by a factor of 2-5.

The corresponding projected number densities for a WDM model are also shown in
Fig. 2. This model comes from the high resolution coco-warm simulation, the
WDM run of the Copernicus Complexio project (Hellwing et al., 2016; Bose et
al., 2016, 2016), which corresponds to a thermal WDM particle of mass 3.3 keV.
This is indistinguishable from a sterile neutrino model of mass 7 keV with
leptogenesis parameter, $L_{6}=8.66$ which corresponds to the coldest sterile
neutrino model consistent with the dark matter decay interpretation of the 3.5
keV X-ray line (Lovell et al., 2016). Ruling out this extreme model would
exclude the entire family of 7 keV sterile neutrinos.

Figure 1: Geometry of the source/lens/observer system. $\theta_{s}$ is the
position of the source and $\theta$ is the position of the image.
$R(z,\theta)$ is the transverse distance corresponding to angle $\theta$ at
redshift $z$. When $z<z_{l}$, $R(z,\theta)$ is simply $D(0,z)\theta$, where
$D(0,z)$ is the comoving distance from observer to redshift $z$; when
$z>z_{l}$, $R(z,\theta)=D(0,z)\theta-\hat{\alpha}D(z_{l},z)$ Figure 2:
Relative contributions of lens subhaloes and interlopers. The dashed bars show
the projected number density of subhaloes in the Einstein ring region of a
lens in a halo of $10^{13}$ $h^{-1}{\rm M_{\odot}}$ at redshift $z_{l}=0.2$.
The solid bars show the projected number density in that region of line of
sight haloes. Blue and black lines are for CDM and red for WDM, as indicated
in the legend. The width of the bars show the range of each mass bin. The
projected subhalo number density is calculated with Eqn. 8

##  3 The number density of perturbing interlopers

The lensing effect of an interloper halo projected onto the Einstein ring of a
lens depends on its mass, structure and redshift. There is a strong degeneracy
between mass and redshift. In this section we calculate the lensing effect of
interlopers by creating mock Einstein ring images using ray-tracing
simulations. The mass model for the lens plus perturber consists of a main
lens and a halo along the line of sight.

We assume that the main lens is at $z_{l}=0.2$ and has a SIS profile with
$\sigma_{v}=350$ $\rm km~{}s^{-1}$. An interloper of mass $M_{\rm int}=5\times
10^{6}h^{-1}{\rm M_{\odot}}$ is placed at $z_{\rm int}=0.18$ and a Gaussian
source is place at redshift $z_{l}=1.0$. The perturber density profile is
assumed to have the NFW form (Navarro et al., 1997) with concentration given
by the median of to mass-concentration relation of Neto et al. (2007). The
brightness distribution of the source galaxy is assumed to be Gaussian with
dispersion $\sigma_{\rm source}=0.05\arcsec$.

We then use a ray-tracing code to generate a lensed image on a plane of
500$\times$500 pixels. The size of each pixel is 0.043”, which is close to the
Hubble telescope imaging resolution. We assume that the uncertainty in the
flux in each pixel is 10% of the mean flux. Once a mock image has been
generated, we use an MCMC minimization method to fit the image with the same
mass model as above in which all parameters, except the redshift, $z_{\rm
int}$, and mass, $M_{\rm int}$, of the interlopers are fixed.

The posterior distribution of $z_{\rm int}$ and $M_{\rm int}$, displayed in
Fig. 3, clearly shows that these two parameters are highly degenerate with the
1-$\sigma$ contour including a very wide range of redshifts and masses. A low
mass interloper in front of the lens can have a similar lensing effect as a
higher mass interloper behind the lens. Since the halo mass function increases
with decreasing mass, the higher the lens redshift, the more important
interlopers become. We find that within a small range of the Einstein radius,
the form of the degeneracy between $z_{\rm int}$ and $M_{\rm int}$ is
independent of the angular position of the interloper.

Figure 3: The posterior distribution of interloper mass, $M_{\rm int}$, and
redshift, $z_{\rm int}$ for an interloper of mass of $5\times 10^{6}h^{-1}{\rm
M_{\odot}}$ placed at $z=0.18$. The contours show the 68% and 95% confidence
levels. There is a clear degeneracy between $z_{\rm int}$ and $M_{\rm int}$;
we have assumed a flux uncertainties of 10% of the mean flux.

If we fix the interloper redshift to be $z_{\rm lens}$ during the fitting, we
can derive a best-fit “effective mass,” $M_{\rm eff}(M_{\rm int},z_{\rm
int},\theta)$, where $M_{\rm int}$ and $z_{\rm int}$ are the true mass and
redshift of the interloper, and $\theta$ is its angular position. In other
words, we can use a subhalo of $M_{\rm eff}$ at redshift $z_{\rm lens}$ to
model an image perturbation caused by an interloper of mass $M_{\rm int}$ at
redshift $z_{\rm int}$.

Note that, in this ray-tracing test, we do not include observational details
such as the PSF, source complexity or noise variation across the the image. If
an interloper with mass, $M_{\rm int}$, and redshift, $z_{\rm int}$, has the
same lensing effect as a subhalo of mass, $M_{\rm eff}$, in this idealized
simulation, it should have the same lensing effect as a subhalo of $M_{\rm
eff}$ in a more realistic simulation with PSF and noise added. Our idealized
ray-tracing simulation therefore encapsulates the key information regarding
the mass-redshift degeneracy inherent in the modelling of the perturber.

We assume that the mass detection limit for subhaloes is $M_{\rm low}$ within
a thin annulus of thickness $\delta\theta$ around the Einstein radius. Then,
any interlopers with $M_{\rm eff}>M_{\rm low}$ can be detected. We refer to
these as “perturbing interlopers”. The projected number density of perturbing
interlopers can be written as:

$\displaystyle\Sigma_{\rm pb}(>M_{\rm low})=$ (5)
$\displaystyle\frac{1}{\theta_{E}\delta\theta}\int_{0}^{z_{s}}n(M_{\rm
int,low}<m<M_{\rm max},z)R(z,\theta_{E})\delta
R(z,\delta\theta)\frac{d\chi(z)}{dz}dz\,,$

where $M_{\rm int,low}$ is defined implicitly by $M_{\rm eff}(M_{\rm
int,low},z)=M_{\rm low}$, $M_{\rm max}=10^{11}h^{-1}{\rm M_{\odot}}$ is a cut
off mass we impose for the maximum halo mass considered for the mass function
in the volume along the line of sight to the lens. The number density of
haloes is dominated by the low mass end, the exact choice of $M_{\rm max}$ has
no effect on the results.

The projected subhalo number density in a CDM universe can be written as:

$\Sigma_{\rm sub,cdm}(m)=\Sigma_{0}\,\left(\frac{m}{h^{-1}{\rm
M_{\odot}}}\right)^{-\alpha}\,,$ (6)

and the cumulative surface density of subhaloes in the mass range,
$[M_{1},M_{2}]$, can be written as

$\Sigma_{\rm
sub,cdm}(M_{1}<m<M_{2})=\left\\{\begin{array}[]{ll}\Sigma_{0}\ln{\frac{M_{2}}{M_{1}}}&\textrm{if
$\alpha=1$}\,,\\\ \\\
\frac{\Sigma_{0}}{1-\alpha}(M_{2}^{1-\alpha}-M_{1}^{1-\alpha})&\textrm{otherwise}\\\
\end{array}\right.\,.$

According to Fig. 2, the projected number density of subhaloes with mass
$10^{6}$ to $10^{7}$ $h^{-1}{\rm M_{\odot}}$ is 1.3/$\rm arcsec^{2}$. Thus, we
have $\Sigma_{0}=3.1\times 10^{5}/\rm arcsec^{2}$, assuming $\alpha=1.9$.

Following Schneider et al. (2012) and Lovell et al. (2014), we write the
subhalo mass function as

$\frac{d\Sigma_{\rm sub,wdm}}{dm}=\frac{d\Sigma_{\rm sub,cdm}}{dm}(1+m_{\rm
c}/m)^{-\beta}\,.$ (7)

and the cumulative mass function, $\Sigma_{\rm sub,wdm}(M1,M2)$, can be
written as:

$\Sigma_{\rm
sub,wdm}=\frac{\Sigma_{0}}{1-\alpha+\beta}\left[F(M_{2},\alpha,\beta,m_{c})-F(M_{1},\alpha,\beta,m_{c})\right]\,,$
(8)

where,

$F(x,\alpha,\beta,m_{c})=\frac{x^{1-\alpha+\beta}}{m_{c}^{\beta}}{\rm{}_{2}F_{1}}\left(\beta,1-\alpha+\beta,2-\alpha+\beta,\frac{-x}{m_{c}}\right)\,.$
(9)

Here $\rm{}_{2}F_{1}$ is the hypergeometric function. In the coco-warm
simulation, $\beta=1.3$ and $m_{\rm c}=1.3\times 10^{8}$ $h^{-1}{\rm
M_{\odot}}$ (Li et al., 2016).

Fig. 4 shows the cumulative number density of perturbing interlopers in CDM
and in the WDM model of coco-warm. For comparison, we overplot the projected
number density of subhaloes in a host halo of $10^{13}h^{-1}{\rm M_{\odot}}$
at redshift $z_{l}=0.2$. Fig. 4 shows that for both CDM and WDM the perturbing
interlopers dominate the distortions in the Einstein ring image. In the CDM
case, the projected number density of perturbing interlopers is $\sim 3$ times
the number density of lensing subhaloes. In the coco-warm case, the excess is
a factor of 2 at $M=10^{9}$ $h^{-1}{\rm M_{\odot}}$, but decreases to 50% at
$M=10^{6}$ $h^{-1}{\rm M_{\odot}}$. Thus, the interlopers act to magnify the
difference in the number of detectable perturbers in the two cases. The
diffferent boost factors between CDM and WDM are due to the differences in the
shapes of the halo mass function in the two models. The total perturber mass
function is an integral over all perturbing interlopers and perturbing
subhaloes. In a CDM universe, the mass function over the mass range of
interest follows a power law whose index is very similar to that of the
subhalo mass function; the total perturber mass function is then just boosted
by a constant factor. The halo and subhalo mass functions in the WDM model
both have a mass cutoff at about $10^{8}$ $h^{-1}{\rm M_{\odot}}$. The
detection limit for interloper haloes varies with redshift so, in this case,
the total perturber mass function does not have exactly the same shape as the
subhalo mass function.

Figure 4: Cumulative number densities of perturbing interlopers and subhaloes
and as a function of the subhalo mass detection limit. The blue dashed line
gives the number density of subhaloes of mass greater than $M_{\rm low}$ in a
host halo of mass of $10^{13}h^{-1}{\rm M_{\odot}}$ at redshift 0.2, derived
by Xu et al. 2015. The red dashed line gives the corresponding number density
of subhaloes in the coco-warm simulation. The blue and red solid lines show
the number density of perturbing interlopers along the line of sight in the
CDM and coco-warm cases respectively. The lower panel shows $[\Sigma_{\rm
pb}(>M)-\Sigma_{\rm sub}(>M)]/\Sigma_{\rm sub}(>M)$.

##  4 Constraints on the identity of dark matter

As we have seen, to predict correctly the distortions in the Einstein ring
image it is necessary to model the total perturber surface density,
$\Sigma_{\rm tot}=\Sigma_{\rm sub}+\Sigma_{\rm pb}$, which includes both
interlopers and subhaloes in the lens. We find that the total surface density
of perturbers in the CDM and WDM models can be described by a formula of the
form of Eqn.6 and Eqn.7 respectively. For CDM, $\Sigma_{\rm tot}$ can be used
with $\alpha=1.9$ and $\log{\Sigma_{0}}=6.2$; for coco-warm , $\Sigma_{\rm
tot}$ can be used with $\alpha=1.9$, $\log{\Sigma_{0}}=6.1$, $\beta=1.3$ and
$\log{(m_{c}/h^{-1}{\rm M_{\odot}})}=8.3$. We can then exploit the difference
in the perturber mass functions to attempt to constrain the identity of dark
matter. Here the key parameter is $m_{c}$ which describes the cut off mass for
the perturber mass function in the WDM model.

To explore the constraining power of a detection of strong lensing
perturbations we adopt a similar methodology to that introduced by Li et al.
(2016). First, we generate mock subhalo detections using the following Monte
Carlo method.

We fix the lens and source redshifts to be $z_{l}=0.2$ and $z_{s}=1.0$
respectively, and assume that the lens galaxy is a SIS with velocity
dispersion, $\sigma_{v}=350$ $\rm km~{}s^{-1}$, which is similar to those of
the most massive lenses in the Sloan Lens ACS Survey(SLACS, Bolton et al.,
2006) lens sample.

For each lens, we randomly sample subhaloes and perturbing interlopers around
the Einstein ring region according to their mass functions.

Following Li et al. (2016) and Vegetti & Koopmans (2009b), we assume that only
perturbers that fall in a thin annulus around the Einstein radius of width
$2\delta\theta=0.6$ arcsec can be detected. We consider two different
detection limits, $M_{\rm low}$: $10^{8}$ $h^{-1}{\rm M_{\odot}}$, the best
current limit using HST imaging (Vegetti et al., 2014), and $10^{7}$
$h^{-1}{\rm M_{\odot}}$, the detection limit that can be reached using Laser
Guide Star (LGS) Adaptive Optics (AO) imaging with Keck (Vegetti et al., 2012)
or a next generation telescope like the TMT, or VLBI (Skidmore et al., 2015;
McKean et al., 2015). We assume that each subhalo detection has a Gaussian
measurement error with standard deviation, $\sigma_{m}=M_{\rm low}/3$. We
generate two sets of mock detection catalogues, with the mass functions
appropriate to CDM and coco-warm respectively.

We then perform an MCMC fit for each mock detection catalogue. There are four
free parameters in the model: $\alpha$, $\Sigma_{0}$, $\beta$ and $m_{c}$.
Given these model parameters, the mean number of detected subhaloes can be
written as:

$\displaystyle\begin{split}&\mu(\alpha,\beta,m_{\rm
c},\Sigma_{0})=4\pi\theta_{\rm E}\delta\theta\\\ &\int^{\infty}_{M_{\rm
low}}\int^{M_{\rm max}}_{M_{\rm min}}\frac{d\Sigma_{\rm
tot}}{dm}\frac{1}{\sqrt{2\pi}\sigma_{m}}\exp\left[\frac{-(m-m^{\prime})^{2}}{2\sigma_{m}^{2}}\right]dm^{\prime}dm\,.\\\
\end{split}$ (10)

The likelihood of finding a set of $n_{s}$ subhaloes of masses, ${\bf
m}\equiv\\{m_{1},m_{2},\dots m_{n_{s}}\\}$, in one Einstein ring system is
then given by:

$\mathcal{L}(n_{s},{\bf m}|{\bf
p,q})=\frac{e^{-\mu}\mu^{n_{s}}}{n_{s}!}\prod^{n_{s}}_{i=1}P(m_{i}|{\bf
p,q})\,,$ (11)

where the vector, ${\bf p}=\\{\Sigma_{0},\alpha,\beta,m_{c}\\}$, contains the
parameters of the model and the vector, ${\bf q}=\\{M_{\rm min},M_{\rm
max},M_{\rm low}\\}$, contains the values of the parameters that define the
minimum and maximum masses cut off we consider for the perturber mass function
and the mass detection limit. The parameters, ${\bf q}$, are fixed during the
fitting process. In this process, we set $M_{\rm min}=10^{6}$, and $M_{\rm
max}=10^{11}$ $h^{-1}{\rm M_{\odot}}$. The exact choice of $M_{\rm min}$ and
$M_{\rm max}$ does not affect the results.

The term $P(m_{i}|{\bf p,q})$ gives the probability density of detecting a
subhalo of measured mass, $m_{i}$:

$P(m_{i}|{\bf p,q})=\frac{\int^{M_{\rm max}}_{M_{\rm min}}\frac{d\Sigma_{\rm
tot}}{dm}\exp\left[\frac{-(m_{i}-m^{\prime})^{2}}{2\sigma_{m}^{2}}\right]dm^{\prime}}{\int^{M_{\rm
max}}_{M_{\rm low}}\int^{M_{\rm max}}_{M_{\rm min}}\frac{d\Sigma_{\rm
tot}}{dm}\exp\left[\frac{-(m-m^{\prime})^{2}}{2\sigma_{m}^{2}}\right]dm^{\prime}dm}$
(12)

The denominator in this equation is a normalization factor. The total
likelihood for $N$ lenses may be written as:

$\mathcal{L}_{\rm tot}=\prod_{j=0}^{N}\mathcal{L}(n_{j},{\bf m_{j}}|\bf
p,q)\,,$ (13)

where $n_{j}$ and ${\bf m}_{j}$ are the number and masses of the perturbers
detected in the $j$th system.

Following Li et al. (2016), we adopt a Gaussian prior for $\alpha$ with
expectation 1.9 and standard deviation 0.1, and a Gaussian prior for $\beta$
with expectation 1.3 and standard deviation 0.1. We adopt flat priors for
$\log(\Sigma_{0})$ and $\log(m_{c}/h^{-1}{\rm M_{\odot}})$ in the ranges
$[1,10]$ and $[4,11]$ respectively.

Fig. 5 shows the posterior distributions of $\Sigma_{\rm tot}(>M_{\rm low})$,
and the cutoff mass, $m_{\rm c}$. The upper panels are for $N=20$ lenses and
the lower panels for $N=100$ lenses. In both cases the detection limit is
assumed to be $M_{\rm low}=10^{7}h^{-1}{\rm M_{\odot}}$. The left panel shows
the result for our CDM mock catalogues and the right panel for the coco-warm
case.

Encouragingly, we find that a detection limit of $10^{7}$ $h^{-1}{\rm
M_{\odot}}$ is sufficient to distinguish between the two dark matter models.
If we live in a CDM universe (left panel), with a sample of only 20 lenses we
are able to rule out $\log(m_{c}/h^{-1}{\rm M_{\odot}})=8.3$ at the $3\sigma$
level. By contrast, if we live in a universe in which the dark matter consists
of 7 kev sterile neutrinos (right panel), with N=20 lenses and $M_{\rm
low}=10^{7}h^{-1}$ $h^{-1}{\rm M_{\odot}}$ we can rule out, at the $3\sigma$
level, all dark matter models with $\log{(m_{c}/h^{-1}{\rm M_{\odot}})}<5$,
which, of course, includes CDM! The constraining power increases with the
number of lens systems. If the number is 100, and the dark matter is as in
coco-warm, we can rule out all dark matter models with $\log{(m_{c}/h^{-1}{\rm
M_{\odot}})}<7.5$ at $3\sigma$.

Fig. 6 shows the constraints on $m_{c}$ and $\Sigma_{\rm tot}$ that can be
obtained for the coco-warm model with $N=100$ and $M_{\rm low}=10^{8}$
$h^{-1}{\rm M_{\odot}}$. Dark matter models with $m_{c}>10^{9}h^{-1}{\rm
M_{\odot}}$ are disfavoured, but the CDM model cannot be ruled out by this
experiment. This agrees with the conclusion of Li16 that the constraining
power on $m_{c}$ is weaker when the detection limit, $M_{\rm
low}>10^{8}h^{-1}{\rm M_{\odot}}$. Above this mass, the slope of the mass
function of perturbers in the coco-warm model is intrinsically similar to that
of CDM. On the other hand, with $N=100$, one can place a tight constraint on
$\Sigma_{\rm tot}$ which would provide a strong hint that the dark matter is
not CDM since, as we can see in the figure, the best-fit $\Sigma_{\rm tot}$ is
far below the prediction of a CDM universe. This demonstrates that the
identity of the dark matter can be strongly constrained by the the total
number of perturbations alone.

In this paper we have assumed a lens model with $\sigma_{v}=350$ km/s and
$z_{s}=1$, which is near the upper envelope of the SLACS sample. More massive
lenses, in combination with more distant sources, produce larger Einstein
rings and these lead to larger volumes for interloper detection. These lenses
should be high priority targets for future high-resolution observations. If we
adopt a configuration similar to the average SLACS sample, with $z_{l}=0.7$
and $\sigma_{v}=275$ km s-1 (Bolton et al., 2006), we require 50% more lenses
to achieve similar constraining power.

An important simplification we have made is to assume a uniform detection
limit for the perturber over the entire Einstein ring. In a real situation the
detection limit varies across the Einstein ring region and a sensitivity map
that specifies the subhalo mass detection limit at each pixel of the image,
like those made by Vegetti et al. (2014), is crucial for constraining the
perturber mass function. Once such a map has been constructed, the strategy
used in this paper can be applied with minor changes. In particular, in eqns.
11-13, one should first calculate the likelihood of detecting $n_{s}$
perturbers in the $i$th pixel of the $j$th lens and then sum the likelihood
over all the pixels of all lenses.

In this study we have also neglected the effects of the galaxy in the lens on
the population of subhaloes orbiting in the same halo. In a recent paper,
Sawala et al. (2016a) calculated the changes in the abundance and spatial
distribution of subhaloes in the mass range $10^{6.5}$ to $10^{8.5}$
$h^{-1}{\rm M_{\odot}}$, in haloes of mass of $10^{12}h^{-1}{\rm M_{\odot}}$,
caused by interaction with the central galaxy. By comparing the hydrodynamical
simulations of the Apostle project of Local Group simulations with their dark
matter only counterparts, they found the reduction in the number of subhaloes
as a function of radial distance due to tidal disruption in the potential well
deepened by the presence of the central galaxy to be approximately independent
of subhalo mass. At halocentric distances $r<50$ kpc, the number of subhalos
is reduced by $\sim 40-50$% and at radii in the range $r=50-200$ kpc by 23%.

The host haloes in the Apostle simulations are an order of magnitude less
massive than the halos we are considering in this study. If we assume that the
reduction in numbers scales with $r/r_{\rm 200}$, we should expect the number
of subhaloes in strong lenses system also to be $\sim 20-50$% smaller than the
number predicted in dark matter only simulations. This effect, however, does
not alter the conclusions in this paper because, as we have seen, the
perturbers of Einstein ring systems are predominantly field haloes along the
line of sight to the lens, rather than subhaloes.

Figure 5: The posterior distribution of $\Sigma_{\rm tot}(>M_{\rm low})$ (in
units of $\Sigma_{\rm tot}$ arcsec-2) and perturber mass function cutoff mass,
$m_{\rm c}$. The contours indicate the 67%, 95% and 99.7% confidence levels.
The left panels show results for CDM while the right panels show results for
the coco-warm model. The upper panels are for $N=20$ and the lower panels for
$N=100$ lenses. The detection limit is assumed to be $M_{\rm
low}=10^{7}h^{-1}{\rm M_{\odot}}$. In the right panels the crosses show the
input values of $m_{c}$ and $\Sigma_{\rm tot}$; in the left panels the dashed
lines indicate the input value of $\Sigma_{\rm tot}$. The arrows mark the
value of $m_{c}$ for the coco-warm model.  Figure 6:  As Fig. 5, but for a
detection threshold, $M_{\rm low}=10^{8}h^{-1}{\rm M_{\odot}}$, and a number
of lenses, $N=100$. The input perturber mass function is from the coco-warm
model. The black dashed line shows the expected value of $\Sigma_{\rm tot}$ in
CDM.

##  5 Summary

The most direct, and potentially conclusive, test of different models for the
dark matter is to measure the mass function of dark matter halos in the low-
mass regime where different models that agree with CMB and large-scale
structure data can be expected to differ. Unfortunately, attempts to infer the
small-mass end of the dark halo mass function from observations of visible
galaxies are hampered by the intrinsically low luminosity of faint objects and
further complicated by uncertainties in modeling baryon effects.

In contrast Einstein rings (and giant arcs) produced by strong gravitational
lensing, offer a clean and powerful means to detect small halos and measure or
constrain the halo mass function. These small halos perturb lensed images and
by modelling these perturbations, it is possible to detect individual haloes
projected onto the image and measure their mass. There is a strong degeneracy
between the mass of a perturber and its redshift. As a result, the lensing
effect of an interloper halo along the line of sight can be modelled as that
produced by a (sub)halo of some effective mass located at the redshift of a
lens.

In this paper we have compared the CDM model with a WDM model whose linear
perturbation power spectrum is that of a thermally produced 3.3 keV particle
and provides a very good approximation to the linear power spectrum of the
coldest possible 7 keV sterile neutrino consistent with a particle decay
interpretation of the recently discovered 3.5 keV line in the X-ray spectra of
galaxies and clusters. Ruling out this model by detecting small halos below
the cutoff mass in its predicted halo mass function would rule out all 7 keV
sterile neutrino models. Similarly, a failure to detect small-mass subhalos
would rule out CDM.

For both CDM and WDM models we have calculated the projected number density of
interlopers and compared it to the projected number of subhaloes. We defined
the ‘perturbing’ interlopers as those that generate a larger lensing signal
than a subhalo of mass, $M_{\rm low}$. We then derived the effective mass
function of perturbers, including both perturbing interlopers and subhaloes.
We find that the total number density of perturbers is 4 times of that of
subhaloes in CDM and 1.5-2 times of that of subhaloes in our WDM model.
Interlopers therefore boost the probability of detection and act to magnify
the difference between CDM and WDM.

We find that a measurement of only 20 strong lensing systems with a detection
threshold of $M_{\rm low}=10^{7}$ $h^{-1}{\rm M_{\odot}}$ is enough to
distinguish between CDM and our WDM model at the 2$\sigma$ level. With a
survey of 100 strong lenses the confidence level increases to 3$\sigma$. If
the threshold mass, $M_{\rm low}=10^{8}$ $h^{-1}{\rm M_{\odot}}$, the
constraint on the cutoff halo mass of our warm dark matter model, $m_{c}$
becomes weaker because the slope of the effective mass function above $10^{8}$
$h^{-1}{\rm M_{\odot}}$ in this model is similar to that in CDM but the
constraint on the total number density of perturbers is tight, thus retaining
discriminating power between the models.

Strong gravitational lensing provides, in principle, a clean test of dark
matter models. The quality of existing data and analysis technique is already
sufficient to detect dark low-mass haloes, too small to have made a galaxy. As
a result this technique is almost unaffected by uncertain baryon effects,
except for the possibility that the disruption of subhaloes orbiting within a
large halo may be enhanced by the concentration of mass induced by the central
galaxy. This kind of processes can be quantified with hydrodynamic simulations
and, once this is achieved, as we shown a conclusive test of the nature of the
dark matter will be possible. In particular lensing measurements forthcoming
in the next few years offer the possibility of ruling out the main current
candidates for the dark matter, CDM and WDM.

## Acknowledgements

RL acknowledges NSFC grant (Nos.11303033,11511130054，11333001), support from
the Newton Fund, Youth Innovation Promotion Association of CAS and YIPA of
NAOC. CSF and SMC acknowledge the European Research Council Advanced
Investigator grant, GA 267291, COSMIWAY. LG acknowledges support from the NSFC
grant (Nos 11133003, 11425312), the Strategic Priority Research Program The
Emergence of Cosmological Structure of the Chinese Academy of Sciences (No.
XDB09000000), and a Newton Advanced Fellowship, as well as the hospitality of
the Institute for Computational Cosmology at Durham University. This work was
supported by the Consolidated Grant [ST/L00075X/1] to Durham from the Science
and Technology Facilities Council. This work used the DiRAC Data Centric
system at Durham University, operated by the Institute for Computational
Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The
DiRAC system is funded by BIS National E-infrastructure capital grant
ST/K00042X/1, STFC capital grant ST/H008519/1, STFC DiRAC Operations grant
ST/K003267/1, and Durham University. DiRAC is part of the National
E-Infrastructure. We thank the participants of the workshop “Dark matter on
the smallest scale” (Lorentz Centre – 4-8 April 2016) for lively discussions
and Simon White for his insightful comments.

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