We investigate constraints on the distribution of dark matter in the neighbourhood of the Galactic centre that may eventually be attained with the high-resolution Event Horizon Telescope (EHT). The shadow of a black hole in vacuum is used to generate a toy model describing how dark matter a ects the size of the shadow of the supermassive black hole located at the Galactic centre. Observations by the EHT may constrain the properties of the dark matter distribution in a possible density spike around the black hole. Current uncertainties due to both the resolution of the telescope and the analysis of stellar orbits prevent one from discerning the e ect of dark matter on the measured size of the shadow. The change in the size of the shadow induced by dark matter can be seen as an additional uncertainty in any test of general relativity that relies on using the angular size of the shadow to estimate the Schwarzschild radius of the black hole.

In the framework of the standard model of cosmology, dark matter comprises about 23 percent of the energy content of the universe. Consequently, unravelling the mystery of the nature of dark matter is one of the most challenging problems of modern cosmology. In this context, astronomical observations can provide significant constraints on the dark matter distribution. The dawning of a new era of telescopes with high resolution and sensitivity has widened the scope of precision experiments that may be used to set constraints on the properties of dark matter. In this work, we focus on one such experiment. We consider an experiment focusing on the Galactic centre, since the centre of the Milky Way has become a favoured region for searches for dark matter signatures. The dark matter density is indeed expected to be higher in the neighbourhood of the Galactic centre, according to N-body simulations. Moreover, if dark matter is indeed made of weakly interacting particles, there is likely to be an enhancement of dark matter around the supermassive black hole Sgr A* located at the centre of the Milky Way.

The very high and still increasing resolution of the Event Horizon Telescope (EHT) is about to allow one to get a picture of the shadow of this black hole. Such an imaging survey may then be used to constrain the properties of a dark matter spike around the centre. This is of particular interest because such a geometrical measurement avoids any knowledge of the precise nature of the dark matter, such as its decay or annihilation properties. One needs a model describing how the dark matter around the centre modifies the size of the shadow to be able to make quantitative comparisons with future EHT pictures.

In Sect. 2, we recall the notion of shadow of a black hole in a vacuum, before presenting a toy model in Sect. 3 that describes how the presence of dark matter around a black hole would affect the bending of light induced by the compact object and how this could be used to extract constraints on the properties of the dark matter distribution. Finally, we analyse the uncertainties involved in the process of inferring the properties of the black hole from the measurement of the size of the shadow.

The shadow of a black hole is closely related to the way that
light rays are bent by the black hole and, more particularly, to the
deflection angle. Starting from the Schwarzschild metric, which
describes the geometry of a non-rotating black hole in a vacuum,
one can show the relation between the impact parameter b and
the closest approach distance rm of a photon

rm
rS
rm
(1)
where rS is the Schwarzschild radius of the black hole. It turns
out that the deflection angle features a divergence for a
particular value of the impact parameter. This value defines the shadow
of the black hole. More precisely, this divergence corresponds
to light rays infinitely bent by the black hole and thus
performing an infinite number of loops around it. Such orbits are
unstable under small perturbations, which results in the photons
eventually crossing the horizon and falling onto the
singularity. Therefore, the shadow represents the minimum impact
parameter of a photon escaping the attraction of the black hole.
Conversely, it is the minimum impact parameter of a photon
coming from infinity for it to end up on the photon sphere

In this section we present a toy model describing how the shadow of a black hole is modified if the singularity is embedded in dark matter. (2) (3) (4)

To determine the behaviour of light rays, one must first derive
the metric describing the geometry of spacetime induced by the
black hole embedded in matter. The matter distribution is
assumed to be spherically symmetric for the purpose of
simplification. We start from the general form of a static and spherically
symmetric metric

A(r)dr2 r2 d 2 + sin2 d'2 ; and determine A and B using the Einstein equations. The dark matter halo is assumed to be collisionless and at rest. Under these assumptions, the first metric coe cient is given by

Consequently, the shadow is the main observable feature of the black hole in a direct imaging survey, and this is what the EHT collaboration aims to observe in the near future, using the technique of Very Long Baseline Interferometry (VLBI). More precisely, the EHT project consists in phasing up millimetre and sub-millimetre telescopes scattered over the world, which together will form an e ective high-resolution Earth-sized telescope (Doeleman 2010). The EHT array currently comprises the Submillimeter Telescope (SMT) in Arizona, the Combined Array for Research in Millimeter-wave Astronomy (CARMA) in California, and three telescopes in Hawaii: the Caltech Submillimeter Observatory (CSO), the James Clerk Maxwell Telescope (JCMT), and the Submillimeter Array (SMA)1. The network is soon to be complemented with the Atacama Large Millimeter/submillimeter Array (ALMA) in Chile. The data recorded simultaneously by all these telescopes are then processed by a dedicated supercomputer. So far, two baselines have been put together, namely the CARMA-SMT and the Hawaii-SMT baselines, in order to achieve an angular resolution of 58 microarcsec ( as) at 230 GHz. However, the angular Schwarzschild radius of the supermassive black hole Sgr A* is 10:3 1:24 as and thus the angular diameter of the shadow is of the order of 53:6 as. This was obtained via rS = 2GM=c2, where c is the speed of light and G the gravitation constant, using the values of the distance of Sgr A* from the Sun, R = 8:28 0:33 kpc, and the mass of Sgr A*, M = 4:3 0:36 106 M as given in Gillessen et al. (2009). The current resolution is therefore not yet su cient to get an image of the shadow of the black hole, but it is quite close. Adding more baselines to the EHT array will thus improve the resolution so that the shadow of Sgr A* should be observed in the very near future. Moreover, the resolution is better at lower wavelengths, so future observations are going to be carried out at an additional, higher frequency, namely 345 GHz. Consequently, the addition of ALMA, along with another baseline between the Plateau de Bure interferometer in France and the South Pole Telescope (SPT), will allow the EHT array to achieve a resolution of 15 as at 345 GHz.

Imaging the shadow of a black hole will allow one to test the prediction of general relativity for the radius of the shadow and study accretion flows in the vicinity of the black hole. But it will also allow setting constraints on the properties of a hypothetical distribution of dark matter surrounding the supermassive black 1 See http://www.eventhorizontelescope.org/ where m(r) =

4 r02 (r0) dr0 is the enclosed mass in the 0 sphere of radius r. The di erential equation that rules the other unknown coe cient is derived by injecting the expression for A back into the Einstein equations and it reads as B0 B = 2m(r) r2 1 2m(r) ! 1

r
3.2. Shadow of a black hole surrounded by a dark matter
spike
A dark matter density profile must now be chosen. Here we
consider dark matter distributed according to a density spike around
the central black hole. More precisely, the galaxy is believed to
be embedded in a halo with a moderately steep profile ( / r
where 0 < < 2) from N-body simulations (see Bringmann
& Weniger (2012) or

In the remainder of this section, we neglect the contribution of the halo itself and focus on the gravitational lensing e ects of the spike, which should be the dominant component of the system, as far as the shadow is concerned. Therefore, the spike is assumed to be surrounded by a vacuum in the calculations involving the shadow. A halo without a spike would have a negligible e ect on the shadow, thanks to a less severe enhancement of the dark matter density at the centre. We next consider the following density profile for the system comprising the black hole of mass MBH (equivalent to a point mass) and the dark matter spike of mass Mspike and radius R, (r) (r) = MBH 4 r2 + where is taken as 7/3, and is determined by the normalization condition that the integral of the spike density be equal to the total mass of the spike. The uncertainty on leads to changes in the size of the shadow, which are negligible with respect to all other uncertainties by several orders of magnitude. Therefore with this model the di erential equation for B takes the form where the mass ratio is q = Mspike=MBH. Considering that R should be much larger than the Schwarzschild radius rS = 2MBH, and q should be smaller than 1, the second term in each bracket is going to be a perturbative term for small radii (r rS), relevant to deriving the radius of the shadow. Consequently, Eq. (6) can be approximately integrated using the fact that the numerator on the right-hand side is the derivative of the denominator, except for the small term containing q and R. This leads to for r < R, where B0 is a constant that can be obtained using the expression of the metric coe cient B in a vacuum: B(r) = 1 rS;tot=r for r > R, where rS;tot = rS(1 + q) is the total Schwarzschild radius corresponding to Mtot = MBH + Mspike. Therefore continuity of the metric at the boundary of the spike r = R leads to B0 = 1. In the following, the metric used to describe the geometry induced by a black hole surrounded by a dark matter spike will be determined by the coe cients where the first equation is exact, and the second equation holds for parameters of the spike such that the perturbative approach is valid. More precisely, to make sure that this approximation was reasonable, we numerically solved Eq. (6) using a Runge-Kutta scheme to compute the actual function B(r), given the boundary condition B(R) = 1 rS;tot=R. Then we computed the product of functions A(r) given in (8) and B(r) obtained numerically. It turned out to be equal to 1 with an error of 10 3 for q = 0:1 and R = 103 rS, which are sensible limits from a physical point of view (since the spike is predicted to be both large and light with respect to the black hole). Such an error will be insignificant because the radius of the shadow will be given with a much greater uncertainty than one percent (see below). Moreover, these values of q and R are the very limits of this perturbative approach, and for most values of the parameters, the error is actually even smaller than 10 3.

With these approximations, the metric can be derived analytically for the density profiles considered here. Therefore, the approximations detailed above allow us to pursue the calculation and obtain an analytic expression for the size of the shadow rshadow. This allows for faster numerical calculations. Then, constraints on the properties of the system can be extracted from EHT pictures by measuring the deviation of the actual shadow with respect to the expected shadow of a black hole in vacuum. Consequently, uncertainties on the measurement of rshadow are going to play a crucial part.

First of all, one needs to compute the size of the shadow for di erent values of q and R. Similar to the case of a black hole in vacuum, the impact parameter b as a function of the closest approach distance rm is given by b(rm) = pB(rm)

rm Then the radius of the shadow is given by minimizing this function with respect to rm. The impact parameter assumes its minimum for a value of rm such that rm(min) = 23 rS BBBBB@B01 + q BBBB@0 rm(mRin) CCCCA13 CCCCCCA1 : (9) (10) This equation can be solved perturbatively, considering that q rm(min)=R 3 1 for realistic models. We solve it by iteration, starting from the zeroth order and injecting the solution back into the right-hand side of Eq. (10). Only four iterations are required to retrieve the solution with a precision higher than 10 3. This gives an analytic expression for the radius of the shadow rshadow = b(rm(min)(q; R)) as a function of q and R.

It turns out that there is a degeneracy in R. More precisely,
the size of the shadow does not depend on the radius of the spike
for R larger than 103 rS. Here we do not consider any lower
values of the radius of the spike because R is often predicted to
be large with respect to rS (for instance R 1 pc according to

Considering the degeneracy in R, the best approach is to plot the resulting curve of rshadow as a function of the mass ratio q, as shown in Fig. 2. It is important to note that here rshadow is expressed in units of the total Schwarzschild radius rS;tot corresponding to the total mass Mtot = MBH + Mspike = MBH(1 + q).

This is the only quantity known from previous studies of the motion of stars at the Galactic centre: Mtot = 4:3 0:36 106 M (Gillessen et al. 2009). This value is in general taken as the before taking the resolution of the EHT into account, the uncertainty due to the mass and distance of Sgr A* is already greater than the width of the range of values of shadow. To observe an e ect of dark matter, the sum of the uncertainty on S;tot and the black hole mass, but dynamical studies cannot separately mea- error due to resolution should be smaller than 5 as. sure Mspike and MBH, so we consider that the mass given in the Shown in Fig. 3 are only the best resolutions achievable by literature is the total mass in the framework of a black hole sur- adding the Hawaii-ALMA and Plateau de Bure-SPT baselines rounded by a dark matter spike. Therefore, this allows one to to the current baselines (CARMA-SMT and Hawaii-SMT), for switch from the unknowns R, MBH, and Mspike to R, Mtot, and q, two observing frequencies at 230 GHz and 345 GHz. Finally, where Mtot is already known. we consider a resolution of 1 as, which would be ideal to carry

Considering that the size of the shadow obtained by solv- out a precise measurement of the size of the shadow and observe ing (10) is expressed in units of the Schwarzschild radius rS of a possible change induced by dark matter. For now, a resoluthe black hole alone, and as rS;tot = rS(1 + q), the value of rshadow tion as high as 15 as, despite being huge by astronomical stangiven by the calculation had to be divided by (1 + q), in order dards, will only allow one to see the shape of the shadow. This to be expressed in terms of rS;tot (which is a known length scale, is the primary goal of the EHT since the collaboration aims to unlike rS). Moreover, expressing rshadow in terms of rS;tot allows shed light on physical processes taking place in the vicinity of us to break a degeneracy, since rshadow in units of rS is very close the black hole, such as accretion flows. The following step is to to 2.6 for all values of q and R in the region of interest. The aim measure the size of the shadow to obtain a direct estimate of the is then to set constraints on the remaining unknowns q and R. angular Schwarzschild radius of the black hole and possibly see

The quantity of interest from the observational point of view the e ect of dark matter. However, with a resolution of 15 as, turns out to be the angular diameter of the shadow shadow = the size of the shadow will be measured with an error of almost 2rshadow=R . Shown in Fig. 3 is the angular diameter of the 30 percent. Therefore, with such a large error, discerning a deshadow derived from the previous calculation. This graph pro- viation from the value of the size of the shadow in vacuum is vides a clear way of constraining the mass ratio from the going to prove a challenging task, but even measuring the size of value of the radius of the shadow. In this study q ranges from the shadow with high enough precision is going to be di cult. 10 3 to 10 1. Going down to lower values would be useless, As a matter of fact, the EHT is going to be able to carry out a since rshadow(q) is almost constant below 10 3. Moreover, the direct measurement of the total Schwarzschild radius, but with spike is likely to have a perturbative e ect, so q should be small an uncertainty of 15 as, i.e. larger than the error coming from enough. That is why q does not go above 10 1 here. Finally, the the measurement using stellar orbits. The existing constraints on range considered here shows the most compelling variations of S;tot are thus not going to be improved from the observation of the size of the shadow: shadow goes down by 5 as between 10 3 the shadow with a resolution of 15 as. Nevertheless, this resand 10 1, which sets the minimum uncertainty required to dis- olution planned for the near future is certainly not the ultimate cern an e ect of dark matter. The size of the shadow is in fact goal, and it can surely be improved even more by adding still mostly sensitive to values of the mass ratio q greater than 10 2. longer baselines, with potential EHT sites in New Zealand and Consequently, measuring the size of the shadow in principle al- Africa. lows one to distinguish between high values of the mass ratio. We now focus on the constraints on the dark matter spike. However, the uncertainty on the shadow due to the error on the On the one hand, the angular size of the shadow can only be total angular Schwarzschild radius, S;tot = 6:43 as, must be known with an error over 7 as, due to the uncertainty on S;tot, taken into account. The angular size of the shadow in this model and even for a resolution as high as 1 as (Fig. 4). As a result, is indeed derived in terms of the total angular Schwarschild ra- the error on the size of the shadow is actually too large for q to dius S;tot = rS;tot=R . This uncertainty comes from the error on be constrained. The uncertainty on S;tot turns out to be critical, the measurement of the distance of Sgr A* from the Sun, R = and even increasing the resolution of the EHT much more would 8:28 0:33 kpc, and the total mass, Mtot = 4:3 0:36 106 M not allow us to set stringent constraints on q. Therefore, plac(Gillessen et al. 2009). Consequently, it turns out that, even ing constraints on the dark matter distribution not only requires improved resolution, but also smaller uncertainties on the total mass and the distance of Sgr A*.

On the other hand, the degeneracy in R means that one cannot set constraints on the size of the spike directly from this study of the shadow. In fact, there is a relation between q and R, but it is independent of the calculation of the shadow. So far, for the purpose of determining how dark matter a ects the shadow, we have neglected the smooth distribution (with an NFW profile) surrounding the spike, since its e ect on the shadow is negligible. However, this underlying distribution can be used to derive a relation between the mass ratio and the radius of the spike, using the continuity of the density at the boundary of the spike: R =

qMtot 6 0Rgal(1 + q) !1=2

(11) R

R

!2 with 0 = + 1

where Rgal = 20 kpc is the ra

Rgal Rgal dius characterising the dark matter halo, R = 8:28 kpc is the distance of the solar system from the Galactic centre and

= 0:3 GeV cm 3 is the local dark matter density at the Sun.

Here we have used R Rgal to obtain a simplified expression.

Since q cannot be properly constrained at the moment, the radius of the spike cannot be constrained either, but Eq. (11) provides a way of translating future constraints on q into constraints on R, with an uncertainty given by the uncertainties on q, Mtot and R .

Although a direct measurement of the e ect of dark matter is made more di cult by large uncertainties, quantifying the change in the shadow induced by a dark matter spike is useful as it turns out to be an additional source of uncertainty when testing the predictions of general relativity by measuring the size of the shadow. 3.4. Comments on some approximations made in this study First of all, throughout this work we have neglected the e ect of stars located around the Galactic centre on the bending of light and, more precisely, on the size of the shadow of the black hole. This seems indeed a sensible approximation since stars around the Galactic centre form a mass distribution that extends much further than the dark matter spike. Therefore, even if the luminous mass around the Galactic centre is significant, its effect on gravitational lensing is actually negligible due to its being too widespread, as can be seen from the previous sections.

Moreover, there is much less severe enhancement of the
density of stars at the centre than what is expected for dark matter

Then we made approximations related to the properties of the black hole itself. A Schwarzschild black hole is indeed an idealized case, and from a more realistic point of view, the supermassive black hole at the Galactic centre should be rotating.

The e ect of the spin of such a Kerr black hole is to compress
the shadow border on one side and to shift its position

Finally, it is worth noticing that this study of the shadow does not depend on any assumption about the nature of dark matter.

The model only depends on a dark matter mass distribution, regardless of the nature of the dark matter. Therefore, if a positive e ect on the shadow were to be detected, this may be a sign that dark matter is not an artefact of the way observations are interpreted.

In this work, we have focused on an original way of setting constraints on the properties of dark matter by observing the Galactic centre. The study of the shadow of the central black hole embedded in a spike of dark matter in principle allows one to constrain the properties of the spike, using future observations carried out by the Event Horizon Telescope. The EHT is indeed close to reaching a high enough resolution to take a picture of the shadow. We have presented a model that describes the e ect of a dark matter spike on the shadow of the black hole Sgr A*. Values predicted by this model are then to be compared to the actual value of the size of the shadow measured by the EHT, in order to discern the e ect of dark matter and constrain the mass and the radius of the spike. We quantified the various uncertainties involved in the problem. It turns out that the uncertainties on the mass and distance of Sgr A* that translate into an error S;tot on the shadow, along with the resolution of the EHT array, make it impossible to set relevant constraints on the dark matter distribution. Consequently, increasing the resolution is not su cient to take advantage of a direct observation of the shadow, but the focus should also be on reducing the uncertainties on S;tot. This is crucial when trying to discern deviations from the case of a black hole in vacuum. Nevertheless, this apparent drawback of the method can be turned into an advantage, because this method allows one to compute the scale of variation of the size of the shadow, 5 as. This proves to be an additional uncertainty when estimating the angular Schwarzschild radius of Sgr A* from the size of the shadow, which is one goal of the EHT.

Acknowledgements. We would like to thank Céline Boehm, Alexander Belikov, and Timur Delahaye for valuable comments. We are also indebted to Kaiki Inoue, Pierre Fleury, and Clément Ranc for fruitful discussions. This research was supported in part by ERC project 267117 (DARK) hosted by Université Pierre et Marie Curie ? Paris 6.