Shadow of a black hole surrounded by dark matter

We consider a simple spherical model consisting of a Schwarzschild black hole of mass $M$ and a dark matter of mass $\Delta M$ around it. We derive the general formula for the radius of black-hole shadow in this case. It is shown that the distortion of the shadow is not negligible once the effective radius of the dark matter halo is of order $\sim \sqrt{3 M \Delta M}$. For this to happen, for example, for the galactic black hole, the dark matter must be abnormally concentrated near the black hole. For small deviations from the Schwarzschild limit, the dominant contribution into the position of a shadow is due to the dark matter under the photon sphere, but at larger deviations from the Schwarzschild shadow, the matter outside the photon sphere cannot be ignored.


I. INTRODUCTION
Recent observations of black holes in the electromagnetic spectrum succeeded in observing the first image of the black hole in the center of galaxy M87 [1,2]. Although the first image of the black hole does not allow one to identify the black hole geometry clearly, the principal strategy for the improvement of measurements should lead to much higher resolution in the future [3]. Therefore it is difficult to underestimate theoretical efforts to calculate forms of shadows cast by black holes and black-hole mimickers in various theories of gravity and astrophysical environments [4]- [32].
At the same time, it is believed that 85% of mass in the Universe consists of the invisible dark matter [33]. The abnormally high velocities of stars at the outskirts of galaxies imply that visible disks of galaxies are immersed in a much larger roughly spherical halo of dark matter [34,35]. The dark matter does not interact with the electromagnetic field and therefore the propagation of light is possible inside the dark matter halo. The natural question would be to learn whether the black hole shadow could be distorted because of the tidal forces induced by the invisible matter. A few attempts in this direction have been performed in [36][37][38] and a similar work was done for the dark energy in [39]. However, in all of the above works one or the other particular equation of state for the dark matter or dark energy was assumed, so that the results look highly model-dependent. For example, in the case of the dark energy [39], one particular solution from the family of solutions obtained by Kiselev [40] was analyzed as to possible consequences for black hole shadows.
In our opinion, before considering particular models for the invisible matter, a simpler question must be answered: Can dark matter deform the black hole geometry * roman.konoplya@gmail.com so strongly, that the shadow would change seemingly? In order to answer this question we should imply only some most basic and simple features of the dark matter: that it has a kind of an effective mass and does not interacting with the electromagnetic field, so that its influence on the black-hole shadow is only through the deformation of the background geometry. For this purpose we will consider the spherically symmetric configuration, consisting of the Schwarzschild black hole and a spherical halo of dark matter with a given mass around it. This model was considered for testing the gravitational response (such as quasinormal modes [44] or echoes [45]) of black holes in the astrophysical environment [41][42][43].
Here we will use the above framework for estimation of the effect of dark matter on the size of the shadow of a black hole. The paper is organized as follows. Sec. II introduces the mass function and the essential properties of the space-time under consideration. Sec. III briefly relates the deduction of the formula for a shadow of an arbitrary spherically symmetric background. In Sec. IV we derive the main results of the paper, devoted to the general formula for the radius of the shadow in the presence of dark matter. Finally, in Sec. V we summarize the obtained results and discuss the open questions.

II. MODELLING DARK MATTER
There are various approaches to modelling dark matter in General Relativity, taking into account current cosmological observations. Here we will try to explore a more agnostic approach and will use the two facts: • The dark matter is invisible matter which does not interact with an electromagnetic field. This way it should not hamper propagation of light rays. This statement is certainly not true for some models of barionic dark matter which may absorb light on its way to the observer. If one suppose that the heavy barionic dark matter intensively absorbs light, any conclusion depends on a particular distribution and equation of state of the dark matter.
• The dark matter possesses some energy which can be modelled as an additional effective mass in the mass function of a black hole.
Therefore, we choose the metric for the above configuration in the following way where and the mass function is given by This way m(r) and m ′ (r) are continuous functions (see Fig. 1). Here ∆M > 0 (∆M < 0) corresponds to positive (negative) energy density of matter. Although our primary motivation is to study positive ∆M , we will also consider effect of the effectively negative mass, having in mind, as a by-product, possible exotic matter with a negative kinetic term or repulsion.
Thus, the dark matter is situated in the region which begins from r = r s > 2M and finishes at r = r s + ∆r s , while the event horizon is still located at r = 2M . One should have in mind that for ∆M > 0 and some fixed values of the r s and ∆r s , the allowed values of ∆M are constrained, because if ∆M is too large, it simply increases the total mass of the black hole and the radius of the event horizon. As an example, from fig. (2) one can see that the bottom metric is approaching the state where the radius of the event horizon becomes larger at δM/M ≈ 45 for given r s and ∆r s . The absolute value of the negative ∆M is not constrained in this way.

III. GENERAL FORMULA FOR THE RADIUS OF SHADOW OF SPHERICALLY SYMMETRIC BLACK HOLES
Here, following [46], we will briefly present the deduction of the formula for a shadow of an arbitrary spherically symmetric black hole. The metric of the most general spherically symmetric spacetime can be written in the following way In spherical spaces, each plane is equatorial and one can choose ϑ = π/2, so that p ϑ = 0. The Hamiltonian for light rays is The light rays are the solutions to the equations of motion:ṗ so thatṫ From H = 0 it follows that Here a dot designates derivatives with respect to an affine parameter and a prime is for derivatives with respect to r. The momenta p t and p ϕ are constants of motion; ω 0 := −p t . From (7) and (8) one finds that Using p r from (9), we have where, following [46], the function h(r) is defined as follows: Along a circular light orbit the radial velocity and acceleration must be zero, that isṙ = 0 andr = 0. From (8) it follows that p r = 0, while from (9) we find Differentiating (8) with respect to the affine parameter givesṗ Then, the requirement of zero radial velocity and acceleration leads toṗ r = 0, and we obtain From (13) and (15) it follows that Then, the radius of a photon sphere is a solution to the equation Following the designations of [46], we will use r O for the position of the observer and α for the angle respectively the radial direction. Then, we have The equation (11) can be rewritten in terms of the minimal radius R as follows  Then we have and consequently The angular radius of the shadow is then determined by With these formulas at hand we are ready to find the radius of the shadow for a black hole metric discussed in the previous section.

IV. SHADOW OF A BLACK HOLE SURROUNDED BY DARK MATTER
Depending on the position of the photon sphere relatively the position of the dark matter halo, there are three qualitatively different situations: 1. When the dark matter is distributed in such a way that the photon sphere lies inside the dark matter configuration, that is, between the even horizon r 0 and the beginning of the dark matter layer r s , then for spherically symmetric configuration the observer will see the non-distorted Schwarzschild shadow, and the photon sphere is simply 2. When the photon sphere lies outside the dark matter configuration from the side of the observer, that is between the observer and the dark matter, then the observed photon sphere has simply an added mass of the dark matter: This situation does not look realistic as then it would mean that the dark matter exists only in the proximity of the black hole and, at the same time, does not fall onto the black hole.
3. The only non-trivial situation occurs when the closest to the horizon photon orbit is placed inside the dark matter configuration. In that case, solution of the equation (17) for the metric functions defined in (1,2) give the following expression for the radius of the photon sphere: where K = 12∆Mr s ∆r s (∆r s − 3M ) −18M ∆M∆r 2 s + 3∆Mr 2 s (4∆r s + 3∆M) + ∆r 4 s The function h(r) is given by The matter under the photon sphere should then move towards the horizon and a more realistic model should include non-static configuration of matter. However, here we are interested in a robust description of the deviation of the black-hole shadow owing to the total energy of dark matter, that is, the effect of the effective mass of the black-hole environment. From fig. (3) one can see that once the mass of the dark matter is fixed and the distance ∆r s over which this mass is distributed is increased, then the diminished density of the dark matter leads to larger values of the radius of the photon sphere. In the limit of zero density this radius approaches its Schwarzschild value r = 3M . Although the radius of the photon sphere decreases when the mass ∆M is increasing (see fig. (4)), from fig. (5), one can notice that the effect for the shadow is opposite: larger masses of dark matter correspond to larger radii of shadows.
The analytical expression for the radius of shadow as a function of M , ∆M , r s and δr s is rather cumbersome, but a concise expression can be found for sufficiently large ∆r s , that is, for the astrophysically most expected situation in which the cloud of dark matter is distributed over the whole halo rather then concentrated in a single place. The Taylor expansion for large ∆r s gives Suppose that the dark matter surrounds the black hole right from its event horizon. Then, r s = 2M and the above formula is reduced to the following form That means that for the black hole shadow to be considerably distorted by dark matter, one should have which is not fulfilled for the central black hole in our galaxy, because from the estimations of the mass of dark matter halo in our galaxy, we know that ∆M ≈ 6 · 10 11 − 3·10 12 M Sun [34], while the mass of the galactic black hole is M ≈ 4.3 · 10 6 M Sun . This leads to values of ∆r s which are many orders smaller than the characteristic sizes of galaxies or dark matter halos.
It is natural to suppose that one can use the fact that the gravitational force of a spherically symmetric source acts only on the internal ball at any given radius. Then, the radius of the shadow for a given value of the photon sphere could simply be calculated from the supposition that the dark matter only inside the photon sphere acts, while the dark matter outside the photon sphere can be completely ignored. Then, the Schwarzschild formula for the radius of the shadow sinα sh r O = 3 √ 3M could simply be altered by adding the fraction of ∆M which lies inside the photon sphere. In other words, using the expression for the mass function (2) the radius of the shadow would be However, the roots given by this equations differs from the one obtained earlier via direct differentiation of h(r) 2 , as can be seen on fig. (6). In this way we can see that the black-hole shadow depends not only on the matter under the photon sphere, but also on the of matter outside it. However, once ∆r s is large, the Taylor expansion of the above expression produces exactly the same first three terms, but differs in higher order terms: Therefore, the matter under the photon sphere describes relatively small deviations from the Schwarzschild limit perfectly and even for strong deviations, when ∆M/M is so large, that the metric function is close to the creating a new position for the event horizon (for example, as on fig. (2) for ∆M/M ≈ 44), the difference in the position of shadow calculated from the correct formula and (31) is only about a few percents; though the full effect of the dark matter is about twice as large as a few percents. This confirms that once the deviation of the shadow from its Schwarzschild limit is not small, the matter outside the photon sphere cannot be ignored, when calculating the full effect.

V. DISCUSSIONS
Here we found an analytical expression for the radius of the black hole shadow, supposing a simple spherical configuration of dark matter around it. A robust estimates show that the dark matter is unlikely to manifest in the shadows of galactic black holes, unless its concentration near the black hole is abnormally high. Furthermore, if one believes that such a high concentration of dark matter is possible, he could study more accurate models for the distribution of dark matter, include rotation of a black hole and dark matter and consider various equations of state. High deviations from spherical distribution of dark matter in the halo (if confirmed) would apparently distort the shape of the shadow as well. It is worthwhile noticing that when the deviation of the radius of shadow from its Schwarzschild value is small, the dominant contribution into the position of a shadow is due to the dark matter under the photon sphere. Nevertheless, this is not so for large deviations from Schwarzschild limit and the matter outside the photon sphere cannot be ignored in that case.

ACKNOWLEDGMENTS
The author acknowledges the support of the grant 19-03950S of Czech Science Foundation (GAČR) and Alexander Zhidenko for useful discussions. This publication has been prepared with the support of the "RUDN University Program 5-100".