# Weak deflection angle of a dirty black hole

Reggie C. Pantig<EMAIL_ADDRESS>Physics Department, De La Salle
University-Manila, 2401 Taft Ave., 1004 Manila Philippines  Emmanuel T.
Rodulfo<EMAIL_ADDRESS>
###### Abstract

In this paper, we present the weak deflection angle in a Schwarzschild black
hole of mass $m$ surrounded by the dark matter of mass $M$ and thickness
$\Delta r_{s}$. The Gauss-Bonnet theorem, formulated for asymptotic
spacetimes, is found to be ill-behaved in the third-order of $1/\Delta r_{s}$
for very large $\Delta r_{s}$. Using the finite-distance for the radial
locations of the source and the receiver, we derived the expression for the
weak deflection angle up to the third-order of $1/\Delta r_{s}$ using Ishihara
(et al.) method. The result showed that the required dark matter thickness is
$\sim 2\sqrt{3mM}$ for the deviations in the weak deflection angle to occur.
Such thickness requirement is better by a factor of 2 as compared to the
deviations in the shadow radius ($\sim\sqrt{3mM}$). It implies that the use of
the weak deflection angle in detecting dark matter effects in one’s galaxy is
better than using any deviations in the shadow radius.

##  I Introduction

One of the fruitful achievements of the human mind is the general theory of
relativity by Albert Einstein, which relates the phenomena of gravitation to
the geometry of spacetime. One consequence of the theory is the existence of
black holes that have long remained to be a theoretical construct, until the
release of the first image of the black hole shadow at the center of the M87
galaxy on April 10, 2019 Akiyama  _et al._ (2019).

At present, dark matter is another entity that remains so mysterious and
elusive. The $\Lambda$CDM model of cosmology suggests that the content of our
universe is made up of 27$\%$ dark matter, which constitutes 85$\%$ of the
total mass Jarosik  _et al._ (2011). Using Earth-based laboratories, several
scientists attempted to detect dark matter by direct means, but they gave
inconsistent results with each other. (see Bernabei  _et al._ (2010) and
Aprile  _et al._ (2010)). Even long before these experiments, indirect search
through dark matter annihilation also revealed null results Desai  _et al._
(2004); Peirani  _et al._ (2004). It proves that dark matter detection is more
difficult compared to gravitational wave detection Abbott  _et al._ (2016).

Emerging theoretical efforts have shown alternative means of possible dark
matter detection using the changes in the silhouette of a black hole. The
spacetime of pure dark matter and black hole were combined rigorously by Xu in
Ref. Xu  _et al._ (2018), and this formalism of using dark matter density
profiles to black hole spacetime is applied immediately to our own and M87
galaxy Hou  _et al._ (2018a); Jusufi  _et al._ (2019). Several studies are
also present that analyze the black hole shadow and deflection angle under the
influence of dark matter modeled as a perfect fluid Hou  _et al._ (2018b);
Haroon  _et al._ (2019). However, Konoplya in Ref. Konoplya (2019) considered
these black hole metrics as highly model-dependent. Using a less model-
dependent and agnostic view on dark matter, he estimated the condition for
dark matter to have some notable effect on the shadow radius.

Not only the study of shadow from various black hole models has gained much
attention from many researchers, but also the study of gravitational lensing
(GL). GL has proved useful in probing many astrophysical objects. Long ago, it
is used to probe the coevolution of supermassive black holes (SMBH) and
galaxies Peng  _et al._ (2006). It is also used to probe exotic entities like
dark matter Trimble (1987); Metcalf and Madau (2001); Metcalf and Zhao (2002)
that permeates a whole galaxy, and even galaxy clusters Hoekstra and Jain
(2008); Ellis (2010). Astrophysical black holes, which can also be
approximately described by the Schwarzschild metric for useful estimates, are
studied extensively in terms of lensing and the relativistic images that it
produces Virbhadra and Ellis (2000); Virbhadra (2009).

Perhaps the most popular way to obtain the weak deflection angle is by using
the Gauss-Bonnet theorem (GBT) introduced in Ref. Gibbons and Werner (2008).
Since then, the GBT has been proven very useful in calculating the weak
deflection angle of various black hole models that demonstrate asymptotic
behavior Övgün  _et al._ (2018a, b); Övgün (2018); Övgün  _et al._ (2019a);
Övgün (2019a); Jusufi and Övgün (2018); Javed  _et al._ (2019). Other notable
studies also existed that includes dark matter, phantom matter, and
quintessential energy, Övgün (2019b); Övgün  _et al._ (2019b); Haroon  _et
al._ (2019); De Leon and Vega (2019) on their analysis of the weak deflection
angle. Gravitational lensing by exotic matter or energy, which possibly
deviate from some well-known equation of state, is also explored in Kitamura
_et al._ (2014); Nakajima  _et al._ (2014); Izumi  _et al._ (2013).

Calculation of the weak deflection angle for non-asymptotic spacetimes seems
problematic using the GBT as far as its asymptotic form is concerned.
Recently, an improved analysis in Ref. Ishihara  _et al._ (2016) made the
calculation possible by considering the finite distances of the source and
receiver. However, the correspondence between the GBT and finite-distance
involves the brute force of evaluating integrals, which contain the orbit
equation. This method, whether the spacetime is asymptotic or not, axially-
symmetric or not Ono  _et al._ (2019); Li and Övgün (2020); Zhang  _et al._
(2019), is also useful in strong gravitational lensing regime Ishihara  _et
al._ (2017); Azreg-Aïnou  _et al._ (2017).

It is interesting to investigate the effect of dark matter configuration used
in Ref. Konoplya (2019) on a different black hole phenomenon: weak
gravitational lensing. In particular, this paper seeks to derive the
expression for the weak deflection angle caused by the main property of dark
matter - its mass. Although that the deviation in the shadow radius already
gives us the idea that null geodesics are affected, it is interesting to find
out whether if the deviation in the black hole’s weak deflection angle offers
a better condition for dark matter detection.

We organize the paper as follows: Sect. II introduces the dirty Schwarzschild
metric alongside with the description of the simple dark matter model in
consideration. In Sect. III, we present three possible cases that show the
different positions of the source and receiver relative to the dark matter
distribution and utilize the GBT. In Sect. IV, we proceed to calculate the
weak deflection angle by assuming some finite-distance of the source and the
receiver using the Ishihara (et al.) method. Lastly, in Sect. V, we summarize
the results and indicate some possible future research direction. The metric
signature in this study is +2, and we use $G=c=1$.

##  II Dirty Schwarzschild black hole

The term ”dirty” black hole has its roots in Ref. Visser (1992, 1993); Macedo
_et al._ (2016); Leung  _et al._ (1997); Krauss  _et al._ (1996) which
describes a black hole surrounded by some astrophysical environment. These
dirty black holes can be categorized as follows: (a) those that came from a
specific Lagrangian, or a certain field theory which results to a metric as
the Einstein field equation is solved (for examples, see Refs. Shapere  _et
al._ (1991); Dowker  _et al._ (1992); Gibbons and ichi Maeda (1988); Allen
_et al._ (1990); Galt’sov and Ershov (1989); Lahiri (1992)), (b) generic black
hole metrics with sufficient generality Nielsen and Birnholz (2019) that came
from some hypothetical configuration (or derived from empirical data) of
astrophysical environment, and (c) dirty black holes which came from solutions
to a non-Einsteinian theory such as pseudo-complex general relativity Moffat
(1979); Hess and Greiner (2009); Mann and Moffat (1982).

Here, we give a brief overview and describe the dirty Schwarzschild black hole
used in Ref. Konoplya (2019), where the author studied dark matter effects on
the photonsphere and black hole shadow. The astrophysical environment in
consideration is a spherical shell of dark matter described only by its mass
M, inner radius $r_{s}$, and thickness $\Delta r_{s}$ (the subscript $s$
denotes shell). Further, the dark matter mass $M$ is treated as an additional
effective mass to the black hole while maintaining its non-interaction with
the electromagnetic field. Since the physical parameters of the dark matter
shell are hypothetical, such a dirty black hole falls to the second category
described above. The generic metric produced still has sufficient generality
because the black hole itself came from the vacuum solution of the Einstein
equation.

One can then assume a piecewise function to impose three domains Konoplya
(2019):

$\mathcal{M}(r)=\begin{cases}m,&r<r_{s};\\\ m+MG(r),&r_{s}\leq r\leq
r_{s}+\Delta r_{s};\\\ m+M,&r>r_{s}+\Delta r_{s}\end{cases}$ (1)

where

$G(r)=\left(3-2\frac{r-r_{s}}{\Delta r_{s}}\right)\left(\frac{r-r_{s}}{\Delta
r_{s}}\right)^{2}.$ (2)

The expression for $G(r)$ is chosen so that $\mathcal{M}(r)$ and
$\mathcal{M}^{\prime}(r)$ are continuous (see Fig. 1). Note also the
possibility of $M<0$.

Figure 1: Example of a choice for mass function. Here, $M=10m$, $r_{s}=2m$,
$\Delta r_{s}=20m$, $m=1$.

The Schwarzschild metric is one of the famous, yet simplest, vacuum solution
of the Einstein field equations. Consider surrounding it with a spherical
shell of dark matter whose parameters are described by Eq. (2). Then we have

$d{s}^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$ (3)

where the metric function $f(r)$ is now given by

$f(r)=1-\frac{2\mathcal{M}(r)}{r}.$ (4)

The general scenario is depicted in Fig. 2 where $r_{s}\neq r_{h}$, which
makes the piecewise function in Eq. (1) very clear. Considering the first
domain and if $r$ is between $r_{h}$ and $r_{s}$, the dark matter outside $r$
has no bearing to any black hole phenomena that we want to analyze. The
situation is completely equivalent as if there’s no dark matter surrounding
the black hole. Suppose that $r$ describes the photonsphere radius $r_{ph}$,
then we simply have the trivial expression $r_{ph}=3m$.

Figure 2: A representation of a black hole surrounded by a thin shell of dark
matter where $r_{s}\neq r_{h}$.

The third domain of Eq. (1) seems an innocent-looking condition where the
parameters $r_{s}$ and $\Delta r_{s}$ does not exist. The domain can be
interpreted in two ways: (1) an isolated black hole surrounded by some dark
matter shell, and (2) a black hole at the center of a galaxy (combined mass is
$m$) where both are surrounded by dark matter halo. Taking the example of
deriving the photonsphere radius, the former case gives $r_{ph}=3(m+M)$ as the
null geodesic has no way to cross the outer radius of the dark matter shell.
The study of the deflection angle is more applicable to the latter case, but
the third domain’s applicability rests on the assumption that the null
geodesic never crosses the region of dark matter halo. For this to happen, the
location of the source and the receiver must be tremendously remote from the
lensing galaxy. It is also easy to see why the first and third domains should
satisfy the Einstein equation.

For the second domain, the spacetime where the $r$-coordinate is located is
now different due to the introduction of dark matter mass $M$ and its physical
dimensions. Since the configuration is hypothetical, we don’t know the
specific field theory, which may lead to the stress-energy tensor that
produces such expression of the metric if the Einstein equation is solved. It
is also true for black hole spacetime fused with dark matter models whose
density profiles came from observational data Xu  _et al._ (2018); Hou  _et
al._ (2018a); Jusufi  _et al._ (2019), hence they can be considered as dirty
black holes. It is possible, however, to satisfy the Einstein equation. The
key is to express the Einstein tensor in terms of appropriately chosen
orthogonal bases along with the stress-energy tensor Azreg-Aïnou (2014).

Assuming that the new spacetime expressed through the second domain of Eq. (1)
satisfies the Einstein equation, one can proceed with the analysis of black
hole properties. It is found out that it gives a non-trivial expression for
the photonsphere radius inside the dark matter shell, and is also true even
with the approximation $\Delta r_{s}>>r_{s}$ Konoplya (2019). Nevertheless, it
gave relevant insights to the effect of dark matter on the photonsphere
radius, and as a consequence, to the deviations in the shadow radius. Note
that being in the piecewise relation, the equation for the second domain
cannot be applied for values of $r$ beyond its lower and upper bounds. An
observer inside the dark matter shell has a different reality condition
compared to being outside it, or as the first domain applies.

Looking at Eq. (2) again, the function allows $r_{s}$ to have values smaller
than $r=2m$. For simplicity, one can set $r_{s}=2m$ and assume that the dark
matter shell is static. By being static, it means that dark matter is not
affected by the radial pull of the black hole. Nevertheless, it amplifies the
change in the black hole’s geometry, which then manifests to the dynamics of
null and time-like particles.

The idea of an astrophysical environment that surrounds a black hole is common
in several studies Konoplya  _et al._ (2019); Cardoso and Pani (2017). The
reasons include (1) verify whether the deviation observed is due to the new
physics happening near the horizon, or due to some effect of the astrophysical
environment; and (2) directly examine and study the influence of astrophysical
environment to the geometry of black hole. Only recently that the mass
function in Eq. (1) has been used to make some estimates of dark matter
effects to the radius of the black hole shadow Konoplya (2019).

##  III Deflection angle in a dirty black hole using the Gauss-Bonnet theorem

Consider $D$ as a freely orientable two-dimensional curved surface described
by the Gaussian curvature $K$, and $dS$ is its infinitesimal area element (See
Fig. 3). It contains several boundaries that are differentiable curves,
denoted by $\partial D_{a}$ ($a=1,2,...,N$), with geodesic curvature
$\kappa_{g}$. Such boundary also contains the line element $\ell$ where its
sign remains consistent with the surface orientation. Let $\theta_{a}$
represents the exterior or jump angles along the vertices of the surface. Then
the Gauss-Bonnet theorem states that Ishihara  _et al._ (2016, 2017); Do Carmo
(2016); Klingenberg (2013)

$\iint_{D}KdS+\sum\limits_{a=1}^{N}\int_{\partial
D_{a}}\kappa_{g}d\ell+\sum\limits_{a=1}^{N}\theta_{a}=2\pi.$ (5)

Figure 3: Schematic diagram of a curved surface for Gauss-Bonnet theorem. The
interior angles are denoted by $\varepsilon$ while the exterior (or ”jump”)
angles are $\theta$.

In the weak lensing regime, the Gauss-Bonnet theorem proves to be a powerful
tool in calculating the weak deflection angle $\hat{\alpha}$ as long as the
spacetime is asymptotically flat. If the spacetime is static, and spherically
symmetric (SSS), $\kappa_{g}=0$ along the boundary curves and the second term
in Eq. (5) vanishes. Assuming asymptotic flatness of the spacetime being
considered, the weak deflection angle can be derived as Gibbons and Werner
(2008); Javed  _et al._ (2019); Ishihara  _et al._ (2016, 2017); Werner
(2012); Ono  _et al._ (2017)

$\hat{\alpha}=-\iint_{{}_{R}^{\infty}\square_{S}^{\infty}}KdS$ (6)

where $K$ is the Gaussian optical curvature that is integrated over the
quadrilateral ${}_{R}^{\infty}\square_{S}^{\infty}$, $dS$ is the surface area
element. These quantities that are important in GBT, are defined as follows:

$K=\frac{R_{r\phi r\phi}}{\gamma},$ (7) $dS=\sqrt{\gamma}drd\phi$ (8)

where $\gamma$ denotes the determinant of the spatial metric $\gamma_{ij}$ as
$i$ and $j$ run from 1 to 3. The spatial metric for an SSS spacetime can be
found by considering the null condition $ds^{2}=0$, and solving for $dt$
yields Övgün  _et al._ (2018a)

$dt=\sqrt{\gamma_{ij}dx^{i}dx^{j}}.$ (9)

Rewriting Eq. (3) as

$d{s}^{2}=A(r)dt^{2}+B(r)dr^{2}+C(r)d\theta^{2}+D(r,\theta)d\phi^{2},$ (10)

the definition of $\gamma_{ij}$ is given by

$\gamma_{ij}dx^{i}dx^{j}=\frac{1}{A(r)}\left(B(r)dr^{2}+C(r)d\theta^{2}+D(r,\theta)d\phi^{2}\right).$
(11)

Dealing only in equatorial orbits, the Gaussian curvature $K$, being related
to the two-dimensional Riemann tensor, can be written in terms of affine
connection as Haroon  _et al._ (2019); Övgün  _et al._ (2019a)

$K=\frac{1}{\sqrt{\gamma}}\left[\frac{\partial}{\partial\phi}\left(\frac{\sqrt{\gamma}}{\gamma_{rr}}\Gamma_{rr}^{\phi}\right)-\frac{\partial}{\partial
r}\left(\frac{\sqrt{\gamma}}{\gamma_{rr}}\Gamma_{r\phi}^{\phi}\right)\right]$
(12)

and using Eq. (10), $K$ can be written as Ono  _et al._ (2017)

$K=-\sqrt{\frac{A(r)^{2}}{B(r)D(r)}}\frac{\partial}{\partial
r}\left[\frac{1}{2}\sqrt{\frac{A(r)^{2}}{B(r)D(r)}}\frac{\partial}{\partial
r}\left(\frac{D(r)}{A(r)}\right)\right].$ (13)

With the area surface element $dS$ in Eq. (8), we can write Eq. (6) as

$\hat{\alpha}=\int_{0}^{\pi}\int_{r_{o}}^{\infty}\mathcal{K}drd\phi,$ (14)

where the Gaussian curvature term De Leon and Vega (2019) reads

$\mathcal{K}=-\frac{\partial}{\partial
r}\left[\frac{1}{2}\sqrt{\frac{A(r)^{2}}{B(r)D(r)}}\frac{\partial}{\partial
r}\left(\frac{D(r)}{A(r)}\right)\right].$ (15)

Also, $r_{o}$ denotes the radial distance of the photon’s closest approach to
the lensing black hole, and can be obtained through the orbit equation. For an
SSS metric such as the Schwarzschild metric, the photon’s orbit equation in
the equatorial plane reads

$\left(\frac{dr}{d\phi}\right)^{2}=\frac{D(r)(D(r)-A(r)b^{2})}{A(r)B(r)b^{2}}$
(16)

where $b$ is the impact parameter. For convenience and as usually done in
celestial mechanics, we can set $u=\frac{1}{r}$. Thus, the above can be
expressed as

$\left(\frac{du}{d\phi}\right)^{2}\equiv
F(u)=\frac{u^{4}D(u)(D(u)-A(u)b^{2})}{A(u)B(u)b^{2}}.$ (17)

The closest approach $u_{o}$ can be found by iteratively solving $u$ in Eq.
(17) and imposing the boundary condition
$\frac{du}{d\phi}\big{|}_{\phi=\frac{\pi}{2}}=0$. For example, for the
Schwarzschild metric, $u$ is given by

$u=\frac{\sin\phi}{b}+\frac{m}{b^{2}}(1+\cos^{2}\phi).$ (18)

With $u_{o}$, we can now recast Eq. (14) as

$\hat{\alpha}=\int_{0}^{\pi}\int_{0}^{u_{o}}-\frac{\mathcal{K}}{u^{2}}dud\phi.$
(19)

###  III.1 Case 1: $\mathcal{M}(r)=m$

Let $u_{S}$ and $u_{R}$ be the reciprocal of the source’s and receiver’s
radial distance from the lensing object. The use of Eq. (14) assumes that
these radial distances are so far away that the spacetime becomes
approximately Minkowskian. For the first domain in Eq. (1) to hold, the inner
radius $r_{s}$ of the dark matter shell should always larger than the source’s
and receiver’s radial distance (See Fig. 4). With the metric function taking
the form

$f(r)=1-\frac{2m}{r},$ (20)

the Gaussian curvature and the area element in the equatorial plane reads

$\displaystyle K$ $\displaystyle=-\frac{2m}{r^{3}}+\frac{3m^{2}}{r^{4}}$
$\displaystyle dS$ $\displaystyle=r+3m+\mathcal{O}(m^{2})$ (21)

respectively. Using Eq. (19), we find

$\displaystyle\hat{\alpha}$
$\displaystyle=\int_{0}^{\pi}\int_{0}^{\frac{\sin(\phi)}{b}+\frac{m}{b^{2}}(1+\cos^{2}\phi)}-\frac{\mathcal{K}}{u^{2}}dud\phi$
$\displaystyle=\int_{0}^{\pi}\int_{0}^{\frac{\sin(\phi)}{b}+\frac{m}{b^{2}}(1+\cos^{2}\phi)}2m+3m^{2}u+\mathcal{O}\left(m^{3}\right)dud\phi$
$\displaystyle=\frac{4m}{b}+\frac{15\pi
m^{2}}{4b^{2}}+\mathcal{O}\left(m^{3}\right)$ (22)

which is the weak deflection angle of a Schwarzschild black hole up to the
second-order in $m$.

Figure 4: Quadrilateral embedded in curved spacetime while enclosing dark
matter shell.

###  III.2 Case 2: $\mathcal{M}(r)=m+M$

Following the discussion in Sect. II about the possible interpretation of this
case, Fig. 5 shows the use of the third domain of Eq. (1). As we deal only
with a non-rotating black hole, the metric function is trivial:

$f(r)=1-\frac{2(m+M)}{r}.$ (23)

Using Eq. (14), we find

$\displaystyle\hat{\alpha}$
$\displaystyle=\int_{0}^{\pi}\int_{0}^{\frac{\sin(\phi)}{b}+\frac{m+M}{b^{2}}(1+\cos^{2}\phi)}-\frac{\mathcal{K}}{u^{2}}dud\phi$
$\displaystyle=\int_{0}^{\pi}\int_{0}^{\frac{\sin(\phi)}{b}+\frac{m+M}{b^{2}}(1+\cos^{2}\phi)}2(m+M)$
$\displaystyle+3(m+M)^{2}u+\mathcal{O}\left[(m+M)^{3}\right]dud\phi$
$\displaystyle=\frac{4(m+M)}{b}+\frac{15\pi(m+M)^{2}}{4b^{2}}+\mathcal{O}\left[(m+M)^{3}\right]$
(24)

The result in Eq. (III.2) shows clearly how the dark matter mass acts as an
additional effective mass to the black hole - it increases the value of the
weak deflection angle. It clearly shows how $M$ adds up to the spacetime
distortion that affects the path of the null geodesic. It also means that the
photon’s path never enters and leaves the dark matter shell as it travels from
the source to the receiver. Suppose that $M=0$ and we increase $m$, the null
geodesic would only shift to its new orbital equilibrium while being outside
the vicinity of the event horizon.

Figure 5: Geometrical configuration of the quadrilateral when S and R are
beyond the outermost radius of dark matter shell.

###  III.3 Case 3: $\mathcal{M}(r)=m+MG(r)$

In this scenario, the null geodesic should remain inside the dark matter shell
for the second domain in Eq. (1) to hold. Such restriction is to determine how
the weak deflection angle should behave, knowing that the dark matter beneath
or above the null geodesic can affect it. This scenario is the same as the
restriction imposed in Ref. Konoplya (2019) for the photonsphere radius.

Suppose that $\Delta r_{s}$ is fixed where its thickness is not yet comparable
to dark matter halos and accommodates both the source and the receiver inside
the shell. In calculating the weak deflection angle using Eq. (19), the
asymptotic condition must apply, but the problem is that the source and the
receiver locations exceed that of the shell’s outer radius. Hence, given Eq.
(1), such a situation invalidates the use of Eq. (2). Therefore, there is a
necessity for the approximation of $\Delta r_{s}$ that it must be very large.
Such approximation directs the analysis to the scenario involving the source
and the receiver inside a very large dark matter halo. This means there is an
assumption that $\Delta r_{s}>>\frac{1}{u_{S}}$ (and $u_{R}$). See Fig. 6.

Figure 6: Far approximation of S and R (i.e. $u_{S}<<1$ and $u_{S}<<1$) always
implies the approximation that $\Delta r_{s}>>\frac{1}{u_{S}}$ (or $u_{R}$)
for the 2nd domain in Eq. (1) to hold.

The orbit equation in this case is

$\displaystyle F(u)$ $\displaystyle=\frac{1}{b^{2}}+u^{2}(2mu-1)$
$\displaystyle+\frac{2M}{\Delta
r_{s}^{2}}(r_{s}u-1)^{2}\left(\frac{2r_{s}u}{\Delta r_{s}}-\frac{2}{\Delta
r_{s}}+3u\right).$ (25)

Note that there is no coupling between the black hole mass $m$ and dark matter
mass $M$ in Eq. (III.3), unlike the coupling between the spin parameter $a$
and $m$ in Kerr spacetime. The iterative solution of Eq. (III.3) to obtain
$u_{o}$ is then

$u=\frac{\sin\phi}{b}+\frac{m}{b^{2}}(1+\cos^{2}\phi)+\frac{3Mr_{s}^{2}}{b^{2}\Delta
r_{s}^{2}}.$ (26)

Hence, the integrand of Eq. (19), up to the third-order of $1/\Delta r_{s}$,
is expressed by

$\displaystyle\hat{\alpha}$
$\displaystyle=\int_{0}^{\pi}\int_{0}^{u_{o}}-\frac{\mathcal{K}}{u^{2}}dud\phi=\int_{0}^{\pi}\int_{0}^{u_{o}}\bigg{[}2m+\frac{6Mr_{s}^{2}}{\Delta
r_{s}^{2}}$ $\displaystyle-\frac{6mMr_{s}(2-3r_{s}u)}{\Delta
r_{s}^{2}}-\frac{4M}{\Delta r_{s}^{3}}\left(\frac{1}{u^{3}}-r_{s}^{3}\right)$
$\displaystyle-\frac{12mMr_{s}^{2}(1-r_{s}u)}{\Delta
r_{s}^{3}}+\mathcal{O}\left(m^{2},\frac{1}{\Delta
r_{s}^{4}}\right)\bigg{]}dud\phi.$ (27)

The result shows divergence in the 4th term if one attempts to evaluate the
integral. It means that in the third-order of $1/\Delta r_{s}$, if one insists
to be more precise in their calculations, the spacetime is not asymptotically
flat and GBT cannot be used. The same case also occur in other non-asymptotic
spacetime such as those that involved the cosmological constant such as the
Kottler spacetime Kottler (1918). However, the cosmological constant has no
adjustable parameters where an approximation can be made possible.

The condition $\Delta r_{s}>>\frac{1}{u}$ gives the approximate expression if
$\Delta r_{s}$ is very large, at least comparable to the known size of dark
matter halos. Therefore, the manifestation of dark matter effects to the weak
deflection angle can be seen to begin with the second-order of $1/\Delta
r_{s}$. Fortunately, there is asymptotic flatness and proceeding with the GBT
calculation, we find

$\displaystyle\hat{\alpha}$
$\displaystyle=\int_{0}^{\pi}\int_{0}^{u_{o}}-\frac{\mathcal{K}}{u^{2}}dud\phi=\int_{0}^{\pi}\int_{0}^{u_{o}}\bigg{[}2m+\frac{6Mr_{s}^{2}}{\Delta
r_{s}^{2}}$ $\displaystyle-\frac{6mMr_{s}(2-3r_{s}u)}{\Delta
r_{s}^{2}}\bigg{]}dud\phi=\frac{4m}{b}+\frac{12Mr_{s}^{2}}{b\Delta r_{s}^{2}}$
$\displaystyle-\frac{24mMr_{s}}{b\Delta r_{s}^{2}}+\frac{15\pi
m^{2}}{4b^{2}}+\frac{39\pi mMr_{s}^{2}}{2b^{2}\Delta r_{s}^{2}}.$ (28)

showing higher-order terms. It is now a matter of question whether the third
term in Eq. (III.3) can also be neglected due to the nature of dark matter
model used in this study. To find out, we will use Ishihara (et al.) method
Ishihara  _et al._ (2016) to purposely show that we can compute the weak
deflection angle up to the third-order in $1/\Delta r_{s}$, and compare the
result to Eq. (III.3) at least in second-order of $1/\Delta r_{s}$. The
formalism developed is very useful in dealing with Kottler spacetime and
Schwarzschild-like solutions in the Weyl conformal gravity Mannheim and
Kazanas (1989) because it assumes that $u_{S}$ and $u_{R}$ are at finite
distance from the black hole. The computation using this method will be the
subject of the next section.

##  IV Deflection angle using finite distance

In this section, we use the method in Ref. Ishihara  _et al._ (2016) to
calculate the weak deflection angle of a black hole under the influence of
dark matter, as demonstrated in Fig. 6. From the GBT, the generalized
correspondence between the deflection angle and the surface integral of the
Gaussian curvature reads

$\hat{\alpha}=\phi_{RS}+\Psi_{R}-\Psi_{S}$
$=\int_{u_{R}}^{u_{o}}\frac{1}{\sqrt{F(u)}}du+\int_{u_{S}}^{u_{o}}\frac{1}{\sqrt{F(u)}}du+\Psi_{R}-\Psi_{S}$
(29)

where $F(u)$ and $u_{o}$ are given by Eq. (III.3) and Eq. (26) respectively.
The angles $\Psi_{S}$ and $\Psi_{R}$ in Eq. (29) are determined through the
inner product of the unit basis vector in the spacetime considered, and the
unit vector with respect to the lensing object. The unit basis vector $e^{i}$,
along the equatorial plane, is given by

$e^{i}=\left(\frac{dr}{dt},0,\frac{d\phi}{dt}\right)=\frac{d\phi}{dt}\left(\frac{dr}{d\phi},0,1\right)$
(30)

while the unit radial vector, which is along the radial direction from the
lens is

$R^{i}=\left(\frac{1}{\sqrt{\gamma_{rr}}},0,0\right).$ (31)

Hence, the inner product suggests the definition

$\cos\Psi\equiv\gamma_{ij}e^{i}R^{j}$
$\cos\Psi=\sqrt{\gamma_{rr}}\frac{A(r)b}{D(r)}\frac{dr}{d\phi}.$ (32)

Using $F(u)$ in Eq. (17), it is easy to see that

$\sin\Psi=\sqrt{\frac{A(r)}{D(r)}}b$ (33)

where it clear that it is favorable to use $\sin\Psi$ than using $\cos\Psi$.

Taylor expansion of $\Delta r_{s}$ as it is very large resulted to an angle
$\Psi$ calculated as

$\Psi=\arcsin(bu\sqrt{1-2mu})$ $-\frac{3bM}{\Delta
r_{s}^{2}}\frac{(r_{s}u-1)^{2}}{\sqrt{1-2mu}\sqrt{b^{2}u^{2}(2mu-1)+1}}$
$-\frac{2bM}{\Delta
r_{s}^{3}}\frac{(r_{s}u-1)^{3}}{u\sqrt{1-2mu}\sqrt{b^{2}u^{2}(2mu-1)+1}}+\mathcal{O}\left(\frac{1}{\Delta
r_{s}^{4}}\right).$ (34)

Continuing, we find that

$\Psi_{R}-\Psi_{S}=(\Psi_{R}^{\text{Schw}}-\Psi_{S}^{\text{Schw}})-\frac{3bM}{\Delta
r_{s}^{2}}\left[\frac{(r_{s}u_{R}-1)^{2}}{\sqrt{1-b^{2}u_{R}^{2}}}+\frac{(r_{s}u_{S}-1)^{2}}{\sqrt{1-b^{2}u_{S}^{2}}}\right]$
$-\frac{2bM}{\Delta
r_{s}^{3}}\left[\frac{(r_{s}u_{S}-1)^{3}}{u_{R}\sqrt{1-b^{2}u_{R}^{2}}}+\frac{(r_{s}u_{S}-1)^{3}}{u_{S}\sqrt{1-b^{2}u_{S}^{2}}}\right]$
$+\frac{3bmM}{\Delta
r_{s}^{2}}\left[\frac{u_{R}\left(2b^{2}u_{R}^{2}-1\right)(r_{s}u_{R}-1)^{2}}{\left(1-b^{2}u_{R}^{2}\right)^{3/2}}+\frac{u_{S}\left(2b^{2}u_{S}^{2}-1\right)(r_{s}u_{S}-1)^{2}}{\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\right]$
$+\frac{2bmM}{\Delta
r_{s}^{3}}\left[\frac{\left(2b^{2}u_{R}^{2}-1\right)(r_{s}u_{R}-1)^{3}}{\left(1-b^{2}u_{R}^{2}\right)^{3/2}}+\frac{\left(2b^{2}u_{S}^{2}-1\right)(r_{s}u_{S}-1)^{3}}{\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\right]+\mathcal{O}\left(\frac{1}{\Delta
r_{s}^{4}}\right)$ (35)

where

$(\Psi_{R}^{\text{Schw}}-\Psi_{S}^{\text{Schw}})=[\arcsin(u_{R}b)+\arcsin(u_{S}b)-\pi]$
$-bm\left[\frac{u_{R}^{2}}{\sqrt{1-b^{2}u_{R}^{2}}}+\frac{u_{S}^{2}}{\sqrt{1-b^{2}u_{S}^{2}}}\right].$
(36)

Note the above expression contains a divergent term in the third-order of
$1/\Delta r_{s}$. Hence, the spacetime caused by the combination of black hole
and dark matter with very large $\Delta r_{s}$ is non-asymptotic and the limit
$u_{S}\rightarrow 0$ and $u_{R}\rightarrow 0$ is not allowed.

We now proceed to calculate the $\phi_{RS}$ part by evaluating the integrals
in Eq. (29). The function $F(u)$ in Eq. (III.3) results to an integrand of the
form

$\frac{1}{\sqrt{F(u)}}=\frac{b}{\sqrt{1-b^{2}u^{2}}}-\frac{b^{3}u^{3}m}{\left(1-b^{2}u^{2}\right)^{3/2}}$
$-\frac{3Mb^{3}}{\Delta
r_{s}^{2}}\frac{u(r_{s}u-1)^{2}}{\left(1-b^{2}u^{2}\right)^{3/2}}+\frac{9b^{5}mMu^{4}}{\Delta
r_{s}^{2}}\frac{(r_{s}u-1)^{2}}{\left(1-b^{2}u^{2}\right)^{5/2}}$
$-\frac{2Mb^{3}}{\Delta
r_{s}^{3}}\frac{(r_{s}u-1)^{3}}{\left(1-b^{2}u^{2}\right)^{3/2}}+\frac{6b^{5}mMu^{3}}{\Delta
r_{s}^{3}}\frac{(r_{s}u-1)^{3}}{\left(1-b^{2}u^{2}\right)^{5/2}}+\mathcal{O}\left(\frac{1}{\Delta
r_{s}^{4}}\right).$ (37)

The evaluation of the integral gives a cumbersome expression and reveals no
divergence especially in the third-order of $1/\Delta r_{s}$ and hence, can be
safely omitted for brevity. Due to Eq. (29), the expression for the source and
receiver are essentially the same, and that is

$\int\frac{1}{\sqrt{F(u)}}du=\arcsin
bu+\frac{m}{b}\frac{\left(b^{2}u^{2}-2\right)}{\sqrt{1-b^{2}u^{2}}}$
$+\frac{9M}{\Delta
r_{s}^{2}}\left[\left(m-\frac{2r_{s}}{3}\right)+\frac{5mr_{s}^{2}}{2b^{2}}\right]\arcsin(bu)-\frac{3M}{b\Delta
r_{s}^{2}}\frac{\left[b^{2}\left(-r_{s}^{2}u^{2}-2r_{s}u+1\right)+2r_{s}^{2}\right]}{\sqrt{1-b^{2}u^{2}}}$
$-\frac{3mMr_{s}}{2b\Delta
r_{s}^{2}}\frac{\left(48r_{s}b^{2}u^{2}+6b^{2}u+15r_{s}^{2}u-32r_{s}\right)}{\left(1-b^{2}u^{2}\right)^{3/2}}+\mathcal{O}\left(\frac{1}{\Delta
r_{s}^{3}}\right)+C$ (38)

where C is a constant. The expression for $\phi_{RS}$ includes the sum of two
evaluated integrals:

$\phi_{RS}=\phi_{RS}^{\text{Schw}}-\frac{9M}{\Delta
r_{s}^{2}}\left[\left(m-\frac{2r_{s}}{3}\right)+\frac{5mr_{s}^{2}}{2b^{2}}\right]\left[\arcsin(bu_{R})+\arcsin(bu_{S})\right]$
$+\frac{3M}{b\Delta
r_{s}^{2}}\left\\{\frac{\left[b^{2}\left(-r_{s}^{2}u_{R}^{2}-2r_{s}u_{R}+1\right)+2r_{s}^{2}\right]}{\sqrt{1-b^{2}u_{R}^{2}}}+\frac{\left[b^{2}\left(-r_{s}^{2}u_{S}^{2}-2r_{s}u_{S}+1\right)+2r_{s}^{2}\right]}{\sqrt{1-b^{2}u_{S}^{2}}}\right\\}$
$\displaystyle+\frac{3mMr_{s}}{2b\Delta
r_{s}^{2}}\bigl{[}\frac{\left(48r_{s}b^{2}u_{R}^{2}+6b^{2}u_{R}+15r_{s}^{2}u_{R}-32r_{s}\right)}{\left(1-b^{2}u_{R}^{2}\right)^{3/2}}$
$\displaystyle+\frac{\left(48r_{s}b^{2}u_{S}^{2}+6b^{2}u_{S}+15r_{s}^{2}u_{S}-32r_{s}\right)}{\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\bigr{]}+\mathcal{O}\left(\frac{1}{\Delta
r_{s}^{3}}\right)$ (39)

where we introduced

$\phi_{RS}^{\text{Schw}}=\pi-\arcsin(u_{R}b)-\arcsin(u_{S}b)-\frac{m}{b}\left[\frac{\left(b^{2}u_{R}^{2}-2\right)}{\sqrt{1-b^{2}u_{R}^{2}}}+\frac{\left(b^{2}u_{S}^{2}-2\right)}{\sqrt{1-b^{2}u_{S}^{2}}}\right].$
(40)

We can finally calculate the deflection angle $\hat{\alpha}$ using Eq. (29).
Combining Eq. (35) and Eq. (IV), we find

$\hat{\alpha}\approx\frac{2m}{b}\left[\sqrt{1-b^{2}u_{R}^{2}}+\sqrt{1-b^{2}u_{S}^{2}}\right]$
$-\frac{9M}{\Delta
r_{s}^{2}}\left[\left(m-\frac{2r_{s}}{3}\right)+\frac{5mr_{s}^{2}}{2b^{2}}\right]\left[\arcsin(bu_{R})+\arcsin(bu_{S})\right]$
$+\frac{6Mr_{s}^{2}}{b\Delta
r_{s}^{2}}\left[\sqrt{1-b^{2}u_{R}^{2}}+\sqrt{1-b^{2}u_{S}^{2}}\right]-\frac{2bM}{\Delta
r_{s}^{3}}\left[\frac{(r_{s}u_{R}-1)^{3}}{u_{R}\sqrt{1-b^{2}u_{R}^{2}}}+\frac{(r_{s}u_{S}-1)^{3}}{u_{S}\sqrt{1-b^{2}u_{S}^{2}}}\right]-$
$\frac{3mM}{2\Delta
r_{s}^{2}}\left\\{\frac{\left[4b^{2}u_{R}+r_{s}(52b^{2}u_{R}^{2}+15r_{s}u_{R}-32)\right]}{b\left(1-b^{2}u_{R}^{2}\right)^{3/2}}+\frac{\left[4b^{2}u_{S}+r_{s}(52b^{2}u_{S}^{2}+15r_{s}u_{S}-32)\right]}{b\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\right\\}$
$+\frac{2bmM}{\Delta
r_{s}^{3}}\left[\frac{\left(2b^{2}u_{R}^{2}-1\right)(r_{s}u_{R}-1)^{3}}{\left(1-b^{2}u_{R}^{2}\right)^{3/2}}+\frac{\left(2b^{2}u_{S}^{2}-1\right)(r_{s}u_{S}-1)^{3}}{\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\right]+\mathcal{O}\left(\frac{1}{\Delta
r_{s}^{4}}\right).$ (41)

The next procedure is to impose the limit $u_{S}\rightarrow 0$ and
$u_{S}\rightarrow 0$ in Eq. (41) which is not problematic in asymptotic
spacetimes. However, this limit is not applicable or allowed in Eq. (41) since
$\hat{\alpha}$ will apparently diverge. Hence, the reciprocal of $u_{S}$ and
$u_{R}$ can be interpreted as finite distances but with very small values. For
the far source and receiver, it is then safe to impose $u_{S}<<1$ and
$u_{R}<<1$ Ishihara  _et al._ (2016); Haroon  _et al._ (2019). Eq. (41) then
becomes

$\hat{\alpha}\approx\frac{4m}{b}+\frac{12Mr_{s}^{2}}{b\Delta
r_{s}^{2}}-\frac{96mMr_{s}}{b\Delta r_{s}^{2}}$ $+\frac{2Mb}{\Delta
r_{s}^{3}}\left[\left(\frac{1}{u_{R}}+\frac{1}{u_{S}}\right)+2m\right]+\mathcal{O}\left(\frac{1}{\Delta
r_{s}^{4}}\right).$ (42)

Comparing this result to Eq. (III.3), we see that the first two terms are
identical, except the third term, and the existence of terms in the third-
order of $1/\Delta r_{s}$ which cannot be derived from the GBT given by Eq.
(14). Dark matter mass, being an additional effective mass, makes the $mM$
term to be interpreted as a higher-order term in mass and hence, can safely be
neglected. The significant contribution to the weak deflection angle only
arises from the first two terms. Moreover, the source and the receiver are
inside the dark matter halo, hence $\Delta r_{s}>>1/u_{S}$ and $\Delta
r_{s}>>1/u_{R}$ is possible. Hence, we are left with

$\hat{\alpha}\approx\frac{4m}{b}+\frac{12Mr_{s}^{2}}{b\Delta r_{s}^{2}}$ (43)

and we see that dark matter has an effect to increase the value of the weak
deflection angle. See Fig. (7). Moreover, the abnormality of such increase in
the weak deflection angle due to the very small $\Delta r_{s}$ might constrain
the value of dark matter mass $M$. The weak deflection angle is asymptotic to
the Schwarzschild case as $\Delta r_{s}$ increases, as expected. Note that if
the mass $M$ is negative, which might represent an exotic matter with negative
kinetic term, the weak deflection angle decreases.

Figure 7: Weak deflection angle as $\Delta r_{s}$ varies. Here, $r_{s}=2m$
which coincides with the event horizon, and $b=1000m$.

We recall the estimate in Ref. Konoplya (2019) that for notable dark matter
effects to occur in the shadow radius, the effective radius of the dark matter
distribution must be in the order of $\Delta r_{s}\sim\sqrt{3mM}$. This result
is found by the same analysis using Fig. 6. For a given value of $M$, it
revealed a very small value of $\Delta r_{s}$, which implies that dark matter
must be concentrated near the black hole to change the shadow radius
considerably. Further analysis also revealed that when the dark matter density
is very low ($\Delta r_{s}$ is very large), the effect of dark matter outside
the photon’s orbit can be safely neglected. Hence, the dark matter mass
beneath the photon’s orbit is the main contributor to any deviations. Such a
previous conclusion synchronizes with the results in Eq. (III.3) and Eq. (42).
Since dark matter acts as an additional effective mass to the black hole, we
can interpret the $mM$ term as a higher-order term in mass, and their coupling
can be ignored.

Surprisingly, Eq. (43) reveals that dark matter effects will occur in the weak
deflection angle (as deviation) when $\Delta r_{s}=2\sqrt{3mM}$. Since there
is an increase in the thickness requirement, dark matter detection via
deviations in the weak deflection angle of a black hole is better compared
when a deviation in shadow radius is used. Unfortunately, however, such an
increase is still irrelevant, at least for the technological capabilities we
have today. Consider an estimate for dark matter mass in our galaxy which is
$M\approx 1.0\times 10^{12}M_{\odot}$ Battaglia  _et al._ (2005) while the
mass of the central black hole is around $m\approx 4.3\times 10^{6}M_{\odot}$.
It gives a required dark matter thickness of $\Delta r_{s}\approx 6.13\times
10^{12}$ m $\approx 2\times 10^{-4}$ pc to see any changes in the weak
deflection angle. Such value is lower in many orders of magnitude even if we
compare it to the core radius of the dark matter halo present in our galaxy
($r_{o}\approx 15.7-17.46$ kpc) de Oliveira  _et al._ (2015).

##  V Conclusion

In this paper, we present an analytic formula for the weak deflection angle
using a simple dark matter model that only incorporates its basic features,
such as mass and physical parameters. It is shown that the GBT, with its form
given by Eq. (19), can be used only to the second-order of $1/\Delta r_{s}$,
but is ill-behaved in the third-order of $1/\Delta r_{s}$. If precision in the
calculation matters, the resulting apparent divergence is found and justified
using Ishihara (et al.) method. The expression containing the second-order in
$1/\Delta r_{s}$ represents the approximate condition where the initial
manifestation of dark matter effect occurs as a deviation in the weak
deflection angle. We found that it is twice that of shadow radius deviations.
Interestingly, if one seeks to detect dark matter using the central black hole
in one’s galaxy, using the weak deflection angle is better than observing
deviations in the shadow radius. Although being better, the deviation is still
very small to be detected by current technology.

Extensions of the current study to non-spherical, or non-static dark matter
distribution, surrounding a more complicated black hole metrics, are left for
future work.

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