The effect of dark matter spikes on galactic black hole ringdown waveforms and
shadows

Ramin G. Daghigh1 and Gabor Kunstatter2

1 Natural Sciences Department, Metropolitan State University, Saint Paul,
Minnesota, USA 55106

2 Physics Department, University of Winnipeg, Winnipeg, MB Canada R3B 2E9

###### Abstract

We investigate the effect of dark matter spikes on the ringdown waveforms and
shadows of the supermassive black holes at the core of the Milky Way and M87.
The dominant effect comes in the form of a redshift factor due to the mass of
the spike and is significant for large supermassive black holes such as M87*.
The subdominant contributions, recently calculated in ApJ 940 33 (2022), give
information about the distribution of matter in the spike. We show that these
subdominant contributions are small as compared to the redshift factor and
likely difficult to detect except perhaps for very large black holes. We also
show that the effect of a dark matter spike, with density near the
observational upper bounds, on the shadow radius of M87* is only an order of
magnitude smaller than is currently accessible with Event Horizon Telescope
data.

##  1 Introduction

Dark matter (DM) halos are known to surround most, if not all, supermassive
black holes at the center of galaxies. It has been proposed that DM spikes
exist at the core of galactic halos [1, 2, 3, 4]. Several papers [5, 6, 7, 8,
9, 10] have recently derived the effects of DM spikes on the gravitational
waves emitted from black holes, but until recently the prospect of detecting
these waves from supermassive black holes has been remote. This has now
changed due to recent experimental developments.

The North American Nanohertz Observatory for Gravitational Waves
(NANOGrav)[11], Parkes Pulsar Timing Array (PPTA)[12], and European Pulsar
Timing Array (EPTA)[13] have found a common background noise at frequencies
around $10^{-8}$ Hz. The very recent observations of Hellings–Downs
correlations by the Chinese Pulsar Timing Array (CPTA) collaboration [14],
EPTA collaboration [15], NANOGrav collaboration [16] and PPTA collaboration
[17] strongly suggest a stochastic gravitational-wave origin of this
background noise. Supermassive black hole binaries are among the main
candidates to produce this background gravitational wave in the nHz range. A
recent paper[18] has looked for signatures of DM spikes surrounding
supermassive black holes in this data. Although these observations are still
very far from being able to measure ringdown waveforms from individual
supermassive black holes, it is nonetheless important to determine accurately
the effect of the DM distributions on these waveforms.

Another important gravitational signature of black holes, namely the radius of
the photon ring, has recently been measured for M87 using the remarkable
images of the central black hole’s shadow obtained by the Event Horizon
Telescope [19]. Moreover, the authors of [20] have used novel statistical
methods to reduce the error in this measurement to approximately $1\%$,
leading to the possibility that measurements of the photon ring might one day
yield information about the DM distribution surrounding supermassive black
holes.

The shadow size of the galactic black hole Sagittarius A* (Sgr A*) in the
presence of a DM spike was investigated in [21].111Similar calculations for
the shadow size of black holes surrounded by DM halos were done in [22, 23].
In a more recent paper [24], the present authors studied the impact of the DM
spike on ringdown waveforms for the supermassive black holes Sgr A* and M87*.
Using an approach similar to that of [24] in solving the Tolman-Oppenheimer-
Volkoff (TOV) equations, the authors of [25] calculated the quasinormal modes
(QNMs) that dominate the gravitational ringdown process of perturbed black
holes surrounded by DM spikes. In these previous papers, the calculations were
done in terms of the Schwarzschild time of the central black hole as measured
by an observer located between the black hole event horizon and the inner
radius of the spike.

In the following, we calculate the effect of DM spikes on the ringdown
waveforms and shadows of the supermassive black holes at the center of Milky
Way and M87 galaxies. We emphasize the importance of expressing the results in
the frame of an asymptotic stationary observer for obtaining the correct
predictions222The authors are grateful to Andrei Frolov for illuminating
discussions in this regard.. Using the Schwarzschild time within the inner
radius of the spike effectively neglects an overall redshift factor that
provides the dominant contribution. Ignoring this redshift, as done in [24],
underestimates considerably the size of the effect.

In the context of shadows, we confirm that the presence of a DM spike
increases the size of the black hole shadow[21, 24]. This calculation has
taken on new significance given the recent results of [20] for M87. We point
out that for the spike density estimated in [24]333We note a typo in Eqs. (30)
and (31) of [24], where the authors estimate the DM spike density based on
observational data. More specifically, the term $\left(1+r_{0}/r\right)$
should be $\left(1+r/r_{0}\right)$. The numerical results, however, are
correct. based on the observational data provided in [26], the increase of the
shadow size due to a DM spike can be of the order of $0.1\%$. While not
detectable with current data, this is just an order of magnitude less than the
proposed error bars of [20].

The paper is organized as follows. Section 2 derives the metric and describes
the subtleties associated with the choice of reference frame. Section 3
derives the wave equation, which is covariant under the corresponding change
of frame. The resulting solution for the waveform is a scalar whose shape
depends on the choice of coordinates. Section 4 presents a discussion of
galactic black holes with DM spikes and their associated ringdown waveforms.
Section 5 reviews the shadow calculations and conclusions are given in Section
6. Appendix A is devoted to deriving an approximate expression for the
discontinuities in the QNM potential that necessarily exist at the boundaries
of a DM spike. The discontinuities are proportional to the change in matter
density at the boundaries and negligibly small for the proposed spike
profiles.

##  2 Deriving the Spacetime Metric

In order to calculate ringdown waveforms that might emerge from compact
objects, one first needs to determine the static background metric based on
the conjectured matter content surrounding the object. We are thinking here of
DM halos/spikes. This requires solving the TOV equations with appropriate
boundary conditions.

One starts with the most general $4$-D spherically symmetric static metric (up
to coordinate transformations)

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-A(r)dt^{2}+B(r)^{-1}dr^{2}+r^{2}d\Omega^{2}$ (1)
$\displaystyle=$
$\displaystyle-e^{\mu(r)}dt^{2}+\left(1-\frac{2M(r)}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$

where we use geometric units with $c=G=1$. For asymptotically flat metrics,
${e^{\mu(r)}\to\text{constant}}$ at spatial infinity. This constant is
normally chosen to be unity thereby fixing the time coordinate to be the
proper time of a stationary asymptotic observer. This is the correct choice
when comparing waves emitted by distant compact objects, despite the fact that
it has not always been used in recent discussions of ringdown waveforms.

In order to emphasize the importance of choosing the observer’s frame, we
outline the derivation, using the TOV equations, of the metric in the
spacetime containing an extended but finite spherical shell of matter
surrounding a black hole. See [24] for more details on the derivation . We
denote the inner radius of the shell by $r_{\text{in}}$ and the outer radius
by $r_{\text{out}}$.

  1. 1.

Assume a perfect fluid stress tensor for the shell of matter

$\displaystyle T^{\mu}_{\nu}=\hbox{diag}(\rho(r),p(r),p(r),p(r)).$ (2)

This yields the TOV equations (see [27]) in the shell region:

$\displaystyle G_{tt}$ $\displaystyle=0$
$\displaystyle\implies\frac{dM(r)}{dr}=4\pi r^{2}\rho(r)$ (3) $\displaystyle
G_{rr}$ $\displaystyle=0$
$\displaystyle\implies\frac{d\mu(r)}{dr}=2\frac{M(r)+4\pi
r^{3}p(r)}{r\left[r-2M(r)\right]}$ (4) $\displaystyle\partial_{\nu}T^{r\nu}$
$\displaystyle=$ $\displaystyle
0\implies\frac{dp(r)}{dr}=-[\rho(r)+p(r)]\frac{M(r)+4\pi
r^{3}p(r)}{r\left[r-2M(r)\right]}$ (5)

  2. 2.

We have three equations in four unknowns $[\mu(r),M(r),\rho(r),p(r)]$, so they
need to be supplemented by a fourth equation. Normally this is taken to be the
equation of state relating $\rho$ to $p$. In the cases we are interested in,
such as DM spikes, theoretical calculations [2] predict a particular density
profile for the shell. This provides the extra equation. Therefore, there is
no freedom left to specify the equation of state. For simplicity, we assume
that the pressure is negligible when solving for $\mu(r)$ in Eq. (4).444An
alternative assumption is that of zero radial pressure, but non-zero
transverse pressure which describes an anisotropic fluid. This yields an exact
solution for the metric in the spike region[28].  The validity of this
assumption in the case of DM spikes was proven in [24].

  3. 3.

Given the density profile, one can obtain the mass function by integrating Eq.
(3) from $r_{\text{BH}}$ to spatial infinity with the boundary condition that
$M(r<r_{\text{in}})=M_{\text{BH}}$, where we use $r_{\text{BH}}$ and
$M_{\text{BH}}$ to indicate the horizon radius and mass of the central black
hole respectively.

  4. 4.

Given the mass function obtained above, one integrates Eq. (4) to get the
second metric function $\mu(r)$.

  5. 5.

Since the TOV equations are first order, each requires the specification of a
single boundary condition in order to provide a unique solution. For Eq. (3),
the correct boundary condition is that $M(r<r_{\text{in}})=M_{\text{BH}}$.
Equivalently, one can choose ${M(r\geq
r_{\text{out}})=M_{\text{total}}=M_{\text{BH}}+M^{\text{shell}}_{\text{total}}}$,
where $M^{\text{shell}}_{\text{total}}$ is the total mass of the shell. Since
the tangential metric must be continuous at the boundaries for the surface
geometry on the boundary to be uniquely defined, there is no freedom in the
choice of boundary conditions.

  6. 6.

A potential subtlety does arise in the choice of boundary condition for
$g_{{t}{t}}(r)$ (and consequently for $\mu(r)$) in Eq. (4), which in turn
determines the time coordinate. The obvious choice is that $g_{{t}{t}}(r)\to
1$ ($\mu(r)\to 0$) far from the black hole so that ${t}$ is the proper time of
an asymptotic observer at rest relative to the black hole. This is presumably
the frame in which the measurements of the ringdown waveform and shadow are
made. On the other hand, it is useful to consider the ringdown waveforms
caused by outgoing waves that originate in the vacuum region
$r<r_{\text{in}}$, where the initial data are specified. Such outgoing
boundary conditions are preferred for two reasons: first, they are in some
sense more physical since the ringdown waveforms from the merger of binary
black holes do originate from near the horizon. Secondly, numerical
integrations of the wave equation yield better results (i.e. a larger number
of reliable oscillations before errors set in) than simulations that start
from ingoing waves. When starting with outgoing initial data, as done in [24],
it seems natural to impose the boundary condition on the metric in the shell’s
interior, namely

$g_{tt}(r<r_{\text{in}})=1-\frac{2M_{BH}}{r}\,,$ (6)

which is the Schwarzschild metric of the central black hole. We emphasize the
fact that although this $t$ is not the proper time of a stationary asymptotic
observer in the presence of the spike, it nonetheless does have a physical
interpretation as explained below.

Following [24] we integrate outward starting at $r_{\text{in}}$ with the
boundary condition (6), which fixes the time coordinate $t$ to be the
Schwarzschild time of the vacuum metric associated with the central black
hole. In this case, $t$ is what would be the proper time of a stationary
observer at infinity had the DM spike not been present. This is not the same
as the actual proper time of an asymptotic observer when the spike is present.
It is the latter proper time with respect to which measurements will be taken.
We can rewrite Eq. (6) in terms of $\mu(r)$ as

$\mu(r<r_{\text{in}})=\ln\left(1-\frac{2M_{\text{BH}}}{r}\right),$ (7)

where clearly $\mu(r)\rightarrow 0$ as $r\rightarrow\infty$.

The integration of Eq. (4) proceeds to larger radii with the assumption that
the metric function be continuous at both shell boundaries. This is required
to ensure that the shell boundary has a well-defined geometry.555Note that
$\mu(r)$ is not necessarily smooth at the boundaries. This introduces
discontinuities in the QNM potential, which can be shown to be small in the
present context. For details, see Appendix A. In the shell region
$r_{\text{in}}\leq r<r_{\text{out}}$, we find the metric function to be

$\mu(r)=\ln\left(1-\frac{2M_{\text{BH}}}{r_{\text{in}}}\right)+\int_{r_{\text{in}}}^{r}dr\frac{2M(r)}{r(r-2M(r))}~{}.$
(8)

To obtain a solution that is continous at the outer radius of the shell we
extend the integral in Eq. (8) past $r_{\text{out}}$ as follows:

$\displaystyle\mu(r\geq r_{\text{out}})$ $\displaystyle=$
$\displaystyle\ln\left(1-\frac{2M_{\text{BH}}}{r_{\text{in}}}\right)+\int_{r_{\text{in}}}^{r_{\text{out}}}dr\frac{2M(r)}{r(r-2M(r))}+\int_{r_{\text{out}}}^{r}dr\frac{2M_{\text{total}}}{r(r-2M_{\text{total}})}$
(9) $\displaystyle=$
$\displaystyle\ln\left(1-\frac{2M_{\text{total}}}{r}\right)+\ln(\mathcal{C})~{},$

where

${\mathcal{C}}=\frac{\left(1-\frac{2M_{\text{BH}}}{r_{\text{in}}}\right)}{\left(1-\frac{2M_{total}}{r_{\text{out}}}\right)}\exp{\int_{r_{\text{in}}}^{r_{\text{out}}}dr\frac{M(r)}{r(r-2M(r))}}.$
(10)

One thereby obtains a metric on the exterior that is asymptotically flat but
for which $\mu(r)$ does not go to zero (i.e. $g_{tt}(r)\not\rightarrow 1$) at
infinity:

$ds^{2}=-{\mathcal{C}}\left(1-\frac{2M_{\text{total}}}{r}\right)dt^{2}+\left(1-\frac{2M_{\text{total}}}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(11)

The constant $\mathcal{C}\neq 1$ arises due to the presence of the matter
shell so that $t$ is not the proper time of an asymptotic observer. It turns
out that the constant $\mathcal{C}$ provides the dominant contribution to the
change in ringdown waveform caused by the presence of the shell. The
relationship between the time $t$ in (11) and the proper time $\tilde{t}$ of
an asymptotic observer is

$\displaystyle\tilde{t}=\sqrt{\mathcal{C}}t~{}.$ (12)

This is simply a redshift factor that is a consequence of the extra mass in
the system provided by the shell. For positive definite shell densities
$\mathcal{C}$ is greater than one, so that in terms of the rescaled time
$\tilde{t}$, the waveform appears stretched. For example, if instead of an
extended shell one had a thin shell of radius $a$ and mass $M_{s}$ surrounding
the black hole, the redshift factor would be

$\displaystyle\sqrt{\mathcal{C}}$ $\displaystyle=$
$\displaystyle\sqrt{1+\frac{\frac{2M_{s}}{a}}{1-\frac{2(M_{0}+M_{s})}{a}}}$
(13) $\displaystyle\approx$ $\displaystyle 1+\frac{M_{s}}{a}\,,\quad
a>>M_{s}~{}.$

One can remove the effect of the redshift by dividing $g_{tt}$ by
${\mathcal{C}}$, as done by[28] in studying the effect of halos around
galactic black holes, in order to compare asymptotic waveforms for systems
with approximately the same mass. This is equivalent to working with the
boundary condition (6) as done in [24] and yields an effect that is
considerably smaller than the redshift contribution, in agreement with the
results of [28] in the case of halos.

##  3 Wave Equation

In terms of the Schwarzschild time $t$ of the vacuum metric of the central
black hole, the QNM wave equation has the general form

$\left(\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial
r_{*}^{2}}+V(r(r_{*}))\right)\Psi(t,r_{*})=0,$ (14)

where $r_{*}$ is the tortoise coordinate defined as

$dr_{*}=\frac{dr}{\sqrt{A(r)B(r)}}.$ (15)

For the particular case of massless scalar perturbations, the QNM potential is

$V(r)=A(r)\frac{l(l+1)}{r^{2}}+\frac{1}{2r}\frac{d}{dr}\left[A(r)B(r)\right]~{}.$
(16)

With the given potential, one then solves (14) for the wave $\Psi(t,r_{*})$
with the initial condition of an outgoing wave

$\displaystyle\Psi(0,r_{*})$ $\displaystyle=$ $\displaystyle f(r_{*})$
$\displaystyle\left.\partial_{t}\Psi(t,r_{*})\right|_{t=0}$ $\displaystyle=$
$\displaystyle-\partial_{r_{*}}\Psi(0,r_{*})~{},$ (17)

where $f(r_{*})$ is usually taken to be a Gaussian function.

As explained earlier, in order to find the waveform measured by an asymptotic
observer, one can first determine the waveform in terms of the Schwarzschild
time $t$ of the interior geometry ($r<r_{\text{in}}$) by solving Eq. (14) with
initial conditions (17) and then rescale the time coordinate to the proper
time of an asymptotic observer, $\tilde{t}=\sqrt{\mathcal{C}}t$.
Alternatively, one can do the entire calculation in terms of $\tilde{t}$. In
the latter case, one calculates ${\tilde{\mu}}(r)$ by integrating inward from
$r_{\text{out}}$ with the boundary condition

$\tilde{\mu}(r\geq
r_{\text{out}})=\ln\left(1-\frac{2M_{\text{total}}}{r}\right).$ (18)

The integration of Eq. (4) then proceeds to the shell region of
$r_{\text{in}}\leq r<r_{\text{out}}$, where

$\tilde{\mu}(r)=\ln\left(1-\frac{2M_{\text{total}}}{r_{\text{out}}}\right)+\int_{r_{\text{out}}}^{r}dr\frac{2M(r)}{r(r-2M(r))}.$
(19)

Combining Eqs. (8), (10), and (19), one finds that
$\tilde{\mu}(r)=\mu(r)-\ln(\mathcal{C})$, which means

$\displaystyle\tilde{A}(r)=A(r)/\mathcal{C}.$ (20)

Therefore, the conversion of the outward integration to an inward integration
suitable for an asymptotic observer can be reduced to a simple rescaling. The
tortoise coordinate for the asymptotic observer is

$d\tilde{r}_{*}=\frac{dr}{\sqrt{\tilde{A}(r)B(r)}}=\sqrt{\mathcal{C}}dr_{*}.$
(21)

In order to verify the covariance of the wave equation under time rescales, we
note that

$\tilde{V}(r)=\tilde{A}(r)\frac{l(l+1)}{r^{2}}+\frac{1}{2r}\frac{d}{dr}\left[\tilde{A}(r)B(r)\right]=\frac{V(r)}{\mathcal{C}}.$
(22)

Putting these transformations together we find, as expected, that

$\mathcal{C}\left(\frac{\partial^{2}}{\partial\tilde{t}^{2}}-\frac{\partial^{2}}{\partial\tilde{r}_{*}^{2}}+\tilde{V}(\tilde{r}_{*})\right)\tilde{\Psi}(\tilde{t},\tilde{r}_{*})=0~{},$
(23)

where

$\displaystyle\tilde{V}(\tilde{r}_{*})=\frac{V(r(\tilde{r}_{*}))}{\mathcal{C}}.$
(24)

Eq. (23) is of the same form as the wave equation (14). This is all simply a
manifestation of the fact that the field is a scalar, so that the wave
equation is covariant and the solutions are related by

$\displaystyle\tilde{\Psi}(\tilde{t},\tilde{r}_{*})=\Psi(t,r_{*})=\Psi(\frac{1}{\sqrt{\mathcal{C}}}\tilde{t},\frac{1}{\sqrt{\mathcal{C}}}\tilde{r}_{*})~{}.$
(25)

In the frame of the asymptotic observer, the same initial conditions as in Eq.
(17) above take the form

$\displaystyle\tilde{\Psi}(0,\tilde{r}_{*})$ $\displaystyle=$
$\displaystyle{\tilde{f}}(\tilde{r}_{*})=f(r_{*}(\tilde{r}_{*}))=f(\frac{1}{\sqrt{\mathcal{C}}}\tilde{r}_{*})$
$\displaystyle\left.\sqrt{\mathcal{C}}\partial_{\tilde{t}}\tilde{\Psi}(\tilde{t},\tilde{r}_{*})\right|_{\tilde{t}=0}$
$\displaystyle=$
$\displaystyle-\sqrt{\mathcal{C}}\partial_{\tilde{r}_{*}}\tilde{\Psi}(0,\tilde{r}_{*})=\sqrt{\mathcal{C}}\partial_{\tilde{r}_{*}}f\left(\frac{1}{\sqrt{\mathcal{C}}}\tilde{r}\right)~{}.$
(26)

Thus Gaussian initial data specified in terms of $r_{*}$ change the width by a
factor of $\sqrt{\mathcal{C}}$ under the frame transformation. Of course, both
forms correspond to the same initial data in terms of the areal radius.
Although it is important to compare apples to apples with respect to initial
data, numerical calculations are not sensitive to small changes to the initial
wave due to the fact that black holes oscillate with natural vibrational modes
(QNMs) that have unique frequencies and damping rates.

##  4 Galactic Black Holes with Dark Matter Spikes

As we mentioned earlier, the results in [24] are obtained in the Schwarzschild
coordinates in the region $r<r_{\text{in}}$. In this section, we examine the
same waveforms as seen by an asymptotic stationary observer.

Figure 1: Scalar QNM potential as a function of radial coordinate for $l=2$
and $\gamma_{\text{sp}}=7/3$ for Sgr A* with the mass of
$M_{\text{BH}}=4.1\times 10^{6}M_{\odot}$ surrounded by a DM spike. $V(r)$, in
dashed red, is the potential corresponding to $t$ and $\tilde{V}(r)$, in solid
brown, is the potential viewed by an asymptotic observer. We choose
$\rho_{\text{sp}}=6.7\times 10^{-20}$ g cm-3 ($\approx 840$ times the expected
value), $r_{\text{b}}=2r_{\text{BH}}$ and $R_{\text{sp}}=0.235$ kpc. With
these values, $\mathcal{C}=1.172$. For comparison, we include the
Schwarzschild potential, $V_{S}(r)$, in dotted green for
$M_{\text{BH}}=4.1\times 10^{6}M_{\odot}$. All our variables are expressed in
terms of black hole parameters.

Figure 2: In solid brown, ringdown waveform $\tilde{\Psi}$ (left) and
$\ln|\tilde{\Psi}|$ (right) as a function of time $\tilde{t}$ for the
potential, $\tilde{V}(r)$, shown in solid brown in Figure 1. For numerical
purposes, we detect the waveform at $r=80~{}r_{\text{BH}}$. For comparison, we
include the Schwarzschild ringdown waveform in dotted green. All our variables
are expressed in terms of black hole parameters. Figure 3: Scalar QNM
potential as a function of radial coordinate for $l=2$ and
$\gamma_{\text{sp}}=7/3$ for M87* with the mass of $M_{\text{BH}}=6.4\times
10^{9}M_{\odot}$ surrounded by a DM spike. $V(r)$, in dashed red, is the
potential corresponding to $t$ and $\tilde{V}(r)$, in solid brown, is the
potential viewed by an asymptotic observer. We choose
$\rho_{\text{sp}}=1.8\times 10^{-22}$ g cm-3 ($8.4$ times the expected value),
$r_{\text{b}}=2r_{\text{BH}}$ and $R_{\text{sp}}=4.26$ kpc. With these values,
$\mathcal{C}=1.031$. For comparison, we include the Schwarzschild potential,
$V_{S}(r)$, in dotted green for $M_{\text{BH}}=6.4\times 10^{9}M_{\odot}$. All
our variables are expressed in terms of black hole parameters.

Figure 4: In solid brown, ringdown waveform $\tilde{\Psi}$ (left) and
$\ln|\tilde{\Psi}|$ (right) as a function of time $\tilde{t}$ for the
potential, $\tilde{V}(r)$, shown in solid brown in Figure 3. For numerical
purposes, we detect the waveform at $r=80~{}r_{\text{BH}}$. For comparison, we
include the Schwarzschild ringdown waveform in dotted green. All our variables
are expressed in terms of black hole parameters.

To see how the DM spike influences the shape of the QNM potential given in Eq.
(16), in Figures 1, 3, and 5 we plot the Schwarzschild potential together with
the potentials corresponding to the time coordinates $t$ and $\tilde{t}$,
respectively. Figure 1 shows the potential for Sgr A* with a DM density at the
outer edge of the spike, $\rho_{\text{sp}}$, approximately $840$ times the
expected value given in Table $1$ of [24] for the case of
$\gamma_{\text{sp}}=7/3$. Note that while the potential for the coordinate $t$
closely follows the Schwarzschild case, the potential for the asymptotic
observer is very different. Figures 3 and 5 show the potential for M87* with
$\rho_{\text{sp}}\approx 8.4$ and $\rho_{\text{sp}}\approx 84$ times the
expected value, provided in Table $2$ of [24] for the case of
$\alpha_{\gamma}=1.94$, respectively.

The ringdown waveforms for the potentials in Figures 1, 3, and 5 are shown in
Figures 2, 4, and 6 respectively. These waveforms are caused by the outgoing
Gaussian pulse used in [24] that initiates near the event horizon (inside the
photon sphere) and gets detected after being transmitted through the QNM
potential. In these figures, we also provide the ringdown waveform for the
Schwarzschild case for comparison. Note that for the more massive black hole
M87*, less density at the outer edge of the spike is required to achieve an
observable signal. In Figures (3) and (4), the change in the potential and
ringdown waveform due to the presence of DM spike is visible in the asymptotic
frame even for a density as low as $\approx 8.4$ times the expected
$\rho_{\text{sp}}$. This is not the case for an observer who uses the time
coordinate $t$.

Figure 5: Scalar QNM potential as a function of radial coordinate for $l=2$
and $\gamma_{\text{sp}}=7/3$ for M87* with the mass of
$M_{\text{BH}}=6.4\times 10^{9}M_{\odot}$ surrounded by a DM spike. $V(r)$, in
dashed red, is the potential corresponding to $t$, and $\tilde{V}(r)$, in
solid brown, is the potential viewed by an asymptotic observer. We choose
$\rho_{\text{sp}}=1.8\times 10^{-21}$ g cm-3 ($84$ times the expected value),
$r_{\text{b}}=2r_{\text{BH}}$ and $R_{\text{sp}}=4.26$ kpc. With these values,
$\mathcal{C}=1.371$. For comparison, we include the Schwarzschild potential,
$V_{S}(r)$, in dotted green for $M_{\text{BH}}=6.4\times 10^{9}M_{\odot}$. All
our variables are expressed in terms of black hole parameters.

Figure 6: In solid brown, ringdown waveform $\tilde{\Psi}$ (left) and
$\ln|\tilde{\Psi}|$ (right) as a function of time $\tilde{t}$ for the
potential, $\tilde{V}(r)$, shown in solid brown in Figure 5. For numerical
purposes, we detect the waveform at $r=70~{}r_{\text{BH}}$. For comparison, we
include the Schwarzschild ringdown waveform in dotted green. All our variables
are expressed in terms of black hole parameters.

Figure 7: In solid brown, ringdown waveform $\tilde{\Psi}$ (left) and
$\ln|\tilde{\Psi}|$ (right) as a function of time $\tilde{t}$ for the
potential, $\tilde{V}(r)$, shown in solid brown in Figure 5. Unlike all the
other figures, the ringdown is produced by an ingoing pulse starting at
$r\approx 41~{}r_{\text{BH}}$. For numerical purposes, we detect the waveform
at $r=70~{}r_{\text{BH}}$. For comparison, we include the Schwarzschild
ringdown waveform in dotted green. All our variables are expressed in terms of
black hole parameters.

Finally, in Figure 7, we generate the ringdown waveform produced by the same
Gaussian pulse that moves inward from a far distance outside the potential
$\tilde{V}(r)$ shown in Figure 5 and reflects back to generate the ringdown
waveform. The reflected wave produces considerably fewer oscillations than the
transmitted wave shown in Figure 6. However, the transmitted and reflected
waves are consistent in terms of the observed impact of the DM spike.

Note that it is not possible numerically to generate these ringdown waveforms
for an observer at a region outside the spike, since $R_{\text{sp}}$ is of the
order of at least $10^{7}$ times the black hole radius. However, the potential
drops very quickly with distance. For example, at $r=50~{}r_{\text{BH}}$ the
potential is three orders of magnitude smaller than its value at the peak.
This allows us to generate the ringdown waveform for an observer located at a
radius close to the black hole (roughly around $80~{}r_{\text{BH}}$). Since
the potential is already relatively small at this radius, we do not expect any
significant modification in the waveform as it travels all the way to the
region outside the spike.

It is evident from the results in this section that the impact of the the DM
spike on the ringdown waveform is more pronounced in the frame of an
asymptotic observer, compared to the results presented in [24]. More
specifically, to generate roughly the same modification in the ringdown
waveform, we need an order of magnitude less DM density in terms of the proper
time $\tilde{t}$ of an asymptotic observer compared to an observer who uses
$t$. We can, therefore, conclude that if a significant gravitational wave
detection associated with perturbations of a black hole with a mass comparable
to M87* occurs, it might provide the means to detect the presence of a DM
spike or at least put a model dependent bound on its parameters. More massive
galactic black holes could likely yield even clearer signals, as argued in
[24].

##  5 Shadow Radius

We summarize the shadow calculations as follows.666For a good review of black
hole shadow calculations, see [29]. Consider the action for particle geodesics

$S=\int_{\lambda_{0}}^{\lambda_{f}}L(x,\dot{x})d\lambda=\frac{1}{2}\int_{\lambda_{0}}^{\lambda_{f}}g_{\mu\nu}\dot{x}^{\nu}\dot{x}^{\mu}d\lambda,$
(27)

where $L$ is the Lagrangian and dot represents derivative with respect to the
affine parameter $\lambda$. For the general static spherically symmetric
spacetime in Eq. (1), the Lagrangian is

$L(x,\dot{x})=\frac{1}{2}\left[-A(r)\dot{t}^{2}+B(r)^{-1}\dot{r}^{2}+r^{2}(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2})\right].$
(28)

The spherical symmetry allows us to take $\theta=\pi/2$ without the loss of
generality. The $t$ and $\phi$ components of the Euler-Lagrange equation

$\frac{d}{d\lambda}\left(\frac{\partial
L}{\partial\dot{x}^{\mu}}\right)-\frac{\partial L}{\partial x^{\mu}}=0$ (29)

give us two constants of motion $E=A(r)\dot{t}$ and $L=r^{2}\dot{\phi}$, which
correspond to energy and angular momentum respectively.

For null geodesics we have $g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$. Hence

$-A(r)\dot{t}^{2}+B(r)^{-1}\dot{r}^{2}+r^{2}\dot{\phi}^{2}=0.$ (30)

After combining the constants of motion with the above equation and using the
fact that $\dot{r}^{2}/\dot{\phi}^{2}=(dr/d\phi)^{2}$, one obtains the orbit
equation for null geodesics,

$\left(\frac{dr}{d\phi}\right)^{2}=r^{2}B(r)\left(\frac{r^{2}}{A(r)}\frac{1}{b^{2}}-1\right),$
(31)

where $b=L/E$ is the impact parameter defined as the perpendicular distance,
measured at infinity, between the geodesic and the parallel line that passes
through the center of black hole in an asymptotically Minkowski
spacetime777Specifically, the spacetime must be asymptotically flat and the
metric must be asymptotically Minkowski so that $A(r)\to 1$ as $r\to\infty$. .
The closest distance, $R$, of the light ray to the black hole is the turning
point of the geodesic, where the condition $dr/d\phi|_{R}=0$ has to hold. This
condition gives us a relation between $R$ and $b$,

$b^{2}=\frac{R^{2}}{A(R)}.$ (32)

When $R=r_{\text{ph}}$, where $r_{\text{ph}}$ is the radius of the photon
sphere, the light ray that is coming from infinity will orbit around the black
hole. In this situation, in addition to $dr/d\phi|_{r_{\text{ph}}}=0$, the
condition $d^{2}r/d\phi^{2}|_{r_{\text{ph}}}=0$ should also hold. Combining
these two conditions with Eq. (31) gives

$\frac{1}{A(r)}\frac{dA(r)}{dr}=\frac{2}{r},$ (33)

which can be used to determine the radius of the photon sphere. It is easy to
show that for the Schwarzschild spacetime, where $A(r)=1-2M_{\text{BH}}/r$,
Eq. (33) gives $r_{\text{ph}}=3M_{\text{BH}}$. Once we determine
$r_{\text{ph}}$, we can use Eq. (32) to find the corresponding impact
parameter

$b_{\text{ph}}=\frac{r_{\text{ph}}}{\sqrt{A(r_{\text{ph}})}}~{}.$ (34)

The shadow radius, $r_{\text{sh}}$, turns out to be equal to $b_{\text{ph}}$,
because the rays with $b<b_{\text{ph}}$ cannot escape the black hole. It is
important, however, to note that $b$ only corresponds to the impact parameter
at infinity if we assume the black hole spacetime is asymptotically Minkowski.
For asymptotically flat black holes, $r_{\text{sh}}$ does not need to be equal
to $b_{\text{ph}}$. This can be explained as the following. If the distance
between the observer and the center of the black hole is $r_{\text{o}}$ and
the angle between $r_{\text{o}}$ and the geodesic at infinity is $\alpha$,
then we can write the relation

$\sin\alpha=\frac{\mathcal{I}}{r_{\text{o}}},$ (35)

where $\mathcal{I}$ is the impact parameter. Note that at the observer
position, we also have

$\sin\alpha\sim\frac{\sqrt{r^{2}B(r)}d\phi}{dr}\bigg{|}_{r_{\text{o}}}\sim\left(b^{2}\frac{A(r_{\text{o}})}{r_{\text{o}}^{2}}\right)^{1/2}=\left(\frac{R^{2}}{A(R)}\frac{A(r_{\text{o}})}{r_{\text{o}}^{2}}\right)^{1/2},$
(36)

where we use Eq. (31) and the small angle approximation. The above two
equations give

$\mathcal{I}\sim b\sqrt{A(r_{\text{o}})}.$ (37)

It is clear that $\mathcal{I}=b$ only when $A(r_{\text{o}})=1$. Finally, the
black hole shadow size is given by

$r_{\text{sh}}=\mathcal{I}_{\text{ph}}\sim
b_{\text{ph}}\sqrt{A(r_{\text{o}})}=r_{\text{ph}}\sqrt{\frac{A(r_{\text{o}})}{A(r_{\text{ph}})}}.$
(38)

Note that the shadow radius is invariant under time rescalings, so that one
can obtain the correct result using either $\tilde{t}$ or $t$.

We now explore how the shadow of a black hole changes in the presence of a DM
spike. In an asymptotically Minkowski space, $A(r_{\text{o}})\approx 1$. For
the Schwarzschild case, we find the radius of the shadow to be

$r_{\text{sh}}^{\text{Schw}}\sim
r_{\text{ph}}\sqrt{\frac{1}{A(r_{\text{ph}})}}=3\sqrt{3}M_{\text{BH}},$ (39)

where we use the fact that $r_{\text{ph}}=3M_{\text{BH}}$. In the presence of
a DM spike, the asymptotic observer will measure the radius to be

$r_{\text{sh}}^{\text{sp}}\sim
r_{\text{ph}}\sqrt{\frac{1}{\tilde{A}(r_{\text{ph}})}}=r_{\text{ph}}\sqrt{\frac{1}{A(r_{\text{ph}})/\mathcal{C}}}=3\sqrt{3\mathcal{C}}M_{\text{BH}}.$
(40)

Therefore the black hole shadow radius will appear to be $\sqrt{\mathcal{C}}$
times larger due to the presence of DM. For the shadow radius of M87*, bounds
on the DM spike density produce an effect888$\mathcal{C}\approx 1.004$ for the
M87* spike profile provided in Table 2 of [24] for the case of
$\alpha_{\gamma}=1.94$. This leads to an approximately $0.2\%$ enlargement of
the shadow diameter.  that is an order of magnitude smaller than is accessible
with current Event Horizon Telescope data [20].

##  6 Conclusion

We have studied the effect of a DM spike on the ringdown waveforms associated
with galactic black holes. For the range of spike parameters proposed for Sgr
A* and M87*, the predominant effect of the spike as seen from infinity comes
in the form of a redshift factor. Removing this redshift factor is equivalent
to working in the Schwarzschild time $t$ of an observer measuring the geometry
below the inner radius of the spike. The residual effect once this redshift is
removed has been shown[24] to be too small to be detectable. However, the
impact of the redshift on the asymptotic ringdown waveform is significant.
Specifically, it should be observable for M87* for densities within one or two
orders of magnitude of the maximum value suggested by current
observations[30]. We conclude that if it becomes possible to detect
gravitational waves associated with perturbations of a black hole with a mass
comparable to M87*, it might provide the means to detect the presence of a DM
spike or put a model dependent bound on its parameters. Our results suggest
that more massive galactic black holes could yield even clearer signals,
implying that the effect of DM spikes on the ringdown waveforms of
supermassive black holes is worthy of further study.

We also showed that the presence of a DM spike will increase the black hole
shadow radius by a factor of $\sqrt{\mathcal{C}}$. For the shadow radius of
M87*, bounds on the DM spike density produce an effect that is an order of
magnitude smaller than is accessible with current Event Horizon Telescope data
[20]. However, more massive black holes can significantly restrict the upper
bound of the spike parameters and may produce a detectable increase in the
shadow size within the proposed error bars of [20].

Acknowledgments

We thank Andrei Frolov for helpful discussions. G.K. gratefully acknowledges
that this research was supported in part by a Discovery Grant from the Natural
Sciences and Engineering Research Council of Canada.

##  Appendix A Discontinuity in Potential

Here, we derive the discontinuities in the QNM potential that exist at the
boundaries of the matter shell. To do this, we first rewrite Eq. (16) as

$\displaystyle\frac{V(r)}{A(r)}$ $\displaystyle=$
$\displaystyle\frac{l(l+1)}{r^{2}}+\frac{1}{2r}\left[B(r)\frac{A^{\prime}(r)}{A(r)}+B^{\prime}(r)\right]$
(41) $\displaystyle=$
$\displaystyle\frac{l(l+1)}{r^{2}}+\frac{1}{2r}\left[\left(1-\frac{2M(r)}{r}\right)\mu^{\prime}(r)+\left(\frac{-2M^{\prime}(r)}{r}+\frac{2M(r)}{r^{2}}\right)\right]~{}.$
(42)

Using the TOV equations (3) and (4), we can write

$\displaystyle\frac{V(r)}{A(r)}$ $\displaystyle=$
$\displaystyle\frac{l(l+1)}{r^{2}}+\frac{1}{2r}\left[\left(1-\frac{2M(r)}{r}\right)\left[\frac{2M(r)}{r(r-2M(r))}\right]+\left(\frac{-8\pi
r^{2}\rho(r)}{r}+\frac{2M(r)}{r^{2}}\right)\right]$ (43) $\displaystyle=$
$\displaystyle\frac{l(l+1)}{r^{2}}+\frac{1}{2r^{2}}\left[\frac{2M(r)}{r}+\left({-8\pi
r^{2}\rho(r)}+\frac{2M(r)}{r}\right)\right]$ $\displaystyle=$
$\displaystyle\frac{l(l+1)}{r^{2}}+\frac{1}{r^{3}}\left[2M(r)-4\pi
r^{3}\rho(r)\right]~{}.$

Therefore,

$\displaystyle
V(r)=A(r)\left[\frac{l(l+1)}{r^{2}}+\frac{1}{r^{3}}\left(2M(r)-4\pi
r^{3}\rho(r)\right)\right]$ (44)

and since only $\rho$ is discontinuous, the discontinuity in $V(r)$ is

$\displaystyle\Delta V(r)$ $\displaystyle=$ $\displaystyle-4\pi
A(r)\Delta\rho(r)~{}.$ (45)

Thus, the discontinuities in the QNM potential are proportional to the change
in matter density at the boundaries, and consequently negligibly small for
proposed DM spike profiles [21, 24, 30].

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