# More Excitement Across the Horizon

María R. Preciado-Rivas<EMAIL_ADDRESS>Institute for Quantum
Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario,
N2L 3G1, Canada  Manar Naeem<EMAIL_ADDRESS>Institute for Quantum
Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario, N2L 3G1, Canada  Robert B. Mann<EMAIL_ADDRESS>Institute for
Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario, N2L 3G1, Canada  Perimeter Institute for Theoretical Physics,
Waterloo, Ontario, N2L 2Y5, Canada  Jorma Louko<EMAIL_ADDRESS>School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD,
UK

(February 2024)

###### Abstract

An Unruh-DeWitt (UDW) detector falling radially into a four-dimensional
Schwarzschild black hole, coupled linearly to a massless scalar field that has
been prepared in the Hartle-Hawking or Unruh state, has recently been shown to
exhibit a local extremum in its transition probability near the black hole
horizon [K.K. Ng et al., New J. Phys. 24 (2022) 103018]. We show that a
similar phenomenon is present in the transition rate of a UDW detector falling
radially into a spinless Bañados-Teitelboim-Zanelli (BTZ) black hole, with the
scalar field prepared in the Hartle-Hawking state. We give extensive numerical
results as a function of the detector’s energy gap, the black hole’s mass, and
the detector’s drop-off radius. Our results suggest that the effect is robust,
motivating a search for a similar effect in other black hole spacetimes, and
calling for an explanation of the physical origin of the effect.

##  I Introduction

While a quantum theory of gravity continues to elude us, the field of
relativistic quantum information has emerged as a powerful tool for
investigating quantum effects in curved spacetime. Within this framework,
models for particle detectors provide us with an operational approach for
probing quantum fields in scenarios with no distinguished notion of particle.
The simplest model is the Unruh-DeWitt (UDW) particle detector [1, 2], which
is a two-level quantum system that locally interacts with a quantum scalar
field. One quantity of interest is the transition probability between its two
levels; another is the derivative of this probability with respect to the
total detection time, known as the transition rate. One well-known result
obtained from this model is that a uniformly linearly accelerated detector
experiences the Minkowski vacuum as thermal, with a temperature proportional
to the proper acceleration, a phenomenon known as the Unruh effect [1, 2, 3].
Another result is the degradation of the entanglement between two detectors
that are in relative non-inertial motion [4, 5].

In a similar manner, one can examine the response function of a detector in
black hole spacetimes. For example, a static detector in the exterior of a
Schwarzschild black hole responds thermally to the Hartle-Hawking(-Israel)
state [6, 7]; similarly, the transition rate of a stationary detector co-
rotating with the Bañados-Teitelboim-Zanelli (BTZ) black hole thermalizes at
the Hawking temperature [8].

Much less is known about the response of detectors that freely fall into a
black hole and cross its horizon. The general expectation is that the
transition rate will be smooth and the detector will not thermalize. A
detector freely falling through a stationary cavity in a Schwarzschild
background was shown to probe spacetime curvature when compared to that of an
equivalently accelerated detector travelling through an inertial cavity in the
absence of curvature [9]. Other studies have taken place in more simplified
lower-dimensional settings. A numerical study of the transition rate of a UDW
detector freely falling toward a $(1+1)$ dimensional Schwarzschild black hole
provided evidence of how its thermal properties are gradually lost during the
infall [10, 11], and the transition rate of a detector infalling toward a
Cauchy horizon in $(1+1)$ dimensions was shown to diverge for a field in both
the Unruh and Hartle-Hawking-Israel states [12]; in both cases the detector
was taken to couple to the momentum of the field, to cure infrared ambiguities
that occur in $(1+1)$ dimensions. Coupling to field momentum was also employed
in a study of the behaviour of entanglement and mutual information acquired by
two freely falling detectors in $(1+1)$ dimensional Schwarzschild spacetime,
finding that correlations can be harvested even when the detectors are
causally disconnected by the event horizon [13]. In $(2+1)$-dimensions, the
transition rate of a detector radially falling toward a static BTZ black hole
was computed [8], but the detector was switched on and off in the region
exterior to the black hole; no regions in parameter space were found where the
transition rate was thermal.

Recently, the response of a detector freely falling toward and across the
horizon of a $(3+1)$-dimensional Schwarzschild black hole was considered [14].
Taking the detector’s initial position to be at infinity with zero initial
velocity, the response function, a multiple of the transition probability, was
numerically calculated for a field in the Hartle-Hawking(-Israel) and Unruh
states. Surprisingly, a small but discernible local extremum in the response
appears as the detector crosses the horizon of the black hole for both field
states. The locus of the peak or dip in the response function depends on the
interaction duration and the detector’s energy gap.

It is of significant interest to understand the extent to which this
phenomenon takes place for other black holes. To this end, we investigate the
transition rate (the temporal derivative of the response function) of a freely
falling UDW detector into a static BTZ black hole in $(2+1)$ dimensions. This
setting affords several advantages. First, it extends previous work [8] on
freely falling detectors in this setting, allowing us to calibrate the
detector’s transition rate at large distances from the horizon with that
across the horizon. More importantly, the response can be computed as a sum
over images instead of the mode sum necessary in the Schwarzschild case [14],
a considerable technical advantage that allows us to explore a broader range
of parameter space.

We find that the BTZ case exhibits similar, but much richer, behaviour to that
of the Schwarzschild case [14]. The transition rate exhibits the general
feature of slow oscillation as a function of proper time as the detector
approaches the black hole. However, near and beyond the horizon the rate
sharply peaks and then drops before rapidly increasing near the black hole’s
singularity. Furthermore, there are specific times at which the transition
rate is not differentiable – we refer to these as “glitches”. Both features
depend on the mass of the black hole and so serve as a kind of ‘early warning
system’ as to whether the detector will continue to freely fall in AdS3
spacetime or across an event horizon in a spacetime of the same local
geometry.

The outline of our paper is as follows. In section II we present the basic
formalism needed to compute the transition rate of a detector freely falling
in both the BTZ spacetime and an AdS3 spacetime of the same local geometry. We
present in section III the results of our study, illustrating with a number of
figures how the transition rate depends on the mass of the black hole, the
initial conditions of the trajectory, the energy gap of the detector, and the
boundary conditions of the field at asymptotic infinity. In section IV, we sum
up our work, drawing conclusions about the results and suggesting new avenues
for future study.

##  II Setup

We consider the Unruh-DeWitt model for a detector coupled to a scalar field.
Since much of the formalism has been previously presented [8], we shall only
recapitulate the essential tools needed for our study.

A UDW detector is a pointlike qubit having two energy levels, denoted by
$|g\rangle$ and $|e\rangle$, with the respective energy eigenvalues $0$ and
$E$. The quantity $E$ may be positive or negative; in the former case case
$|g\rangle$ is the ground state and $|e\rangle$ is the excited state, while in
the latter case the roles are reversed. The detector moves on a timelike
worldline $\mathsf{x}(\tau)$, parametrized by its proper time $\tau$ and
interacts with a real massless scalar field $\phi$ via the interaction
Hamiltonian

$H_{\text{int}}=\lambda\chi(\tau)\mu(\tau)\phi\bigl{(}\mathsf{x}(\tau)\bigr{)}\
,$ (1)

where
$\mu(\tau)=\ket{e}\bra{g}e^{\mathsf{i}E\tau}+\ket{g}\bra{e}e^{-\mathsf{i}E\tau}$
is the detector’s monopole moment operator, $\lambda$ is a coupling constant,
and $\chi$ is the switching function, specifying how the detector is switched
on and off.

In first-order perturbation theory, the probability of the detector to make a
transition from the state $|g\rangle$ to the state $|e\rangle$ is proportional
to the response function, given by

$\mathcal{F}=\int\mathrm{d}\tau^{\prime}\,\mathrm{d}\tau^{\prime\prime}\,\mathrm{e}^{-\mathsf{i}E(\tau^{\prime}-\tau^{\prime\prime})}W(\tau^{\prime},\tau^{\prime\prime})\,,$
(2)

where $W$ is the pullback of field’s Wightman function in the state in which
the field was initially prepared [15, 16, 17, 18]. In $(2+1)$ spacetime
dimensions, we may take $\chi$ to be the characteristic function of a time
interval, since $\mathcal{F}$ remains well defined for this $\chi$ despite the
nonsmoothness at the switch-on and switch-off moments, and we may further
consider the detector’s transition rate, given by (a multiple of) the
derivative of $\mathcal{F}$ with respect to the switch-off moment. The
transition rate can be written as [8]

$\dot{\mathcal{F}}_{\tau}=\frac{1}{4}+2\int^{\Delta\tau}_{0}\,\mathrm{d}s\operatorname{Re}\left[\mathrm{e}^{-\mathsf{i}Es}W(\tau,\tau-s)\right]\,,$
(3)

where $\tau$ denotes the switch-off moment and $\Delta\tau$ denotes the total
proper time that the detector operates. The transition rate has a measurable
interpretation in terms of an ensemble of identical detectors, all following
the trajectory $\mathsf{x}(\tau)$ and switched off at distinct moments [18,
19].

We are interested in the transition rate of a detector on a radially in-
falling geodesic in a spinless nonextremal BTZ black hole spacetime [20, 21].
Outside the horizon, the line element is

$\displaystyle\text{d}s^{2}$
$\displaystyle=-f(r)\text{d}t^{2}+\dfrac{\text{d}r^{2}}{f(r)}+r^{2}\text{d}\varphi^{2}\,,$
(4)

where $f(r)=r^{2}/\ell^{2}-M$, $t\in\mathbb{R}$, $\varphi\in[0,2\pi)$, the
dimensionless parameter $M$ is the mass of the black hole, where
nonextremality means $M>0$. The exterior is at $r\in(r_{h},\infty)$, where
$r=r_{h}\equiv\ell\sqrt{M}$ is the location of the event horizon. The metric
(4) is a vacuum solution to the Einstein field equations with cosmological
constant $\Lambda=-1/\ell^{2}$, where $\ell\;(>0)$ is the AdS length.

By setting $\varphi\to y\in(-\infty,\infty)$ one obtains AdS3-Rindler
spacetime, where the parameter $M$ in (4) can be se to unity without loss of
generality [22, 23]. Conversely, the BTZ spacetime can be obtained from
AdS3-Rindler by identifying $y\to\phi\in(0,2\pi\sqrt{M})$, and then rescaling
$\phi\to\sqrt{M}\varphi$ and the radial and time coordinates to obtain (4);
its local geometry is therefore that of anti-de Sitter spacetime. This fact
will be important when one considers a quantum field in BTZ spacetime, in
which the correlation functions can be written as a sum of correlators in
AdS3.

The radial timelike geodesic followed by the detector is given in BTZ
coordinates by [8, 24]

$\displaystyle t$
$\displaystyle=\left(\ell/\sqrt{M}\right)\operatorname{arctanh}{\left(\frac{\tan{\tilde{\tau}}}{\sqrt{q^{2}-1}}\right)},$
(5) $\displaystyle r$ $\displaystyle=\ell\sqrt{M}q\cos{\tilde{\tau}},$
$\displaystyle{\varphi}$ $\displaystyle={\varphi_{0}},$

where $M$ is the mass of the black hole, $q>1$, $\varphi_{0}$ is a constant
value of $\varphi$, and $\tilde{\tau}$ is a dimensionless affine parameter
such that the proper time equals $\tilde{\tau}\ell$. $r$ reaches its maximum
value $q\ell\sqrt{M}$ at $\tilde{\tau}=0$ with $t=0$, as depicted in Fig. 1.

Figure 1: Penrose diagram indicating the trajectory followed by a UDW detector
free falling into a BTZ black hole

We consider a massless conformally coupled quantum scalar field
$\hat{\phi}(\mathsf{x})$ satisfying the Klein-Gordon equation,

$\displaystyle(\square-R/8)\hat{\phi}(\mathsf{x})=0\,,$ (6)

where $\square$ is the d’Alembert operator and $R$ is the Ricci scalar. Since
the BTZ spacetime is a $\mathbb{Z}$ quotient of an open set in AdS3 [20, 21],
we can consider field correlations in terms of the Wightman function
$W(\mathsf{x},\mathsf{x}^{\prime})\coloneqq\braket{0}{\hat{\phi}(\mathsf{x})\hat{\phi}(\mathsf{x}^{\prime})}{0}$
in AdS3 spacetime. $W(\mathsf{x},\mathsf{x}^{\prime})$ is the two-point
correlation function in a vacuum state $\ket{0}$ of the field. We take the
field to be in the Hartle-Hawking vacuum state so that the BTZ black hole is
in thermal equilibrium with a heat bath at asymptotic infinity. We then have
[25, 26]:

$\displaystyle
W_{\text{BTZ}}(\mathsf{x},\mathsf{x}^{\prime})=\sum_{n=-\infty}^{\infty}W_{\text{ADS}}(\mathsf{x},\Gamma^{n}\mathsf{x}^{\prime})$
(7)
$\displaystyle=\dfrac{1}{4\pi\sqrt{2}\ell}\sum_{n=-\infty}^{\infty}\left[\dfrac{1}{\sqrt{\sigma_{\epsilon}(\mathsf{x},\Gamma^{n}\mathsf{x}^{\prime})}}-\dfrac{\zeta}{\sqrt{\sigma_{\epsilon}(\mathsf{x},\Gamma^{n}\mathsf{x}^{\prime})+2}}\right]$

where $\Gamma:(t,r,\varphi)\mapsto(t,r,\varphi+2\pi)$ in the exterior. The
parameter $\zeta\in\\{-1,0,1\\}$ specifies the boundary conditions of the
field at asymptotic infinity: Neumann, transparent, or Dirichlet,
respectively. The quantity

$\displaystyle\sigma_{\epsilon}(\mathsf{x},\Gamma^{n}\mathsf{x}^{\prime})$
$\displaystyle=\dfrac{rr^{\prime}}{r_{h}^{2}}\cosh\left[\dfrac{r_{h}}{\ell}(\Delta\varphi-2\pi
n)\right]-1$ (8) $\displaystyle-\dfrac{\sqrt{(r^{2}-r_{h}^{2})(r^{\prime
2}-r_{h}^{2})}}{r_{h}^{2}}\cosh\left(\dfrac{r_{h}}{\ell^{2}}\Delta
t-\mathsf{i}\epsilon\right)$

is the squared geodesic separation (scaled by the square of the AdS length)
between the two points in the covering space, with
$\Delta\varphi\coloneqq\varphi-\varphi^{\prime},\Delta t\coloneqq
t-t^{\prime}$, and the notation encodes the distributional character of the
Wightman function as the limit $\epsilon\to 0_{+}$. Inserting (7) into (3) we
obtain [8]

$\dot{\mathcal{F}}_{\tau}(E)=1/4+\frac{1}{2\pi\sqrt{2}}\sum_{n=-\infty}^{\infty}\int_{0}^{\Delta\tilde{\tau}}\mathrm{d}\tilde{s}\operatorname{Re}\left[\frac{\mathrm{e}^{-\mathsf{i}\tilde{E}\tilde{s}}}{\sqrt{-1+K_{n}\cos\tilde{\tau}\cos(\tilde{\tau}-\tilde{s})+\sin\tilde{\tau}\sin(\tilde{\tau}-\tilde{s})}}\right.\\\
\left.-\zeta\frac{\mathrm{e}^{-\mathsf{i}\tilde{E}\tilde{s}}}{\sqrt{1+K_{n}\cos\tilde{\tau}\cos(\tilde{\tau}-\tilde{s})+\sin\tilde{\tau}\sin(\tilde{\tau}-\tilde{s})}}\right],$
(9)

where

$K_{n}:=1+2q^{2}\sinh^{2}{\left(n\pi\sqrt{M}\right)},$ (10)

with $\tilde{\tau}:=\tau/\ell$, $\Delta\tilde{\tau}:=\Delta\tau/\ell$, and
$\tilde{E}:=E\ell$. The detector is sharply switched on at a time
$\tau_{0}=\tau-\Delta\tau$, where $\tau$ corresponds to the moment it is
sharply switched off and observed.

Several observations are in order.

First, the $n\neq 0$ terms in (9) are invariant under $n\to-n$. The $n\neq 0$
part of the sum hence reduces to twice the sum over $n\geq 1$.

Second, while we have above arrived at the transition rate formula (9) using
the expression (8), which is valid only in the black hole exterior, (9) holds
over the detector’s full trajectory, even when the detector operates already
before exiting the white hole and/or after entering the black hole. This
follows from the existence of a global analytic chart, by analytic
continuation in (9).

Third, the transition rate decomposes as

$\dot{\mathcal{F}}_{\tau}=\dot{\mathcal{F}}_{\tau}^{n=0}+\dot{\mathcal{F}}_{\tau}^{n\neq
0}$ (11)

where $\dot{\mathcal{F}}_{\tau}^{n=0}$ consists of the $n=0$ term, and
$\dot{\mathcal{F}}_{\tau}^{n\neq 0}$ consists of the sum over $n\neq 0$ terms.
$\dot{\mathcal{F}}_{\tau}^{n=0}$ depends only on $E\ell$ and
$\Delta\tau/\ell$. That is, it does not depend on the mass $M$ or the switch-
on time $\tau_{0}$ (or switch-off time $\tau$) and is given by [8]

$\dot{\mathcal{F}}_{\tau}^{n=0}(E)=\frac{1}{4}-\frac{1}{4\pi}\int^{\Delta\tilde{\tau}}_{0}\,d\tilde{s}\left[\frac{\sin{\left(\tilde{E}\tilde{s}\right)}}{\sin{\left(\tilde{s}/2\right)}}+\zeta\frac{\cos{\left(\tilde{E}\tilde{s}\right)}}{\cos{\left(\tilde{s}/2\right)}}\right].$
(12)

which is the transition rate in AdS spacetime.

Fourth, the square roots in (9) are positive for positive arguments, and they
are analytically continued to negative arguments by giving $\tilde{s}$ a small
negative imaginary part [8, 27]. This follows from the distributional
definition of the Wightman function, encoded in the $i\epsilon$ in (8).

From now on we specialise to the case $\tau_{0}=0$, in which the detector is
switched on at the moment when $r$ takes its largest value, as illustrated in
Figure 1; the numerical results in the rest of the paper will be for this
case. Then (9) holds with $0<\tilde{\tau}<\pi/2$ and
$\Delta\tilde{\tau}=\tilde{\tau}$, so that the integration range of
$\tilde{s}$ is from $0$ to $\tilde{\tau}$. The $n=0$ term is given by (12).
What remains is to find the branches of the square roots in the $n\neq 0$
terms. In the part involving $\zeta$, the function under the square root is
positive for all $\tilde{s}$ in the integration interval, so the square root
is positive. In the part not involving $\zeta$, the function under the square
root is positive at $\tilde{s}=0$. It has in the integration interval at most
one zero, denoted by $\tilde{s}^{*}$, and the derivative of the function at
$\tilde{s}^{*}$ is negative. It follows that the square root is positive for
$\tilde{s}<\tilde{s}^{*}$ and positive imaginary for
$\tilde{s}>\tilde{s}^{*}$. At $\tilde{s}=\tilde{s}^{*}$, the square root
vanishes, so that the integrand is singular, but this singularity is of the
inverse square root type and hence integrable, rendering the integral well
defined.

Finally, we note that the singularities in the integrand coincide with an
endpoint of the integration interval when $\Delta\tau$ takes the value

$\Delta\tilde{\tau}_{n}:=\arccos(1/K_{n})\,,$ (13)

both in the special case $\tau_{0}=0$ and more generally. At these values of
$\tau$ the transition rate is still well defined but not differentiable, and
the response function $\mathcal{F}$ in (2) is not twice differentiable, as a
function of the switch-off moment. These “kinks” or “glitches” will feature
prominently in our numerical results.

##  III Results

In this section, we systematically arrange our results to facilitate the
examination of the transition rate’s dependence on various factors, namely the
boundary conditions of the Wightman function at infinity given by $\zeta$, the
detector energy gap $E$, the mass $M$ of the black hole, and the detector’s
initial position with respect to the black hole’s event horizon, given by $q$.

###  III.1 Transition rate for different boundary conditions

In Fig. 2, we show the transition rate $\dot{\mathcal{F}}_{\tau}$ as a
function of the detector’s total proper time $\Delta\tau/\ell$ for the three
boundary conditions, $\zeta=-1$, 0 and 1, with $\tau_{0}=0$, $q=100$,
$M=10^{-4}$, and energy gaps (a) $E\ell=-5$ or (b) $E\ell=20$. These
parameters are the same as those used in the previous study of a freely
falling detector in BTZ spacetime [8] but (unlike that study) include the
transition rate for $\Delta\tau/\ell\gtrsim\arccos{(1/q)}$, that is, near and
beyond the horizon crossing. For $\Delta\tau/\ell\lesssim\arccos{(1/q)}$, we
find close agreement with the results of [8], given by the orange dashed
curves.

Furthermore, we observe that the transition rate for a detection time shorter
than the first glitch $\Delta\tau_{1}/\ell$ has a behaviour remarkably
distinct from that after the first glitch. We first discuss the transition
rate for $\Delta\tau/\ell\lesssim\Delta\tau_{1}/\ell$ (with
$\Delta\tau_{1}/\ell\approx 1.52257$ for $q=100$ and $M=10^{-4}$).

Figure 2: The transition rate $\dot{\mathcal{F}}_{\tau}$ of a detector freely
falling into a BTZ black hole, shown by the blue, solid curves with $q=100$,
$M=10^{-4}$, $\tau_{0}=0$, and (a) $E\ell=-5$ or (b) $E\ell=20$ calculated
with $n=356$ or $n=320$ terms of the image sum, respectively. For comparison,
we plot the AdS3 case, given by the zeroth term
$\dot{\mathcal{F}}_{\tau}^{n=0}$ (green, dotted curve). We consider the
Wightman function using Neumann (thinnest), transparent, and Dirichlet
(thickest) boundary conditions. The results of [8] are shown in orange dashed
curves. The total time for the detector to reach the event horizon of the BTZ
black hole is indicated with a dashed vertical line at
$\Delta\tau/\ell=\arccos{(0.01)}$. The values of $\Delta\tau_{n}/\ell$ for
$n=1,2,3$ are shown with crosses inside the insets, which are close-ups of the
transition rate near the horizon crossing.

In the case of the Dirichlet boundary condition, $\zeta=1$, the transition
rate bears a strong resemblance to the zeroth term
$\dot{\mathcal{F}}_{\tau}^{n=0}$, which is the transition rate of a stationary
detector in pure AdS3. This agreement is also present for other energy gaps
(as we illustrate in Figs. A1 and A2 of the appendix and is commensurate with
previous work [8]). For the boundary conditions $\zeta=0$ and $-1$, by
contrast, we observe a notable difference between the transition rate and the
zeroth term $\dot{\mathcal{F}}_{\tau}^{n=0}$. This dissimilarity increases as
the magnitude of the energy gap $|E\ell|$ decreases, becoming appreciable when
$|E\ell|=5$ as seen in Figs. 2(a) and A2(b).

We observe in Fig. 2 (as well as in Figs. A1 and A2 of the appendix) that the
transition rate undergoes more oscillations as $|E\ell|$ increases. This
characteristic is expected since $E\ell$ is the argument of a complex
exponential function in Eq. (9). We also note that the transition rates for
$\zeta=-1$ and $\zeta=0$ are in-phase, whereas they are out-of-phase for
$\zeta=1$. Another way we can describe the behaviour in this regime is that
the $\zeta=0$ and $\zeta=-1$ boundary conditions have the effect of increasing
the magnitude of the oscillations in the transition rate when compared to the
oscillations in the rate calculated with $\zeta=1$.

We now turn our attention to the transition rate near and beyond the first
glitch, which is shown in more detail in the insets of Fig. 2. The transition
rate differs considerably from the AdS3 case (the zeroth term
$\dot{\mathcal{F}}_{\tau}^{n=0}$) for detection times
$\Delta\tau/\ell\gtrsim\Delta\tau_{1}/\ell$, with the difference being more
pronounced as more kinks $\Delta\tau_{n}/\ell$ are reached or the black hole
singularity at $\pi/2$ is approached. To investigate this difference, we plot
in Fig. 3 the individual contributions from the first few terms in the image
sum using the same parameters as in Fig. 2 and $\zeta=1$. We observe that
beyond the $n$-th glitch, $\Delta\tau_{n}/\ell$, the contribution from the
$n$-term rapidly becomes appreciable and comparable to the zeroth term, which
gives rise to the divergence near the black hole singularity that we see in
Fig. 2. Note also that the $n=0$ term is very similar across the three
boundary conditions; hence the difference in the transition rate stems from
the contributions of the $|n|\geq 1$ terms.

We also see in Fig. 3 that there is dissimilar behaviour in the contributions
for negative and positive gaps. The terms for negative gaps are oscillatory
with a pseudo-period depending on the magnitude of the gap, whereas the terms
for positive gaps exhibit a dip right after the glitch but do not oscillate.
It is remarkable that the contribution of the $n$-th term oscillates when
$\Delta\tau/\ell\lesssim\Delta\tau_{n}/\ell$ for any gap (not shown). However,
the amplitude of these oscillations becomes negligible far from the glitch
when compared to the amplitude of the zeroth term. This is as expected,
considering that the transition rate in this interval resembles the pure AdS3
term.

For $\Delta\tau/\ell\gtrsim\Delta\tau_{1}/\ell$, the contributions add up in
such a way that the transition rate diverges as $\Delta\tau/\ell\to\pi/2$. For
$\zeta=0$ and $\zeta=-1$, the terms are large enough that the transition rate
for these boundary conditions has a positive deviation from the rate obtained
with $\zeta=1$. This deviation becomes more pronounced as the total detection
time approaches $\pi/2$, as depicted in Fig. 2 (and Figs. A1 and A2 of the
appendix). Overall, the transition rates are notably numerically different for
the three types of boundary conditions, as previously observed [8], but the
qualitative behaviour of the transition rate remains consistent across the
various boundary conditions. Therefore, we confine the results presented in
the main text hereafter to the case $\zeta=1$.

Figure 3: Individual contributions from the first six terms of the image sum
for the transition rate calculated using the same parameters as Fig. 2 and
$\zeta=1$. The horizon crossing is shown with a dashed vertical line at
$\Delta\tau/\ell=\arccos(0.01)=1.5608$.

Of particular note is the transition rate in Fig. 2(a), which attains a
prominent local maximum followed by a local minimum before the horizon
crossing at $\Delta\tau/\ell=\arccos(0.01)$; that is, a peak and a dip appear
at switch-off times for which the detector has not yet reached the event
horizon. Specifically, the local maximum/minimum feature is in between the
glitches $\Delta\tau_{1}/\ell$ and $\Delta\tau_{2}/\ell$. In the close-up near
the horizon crossing shown in the inset, we observe that the transition rate
attains another, less prominent, local maximum followed by another local
minimum. The aforementioned feature in the transition rate for a freely
falling detector in a BTZ black hole resembles the local extremum that was
observed in the Schwarzschild case [14].

###  III.2 Transition rate for different energy gaps

In the previous subsection, we briefly discussed the effect of the detector’s
energy gap on the transition rate, considering the various boundary
conditions. In this subsection, we delve into a more detailed discussion by
exploring additional values of the energy gap.

In Fig. 4, we present the transition rate varying the detector’s energy gap
for fixed black hole mass $M=10^{-4}$, $q=100$, $\tau_{0}=0$ and $\zeta=1$. We
observe that the transition rate starts increasing (decreasing) for $E\ell<0$
($E\ell>0$), attaining a first local maximum (minimum) that is located at
shorter $\Delta\tau/\ell$ for larger $|E\ell|$. As also noted in the previous
subsection, the number of oscillations in
$\Delta\tau/\ell\lesssim\Delta\tau_{1}/\ell\approx 1.52257$ that follow the
first extremum increases when the absolute value of $E\ell$ increases, but the
amplitude of these oscillations gradually diminishes. For these detection
times, the transition rate oscillates around $\sim 0.5$ for $E\ell<0$ and
$\sim 0$ for $E\ell>0$.

The results in Fig. 4 are in accord with the intuition that a wider energy gap
in an excited detector leads to a stronger tendency for decay to the ground
state. Conversely, a narrower energy gap in a detector in the ground state
makes it more susceptible to excitation. The fact that the deexcitation
transition rate remains positive across the entire detection time domain
indicates that the response function is monotonically increasing with respect
to the detection time. Conversely, the occurrence of a negative excitation
transition rate within certain intervals of the detection time suggests that
the response function exhibits oscillatory behaviour, displaying local extrema
at the points where the transition rate crosses zero.

Figure 4: Transition rate $\dot{\mathcal{F}}_{\tau}$ calculated with
$M=10^{-4}$, $q=100$, $\tau_{0}=0$, $\zeta=1$ and varying the detector’s
energy gap. The image sum for $E\ell=\pm 1,\,\pm 2,\,\pm 3,\pm 4,\,\pm
5,\,-8,\,10,\,\pm 14$ and $\pm 20$ is calculated with
$n=388,\,361,\,333,\,349,\,356,\,337,\,333,\,327$ and $320$ terms,
respectively. The AdS case (zeroth term $\dot{\mathcal{F}}_{\tau}^{n=0}$) for
each gap is shown with dotted curves of the same colour.

With respect to $\Delta\tau/\ell\gtrsim\Delta\tau_{1}/\ell$, we note that the
behaviour of the transition rate for negative gaps differs significantly from
that for positive gaps, as observed in the insets of Fig. 4 (and Fig. 3 of the
previous subsection). As the magnitude of $E\ell$ increases, the transition
rate for $E\ell<0$ experiences a larger number of oscillations of diminishing
amplitude for fixed $\Delta\tau/\ell$. On the other hand, for $E\ell>0$, the
transition rate features dips or peaks in between the glitches
$\Delta\tau_{n}/\ell$ but no appreciable oscillations. We can trace this
difference back to the sign of the terms of the image sum, whose contributions
either have the same sign and add up, resulting in local maxima and minima for
negative gaps, or have different signs and cancel, resulting in a function
without oscillations for positive gaps. In both cases, however, the overall
trend of the transition rate is to increase as more glitches
$\Delta\tau_{n}/\ell$ are reached, or the singularity at $r=0$ is approached
by the detector.

The transition rate for negative and large enough energy gaps becomes more
intricate as $\Delta\tau/\ell$ increases due to the fact that the contribution
of the $n$-th term of the image sum, which becomes significant for detection
times larger than $\Delta\tau_{n}/\ell$, has a quasiperiod different from that
of previous terms. Another remarkable trait for negative gaps is the
alternating order of the local maxima and minima in the transition rate after
each $\Delta\tau_{n}/\ell$ as $|E\ell|$ increases. Finally, another feature
that can be retrieved from Fig. 4 is that for smaller $|E\ell|$, the earlier
the onset of the deviation of the transition rate (solid curves) from the
zeroth term (dotted curves). Such deviation occurs at detection times for
which the detector has not yet reached the event horizon in all the cases
shown.

We see in Fig. A3, and in the previous subsection, that the discrepancy
between the transition rate and the pure AdS3 term is more noticeable for the
boundary condition $\zeta=-1$ and is present for most of the detection time
range.

###  III.3 Transition rate for different masses

Figure 5: Transition rate $\dot{\mathcal{F}}_{\tau}$ as a function of
$\Delta\tau/\ell$ for energy gap $E\ell=-5$, boundary condition $\zeta=1$ and
black hole mass $M\leq 10^{-6}$ (top-left), $2\times 10^{-6}\leq M\leq
10^{-5}$ (top-right), $2\times 10^{-5}\leq M\leq 3\times 10^{-4}$ (bottom-
left) or $M\geq 4\times 10^{-4}$ (bottom-right). The corresponding zeroth term
(the AdS 3 case) is also shown with grey, dotted curves. The horizon crossing
is shown with a dashed vertical line at $\Delta\tau/\ell=\arccos{(0.01)}$. The
total detection time ranges from 0 to $\pi/2$ in the main plots and from
$1.51$ to $\pi/2$ in the insets, except in the last inset, where it ranges
from $1.555$ to $\pi/2$. $n=11236$, 7946, 5026, 3553, 2513, 1779, 1454, 1257,
1124, 796, 506, 356, 252, 218, 195, 70 and 25 terms of the image sum are used
in order of increasing mass.

With fixed energy gap $E\ell=-5$, $q=100$, $\tau_{0}=0$ and $\zeta=1$, we
calculate the transition rate for a range of black hole mass, plotting the
results in Fig. 5. We find that the rate is quite sensitive to the magnitude
of the black hole mass. More specifically, the transition rate calculated with
$M=10^{-7}$, the smallest mass we explored, is punctuated by sharp local
maxima and minima, corresponding to several glitches $\Delta\tau_{n}/\ell$
occurring before the horizon crossing. For this particular mass, the image sum
was truncated at the $n=11236$ term. We observe that, as the black hole mass
increases, the first glitch $\Delta\tau_{1}/\ell$ appears at a longer
detection time, the number of glitches $\Delta\tau_{n}/\ell$ before the
horizon crossing decreases, and fewer terms in the image sum are necessary to
calculate a converged transition rate. We also observe that there is a clear
difference between the transition rate $\dot{\mathcal{F}}_{\tau}$ and the
zeroth term $\dot{\mathcal{F}}_{\tau}^{n=0}$ for $M=10^{-7}$. However, this
deviation from the zeroth term for
$\Delta\tau/\ell\lesssim\Delta\tau_{1}/\ell$ becomes negligible as the black
hole mass gets bigger. These results suggest that the smaller the mass of the
black hole, the sooner a BTZ black hole spacetime can be distinguished from
AdS3 spacetime with the same constant curvature.

In Fig. 5 (top-left), the broader picture is similar across the range of mass
$10^{-7}\leq M\leq 10^{-6}$. The transition rate starts increasing, as
observed for $E/\ell<0$ in the previous subsection, and reaches a peak at
$\sim\pi/8$. It then attains a local minimum at $\sim 5\pi/16$ and later grows
rapidly, increasing significantly as the detector crosses the horizon. For
masses $M\geq 2\times 10^{-6}$, the transition rate has a close resemblance to
the $n=0$ term until reaching the first glitch $\Delta\tau_{1}/\ell$.

For the range of mass $2\times 10^{-6}\leq M\leq 10^{-5}$ (Fig. 5, top-right),
the transition rate has a pronounced dip after the first glitch. By contrast,
it has a prominent peak after the first glitch for $M\geq 2\times 10^{-5}$,
Fig. 5 (bottom-left and bottom-right). The dip (or peak) is enhanced as the
mass increases.

In Fig. 5 (bottom-right), we find that the first glitch in the transition rate
is at the horizon crossing for $M\approx 5.00714\times 10^{-4}$. For larger
masses, the peak after the first glitch is entirely inside the horizon, and
the transition rate has the same shape as for small masses, but the
oscillations are squeezed closer to the singularity. However, we still observe
a noticeable difference in the transition with respect to the $n=0$ term, even
for masses as big as $M=4\times 10^{-3}$. It is remarkable that this
difference is noticeable for detection times for which the detector has not
yet reached the event horizon.

For other boundary conditions ($\zeta=0,-1$; see Fig. A4), the peak at $\pi/8$
and the dip at $5\pi/16$ get a larger amplitude, becoming more apparent.
Moreover, the dip at $5\pi/16$ reaches negative values for small masses and/or
becomes a global minimum. On the other hand, the transition rate with these
boundary conditions gets a positive shift that increases with the detection
time for $\Delta\tau\gtrsim 3\pi/8$. As a result, the dips become subtler, and
the peaks are enhanced for a long enough detection time with these boundary
conditions.

###  III.4 Transition rate varying $q$

In Fig. 6, we explore the transition rate with fixed $E\ell=-5$, $\tau_{0}=0$,
$\zeta=1$, $M=10^{-4}$ and varying $q$, the parameter that determines the
detector’s initial radial position with respect to the black hole’s event
horizon. We observe that fixing the black hole mass $M$ and varying $q$ has
the same effect on the transition rate as fixing $q$ and varying $M$ (see Fig.
5). Namely, the first glitch is located at a longer detection time, the number
of glitches before the horizon crossing decreases, fewer terms are needed to
calculate the image sum, and the deviation from the pure AdS3 term is reduced
as the parameter $q$ increases. This similar effect is expected since an
increasing function of $q$ is multiplying an increasing function of $M$ in the
second term of $K_{n}$ of Eq. (10), and the parameters $q$ and $M$ affect the
transition rate only through $K_{n}$. However, increasing $q$ also increases
the total proper time for the detector to reach the event horizon. Finally,
the results in Fig. 6 imply that the closer the detector is initially located
with respect to the event horizon, the sooner it can discern between being in
an AdS3 spacetime and a BTZ black hole.

Figure 6: Transition rate $\dot{\mathcal{F}}_{\tau}$ as a function of
$\Delta\tau/\ell$ for $\tau_{0}=0$, $E\ell=-5$, $\zeta=1$, $M=10^{-4}$ and
varying $q$. The image sum in the transition was truncated at the term
$n=451$, 429, 400, 385, 378, 356, 321, 305 and 281 in order of increasing $q$.
The different horizon crossings, at $\Delta\tau/\ell=\arccos(1/q)$, are
indicated with vertical dashed lines in the same colour as the transition
rates for each value of $q$.

Given that a short initial distance paired with a large mass produces a
transition rate that resembles that of a large initial distance paired with
some small mass, we investigate combinations of $q$ and $M$ that produce
similar transition rates. For this purpose, we compare in Fig. 7 transition
rates that have the first glitch, $\Delta\tau_{1}/\ell$, at the same total
detection time. One can find $M_{2}$ and $q_{2}$ that will produce the same
first glitch in the transition rate as $M_{1}$ and $q_{1}$; however, the other
glitches, $\Delta\tau_{n}/\ell$ for $n>1$, are not guaranteed to be equal.
This parameter calibration results in transition rates that are qualitatively
the same, regarding the positions of their peaks and dips, at least for a
detection time close to $\Delta\tau_{1}/\ell$. As a consequence, the peaks and
dips (or glitches) can appear in the transition rate before or after (and
close or far from) the proper time at which the detector reaches the event
horizon, as the horizon crossing depends on $q$ through
$\Delta\tau/\ell=\arccos{(1/q)}$. This observation implies that, without
independent determination of the black hole mass, the approximate positions of
the peaks or dips in the transition rate cannot be used to establish an early
warning system indicating when Alice, an observer freely falling with a
detector into a BTZ black hole, will cross the event horizon. However, she
could still discern a BTZ black hole spacetime from an AdS3 spacetime based on
the presence or absence of these peaks, albeit possibly after the horizon is
crossed.

Figure 7: Transition rates $\dot{\mathcal{F}}_{\tau}$ as a function of
$\Delta\tau/\ell$ for $\tau_{0}=0$, $E\ell=-5$ and $\zeta=1$, varying both $M$
and $q$ such that the first glitch occurs at the same time. The horizon
crossing, $\Delta\tau/\ell=\arccos(1/q)$, is indicated with a vertical solid
line in the same colour as the transition rate for each value of $q$.

In the case that Alice can measure the transition rate with infinite
precision, then not only could she distinguish the difference between AdS3
spacetime and a BTZ black hole spacetime, but she could also determine, at any
time, when she is crossing the event horizon by estimating the value of $q$
and the black hole’s mass $M$, distinguishing between any of the cases
presented in Fig. 7. Therefore, Alice could establish an early warning system
that can alert her that she is entering the black hole by ideally measuring
the transition rate of the detector.

We now consider a scenario where these ideal measurements of the detector’s
transition rate are inaccessible, and only the time at which each glitch
occurs ($\Delta\tau_{n}/\ell$) is available. In this situation, Alice can
still estimate the mass of the black hole and the location of its event
horizon. Both $M$ and $q$ can be determined by having information about two of
the glitches in the transition rate, as described by equations (10) and (13).
Let us assume that Alice can measure the first and second glitches. With this
information, she can establish a “weaker” early warning system given that the
second glitch happens before the detector reaches the event horizon, that is,

$\Delta\tau_{2}/\ell<\arccos{(1/q)}.$ (14)

For fixed $q$, (14) is satisfied when

$0<\sqrt{M}<\frac{1}{2\pi}\operatorname{arcsinh}{\left(\frac{q-1}{2q^{2}}\right)^{1/2}},$
(15)

and for fixed mass $M$, (14) is satisfied if

$0<\sqrt{M}<\frac{1}{2\pi}\operatorname{arcsinh}{\frac{1}{2\sqrt{2}}}\quad\text{and}\quad
q_{-}<q<q_{+},$ (16)

where

$q_{\pm}=\frac{1\pm\sqrt{1-8\sinh^{2}(2\pi\sqrt{M})}}{4\sinh^{2}(2\pi\sqrt{M})}.$
(17)

This means that the weaker early warning system is constrained to work for a
black hole mass below the critical value
$M_{*}=\left[1/(2\pi)\operatorname{arcsinh}{\left(1/(2\sqrt{2})\right)}\right]^{2}\approx
3.0425\times 10^{-3}$, and within a limited range of the detector’s initial
position.

##  IV Conclusion

We have numerically calculated the transition rate of an Unruh-DeWitt detector
coupled to a conformal scalar field as the detector falls radially into a
spinless BTZ black hole. We have been able to explore a broad range of values
of the black hole mass, the detector’s energy gap and initial position, and
scalar field boundary conditions at asymptotic infinity. The problem is
technically tractable due to the relatively simple calculation of the Wightman
function of the BTZ black hole as the image sum of the corresponding AdS3
spacetime Wightman function. We found that the transition rates have the same
qualitative behaviour across different boundary conditions (though differing
quantitatively), and we have observed that the transition rate differs
significantly for detectors with positive and negative energy gaps.

We discovered that the detector’s transition rate is nondifferentiable at
certain discrete values of the detector’s proper time; we refer to these
points of nondifferentiability as glitches. There is one glitch in the
transition rate for each term ($n\geq 1$) of the image sum in the BTZ Wightman
function, and it corresponds to the proper time of the detector at which the
integrand in the transition rate’s final expression diverges. As a result, the
transition rate is not smooth at these glitches. The contribution from any
term ($n\geq 1$) to the transition rate is negligible for detection times
shorter than the $n$-th glitch but rapidly becomes prominent for detection
times larger than the glitch, exhibiting oscillations that are determined by
the magnitude and the sign of the detector’s energy gap. Consequently, the
transition rate for a detector freely infalling into a BTZ black hole
resembles that of a detector in an AdS3 spacetime for detection times shorter
than the first ($n=1$) glitch. After the first glitch, the transition rate
presents a richer and more complex behaviour.

The glitches are fully determined by the black hole’s mass and the detector’s
initial position relative to the event horizon. Adjusting these parameters
changes the occurrence of glitches in the transition rate. We found that
increasing the black hole’s mass and the detector’s initial position results
in the BTZ transition rate resembling the AdS3 transition rate for a longer
period of time (as the first glitch appears at a later detection time) and in
the number of glitches before the horizon being smaller (requiring fewer terms
to obtain a converging transition rate). Conversely, for smaller mass or
closer initial position, the transition rate starts to differ from that of
pure AdS3 at shorter detection times.

Our study was motivated by the observation that the response function of a
detector interacting with the Hartle-Hawking(-Israel) state and freely falling
in a Schwarzschild black hole has a non-monotonic behaviour and presents a
small but discernible local extremum as the detector’s crosses the event
horizon [14]. Our results indicate that a similar phenomenon is present for
the BTZ black hole, but with much richer behaviour than in the Schwarzschild
case. Specifically, more than one local extrema can be present across the
horizon for certain ranges of the black hole’s mass, the detector’s initial
position and energy gap. The detection time at which these extrema appear in
the transition rate will be determined by the black hole’s mass and the
detector’s relative position with respect to the event horizon. Meanwhile, the
number and amplitude of these extrema are determined by the magnitude and sign
of the detector’s energy gap.

Finally, we have suggested that an observer with a detector that is freely
falling can discern between being in an AdS3 spacetime and a BTZ black hole
spacetime by measuring the detector’s transition rate. Moreover the observer
would be able to know, in their proper time, when they are crossing the black
hole’s event horizon, having in this way a sort of an early warning system. We
also proposed a second version of the warning system that would depend only on
the time at which the first two glitches happen. If the second glitch occurs
before the time of the horizon crossing, then an observer can know the
position of the event horizon before they cross it. This weaker version of the
early warning system is valid only for masses smaller than a certain critical
mass and for a limited range of the detector’s initial position.

Our results are somewhat limited insofar as they do not allow for a direct
comparison between the Schwarzschild and BTZ black holes. One limitation is
that the detector’s initial conditions are different. Another is that the BTZ
spacetime has constant curvature whereas the Schwarzschild case does not. Yet
another limitation is that the response function was calculated for the
Schwarzschild spacetime [14], whereas we computed the transition rate for the
BTZ spacetime. We leave the task of comparing both response functions instead
of the transition rate subject to future work.

Our results suggest a number of new research directions. One is to consider
the effects of hidden topology – for example, comparing the transition rate of
an infalling detector for a BTZ black hole to its corresponding geon
counterpart. Adding effects of rotation should be straightforward, since the
curvature remains constant. Another interesting problem is to study what
happens to two maximally entangled detectors when one (or both) falls into a
black hole. More challenging problems will involve black hole spacetimes that
do not have constant curvature.

## Acknowledgments

This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada. MRPR acknowledges that this research was
undertaken thanks in part to the funding from the Mike and Ophelia Lazaridis
Graduate Fellowship. The work of JL was supported by United Kingdom Research
and Innovation Science and Technology Facilities Council [grant number
ST/S002227/1]. For the purpose of open access, the authors have applied a CC
BY public copyright licence to any Author Accepted Manuscript version arising.

*

##  Appendix A Additional results

In this appendix, we present additional results that complement the discussion
in Section III. In particular, we show the transition rate as a function of
the detector’s proper time $\Delta\tau/\ell$ for the three boundary conditions
of the Wightman function, $\zeta=-1,\,0,$ and $1$, with the same parameters
used in Fig. 2 but different energy gaps, specifically $E\ell=-1$ and
$E\ell=-20$ in Fig. A1, and $E\ell=1$ and $E\ell=5$ in Fig. A2.

Figure A1: Zeroth term $\dot{\mathcal{F}}_{\tau}^{n=0}$ (green, dotted curves)
and transitions rate $\dot{\mathcal{F}}_{\tau}$ (blue, solid curves) for the
same parameters as Fig. 2 and (a) $E\ell=-1$ or (b) $E\ell=-20$ calculated
with $n=388$ or $n=320$ terms of the image sum, respectively.

Figure A2:  Zeroth term $\dot{\mathcal{F}}_{\tau}^{n=0}$ (green, dotted
curves) and transitions rate $\dot{\mathcal{F}}_{\tau}$ (blue, solid curves)
for the same parameters as in Fig. 2 and (a) $E\ell=1$ or (b) $E\ell=5$
calculated with $n=388$ or $n=356$ terms of the image sum, respectively.

We first discuss the transition rate for detection times shorter than the
first glitch $\Delta\tau_{1}/\ell$ (with $\Delta\tau_{1}/\ell\approx 1.52257$
for $q=100$ and $M=10^{-4}$). We observe in Figs. A1 and A2 that the Dirichlet
boundary condition ($\zeta=1$) bears a strong resemblance to the corresponding
pure AdS3 transition rate, that is, the $n=0$ term, across the different
energy gaps shown. This is due to partial cancellations in the terms of
equation (9). For the other boundary conditions, $\zeta=0$ and $\zeta=-1$, it
can be argued the transition rate presents the same qualitative behaviour as
the pure AdS3 transition rate. However, there is a quantitative dissimilarity
for these boundary conditions that increases as the magnitude of the energy
gap $|E\ell|$ decreases. We see in Fig. 2(a) and Fig. A2(b) that the
difference between these transitions rate becomes noticeable for $|E\ell|=5$.

We now discuss the transition rate near and beyond the first glitch, which is
shown in more detail in the insets of Figs. A1 and A2. As pointed out before,
the transition rate qualitatively differs from the pure AdS3 case (the zeroth
term) for detection times near and beyond the first glitch,
$\Delta\tau/\ell\gtrsim\Delta\tau_{1}/\ell$, across the different boundary
conditions and energy gaps. This difference is explained in the discussion of
Fig. 3. While the transition rate calculated with a negative gap oscillates
with a pseudo-period proportional to the magnitude of the gap, as observed in
Fig. A1, the transition rate calculated with a positive gap has dips after
each glitch but does not oscillate, as observed in Fig. A2.

Another noteworthy result is that the transition rates for a freely falling
detector in a BTZ black hole calculated with a negative energy gap, as in Fig.
A1, also exhibit the features of the transition rate in Fig. 2(b), that is,
they present oscillations that resemble the local extremum obtained in the
Schwarzschild black hole case [14].

Since the discussion of the results in the main text was limited to the
$\zeta=1$ case when studying the transition rate for different detector energy
gaps in Section III.2, black hole mass in Section III.3, and initial parameter
$q$ in Section III.4, we present here the transition rate calculated with the
boundary condition $\zeta=-1$ for completeness. The transition rate for
$\zeta=0$ (not shown) is always between these two results.

In Fig. A3, we show the transition rate varying the detector’s energy gap and
using the same parameters as in Fig. 4 but with boundary condition $\zeta=-1$.
While the overlap between the zeroth term (dotted curves) and the transition
rate (solid curves) lasts up to detection times close to the first glitch in
the results for the boundary condition $\zeta=1$ in Fig. 4, we observe that
there is little to no overlap between the zeroth term and the transition rate
for $\zeta=-1$ across the different energy gaps in Fig. A3. This observation
agrees with what was discussed in the sections regarding the transition rate
for the different boundary conditions. Overall, the changes produced in the
transition rate with $\zeta=-1$, as the magnitude and sign of the energy gap
vary, are similar to the ones discussed for the transition rate with
$\zeta=1$.

Figure A3:  Transition rate $\dot{\mathcal{F}}_{\tau}$ as a function of
$\Delta\tau/\ell$ for the same parameters as Fig. 4 but boundary condition
$\zeta=-1$. The pure AdS3 (zeroth term $\dot{\mathcal{F}}^{n=0}_{\tau}$) for
each gap is shown with dotted curves of the same colour.

In Fig. A4, we depict the transition rate for a range of black hole mass. We
use the same fixed parameters as in Fig. 5 but boundary condition $\zeta=-1$.
We can observe that the magnitude of the oscillations with local maximum at
$\pi/8$ and minimum at $5\pi/16$ is considerably greater for the $\zeta=-1$
case. As a result, the sharp peaks corresponding to the glitches in the
transition rate are less noticeable in Fig. A4 than in Fig. 5. As discussed in
the main text, for $\Delta\tau/\ell\gtrsim 3\pi/8$, a positive shift for
$\zeta=-1$ in the transition rate that increases with the detection time
causes the peaks to become more apparent and the dip to become more subtler
when compared to the $\zeta=1$ case.

Figure A4:  Transition rate $\dot{\mathcal{F}}_{\tau}$ as a function of
$\Delta\tau/\ell$ for the same parameters as Fig. 5 but boundary condition
$\zeta=-1$. The corresponding zeroth term (the AdS3 case) is also shown with
grey, dotted curves.

Finally, in Fig. A5, we present the transition rate by varying $q$, calculated
with the same parameters as in Fig. 6 but for $\zeta=-1$. As also observed in
the transition rate calculated for variable black hole mass, the effect of the
changing the boundary condition from $\zeta=1$ to $\zeta=-1$ is that the
amplitude of the first peak and dip located at $\Delta\tau/\ell=\pi/8$ and
$\Delta\tau/\ell=5\pi/16$ gets larger.

Figure A5: Transition rate $\dot{\mathcal{F}}_{\tau}$ as a function of
$\Delta\tau/\ell$ for the same parameters as Fig. 6 but boundary condition
$\zeta=-1$.

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# SRG/eROSITA Survey in the Lockman Hole: Classification of X-ray Sources

M.I. Belvedersky  1,2<EMAIL_ADDRESS>S.D. Bykov  3, 4  M.R. Gilfanov  1,4
1Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32,
117997 Moscow, Russia 2National Research University Higher School of Economics
(HSE), Moscow, Russia 3Kazan Federal University, Department of Astronomy and
Satellite Geodesy, 420008 Kazan, Russia 4Max Planck Institute for
Astrophysics, Karl-Schwarzschild-Str 1, Garching b. Muenchen D-85741, Germany

###### Abstract

We have classified the point-like X-ray sources detected by the SRG/eROSITA
telescope in the deep Lockman Hole survey. The goal was to separate the
sources into Galactic and extragalactic objects. In this work have used the
results of our previous cross-match of X-ray sources with optical catalogs. To
classify SRG/eROSITA sources we have used the flux ratio $F_{\rm x}/F_{\rm o}$
and information about the source optical extent. As a result, of the 6885
X-ray sources in the eROSITA catalog 357 sources have been classified as
Galactic and 5929 and as extragalactic. 539 out of 6885 have been treated as
hostless, i.e., having no optical counterparts in the optical catalogs under
consideration. 60 have remained unclassified due to the insufficient
reliability of optical photometry. Recall and precision for the extragalactic
sources are 99.9 and 98.9% (respectively) and 91.6 and 99.7% for the Galactic
sources. Using this classification, we have constructed the curves of
cumulative number counts for the Galactic and extragalactic sources in the
Lockman Hole field. The code that accompanies this paper is available at
https://github.com/mbelveder/ero-lh-class.git.

###### keywords:

X-ray sky surveys, Lockman Hole, AGN, quasars, X-ray active stars, source
counts.

20224812755[766]

15.11.2022

##  1 INTRODUCTION

The Spectrum-RG (SRG) orbital X-ray observatory was launched on July 13, 2019,
and is currently in a halo orbit around the libration point L2 of the
Sun–Earth system (Sunyaev et al., 2021). There are two X-ray telescopes with
grazing-incidence optics onboard the observatory: the eROSITA telescope
(Predehl et al., 2021) sensitive in the 0.2–9.0 keV energy band and the
Mikhail Pavlinsky ART-XC telescope (Pavlinsky et al., 2021) sensitive in the
4–30 keV energy band. It is expected that during its 4-year all-sky survey
SRG/eROSITA will detect $\sim 4$ million X-ray sources of various types in the
entire sky and will provide a huge volume of data to solve a wide range of
problems in astrophysics and cosmology (Prokopenko & Gilfanov, 2009; Merloni
et al., 2012; Kolodzig et al., 2013a, b). ART-XC investigates hard X-ray
objects (Mereminskiy et al., 2018, 2019).

To study the populations of objects of various types detected in X-ray
surveys, first of all it is necessary to separate the objects of our Galaxy
(stars, compact objects) and extragalactic objects (galaxies, active galactic
nuclei – AGNs, quasars). The problem is complicated by the fact that the data
on the distances to X-ray objects are often unavailable (Salvato et al.,
2019). Therefore, we to use additional multiwavelength information (for
example, optical spectra or photometry) to determine whether a particular
source is Galactic or extragalactic. Some existing methods include the
diagnostics based on the spectral energy distribution (SED) (Salvato et al.,
2009; Marchesi et al., 2016), the classification by positions on the
color–magnitude or color–color diagrams (Maccacaro et al., 1988; Brusa et al.,
2007, 2010; Xue et al., 2011; Civano et al., 2012; Salvato et al., 2018,
2022), and the identification of sources with proper motions based on data
from astrometric missions such as Gaia (Gaia Collaboration et al., 2016,
2022). For example, a multilevel scheme in which the main factors of the
classification into Galactic and extragalactic objects are the object extent
in an optical image, its proper motion, and the relation between the optical
($grz$), infrared ($W1$), and X-ray (0.5–2 keV) fluxes is used in Salvato et
al. (2022) to solve this problem. To evaluate the quality of the
classification schemes, it is necessary that some of the objects have a known
class through spectroscopy or other reliable methods.

The goal of this paper is to classify the point X-ray sources detected by
eROSITA during the Lockman Hole survey. For this task, apart from X-ray data,
we use data from the Gaia astrometric satellite and the DESI LIS photometric
survey. Our method is distinguished by the simplicity of interpretation and
can be used to construct more complex classification algorithms, including
those for X-ray all-sky survey data.

The paper is structured as follows. In Section 2 we describe the data used. In
Section 3 we present the sample of sources with reliable information about
their distances and our algorithm for the classification of sources with
unknown distances. We describe and discuss the results in Section 4 and reach
the conclusions in Section 5. The magnitudes used are in the AB system.

##  2 DATA

###  2.1 X-ray data

The SRG/eROSITA Lockman Hole survey was performed in October 2019 during the
verification observations of the SRG observatory. The survey characteristics,
the procedure for constructing the catalog of X-ray sources, and the catalog
itself are presented in M. Gilfanov et al (2022). The catalog of X-ray sources
includes 6885 point-like sources with a detection likelihood (DET_LIKE) $>$
10, roughly corresponding to a 4$\sigma$ significance.

###  2.2 Search for Optical Counterparts of X-ray Sources

The most important step before the classification of X-ray sources is their
cross-match with optical catalogs. This problem is not trivial, since the
error circle of X-ray sources often include more than one optical object; the
exact number depends on the depth of the optical catalog and the sky density
of sources in the area. We chose the DESI Legacy Imaging Surveys DR9 catalog
(Dey et al. 2019; hereafter DESI LIS) as a reference optical catalog. The
eROSITA X-ray sources were cross-matched with optical objects from the DESI
LIS catalog in Bykov et al. (2022). For this purpose, we used a neural network
model to characterize the photometric attributes of optical counterparts to
X-ray sources and field objects (not counterparts) in combination with the
NWAY code (Salvato et al., 2018). The Nway algorithms are based on the
Bayesian formalism and allow any additional (apart from positional and
photometric) prior information to be used to search for counterparts. In Bykov
et al. (2022) for this purpose we used predictions of the classifier trained
on a specially prepared sample. As a result, we found the most probable
optical DESI LIS counterpart for each point-like SRG/eROSITA source.

For each X-ray source the Lockman Hole catalog provides the parameter
$p_{\rm\text{(}any)}$ that characterizes our confidence in the presence of an
optical counterpart (Bykov et al., 2022). The closer this parameter to unity,
the more reliable the assertion that the X-ray source has a counterpart in the
optical catalog under consideration. The objects with low
$p_{\rm\text{(}any)}$ are likely hostless, i.e., they have no counterpart in
the optical catalog. Following the recommendation given in Bykov et al.
(2022), we chose an upper boundary for the classification of the source as
hostless, $p_{\rm\text{(}any,0)}$ = 0.12. With this choice 539 sources (7.8%)
are classified as hostless. We plan to study these objects in the future. Here
we will investigate the sources classified as having a counterpart in the DESI
LIS catalog, their total number is 6346. The classification precision and the
choice of the $p_{\rm\text{(}any)}$ threshold are discussed in detail in Bykov
et al. 2022.

###  2.3 Photometric, Astrometric, and Spectroscopic Data

Photometric information about the counterparts was taken from the DESI LIS
catalog. This catalog includes the surveys performed with three telescopes:
BASS (in the $g$, $r$ bands), MzLS ($z$), and DECaLS ($g$, $r$, $z$). For the
Lockman Hole we used the BASS and MzLS data, since the DECaLS survey was not
conducted in this region. All of the fluxes were corrected for interstellar
absorption.

Information about the proper motions of the objects was taken from the Gaia
DR3 catalog (hereafter Gaia) Gaia Collaboration et al. 2016, 2022.

Spectroscopic information is available for $\sim$ 1/3 of the X-ray catalog.
For them we used the spectral classification and redshifts from the Sloan
Digital Sky Survey DR16 (Ahumada et al. 2020, hereafter SDSS) and The Million
Quasars Catalogue111https://heasarc.gsfc.nasa.gov/W3Browse/all/milliquas.html
v7.5 (Flesch 2021, hereafter Milliquas) catalogs. The SDSS DR16 catalog is the
fourth release of the fourth SDSS phase (SDSS-IV). The Milliquas catalog is a
composite one and contains information about quasars and AGNs from various
sources published up until April 30, 2022. We additionally used the Simbad
database (Wenger et al., 2000), which also contains the information collected
from various sources. Note that the Simbad database is not a catalog and,
therefore, the information from it will have a low priority for us compared to
other sources of information.

##  3 CLASSIFICATION OF X-RAY SOURCES

###  3.1 Cross-Match with the Gaia Catalog and Spectrometric Catalogs

To classify X-ray sources we used the coordinates of their most probable
optical counterparts from the DESI LIS catalog. For the classification of
Galactic sources we used information about the source parallaxes and proper
motions from the Gaia DR3 catalog. The optical counterparts were cross-matched
with Gaia sources; the search was conducted within a $None$ radius. A
counterpart in the Gaia catalog was found for 2475 DESI LIS objects. There
were 2 Gaia sources in the search circle in 8 cases, the nearest one was
chosen as a counterpart. The sources for which the signal-to-noise (S/N)
exceeded 5 at least for one of the following parameters were classified as
Galactic:

  * •

pmra, the proper motion in right ascension;

  * •

pmdec, the proper motion in declination;

  * •

pm, the total proper motion
($\text{pm}=\sqrt{\text{pmra}^{2}+\text{pmdec}^{2}}$);

  * •

parallax, the absolute stellar parallax.

We found a total of 338 Galactic sources. Following the recommendation of the
DESI team, no correction for the proper motion of the Gaia sources was made
due to the small (six months) difference between the reference epochs of the
DESI and Gaia DR3 catalogs (D. Lang, the DESI team, personal communication).

Then we identified the sources for which there was an optical spectroscopy.
For this purpose, we used the SDSS, Milliquas, and Simbad spectroscopic data.
The search for matches was performed with a search radius of $None$. In the
case of SDSS and Milliquas, there were no more than one source in the search
circle. For Simbad there were two and three sources in the search circle in 72
and 21 cases, respectively. At the same time, only four objects with ambiguous
matches in Simbad were not cross-matched with the Gaia, SDSS, or Milliquas
catalogs. For these objects, as in the case of Gaia, we chose the nearest
source. The final classification of objects was made with the following
priority: Gaia, SDSS, Milliquas, and Simbad. This means that if the object
under consideration was not classified as Galactic (according to the Gaia
data) then its class was set in accordance with the SDSS data222Among the 144
galaxies found using the SDSS catalog, 7 had the AGN subclass. We designated
these objects as quasars. and so on. A priority higher than Milliquas was
given to the SDSS catalog. Although Milliquas contains only reliable
spectroscopic measurements, it was compiled from the catalogs produced by
various authors (i.e., it is methodologically inhomogeneous). In our case, the
result of the final classification changes insignificantly when a higher
priority than SDSS is assigned to the Milliquas catalog. Thus, in total, we
determined the classes for 2884 of the 6346 objects.

###  3.2 $F_{\rm x}/F_{\rm opt}$ – Color Diagram

To classify the remaining objects, we investigated the characteristics of the
objects with a known class (see previous subsection). Of all the combinations
of parameters, the logarithm of the ratio of the X-ray (0.5 – 2 keV) and
optical fluxes ($z$) in combination with the optical ($r$ – $z$) color and the
information from DESI LIS about the object optical extent turned out to be
astrophysically most informative and interesting for our goal.

In Fig. 1 the X-ray sources are plotted on the $\lg(F_{\rm x}/F_{\rm o})$ –
optical color ($r$ – $z$) plane. The objects whose nature was confirmed based
on the Gaia measurements and those who have optical spectroscopy are
designated by different colors. The sources that we failed to classify based
on the spectroscopic data and the information about their proper motions are
marked by the gray circles. The black circumferences mark the sources that are
extended ones, according to the DESI LIS catalog
($\text{type}\neq\text{PSF}$).

Figure 1:  The $\lg(F_{\rm x}/F_{\rm o})$–optical color diagram for the
SRG/eROSITA point-like X-ray sources in the Lockman Hole. The difference
between the DESI LIS $r$ and $z$ magnitudes is plotted along the vertical
axis. The logarithm of the ratio of the 0.5–2 keV X-ray flux to the optical
DESI LIS $z$ flux is plotted along the horizontal axis. The objects classified
based on the Gaia data or optical spectroscopy are indicated by different
colors, according to the legend. The gray circles and the black circumferences
designate the objects of unknown nature and the optically extended objects,
respectively. Most of the Galactic objects are located in the left part of the
diagram (leftward of the boundary marked by the solid vertical line). The
extragalactic objects are located predominantly in the right part of the
diagram, but some of them penetrate into the left part and are concentrated in
the triangular region bounded by the dashed line.

The following conclusions can be drawn from an analysis of Fig. 1.

  1. 1.

It can be seen that the Galactic objects (red circles) are located
predominantly in the left part of this diagram, in the region of low
$\lg(F_{\rm x}/F_{\rm o})$, while the extragalactic ones (green and blue
circles) are located on the right. This is an expression of the well-known
fact that the stars, on average, have lower ratios $\lg(F_{\rm x}/F_{\rm o})$
than the AGNs and quasars (Maccacaro et al., 1988; Marchesi et al., 2016;
Salvato et al., 2018; Hasinger et al., 2021). Note that 57 Galactic sources
turned out to be beyond the lower boundary in Fig. 1. All these sources are
brighter than 15 mag in the $g$ band and may be overexposed in the DESI LIS
images, so that their color determined from the DESI LIS data may be
distorted. The Gaia data unambiguously point to their Galactic nature, while
the possible distortions of their optical color do not affect the
classification of sources.

Fig. 2 presents the $\lg(F_{\rm x}/F_{\rm o})$ distributions for the Galactic
(red solid line) and extragalactic (the green dash–dotted and blue dashed
lines for the galaxies and quasars, respectively) sources. Based on Figs. 1
and 2, we introduced an arbitrary boundary, $\lg(F_{\rm x}/F_{\rm o})$ = -1.4,
separating the stars and quasars. This boundary is indicated in Fig. 1 by the
solid vertical line.

Figure 2: The $\lg(F_{\rm x}/F_{\rm o})$ distributions for the Galactic (red
solid line) and extragalactic (the green dash-dotted and blue dashed lines for
the galaxies and quasars, respectively) sources. The distribution of the
sources without a class is indicated by the dotted line. The classification of
sources is based on the Gaia catalog, the spectroscopic catalogs, and the
Simbad database, as described in the text. It can be seen that the ratio of
the X-ray and optical fluxes separates satisfactorily the Galactic sources and
quasars. At the same time, additional information is required to separate the
Galactic sources and galaxies.

  2. 2.

A small fraction of the extragalactic objects (mostly galaxies333Note that the
eROSITA X-ray sources classified as “galaxy”, as a rule, have an active
nucleus of low or moderate luminosity, so that the emission from the stellar
population plays a prominent role in the optical band. Their emission in the
X-ray band is determined by accretion onto a supermassive black hole.)
penetrate into the left part of the diagram near $r-z\sim 0.5$. In Fig. 1 this
region is marked by the dashed line in the form of a triangle. However, it can
be seen that some of the extragalactic objects can also be located below the
designated region. It can also be seen that the triangular region is uniformly
filled with stars: their density in it does not decrease compared to other
parts of the left half of the diagram.

It is important to note that 81% of the unclassified objects in the left part
of the diagram are optically extended ones (gray circles with black boundary),
suggesting their extragalactic nature. This can be explained by the fact that
the extragalactic objects penetrating into the left part of the diagram are,
as a rule, relatively nearby galaxies that are comparatively easily resolved
by modern optical telescopes. This is illustrated by Fig. 3, which shows the
redshift distribution of the extragalactic objects from different parts of the
$\lg(F_{\rm x}/F_{\rm o})$–color diagram.

Figure 3: Redshift distribution of the extragalactic objects from different
parts of the diagram in Fig. 1. The distributions for the galaxies and quasars
are presented on the upper and lower panels, respectively. On both panels the
lines and the shaded columns designate the distributions of all sources of the
specified type and the distributions of the corresponding types of objects
from the left part of the diagram in Fig. 1, respectively ($\lg(F_{\rm
x}/F_{\rm o})<-1.4$). It can be seen that low-redshift objects fall into the
left part of the diagram in Fig. 1.

  3. 3.

A small number (24) of Galactic objects are located in the right part of the
$\lg(F_{\rm x}/F_{\rm o})$–color diagram. These Galactic objects have high
$\lg(F_{\rm x}/F_{\rm o})$ atypical for stars and, apparently, are cataclysmic
variables and/or active binaries etc. Some of them located near the boundary
can be M-type stars. Interestingly, half (12) of the objects with high
$\lg(F_{\rm x}/F_{\rm o})$ were identified based on the spectroscopic data
rather than the Gaia measurements. One of the next papers in the series of
publications on the Lockman Hole will be devoted to a detailed study and
classification of these objects.

###  3.3 Classification Algorithm

Using our analysis of the diagram in Fig. 1, we formulated an algorithm for
the classification of point-like X-ray sources from the Lockman Hole catalog.
This algorithm does not claim to be universal and was tailored to the specific
parameters of the deep Lockman Hole survey with the eROSITA telescope. Second,
our goal at this stage is the binary classification of the X-ray sources in
the Lockman Hole into Galactic and extragalactic sources. The classification
algorithm based on the ideas presented here will be generalized in our
succeeding papers.

The classification scheme includes the following four steps:

  1. 1.

The Galactic objects were classified based on the Gaia data (338) or the
classification of the spectroscopic catalogs (13).

  2. 2.

The extragalactic sources were classified based on the classification of the
spectroscopic catalogs and Simbad (2339 quasars, 194 galaxies).

The sources to which we failed to assign a class based on the Gaia data or the
spectroscopic catalogs were processed as follows:

  1. 3.

All of the optically extended sources (in the DESI LIS catalog
$\text{type}\neq\text{PSF}$) were classified as extragalactic (1804 sources).

  2. 4.

The point sources were classified in accordance with $\lg(F_{\rm x}/F_{\rm
o})$. The sources were classified as extragalactic at $\lg(F_{\rm x}/F_{\rm
o})>-1.4$ (1592 sources) and as Galactic at $\lg(F_{\rm x}/F_{\rm o})<-1.4$ (6
sources).

Step 4 of this scheme is based on the fact that the extragalactic sources
account for more than 99.9% of all the sources with a known class rightward of
the vertical line in Fig. 1 (i.e., with $\lg(F_{\rm x}/F_{\rm o})>-1.4$),
while 99.7% of all the point sources leftward of this line are Galactic. Note
that this classification leads to the “loss” of Galactic objects with high
$\lg(F_{\rm x}/F_{\rm o})$. However, within the binary classification, which
is our goal, this loss is insignificant, as follows from the above numbers.
This is discussed in more detail in the next section. The number of extended
and point sources for each class is given in Table 1.

Source type | Point | Extended
---|---|---
Galactic | 333 | 5
Galactic (non-Gaia) | 12 | 1
Galaxy | 28 | 166
Quasar | 1886 | 453
Unknown | 1626 | 1836

Table 1: The number of optically extended and point sources separated into
classes. Almost all of the Galactic objects are point ones. The optically
extended Galactic objects are discussed in 4.2.

Figure 4 presents the $g$ magnitude distributions for the sources mentioned in
different steps of the classification algorithm. The unclassified objects
(gray dotted line) are, on average, less bright in the optical range than
those to which we managed to assign a class based on the Gaia catalog and the
spectroscopic catalogs (for more details, see Subsection 3.4).

Of the 6885 sources from the catalog, we considered 6346 sources with a
counterpart ($p_{\rm any}>0.12$). For 2884 of them we managed to assign a
class based on the extraneous catalogs or the Simbad database (see Subsection
3.1). Of the remaining 3462 objects, 60 have an optical $z$ flux in the DESI
LIS catalog with S/N $<$ 3\. These objects were not classified, since $F_{\rm
x}/F_{\rm o}$ for them were deemed unknown. We classified 3402 sources
according to the rules described above. Table 2 gives the final number of
extragalactic and Galactic sources.

| Classification
---
method
Extragalactic | Galactic
| Based on catalogs
---
(2884)
2533 | 351
| According to rules
---
(3402)
3396 | 6
| Total
---
(6286)
5929 | 357

Table 2: The number of extragalactic and Galactic sources among the objects to
which we assigned a class based on the extraneous catalogs or the Simbad
database (see Subsection 3.1) and those classified according to the rules
described in Subsection 3.3 Figure 4:  The $g$ magnitude distributions for the
Galactic and extragalactic sources described in Subsection 3.3. The Galactic
objects (red solid line) are, on average, brighter than the extragalactic ones
(the blue and green lines for the quasars and galaxies, respectively). The
black dash–dotted line indicates the distribution for the optically extended
sources. The unclassified objects (gray dotted line) are, on average, less
bright than those to which we managed to assign a class based on the Gaia
catalog, the spectroscopic catalogs, and Simbad.

###  3.4 Classification Quality Metrics

To estimate the classification quality, we used the sample of 2864 objects to
which we managed to assign a class based on the Gaia catalog, the
spectroscopic catalogs, and the Simbad database and the optical $z$ flux from
which was determined reliably (hereafter the validation sample).

However, it is worth noting that the validation sample constructed in this way
is not quite representative, since it is subject to selection effects. This is
primarily because for the spectroscopic SDSS programs the objects satisfying
certain criteria in accordance with the peculiar goals of these observing
programs were chosen. In addition, reliable (and, on average, brighter)
targets are usually chosen for spectroscopic observations. For this reason,
the validation sample contains, on average, brighter sources (see Fig. 4).
Larger validation samples are required to achieve the necessary accuracy of
determining the quality metrics, whereas there exist a limited number of
sources of spectroscopic information for a large number of objects. Therefore,
the construction of a representative validation sample is a general difficulty
that the creators of the object classification methods run into; this problem
is beyond the scope of our paper.

Our validation sample allows the quality of the binary classification into
Galactic and extragalactic objects to be estimated. For this purpose, we
applied steps 3 and 4 of the classification algorithm to the sources from this
sample in such a way as if the classes of these sources are unknown to us. The
results of our comparison of the predicted and real classes are discussed
below.

Errors of two types can be made during the classification: we can erroneously
call an extragalactic source a Galactic one and vice versa. Let us designate
the extragalactic objects as a positive class. The Galactic object erroneously
classified as extragalactic will then give a false positive (FP) result. The
Galactic object classified as extragalactic will give a false negative (FN)
result. The recall for the classification of extragalactic sources is
calculated from the formula TP/(TP + FN), while the precision is calculated as
TP/(TP + FP), where TP (true positive) is the number of correctly classified
extragalactic sources. The metrics for the Galactic sources as a positive
class are calculated in a similar way.

On the validation sample we managed to correctly classify 2529 extragalactic
objects and 306 Galactic ones. At the same time, 28 Galactic objects were
erroneously classified as extragalactic and one extragalactic object was
classified as Galactic. Thus, the recall and the precision are

  * •

$99.96\pm 0.04\%$ and $98.90\pm 0.2\%$, respectively, for the selection of
extragalactic objects,

  * •

$91.62\pm 0.5\%$ and $99.67\pm 0.1\%$, respectively, for the selection of
Galactic objects.

The statistical uncertainties given above were calculated under the assumption
of a binomial distribution.

We also repeated the above calculations by dividing the sample into two halves
and using one of them to choose the parameters of the classification algorithm
and the other one to determine the recall and precision metrics and obtained
results similar to those described above.

In accordance with the discussion in the previous subsection (see the
discussion after step 4 of the classification scheme), 28 of the 29
incorrectly classified objects are accounted for by the cases where a Galactic
object is erroneously classified as extragalactic. Among them there are almost
all (12 of 13) of the objects from the validation sample that were found to
belong to the Galactic ones not based on the Gaia data (see Subsection 4.2 and
Table 4). They all have atypically high values of $F_{\rm x}/F_{\rm o}$.

##  4 DISCUSSION

###  4.1 Curves of Cumulative Number Counts (logN–logS) for the Galactic and
Extragalactic Sources

Figure 5:  The curves of cumulative source number counts log$N$–log$S$ in the
Lockman Hole. The curve was normalized to square degree. The red dashed line
indicates the curve for all point X-ray sources from the catalog by Gilfanov
et al. (2023, in preparation). The blue and green solid lines indicate the
curves of cumulative number counts for the sources classified as extragalactic
and Galactic, respectively. The dotted line indicates the dependence $\propto
F_{X}^{-1}$, normalized to the number of Galactic sources at fluxes $\gtrsim
2\times 10^{-14}$ erg/s/cm2. The semitransparent regions around the curves
indicate the statistical uncertainties calculated under the assumption of a
Poisson distribution. The black line indicates the curve of cumulative source
number counts from Georgakakis et al. (2008).

Figure 5 presents the curves of cumulative number counts444Recall that the
curve of cumulative source number counts (the so-called log$N$–log$S$ curve)
shows the flux dependence of the number of sources (as a rule, normalized to
unit solid angle) with a flux higher than the specified one. for the sources
classified as Galactic and extragalactic and the full curve of cumulative
number counts for all sources. For comparison, the figure presents the curve
of cumulative number counts from the Chandra data in Georgakakis et al.
(2008). For the eROSITA sources the counts were converted to the 0.5–2.0 keV
flux corrected for interstellar absorption by assuming a median value of
$NH=7\times 10^{19}$ cm-2 and a slope of the power-law spectrum $\Gamma=1.9$.

The incompleteness of the curves of cumulative number counts presented in Fig.
5 was not corrected in this paper. It follows from the curve of the effective
Lockman Hole survey area given in (M. Gilfanov et al, 2022) that the
incompleteness effects begin to play a role at a $\sim(2-3)\times 10^{-14}$
erg/s/cm2. In Fig. 5 this is reflected in the flattening of the curves of
cumulative number counts at lower fluxes.

Extragalactic sources (AGNs and quasars) account for most of the sources
detected by eROSITA, while the fraction of Galactic sources (stars) at fluxes
$\sim 10^{-14}$ erg/s/cm2 is $\sim 10\%$. On the whole, this value is typical
for fields at high Galactic latitudes, but it can change with location in the
sky. A difference in the slopes of the curves of cumulative number counts for
these two populations is also clearly seen. The curve of cumulative number
counts for the extragalactic sources follows the law $\propto F_{X}^{-3/2}$,
as would be expected in this range of fluxes, whereas the curve of cumulative
number counts for the Galactic sources is flatter and follows the dependence
$\propto F_{X}^{-1}$. Note that this slope coincides remarkably well with the
value expected for the population of sources in the disk, but this analogy
seems to be inapplicable here.

Georgakakis et al. (2008) did not separate the sources into extragalactic and
Galactic ones. Accordingly, their curve of cumulative number counts agrees
well with the full eROSITA curve of cumulative number counts. For example, the
number density of sources at fluxes $F_{\rm x}>2\times 10^{-14}$ is $33.2\pm
1.1$ and $33.6\pm 1.4$ sources/deg2 from the eROSITA and Chandra data,
respectively.

###  4.2 Peculiar Sources

Here we provide the lists of objects that for some reasons are unusual and
were revealed during the classification of X-ray sources in the Lockman Hole.

Table 3 lists the Gaia Galactic sources selected in Subsection 3.1 and lying
in Fig. 1 in the region of extragalactic objects ($\lg(F_{\rm x}/F_{\rm
o})>-1.4$). These Galactic sources have an atypically large ratio of the X-ray
flux to the optical one and can be candidates for cataclysmic variables. The
source SRGe J103224.9+572814 is worth noting separately: according to the Gaia
catalog, it has a significant proper motion ($12.5\pm 0.5$ mas yr-1), while
being a quasar ($z=2.43\pm 0.00014$), according to the spectroscopic catalogs.
For more details on such objects, see, e.g., Souchay et al. (2022) and
Khamitov et al. (2022).

Table 4 lists the objects that were classified as Galactic based on the
spectroscopic catalogs or the Simbad database (without using Gaia data). Most
of them, just as the objects from Table 3, have $\lg(F_{\rm x}/F_{\rm o})$
typical for extragalactic objects.

The optical extent of some of the Gaia Galactic objects (marked by the red
circles with the black boundary in Fig. 1, their list is given in Table 5) can
be associated with their relatively large proper motions and, as a
consequence, the source position displacement between the optical images taken
at different times. In turn, the proper motions can be caused by the galaxy
photocenter displacement due to, for example, the motion of the jets or
microlensing. The errors of the Gaia astrometric solutions cannot be ruled out
completely either. The recent paper by Khamitov et al. (2022) is devoted to
searching for and studying such objects in the eROSITA X-ray catalog.

| Source | SDSS class | MILQ class | Simbad class | Color | lg(Fx/Fo)
---|---|---|---|---|---|---
1 | SRGe J103906.0+553044 | - | - | Star | -3.62 | -1.37
2 | SRGe J105633.1+583529 | - | - | Star | -1.82 | -1.30
3 | SRGe J104522.1+555738 | - | QSO | Radio Source | -0.56 | -1.27
4 | SRGe J103618.3+581246 | - | - | - | -0.70 | -1.21
5 | SRGe J104007.7+595700 | - | - | Long-Period Variable Candidate | -2.56 | -1.15
6 | SRGe J103453.6+553633 | - | - | - | 1.81 | -1.14
7 | SRGe J105310.4+575437 | - | - | - | 1.76 | -1.01
8 | SRGe J103224.9+572814 | QSO | QSO | Quasar | 1.11 | -0.93
9 | SRGe J105848.1+592918 | - | - | HII Region | -0.60 | -0.82
10 | SRGe J102411.6+561606 | - | - | - | -0.36 | -0.64
11 | SRGe J110137.5+572926 | - | - | Star | -2.87 | 0.03
12 | SRGe J104325.5+563300 | - | - | Cataclysmic Binary | -0.32 | 0.13

Table 3:  The Gaia Galactic sources (designated by the red circles in Fig. 1)
with $\lg(F_{\rm x}/F_{\rm o})>-1.4$. The colors below –1 may be distorted and
are discussed in Subsection 3.2.

| Source | SDSS class | SDSS subclass | MILQ class | Simbad class | Color | lg(Fx/Fo)
---|---|---|---|---|---|---|---
1 | SRGe J103049.0+563614 | - | - | - | Star | -0.02 | -0.27
2 | SRGe J103606.5+573624 | STAR | K5 | STAR | - | 0.34 | -0.08
3 | SRGe J104809.2+581036 | - | - | - | Star | -0.41 | -3.59
4 | SRGe J105106.4+552342 | STAR | O9.5Iae | STAR | - | 0.08 | 0.31
5 | SRGe J105237.2+600836 | - | - | - | Star | 0.24 | -0.03
6 | SRGe J103256.5+574818 | - | - | - | Star | 0.78 | 0.53
7 | SRGe J103511.4+600547 | - | - | STAR | - | 0.16 | 0.05
8 | SRGe J105118.5+551934 | STAR | O9.5Iae | STAR | Blue Object | -0.15 | -0.12
9 | SRGe J104636.2+555934 | - | - | - | Star | -1.43 | -0.01
10 | SRGe J103426.1+575524 | - | - | - | Star | - | 0.46
11 | SRGe J105432.9+590945 | - | - | - | Star | 0.7 | -0.29
12 | SRGe J103744.3+571156 | STAR | O8e | QSO | BL Lac | 0.49 | 0.43
13 | SRGe J104429.8+595354 | - | - | - | Star | 0.14 | -0.38

Table 4:  The objects that were classified as Galactic based on information
from the spectroscopic catalogs or the Simbad database (not based on Gaia
data). Marked by the red circumferences in Fig. 1. The colors below –1 may be
distorted and are discussed in Subsection 3.2.

| Source | GAIA PM (mas/yr) | GAIA PM SNR | lg(Fx/Fo)
---|---|---|---|---
1 | SRGe J105146.0+554552 | 144.64 | 204.3 | 1.91
2 | SRGe J104425.9+565729 | 24.21 | 69.3 | 1.46
3 | SRGe J105603.0+595317 | 5.58 | 14.4 | -0.24
4 | SRGe J105130.7+573439 | 63.28 | 310.6 | 2.00
5 | SRGe J102411.6+561606 | 9.38 | 602.8 | -0.36

Table 5:  The Gaia Galactic objects that are extended objects according to the
DESI LIS catalog. Marked by the red circumferences with the black boundary in
Fig. 1.

##  5 CONCLUSIONS

We classified the X-ray sources in the SRG/eROSITA catalog produced based on
the Lockman Hole survey (M. Gilfanov et al, 2022). For this purpose, we used
the results of the crossmatch of X-ray sources with objects from the optical
DESI LIS catalog made in Bykov et al. (2022). Of the 6885 X-ray sources, we
managed to assign a class to 2884 sources after their cross-match with the
Gaia DR3, SDSS DR16, MILLIQUAS catalogs, and the Simbad database. For the
remaining sources we performed the classification based on information about
the object optical extent and the ratio $F_{\rm x}/F_{o}$ of the X-ray and
optical ($z$) fluxes. As a result, we classified 357 and 5929 sources as
Galactic and extragalactic, respectively. We estimated the selection precision
and recall for the extragalactic (99.9 and 98.9%, respectively) and Galactic
(91.6 and 99.7%) sources. No classification was made for the 60 sources for
which the DESI LIS photometry had an insufficient S/N. Using the cross-match
results from Bykov et al. (2022), we classified 539 sources as hostless, i.e.,
having no optical counterparts in the DESI LIS survey. The results of this
paper will be used in our subsequent papers to construct the luminosity
function of AGNs and quasars in the Lockman Hole field. The code that
accompanies this paper is available at https://github.com/mbelveder/ero-lh-
class.git.

## ACKNOWLEDGMENTS

This study is based on observations with the eROSITA telescope onboard the SRG
observatory. The SRG observatory was built by Roskosmos in the interests of
the Russian Academy of Sciences represented by the Space Research Institute
(IKI) within the framework of the Russian Federal Space Program, with the
participation of the Deutsches Zentrum fur Luft- und Raumfahrt (DLR). The
SRG/eROSITA X-ray telescope was built by a consortium of German institutes led
by MPE, and supported by DLR. The SRG spacecraft was designed, built, launched
and is operated by the Lavochkin Association and its subcontractors. The
science data are downlinked via the Deep Space Network Antennae in Bear Lakes,
Ussurijsk, and Baykonur, funded by Roskosmos. The eROSITA data used in this
paper were processed with the eSASS software developed by the German eROSITA
consortium and the proprietary data reduction and analysis software developed
by the Russian eROSITA consortium.

The following software was used: NumPy (Harris et al., 2020), Matplotlib
(Hunter, 2007), SciPy (Virtanen et al., 2020), Pandas(The pandas development
team, 2020), AstroPy (Astropy Collaboration et al., 2018). We used the SIMBAD
data base maintained by the Strasbourg Astronomical Data Center
(CDS555https://simbad.u-strasbg.fr/simbad, Strasbourg, France).

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