11institutetext: Porter School of the Environment and Earth Sciences, Raymond
and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv,
6997801, Israel
11email<EMAIL_ADDRESS>22institutetext: Department of Particle Physics
and Astrophysics, Weizmann Institute of Science, Rehovot 7610001, Israel

# Model Independent Periodogram for Scanning Astrometry

A. Binnenfeld Model Independent Periodogram for Scanning AstrometryModel
Independent Periodogram for Scanning Astrometry S. Shahaf Model Independent
Periodogram for Scanning AstrometryModel Independent Periodogram for Scanning
Astrometry S. Zucker Model Independent Periodogram for Scanning
AstrometryModel Independent Periodogram for Scanning Astrometry

(Accepted XXX. Received YYY)

We present a new periodogram for periodicity detection in one-dimensional
time-series data from scanning astrometry space missions, like Hipparcos or
Gaia. The periodogram is non-parametric and does not rely on a full or
approximate orbital solution. Since no specific properties of the periodic
signal are assumed, the method is expected to be suitable for the detection of
various types of periodic phenomena, from highly eccentric orbits to periodic
variability-induced movers. The periodogram is an extension of the phase-
distance correlation periodogram (PDC) we introduced in previous papers based
on the statistical concept of distance correlation. We demonstrate the
performance of the periodogram using publicly available Hipparcos data, as
well as simulated data. We also discuss its applicability for Gaia epoch
astrometry, to be published in the future data release 4 (DR4).

###### Key Words.:

methods: data analysis – methods: statistical – astrometry – binaries: general
– planets and satellites: detection

##  1 Introduction

Major astrometric survey space missions such as Hipparcos and Gaia provide
position and parallax measurements for a large number of celestial objects
(Perryman et al., 1997; Gaia Collaboration et al., 2016). Both missions
monitor the sky following a complex scanning law, yielding a sparse unevenly-
sampled astrometric time series of the various targets (van Leeuwen & Evans,
1998; Gaia Collaboration et al., 2016). This scanning is performed along
reference great circles (RGC), gradually covering the entire celestial sphere.
Thus, in each epoch, an object location is conveyed by two quantities: the
scanning angle $\theta$, and the abscissa $v$.

In its original meaning, the Latin word ‘abscissa’ refers to the horizontal
coordinate in a two-dimensional Cartesian system. In the context of scanning
astrometry, the word abscissa refers to one-dimensional positional coordinates
that are measured along the scanning direction of the instrument. While
scanning astrometry provides well-constrained abscissae, positional
measurements in the cross-scan direction, perpendicular to the scanning
direction, are obtained with significantly lower precision. As a result,
scanning astrometry data are often considered one-dimensional positional
measurements. The information they convey is only partial information
regarding the astrometric position, while the actual position lies somewhere
on an imaginary line perpendicular to the scanning direction. Note that the
abscissa is often specified with respect to the catalogue nominal position.
Fig. 1 illustrates the geometric setting described above.

Full astrometric solutions for the observed targets include several parameters
fitted to minimize the model errors in terms of the abscissa residuals (see
Perryman et al., 1997; ESA, 1997). In the simplest cases, the basic five-
parameter solution (right ascension, declination, two-dimensional proper
motion vector, and parallax) is sufficient to model the target’s proper motion
combined with the apparent motion induced by the motion of Earth. When
considering the astrometric motion of a binary system, seven or nine
parameters are required to account for the target’s acceleration. In some
cases, if the orbital period is sufficiently short, a twelve-parameter model
is fitted, to fully account for the Keplerian motion of the system. Solving
for those parameters is more complicated and might even be quite
computationally demanding and prone to errors of various kinds (e.g. van
Leeuwen, 2007).

With the very high angular precision of modern instruments such as Gaia, the
situation might become even more complicated when the binary system includes
compact objects, and relativity comes into effect through light deflection and
delays (e.g. Halbwachs, 2009). Apparent periodic astrometric motion can also
occur in unresolved visual double stars, in which one of the components is a
periodically pulsating star. In these cases, commonly known as VIMs
(Variability Induced Movers), the photocenter of the two blended stars
undergoes a periodic displacement, which is related to the photometric
periodic variability (Wielen, 1996).

In other types of astronomical observations, such as radial velocities (RV) or
photometry, the measured quantity is simply a scalar. The common procedure in
this context is to first look for periodicity. If such periodicity is indeed
found and a crude estimate of the period is obtained, then a full
characterization of the variability is attempted, such as a Keplerian RV
model, an eclipsing-binary model, a Cepheid light-curve model, etc. The most
common approach to the first step, detecting the periodicity, is using a
periodogram. A periodogram scans a grid of trial periods (or frequencies) and
assigns each one a score. This score quantifies the plausibility that a
periodicity exists in the data, assuming this particular trial period.

The most commonly used periodogram for scalar data is probably the Generalized
Lomb-Scargle (GLS) periodogram, which fits a sinusoidal function to the data
and quantifies the goodness of fit (Zechmeister & Kürster, 2009). The
underlying assumption is that any periodic function can be described as a sum
of harmonics, and assuming the first harmonic is the dominant one, the
function can be approximated by a sinusoid. However, the case of abscissa data
resulting from scanning astrometry is more complicated and requires special
treatment. This is mainly related to the fact that every abscissa is measured
along a different scanning direction. Therefore, the data cannot be treated in
the same way as simple scalar data.

Figure 1: Definition of the abscissa residual $v$ for a single-epoch position
($\alpha_{m}$, $\delta_{m}$), with respect to the nominal catalogue position
($\alpha_{0}$, $\delta_{0}$) and the RGC orientation $\theta$. The actual
position of the observed target lies somewhere along the blue dotted arrowed
segment. The figure is based on one originally published by van Leeuwen &
Evans (1998).

In this work, we present a non-parametric method to detect periodicity in
time-series of one-dimensional astrometric data. This periodogram does not
search for a specific type of periodic astrometric motion, and it is based on
the concept of phase distance correlation (PDC). Zucker (2018) introduced the
PDC periodogram as a new method to detect periodicity in time-series data.
Essentially, for each trial period, PDC quantifies the statistical dependence
between the measured quantity and the phase (according to the trial period),
using the recently introduced distance correlation (Székely et al., 2007).

Distance correlation is a measure of the statistical dependence between random
variables (as opposed to the correlation coefficient, which is a measure of
linear dependence). One of its merits is that the two random variables need
not be of the same dimensions and can even be drawn from general metric
spaces, not necessarily Euclidean (Lyons, 2013). Lyons showed that if the two
metrics involved are of ‘strong negative type’(see Zinger, Kakosyan, &
Klebanov, 1992), distance correlation is guaranteed to be applicable as a
measure of statistical dependence.

The PDC periodogram, as introduced by Zucker (2018), calculates for each trial
period, the distance correlation between the analyzed quantity (such as RV)
and the phase according to the trial period. As Euclidean spaces are
themselves of strong negative type (Lyons, 2013), Zucker (2019) further
formulated and demonstrated the application of the PDC periodogram to two-
dimensional astrometry, using the regular Euclidean metric to quantify
distances between different measurements.

In the next section, we discuss the specific adaptation of the PDC concept to
scanning astrometry. Sect. 3 demonstrates the application of the new
periodogram to Hipparcos data, as well as additional simulated data. We
conclude in Sect. 4 and discuss the method and its potential for future
studies, particularly in the context of Gaia data.

##  2 PDC for one-dimensional scanning astrometry

###  2.1 A new metric

In order to extend the PDC concept to data from scanning astrometry, we had to
define a suitable metric for the set of observations, using the above-
mentioned parameters: scanning angle $\theta$ and abscissa $v$. This metric
should then enable estimating the distance correlation measure used to compute
the PDC periodogram. After experimenting with a few alternative metrics, we
chose the following, which is based on the concept of Energy Distance.

Energy distance is a statistical distance between probability distributions,
first introduced by Gábor Székely (Székely & Rizzo, 2013). Note that energy
distance is a metric of strong negative type (Rizzo & Székely, 2016), and
therefore satisfies the sufficient condition for the applicability of distance
correlation as an independence test, as mentioned in the previous section.

As shown in Sect. 1, the actual astrometric position of the target can be
assumed to lie on a line segment whose length is constrained by the width of
the scanning sensor (Fig. 1). We do not have a reason to assume that there is
a preference for one particular area of the detector over another, and
therefore all points on the segment are a priori equally probable.

In a Bayesian inference setting, we could set a uniform prior for the position
along the width of the detector. However, our approach is based on distance
correlation and therefore relies on our ability to define a metric between
pairs of measurements. To qualitatively reflect the logic of an uninformative
prior, we choose to represent each segment only by the position of its edges.
This choice defines a one-to-one correspondence between the set of all
segments and the set of all unordered pairs of points on the plane.

For the set of all pairs of points on the plane we can use the Energy Distance
metric, in the following way: let $U$ and $W$ be two segments in the plane,
which can be represented by their endpoints: $\bm{u_{1}}$, $\bm{u_{2}}$ and
$\bm{w_{1}}$,$\bm{w_{2}}$, respectively. We can then consider each segment as
a sample of two points and calculate the energy distance between the two
samples corresponding to $U$ and $W$ (Székely & Rizzo, 2013):

$\mathcal{D}(U,W)=\frac{1}{2}\sum_{i=1}^{2}\sum_{j=1}^{2}{|\bm{u_{i}}-\bm{w_{j}}|}-\frac{1}{2}|\bm{u_{1}}-\bm{u_{2}}|-\frac{1}{2}|\bm{w_{1}}-\bm{w_{2}}|$
(1)

Note that $|\bm{u_{1}}-\bm{u_{2}}|$ is actually the length of the segment $U$.
Now, let the two segments be defined by the abscissae $v_{1}$ and $v_{2}$ and
the scanning directions $\theta_{1}$ and $\theta_{2}$ (see Fig. 1), and assume
that the width of the scanning field of view, and therefore the length of the
segments, is $L$. The segments are then uniquely defined by the following two
sets of vectors on the plane:

$\begin{gathered}U=\left\\{\begin{pmatrix}v_{1}\sin{\theta_{1}}-\frac{1}{2}L\cos{\theta_{1}}\\\
v_{1}\cos{\theta_{1}}+\frac{1}{2}L\sin{\theta_{1}}\\\
\end{pmatrix},\begin{pmatrix}v_{1}\sin{\theta_{1}}+\frac{1}{2}L\cos{\theta_{1}}\\\
v_{1}\cos{\theta_{1}}-\frac{1}{2}L\sin{\theta_{1}}\\\
\end{pmatrix}\right\\}\\\ \\\
W=\left\\{\begin{pmatrix}v_{2}\sin{\theta_{2}}-\frac{1}{2}L\cos{\theta_{2}}\\\
v_{2}\cos{\theta_{2}}+\frac{1}{2}L\sin{\theta_{2}}\\\
\end{pmatrix},\begin{pmatrix}v_{2}\sin{\theta_{2}}+\frac{1}{2}L\cos{\theta_{2}}\\\
v_{2}\cos{\theta_{2}}-\frac{1}{2}L\sin{\theta_{2}}\\\
\end{pmatrix}\right\\}\end{gathered}$ (2)

We can now substitute the endpoints of the segments defined in Eq. 2 into the
definition of the energy distance (Eq. 1), and get the following expression:

$\mathcal{D}(U,W)=\frac{1}{2}\biggl{(}\,\Bigl{\\{}v_{1}^{2}-2v_{1}v_{2}\cos(\theta_{1}-\theta_{2})+v_{2}^{2}\\\
+L(v_{1}+v_{2})\sin(\theta_{1}-\theta_{2})+\frac{1}{2}L^{2}\Bigl{[}\,1+\cos(\theta_{1}-\theta_{2})\Bigr{]}\,\Bigr{\\}}^{\frac{1}{2}}\\\
+\Bigl{\\{}(v_{1}^{2}-2v_{1}v_{2}\cos(\theta_{1}-\theta_{2})+v_{2}^{2}\\\
+L(v_{1}-v_{2})\sin(\theta_{1}-\theta_{2})+\frac{1}{2}L^{2}\Bigl{[}\,1-\cos(\theta_{1}-\theta_{2})\Bigr{]}\,\Bigr{\\}}^{\frac{1}{2}}\\\
+\Bigl{\\{}v_{1}^{2}-2v_{1}v_{2}\cos(\theta_{1}-\theta_{2})+v_{2}^{2}\\\
-L(v_{1}-v_{2})\sin(\theta_{1}-\theta_{2})+\frac{1}{2}L^{2}\Bigl{[}\,1-\cos(\theta_{1}-\theta_{2})\Bigr{]}\,\Bigr{\\}}^{\frac{1}{2}}\\\
+\Bigl{\\{}(v_{1}^{2}-2v_{1}v_{2}\cos(\theta_{1}-\theta_{2})+v_{2}^{2}\\\
-L(v_{1}+v_{2})\sin{(\theta_{1}-\theta_{2})}+\frac{1}{2}L^{2}\Bigl{[}\,1+\cos(\theta_{1}-\theta_{2})\Bigr{]}\,\Bigr{\\}}^{\frac{1}{2}}\biggr{)}\,-L$
(3)

Thus we have obtained a metric for the set of astrometric measurements,
without any further assumptions concerning the observed one-dimensional object
position (i.e. model-independent). We will use it in the following section to
formulate the PDC for one-dimensional astrometric measurements.

In developing the above metric we assumed there is no information available
about the location of the object in the direction perpendicular to the
scanning direction. While some negligible information might be effective, we
preferred to follow the guidelines proposed by Lindegren & Bastian (2010) who
recommended treating measurements of space scanning astrometry as one-
dimensional.

###  2.2 Astrometric PDC periodogram formulation

Following Zucker (2018) and Zucker (2019), let us define a distance matrix
based on the metric we have introduced in Eq. 3. For each pair of astrometric
measurements ($i$ and $j$), the entry in the distance matrix would then be:

$a_{ij}=\mathcal{D}(i,j)\ .$ (4)

For each trial period $P$ we define a phase distance matrix:

$\begin{split}\phi_{ij}&=(t_{i}-t_{j})\mod P\ ,\\\
b_{ij}&=\phi_{ij}(P-\phi_{ij})\ .\end{split}$ (5)

Now we apply $\mathcal{U}$-centering to the two matrices, which will allow an
unbiased estimator of the distance correlation (Székely & Rizzo, 2014):

$A_{ij}=\begin{cases}\begin{split}a_{ij}-\frac{1}{N-2}\sum\limits_{k=1}^{N}a_{ik}-\frac{1}{N-2}\sum\limits_{k=1}^{N}a_{kj}\\\
+\frac{1}{(N-1)(N-2)}\sum\limits_{k,l=1}^{N}a_{kl}\end{split}&\text{if $i\neq
j$ ,}\\\ \\\ 0&\text{if $i=j$ .}\end{cases}$ (6)

A similar procedure is applied to obtain the matrix $B_{ij}$ from $b_{ij}$.
Using the $\mathcal{U}$-centered matrices the unbiased estimator of the
distance correlation can now be computed via the expression:

$D=\frac{\sum\limits_{ij}A_{ij}B_{ij}}{\sqrt{(\sum\limits_{ij}A^{2}_{ij})(\sum\limits_{ij}B^{2}_{ij})}}\
.$ (7)

The significance of a period detected by PDC can be assessed using a
permutation test, which we have indeed used in previous papers (Binnenfeld et
al., 2020, 2022): we created a sample of $D$ values by randomly shuffling the
assignment of measurements to phases, and recalculating $D$ for this random
allocation of phases. Then we could obtain a threshold value matching a
desired level of false-alarm probability (FAP).

In this paper, we use instead a chi-square test tailored for distance
correlation, which was recently proposed by Shen et al. (2022). It is very
fast to compute and it spares the need for the computationally heavy
permutation test.

##  3 Examples

###  3.1 Hipparcos intermediate astrometric data

The launch of the Hipparcos Space Telescope in 1989 was a major breakthrough
in astronomy. In its four-year mission it measured positions, parallaxes, and
proper motions for about 120,000 stars, and its precision of up to a few mili-
arc-seconds (mas) was unprecedented at the time. The Hipparcos Catalogue was
published in 1997 (Perryman et al., 1997).

To test and demonstrate our newly developed periodogram, we used the publicly
available Hipparcos intermediate astrometric data (IAD), which include the
individual abscissa residuals (after subtracting the basic astrometric model)
for each source in the catalogue, and are publicly available. Two different
versions of the data are offered, produced by the two different data analysis
consortia FAST (Kovalevsky et al., 1992) and NDAC (Lindegren et al., 1992).

We have experimented with various possible values of $L$, the length of the
segments on which the actual astrometric position of the target can be assumed
to lie. We found that values smaller than a few milli-arc-seconds produced
very similar periodograms. Thus, we opted to use $L=1$ mas in our
demonstrations.

####  3.1.1 Binary stars

Figure 2: Astrometric PDC periodograms for selected Hipparcos binary-star
targets from the catalogue of astrometric orbits published by Goldin & Makarov
(2007). More information on the targets can be found in Table 1. The dashed
vertical lines mark the published orbital frequencies, and the dotted
horizontal lines correspond to a FAP level of $10^{-3}$.

To demonstrate the capability of our astrometric PDC method to detect periodic
orbits of binary stars, we used a catalogue of astrometric orbits from
Hipparcos published by Goldin & Makarov (2007). The stars included in this
catalogue had been classified in the original Hipparcos catalogue as
’stochastic binaries’, i.e., binary stars whose orbits could not have been
characterized by the Hipparcos teams.

Fig. 2 presents six example periodograms selected out of many positive results
for the catalogue items, all based on NDAC data. In all of the PDC
periodograms we obtained, a significant peak appears at the expected
frequency, corresponding to the period of the published solution (marked in
dashed vertical lines). We also mark in dotted horizontal lines the value
corresponding to a FAP level of $10^{-3}$. More information on the presented
targets can be found in Table 1.

Table 1: Details of the binary stars used for demonstration (Fig. 2).
$\begin{array}[]{llcccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\textrm{HIP}&\textrm{Designations}&\textrm{\\#
epochs}&\textrm{Published period}&\textrm{Eccentricity}&\textrm{PDC period}\\\
&&&\textrm{[days]}&&\textrm{[days]}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\textrm{HIP
21965}&\textrm{HD 30051}&38&709^{+26}_{-23}&0.29^{+0.30}_{-0.17}&746\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\textrm{HIP 30342}&\nu\,\textrm{
Pictoris}&37&452^{+13}_{-16}&0.20^{+0.35}_{-0.19}&435\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\textrm{HIP 32307}&\textrm{HD
49377}&41&889^{+53}_{-56}&0.15^{+0.25}_{-0.16}&909\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\textrm{HIP 38018}&\textrm{HD
61994}&56&547^{+8}_{-9}&0.52^{+0.11}_{-0.09}&549\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\textrm{HIP 76006}&\textrm{HD
138525}&49&572^{+21}_{-21}&0.51^{+0.32}_{-0.27}&588\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\textrm{HIP 113136}&\delta\textrm{
Aquarii}&24&483^{+20}_{-19}&0.12^{+0.25}_{-0.15}&485\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\hline\cr\end{array}$

####  3.1.2 Brown dwarfs

Figure 3: Astrometric PDC periodograms for selected brown-dwarf hosting stars.
More information on the targets can be found in Table 2. The dashed vertical
lines mark the published orbital frequencies, and the dotted horizontal line
corresponds to a FAP level of $10^{-3}$.

Following the positive results obtained for binary-star systems, we found our
method to be sensitive enough to detect astrometric orbits induced by brown-
dwarf companions as well.

Fig. 3 presents three examples of the results obtained by applying our method
to the Hipparcos IAD of known brown-dwarf hosting stars. We applied it to the
FAST data of HIP 13769 and HIP 113718, and to the NDAC data of HIP 62145.
Their orbital parameters, listed in Table 2, were published by Tokovinin et
al. (1994), Halbwachs et al. (2000), and Reffert & Quirrenbach (2011).

In all of the presented periodograms pertaining to those candidates, a
significant peak appears close to the expected frequency, matching the orbital
period of the known companion (marked in dashed vertical lines). FAP levels of
$10^{-3}$ are marked in dotted horizontal lines.

Table 2: Details for the brown-dwarf hosting stars used for demonstration.
$\begin{array}[]{llccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\textrm{HIP}&\textrm{Designations}&\textrm{\\# of
epochs}&\textrm{Published period}&\textrm{eccentricity}&\textrm{Min.
mass}&\textrm{PDC period}\\\
&&&\textrm{[days]}&&[M_{\textrm{J}}]&\textrm{[days]}\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\textrm{HIP
13769}&\textrm{HD
18445}&26&554.58^{+1.25}_{-1.25}&0.558^{+0.067}_{-0.067}&44&562\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\textrm{HIP 62145}&\textrm{HD
110833}&47&572^{+21}_{-21}&0.51^{+0.32}_{-0.27}&17&588\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\textrm{HIP 113718}&\textrm{HD
217580}&18&454.66^{+0.94}_{-0.94}&0.52^{+0.022}_{-0.022}&67&442.5\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$

###  3.2 Simulated data

Figure 4: Astrometric PDC periodograms for a simulated system with a 100-day
orbital period and a 60-day satellite rotation period. The dashed vertical
line marks the simulated orbital frequency, and the dotted horizontal line
corresponds to a FAP level of $10^{-3}$. Note that an additional peak matching
the period of $\sim 30$ days appears in the periodogram, and it is probably
related to half the satellite rotation period.

The above results demonstrate that our newly developed periodogram is
effective for periodicity detection in Hipparcos IAD. The data are given with
respect to the catalogue position, and therefore they are corrected for the
proper motion and parallactic motion of the observed star. However, in
principle, especially in the presence of some astrophysical periodicity, the
correction might not be perfect, and some residual effects of the parallax and
the proper motion might still be present. To demonstrate the periodogram
suitability in those conditions, we have further tested it using a simulation.
In addition, the scanning law of the telescope might be affected by the motion
of the satellite and as a result, the scanning angle might not be completely
random. We included such an effect in our simulation as well.

We randomly drew $35$ epochs from a uniform distribution over an interval of
three years. We used them to sample the astrometric signature of a Keplerian
orbit with a semi-major axis of $5\,\textrm{mas}$, high eccentricity
($e=0.8$), and a period of $100$ days. We added to it a circular
$5\,\textrm{mas}$ orbit with a period of $365$ days and a $60^{\circ}$
inclination, to simulate a typical parallax signal, and also a linear trend
simulating the effect of a linear $3\,\textrm{mas\,year}^{-1}$ proper motion.
We further added white noise to the data at a signal-to-noise ratio of
$\textrm{SNR}=75$. The astrometric position at each epoch was then translated
into the scanning angle and abscissa representation, with the scanning angle
following a simulated $60$ days satellite rotation period.

Fig. 4 shows the results of applying the astrometric PDC to the simulated data
described above. A prominent peak indeed appears at the simulated orbital
period. An additional peak matching a $\sim 30$ day period appears in the
periodogram, and it is probably due to interaction between the proper motion
and the sampling window function caused by the satellite simulated 60-day
rotation period. The residual parallax and proper motion, at the levels we
introduced them, do not seem to affect the periodogram ability to detect the
orbital period of the system, though a significant peak does appear, as might
be expected, around a one-year period, probably related to the residual
parallactic motion.

###  3.3 Comparison with Delisle & Ségransan (2022)

In a recent publication, Delisle & Ségransan (2022) presented an astrometric
periodogram based on a linearized Keplerlian model fitted to a set of one-
dimensional astrometric measurements. Their approach relies on analytical
approximations and, therefore, is probably superior to ours in terms of its
computational cost. They also show that the statistical efficiency of their
method degrades slowly as the orbital eccentricity increases, so that a
${\sim}10\%$ loss in power is only achieved for $e\,{\gtrsim}\,0.8$.
Nevertheless, their approach is still model-dependent; for example, it might
be suboptimal for detecting extremely eccentric orbits or variability-induced
motion. When considering the wide range of variability sources, astrophysical
or otherwise, the non-parametric approach presented in this work might prove
valuable, and complementary to other techniques.

Table 3: Comparison of the period detection by Delisle & Ségransan (2022) to
those performed using the astrometric PDC periodogram for HIP 117622 and HIP
12726. We used the literature parameters published in Goldin & Makarov (2007).
$\begin{array}[]{lccccccccc}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\lx@intercol\text{HIP}\hfil\lx@intercol&\lx@intercol a_{0}\text{
[mas]}\hfil\lx@intercol&\lx@intercol\hfil
e\hfil\lx@intercol&\lx@intercol\textrm{literature period
[d]}\hfil\lx@intercol&\lx@intercol\textrm{D\&S period
[d]}\hfil\lx@intercol&\lx@intercol\text{D\&S
FAP}\hfil\lx@intercol&\lx@intercol\hfil\textrm{PDC period
[d]}\hfil\lx@intercol&\lx@intercol\hfil\text{PDC FAP}\hfil\lx@intercol\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\textrm{117622}&12.1^{+2.2}_{-1.2}&0.25^{+0.25}_{-0.16}&1008^{+161}_{-41}&1049&3\times
10^{-5}&943&5\times 10^{-3}\\\ \\\
\textrm{12726}&11.7^{+3.8}_{-1.6}&0.61^{+0.16}_{-0.13}&536^{+12}_{-12}&570&1\times
10^{-3}&572&1\times 10^{-4}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\end{array}$

To demonstrate their method, Delisle & Ségransan (2022) used two systems from
the Hipparcos database that exhibited astrometric periodicity: HIP 117622 and
HIP 12726. We compared the results they obtained in detecting the periodic
signals known from the literature to those of the astrometric PDC periodogram.
As can be seen in Table 3, the astrometric PDC periodogram successfully
detects the known periodicities, with corresponding FAP values comparable to
those of Delisle & Ségransan (2022) for the moderately eccentric HIP 117622
($e=0.25$), and an order of magnitude smaller for the more eccentric HIP 12726
($e=0.61$).

##  4 Conclusion

In this paper, we have presented the astrometric PDC method and demonstrated
that it can be used to detect and quantify periodicity in the Hipparcos
catalogue. It offers a new approach to studying data from scanning astrometry
while avoiding complex, model-dependent solutions characterizing the
variability, such as Keplerian models. We believe this approach can pave the
way to new discoveries and insights.

Considered by many to be Hipparcos successor, Gaia was launched in 2014, with
its latest data release including more than $1.8$ billion stars with an
unparalleled high precision (DR3, Gaia Collaboration et al., 2022b).

The epoch astrometry data of Gaia is expected to become available to the
community as part of the future data release 4 (DR4). Previous data releases
have already shown its potential in the classification of stellar multiplicity
(Gaia Collaboration et al., 2022a; Halbwachs et al., 2022), and also in the
detection of substellar and planetary mass companions (Holl et al., 2022).
Therefore, we expect our periodogram to be useful for the analysis of the Gaia
astrometric data, potentially discovering signals that cannot be identified
using standard analysis methods.

In a recent paper (Binnenfeld et al., 2022), we have introduced the partial
phase distance correlation periodograms, which allow accounting for ’nuisance’
parameters in order to eliminate spurious peaks related to it. In the case of
one-dimensional astrometry, the angle $\theta$ describing the RGC orientation
may be considered a nuisance, since it is determined arbitrarily by the
satellite orbit, and not related to the examined astrophysical phenomenon (see
Holl et al., 2022). Therefore, we plan to explore reducing its effect using
the partial distance correlation periodograms.

Using Hipparcos photometry as a nuisance variable to the astrometry may be
informative as well. It might help in distinguishing orbital movement from
luminosity variability in the case of VIMs. Gaia DR4 will also include various
other types of epoch data, such as radial velocities and photometry in
different bands, which we may be able to use in a similar way.

Hipparcos data and its astrometric solutions were extensively researched in
search of stellar multiplicity, and also for orbital companions such as brown
dwarfs and massive planets (e.g. Mazeh et al., 1999; Zucker & Mazeh, 2000,
2001; Sozzetti & Desidera, 2010; Snellen & Brown, 2018). Nevertheless, we
believe that due to the unique qualities of our new method, using it to
carefully analyze the Hipparcos catalogue in its entirety can potentially
reveal many new astrometric orbits, especially those of high eccentricity.

Another potential context in which our newly developed tool can contribute
significantly is the Nancy Grace Roman Space Telescope (NGRST, known before as
WFIRST; Spergel et al., 2013), currently expected to launch by 2027.

We provide our Python implementation of the periodogram in the form of a
public GitHub repository111PDC and its extensions, including USuRPER, partial
distance correlation periodograms, and the periodogram presented in this work,
are all available as part of the SPARTA package (Shahaf et al., 2020), at
https://github.com/SPARTA-dev/SPARTA..

###### Acknowledgements.

We thank the anonymous referee for their wise comments that helped to improve
the manuscript. This research was supported by the Ministry of Innovation,
Science & Technology, Israel (grant 3-18143). The research of SS is supported
by a Benoziyo prize postdoctoral fellowship. The analyses done for this paper
made use of the code packages: NumPy (Harris et al., 2020), SciPy (Virtanen et
al., 2020) and SPARTA (Shahaf et al., 2020).

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# Point mass Cosmological Black Holes

Javad T. Firouzjaee  School of Astronomy, Institute for Research in
Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran
<EMAIL_ADDRESS><EMAIL_ADDRESS>Touhid Feghhi  School of
Astronomy, Institute for Research in Fundamental Sciences (IPM), P. O. Box
19395-5531, Tehran, Iran

###### Abstract

Abstract: Real black holes in the universe are located in the expanding
accelerating background which are called the cosmological black holes. Hence,
it is necessary to model these black holes in the cosmological background
where the dark energy is the dominant energy. In this paper, we argue that
most of the dynamical cosmological black holes can be modeled by point mass
cosmological black holes. Considering the de Sitter background for the
accelerating universe, we present the point mass cosmological background in
the cosmological de Sitter space time. Our work also includes the point mass
black holes which have charge and angular momentum. We study the mass,
horizons, redshift structure and geodesics properties for these black holes.

###### Contents

  1. I Introduction
  2. II From dynamical black holes to the point mass black holes
  3. III What is the dynamical phase of the Sgr A∗ and M87
  4. IV Point mass CBH
  5. V Charged Black hole
  6. VI Kerr- de Sitter Black hole
  7. VII Cosmological black holes properties
    1. VII.1 Misner-Sharp mass
    2. VII.2 Redshift
    3. VII.3 Black hole boundary
    4. VII.4 Circular orbits for CBHs
  8. VIII Conclusion
  9. A The CBH’s orbits effective potential

##  I Introduction

After Hubble discovery of the cosmic expansion, people needed to make black
hole models which are embedded in the expanding Friedmann-Lemaître-Robertson
Walker background. The 1933 McVittie solution McVittie:1933zz  was the first
attempt to model the point mass black hole in the cosmological background. The
question arises to whether the effects of the cosmological expansion on
gravitating local systems, such as the solar or galaxy system, accompanying
the expansion of the universe. The McVittie black hole is embedded in a
general FLRW background, so that the region between the black hole horizon and
the cosmological horizon is evolving. Even though this model was
comprehensively studied in different aspects of black hole Nolan:1999kk ;
Kaloper:2010ec ; Lake:2011ni , the McVittie metric cannot describe a black
hole evolving in an FLRW universe. The collection of known exact solutions
with respect to the point-mass modeling were investigated in Point mass-CBH .

The main criticism for some these metrics is that they manufacture the
geometry and attribute it to the black hole models. For example, if one
replaces the constant mass in the Schwarzschild metric, $m$, with a specific
function of $m(t,r)$, the energy condition might be violated. With this
action, matter on the right-hand side of the Einstein equation will be
restricted, and the black hole horizon and singularity change their position
and properties generally. The standard way to derive a metric is to know its
matter field and its symmetry. Along this way Lemaître-Tolman-Bondi (LTB)
model was introduced which describe the perfect fluid collapse in the
spherically symmetric space time LTB . Apart from its dynamical nature, the
FLRW is a special case of this metric and can be modeled as a background.
Since the geometry is not static, the need for a local definition of black
holes and their boundaries (horizons) have led to concepts such as an isolated
horizon Ashtekar:2000sz , Ashtekar and Krishnan’s dynamical horizon
Ashtekar:2002ag , and Booth and Fairhurst’s slowly evolving horizon
Booth:2003ji . Inspired by these LTB metric properties, the cosmological black
hole (CBH) can be built Firouzjaee-2010 ; Firouzjaee-penn ; Moradi-2015  that
its singularity and horizon is formed during the collapse.

In many respects, a CBH shows different properties relative to the stationary
black holes. The first evident difference is that in the CBH has one more
horizon except black hole horizons (event and apparent horizon) which called
the cosmological horizon Gibbons:1977mu ; book-faraoni ; Helou:2016xyu , and
the causal structure of the black hole will be different from stationary one.
Second, the mass definition of the black hole will need to be extended to the
quasi-local masses Firouzjaee:2010ia  rather than like a ADM mass. Even though
the Hawking radiation from the stationary black hole is revised for CBH case
radiation-CBH .

The first CBH application is in the primordial black hole modeling PBH-CBH .
These black holes form when the FLRW background density perturbations exceed
some threshold values in the radiation dominated era. One can use McVittie CBH
to model the cosmological defect in the inflationary phase McVittie-inflation
. The second CBH application is to model the structure in the matter dominated
era CBH-structure formation  where these models were used to investigate the
dark energy, virialization, rotation curve and week-lensing, etc.

In this paper, in section II we infer the point mass CBH form a big class of
CBH which can be used for the cosmological structure. Section III, consider
the validity of the point mass approximation for astrophysical black holes.
IV, V and VI is devoted to make a point mass CBH in presence of mass, mass-
charge and mass-angular momentum. Then, in Section VI we study these black
holes mass, horizon, redshift structures and geodesics properties. The
conclusion and discussions are given in Section VIII.

##  II From dynamical black holes to the point mass black holes

The main feature of a dynamical black hole in astrophysics is its matter flux
which helps the black hole formation and its growth. It was shown
Firouzjaee:2010ia ; Moradi-2015  that the black hole apparent horizon growth
is proportional to the matter flux which falls to it and eventually tends to
the black hole event horizon when all matter flux is absorbed by black hole.
Here we are interested in the case that the black hole evolves slowly due to
the matter flux. First we introduce the dynamical horizon properties
Ashtekar:2002ag  which is a general case slowly evolving horizon Booth:2003ji
and then we consider the slowly evolving limitation. Geometry of the dynamical
horizon $H$ is expressed by the unit normal to $H$ by $\hat{\tau^{a}}$;
$g_{ab}\hat{\tau^{a}}\hat{\tau^{b}}=-1$. The unit space-like vector orthogonal
to $S$ (a point on apparent horizon which has 2-sphere topology) and tangent
to $H$ is represented by $\hat{r^{a}}$. The rescaling freedom in the choice of
null normals will be fixed via $l^{a}=\hat{\tau^{a}}+\hat{r^{a}}$ and
$n^{a}=\hat{\tau^{a}}-\hat{r^{a}}$. We introduce the area radius $R$, a
function which is constant on each $S$ and satisfies $a_{S}=4\pi R^{2}$. Now,
the 3-volume $d^{3}V$ on $H$ can be decomposed as $d^{3}V=|\partial
R|^{-1}dRd^{2}V$ where $\partial$ denotes the gradient on $H$. Hence, as we
will see, our calculations will simplify if we choose $N_{R}=|\partial R|$
which is amplitude of normal vector of the $R=const$ surface.

We define the flux of energy associated with $\xi_{(R)}^{a}=N_{R}\ell^{a}$
across $\Delta H$ as:

$\displaystyle\mathcal{F}^{(R)}_{\rm matter}:=\int_{\Delta
H}T_{ab}\hat{\tau^{a}}\xi_{(R)}^{b}d^{3}V=$
$\displaystyle\frac{1}{G}(M(r_{2})-M(r_{1})).$ (1)

In the spherically symmetric case $M(r)$ is the Misner-Sharp mass
Firouzjaee:2010ia . With calculating this quantity on the apparent horizon

$\frac{dM}{dt}|_{AH}=(M^{\prime}\frac{dr}{dt}+\dot{M}).$ (2)

Here $\dot{}$ and ′ are partial differentials relative to $t$ and $r$
respectively. Take a collapsing ideal fluid within a compact spherically
symmetric spacetime region described by the following metric in the comoving
coordinates $(t,r,\theta,\varphi)$:

$ds^{2}=-e^{2\nu(t,r)}dt^{2}+e^{2\psi(t,r)}dr^{2}+R(t,r)^{2}d\Omega^{2}.$ (3)

Assuming the stress-energy tensor for the perfect fluid in the form

$\displaystyle T^{t}_{t}=-\rho(t,r),~{}~{}T^{r}_{r}=p_{r}(t,r),~{}~{}$
$\displaystyle
T^{\theta}_{\theta}=T^{\varphi}_{\varphi}=p_{\theta}(t,r)=w\rho(t,r),$ (4)

with the weak energy condition

$\rho\geq 0~{}~{}\rho+p_{r}\geq 0~{}~{}\rho+p_{\theta}\geq 0,$ (5)

where $w$ describes the equation of state which is a barotropic equation of
state between energy density, $\rho$, and pressure, $p$. Einstein equations
give,

$\displaystyle\rho=\frac{2M^{\prime}}{R^{2}R^{\prime}}~{},~{}~{}p_{r}=-\frac{2\dot{M}}{R^{2}\dot{R}},$
(6)

where $M$, Misner-Sharp mass, is defined by

$\displaystyle
e^{-2\psi}(R^{\prime})^{2}-e^{-2\nu}(\dot{R})^{2}=1-\frac{2M}{R}.$ (7)

If a collapsing metric is build by this metric, one can show that the apparent
horizon will form at $R=2M$ surface Firouzjaee-penn . We can define the matter
flux into the apparent horizon matter flux relative to fluid 4-velocity

$D_{t}M=\frac{1}{e^{\nu}}\frac{dM}{dt}|_{AH}=\frac{1}{e^{\nu}}(M^{\prime}\frac{dr}{dt}+\dot{M}).$
(8)

Using the Einstein equation Firouzjaee-penn ; Helou:2016xyu  we get

$D_{t}M=4\pi R_{H}^{2}(-U)\frac{\rho+p}{1-8\pi R_{H}^{2}\rho}$ (9)

where $U=\dfrac{\dot{R}}{e^{\nu}}$ and the $R_{H}$ the areal radius on the
apparent horizon. At first glance, we can see that the matter (flux) on the
apparent becomes zero when density becomes zero on the apparent horizon. We
define this quantity which characterize the slowly evolving horizon
Booth:2003ji  when

$\epsilon^{2}=\alpha\theta_{n}^{2}R_{H}^{2}\ll 1,$ (10)

where $\theta_{n}$ is the ingoing null geodesics expansion and alpha is a
constant. This quantity in the spherically symmetric space time becomes
Firouzjaee-penn

$\epsilon^{2}=4\pi R_{H}^{2}\frac{(1+w)\rho}{1-4\pi R_{H}^{2}(1-w)\rho}$ (11)

To have an slowly evolving black hole with mass $M=2R_{H}$ the energy density
for this black hole, $\rho$, must be small. In the special case the
$\epsilon=\rho=0$ we get the isolated horizon Ashtekar:2000sz  which the
matter flux is zero.

Proposition: If a dynamical black hole evolves and finally its horizon becomes
isolated, the space time geometry around the black hole horizon becomes
Schwarzschild metric with the same Misner-Sharp mass.

If the horizon becomes isolated i.e. $\epsilon=0$ then from Eq. (11) the
energy density becomes zero, $\rho=p=0$. Hence we have underdensity (vacuum)
around the black hole horizon. On the other hand from uniqueness theorem we
know that the vacuum solutions of the Einstein equation around a spherically
symmetric mass distribution have Schwarzschild form

$ds^{2}=-(1-\frac{2C}{R})dt^{2}+(1-\frac{2C}{R})^{-1}dR^{2}+R^{2}d\Omega^{2}.$
(12)

Using the fact the $M=2R_{H}$ is the apparent horizon for Schwartzchild metric
(as boundary condition), we get $C=M$. Therefore, in the isolated horizon case
the space time geometry around horizon becomes Schwarzschild with the Misner-
Sharp mass, $C=M=\frac{R_{H}}{2}$.

Located in the cosmological background, the CBH collapsing part separates from
the expanding part and the density between them decreases. This underdense
region is usually called void in the cosmology. As a result, the matter flux
can not exist forever and after sometime it decreases Firouzjaee:2010ia ;
Moradi-2015 . In this case as inferred above taking the black hole as a point
mass is a good model.

##  III What is the dynamical phase of the Sgr A∗ and M87

To know how the point mass approximation is correct for the real astrophysical
black hole, let us study the dynamical growth of the two famous black holes
Sgr A∗ and M87. Sgr A∗ is a supermassive black hole in our galaxy’s center.
From the radio source of the galactic center, the Sgr A∗ mass is estimated
$M\sim 10^{6}M_{\odot}$. From the the X-ray and infrared emission of the
Galactic center sgr , one provides an upper limit on the Sgr A∗ black hole
mass accretion rate as $\dot{M}<10^{-5}M_{\odot}yr^{-1}$. For an astrophysical
event such as lensing for light the typical lenght (time) scale is about
$kpc\sim 10^{19}m$ which is equivalent to $10^{3}yr$ for passing light of this
length Ghez:2008ms . The black hole mass growth for this time scale is $\delta
M<10^{-2}M_{\odot}$. Since the black hole mass is proportional to its radius
the change of the black hole horizon becomes

$\frac{\delta R_{H}}{R_{H}}=\frac{\delta M}{M}<10^{-8}.$ (13)

As a result, in practice the black hole remains static relative to the
astrophysical time scale and we can make the point mass approximation in many
cases for the black holes physics study as the astrophysical text book do bh-
book .

Another supermassive black hole is M87 in cluster Virgo fifty million light-
years away which is the most massive black holes known and has been the
subject of several stellar and gas-dynamical mass measurements. This
suppermassive black hole mass is $M\sim 10^{9}M_{\odot}$ and the mass
accretion rate of this black hole is approximated by
$\dot{M}<10^{-4}M_{\odot}yr^{-1}$ m87 . The typical distance of the Virgo
cluster from us is about $10^{22}m$ which is equivalent to $10^{6}yr$ for
coming light. At this time scale the black hole mass growth will be $\delta
M<10^{2}M_{\odot}$ and equivalently the the black hole horizon growth becomes

$\frac{\delta R_{H}}{R_{H}}=\frac{\delta M}{M}<10^{-7}.$ (14)

Similar to the Sgr A∗, in practice this black hole remains static relative to
the astrophysical time scale Owen:2000vi  and we can make the point mass
approximation in many cases for the black hole’s physics study.

##  IV Point mass CBH

In the last section we inferred that since the matter and matter flux around
the black hole decrease and the black hole mass is bigger than the total
matter around it, the point mass black hole can be a good model for CBH. On
the other side, the cosmological constant is the best model to describe the
cosmic acceleration which we called it dark energy in the matter sector. The
de Sitter metric is the Einstein equation solution with the cosmological
constant. Our analysis based on two paradigm: first, the dark energy is the
main cosmological matter (about 70 %) and all other matter (dark matter and
baryonic matter) located in the matter flux which have fallen in the black
hole; second at last time of evolution the matter flux becomes zero
Firouzjaee-2010 ; Moradi-2015 ; Firouzjaee-penn  and we get a point mass black
hole. In this case, it is sufficient to find the point mass black hole in the
cosmological de Sitter background.

One way to make a cosmological black hole is to add the FLRW scale factor as a
conformal coefficient for the point mass black hole metric. Sultana and Dyer
sultana  have introduced a metric which FLRW scale factor is a conformal
coefficient of the Schwarzschild metric. However that metric violate the
energy condition and does not describe the ordinary matter. To do similar
analysis with the de Sitter line element Culetu  consider the de Sitter metric

$ds^{2}=-d\tau^{2}+e^{2\sqrt{\frac{\Lambda}{3}}\tau}(dr^{2}+r^{2}d\Omega^{2}).$
(15)

By the coordinate transformation
$t=\sqrt{\frac{\Lambda}{3}}e^{-\sqrt{\frac{\Lambda}{3}}\tau}$, the metric (15)
will be transformed into conformally flat form

$ds^{2}=\frac{\Lambda}{3t^{2}}[-dt^{2}+dr^{2}+r^{2}d\Omega^{2}].$ (16)

If we embed a black hole in this space time and write the conformal
Schwarzschild metric, we will have

$ds^{2}=\frac{\Lambda}{3t^{2}}[-(1-\frac{2m}{r})dt^{2}+\frac{1}{(1-\frac{2m}{r})}dr^{2}+r^{2}d\Omega^{2}].$
(17)

By calculating the Einstein tensor for this line element we see that the
strong energy condition is violated, so this metric is not proper for our
study and instead we consider the general form of Schwarzschild de Sitter
metric.

One may think that the Schwarzschild-de Sitter is a good metric for the CBH,
but Schwarzschild-de Sitter has been written in the static coordinate.
Consequently it is needed find the Schwarzschild-de Sitter metric in the
cosmological coordinate to present the point mass CBH. Since the standard
cosmological metrics are written in the synchronous coordinate, we first have
to transform the Schwarzschild-de Sitter to the synchronous coordinates. We
know that the Schwarzschild-de Sitter metric is given by

$ds^{2}=-\Phi dt^{2}+\Phi^{-1}dR^{2}+R^{2}d\Omega^{2},$ (18)

where

$\Phi=1-\frac{\Lambda}{3}R^{2}-\frac{2M}{R}.$ (19)

By these coordinate transformations podolski

$\begin{split}&d\tau=dt-\frac{\sqrt{1-\Phi}}{\Phi}dR,\\\
&dr=-dt+\frac{1}{\Phi\sqrt{1-\Phi}}dR,\end{split}$ (20)

metric (18) will be

$ds^{2}=-d\tau^{2}+(\frac{2M}{R}+\frac{\Lambda}{3}R^{2})dr^{2}+R^{2}d\Omega^{2}.$
(21)

To find $R$ as a function of $r$ and $\tau$ we can use

$\int d\tau+dr=\int\frac{dR}{\sqrt{1-\Phi}}.$ (22)

Therefore, we can write

$\tau+r=\int\frac{dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2M}{R}}}=\frac{2}{\sqrt{3\Lambda}}\ln(\Lambda
R^{\frac{3}{2}}+\sqrt{6M\Lambda+\Lambda^{2}R^{3}}).$ (23)

Hence, we can write $R$

$R=\frac{e^{-\sqrt{\frac{\Lambda}{3}}(r+\tau)}(e^{\sqrt{3\Lambda}(r+\tau)}-6\Lambda
M)^{\frac{2}{3}}}{2^{\frac{2}{3}}\Lambda^{\frac{2}{3}}}.$ (24)

In the limit where the black hole mass tend to zero, the metric
$\displaystyle{\lim_{M\to 0}ds^{2}}$ will be

$\begin{split}ds^{2}=-d\tau^{2}+\frac{e^{2\sqrt{\frac{\Lambda}{3}}\tau}}{(2\Lambda)^{\frac{4}{3}}}[\frac{\Lambda}{3}e^{2\sqrt{\frac{\Lambda}{3}}r}dr^{2}+e^{2\sqrt{\frac{\Lambda}{3}}r}d\Omega^{2}].\end{split}$
(25)

By redefining $S=\frac{e^{2\sqrt{\frac{\Lambda}{3}}r}}{(2\Lambda)^{2/3}}$ we
get

$ds^{2}=-d\tau^{2}+e^{2\sqrt{\frac{\Lambda}{3}}\tau}[dS^{2}+S^{2}d\Omega^{2}],$
(26)

which is de Sitter metric.
With calculating the Ricci scalar, we can see that the singularity is located
at $r+\tau=\frac{1}{\sqrt{3\Lambda}}\ln(6\Lambda M)$ that is equivalently
$r=0$.

The Penrose-Carter diagram of these black holes can be seen in the
Gibbons:1977mu . These black holes properties will be discussed in Section VI.

##  V Charged Black hole

It is usually assumed that the charged black hole can not appear during the
gravitational collapse. But a highly magnetized plasma accretes onto the black
hole, the charge to mass ratio can be big in some case. In particular, in the
merging of a binary system of neutron stars, it is expected at the final steps
of a gravitational collapse to a charged black hole ccbh . If we add charge to
a point mass we get the Reissner Nordstrom solution. If we solve the Einstein
equation with cosmological constant and a point mass with charge we get the de
Sitter-Reissner Nordstrom solution. Here we use Carter’s spherically symmetric
three parameter (M,$\Lambda$,Q) solution to Einstein’s equations where Q is
the electric charge of black hole carter . The metric in static coordinates is

$ds^{2}=-\Phi dt^{2}+\Phi^{-1}dR^{2}+R^{2}d\Omega^{2},$ (27)

where

$\Phi=1-\frac{\Lambda}{3}R^{2}-\frac{2M}{R}+\frac{Q^{2}}{R^{2}}.$ (28)

By coordinate transformations given by (20), the metric (27) will be

$ds^{2}=-d\tau^{2}+(\frac{\Lambda}{3}R^{2}+\frac{2M}{R}-\frac{Q^{2}}{R^{2}})dr^{2}+R^{2}d\Omega^{2}.$
(29)

We can write

$\tau+r=\int\frac{dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2M}{R}-\frac{Q^{2}}{R^{2}}}}.$
(30)

If we define
$G(R)=\int\frac{dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2M}{R}-\frac{Q^{2}}{R^{2}}}}$
then $R=G^{-1}(\tau+r)$. We can calculate the inverse function numerically, as
if we draw the function$r+\tau=G(R)$ and change the variables $(R,G(R))$ to
$(G(R),R)$ and then use numerical methods to find the equation of drawn line.

Because of existence of three free parameters $\Lambda$ , $Q$ and $M$, this
integral requires tedious calculations (if an analytical solution exists), and
even after finding the integral, it would be much harder to find the inverse
function $R$, which will give the solution. Hence, we need to suppose some
simplifying assumptions. At large ’$R$’ where
$\frac{\Lambda}{3}R^{2}\gg\frac{2M}{R}$ and
$\frac{\Lambda}{3}R^{2}\gg\frac{Q^{2}}{R^{2}}$ or equivalently $R\gg
Max(\sqrt[3]{\frac{6M}{\Lambda}},\sqrt[4]{\frac{3Q^{2}}{\Lambda}})$

$\begin{split}\tau+r&=\int\frac{dR}{\sqrt{\frac{\Lambda}{3}*R^{2}+\frac{2M}{R}-\frac{Q^{2}}{R^{2}}}}\simeq\sqrt{\frac{3}{\Lambda}}\int
R^{-1}[1-\frac{3M}{\Lambda R^{3}}+\frac{3Q^{2}}{2\Lambda R^{4}}]dR\\\
&=\sqrt{\frac{3}{\Lambda}}(\ln(R)+\frac{M}{\Lambda
R^{3}}-\frac{3Q^{2}}{8\Lambda R^{4}}).\end{split}$ (31)

As a result

$R=\exp^{x_{i}}.$ (32)

Where $x_{i}$ represents the roots of equation

$8(r+\tau)-\sqrt{\frac{3}{\Lambda}}(\frac{8\Lambda
xe^{4x}+8Me^{x}-3Q^{2}}{\Lambda e^{4x}})=0$ (33)

For $\displaystyle{\lim_{Q,M\to 0}ds^{2}}$, according to (32),
$R=\exp^{\sqrt{\frac{\Lambda}{3}}(r+\tau)}$ and the line element (29) will be
(25) and it represents de Sitter metric.

It might be interesting to consider the case $M=0$ (Q $\neq 0$)

$\int\frac{dR}{\sqrt{\frac{\Lambda}{3}R^{2}-\frac{Q^{2}}{R^{2}}}}=\frac{1}{2\sqrt{\frac{\Lambda}{3}}}\ln(\Lambda
R^{2}+\sqrt{\Lambda^{2}R^{4}-3\Lambda Q^{2}}).$ (34)

Then

$R=\frac{e^{-\sqrt{\frac{\Lambda}{3}}(r+\tau)}\sqrt{e^{4\sqrt{\frac{\Lambda}{3}}(r+\tau)}+3\Lambda
Q^{2}}}{\sqrt{2\Lambda}}$ (35)

The singularities are located at $R=0$ and
$R=\sqrt[4]{\frac{3Q^{2}}{\Lambda}}$.

For the region $\frac{Q^{2}}{2M}\ll R\ll\sqrt[3]{\frac{6M}{\Lambda}}$ ( if
$\sqrt[3]{\Lambda}Q^{2}\ll M$ ), the equation(30) will be

$\begin{split}r+\tau=&\int\frac{dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2M}{R}-\frac{Q^{2}}{R^{2}}}}=\int\frac{\sqrt{R}}{\sqrt{2M}}[1-\frac{Q^{2}}{2MR}+\frac{\Lambda}{6M}R^{3}]^{-\frac{1}{2}}dR\\\
&\simeq\frac{1}{\sqrt{2M}}(\frac{2}{3}R^{\frac{3}{2}}+\frac{Q^{2}}{2M}R^{\frac{1}{2}}-\frac{\Lambda}{54M}R^{\frac{9}{2}})\end{split}$
(36)

and we will have $R=x_{i}$, where $x_{i}$ are the roots of this equation

$-5832M(r+\tau)^{2}+729M^{2}Q^{2}x+1944MQ^{2}x^{2}+1296x^{3}-54\Lambda
M^{2}Q^{2}x^{5}-72\Lambda Mx^{6}+\Lambda^{2}M^{2}x^{9}=0$ (37)

The Penrose-Carter diagram of these black holes can be seen in the
Belhaj:2009ii .

##  VI Kerr- de Sitter Black hole

In this section we want to describe the point mass CBH with the angular
momentum. In Boyer-Lindquist like coordinates employed by Carter, the Kerr-de
Sitter line element will be carter :

$\begin{split}ds^{2}=&(R^{2}+a^{2}cos^{2}\Theta)[\frac{dR^{2}}{\Delta_{R}}+\frac{d\Theta^{2}}{1+\frac{\Lambda}{3}a^{2}cos^{2}\Theta}]+sin^{2}\Theta\frac{(1+\frac{\Lambda}{3}a^{2}cos^{2}\Theta)}{R^{2}+a^{2}cos^{2}\Theta}[\frac{adt-(R^{2}+a^{2})d\varphi}{1+\frac{\Lambda}{3}a^{2}}]^{2}\\\
&-\frac{\Delta_{R}}{(R^{2}+a^{2}cos^{2}\Theta)}[\frac{dt-asin^{2}\Theta
d\varphi}{1+\frac{\Lambda}{3}a^{2}}]^{2},\end{split}$ (38)

where ’a’ is Kerr parameter and

$\Delta_{R}=(R^{2}+a^{2})(1-\frac{\Lambda}{3}R^{2})-2MR.$ (39)

The Kerr-de sitter metric is rather complicated , so finding a coordinate in
which metric becomes in the form of $ds^{2}=\Phi
dt^{2}-\Phi^{-1}dR^{2}-R^{2}d\Omega^{2}$ is elusive. In this way, to simplify
metric, we first take $\Theta=0$ polar cut of the metric and due to
axisymmetry $\varphi$ can be set to equal any value from 0 to $2\pi$. With
these assumptions the metric (38) will be

$ds^{2}=\frac{R^{2}+a^{2}}{\Delta_{R}}dR^{2}-\frac{\Delta_{R}}{R^{2}+a^{2}}[\frac{dt}{1+\frac{\Lambda}{3}a^{2}}]^{2}.$
(40)

If we suppose $dt^{{}^{\prime}}=\frac{dt}{1+\frac{\Lambda}{3}a^{2}}$ we will
have

$ds^{2}=\frac{R^{2}+a^{2}}{\Delta_{R}}dR^{2}-\frac{\Delta_{R}}{R^{2}+a^{2}}(dt^{{}^{\prime}})^{2}.$
(41)

Similar to the last section if we use coordinate transformations (20) the line
element (41) will be

$ds^{2}=-d\tau^{2}+(\frac{\Lambda}{3}R^{2}+\frac{2MR}{R^{2}+a^{2}})dr^{2},$
(42)

Where

$\int d\tau+dr=\int\frac{dR}{\sqrt{1-\Phi}},$ (43)

And

$\Phi=\frac{\Delta_{R}}{R^{2}+a^{2}}.$ (44)

We can write

$\tau+r=\int\frac{dR}{\sqrt{1-\frac{\Delta_{R}}{R^{2}+a^{2}}}}=\int\frac{dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2MR}{R^{2}+a^{2}}}}.$
(45)

Since this integral calculation is complex, we will consider two simple cases
$R\ll a$ and $R\gg a$.

For $R\ll a$ we will have

$\tau+r\simeq\int\frac{dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2MR}{a^{2}}}}$
(46)

Hence we get

$\tau+r=\sqrt{\frac{3}{\Lambda}}ln(\sqrt{\frac{\Lambda}{3}}R+\frac{M}{a^{2}}\sqrt{\frac{3}{\Lambda}}+\sqrt{-(\frac{M}{a^{2}}\sqrt{\frac{3}{\Lambda}})^{2}+(\sqrt{\frac{\Lambda}{3}}R+\frac{M}{a^{2}}\sqrt{\frac{3}{\Lambda}})^{2}}).$
(47)

To find ’$R$’ we must find the inverse form of above function, so it will be

$R=\frac{\sqrt{3}}{2\Lambda}(exp(\sqrt{\frac{\Lambda}{3}}(r+\tau))-2\frac{M}{a^{2}}\sqrt{\frac{3}{\Lambda}}+(\frac{M}{a^{2}}\sqrt{\frac{3}{\Lambda}})^{2}exp(-\sqrt{\frac{\Lambda}{3}}(r+\tau))).$
(48)

For $R\gg a$ and $R\gg M$ we will have

$\begin{split}\tau+r=&\int\frac{dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2MR}{R^{2}+a^{2}}}}\simeq\int\frac{(1+\frac{a^{2}}{2R^{2}})dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2M}{R}}}\\\
&=\sqrt{\frac{3}{\Lambda}}[ln(R)+\frac{M}{\Lambda
R^{3}}]+\frac{a^{2}}{2}\sqrt{\frac{3}{\Lambda}}[-\frac{1}{2R^{2}}+\frac{3M}{5\Lambda
R^{5}}],\end{split}$ (49)

Since $R\gg a$ and $R\gg M$, if we neglect $\frac{1}{R^{5}}$ term, then we
will have $R=e^{x_{i}}$, where $x_{i}$ are the roots of equation

$-4\sqrt{3}\Lambda^{\frac{3}{2}}(r+\tau)e^{3x}+12\Lambda xe^{3x}-3\Lambda
a^{2}e^{x}+12M=0$ (50)

If we only suppose $R\gg a$ (not $R\gg M$) we will have

$\begin{split}\tau+r=&\int\frac{dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2Mr}{R^{2}+a^{2}}}}\simeq\int\frac{(1+\frac{a^{2}}{2R^{2}})dR}{\sqrt{\frac{\Lambda}{3}R^{2}+\frac{2M}{R}}}\end{split}.$
(51)

Hence if we define

$\displaystyle
G(R)=\tau+r=\sqrt{\frac{4}{3\Lambda}}arcsinh(\sqrt{\frac{3\Lambda}{8M}}\frac{2}{3}R^{\frac{3}{2}})+$
(52)
$\displaystyle\frac{1}{4a^{2}}(\frac{3}{2})^{-\frac{2}{3}}\frac{-3(\frac{-\Lambda(\frac{4}{9}R^{3})}{M})^{\frac{5}{6}}(8M+3\Lambda(\frac{4}{9}R^{3}))+2*3^{\frac{1}{6}}\Lambda(\frac{4}{9}R^{3})\sqrt{8+\frac{3\Lambda(\frac{4}{9}R^{3})}{M}}Beta[-\frac{3\Lambda\frac{4}{9}R^{3}}{M},\frac{5}{6},\frac{1}{2}]}{2M(\frac{4}{9}R^{3})^{\frac{1}{6}}(\frac{-\Lambda(\frac{4}{9}R^{3})}{M})^{\frac{5}{6}}\sqrt{(8M+3\Lambda(\frac{4}{9}R^{3}))}},$

where $Beta[x,a,b]$ is incomplete beta function. Then we get

$R=G^{-1}(r+\tau).$ (53)

Therefore, we present point mass CBH with angular momentum with finding the
$G$ function.

The Penrose-Carter diagram of these black holes can be seen in the
kerrdesitter .

##  VII Cosmological black holes properties

From the last sections discussion a question arises as to whether we can see
the trace of the cosmological constant (which play the cosmological
acceleration role) in the observation. To make an estimate consider the
Schwarzschild-de Sitter metric in the stationary coordinate (19). Since the
$R$ value is invariant due to the transformation to the cosmological
coordinate, we can compare the two terms $\frac{\Lambda}{3}R^{2}$ from
cosmological constant and $\frac{2M}{R}$ from the black hole. In cosmology the
value $\frac{\Lambda}{3}=H_{0}^{2}$ where the $H_{0}=67.74\pm
0.46km/s/Mpc=2.195\pm 0.01510^{-18}s^{-1}$ is the Hubble constant at the
present time. For a black hole with the sun mass $m_{\odot}$ at scale
$R\gtrsim 10^{18}m\simeq 100pc$ the cosmological constant term cannot be
negligible. In this scale some phenomena of black hole like lensing mojahed
and Cosmic Microwave Background distortion from primordial black hole are
important and we have to use the cosmological constant model to describe them.
In this part we investigate what happen for black hole mass, boundary and
redshift structure if someone take the cosmological coordinate.

###  VII.1 Misner-Sharp mass

The spherically symmetric metric can be written

$ds^{2}=\gamma_{ab}dx^{a}dx^{b}+R^{2}d\Omega^{2}.$ (54)

where $a,b:(t,r)$. In this form the Misner-Sharp mass which is defined for the
spherically symmetric space time Firouzjaee:2010ia  becomes

$M_{ms}=\frac{R}{2}(1-\gamma^{ab}\partial_{a}R\partial_{b}R)$ (55)

Since $\gamma^{ab}\partial_{a}R\partial_{b}R$ is invariant under the
coordinate transformation $(t,r)\rightarrow(t^{\prime},r^{\prime})$ the
Misner-Sharp mass is also invariant under these transformation.
Since the coordinate transformation from the stationary Schwarzschild-de
Sitter coordinate to the cosmological coordinate has only the $(t,r)$ part, as
a result, spherically symmetric cosmological black holes the Misner-Sharp will
be the same Schwarzschild charge de Sitter mass:

$M_{ms}=\frac{R}{2}(1-\gamma^{ab}\partial_{a}R\partial_{b}R)=m+\dfrac{\Lambda}{6}R^{2}-\dfrac{Q}{2R}.$
(56)

One can also calculate the matter flux for above cosmological black hole in
the spherically symmetric case. It can be shown from equation (8) that the
matter flux is zero for these CBHs. This verifies that these CBH are point
mass CBHs.

###  VII.2 Redshift

If an emitter sends a light ray to an observer with null vector $k^{\mu}$, the
relative light redshift of the emitted ($e$) frequency, ($w$), that is
calculated by observer ($o$) with 4-velocity $u^{\mu}$ is,

$1+z=\frac{w_{e}}{w_{o}}=\frac{(k_{\mu}u^{\mu})_{e}}{(k_{\mu}u^{\mu})_{o}},$
(57)

where the light null geodesic $k^{\mu}$ is affinary parameterized. It can be
shown that the affinnary null geodesics equation

$k^{\nu}k^{\mu}_{;\nu}=0$ (58)

will not change from coordinate transformation (because this relation is a
covariant tensorial relation). As a result, the affine parameterized equation
will remain the same. Hence, it results that the redshift properties of a
spacetime for an observer will not change due to the coordinate
transformation.
Consequently, the infinite redshift surface for the above CBHs will be the
same infinite redshift surface in the stationary coordinate.

###  VII.3 Black hole boundary

The event horizon is usually used to define the black hole boundary in
textbooks. Since the event horizon definition is a global quantity and we have
to know all information about the spacetime evolution finally, we need to
define a quasi-local quantity which can be applied for real dynamical black
hole in numerical relativity. To solve this problem people use the isolated on
dynamical horizon (which are a type of the apparent horizon) to define the
black hole boundary in a dynamical case Ashtekar:2002ag ; Firouzjaee-penn ;
Booth:2003ji .

Black hole boundary definition: A smooth, three-dimensional, space-like sub-
manifold (possibly with boundary) $H$ of space-time is said to be a trapping
horizon if it can be foliated by a family of closed 2-manifolds such that on
each leaf $S$ the expansion $\theta_{(\ell)}$ of one null normal $\ell^{\mu}$
vanishes; and the expansion $\theta_{(n)}<0$ of the other null normal
$n^{\mu}$ is negative. This surface separates the trapped surface,
$\theta_{(n)},\theta_{(\ell)}<0$, from untrapped one
$\theta_{(n)}<0,~{}\theta_{(\ell)}>0$. Similarly, one can define the
cosmological horizon as a three-dimensional surface where the expansion
$\theta_{(n)}$ of one null normal $n^{\mu}$ vanishes and $\theta_{(\ell)}>0$
on both sides of this surface.

In the spherically symmetric space time the black hole boundary is located on
the apparent horizon $\theta_{(\ell)}=0$. In terms of normal metric components
it becomes $\gamma^{ab}\partial_{a}R\partial_{b}R=0$.

Since $\gamma^{ab}\partial_{a}R\partial_{b}R$ is invariant under the
coordinate transformation $(t,r)\rightarrow(t^{\prime},r^{\prime})$ the black
hole boundary or apparent horizon is invariant under these transformation.

A spacetime with a point mass CBHs have two horizons. The first is
cosmological horizon and the second horizon is black hole horizon (event or
apparent horizon). For the point mass CBH Gibbons:1977mu  there are also two
black hole and cosmological horizon. Since the coordinate transformation do
not change the horizons location, it is sufficient to find roots of $\Phi$ in
the Schwarzschild-de Sitter spacetime. Note that the infinite redshift surface
is the same black hole horizon surface in the stationary coordinate. Hence,
the infinite redshift surface will be the same black hole apparent horizon for
these CBHs. It can easily be shown that the expansion of outgoing null
geodesics is proportional to $\Phi$, which $\Phi=0$ determines the position of
horizons. We can depict the roots of
$\Phi=1-\frac{\Lambda}{3}R^{2}-\frac{2m}{R}$ for $\Lambda=10^{-52}m^{-2}$ as a
function of $x=\frac{m}{m_{\odot}}$ where $m_{\odot}$ is the sun mass. There
are three real roots for $\Phi=0$ if $\Lambda m^{2}\leqslant\frac{1}{9}$ Guven
, where one of them is negative, so we consider
$R_{C}=\frac{2}{\sqrt{\Lambda}}\cos(\frac{\cos^{-1}(-3\sqrt{\Lambda}m)}{3})$
and
$R_{H}=\frac{2}{\sqrt{\Lambda}}\cos(\frac{\cos^{-1}(-3\sqrt{\Lambda}m)}{3}+\frac{4\pi}{3})$
which are cosmological and black hole event horizons respectively.

Two cosmological and black hole horizons are depicted in Fig.(1) and Fig.(2).

Figure 1: de Sitter event horizon as a function of mass. Dash line represents
de Sitter event horizon for $m=0$.

It can be seen that using standard value of dark energy, the astrophysical
black hole with mass $m<10^{10}m_{\odot}$ can not change the cosmological
horizon place. On the other side, the standard value of cosmological constant
does not change the black hole horizon place significantly.

Figure 2: Black hole event horizon as a function of mass. Dash line represents
black hole event horizon for $\Lambda=0$.

###  VII.4 Circular orbits for CBHs

In cosmology and astrophysics, many events and physics such as microlensing,
dark matter rotation curve, missing satellite, stars velocity dispersion and
etc. Emerge from the studying of object orbits where most of them are
simplified by circular orbits desitter-geo . In this part, we study the
possibility of the circular orbits around the CBHs (without charge and angular
momentum).

Circular orbits are characterized by a constant radius. In a stationary
coordinate like Schwarzschild de Sitter coordinate there is well defined
coordinate $R$ which is the areal coordinate, but in the cosmological
coordinate we have two comoving and areal radius. Since the angular distance
is proportional to the areal radius we choose the areal radius as a circular
orbits radius. As a consequence of time and radius coordinate transformation
from stationary Schwarzschild-de Sitter to the CBH cosmological coordinate,
the areal coordinate will not changes. As a result, the circular coordinate in
both frames are the same, but are labeled by different coordinates. The detail
of the orbits effective potential can be seen in the Appendix A.

##  VIII Conclusion

Real black holes in the universe, called the cosmological black holes, are
located in the expanding accelerating background. These black holes are
generically dynamical and they are sourced by baryons and dark matter. It’s
been shown that the background expansion leads to voids. These voids are
formed between the black hole and the expanding background and prevent the
black hole’s matter flux from increasing. After this phase of the black hole
evolution, the black holes can be approximated as a point mass. In this case,
the black hole mass is much greater than the matter flux around it. In this
paper, we argue that most of the dynamical cosmological black holes can be
modeled by point mass cosmological black holes finally. The point mass
cosmological background is modeled by the de Sitter space-time. We find the
Schwarzschild-de Sitter metric in the cosmological coordinate and infer that
this metric is the best candidate for the point mass CBH. This metric locally
describes a point mass black hole and at large distance reduces to the
cosmological de Sitter space-time. We also find the point mass black holes
solution with charge and angular momentum. We show that the mass, horizons and
redshift structure of these black holes will not change due to the coordinate
transformation from stationary coordinate to the cosmological coordinate. From
studying the effective potential for different geodesics cases it has been
shown that the stable circular orbits can be exist similar to the stationary
case.

Future work Fagihi  will concentrate on studying the astrophysical and
cosmological observables of these black holes and compare them with stationary
ones.

Acknowledgments:

We would like to thank Alireza Allahyari for useful comments.

##  Appendix A The CBH’s orbits effective potential

Here we want to study the form of effective potential in Schwarzschild-de
Sitter spacetime to study the circular orbits. We can use Euler Lagrange
Equations with metric (18) which is spherically symmetric and so we can choose
$\theta=\frac{\pi}{2}$, hence $d\Omega^{2}=d\varphi^{2}$ circular . The Euler
Lagrange equations will result:

$\begin{split}&\epsilon=-\Phi\dot{t}^{2}+\Phi^{-1}\dot{R}^{2}+R^{2}\dot{\varphi}^{2}\\\
&\dot{t}=\frac{E}{\Phi}\\\ &\dot{\varphi}=\frac{L}{R^{2}},\end{split}$ (59)

where dot is partial differential relative to affine parameter and E and L are
constants. $\epsilon=-1,0,1$ represents timelike, null and spacelike geodesics
respectively. From the equation (59) we can introduce effective potential
V(R), where

$V(R)=E^{2}-\dot{R}^{2}=\Phi(R)(\frac{L^{2}}{R^{2}}-\epsilon)$ (60)

We can depict V(R) as a function of $\frac{M}{R}$ for different values of $L$
and $\alpha$ where $\alpha=9\Lambda M^{2}$ and $0\leqslant\alpha\leqslant 1$,
that $\alpha=1$ is the extreme case podolski .

Figure 3:  Effective potential of null geodesics as a function of
$\frac{R}{M}$ for different values of $\alpha$. Here we have supposed $L=10M$

Figure 4:  Effective potential of null geodesics as a function of
$\frac{R}{M}$ for different values of L. Here we have supposed
$\alpha=\frac{1}{10}$

If we suppose a massless particle at infinity has the initial velocity
$v^{R}_{0}=\sqrt{E^{2}+V(\infty)}$, as the massless particle moves toward
central object, it decelerates to $\sqrt{E^{2}-V(R)}$ and if $E^{2}>V_{max}$,
the zero mass particle will accelerate toward singularity otherwise $\dot{r}$
will change sign and the particle will be reflected to infinity. According to
figure (4) and (4) as we decrease $\Lambda$ or increase L, it becomes harder
for these particles to fall in to singularity. We can also study the circular
orbits for mass less particles. We know that if $\frac{\partial V}{\partial
R}=0$ and $\dot{R}=0$, then we will have circular orbits and the stability
criterion is $\frac{\partial^{2}V}{\partial R^{2}}>0$. Hence we can write

$\begin{split}&\frac{\partial V}{\partial R}=0\rightarrow R=3M\\\
&\dot{R}=0\rightarrow E^{2}=V(R).\end{split}$ (61)

which has the same radius ( $R=3M$) as the $\Lambda=0$ case bh-book .

According to figure (4), $(\frac{\partial^{2}V}{\partial R^{2}})_{|R=3M}<0$,
that means there is no stable circular orbit for mass less particles except
when $E=L=0$ and $R=3M$. By equation (61), if we define critical impact
parameter as $b=\frac{L}{E}$, then we can write:

$b_{critic}=\frac{L}{E}=(\sqrt{\frac{R^{2}}{\Phi(R)}})_{|R=3M}=\frac{3\sqrt{3}M}{\sqrt{1-\alpha}}.$
(62)

If the impact parameter of these mass less particles is greater than critical
value ($b_{critic}$), then these particles will be captured. This definition
leads to a relation for capture cross section for massless particles from
infinity:

$\sigma_{null}=\pi b^{2}=\frac{27\pi M^{2}}{1-\alpha}.$ (63)

We can do the same calculations for the timelike circular orbits. In this case
we have

$\frac{\partial V}{\partial R}=0\rightarrow
L^{2}=\frac{MR^{2}-\frac{\Lambda}{3}R^{5}}{R-3M}.$ (64)

By equation (64) and $\dot{R}=0$ one can easily show:

$E^{2}=V(R)=\frac{(1-\frac{\alpha}{27M^{2}}R^{2}-\frac{2M}{R})^{2}}{R(R-3M)}.$
(65)

According to equation (65) it’s clear that circular orbits exist for $R>3M$,
but figure (6) tells us that effective potential is convex for the given
values of cosmological constant, so there is no circular orbit for time like
case at these values. The value of $\alpha$ for a black hole with mass
$10^{6}$ sun mass and $\Lambda=10^{-52}s^{-1}$ is in order of $10^{-32}$, in
this case we can have stable circular orbit and bound orbit ( figure (10) ).
Figure (6) also represents that for a same value of central mass if we
increase the cosmological constant, $\dot{R}$ increases. For a same value of
$\Lambda$, figure (6) shows that any variation of L in the timelike case is
similar to null one.

Figure 5:  Effective potential of timelike geodesics as a function of
$\frac{R}{M}$ for different values of $\alpha$. Here we have supposed $L=10M$.

Figure 6:  Effective potential of timelike geodesics as a function of
$\frac{R}{M}$ for different values of L. Here we have supposed
$\alpha=\frac{1}{10}$.

Figure 7:  Effective potential of spacelike geodesics as a function of
$\frac{R}{M}$ for different values of $\alpha$. Here we have supposed $L=10M$.

Figure 8:  Effective potential of spacelike geodesics as a function of
$\frac{R}{M}$ for different values of L. Here we have supposed
$\alpha=\frac{1}{10}$.

Figure (8) shows that the spacelike case inside the de Sitter horizon
resembles the null and timelike cases, but for outside de sitter horizon as we
increase $\Lambda$ (for same vale of central mass), $\dot{R}$ decreases.

Figure (10) and (10) represent effective potential for a specific black hole
with mass $10^{6}M_{\odot}$ and $L=5M$ and $3M$. We have also used present
value of $\Lambda=10^{-52}s^{-1}$. We can see that only for some values of L
we can have timelike circular orbits and bound orbits. For the extreme case
you can see podolski .

Figure 9:  Effective potential of spacelike,null and timelike geodesics as a
function of $\frac{R}{M}$ for a black hole with mass $10^{6}M_{\odot}$ and
$\Lambda=10^{-52}s^{-1}$. Here we have supposed $L=5M$.

Figure 10:  Effective potential of spacelike,null and timelike geodesics as a
function of $\frac{R}{M}$ for a black hole with mass $10^{6}M_{\odot}$ and
$\Lambda=10^{-52}s^{-1}$. Here we have supposed $L=3M$.

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