11institutetext: Moscow Institute of Physics and Technology, Moscow Region,
Dolgoprudny, 141700, Russia 22institutetext: LLC Integrant, Moscow, 127495,
Russia 33institutetext: Federal Research Center “Computer Science and Control”
of the Russian Academy of Sciences, Moscow, 117312, Russia
44institutetext: AIRI (Artificial Intelligence Research Institute), Moscow,
105064, Russia
44email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# HPointLoc: Point-based Indoor Place Recognition using Synthetic RGB-D
Images††thanks: Supported in part of HPointLoc dataset preparation by a grant
of Russian Science Foundation No. 20-71-10116 and in part of PNTR framework
development by a grant for research centers in the field of artificial
intelligence, provided by the Analytical Center for the Government of the
Russian Federation in accordance with the subsidy agreement (agreement
identifier 000000D730321P5Q0002) and the agreement with the Moscow Institute
of Physics and Technology dated November 1, 2021 No. 70-2021-00138.

Dmitry Yudin 1144 0000-0002-1407-2633 Yaroslav Solomentsev 1122
10000-0002-2664-7086 Ruslan Musaev 11 0000-0002-0679-7810 Aleksei Staroverov
1144 0000-0002-4730-1543 Aleksandr I. Panov 113344 0000-0002-9747-3837

###### Abstract

We present a novel dataset named as HPointLoc, specially designed for
exploring capabilities of visual place recognition in indoor environment and
loop detection in simultaneous localization and mapping. The loop detection
sub-task is especially relevant when a robot with an on-board RGB-D camera can
drive past the same place (“Point”) at different angles. The dataset is based
on the popular Habitat simulator, in which it is possible to generate
photorealistic indoor scenes using both own sensor data and open datasets,
such as Matterport3D. To study the main stages of solving the place
recognition problem on the HPointLoc dataset, we proposed a new modular
approach named as PNTR. It first performs an image retrieval with the Patch-
NetVLAD method, then extracts keypoints and matches them using R2D2, LoFTR or
SuperPoint with SuperGlue, and finally performs a camera pose optimization
step with TEASER++. Such a solution to the place recognition problem has not
been previously studied in existing publications. The PNTR approach has shown
the best quality metrics on the HPointLoc dataset and has a high potential for
real use in localization systems for unmanned vehicles. The proposed dataset
and framework are publicly available: https://github.com/metra4ok/HPointLoc.

###### Keywords:

Visual Place Recognition, Indoor Localization, Synthetic Image, RGB-D Image,
Deep Learning, Dataset

##  1 Introduction

Place recognition based on camera images is an important task for navigation
of a robot or an unmanned vehicle [28]. This can significantly reduce the cost
and simplify the determining of the agent spatial pose in a 3D scene.

Generally, visual localization has three key stages. The first stage is
retrieving for a given query image the most similar image from a previously
known database images based on global embeddings [3, 10, 22]. At the second
stage, key points are extracted on the query image and on the retrieved image
and matching them. Some modern methods combine these procedures into a single
model [30]. Camera pose optimization is performed in the third step so that
key points on the retrieved image coincide in location with the key points on
the query image. Sometimes this is done in 2D [15, 18], sometimes directly in
3D [38, 41].

The modern trend is learning neural network-based methods that allow these
stages to be performed. For them, it is important to have a diverse and large-
scale dataset. It should contains, in addition to images, information about
the exact camera position with 6 degrees of freedom (6DoF), and also data
about distances corresponding to each pixel (depth map).

This paper focuses on the development of the dataset for indoor localization
that can later be used for robot’s intelligent navigation in the
photorealistic simulator Habitat [27]. This will help to investigate
quantitatively and qualitatively loop detection methods of a robot’s movement
when it enters the vicinity already visited place (“Points”, see Fig. 1).

Another contribution is the development of a new modular approach named as
PNTR. It first performs an image retrieval with the Patch-NetVLAD method, then
extracts keypoints and matches them using R2D2, LoFTR or SuperPoint with
SuperGlue, and finally performs a camera pose optimization step with TEASER++.

##  2 Related Work

Image Retrieval. The search problem for the closest image in the database can
be reformulated as a ranking problem. The solution requires finding
informative and compact local and global descriptors of the images.

The common “classical” approaches obtain global features (embeddings) by
aggregating the local descriptors using the bag of words (BoW) scheme (DBoW2,
DBoW3, FBoW) or vectors of locally aggregated descriptors (VLAD).

In the last few years, new approaches based on deep neural networks have been
released: NetVLAD[3], distilled model HF-Net[25], Ap-Gem[22] approach with
differentiable rank loss function, graph-based approach GraphVLAD[40]. They
had surpassed the classical ones by feature learning for the specific
problems. The similar image candidates can be also re-ranked by analyzing
statistics of geometrically correct matchings of local descriptors for image
patches as in Patch-NetVLAD[11].

To improve the matching of embeddings, some approaches utilize semantic
information. They demonstrate good performance on popular benchmarks [20, 21,
37]. However, in spite of notable achievements of these neural networks, there
still is a problem with the extraction of invariant semantic descriptors of
images which have only low-quality semantic/instance segmentation. Such
segmentation is often inherent in real-time neural networks that generate a
lot of noise in the resulting masks.

Figure 1: Illustration of the localization problem by RGB-D query image. It is
necessary to determine the pose of the corresponding camera relative to the
camera poses from known database, consisting of a regular grid of groups of 6
cameras (“Points”)

Local Feature Extraction and Matching. In the next stage of visual
localization, we should find (detect and describe) common local features
(keypoints) on pairs of images. The difficulties of this problem include
getting informative and compact local descriptors and providing geometrical
verification of detected keypoints.

The classical local features extraction approaches utilize scale-invariant
feature transform based on local gradients [17], BoW [10], etc. These methods
have achieved remarkable performance, but analogously to image retrieval
methods, modern neural network approaches have outperformed them by finding
more robust local features. The recent approaches demonstrate good results due
to appropriate loss functions, neural network architectures (CNN,
Transformers), and training schemes (Siamese networks). For example, the
advantage of end-to-end training allows us to simultaneously train keypoint
detectors and descriptors: SuperPoint [8], R2D2 [23], D2-Net [9], etc.

The goal of the feature matching is to find the same keypoint descriptors in
both query and retrieved images. Metric learning is the most common training
scheme for this purpose. Models using attention mechanisms (LoFTR [30]) and
graphs (SuperGlue [26]) provide highly accurate keypoint matching. In the
inference mode, models can use the k-nearest neighbors algorithm for fast and
robust matching (Faiss[12]).

Camera Pose Optimization is the final stage of the visual localization. The
main problem is to find such transformation (SE3 rotation matrix and
translation vector), which minimizes the pairwise distance between point
clouds. The most popular methods for this task are the classical optimization
approaches such as easy PnP + Ransac[18], graph-based g2o[15], 3D point cloud-
based ICP[41], reliable to outliers TEASER++[38], neural network-based methods
PoseNet[13], DCP [34], etc.

Complex Localization Approaches. One of the most popular three-stage approach
using neural networks is Hierarchical-Localization [25]. HF-Net[25] is an
image retrieval method in this pipeline, SuperPoint[8] is a keypoint
extracting method, SuperGlue[26] is used for keypoint matching and PnP[14]
with RANSAC[18] are for pose optimization. Another state-of-the-art method on
long-term visual localization benchmark[1] is Kapture framework using Ap-GeM
[22] and R2D2 [23].

One of the most popular classic non-neural network visual localization method
is the combination of ORB[24] and DBoW2[10], which is widely used in popular
SLAM methods, in particular, ORB-SLAM2, OpenVSLAM, etc.

Table 1:  Open datasets for visual place recognition and localization  Dataset | Year | Scene | Synth. |  Pose info. | RGB |  Depth info. |  Sem. segm. |  Inst. segm. |  Frames
---|---|---|---|---|---|---|---|---|---
Pittsburgh [19] | 2015 | outdoor |  |  GPS | $\times$ |  |  |  |  278k
Landmarks [19] | 2016 | outdoor |  |  Label | $\times$ |  |  |  |  10k
Google-Landmarks [19] | 2017 | outdoor |  |  GPS | $\times$ |  |  |  |  1.2M
Nordland [19] | 2018 | outdoor |  |  GPS | $\times$ |  |  |  |  143k
CMU-Seasons [1] | 2018 | outdoor |  |  6DoF Pose | $\times$ |  laser scan. |  |  |  82.5k
Aachen Day-Night [1] | 2018 | outdoor |  |  6DoF Pose | $\times$ |  stereo |  |  |  7.5k
RobotCar Seasons [1] | 2018 | outdoor |  |  6DoF Pose | $\times$ |  laser scan. |  |  |  38k
Tokyo 24/7 [19] | 2018 | outdoor |  |  GPS | $\times$ |  stereo |  |  |  2.8M
Argoverse Stereo [7] | 2019 | outdoor |  |  6DoF Pose | $\times$ |  stereo/laser scan. |  |  |  6,6k
NuScenes-lidarseg [5] | 2020 | outdoor |  |  6DoF Pose | $\times$ |  laser scan. |  $\times$ |  |  40k point clouds
Waymo Perception [31] | 2020 | outdoor |  |  6DoF Pose | $\times$ |  stereo/laser scan. |  |  |  390k
KITTI360 [36] | 2020 | outdoor |  |  6DoF Pose | $\times$ |  laser scan. |  $\times$ |  $\times$ |  4$\times$83k
Mapillary SLS [19] | 2020 | outdoor |  |  6DoF Pose | $\times$ |  |  |  |  1.68M
TUM Indoor [16] | 2012 | indoor |  |  6DoF Pose | $\times$ |  laser scan. |  |  |  7k
7-scenes [19] | 2013 | indoor |  |  6DoF Pose | $\times$ |  RGB-D cam. |  |  |  17k
Baidu [39] | 2017 | indoor |  |  6DoF Pose | $\times$ |  laser scan. |  $\times$ |  |  2k
Matterport3D [6] | 2017 | indoor | $\times$ |  Need to gen. | $\times$ |  3D Models |  $\times$ |  $\times$ |  194k
TUM-LSI [16] | 2017 | indoor |  |  6DoF Pose | $\times$ |  laser scan. |  |  |  220
ScanNet [16] | 2017 | indoor |  |  6DoF Pose | $\times$ |  RGB-D cam. |  $\times$ |  $\times$ |  2.4M
2D-3D-Semantics [4] | 2017 | indoor |  |  6DoF Pose | $\times$ |  RGB-D cam. |  $\times$ |  |  70k
Gibson [35] | 2017 | indoor | $\times$ |  Need to gen. | $\times$ |  3D Models |  |  |  572 3D Scenes
Inloc [1] | 2018 | indoor |  |  6DoF Pose | $\times$ |  3D point clouds |  |  |  10k
Replica [29] | 2019 | indoor | $\times$ |  Need to gen. | $\times$ |  3D Models |  $\times$ |  $\times$ |  18 3D Scenes
ROI10 [33] | 2020 | indoor |  |  6DoF Pose | $\times$ |  RGB-D cam. |  |  |  250k
HM3D [2] | 2021 | indoor | $\times$ |  Need to gen. | $\times$ |  3D Models |  |  |  1000 3D scenes
Naver Labs [16] | 2021 | indoor |  |  6DoF Pose | $\times$ |  laser scan. |  $\times$ |  |  100k
HPointloc (our) | 2022 | indoor | $\times$ |  6DoF Pose | $\times$ |  RGB-D cam. |  $\times$ |  $\times$ |  76k

Datasets. There are many known datasets used to solve the visual localization
problem (see Table 1). The most popular of them are presented in the long-term
visual localization challenge [1], as well as in the survey publications [16,
19]. One of the drawbacks of most of them is the lack of data on instance or
semantic segmentation, or depth maps, while this information is very important
for improving existing localization approaches.

If we take the InLoc dataset [32] as an example, then it includes not dense
depth maps, but sparse point clouds. Using them we can obtain an estimate of
the depths for the found keypoints, but in our formulation of the problem, we
do not consider such a case.

Of particular note are datasets that contain 3D models of a scene, rather than
separate frames (Matterport3D[6], Gibson[35], Replica[29], HM3D[2]). They are
usually integrated into simulators of the movement of intelligent agents, for
example, into the popular and computationally efficient Habitat[27]. At the
same time, to obtain a reproducible and useful result based on them, it is
necessary to have sufficiently large and diverse samples from these datasets
containing frames and the corresponding 6DoF camera poses.

In our work, we propose filling the gap among such datasets, specialized in
the study of a specific problem — localization of an intelligent agent in the
vicinity of some key point belonging to a regular grid in a 3D indoor
environment.

Figure 2:  Scheme of the proposed framework for indoor localization using
RGB-D images. Green text refers to the methods used in the PNTR approach.

##  3 The Point-based Approach to Indoor Place Recognition

In this paper, we solve the problem of developing an approach to estimating
the 6DoF pose $P_{q}$ of an intelligent agent from the image $I_{q}$ of its
RGB-D camera (query) in the vicinity of the poses of the cameras $P^{i}_{db}$
from the database (see Figure 1), forming a regular grid of groups of 6
cameras (named as “Points”), the images $I^{i}_{db}$ of which cover 360∘
environment. This formulation required the development of a new special
dataset, named as HPointLoc. The dataset was generated automatically based on
the Habitat simulator[27]. This will provide the perspective to use the
results for navigation and learning the behavior of an intelligent agent
(robot) in real time in this photorealistic environment. On such a
“Point”-based dataset, it is advisable to evaluate the quality of the state-
of-the-art methods of visual place recognition. For their study, we proposed
the modular PNTR approach, which is based on trainable methods. Its modules
are described in more detail later in the paper.

Image Retrieval Module. When developing the approach, various classical image
retrieval models based on a bag of words and neural network approaches are
investigated. They form global feature vectors (embeddings) $E_{q}$ and
$E^{i}_{db}$ for the query image $I_{q}$ and images from the database
$I^{i}_{db}$. Based on embeddings, the most similar image in the sense of the
similarity metric $S$ is determined:

$top1=\underset{i}{\operatorname{argmin}}\left(S\left(E_{q},E^{i}_{db}\right)\right).$
(1)

We consider the classic approaches DBoW2 [10] and FBoW [42] based on the
formation of global descriptors by image keypoints extracted with ORB method.
The main attention is paid to neural network-based approaches of image
retrieval: NetVLAD[3], AP-GeM[22], HF-Net[25] and the Patch-NetVLAD[11] (its
speed up configuration (s) is included in our PNTR approach).

Local Feature Extraction and Matching Module. At the next stage of extraction
and matching of local features, $k$ matches of keypoints on the query image
$F^{j}_{q}$ and the most similar from the base $F^{j}_{top1}$ are found,
$j=1..k$. For each of these points, the distance to the camera in meters
(depth) is known: $D^{j}_{q}$ and $D^{j}_{top1}$, $j=1..k$ respectively.

We examined the extraction and matching of image keypoints based on the
classic popular approaches ORB with DBoW2 [10] or FBoW [42] and the popular
neural networks (used in our PNTR approach) SuperPoint [8] with SuperGlue
[26], R2D2 [23] with feature matching based on the Kapture toolbox, as well as
monolithic LoFTR [30].

Database Preparation Module. The database used to solve the localization
problem is an array of $i\in[1,N]$ camera poses $P^{i}_{db}$, the
corresponding images $I^{i}_{db}$, depth maps $D^{i}_{db}$, and segmentation
masks $M^{i}_{db}$ (the latter may not be used in the approach). In addition,
extracted and pre-calculated global image embeddings $E^{i}_{db}$ and
information about local features $F^{j}_{db}$ on them are additionally entered
into the database.

Camera Pose Optimization Module. Further, the final estimation of the
4$\times$4 matrix of the camera pose $P_{q}$ is carried out. This pose
corresponds to the query image $I_{q}$. In our case, the 4$\times$4 matrix of
the camera pose $P^{top1}_{db}$ is known for retrieved image $I^{top1}_{db}$
from database. An optimization problem is solved for calculating the relative
pose of the camera $P^{q}_{top1}$ (also a 4$\times$4 matrix) based on
minimizing some functional $L$ that takes into account the 2D coordinates $F$
and the depth $D$ of matched keypoints in two images:

$P^{q}_{top1}=\underset{P^{q}_{top1},j\in[1,k]}{\operatorname{argmin}}\left(L\left(F^{j}_{q},D^{j}_{q},F^{j}_{top1},D^{j}_{top1}\right)\right).$
(2)

The final pose is defined as $P_{q}=P^{top1}_{db}P^{q}_{top1}.$

The widely used classical approaches g2o[15], ICP[41] and the newer and highly
accurate Teaser++[38] (used in the proposed PNTR approach) are investigated as
methods for optimizing pose in our framework.

To quantitatively evaluate the quality of the overall framework, we use Recall
metric with different thresholds. It is calculated as the fraction of query
images which localization errors do not exceed the specified threshold,
respectively for distance (translation)
$\epsilon_{t}\in\\{0.25m,0.5m,1m,5m\\}$ and rotation
$\epsilon_{r}\in\\{2^{\circ},5^{\circ},10^{\circ},20^{\circ}\\}$. Such
thresholds were chosen to assess the prospects for solving the problem of loop
detection and place recognition in selected indoor environment using the
developed approach.

##  4 HPointLoc Dataset

The dataset consists of 49 scenes from the Matterport3D dataset and is
intended for training computer vision algorithms or testing them. It was
formed according to the following algorithm.

For each scene, a regular grid with $N$ key poses was generated
$({x}_{id},{y}_{id},{z}_{id}$, ${yaw}_{id},{pitch}_{id},{roll}_{id})$ with a
distance of 2 meters in $x$ and $y$ axes for one scene. A key pose has a
unique ${id}$ and field ${status=1}$. An example of a key pose is marked in
red in Fig. 3a. For each key pose, the two steps were taken:

1\. Generation of five more key pose orientations with 60 degrees step,
simulating a 360 degree key pose camera. The view angle of each frame is 90
degrees. An example of six key images is shown in Fig. 3d.

2\. Generation of $M=50$ random poses in a radius $r=0.5m$ relative to the key
pose $({x}_{id},{y}_{id},{z}_{id})$ with a random orientation (shown in green
in Fig. 3a). If the generated pose does not fall into the free area, then it
is discarded; otherwise we add a new record to the dataset.

Table 2: Summary for HPointLoc Dataset.  | Points | Poses | Categories | Instances | Maps
---|---|---|---|---|---
HPointLoc-Val | 23 | 1088 | 33 | 3266 | 1
HPointLoc-All | 1757 | 86678 | 41 | 488717 | 49
Table 3: Statistics for semantic instances in the proposed HPointLoc-Val and HPointLoc-All datasets | Instances per categ. |  | Instances per categ. |  | Instances per categ.
---|---|---|---|---|---
Category | Val | All | Category | Val | All | Category | Val | All
appliances | 0 | 1077 | cushion | 15 | 5600 | shelving | 18 | 4174
bathtub | 11 | 621 | door | 1174 | 44500 | shower | 4 | 2190
beam | 0 | 981 | fireplace | 0 | 1022 | sink | 6 | 1924
bed | 28 | 5025 | floor | 0 | 52274 | sofa | 6 | 3871
blinds | 0 | 162 | furniture | 8 | 376 | stairs | 15 | 3557
board_panel | 3 | 145 | gym equipment | 10 | 105 | stool | 0 | 7660
cabinet | 26 | 5383 | lighting | 184 | 8797 | table | 15 | 11886
ceiling | 249 | 28143 | mirror | 16 | 1752 | toilet | 3 | 271
chair | 48 | 27946 | misc | 180 | 47346 | towel | 17 | 482
chest_of_drawers | 8 | 2123 | objects | 67 | 28902 | tv_monitor | 10 | 1190
clothes | 0 | 191 | picture | 115 | 18945 | void | 10 | 14445
column | 0 | 4255 | plant | 2 | 4330 | wall | 601 | 107291
counter | 0 | 1521 | railing | 0 | 8157 | window | 110 | 19041
curtain | 11 | 8129 | seating | 6 | 2927 |  |  |
Figure 3: Explanations to the HPointLoc dataset content: a - the generation of
query and database camera poses based on maps from the Matterport3D dataset in
the Habitat environment; b - scene map in the HPointLoc-Val dataset; c -
another scene map in the HPointLoc-All dataset, the positions of the camera
centers are shown by colored dots in the circles corresponding to the
“Points”; d - examples of 6 RGB-images (upper row), depth map (middle row) and
map of object instance segmentation (lower row), corresponding to one “point”
of scene map of the HPointLoc dataset. Six RGB-images from the “point” cover
overlapping areas with a 360-degree view

Each item in the dataset contains the following data: RGB and depth images,
instance segmentation of the frame with 41 classes (see Table 3) and GPS with
Compass data (orientation of the camera is described with a quaternion).

RGB images have 256$\times$256 size and camera viewing angle 90∘. Gaussian
noise model was also added with factor intensity of 0.02, a mean value of 0
and a sigma parameter of 1.

Depth images also have 256$\times$256 size. The depth values lie in the range
from 0 to 1, where 0 is a minimum possible depth (0 m) and 1 — maximum
possible (10 m). Such maximum depth is chosen because most of the commercial
RGB-D cameras, such as Intel RealSense, and ZED, have similar restrictions.
The RGB-D camera stands at 1.25 meters high above the floor.

Examples of top-view scene maps with marked frame poses are shown in Fig. 3b
and 3c. Thus, in each spherical “Point” with a radius of 0.5 meters, there are
6 key images with the prior known data for localization methods (images in the
database) and several dozen images (query images), which need to be localized.
It is assumed in the experiments that the pose of query images by which it is
necessary to localize is not known to the localization method, but all other
data, including the depth map, are known. A summary of the dataset contents is
shown in Table 2. The dataset is split into two parts: the validation
HPointLoc-Val, which contains only one scene, and the complete HPointLoc-All
dataset, containing all 49 scenes, including HPointLoc-Val. We use these
datasets only to validate the proposed approaches, not for training.

##  5 Experimental Results

Table 4: Localization quality based on the pose of top-1 retrieved image (without pose optimization) on the HPointLoc-Val dataset Image retrieval method |  Embedding size | (5m,20°) | (1m,10°) | (0.5m,5°) | (0.25m,2°) | (5m) | (1m) | (0.5m) | (0.25m)
---|---|---|---|---|---|---|---|---|---
FBoW+ORB |  $\sim$1000000 | 0.527 | 0.269 | 0.133 | 0.029 | 0.911 | 0.864 | 0.864 | 0.476
DBoW2+ORB |  $\sim$1000000 | 0.542 | 0.269 | 0.133 | 0.028 | 0.916 | 0.877 | 0.877 | 0.477
AP-GeM |  2048 | 0.559 | 0.282 | 0.138 | 0.032 | 0.964 | 0.893 | 0.893 | 0.493
NetVLAD |  32768 | 0.584 | 0.291 | 0.141 | 0.032 | 0.971 | 0.919 | 0.919 | 0.502
HF-Net |  4096 | 0.600 | 0.291 | 0.138 | 0.032 | 0.975 | 0.934 | 0.934 | 0.505
Patch-NetVLAD(s) |  512$\times$100 patches | 0.590 | 0.296 | 0.140 | 0.030 | 0.983 | 0.963 | 0.963 | 0.515
Patch-NetVLAD(p) |  4096$\times$300 patches | 0.576 | 0.292 | 0.138 | 0.028 | 0.979 | 0.961 | 0.961 | 0.517
Table 5: Quality of various visual localization methods on the HPointLoc-Val dataset Approach | (5m,20°) | (1m,10°) | (0.5m,5°) | (0.25m,2°) | (5m) | (1m) | (0.5m) | (0.25m)
---|---|---|---|---|---|---|---|---
DBoW2+ORB+g2o | 0.844 | 0.812 | 0.78 | 0.726 | 0.905 | 0.857 | 0.841 | 0.796
FBoW+ORB+g2o | 0.801 | 0.766 | 0.719 | 0.64 | 0.903 | 0.826 | 0.78 | 0.719
AP-GeM+R2D2(K)+ICP | 0.961 | 0.862 | 0.805 | 0.733 | 0.965 | 0.871 | 0.828 | 0.777
AP-GeM+R2D2(K)+ TEASER++ | 0.939 | 0.904 | 0.881 | 0.877 | 0.958 | 0.906 | 0.882 | 0.880
Ap-GeM+LofTR+g2o | 0.907 | 0.868 | 0.837 | 0.746 | 0.963 | 0.902 | 0.891 | 0.861
Ap-Gem+LofTR+ICP | 0.903 | 0.846 | 0.825 | 0.762 | 0.961 | 0.877 | 0.862 | 0.807
AP-GeM+LoFTR+ TEASER++ | 0.892 | 0.854 | 0.831 | 0.766 | 0.964 | 0.899 | 0.884 | 0.823
NetVLAD+SP+SuperGlue+g2o | 0.917 | 0.891 | 0.874 | 0.845 | 0.967 | 0.919 | 0.906 | 0.887
NetVLAD+SP+SuperGlue+ICP | 0.944 | 0.893 | 0.868 | 0.794 | 0.969 | 0.909 | 0.893 | 0.842
NetVLAD+SP+SuperGlue+TEASER++ | 0.924 | 0.892 | 0.872 | 0.854 | 0.966 | 0.918 | 0.906 | 0.888
NetVLAD+R2D2(K)+ICP | 0.968 | 0.901 | 0.855 | 0.788 | 0.969 | 0.907 | 0.877 | 0.827
NetVlad+R2D2+TEASER++ | 0.941 | 0.916 | 0.907 | 0.905 | 0.967 | 0.918 | 0.911 | 0.908
NetVLAD+LoFTR+g2o | 0.921 | 0.891 | 0.876 | 0.78 | 0.968 | 0.92 | 0.916 | 0.892
NetVLAD+LoFTR+ICP | 0.917 | 0.875 | 0.858 | 0.799 | 0.971 | 0.9 | 0.892 | 0.843
NetVLAD+LoFTR+ TEASER++ | 0.908 | 0.881 | 0.865 | 0.806 | 0.971 | 0.919 | 0.914 | 0.857
HF-Net+SP+SuperGlue+g2o | 0.925 | 0.903 | 0.881 | 0.852 | 0.972 | 0.936 | 0.92 | 0.896
HF-Net+R2D2(K)+ICP | 0.974 | 0.917 | 0.869 | 0.791 | 0.975 | 0.925 | 0.894 | 0.834
HF-Net+R2D2(K)+ TEASER++ | 0.953 | 0.936 | 0.923 | 0.92 | 0.973 | 0.938 | 0.926 | 0.923
Patch-NetVLAD(s)+SP+SuperGlue+ g2o | 0.946 | 0.924 | 0.908 | 0.883 | 0.982 | 0.96 | 0.947 | 0.924
Patch-NetVLAD(s)+ R2D2(K)+ICP | 0.98 | 0.94 | 0.896 | 0.808 | 0.984 | 0.952 | 0.925 | 0.852
PNTR with SP+SuperGlue | 0.947 | 0.925 | 0.899 | 0.882 | 0.975 | 0.945 | 0.934 | 0.914
PNTR with R2D2(K) | 0.964 | 0.957 | 0.952 | 0.945 | 0.98 | 0.958 | 0.953 | 0.951
PNTR with LoFTR | 0.942 | 0.919 | 0.900 | 0.834 | 0.977 | 0.953 | 0.948 | 0.892
Table 6: Quality of various visual localization methods on the HPointLoc-All dataset Approach | (5m,20°) | (1m,10°) | (0.5m,5°) | (0.25m,2°) | (5m) | (1m) | (0.5m) | (0.25m)
---|---|---|---|---|---|---|---|---
DBoW2+ORB (top-1 db) | 0.592 | 0.303 | 0.150 | 0.033 | 0.903 | 0.882 | 0.881 | 0.498
FBoW+ORB2 (top-1 db) | 0.547 | 0.285 | 0.142 | 0.032 | 0.825 | 0.799 | 0.797 | 0.464
NetVLAD (top-1 db) | 0.643 | 0.317 | 0.158 | 0.034 | 0.957 | 0.877 | 0.876 | 0.487
AP-GeM (top-1 db) | 0.635 | 0.311 | 0.156 | 0.034 | 0.944 | 0.813 | 0.812 | 0.452
HF-Net (top-1 db) | 0.646 | 0.318 | 0.158 | 0.034 | 0.955 | 0.879 | 0.878 | 0.487
Patch-NetVLAD(s) (top-1 db) | 0.644 | 0.320 | 0.159 | 0.034 | 0.969 | 0.944 | 0.942 | 0.516
DBoW2+ORB2+g2o | 0.876 | 0.857 | 0.648 | 0.404 | 0.901 | 0.870 | 0.661 | 0.419
FBoW+ORB2+g2o | 0.776 | 0.748 | 0.571 | 0.357 | 0.820 | 0.769 | 0.590 | 0.377
NetVLAD+SP+SuperGlue+ g2o | 0.925 | 0.867 | 0.647 | 0.406 | 0.950 | 0.877 | 0.656 | 0.414
NetVLAD+SP+SuperGlue+ICP | 0.935 | 0.853 | 0.803 | 0.426 | 0.956 | 0.876 | 0.844 | 0.476
NetVLAD+R2D2(K)+ICP | 0.944 | 0.842 | 0.748 | 0.341 | 0.956 | 0.877 | 0.846 | 0.474
NetVLAD+LoFTR+ICP | 0.934 | 0.863 | 0.829 | 0.458 | 0.956 | 0.877 | 0.849 | 0.480
NetVLAD+LoFTR+TEASER++ | 0.933 | 0.865 | 0.829 | 0.445 | 0.956 | 0.877 | 0.849 | 0.479
Ap-Gem+LofTR+ICP | 0.907 | 0.797 | 0.765 | 0.423 | 0.943 | 0.813 | 0.787 | 0.445
AP-GeM+LoFTR+ TEASER++ | 0.907 | 0.799 | 0.765 | 0.412 | 0.943 | 0.813 | 0.787 | 0.445
HF-Net+SP+SuperGlue+g2o | 0.924 | 0.868 | 0.647 | 0.405 | 0.949 | 0.878 | 0.656 | 0.413
HF-Net+SP+SuperGlue+ICP | 0.94 | 0.861 | 0.817 | 0.439 | 0.955 | 0.879 | 0.85 | 0.479
HF-Net+SP+SuperGlue+TEASER++ | 0.935 | 0.854 | 0.803 | 0.425 | 0.955 | 0.877 | 0.845 | 0.475
HF-Net+R2D2(K)+ICP | 0.943 | 0.845 | 0.752 | 0.342 | 0.955 | 0.879 | 0.848 | 0.475
HF-Net+LoFTR+TEASER++ | 0.933 | 0.867 | 0.831 | 0.445 | 0.955 | 0.879 | 0.851 | 0.480
Patch-NetVLAD(s)+SP+SuperGlue+g2o | 0.956 | 0.934 | 0.693 | 0.433 | 0.967 | 0.942 | 0.701 | 0.440
Patch-NetVLAD(s)+R2D2(K)+ ICP | 0.964 | 0.931 | 0.876 | 0.464 | 0.968 | 0.944 | 0.911 | 0.507
PNTR with SP+SuperGlue | 0.952 | 0.919 | 0.863 | 0.451 | 0.968 | 0.942 | 0.905 | 0.502
PNTR with R2D2(K) | 0.961 | 0.938 | 0.906 | 0.503 | 0.969 | 0.943 | 0.913 | 0.509
PNTR with LoFTR | 0.959 | 0.936 | 0.895 | 0.472 | 0.969 | 0.944 | 0.913 | 0.508

Localization Quality Estimation. The experiments were held on the HPointLoc
dataset. All the methods we used were already pre-trained. According to Fig. 2
and Eq.1, the image retrieval method searches for the closest similar image.
After that, we optimize the relative pose of the retrieved image to query
image (Eq.2). Knowing the absolute pose of the image from the database, we get
the final pose of the query. The quality metrics we used are generally
recognized and observable on the benchmark [1]. Accuracy metrics of true
localizations with the optimization step on the HPointLoc-Val dataset are
given in Table 4. Pose optimization in this case is not accomplished: in other
words, the result pose is equivalent to the database image that was accepted
as the most similar. From Table 4, we can see that at all translation error
thresholds the best method is Patch-NetVLAD with speed up configuration (s).
The angle error should be taken into account secondary since the frame pose is
not optimized.

The localization results of different methods on the HPointLoc-Val dataset are
given in the Table 5. SuperPoint is abbreviated as SP. As we can see, the
highest quality for image retrieval, keypoints matching, camera pose
optimization is for the proposed PNTR approach with the R2D2 feature
extraction and matching based on procedure form the Kapture tool (symbol ‘K’).

The localization results of different methods on the HPointLoc-All are given
in Table 6. We can see that all metrics are worse in comparison with the
evaluation on the HPointLoc-Val dataset. This directly follows from the
degradation of the image retrieval methods.

Figure 4:  Typical problems that sometimes arise in all image retrieval
methods (for example, the NetVLAD) on the HPointLoc Dataset: a — the selection
of an image that looks like a query occurs with an error caused by the lack of
explicit usage of semantics (information about the presence of objects, about
their color, etc.); b — the selected top-1 image contains the same scene as
the query image, but is more than 1 meter away from it and generates a
localization error.

In the course of the experiments, we detect problems that sometimes arise in
all of the considered image retrieval methods. In Fig. 4, they are shown using
the NetVLAD method as an example and demonstrate the need to explicitly use
semantics and the complexity of choosing a top-1 image containing the same
scene as the query image, but the corresponding camera is located at a
considerable distance from the camera for images from the database.

Table 7: Average execution time of different localization stages, sec Image retrieval | Camera Pose Optimization
---|---
Method | Embedding extr. | Embedding match. | Method | Avg. time
NetVLAD | 0.00698 | - | g2o | 0.00074
AP-GeM | 0.01542 | 0.00013 | ICP | 0.00158
HF-Net | 0.09049 | - | Teaser++ | 0.00826
Patch-NetVLAD(s) | 0.01843 | 0.1039 |  |
Patch-NetVLAD(p) | 0.26415 | 2.28863 |  |
Local feature extraction and matching | Overall PNTR approach
Method | Feature extr. | Feature match. | Method | Avg. time
ORB+DBoW2 | 0.00407 | 0.00013 | PNTR with SuperPoint+SuperGlue | 0.13565
ORB+FBoW | 0.00092 | 0.00007 | PNTR with R2D2 | 0.19207
SuperPoint+SuperGlue | 0.0025 | 0.00256 | PNTR with LoFTR | 0.15189
R2D2(K) | 0.0585 | 0.00298 |  |
LoFTR | 0.0213 | - |  |

Time Performance Estimation. Average execution time of the main stages of
localization implemented in the proposed framework is shown in Table 7. The
performance was evaluated on a workstation with NVidia RTX2080Ti 11Gb GPU, AMD
Threadripper 1900X (8 cores, 3.8GHz) CPU, 64Gb RAM.

It should be noted that the Patch-NetVLAD (s) configuration is more than 20
times faster than the Patch-NetVLAD (p) configuration, and their quality
metrics are almost the same. NetVLAD leads in speed, which is 10 times faster
than the almost identical in quality HF-Net method and than the significantly
better Patch-NetVLAD (s).

Among the considered methods of feature extraction, the ORB FBoW method is the
leader, which is almost five times faster than the classical ORB + DBoW2
method, but is significantly inferior in quality. LoFTR, the one-stage method
for extracting and matching points, has the highest performance among neural
network approaches, which slightly exceeds the total performance of the
SuperPoint+SuperGlue method combination. Among the considered optimization
methods, the fastest is the g2o method, which is twice as fast as ICP and
almost 10 times faster than TEASER++. The total performance of the most
accurate groups of methods, except the slow Patch-NetVLAD (p), is 5-10 FPS,
which indicates the potential for their practical use in a parallel global
localization stream to solve the problem of loop closure in SLAM methods.

##  6 Conclusions

In this work, we have shown that the proposed HPointLoc dataset allows us to
visually assess the quality and performance of various approaches for solving
the place recognition problem in a photorealistic indoor environment using
RGB-D images. This is achieved primarily by creating a regular grid of
“Points” during dataset preparation. This can be done for any real indoor
environment whose model can be imported into the Habitat simulator.

On the small subset HPointLoc-Val, localization methods exhibit similar
behavior, so it is sufficient to use it for quick evaluation and comparison of
algorithms. The experiment results have revealed the limitations of the
considered localization methods. It leads to the need of explicitly taking
into account the semantics of the scene and the importance of a correct
interpretation of the localization error when using one or the other image
retrieval method.

State-of-the-art results were obtained for the proposed PNTR approach, which
showed the highest quality on the developed dataset, but is rather slow and
provides an image processing speed of 5 FPS. Nevertheless, it can be still
used for loop detection in SLAM algorithms in a parallel stream to the main
visual tracking procedure.

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# Gravitational axial perturbations of Schwarzschild-like black holes in dark
matter halos

Chao Zhang a,b,c<EMAIL_ADDRESS>Tao Zhua,b corresponding author:
<EMAIL_ADDRESS>Anzhong Wangd<EMAIL_ADDRESS>a Institute for
Theoretical Physics and Cosmology, Zhejiang University of Technology,
Hangzhou, 310032, China
b United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310032, China
c College of Information Engineering, Zhejiang University of Technology,
Hangzhou, 310032, China
d GCAP-CASPER, Physics Department, Baylor University, Waco, Texas 76798-7316,
USA

###### Abstract

Gravitational waves emitted by distorted black holes, such as those arising
from the coalescence of binary black holes, or a falling compact star into a
supermassive black hole, carry not only information about the corresponding
spacetime but also the information about the environment surrounding the black
holes. In this paper, we study the effects of the dark matter halos with three
different density profiles on the gravitational axial perturbations of a
Schwarzschild-like black hole. For this purpose, we first consider modified
Schwarzschild black holes with three different dark matter profiles and derive
the equation of motion of the axial perturbations of the modified
Schwarzschild metric. It is shown that by ignoring the dark matter
perturbations, a Regge-Wheeler-like master equation with a modified potential
for the axial perturbation can be obtained explicitly. Then we calculate the
complex frequencies of the quasinormal modes of the Schwarzschild-like black
hole in the dark matter halos by applying the sixth-order WKB method. The
corresponding gravitational wave spectra with the effects of the dark matter
halos have also been discussed.

##  I Introduction

Black holes (BHs) are one of the most mysterious phenomena in the Universe.
The existence of BHs provides us a perfect way to test gravitational effects
under extremely strong gravity, such as the formation of gigantic jets of
particles and disruption of neighboring stars Kimet2019 . On the other hand,
from the theoretical point of view, BHs are also unique labs to test modified
theories of gravity that are different from general relativity (GR) (see,
e.g., Xiang2019 ; Chao2020 ; Chao2020b ).

The outbreak of interest on BHs has further gained momenta after the detection
of the shadow of the M87 central BH with the Event Horizon Telescope (EHT)
EHTL1 ; EHTL2 ; EHTL3 ; EHTL4 ; EHTL5 ; EHTL6 ; EHTL7 ; EHTL8 . In fact, it is
widely believed that the central region of many galaxies contain supermassive
BHs. The shadow image of the M87 central BH captured by EHT is in good
agreement with the prediction of the spacetime geometry of a BH described by
the Kerr metric Kerr1963 . Nonetheless, since it is believed that up to $90\%$
of the matter in a host galaxy is made up by dark matter (DM) Kimet2020 , it
is natural to expect that the DM halo surrounding a central BH will bring
small deviations to the Kerr metric.

Although DM has not been directly detected, many evidences imply its
existence. In addition, because of the rotation curves of galaxies, the mass-
light ratio of elliptical galaxies, the cosmic microwave background radiation,
and the large-scale structure of the cosmos, a lot of research has been done
on DM and proposed various DM models Zhaoyi2020 . Of course, there are many
open questions related to DM, for instance, the spatial density distribution
of DM in galaxies, solutions of Einstein’s field equations when considering
the DM background, and more.

In this paper, we focus on the effects of DM halos on gravitational waves
(GWs) emitted from the central BH located in a galaxy. The detection of the
first GW from the coalescence of two massive BHs by advanced LIGO marked the
beginning of a new era, the GW astronomy Ref1 . Following this observation,
about 90 GW events have been identified by the LIGO/Virgo/KAGRA scientific
collaborations (see, e.g., GWs ; GWs19a ; GWs19b ; GWsO3b ). In the future,
more ground- and space-based GW detectors will be constructed Moore2015 ;
Aso2013 ; Liu2020 ; Taiji2 , which will enable us to probe signals with a
wider frequency band. This triggered the interests on the quasinormal modes
(QNMs) from GWs, including those from the late-merger and ringdown stages of
coalescence Berti18  as well as those from supermassive BHs, e.g, central BHs
in a galaxy. The detection of QNMs from the ringdown stage will provide a
unique way to probe the matter environment surrounding a BH. For example, it
is shown that the shift from GR on the QNMs due to the surrounding ultralight
bosons might be detectable in the future by the observational data from LISA-
like missions Chung:2021roh . The effects of the dark matter or matter
distributing around BHs on the QNMs have also been considered in
Cardoso:2021wlq ; Liu:2021xfb ; Konoplya .

From the theoretical point of view, QNMs are eigenmodes of dissipative
systems. The information contained in QNMs provide the keys for revealing
whether BHs are ubiquitous in our Universe, and more important whether GR is
the correct theory to describe the event even in the strong field regime.
Readers may find more details in Berti2009 . Basically, the QNM frequency
$\omega$ contains two parts, the real part and the imaginary part. Its real
part gives the frequency of vibration while its imaginary part provides the
damping time. In other words, the frequency we are going to calculate is a
complex number (although it could be purely imaginary in certain
circumstances).

In fact, in GR according to the no-hair theorem, an isolated BH is completely
characterized by only three quantities, mass, spin angular momentum and
electric charge. Astrophysically, we expect BHs to be neutral, so they ought
to be described by the Kerr solution. Then, the quasinormal frequencies and
damping times will depend only on the mass and angular momentum of a BH.
Therefore, to extract the physics from QNMs, at least two QNM signals are
needed. This will require the signal-to-noise ratio (SNR) to be of the order
100. Although such high SNRs are not achievable right now, it has been shown
that they may be achievable once the advanced LIGO and Virgo reach their
designed sensitivities. In any case, it is certain that they will be detected
by the ground-based third-generation detectors, such as Cosmic Explorer or the
Einstein Telescope, as well as the space-based detectors, including LISA,
TianQin Shi2019 , Taiji Taiji2 , and DECIGO Moore2015 .

For this purpose, we first consider modified Schwarzschild black holes with
three different DM profiles and derive the equation of motion of the axial
perturbations 111Normally, the GW spectrum of QNMs can be studied via black
hole perturbations. In the background spacetime which is static and
spherically symmetric, the metric perturbations can decouple into two
independent parts, the polar one and the axial one. In this paper we only
consider the latter case. As we discuss later, the axial perturbation
considered in this paper is equivalent to the odd-parity perturbation, which
corresponds to the parity of $(-1)^{l}$, with $l$ being an index inherited
from spherical harmonics Cardoso2001 . of the modified Schwarzschild metric.
It is shown that by ignoring the DM perturbations, a Regge-Wheeler (RW)-like
master equation with a modified potential for the axial perturbation can be
obtained explicitly. Then we calculate the frequencies of the QNMs of the
Schwarzschild-like black hole in the DM halos by applying the sixth-order WKB
method. The corresponding GW spectra with the effects of the DM halos have
also been discussed.

Here, we consider the QNMs of the axial metric perturbations of a
Schwarzschild-like BH surrounded by the DM halos. Several different background
metrics are investigated by considering three different DM profiles. These
metrics for different dark matter halo profiles can be found in Kimet2019 ;
xu_JCAP . Note that recently the metric of a black hole immersed in dark
matter spike has also been derived Xu:2021dkv . In addition, the Sgr
$\text{A}^{\ast}$ black hole (located in the center of Milky Way Galaxy) and
the M87 galactic central black hole are what we focus on. In other words, the
structure we consider is a BH located at the center of a galaxy. By comparing
the resultant $\omega$’s with their counterparts for the Schwarzschild case,
we see the influence of DM halo on QNMs as well as GWs from central BHs. At
the same time, we will also estimate the detectability of these deviations
from the Schwarzschild case. Finally, by varying some profile-dependent
constants, one can find out how these $\omega$’s are deviating from the
Schwarzschild case with the effects of the DM halos.

The rest of this paper is organized as follows. Sec. II shows some basic
information of the three density profiles of DM halos that we investigate for
the calculations of QNMs. After that, in Sec. III we show how to derive a
Regge-Wheeler-like master equation from the axial perturbation and Einstein’s
field equations. At the same time, we show a brief analysis of the effective
potentials under different DM profiles (including that from the Schwarzschild
case). Section IV contains two parts. In the first part, we present some
resultant $\omega$’s. Some concluding remarks are given by comparing them with
their counterparts in the Schwarzschild case. For the second part, we test the
effects of model-dependent constants. Some results are shown together with an
analysis. Finally, Sec. V provides our main conclusions as well as some
outlooks for future work.

Through out the paper, we adopt the unit system so that $c=G_{N}=1$, where $c$
is the speed of light, while $G_{N}$ stands for the gravitational constant
observed on Earth. In this way, we still have one degree of freedom to choose
the unit for length. This is done later by setting ${r_{\rm MH}}=1$, where
$r_{\rm MH}$ means the radius of the metric horizon (MH) of the BH that we are
focusing on. In this paper, all the greek indices run from 0 to 3. Other usage
of indices is indicated explicitly when it is necessary.

##  II Black hole solutions in dark matter halo

Normally, the black hole spacetimes are not clean and are affected by the
surrounding matters. In this section, we consider the spherically symmetric
static black hole solutions with several different DM halo profiles. In
general, a Schwarzshild BH in the DM halo is described by the metric xu_JCAP

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-G(r)dt^{2}+\frac{1}{G(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2},$

where $G(r)$ denotes the function that describes the effects of the DM halos
and BH on the metric and it reduces to the Schwarzschild solution, viz.,
$G(r)=1-2M/r$ CarrollB , when the DM is absent. For different profiles of the
DM halo, the function $G(r)$ is different xu_JCAP . In the following, we
present the function $G(r)$ for each profile individually.

###  II.1 URC profile

In the universal rotation curve (URC) profile of the DM halo, the distribution
of the DM is described by URC1  (see also halo_review  for a review)

$\displaystyle\rho(r)=\frac{\rho_{0}r_{0}^{3}}{(r+r_{0})(r^{2}+r_{0}^{2})},$
(2.2)

where $\rho_{0}$ is the central density and $r_{0}$ is the characteristic
radius of the DM halo. According to the observations on the M87 galaxy, the
best fit values for the parameters of the URC profile are $\rho_{0}=6.9\times
10^{6}\text{M}_{\odot}/{\rm kpc}^{3}$ and $r_{0}=91.2\;{\rm kpc}$ Salucci_M87
. While in the Milky Way Galaxy, we have $\rho_{0}=5.2\times
10^{7}\text{M}_{\odot}/{\rm kpc}^{3}$ and $r_{0}=7.8\;{\rm kpc}$ dark_matter .
With this halo profile, the function $G(r)$ in the metric (II) is given by
Kimet2019 ; kimet_shadow

$\displaystyle G(r)$ $\displaystyle=$ $\displaystyle
e^{-2\pi^{2}\rho_{0}r_{0}^{2}}\left(1+\frac{r^{2}}{r_{0}^{2}}\right)^{-\frac{2\,\rho_{0}r_{0}^{3}\pi}{r}(1-\frac{r}{r_{0}})}$
$\displaystyle\times\left(1+\frac{r}{r_{0}}\right)^{-\frac{4\,\rho_{0}r_{0}^{3}\pi}{r}(1+\frac{r}{r_{0}})}$
$\displaystyle\times\exp\left[\frac{4\,\rho_{0}r_{0}^{3}\pi\arctan(\frac{r}{r_{0}})(1+\frac{r}{r_{0}})}{r}\right]-\dfrac{2M}{r}.$

Here $M=6.5\times 10^{9}~{}\text{M}_{\odot}$ for the M87 central BH and
$M=4.3\times 10^{6}~{}\text{M}_{\odot}$ for the Sgr $\text{A}^{\ast}$ BH.

###  II.2 The CDM halo with NFW profile

The cold dark matter (CDM) halo with a Navarro-Frenk-White (NFW) profile is
obtained by $N$-body simulations, which has a universal spherically averaged
density profile NFW ; Kimet2019

$\displaystyle\rho(r)=\frac{\rho_{0}}{(r/r_{0})(1+r/r_{0})^{2}},$ (2.4)

where $\rho_{0}$ is the density of the Universe at the moment when the halo
collapsed and $r_{0}$ is the characteristic radius. According to the
observations on the Milky Way Galaxy dark_matter , the best fit values for the
parameters $\rho_{0}$ and $r_{0}$ for the NFW profile are $\rho_{0}=5.23\times
10^{7}\text{M}_{\odot}/{\rm kpc}^{3}$ and $r_{0}=8.1\;{\rm kpc}$. On the other
hand, for the M87 galaxy, we have $\rho_{0}=0.008\times
10^{7.5}~{}\text{M}_{\odot}/\text{kpc}^{3}$ (see Oldham2016 ) and
${r_{0}}=130~{}\text{kpc}$ Kimet2019 . With this halo profile, the function
$G(r)$ in the metric (II) is given by xu_JCAP

$\displaystyle G(r)=\left(1+\frac{r}{{r_{0}}}\right)^{-\frac{8\pi
G_{N}\rho_{0}r_{0}^{3}}{c^{2}r}}-\frac{2G_{N}M}{c^{2}r}.$ (2.5)

Here $M=4.3\times 10^{6}~{}\text{M}_{\odot}$ is the mass of Sgr
$\text{A}^{\ast}$ BH, and $M=6.5\times 10^{9}~{}\text{M}_{\odot}$ is the mass
of M87 central BH.

###  II.3 The SFDM model

For the scalar field dark matter (SFDM) model Xian2018 , the energy density
profile for a DM halo is given by

$\displaystyle\rho(r)=\frac{\rho_{0}\sin(\pi r/r_{0})}{\pi r/r_{0}},$ (2.6)

where $\rho_{0}$ is the central density and $r_{0}$ is the radius at which the
pressure and density are zero. In the Milky Way Galaxy, we have
$\rho_{0}=3.43\times 10^{7}\text{M}_{\odot}/{\rm kpc}^{3}$ and
$r_{0}=15.7\;{\rm kpc}$ Xian2018 . With this halo profile, the function $G(r)$
in the metric (II) is given by

$\displaystyle G(r)=\exp\left[-\frac{8G_{N}\rho_{0}R^{2}}{\pi}\frac{\sin(\pi
r/r_{0})}{\pi r/r_{0}}\right]-\frac{2G_{N}M}{c^{2}r}.$

Here $M=4.3\times 10^{6}~{}\text{M}_{\odot}$ is the mass of Sgr
$\text{A}^{\ast}$ BH.

##  III Regge-Wheeler-like equation for axial metric perturbations

In this section, we consider the linear gravitational perturbations
$h_{\mu\nu}$ around Schwarzshild-like solutions. Let us first start with a
general form of a spherically symmetric spacetime, given by Cruz2019

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-G(r)dt^{2}+F^{-1}(r)dr^{2}+H(r)d\Omega^{2},$ (3.1)

where

$\displaystyle d\Omega^{2}$ $\displaystyle=$ $\displaystyle
d\theta^{2}+\sin^{2}\theta d\phi^{2}.$ (3.2)

Of course, for our case [cf. (II)], we have $G=F$ and $H=r^{2}$. For metric
perturbations, let us start by describing the geometry of a linearly perturbed
spherically symmetric background $\bar{g}_{\mu\nu}$

$\displaystyle{g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu},}$ (3.3)

where

$\displaystyle\bar{g}_{\mu\nu}={\rm
diag}\left(-G(r),\frac{1}{F(r)},H(r),H(r)\sin^{2}\theta\right),~{}~{}$ (3.4)

where $h_{\mu\nu}$ denotes the linear perturbations of the background metric
$\bar{g}_{\mu\nu}$. In general, the perturbation $h_{\mu\nu}$ can be split
into pieces that transform as scalars, vectors, and tensors with respect to
the symmetry of the spacetime. However, in two-dimensional maximally symmetric
space $S^{2}$, it can be shown that the tensor perturbations with transverse-
traceless conditions are identically zero cai_generalized_2013 ;
takahashi_hawking_2010 ; takahashi_master_2010 . Thus, the metric
perturbations can be split as scalar and vector perturbations, i.e.,
$h_{\mu\nu}=h^{\rm S}_{\mu\nu}+h^{\rm V}_{\mu\nu}$. Here, we note that the
scalar perturbation is also called a polar-type perturbation (or even-parity
perturbation), while the vector one is called an axial perturbation (or odd-
parity perturbation) Berti2009 . In this paper, for simplicity, we only focus
on the axial perturbations of the Schwarzshild-like solutions with different
dark matter halos.

We parametrize the axial perturbations in the form of Thomp2017

$\displaystyle h_{\mu\nu}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\begin{pmatrix}0&0&-C_{lm}\csc\theta\partial_{\varphi}&C_{lm}\sin\theta\partial_{\theta}\\\
0&0&-J_{lm}\csc\theta\partial_{\varphi}&J_{lm}\sin\theta\partial_{\theta}\\\
sym&sym&-W_{lm}\csc\theta\big{(}\cot\theta\partial_{\varphi}-\partial_{\theta}\partial_{\varphi}\big{)}&sym\\\
sym&sym&\frac{1}{2}W_{lm}\big{(}\csc\theta\partial^{2}_{\varphi}+\cos\theta\partial_{\theta}-\sin\theta\partial^{2}_{\theta}\big{)}&W_{lm}\big{(}\cos\theta\partial_{\theta}-\sin\theta\partial_{\theta}\partial_{\varphi}\big{)}\end{pmatrix}Y_{lm}(\theta,\varphi)\epsilon,$

where $C_{lm}$, $J_{lm}$, and $W_{lm}$ are functions of $t$ and $r$. The
$Y_{lm}(\theta,\varphi)$ stands for the spherical harmonics Zettilib , and the
$l$ as well as $m$ in the index are integers. Here, $\epsilon$ is a real
number, and $|\epsilon|\ll 1$.

From now on, we set $m=0$ in (III) so that
$\partial_{\varphi}Y_{lm}(\theta,\varphi)=0$. As of now, the background has
the spherical symmetry, and the corresponding linear perturbations do not
depend on $m$ Regge57 ; Thomp2017 . In addition, by adopting the RW gauge
Thomp2017 , we set $W_{lm}=0$.

The Einstein equations are given by

$\displaystyle E_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}={8\pi
G_{N}T_{\mu\nu},}$ (3.6)

where $R_{\mu\nu}$ and $R$ are the Ricci tensor and Ricci scalar, respectively
CarrollB , and $T_{\mu\nu}$ denotes the energy-momentum tensor due to the
presence of DM. For simplicity, here, we ignore the perturbation of the DM
since its effects are expected to be negligible in comparing to the effects of
DM from the modified background geometry. Then, expanding $E_{23}$ and
$E_{13}$ to leading order of $\epsilon$, we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\frac{{\dot{C}}}{G}+\frac{\left(GF^{\prime}+FG^{\prime}\right)}{2G}J+FJ^{\prime},$
(3.7) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{H^{\prime}}{H}{\dot{C}}-{\dot{C}}^{\prime}+{\ddot{J}}$
(3.8)
$\displaystyle+\frac{1}{2}\left[F^{\prime}G^{\prime}+\frac{G\left\\{H\left[F^{\prime}H^{\prime}+2\left(FH^{\prime\prime}+l^{2}+l-2\right)\right]-F\left(H^{\prime}\right)^{2}\right\\}}{H^{2}}-\frac{F\left(G^{\prime}\right)^{2}}{G}+F\left(2G^{\prime\prime}+\frac{G^{\prime}H^{\prime}}{H}\right)\right]J,$

where a dot stands for the time derivative, while a prime stands for the
derivative with respects to $r$. Notice that we have dropped the $lm$ in the
subscripts of $C$ and $J$ for simplicity. After that, by combining (3.7) and
(3.8), we obtain the master equation

$\displaystyle\frac{d^{2}\Psi(t,x)}{dx^{2}}-\left[\frac{d^{2}}{dt^{2}}+V_{\rm
eff}(r)\right]\Psi(t,x)$ $\displaystyle=$ $\displaystyle 0,$ (3.9)

where

$\displaystyle\Psi$ $\displaystyle\equiv$
$\displaystyle\left(\frac{H}{GF}\right)^{-1/2}J,$ (3.10)
$\displaystyle\frac{dr}{dx}$ $\displaystyle=$ $\displaystyle\sqrt{FG},$ (3.11)

and the effective potential is given by

$\displaystyle V_{\rm eff}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4}\left[2F^{\prime}G^{\prime}+\frac{G\left\\{H\left[F^{\prime}H^{\prime}+2FH^{\prime\prime}+4\left(l^{2}+l-2\right)\right]+F\left(H^{\prime}\right)^{2}\right\\}}{H^{2}}-\frac{2F\left(G^{\prime}\right)^{2}}{G}+F\left(4G^{\prime\prime}+\frac{G^{\prime}H^{\prime}}{H}\right)\right].~{}~{}~{}~{}~{}$
(3.12)

By assuming $\Psi=e^{-i\omega t}\Psi(x)$, Eq. (3.9) could be written as

$\displaystyle\frac{d^{2}\Psi(x)}{dx^{2}}+\left[\omega^{2}-V_{\rm
eff}(r)\right]\Psi(x)$ $\displaystyle=$ $\displaystyle 0.$ (3.13)

Table 1: Summary of the cases that we consider for the calculations of QNMs. Case |  | Galaxy |  | $G(r)$ |  | $\rho_{0}~{}\left(M_{\odot}/{\rm kpc}^{3}\right)$ |  | $r_{0}~{}({\rm kpc})$
---|---|---|---|---|---|---|---|---
Schwarzschild |  | N/A |  | $1-\frac{2M}{r}$ |  | $0$ |  | N/A
Case 1 |  | M87 |  | (II.1) |  | $6.9\times 10^{6}$ |  | 91.2
|  | Milky Way |  | (II.1) |  | $5.2\times 10^{7}$ |  | 7.8
Case 2 |  | M87 |  | (2.5) |  | $0.008\times 10^{7.5}$ |  | 130
|  | Milky Way |  | (2.5) |  | $5.23\times 10^{7}$ |  | 8.1
Case 3 |  | Milky Way |  | (II.3) |  | $3.43\times 10^{7}$ |  | 15.7

Following Sec. II, we list the cases that we consider for the calculations of
QNMs in Table 1 and provide some basic information for each case. They are
referred to as case 1, 2 and 3, respectively. Note that the master equations
and QNMs for case 2 and case 3 have also been studied in Liu:2021xfb  with
relatively larger values of $r_{0}$ and $\rho_{0}$. The Schwarzschild case is
also shown.

Figure 1:  Behaviors of $V_{\rm eff}$ for different cases listed in Table 1,
in which the data for the Milky Way is selected for plotting. In addition, we
have chosen $l=2$. Note that there is an inserted figure showing the
amplification of the region around the stationary points of these curves. Also
note that here we are using the unit system so that $c=G_{N}={r_{\rm MH}}=1$.

To find the difference between different cases, the $V_{\rm eff}$’s for each
case listed in Table 1 are plotted in Fig. 1. To make them at the same
starting line, the data for the Milky Way are selected for plotting. In
addition, as an example, we have set $l=2$ for this plot. Notice that, as
noted earlier, here we are using the unit system so that $c=G_{N}={r_{\rm
MH}}=1$. From Fig. 1, it is very clear that the deviation on $V_{\rm eff}$’s
between each two cases in Table 1 is quite small since these curves are almost
overlapped. Actually, it is because of this, that an amplification of the
region around the stationary points of these curves of $V_{\rm eff}$’s is
inserted, so that readers can find more details. That implies we may obtain
quite similar QNMs from these cases in the end. As we see, this is indeed the
case.

##  IV QNMs of Schwarzshild-like black hole in DM halos

With the master equation given by (3.13), we are ready to solve for the
corresponding $\omega$, viz., the QNM, for a specific set of $\\{G,F,H\\}$,
including the cases listed in Table 1 (as well as their extensions by varying
$\rho_{0}$ or $r_{0}$). Note that the QNMs for cases 2 and 3 have also been
considered in Liu:2021xfb  with larger values of $\rho_{0}$ and $\rho_{0}$.
Notice that, $\omega$, in general, is a complex number, often written as
$\omega_{lmn}$ Berti2009 , where $l$ and $m$ are inherited from spherical
harmonics while $n$ is the overtone index. However, since we have set $m=0$,
it is left with two indices only, i.e., $l$ and $n$ (For simplicity, before
stimulating any confusions, we write $\omega_{ln}$ as $\omega$).

Once again, for all the cases mentioned above, we adopt the unit system so
that $c=G_{N}=r_{\rm MH}=1$. Here, the radius of the MH is given by $r_{\rm
MH}=(2G_{N}M)/c^{2}$ CarrollB . In this way, the units for mass, time, and
length are totally fixed.

###  IV.1 Calculations of QNMs for the cases in Table 1

So far, we have obtained the desired master equation [cf. (3.13)], and we have
several background metrics (see Table 1). In addition, we also know the two
boundary conditions, namely, the pure in-going wave at the MH and pure out-
going wave at the spatial infinity Chandra1975 . With everything in hand, and
given a set of $\\{l,n\\}$, we can find the corresponding $\omega_{ln}$, in
principle.

QNMs in GR with the Schwarzschild case have been studied extensively. In this
procedure, several different techniques of calculations were developed, for
instance, the Wentzel-Kramers-Brillouin (WKB) approach Will1985 ; Will1987 ;
Konoplya2003 ; Jerzy2017 , the continued fraction method Leaver1985 , shooting
method Chandra1975 , etc.Kono2011 ; Gund1994 ; Bin2004 . After scrutinizing
these methods, we noticed that the WKB method is probably the most convenient
way for our current problem. Therefore, we use the sixth-order WKB method to
carry out our calculations.

The formula of $\omega$ from the sixth-order WKB method is given by

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\sqrt{-i\left[\left(n+\frac{1}{2}\right)+\sum_{k=2}^{6}\Lambda_{k}\right]\sqrt{-2V_{0}^{\prime\prime}}+V_{0}},~{}~{}~{}~{}$
(4.1)

where

$\displaystyle V_{0}\equiv\left.V_{\rm eff}\right|_{r=r_{\text{max}}},\quad
V_{0}^{\prime\prime}\equiv\left.\frac{dV_{\rm
eff}}{dr^{2}}\right|_{r=r_{\text{max}}},$ (4.2)

with $V_{\rm eff}(r=r_{\text{max}})$ giving the maximum of $V_{\rm eff}$ on
$r\in(1,~{}\infty)$. The expressions of $\Lambda_{k}$’s can be found in
Will1985 ; Will1987 ; Konoplya2003 . Note that $n=0,1,2,...$.

Table 2: The QNMs $\omega$’s that calculated from the sixth-order WKB method for BHs with DM halos by adopting the background and factors given in case 1 of Table 1. Note that each $\omega$ here is compared to its Schwarzschild counterpart. $l$ |  | $n$ |  | M87 |  | Milky Way |  |  $\omega$’s of BHs in Schwarzschild case
---|---|---|---|---|---|---|---|---
2 |  | 0 |  | $0.74718-0.17776i$ |  | $0.74724-0.17778i$ |  | $0.74724-0.17778i$
|  | 1 |  | $0.69254-0.54690i$ |  | $0.69259-0.54696i$ |  | $0.69259-0.54696i$
|  | 2 |  | $0.59700-0.95502i$ |  | $0.59704-0.95511i$ |  | $0.59704-0.95512i$
3 |  | 0 |  | $1.19879-0.18539i$ |  | $1.19888-0.18540i$ |  | $1.19889-0.18541i$
|  | 1 |  | $1.16519-0.56252i$ |  | $1.16528-0.56258i$ |  | $1.16528-0.56258i$
|  | 2 |  | $1.10310-0.95799i$ |  | $1.10318-0.95809i$ |  | $1.10319-0.95809i$
4 |  | 0 |  | $1.61823-0.18831i$ |  | $1.61835-0.18833i$ |  | $1.61836-0.18833i$
|  | 1 |  | $1.59313-0.56861i$ |  | $1.59326-0.56866i$ |  | $1.59326-0.56867i$
|  | 2 |  | $1.54527-0.95970i$ |  | $1.54538-0.95979i$ |  | $1.54539-0.95980i$
Table 3: The QNMs $\omega$’s that calculated from the sixth-order WKB method for BHs with DM halos by adopting the background and factors given in case 2 of Table 1. Note that each $\omega$ here is compared to its Schwarzschild counterpart. $l$ |  | $n$ |  | M87 |  | Milky Way |  |  $\omega$’s of BHs in Schwarzschild case
---|---|---|---|---|---|---|---|---
2 |  | 0 |  | $0.74723-0.17778i$ |  | $0.74723-0.17778i$ |  | $0.74724-0.17778i$
|  | 1 |  | $0.69259-0.54695i$ |  | $0.69259-0.54696i$ |  | $0.69259-0.54696i$
|  | 2 |  | $0.59704-0.95511i$ |  | $0.59704-0.95511i$ |  | $0.59704-0.95512i$
3 |  | 0 |  | $1.19888-0.18540i$ |  | $1.19888-0.18540i$ |  | $1.19889-0.18541i$
|  | 1 |  | $1.16528-0.56258i$ |  | $1.16528-0.56258i$ |  | $1.16528-0.56258i$
|  | 2 |  | $1.10318-0.95808i$ |  | $1.10318-0.95809i$ |  | $1.10319-0.95809i$
4 |  | 0 |  | $1.61834-0.18833i$ |  | $1.61835-0.18833i$ |  | $1.61836-0.18833i$
|  | 1 |  | $1.59325-0.56866i$ |  | $1.59325-0.56866i$ |  | $1.59326-0.56867i$
|  | 2 |  | $1.54538-0.95979i$ |  | $1.54538-0.95979i$ |  | $1.54539-0.95980i$
Table 4: The QNMs $\omega$’s that calculated from the sixth-order WKB method for BHs with DM halos by adopting the background and factors given in case 3 of Table 1. Note that each $\omega$ here is compared to its Schwarzschild counterpart. $l$ |  | $n$ |  | M87 |  | Milky Way |  |  $\omega$’s of BHs in Schwarzschild case
---|---|---|---|---|---|---|---|---
2 |  | 0 |  | N/A |  | $0.74724-0.17778i$ |  | $0.74724-0.17778i$
|  | 1 |  | N/A |  | $0.69259-0.54696i$ |  | $0.69259-0.54696i$
|  | 2 |  | N/A |  | $0.59704-0.95512i$ |  | $0.59704-0.95512i$
3 |  | 0 |  | N/A |  | $1.19888-0.18540i$ |  | $1.19889-0.18541i$
|  | 1 |  | N/A |  | $1.16528-0.56258i$ |  | $1.16528-0.56258i$
|  | 2 |  | N/A |  | $1.10319-0.95809i$ |  | $1.10319-0.95809i$
4 |  | 0 |  | N/A |  | $1.61835-0.18833i$ |  | $1.61836-0.18833i$
|  | 1 |  | N/A |  | $1.59326-0.56867i$ |  | $1.59326-0.56867i$
|  | 2 |  | N/A |  | $1.54539-0.95980i$ |  | $1.54539-0.95980i$

In this subsection, we adopt the choices given by Table 1 to carry out the
calculations. The results of $\omega$’s for this part are exhibited in Tables
2 \- 4 for case 1, 2 and 3, respectively. Note that in these tables, the
results for BHs with DM halos are compared with their Schwarzschild
counterparts. It should be mentioned here that during the calculations for
case 1, since (II.1) is quite complicated and will arise technical
difficulties when performing derivatives, the $V_{\rm eff}$ for case 1 is
expanded before processing to (4.1). In other words, we use the polynomial
form of $V_{\rm eff}$ in case 1. To be more specific, we expand it as

$\displaystyle{V_{\rm eff}}$ $\displaystyle=$
$\displaystyle\sum^{99}_{k=0}\beta_{k}\left(r-\frac{79}{50}\right)^{k}.$ (4.3)

In this way, we have $\left|\left.{V_{\rm
eff}}\right|_{\text{original}}-\left.{V_{\rm
eff}}\right|_{\text{polynomial}}\right|\lesssim{\cal{O}}(10^{-19})$ on
$r\in(r_{\rm MH},~{}r_{\text{max}})$ for $l=2,~{}3,~{}4$, and thus, the
accuracy of QNMs is guaranteed.

By looking at Tables 2 \- 4, we immediately notice that the deviations between
the Schwarzschild and non-Schwarzschild cases occur at the fourth digit or
after that. For most of the $\omega$’s in Tables 2 \- 4, these deviations are
very small, just like we anticipated earlier. Considering the fact that our
calculations contain numerical errors, these deviations are quite negligible.
Nonetheless, we can also observe a relatively large deviation on, e.g.,
$\omega_{40}$ of case 1 for the M87 galaxy, which could probably meet the
designed resolution of the TianQin detector (cf. Table II of Shi2019 ). That
means, according to the current results, we may be able to obtain more
constraints for the URC profile by using the observational data from TianQin-
like detectors in the future.

###  IV.2 Tests of the impacts of $\rho_{0}$ and $r_{0}$ on QNMs

In the above subsection, we have considered the impacts of DM halos on QNMs by
adopting the halo parameters $\rho_{0}$ and $r_{0}$ given in Table. 1. It is
worth mentioning here that these parameters are, in general, derived by
fitting the corresponding density profiles with the observational data of the
rotation curves in different galaxies, see dark_matter  for examples. Thus,
these profiles roughly reflect DM distributions for the whole galaxy. They
tend to be accurate in describing the regions far away from the central BH. In
contrast, to the contexts of the matter environment around the central BH, the
halo parameters $\rho_{0}$ and $r_{0}$ are basically free. Moreover, in the
central region around a BH, the baryonic component also makes significant
contributions to the halo parameters.

In addition, the values of $\rho_{0}$ and $r_{0}$ also change from galaxy to
galaxy. In the Milky Way, $r_{0}\sim 10\;{\rm kpc}$ and $\rho_{0}\sim
10^{7}M_{\odot}/{\rm kpc}^{3}$, as presented in Table. 1. Depending on
specific galaxies, it is shown that in the catalog of dark matter halo models
for galaxies in the Spitzer Photometry and Accurate Rotation Curves (SPARC)
database $r_{0}$ can be as large as $\sim 700\;{\rm kpc}$, and $\rho_{0}$ can
be as large as $10^{10}M_{\odot}/{\rm kpc}^{3}$ for a NFW profile Li:2020iib .
In the SPARC, 175 galaxies are included. Thus, it is natural to expect that
the values of $\rho_{0}$ or $r_{0}$ in some galaxies in the Universe could be
even larger than those given in SPARC.

For these three reasons, it is worth investigating how the frequencies of QNMs
shift with freely changing $\rho_{0}$ and $r_{0}$ in different profiles. We do
the calculations based on Table 1. Nevertheless, this time we try different
values of $\rho_{0}$ and $r_{0}$ in addition to those appearing in Table 1. Of
course, the procedures of calculating QNMs for this part are quite similar to
those in the last subsection.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 2: Results of $\omega_{20}$, $\omega_{30}$, and $\omega_{40}$ by
changing the values of $\rho_{0}$ as well as $r_{0}$ in case 1 (cf. Table 1).
Panels (a) and (c) show $\text{Re}(\omega)$, while panels (b) and (d) show
$\text{Im}(\omega)$. Note that in panels (a) and (b) $r_{0}$ is fixed to be
7.8kpc, while in panels (c) and (d) $\rho_{0}$ is fixed to be $5.2\times
10^{7}M_{\odot}/{\rm kpc}^{3}$. Also note that for the above $\omega$’s we are
adopting the unit system so that $c=G_{N}={r_{\rm MH}}=1$.

First of all, in Fig. 2, we plot the results of $\omega_{20}$, $\omega_{30}$,
and $\omega_{40}$ by changing the values of $\rho_{0}$ as well as $r_{0}$ in
case 1 (cf. Table 1). In panels (a) and (b), the $r_{0}$ is fixed to be 7.8
kpc, while in panels (c) and (d) the $\rho_{0}$ is fixed to be $5.2\times
10^{7}M_{\odot}/{\rm kpc}^{3}$. From panels (a) and (b), we observe that both
$\text{Re}(\omega)$ and $\text{Im}(\omega)$ are very close to their limits of
the Schwarzschild case (cf., e.g,, Table 2) at the beginning when
$\rho_{0}\approx 10^{6}M_{\odot}/{\rm kpc}^{3}$. However, these $\omega$’s
will soon deviate from the Schwarzschild case when $\rho_{0}$ is approaching
$\rho_{0}\approx 10^{12}M_{\odot}/{\rm kpc}^{3}$. Similarly, from panel (c)
and (d), we observe that both $\text{Re}(\omega)$ and $\text{Im}(\omega)$ are
very close to their limits of the Schwarzschild case at the beginning when
$r_{0}\approx 1{\rm kpc}$. However, these $\omega$’s will soon deviate from
the Schwarzschild case when $r_{0}$ is approaching $r_{0}\approx 2500{\rm
kpc}$.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 3:  Results of $\omega_{20}$, $\omega_{30}$, and $\omega_{40}$ by changing the values of $\rho_{0}$ as well as $r_{0}$ in case 2 (cf. Table 1). Panels (a) and (c) show $\text{Re}(\omega)$, while panels (b) and (d) show $\text{Im}(\omega)$. Note that in panels (a) and (b) $r_{0}$ is fixed to be $8.1\text{kpc}$, while in panels (c) and (d) $\rho_{0}$ is fixed to be $5.23\times 10^{7}M_{\odot}/{\rm kpc}^{3}$. Also note that for the above $\omega$’s we are adopting the unit system so that $c=G_{N}={r_{\rm MH}}=1$. |
---|---
(a) | (b)
|
(c) | (d)
Figure 4:  Results of $\omega_{20}$, $\omega_{30}$ and $\omega_{40}$ by
changing the values of $\rho_{0}$ as well as $r_{0}$ in Case 3 (cf. Table 1).
Panel (a) and (c) show $\text{Re}(\omega)$ while Panel (b) and (d) show
$\text{Im}(\omega)$. Note that in Panel (a) and (b) $r_{0}$ is fixed to be
$8.1\text{kpc}$, while in Panel (c) and (d) $\rho_{0}$ is fixed to be
$3.43\times 10^{7}\text{M}_{\odot}/{\rm kpc}^{3}$. Also note that for the
above $\omega$’s we are adopting the unit system so that $c=G_{N}={r_{\rm
MH}}=1$.

After that, by mimicking Fig. 2, we plot the results of $\omega_{20}$,
$\omega_{30}$, and $\omega_{40}$ by changing the values of $\rho_{0}$ as well
as $r_{0}$ for case 2 and 3 (cf. Table 1) in Figs. 3 and 4, respectively.
Basically, we could see quite similar behaviors of $\omega$’s in these two
figures, just like what we have seen in Fig. 2. Therefore, we do not carry out
further analysis to these cases.

From Figs. 2, 3, and 4, it is easy to observe that the QNM frequency is
changing as $\rho_{0}$ or $r_{0}$ increasing. When $r_{0}$ is fixed, the real
part of the QNM frequency is decreasing, while the imaginary part is
increasing as $\rho_{0}$ increases. Similar behaviors also appear when one
fixes $\rho_{0}$ and increases $r_{0}$. When $\rho_{0}$ and $r_{0}$ become
larger, the density of DM becomes bigger accordingly. These features indicate
that the real part of QNM frequency becomes lower, while the imaginary part
becomes larger for a denser DM distribution around the BH.

In order to see how the above shifts on QNM frequencies affect the GWs from
the ringdown stages of coalescences, we need to construct the corresponding GW
waveform. The GW emitted during the ringdown stage can be expressed as a
linear combination of damped sinusoids

$\displaystyle h_{+}+ih_{\times}=\frac{M_{z}}{D_{\rm L}}\sum_{lmn}{\cal
A}_{lmn}{\color[rgb]{0,0,0}e^{i(f_{lmn}t+\phi_{lmn})}e^{-t/\tau_{lmn}}}S_{lmn},$

where $M_{z}$ is the redshifted mass of the black hole, $D_{\rm L}$ the
luminosity distance to the source, ${\cal A}_{lmn}$ the mode amplitude,
$\phi_{lmn}$ the phase coefficient, and $S_{lmn}$ the (complex) spin-weighted
spheroidal harmonics of spin weight 2, which depend on the polar and azimuthal
angles. The GW frequency $f_{lmn}=\text{Re}(\omega_{lmn})$ is the real part of
the QNM frequency for $(l,m,n)$ mode, while the damping time $\tau_{lmn}$ is
related to the imaginary part of the QNM frequency via
$\tau_{lmn}=-1/\text{Im}(\omega_{lmn})$. From the above waveform, it is
evident that a denser DM distribution in the central region near a BH leads to
lower GW frequency and longer damping time for GWs during the ringdown stage.
Considering the designed resolution of space-based detectors, such as LISA,
TianQin, and Taiji, these effects may be detectable. This may provide an
approach to probe the matter distribution in the central region of a galaxy.

##  V Conclusion and Outlooks

In this paper, we study the calculations of QNMs for spherically symmetric BHs
[cf. (II)] with DM halos. We are caring about how DM halos influence the QNMs
and thus the GWs from a central BH located in a galaxy. Several different DM
profiles are investigated for the M87 galaxy as well as the Milky Way Galaxy,
which are referred as case 1, 2, and 3, respectively (cf. Table 1). In
addition, we have focused on the axial perturbation on the background
spacetimes, which are parametrized by (III).

The backgrounds we consider are described by Eqs. (II.1), (2.5), and (II.3)
for these three different cases. From their expressions we observe that the
effects of DM halos are determined by two factors, viz., $r_{0}$ and
$\rho_{0}$. By using Einstein’s field equations [cf. (3.6)] with a general
spherically symmetric background metric [cf. (3.1)], a Regge-Wheeler-like
master equation is obtained [cf. (3.13)], for which the RW gauge is adopted
Thomp2017 . With this in hand, and provided a specific set of $\\{l,n\\}$
(recall that we have set $m=0$), we are able to calculate the corresponding
QNMs.

According to the experience with GR, there are many different methods for the
calculations of QNMs. Nonetheless, in practice, we noticed that the most
convenient way for our problem is the WKB method. Therefore, the sixth-order
WKB method [cf. (4.1)] is used here for calculating QNMs. The results of
$\omega$’s are exhibited in Tables 2 \- 4 for case 1, 2, and 3 (cf. Table 1),
respectively. Note that, in there we have selected the unit system so that
$c=G_{N}=r_{\rm MH}=1$.

For most of the $\omega$’s in Tables 2 \- 4, we notice that the deviations
between the Schwarzschild and non-Schwarzschild cases are quite small.
Considering the fact that our calculations contain numerical errors, these
deviations are quite negligible. Nonetheless, we can also observe a relatively
large deviation, which occurs at the fourth digit, e.g., $\omega_{40}$ of case
1 for the M87 galaxy. It is worth mentioning here that such a discrepancy
could be within the designed resolution of space-based GW detectors, such as
LISA, TianQin, and Taiji Shi2019 . Once they are in operation in the future,
one may obtain more constraints for the URC profile by matching the
observational data with the results of QNMs in here. More importantly, this
result implies the possibility of confining DM profiles with GW observations
in the future.

Since the values of $\rho_{0}$ and $r_{0}$ may change from galaxy to galaxy,
we also investigate the impacts of them on QNMs. Indeed, according to the
current observations to the Milky Way Galaxy and M87 galaxy, the resultant
$\rho_{0}$ and $r_{0}$ will lead to quite negligible deviations from the
Schwarzschild case (cf., e.g., Fig. 1). Nonetheless, it is hard to tell what
kind of parameters we will obtain for other galaxies in the Universe. Thus, it
is worth checking what will happen when $\rho_{0}$ and $r_{0}$ are changing
freely. More importantly, in the context of constraints on BH environments,
these parameters are basically free Cardoso:2021wlq . It is not necessary to
assume that they will preserve similar magnitudes in all the occasions. A more
reasonable way is to consider $\rho_{0}$ and $r_{0}$ on a wider range. This
fact stimulates our interests on how these parameters will influence the QNMs
when they are changing freely.

Basically, we study the influence of one of the two parameters $\rho_{0}$ and
$r_{0}$ on QNMs by fixing the other. The corresponding results are shown in
Figs. 2-4. From these figures, it is very clear that the resultant $\omega$’s
will approach their limits in the Schwarzschild case when $\rho_{0}$ and
$r_{0}$ are getting smaller. In contrast, we can see significant deviations on
$\omega$’s from the Schwarzschild case for sufficiently large $\rho_{0}$ and
$r_{0}$. In addition, we observe that these deviations have been large enough
so that we cannot solely attribute them to numerical errors. For an arbitrary
galaxy in the Universe, it is possible to have large enough $\rho_{0}$ and
$r_{0}$ since they are essentially known to be free parameters. Therefore,
according to the results here, a big deviation on QNMs between the
Schwarzschild and non-Schwarzschild cases will occur. It is shown that the
real part of QNM frequencies becomes smaller while the imaginary part becomes
larger for denser DM distribution in the central region around  a BH. This
implies that the larger dark matter density in the central region near a BH
leads to a lower GW frequency and longer damping time for GWs during the
ringdown stage. Finally, by considering the designed resolution of LISA-like
detectors, one may expect such a large deviation to be found in reality once a
galaxy with suitable $\rho_{0}$ and $r_{0}$ is observed someday. By matching
with the results here, these kinds of observations will either confirm our
current understanding to DM or help us put constraints on the current DM
models.

Our work here can be extended in several directions. First of all, here, we
only consider three different DM profiles. It is interesting to extend the
current work to other DM profiles, for instance, BHs surrounded by superfluid
DM and baryonic matter Kimet2020 . On the other hand, since astrophysical BHs,
in general, have nonzero angular momentum, it is also our plan to extend our
work to rotating BHs. In addition, the even-parity perturbations could also be
investigated. Finally, we may test the effects of DM halos on various modified
theories of gravity.

## ACKNOWLEDGEMENTS

This work is supported by the National Key Research and Development Program of
China Grant No. 2020YFC2201503, the Zhejiang Provincial Natural Science
Foundation of China under Grants No. LR21A050001 and No. LY20A050002, the
National Natural Science Foundation of China under Grants No. 11675143 and No.
11975203, and the Fundamental Research Funds for the Provincial Universities
of Zhejiang in China under Grant No. RF-A2019015.

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