# Real-time Multi-modal Object Detection and Tracking on Edge for Regulatory
Compliance Monitoring

Jia Syuen Lim† Ziwei Wang† Jiajun Liu† Abdelwahed Khamis†
Reza Arablouei† Robert Barlow‡ Ryan McAllister‡ Affiliations: †Data61, CSIRO,
‡Agriculture & Food, CSIRO

###### Abstract

Regulatory compliance auditing across diverse industrial domains requires
heightened quality assurance and traceability. Present manual and intermittent
approaches to such auditing yield significant challenges, potentially leading
to oversights in the monitoring process. To address these issues, we introduce
a real-time, multi-modal sensing system employing 3D time-of-flight and RGB
cameras, coupled with unsupervised learning techniques on edge AI devices.
This enables continuous object tracking thereby enhancing efficiency in
record-keeping and minimizing manual interventions. While we validate the
system in a knife sanitization context within agrifood facilities, emphasizing
its prowess against occlusion and low-light issues with RGB cameras, its
potential spans various industrial monitoring settings.

##  1 Introduction

Regulatory compliance auditing of food processing facilities is a mandatory
requirement in the agrifood export industries. However, the majority of these
audits are currently conducted through manual inspections, which are
intermittent and lack traceability. For instance, crucial hygiene regulations
such as handwashing and knife sanitizing, which are essential for preventing
food-borne illnesses and cross-contamination, are often monitored manually,
leading to potential inconsistencies or oversight. Implementing automated
regulatory compliance checkpoints, along with sensor-driven data collection
and machine learning-based analytics, can offer a robust solution. It can not
only reduce labor costs significantly but also enhance quality assurance by
enabling continuous monitoring and systematic record-keeping.

With the growing computational power of modern edge AI devices, the deployment
of deep neural networks (DNNs) for real-time detection of visual objects has
become more prevalent [3]. However, most existing DNN-based algorithms in
computer vision applications are limited to two-dimensional images and lack
the ability to incorporate geometrical knowledge or fine details. Their
performance can degrade due to occlusion and low ambient light settings.
Recent years have witnessed remarkable advancements in compact yet high-
performance 3D time-of-flight (ToF) technology. The increased accessibility of
ToF technology has facilitated the creation of accurate geometrical and
spatial representations of physical objects in complex environments, including
low-light conditions. By integrating information from multiple sensing modes
and leveraging their complementary properties, the drawbacks of perception and
blind spots associated with individual modes can be effectively mitigated.
This integration can lead to improved accuracy and robustness in real-world
applications, such as behavior recognition [4].

Figure 1: The overall architecture of our proposed system.

In this work, we introduce a near-real-time multi-modal sensing system for
object tracking on edge devices across various industrial contexts. The system
integrates a ToF camera with a 3D ToF sensor and an RGB camera. This depth
information enhances discernment between static and dynamic entities. While
our initial evaluation focuses on knife sanitization monitoring, the system’s
broader aim is to address diverse industrial tracking needs. It’s designed to
detect and track objects like knives, reducing the need for manual annotation.
Our prototype employs an unsupervised multi-modal detector to identify object
positions irrespective of positioning or color nuances. We assess its
performance in a lab simulating industrial environments. System specifics are
in section 2, with performance metrics in section 3.

##  2 System Design

The proposed near-real-time detection and tracking system, depicted in Figure
1, comprises two main hardware components: an integrated ToF camera and an
edge AI device. We utilize the Robot Operating System 2 to manage multi-modal
data streams from the ToF camera to the edge AI device. This system setup
allows us to preprocess data and implement our detection and tracking pipeline
in near real-time.

###  2.1 Hardware Components

####  2.1.1 Time-of-flight camera.

We utilize an Intel® RealSense™ ToF camera, which is a compact and lightweight
device equipped with high-resolution ToF sensing technology, which is ideal
for edge deployment. We initialize both its RGB camera and ToF depth sensor to
operate at a frame rate of 30Hz.

####  2.1.2 Edge AI device.

We utilize an NVIDIA® Jetson AGX Orin™ development kit to receive and process
data captured by the ToF camera via USB-C connection.

###  2.2 Software Pipeline

####  2.2.1 Robot Operating System 2 (ROS 2).

We make use of the open-source middleware ROS 2, which provides a range of
services including device control, message-passing, and package management for
robotics software.

####  2.2.2 Preprocessing components.

A series of preprocessing steps are implemented prior to detection modules.

Static environment prescan: Initially, we conduct a comprehensive prescan of
the static environment, ensuring that the knife sterilizer is clear of any
objects, and save this scan as the static background model.

Background subtraction: We apply a coarse-to-fine filtering process to
subtract the static background from the point cloud. First, to minimize the
computational load, we apply a coarse filter to exclude points that fit within
the voxel grid of the background model. Next, we enhance the background
subtraction process by employing a fine filter that utilizes a KDTree search
to recursively isolate points within a specific distance from the background,
assuming that distant points belong to foreground objects.

Accumulation of points: We implement a first-in-first-out (FIFO) queue
mechanism to accumulate point clouds from $k$ successive frames. This approach
allows for the generation of more comprehensive and denser point clouds.

####  2.2.3 3D object detection.

Our 3D detection pipeline can be divided into several components as in the
following.

Random sampling: The uneven distribution of accumulated points can introduce
noise, potentially compromising the performance of our algorithm. To address
this issue, we employ a random subsampling technique to effectively reduce the
unevenness and noise in the point distribution, whilst enhancing the overall
efficiency.

Gaussian kernel density estimation (KDE): Traditional clustering techniques
often encounter difficulties when dealing with closely clumped point clouds,
which is frequently observed in scenarios where knives are densely packed. To
address this, we employ the KDE algorithm to predict the density map of the
point clouds and employ percentile filtering to enhance the separability of
the point clouds.

HDBSCAN [2]: After processing the point cloud by the KDE algorithm, we use the
hierarchical density-based spatial clustering of applications with noise
(HDBSCAN) algorithm to determine the number of knives present.

Simple online and real-time tracking (SORT) [1]: SORT is a real-time algorithm
designed to track multiple objects in a live video stream. We obtain the 3D
cluster centroids from HDBSCAN and project them onto a 2D image plane. The
projected centroids are fed to SORT for subsequent 2D object tracking.

####  2.2.4 Hand detection.

During the knife counting process, occasionally the knife handles may be
occluded by worker’s hands when they come into the scene. We use the MediaPipe
Hand Landmarker tool to detect the presence of hands in live image streams.
Once the presence of any hand is detected, an event notification is dispatched
to the 3D detection node, instructing it to temporarily halt the knife
counting process until no hand is detected.

Figure 2: From left, point clouds of the objects of interest isolated from the
background, the density maps produced by KDE, and clusters identified by
HDBSCAN.

##  3 Experiment and Results

To assess the performance of the proposed system, we conducted an experiment
at a pilot plant within CSIRO, under the approval of the relevant ethics
committee. During the experiment, a ToF camera was set up on a tripod to
capture a top-down view of a knife sterilizer. We designed our experiment to
simulate a range of real-world scenarios. In total, we established five
different scenarios, each representing a different number of knives, from one
to five. For each scenario, we created 50 unique instances, representing
various knife positions within the sterilization bath.

Density-based filtering: The range between the sensor and the object directly
affects the density of the captured point clouds, which in turn provides
valuable information to the KDE algorithm. We observe that density peaks at
the tip of each knife handle as illustrated in Figure 2. Thus, we use
percentile filtering to exclude outliers, primarily the body of the knife
handles. This filtering step further enhances the separability of the point
clouds, enhancing their suitability for unsupervised cluster analysis.

Detection performance: The final detection results, produced by the SORT
algorithm, take the form of bounding boxes. Figure 3 presents a side-by-side
view of a heatmap of the objects of interest alongside the corresponding
detection results. To accurately quantify discrepancies between the system’s
predictions and the ground truth, we employ the mean absolute error (MAE),
root mean square error (RMSE), and normalized RMSE (NRMSE) performance
measures. Corresponding results for varying numbers of objects are
comprehensively presented in Table 1.

No. of objects |  MAE |  RMSE |  NRMSE
---|---|---|---
1 |  0.168 |  0.410 |  0.410
2 |  0.477 |  0.924 |  0.462
3 |  0.655 |  1.263 |  0.421
4 |  0.953 |  1.520 |  0.380
5 |  1.114 |  1.876 |  0.375
Table 1: Detection performance for different numbers of objects.

Figure 3: Example 2D heatmap and detection results.

The evaluation results demonstrate the commendable performance of our system,
even though there is a slight increase in the detection error as the number of
objects increases, which can primarily be attributed to the inherent
challenges associated with point cloud clustering. As the ToF camera scans
more objects, the density of the point cloud for each individual object may
decrease, posing challenges to the object detection process. However, despite
the complexities involved, the system consistently performs well across
different object counts, highlighting its reliability and effectiveness.

## References

  * [1] Bewley, A., Ge, Z., Ott, L., Ramos, F., and Upcroft, B. Simple online and realtime tracking.  In IEEE ICIP (2016), pp. 3464–3468.
  * [2] McInnes, L., Healy, J., and Astels, S. HDBSCAN: Hierarchical density based clustering.  JOSS (2017).
  * [3] Stäcker, L., Fei, J., Heidenreich, P., Bonarens, F., Rambach, J., Stricker, D., and Stiller, C. Deployment of deep neural networks for object detection on edge ai devices with runtime optimization.  In IEEE ICCVW (2021), pp. 1015–1022.
  * [4] Wang, Z., Liu, J., Arablouei, R., Bishop-Hurley, G., Matthews, M., and Borges, P. Multi-modal sensing for behaviour recognition.  In MobiCom (2022), p. 900–902.


# Linearized gravity and soft graviton theorem in de Sitter spacetime

Pujian Mao and Bochen Zhou

Linearized gravity and soft graviton theorem in de Sitter spacetime

Pujian Mao and Bochen Zhou

Center for Joint Quantum Studies and Department of Physics,

School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China

Abstract. We study the linearized gravity theory in the Newman-Unti gauge in
the near horizon region of the de Sitter spacetime. The linearized Einstein
equation involves the cosmological constant. The near horizon symmetry
consists of near horizon supertranslation and near horizon superrotation. We
compute the near horizon supertranslation charge and find the proper near
horizon fall-off conditions which uncover a soft graviton theorem from the
Ward identity of the near horizon supertranslation.

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###### Contents

  1. 1 Introduction
  2. 2 Near horizon symmetries and charges
    1. 2.1 Near horizon form of the dS spacetime in NU gauge
    2. 2.2 Linearization in dS spacetime
    3. 2.3 Near horizon symmetries in NU gauge
    4. 2.4 Near horizon solution space
    5. 2.5 Near horizon supertranslation charge
  3. 3 Near horizon soft graviton theorem
    1. 3.1 Graviton modes in dS spacetime
    2. 3.2 Soft graviton theorem in coordinate space
    3. 3.3 Soft graviton theorem in momentum space
  4. 4 Conclusion and discussion
  5. A Modification of stress tensor

##  1 Introduction

Black holes are recently shown to carry soft hairs [1] which reveals a much
richer structure that black holes can have. The new degrees of freedom are
labelled by the near horizon symmetries [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32]. Black hole horizons can be considered as inner boundary of the spacetime
which share many common features as the null infinity, see, e.g.,
investigations in [33, 34]. At null infinity, a triangle equivalence was
proven [35] which connects asymptotic symmetry, memory effect, and soft
theorem. Inspired by the null infinity triangle relation, black hole memory
effect was subsequently proposed [36, 37, 38, 26] and it is closely related to
the near horizon symmetry. Another branch following the triangle relation is
the soft theorems relevant to near horizon symmetry which have been
investigated for Schwarzschild black hole [39, 40, 41]. Nevertheless, horizons
are not exclusive to black holes. The expanding universe leads to cosmological
horizons which are locally very similar to the black hole horizon. Soft
theorems of gauge theory in de Sitter (dS) spacetime are obtained from the
near horizon symmetry [42]. The computations in [42] are somewhat similar to
the Schwarzschild case [39, 40]. While, the physical motivation and
consequence are very different. Cosmological horizon is the outer boundary of
inside observer. Because the dS cosmological universe is expanding so fast,
there are events which will never be seen by an observer inside. Considering
the fact that we do live in an expanding universe, the near cosmological
horizon analysis can be intuitively understood as the real world asymptotic
analysis near null infinity. Technically, there is well defined flat limit
from the cosmological solution [43, 44, 45]. In the flat limit, the
cosmological horizon becomes the null infinity. Correspondingly, the soft
theorems derived in dS spacetime do recover the flat spacetime ones [42]. The
aim of the present work is to extend the previous study [42] in dS spacetime
to linearized gravity.

For the linearization of Einstein theory in dS spacetime, the linearized
equations of motion also involve the cosmological constant, see, e.g., [46,
47, 48]. This is the main technical difference than previous studies, e.g.,
linearized gravity in Schwarzschild spacetime [41] or gauge theories in dS
spacetime [42]. Now the cosmological constant arises from both the background
spacetime and the perturbative equations of motion. Hence, it is very
questionable that if the nice structure revealed in [41] or [42] can be
derived for linearized gravity in dS spacetime.

In this work, we apply the Newman-Unti (NU) gauge [49] for the linearized
gravity theory in dS spacetime. In the near cosmological horizon region, we
impose traceless fall-off conditions. We compute the near horizon symmetry
which consists of near horizon supertranslation and near horizon
superrotation. The near horizon solution space is specified and the
supertranslation charge is derived. We find that there is an interesting
reduction of the near horizon solution space which is invariant under
supertranslation and leads to a natural split of the supertranslation charge
into soft and hard parts. We found such configuration for the Schwarzschild
black hole case in previous work [41] and considered that as a coincidence.
Now such configuration also arises for the dS spacetime. This may suggest that
there is a common structure in the near horizon region which is subject to a
soft theorem. A soft graviton theorem in coordinate space is derived from the
Ward identity of the near horizon supertranslation in the reduced solution
space. After transforming the soft graviton theorem into the momentum space,
there is a natural flat limit and it recovers the flat space soft graviton
theorem.

This paper is organized as follows. In the next section, we present the
linearized Einstein equation in dS spacetime, compute the near horizon
symmetry and solution space of the linearized theory. We specify a reduction
of the solution space where the supertranslation charge can split into a soft
piece and a hard piece. In section 3, a soft graviton theorem is derived from
the Ward identity of the near horizon supertranslation charge. The soft
theorem has a desired flat limit in the momentum space. The last section is
devoted to conclusion and discussion. There is one Appendix which presents the
details of the modification of the stress tensor that coupled to the gravity
theory.

##  2 Near horizon symmetries and charges

In this section, we will study the linearized Einstein theory in the dS
spacetime. The NU gauge [49] (see also [50, 51]) will be adapted into the
linearized theory. We will perform the standard near horizon analysis. The
near horizon symmetry, solution space, and surface charge will be computed.

###  2.1 Near horizon form of the dS spacetime in NU gauge

We start from the static patch in dS spacetime, which is part of the full dS
spacetime as shown in Fig. 1.

Figure 1: Penrose diagram of de Sitter spacetime

In static coordinates $(t,r,z,\bar{z})$, the line element of the dS spacetime
is

$\displaystyle\mathrm{d}s^{2}=-F(r)\mathrm{d}t^{2}+F(r)^{-1}\mathrm{d}r^{2}+2\Omega(r)^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}\,,$
(1) $\displaystyle
F(r)=\frac{2r}{\ell}-\frac{r^{2}}{\ell^{2}}\,,\qquad\Omega=\ell-r\,,\qquad\gamma_{z\bar{z}}=\frac{2}{(1+z\bar{z})^{2}}\,,$
(2)

where $\ell$ is the dS radius and it is related to the cosmological constant
by $\Lambda=\frac{3}{\ell^{2}}$. Note that we have set the cosmological
horizon at $r=0$. By introducing the retarded time coordinate
$u=t-\frac{1}{2}\ell\log\left(\frac{r}{2\ell-r}\right)$, the line element can
be written as

$\mathrm{d}s^{2}=\bar{g}_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}=-F(r)\mathrm{d}u^{2}-2\mathrm{d}r\mathrm{d}u+2\Omega(r)^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}\,,$
(3)

which covers only the $\mathcal{H}^{-}$ part of the horizon. A similar
analysis could be performed for $\mathcal{H}^{+}$ simply by a time reverse
transformation of dS spacetime near the bifurcation sphere $B$. In the rest of
this paper, we just focus on the $\mathcal{H}^{-}$ part.

###  2.2 Linearization in dS spacetime

We linearize Einstein theory in dS spacetime (3). The metric expands as
$g_{\mu\nu}=\bar{g}_{\mu\nu}+\kappa h_{\mu\nu}$. The inverse metric is
$g^{\mu\nu}=\bar{g}^{\mu\nu}-\kappa h^{\mu\nu}+{\cal O}(\kappa^{2})$, where
the indices are now raised by $\bar{g}^{\mu\nu}$. Up to the first correction
order ${\cal O}(\kappa)$, the connection is given by

$\Gamma^{\alpha}_{\mu\nu}={\bar{\Gamma}}^{\alpha}_{\mu\nu}+\frac{\kappa}{2}\bar{g}^{\alpha\beta}\left(\bar{\nabla}_{\mu}h_{\nu\beta}+\bar{\nabla}_{\nu}h_{\mu\beta}-\bar{\nabla}_{\beta}h_{\mu\nu}\right).$
(4)

We define

$C^{\alpha}_{\mu\nu}=\bar{g}^{\alpha\beta}\left(\bar{\nabla}_{\mu}h_{\nu\beta}+\bar{\nabla}_{\nu}h_{\mu\beta}-\bar{\nabla}_{\beta}h_{\mu\nu}\right),$
(5)

which is very useful for the computation of the curvature tensor,

$\displaystyle{R_{\mu\nu\alpha}}^{\beta}$
$\displaystyle=\partial_{\nu}\Gamma^{\beta}_{\mu\alpha}-\partial_{\mu}\Gamma^{\beta}_{\nu\alpha}+\Gamma^{\tau}_{\mu\alpha}\Gamma^{\beta}_{\nu\tau}-\Gamma^{\tau}_{\nu\alpha}\Gamma^{\beta}_{\mu\tau}$
$\displaystyle={\bar{R}_{\mu\nu\alpha}}^{\quad\,\,\,\beta}+\frac{\kappa}{2}(\bar{\nabla}_{\nu}C^{\beta}_{\mu\alpha}-\bar{\nabla}_{\mu}C_{\nu\alpha}^{\beta})+{\cal
O}(\kappa^{2})$ (6)
$\displaystyle={\bar{R}_{\mu\nu\alpha}}^{\quad\,\,\,\beta}+\frac{\kappa}{2}\bar{\nabla}_{\nu}\left(\bar{\nabla}_{\mu}h^{\beta}_{\alpha}+\bar{\nabla}_{\alpha}h_{\mu}^{\beta}-\bar{\nabla}^{\beta}h_{\mu\alpha}\right)$
$\displaystyle\hskip
56.9055pt-\frac{\kappa}{2}\bar{\nabla}_{\mu}\left(\bar{\nabla}_{\nu}h^{\beta}_{\alpha}+\bar{\nabla}_{\alpha}h_{\nu}^{\beta}-\bar{\nabla}^{\beta}h_{\nu\alpha}\right)+{\cal
O}(\kappa^{2}).$ (7)

The Ricci tensor is defined from the curvature tensor as

$R_{\mu\alpha}={R_{\mu\nu\alpha}}^{\nu}=\bar{R}_{\mu\alpha}+\frac{\kappa}{2}\bar{\nabla}_{\nu}\left(\bar{\nabla}_{\mu}h^{\nu}_{\alpha}+\bar{\nabla}_{\alpha}h_{\mu}^{\nu}-\bar{\nabla}^{\nu}h_{\mu\alpha}\right)-\frac{\kappa}{2}\bar{\nabla}_{\mu}\bar{\nabla}_{\alpha}h_{\nu}^{\nu}+{\cal
O}(\kappa^{2}).$ (8)

Finally, the Ricci scalar is obtained as

$R=g^{\mu\alpha}R_{\mu\alpha}=\bar{R}-\kappa
h^{\mu\alpha}\bar{R}_{\mu\alpha}+\frac{\kappa}{2}\bar{\nabla}_{\nu}\left(\bar{\nabla}_{\mu}h^{\nu\mu}+\bar{\nabla}^{\mu}h_{\mu}^{\nu}-\bar{\nabla}^{\nu}h_{\mu}^{\mu}\right)-\frac{\kappa}{2}\bar{\nabla}_{\mu}\bar{\nabla}^{\mu}h_{\nu}^{\nu}+{\cal
O}(\kappa^{2}).$ (9)

We consider Einstein equation with a cosmological constant
$\Lambda=\frac{3}{\ell^{2}}$,

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\frac{3}{\ell^{2}}g_{\mu\nu}=T_{\mu\nu},$
(10)

where we use the natural unit $8\pi G_{N}=1$. At the linearized order ${\cal
O}(\kappa)$, Einstein equation is reduced to111The expression of the
linearized Einstein equation seems different from [47]. But they are indeed
the same.

$E_{\mu\nu}\equiv\frac{1}{2}\left[\bar{\nabla}_{\tau}\bar{\nabla}_{\mu}h^{\tau}_{\nu}+\bar{\nabla}_{\tau}\bar{\nabla}_{\nu}h^{\tau}_{\mu}-\bar{\nabla}^{2}h_{\mu\nu}-\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}h\right]\\\
-\frac{1}{2}\bar{g}_{\mu\nu}\left(\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}h^{\alpha\beta}-\bar{\nabla}^{2}h\right)-\frac{3}{\ell^{2}}h_{\mu\nu}+\frac{3}{2\ell^{2}}\bar{g}_{\mu\nu}h=T_{\mu\nu},$
(11)

where we apply the relations from the Einstein equation of the dS spacetime

$\bar{R}_{\mu\nu}=\frac{3}{\ell^{2}}\bar{g}_{\mu\nu},\quad\quad\bar{R}=\frac{12}{\ell^{2}},$
(12)

and $h=\bar{g}^{\mu\nu}h_{\mu\nu}$. The linearized equation (11) is invariant
under the gauge transformation

$h_{\mu\nu}\rightarrow
h_{\mu\nu}+\bar{\nabla}_{\mu}\xi_{\nu}+\bar{\nabla}_{\nu}\xi_{\mu}.$ (13)

To verify the gauge invariance, the following relations of the dS spacetime
are applied,

$(\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}-\bar{\nabla}_{\nu}\bar{\nabla}_{\mu})\xi_{\alpha}={\bar{R}_{\mu\nu\alpha}}^{\quad\,\,\,\beta}\xi_{\beta},\quad\quad\bar{R}_{\mu\nu\alpha\beta}=\frac{1}{\ell^{2}}(\bar{g}_{\mu\alpha}\bar{g}_{\nu\beta}-\bar{g}_{\mu\beta}\bar{g}_{\nu\alpha}).$
(14)

###  2.3 Near horizon symmetries in NU gauge

We will work in the adapted NU gauge for linearized theory [51] where the
following conditions are imposed,

$\displaystyle h_{rr}=h_{rz}=h_{r\bar{z}}=h_{ru}=0\,,$ (15) $\displaystyle
T_{rr}=T_{rz}=T_{r\bar{z}}=T_{ru}=0.$

The radial components of the stress tensor $T_{r\mu}$ are set to zero to adapt
to the gauge conditions of the perturbative metric, which can be done by
introducing an auxiliary conserved symmetric 2-tensor [41] as is detailed in
Appendix A.

The residual gauge transformation that preserves the conditions (15) is
generated by

$\displaystyle\xi_{u}=Z(u,z,\bar{z})+r\partial_{u}f(u,z,\bar{z})-F(r)f(u,z,\bar{z}),$
(16) $\displaystyle\xi_{r}=-f(u,z,\bar{z}),$ (17)
$\displaystyle\xi_{A}=\Omega^{2}Y_{A}(u,z,\bar{z})+\Omega
D_{A}f(u,z,\bar{z}),$ (18)

where $D_{A}$ is the covariant derivative with respect to the unit sphere
$\text{d}s^{2}=\gamma_{AB}\text{d}x^{A}\text{d}x^{B}=\frac{4}{(1+z\bar{z})^{2}}\text{d}z\text{d}\bar{z}$.
The capital Latin indices are raised or lowered by the spherical metric
$\gamma^{AB}$ and $\gamma_{AB}$.

We impose the following near horizon fall-off conditions

$h_{z\bar{z}}={\cal O}(r^{2}).$ (19)

The absence of the leading order in $h_{z\bar{z}}$ is to specify a traceless
propagating mode. We impose a stronger traceless condition from a near horizon
symmetry perspective. The fall-off conditions fix the independent symmetry
parameter $Z$ that generates a translation along $r$ and select a
$u$-independent near horizon supertranslation,

$Z=-\frac{1}{2}D_{A}D^{A}f(u,z,\bar{z})-\frac{\ell}{2}D_{A}Y^{A},\quad\quad
f(u,z,\bar{z})=T(z,\bar{z})+\frac{1}{2}\int\text{d}uD_{A}Y^{A}.$ (20)

Now, $T$ characterizes the near horizon supertranslation and $Y^{A}$ is the
near horizon superrotation. In vector form, it is given by

$\displaystyle\xi^{u}=f,$ (21)
$\displaystyle\xi^{r}=\frac{1}{2}D_{A}D^{A}f+\frac{1}{2}\Omega D_{A}Y^{A},$
(22) $\displaystyle\xi^{A}=Y^{A}+\frac{1}{\Omega}D^{A}f.$ (23)

###  2.4 Near horizon solution space

The organizations of Einstein equation in the Bondi gauge [52, 53] and NU
gauge for the case with a cosmological constant [44, 45] are well known which
much simplify the derivation of a solution space. Such configurations are
inherited by the linearized theory. Suppose that the metric components are
given in series expansion near the horizon as

$\displaystyle h_{AB}=N_{AB}+\sum_{i=1}N^{(i)}_{AB}r^{i},$ (24) $\displaystyle
h_{uA}=U_{A}+\Xi_{A}r+\sum_{i=2}U^{(i)}_{A}r^{i},$ (25) $\displaystyle
h_{uu}=V+\Theta r+\sum_{i=2}V^{(i)}r^{i}.$ (26)

The first class of equations of motion is the radial part which completely
determines the $r$-dependence of the trace of $h_{AB}$, $h_{uA}$, and $h_{uu}$
up to integration constants. In particular, $E_{rr}=0$ leads to

$\partial_{r}^{2}\left(\frac{h_{z\bar{z}}}{\ell-r}\right)=0.$ (27)

The near horizon conditions $h_{z\bar{z}}={\cal O}(r^{2})$ yields that
$h_{z\bar{z}}=0$. Then, $E_{rA}=0$ fix all the coefficients $U^{(i)}_{A}$ for
$i\geq 2$. The two leading orders in $g_{uA}$ are integration constants. Since
the near horizon charges only involve the integration constants, we will not
list the expressions for the higher orders. Next, $E_{ur}=0$ determines the
coefficients $V^{(i)}$ for $i\geq 2$, and

$\Theta=\frac{1}{\ell}V+\frac{1}{2\ell^{3}}D_{A}D_{B}N^{AB}+\frac{1}{\ell^{2}}D_{A}U^{A}-\frac{1}{2\ell}D_{A}\Xi^{A},$
(28)

where the leading order $V$ is an integration constant.

The second class of equations of motion is the standard equation which
includes $E_{zz}=T_{zz}$ and $E_{\bar{z}\bar{z}}=T_{\bar{z}\bar{z}}$. They
determine the time evolution of the traceless part of $N^{(i)}_{AB}$ for
$i\geq 1$. The leading order $N_{AB}$ is completely free which we refer to as
news tensor in the near horizon analysis. Once the first two classes of
equations of motion are satisfied, equation $E_{z\bar{z}}=T_{z\bar{z}}$ is
fulfilled automatically. The other three equations are supplementary. The
Bianchi identity at the linearized order guarantees that the supplementary
equations are satisfied except their leading orders which yield the time
evolution of the integration constants,

$\ell^{2}\partial_{u}\Xi_{A}+2\ell\partial_{u}U_{A}-2U_{A}-\ell^{2}D_{A}\Theta\\\
+D_{B}D_{A}U^{B}-D_{B}D^{B}U_{A}+\partial_{u}D^{B}N_{AB}=2\ell^{2}T_{uA},$
(29)

and

$\ell^{2}\partial_{u}V+D_{A}U^{A}-\frac{\ell}{2}D_{A}D^{A}V+\ell\partial_{u}D_{A}U^{A}=\ell^{3}T_{uu}.$
(30)

It is important to point out that $U_{A}$ is also free. We will show that this
extra freedom is very important for deriving a soft graviton theorem from the
near horizon symmetry.

###  2.5 Near horizon supertranslation charge

For the (linearized) Einstein gravity, the surface charge associated to the
near horizon symmetry is defined as [54, 55]

${\mathcal{Q}}_{\xi}=\int_{B}\,\sqrt{-g}\Big{(}h^{\lambda[u}\nabla_{\lambda}\xi^{r]}-\xi^{\lambda}\nabla^{[u}{h^{r]}}_{\lambda}-\frac{1}{2}h\nabla^{[u}\xi^{r]}+\xi^{[u}\nabla_{\lambda}h^{r]\lambda}-\xi^{[u}\nabla^{r]}h\Big{)}\text{d}z\text{d}\bar{z},$
(31)

where we choose the bifurcation 2-sphere $B$ to evaluate the charge. In this
work, we are mainly interested in the soft theorem associated to the near
horizon supertranslation. Inserting the near horizon solution and the near
horizon supertranslation, one can obtain that

${\mathcal{Q}}_{T}=\int_{B}\,\gamma_{z\bar{z}}T\ell\left(V-\frac{1}{2}D_{A}\Xi^{A}\right).$
(32)

The supertranslation charge can be evaluated on the whole horizon applying the
relations in (29) and (30),

$\displaystyle Q_{T}=$
$\displaystyle\frac{1}{2\ell}\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\partial_{u}\left(D^{A}D^{B}N_{AB}\right)\text{d}^{2}z\text{d}u$
(33) $\displaystyle+\ell\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\left(\ell
T_{uu}^{0}-D^{A}T_{uA}^{0}\right)\text{d}^{2}z\text{d}u$
$\displaystyle+\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}TD^{A}\left(\frac{1}{2}D_{A}V-\frac{\ell}{2}D_{A}\Theta-\frac{2}{\ell}U_{A}\right)\text{d}^{2}z\text{d}u.$

The first line of this charge expression has the desired forms as a soft part
of the charge [56, 57] for computing a soft theorem. The second line is in the
form of a hard piece. We will turn off the third line by hand using the extra
freedom from $U_{A}$ following the treatment in [41], for which we set

$\frac{1}{2}D_{A}V-\frac{\ell}{2}D_{A}\Theta-\frac{2}{\ell}U_{A}=0.$ (34)

The transformation laws of the supertranslation on solutions,

$\displaystyle\delta_{T}U_{A}=-\frac{1}{2}D_{A}D_{B}D^{B}T,$ (35)
$\displaystyle\delta_{T}\Xi_{A}=-\frac{2}{\ell}D_{A}T,$ (36)
$\displaystyle\delta_{T}V=-\frac{1}{\ell}D_{B}D^{B}T,$ (37)
$\displaystyle\delta_{T}\Theta=\frac{1}{\ell^{2}}D_{B}D^{B}T,$ (38)
$\displaystyle\delta_{T}N_{AB}=2\ell D_{A}D_{B}T,$ (39)

guarantee that the extra condition in (34) is preserved by a supertranslation
transformation.

The combination on the left hand side of (34) was first noticed in [41] for
deriving a soft graviton theorem from the near horizon analysis in
Schwarzschild spacetime. We believe this can be a universal structure in the
near horizon analysis. The parameters $V$ and $\Theta$ in (34) represent non-
propagating degree of freedom from the curvature of the spacetime. At null
infinity, they do not contribute to the horizon charge. There is a natural
split of propagating degree of freedom and non-propagating degree of freedom.
But in the near horizon region, the degrees of freedom are mixed. Luckily,
there is the extra degree of freedom from $U_{A}$ which could cancel those
terms. The remarkable thing is that the special choice of $U_{A}$ is
supertranslation invariant. Note that in the null infinity case, one
integration constant is turned off by trivial diffeomorphism [52, 53, 58] in
the metric component $g_{uA}$ which corresponds to the special configuration
in (34) in the near horizon case.

Finally, we split the near horizon supertranslation charge into

$\displaystyle Q_{T}=Q^{S}+Q^{H},$ (40)

where

$\displaystyle Q^{S}$
$\displaystyle=\frac{1}{2\ell}\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\partial_{u}\left(D^{A}D^{B}N_{AB}\right)\text{d}^{2}z\text{d}u,$
(41) $\displaystyle Q^{H}$
$\displaystyle=\ell\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\left(\ell
T_{uu}^{0}-D^{A}T_{uA}^{0}\right)\text{d}^{2}z\text{d}u,$ (42)

are the soft and hard charges respectively. In the null infinity analysis [56,
57], it is normally assumed that the long-range magnetic mass aspect vanishes.
Adapted to the near horizon case, this assumption is equivalent to impose

$\displaystyle\left[D_{z}D_{z}N_{\bar{z}\bar{z}}-D_{\bar{z}}D_{\bar{z}}N_{zz}\right]_{\mathcal{H}^{-}_{\pm}}=0.$
(43)

Then we can rewrite the soft charge as

$\displaystyle
Q^{S}=-\frac{1}{\ell}\int_{\mathcal{H}^{-}}\partial_{\bar{z}}T\partial_{u}\left(\frac{\partial_{\bar{z}}N_{zz}}{\gamma_{z\bar{z}}}\right)\text{d}^{2}z\text{d}u.$
(44)

For the hard part of the charge, it consists only the contribution from the
stress tensor of the coupled matter fields. We have introduced an auxiliary
field to modify the stress tensor which is equivalent to change the way of
coupling the matter fields to gravity. As was shown in [41], the soft theorem
derived from the Ward identity of the near horizon supertranslation charge
will be dependent on the way of the matter fields couplings. The choice in
[41] is subject to a similar expression of the soft factor as the null
infinity case. Here, we will follow the same treatment to further use the
freedom in the auxiliary fields to turn off $T_{uA}^{0}$. The details of the
modification is presented in Appendix A. Then, the hard charge becomes

$Q^{H}=\ell^{2}\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\,T_{uu}^{0}\text{d}^{2}z\text{d}u\,.$
(45)

##  3 Near horizon soft graviton theorem

In this section, we will demonstrate that the soft charge creates a low-energy
soft graviton in near horizon states, and the action of the hard charge in
those states leads to a soft factor, which mimics the scenario in flat
spacetime at null infinity.

###  3.1 Graviton modes in dS spacetime

The mode expansion of the perturbative fields is the crucial ingredient for
deriving a soft theorem from asymptotic symmetry, see, e.g. [57, 35]. It is
very convenient to write down the mode expansion of the free field operator in
the isotropic coordinates $(t,x_{1},x_{2},x_{3})$ in curved spacetime [40, 39,
42, 41]. The line element of the dS spacetime in the isotropic coordinates is

$\mathrm{d}s^{2}=-\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}\mathrm{d}t^{2}+\mathrm{\left(\frac{2\ell^{2}}{\ell^{2}+\rho^{2}}\right)^{2}d}\vec{x}\cdot\mathrm{d}\vec{x},$
(46)

where $\rho$ is related to the radial coordinate by

$r=\ell\frac{(\ell-\rho)^{2}}{\ell^{2}+\rho^{2}}.$ (47)

The isotropic coordinates are connected to retarded coordinates
$(u,r,z,\bar{z})$ by

$t=u+\frac{1}{2}\ell\log\left(\frac{r}{2\ell-r}\right),\quad
x_{1}=\rho\frac{z+\bar{z}}{z\bar{z}+1},\quad
x_{2}=\rho\frac{z-\bar{z}}{i(z\bar{z}+1)},\quad
x_{3}=\rho\frac{1-z\bar{z}}{z\bar{z}+1}.$ (48)

In dS spacetime, $\frac{\partial}{\partial t}$ is a timelike Killing vector
which defines a positive energy of a particle as $\omega=-p_{0}$. One can
write the dispersion relation for massless particles in the isotropic
coordinates as

$\displaystyle-\left(\frac{\rho^{2}+\ell^{2}}{\rho^{2}-\ell^{2}}\right)^{2}\omega^{2}+\left(\mathrm{\frac{\ell^{2}+\rho^{2}}{2\ell^{2}}}\right)^{2}|\vec{p}|^{2}=0.$
(49)

Using the covariant measure of one-particle phase space

$\displaystyle\int\frac{\mathrm{d}\omega\mathrm{d}^{3}\vec{p}}{(2\pi)^{3}}\delta(p^{\mu}p_{\mu})\theta(\omega)=\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}\int\frac{\mathrm{d}^{3}\vec{p}}{(2\pi)^{3}2\omega},$
(50)

a free massless scalar field $\phi(x)$ can be written in mode expansion as

$\displaystyle\phi(x^{\mu})=\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}\int\frac{\mathrm{d}^{3}\vec{p}}{(2\pi)^{3}2\omega}\left[a(\vec{p})e^{ip\cdot
x}+a(\vec{p})^{\dagger}e^{-ip\cdot x}\right].$ (51)

The extra factor
$\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}$ is precisely
from the dispersion relation. Here we consider only the correction from the
dispersion relation of the null momentum, a full mode expansion of scalar
field in dS spacetime can be performed perturbatively, see, e.g., in [59]. The
mode expansion can be extended to a free graviton field simply by inserting
the polarization tensor,

$\displaystyle
h_{\mu\nu}=\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}+\rho^{2}}\right)^{2}\sum_{\alpha=\pm}\int\frac{\text{d}^{3}p}{(2\pi)^{3}}\frac{1}{2\omega}\left[\epsilon^{\alpha*}_{\mu\nu}a_{\alpha}(\vec{p})e^{i\vec{p}\cdot\vec{x}}+\epsilon^{\alpha}_{\mu\nu}a^{\dagger}_{\alpha}(\vec{p})e^{-i\vec{p}\cdot\vec{x}}\right],$
(52)

where $\epsilon_{\mu\nu}^{\alpha}$ is a product of the pair polarization
vectors
$\epsilon_{\mu\nu}^{\alpha}=\epsilon^{\alpha}_{\mu}\epsilon^{\alpha}_{\nu}$.

One can parameterize the null momentum as

$\displaystyle
p_{\mu}=\frac{\omega}{1+z\bar{z}}\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\left(\frac{\ell^{2}-\rho^{2}}{2\ell^{2}}(1+z\bar{z}),(z+\bar{z}),-i(z-\bar{z}),(1-z\bar{z})\right),$
(53)

and similarly the polarization vectors as

$\displaystyle\epsilon^{+\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}\frac{\ell^{2}+\rho^{2}}{2\ell^{2}}\left(\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\bar{z},1,-i,-\bar{z}\right),$
(54) $\displaystyle\epsilon^{-\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}\frac{\ell^{2}+\rho^{2}}{2\ell^{2}}\left(\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}z,1,i,-z\right),$
(55)

which satisfies

$\displaystyle\epsilon^{\alpha\mu}\epsilon^{\beta*}_{\mu}=\delta^{\alpha\beta},\qquad
p_{\mu}\epsilon^{\alpha\mu}=0,\qquad\epsilon^{\alpha}_{r}=0.$ (56)

Projecting the polarization vectors to the sphere, we obtain

$\epsilon_{\bar{z}}^{+}=\frac{2\sqrt{2}\ell^{2}\rho}{(1+z\bar{z})\left(\ell^{2}+\rho^{2}\right)},\quad\quad\epsilon_{z}^{-}=\frac{2\sqrt{2}\ell^{2}\rho}{(1+z\bar{z})\left(\ell^{2}+\rho^{2}\right)}.$
(57)

Eventually, the near horizon field is related to the plane wave modes by

$N_{zz}=\frac{4\gamma_{z\bar{z}}}{\pi^{2}}\frac{\ell^{8}(R+\ell)}{\left[\ell^{2}+\left(R+\ell\right)^{2}\right]^{4}}\int\mathrm{d}\omega\left(\frac{r}{2\ell-r}\right)^{-i\ell\omega/2}\\\
\times\sin\left[\frac{2\ell^{2}(R+\ell)\omega}{R(R+2\ell)}\right]e^{\mathrm{-i\omega}u}a_{+}(\omega\hat{x})+c.c.,$
(58)

where $R=\rho-\ell$ and $R=0$ at the horizon. The integration of the three
momentum $\vec{p}$ in the mode expansion in dS spacetime can be found in [42].
Literally, the near horizon field is obtained from the near horizon limit
$R\rightarrow 0$ of the mode expansion which is however singular. One can
introduce a near horizon regularization $R\rightarrow R+i\mathcal{R}$ to deal
with the divergence [40, 39, 42, 41]. Nevertheless, we will keep the radial
parameter for a moment and take the near horizon limit $R\rightarrow 0$ at the
last step. This is the reason we introduce a new radial parameter $R$ instead
of simply using the previous radial parameter $r$.

###  3.2 Soft graviton theorem in coordinate space

A soft graviton theorem can be derived from the Ward identity of
supertranslation,

$\displaystyle\left\langle{\rm out}\right|[Q_{T},\mathcal{S}]\left|{\rm
in}\right\rangle=0\Rightarrow Q^{S}\left|{\rm
in}\right\rangle=-Q^{H}\left|{\rm in}\right\rangle,$ (59)

in the flat spacetime case [57, 35]. In the present analysis, we have omitted
the out-part on $\mathcal{H}^{+}$, which can be easily restored from the
$\mathcal{CPT}$ invariance. We choose $T=\frac{1}{z-w}$ for the
supertranslation parameter, and thus

$\partial_{\bar{z}}T(z,\bar{z})=2\pi\delta^{2}(z-w).$ (60)

With this choice, the soft charge (44) is reduced to

$Q^{S}=-\frac{4\pi^{2}i}{\ell}\lim_{\omega\rightarrow
0^{+}}\left[\omega\frac{\partial_{\bar{z}}\tilde{N}_{zz}}{\gamma_{z\bar{z}}}\right],$
(61)

where $\tilde{N}_{zz}$ is defined by a Fourier relation

$N_{zz}=\int_{-\infty}^{+\infty}\text{d}ue^{i\omega u}\tilde{N}_{zz},$ (62)

which implies that

$\int_{-\infty}^{\infty}\text{d}u\,\partial_{u}{N}_{zz}(u)=2\pi
i\lim_{\omega\to 0}\left[\omega\tilde{{N}}_{zz}(\omega)\right].$ (63)

Inserting the mode expansion (58), one can obtain

$\displaystyle
Q^{S}=-i\frac{\ell^{2}}{R}\frac{\partial_{\bar{z}}\gamma_{z\bar{z}}}{\gamma_{z\bar{z}}}\lim_{\omega\rightarrow
0^{+}}\left[\omega^{2}a_{+}+\omega^{2}a_{-}^{\dagger}\right].$ (64)

Hence, the soft charge $Q^{S}$ acts on the in-state as

$\displaystyle Q^{S}\left|{\rm
in}\right\rangle=-i\frac{\ell^{2}}{R}\frac{\partial_{\bar{z}}\gamma_{z\bar{z}}}{\gamma_{z\bar{z}}}\lim_{\omega\rightarrow
0^{+}}\omega^{2}a_{-}^{\dagger}\left|{\rm in}\right\rangle.$ (65)

The soft charge creates a low-energy soft graviton in the near horizon region.

For a massless scalar field coupled case as described in Appendix A, one has
the commutation relation [42],

$[\partial_{u}\bar{\phi}(u,z,\bar{z}),\phi(g,z,\bar{z})]=-\frac{i}{\ell^{2}\gamma_{z\bar{z}}}\delta(u-g)\delta^{2}(z-w),$
(66)

which yields the action of the hard charge as

$[Q^{H},\phi]=-iT\partial_{u}\phi.$ (67)

The above relation and the special choice of the supertranslation parameter
$T=\frac{1}{z-w}$ determine the action of the hard charge $Q^{H}$ on the in-
state as

$\displaystyle Q^{H}\left|{\rm
in}\right\rangle=\sum_{k=1}^{n}\frac{E_{k}}{(z-z_{k})}\left|{\rm
in}\right\rangle\,.$ (68)

Finally, we can obtain a soft graviton theorem in coordinate space by
inserting (65) and (68) into the Ward identity (59),

$\partial_{\bar{z}}\left[\gamma_{z\bar{z}}\lim_{\omega\rightarrow
0^{+}}a^{\dagger}_{-}\left|{\rm
in}\right\rangle\right]=\lim_{\omega\rightarrow
0^{+}}\left[\frac{\pi}{i}\frac{R}{\ell^{2}\omega}\gamma_{z\bar{z}}\sum_{k=1}^{n}\frac{E_{k}}{\omega(z-z_{k})}\left|{\rm
in}\right\rangle\right].$ (69)

###  3.3 Soft graviton theorem in momentum space

In this subsection, we will use the null parametrization in (53) to rewrite
the soft theorem in momentum space. For each hard particle, their momenta can
be parametrized as

$\displaystyle
q_{k\mu}^{\mathrm{in}}=\frac{E_{k}^{\mathrm{in}}}{1+z_{k}^{\mathrm{in}}\bar{z}_{k}^{\mathrm{in}}}\frac{2\ell^{2}}{\rho^{2}-\ell^{2}}\left(\frac{\ell^{2}-\rho^{2}}{2\ell^{2}}(1+z_{k}^{\mathrm{in}}\bar{z}_{k}^{\mathrm{in}}),(z_{k}^{\mathrm{in}}+\bar{z}_{k}^{\mathrm{in}}),-i(z_{k}^{\mathrm{in}}-\bar{z}_{k}^{\mathrm{in}}),(1-z_{k}^{\mathrm{in}}\bar{z}_{k}^{\mathrm{in}})\right).$
(70)

One can easily verify that

$\displaystyle\partial_{\bar{z}}\left(\gamma_{z\bar{z}}\sum_{k=1}^{n}\frac{[q_{k}\cdot\epsilon^{+}(p)]^{2}}{p\cdot
q_{k}}\right)$
$\displaystyle=-\gamma_{z\bar{z}}\sum_{k=1}^{n}\frac{E_{k}}{\omega(z-z_{k})}\left(1+\frac{\bar{z}_{k}(z-z_{k})}{1+z_{k}\bar{z}_{k}}\right)$
(71)
$\displaystyle=-\gamma_{z\bar{z}}\sum_{k=1}^{n}\frac{E_{k}}{\omega(z-z_{k})},$

where we have used the conversation of a combination of two components of the
total momentum,

$\displaystyle\sum_{k=1}^{n}\frac{E_{k}\bar{z}_{k}}{1+z_{k}\bar{z}_{k}}\propto\sum_{k=1}^{n}\left(q_{k1}-iq_{k2}\right)=0.$
(72)

Applying the relation (71), we arrive at a soft graviton theorem in the
momentum space as

$\displaystyle\lim_{\omega\rightarrow 0^{+}}$ $\displaystyle\langle{\rm
out}|a_{+}\mathcal{S}-\mathcal{S}a^{\dagger}_{-}\left|{\rm in}\right\rangle$
(73) $\displaystyle=\lim_{\omega\rightarrow 0^{+}}i\frac{\pi
R}{\omega\ell^{2}}\left[\sum_{l=1}^{m}\frac{(q_{l}^{\rm
out}\cdot\epsilon)^{2}}{p\cdot q_{l}^{\rm
out}}-\sum_{k=1}^{n}\frac{(q_{k}^{\rm in}\cdot\epsilon)^{2}}{p\cdot q_{k}^{\rm
in}}\right]\langle{\rm out}|\mathcal{S}\left|{\rm in}\right\rangle,$

where the derivative $\partial_{\bar{z}}$ was dropped from both sides. One can
see that the soft factor on the right hand side of (73) contains the flat
spacetime soft factor and a prefactor $i\frac{\pi R}{\omega\ell^{2}}$. A flat
limit can be taken by setting $\frac{\pi iR}{\omega\ell^{2}}=1$. Physically,
this limit can be understood as the fact that the soft limit and the flat
limit of the cosmological constant $\Lambda$ are at the same order tending to
zero. Note that $iR$ is from the near horizon regularization. The orders of
the near horizon limit and the flat limit for the parameter $R=\rho-\ell$ do
not commute. The present work is based on the near horizon analysis. So we
first take the near horizon limit. However this will somehow prevent the flat
limit since the flat limit means the vanish of the cosmological horizon. Thus,
the horizon regularization $R\rightarrow R-i\mathcal{R}$ also ensures a flat
limit for the soft theorem in the momentum space. Now the regularization
involves a minus sign as $\rho\leq\ell$. Then, the flat limit condition
becomes $\Lambda=\frac{\omega}{\pi\mathcal{R}}$.

##  4 Conclusion and discussion

In this paper, we study the linearized gravity theory in the near horizon
region of the dS spacetime. The near horizon symmetry, near horizon solution
space, and near horizon supertranslation charge are obtained. A soft graviton
theorem in dS spacetime is derived from the Ward identity of the near horizon
supertranslation. The soft theorem has a similar structure as the flat
spacetime soft theorem which relies on a fine tuning structure of the near
horizon fall-off conditions. The remarkable feature of this fine tuning
structure resides in its supertranslation invariance. The soft theorem in dS
spacetime has a well defined flat limit which recovers flat spacetime soft
theorem.

To close this paper, we would like to comment on some subtleties and future
directions. In the near horizon analysis of the black hole case, the soft
limit should involve also the mass parameter of the black hole rather than
simply comparing the soft and hard external particles [60, 61]. Here, our
derivation simply involves the soft and hard external particles. We think this
is a less subtle issue in dS spacetime. The reasoning is as follow. One can
consider a perturbative expansion in the inverse of the curvature length scale
$\frac{1}{\ell}$ [59]. The leading contribution should be the flat spacetime
result. This is achieved from the flat limit of the soft theorem in momentum
space (73). It is meaningful to point out that our derivation is from the near
horizon analysis in dS spacetime. The final result is indeed the leading term
of the dS spacetime as it should be. In some sense, we have not considered the
$\frac{1}{\ell}$ corrections from near horizon analysis which is definitely an
interesting direction for future investigation. There are also some other
interesting future directions such as the subleading soft theorem in the low-
energy expansion, i.e., the dS analogue of the investigations in [62, 63, 64,
51, 65]. A more challenging point is the dual interpretation from the point of
view of celestial holography [66, 67]. In particular, there are some recent
studies on the deformations of the soft theorem [68, 69, 70], which may also
be extended to the case of soft theorem in curved spacetime.

## Acknowledgments

The authors thank Geoffrey Compère for useful correspondence on the linearized
Einstein equation. The authors would like to thank Kai-Yu Zhang for useful
discussions and collaborations in relevant research topics. This work is
supported in part by the National Natural Science Foundation of China (NSFC)
under Grants No. 11935009 and No. 11905156.

##  Appendix A Modification of stress tensor

As was shown in [41], one can modify the stress tensor by adding a divergence-
free symmetric rank two tensor. We will modify the stress tensor to satisfy
the gauge condition in (15). In this work, we will limit ourselves to a
massless complex scalar field which is originally minimally coupled to
gravity. Thus, the stress tensor is

$\displaystyle\tilde{T}_{\mu\nu}\equiv\frac{1}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta
g_{\mu\nu}}=\frac{1}{2}\left(\partial_{\mu}\Phi\partial_{\nu}\bar{\Phi}+\partial_{\nu}\Phi\partial_{\mu}\bar{\Phi}\right)-\frac{1}{2}g_{\mu\nu}\bar{\nabla}^{\rho}\Phi\bar{\nabla}_{\rho}\bar{\Phi}\,,$
(74)

where $\mathcal{L}_{M}=\bar{\nabla}_{\mu}\Phi\bar{\nabla}^{\mu}\bar{\Phi}\,$.
We assume that the scalar fields are given in the form of near horizon
expansions as

$\displaystyle\Phi=\phi+\sum_{n=1}^{\infty}r^{n}\Phi^{(n)}\,,\quad\bar{\Phi}=\bar{\phi}+\sum_{n=1}^{\infty}r^{n}\bar{\Phi}^{(n)}\,.$
(75)

We construct a modified stress tensor by
$T_{\mu\nu}=\tilde{T}_{\mu\nu}-\Upsilon_{\mu\nu}$, where $T_{\mu\nu}$
satisfies the gauge conditions in (15) and $\Upsilon_{\mu\nu}$ is an arbitrary
tensor. Then we will use the divergence-free conditions
$\bar{\nabla}^{\mu}\Upsilon_{\mu\nu}=0$ to fix it. From
$\bar{\nabla}^{\mu}\Upsilon_{\mu r}=0$, one obtains

$\gamma^{AB}\Upsilon_{AB}=\Omega^{3}\partial_{u}\tilde{T}_{rr}-\Omega
D^{A}\tilde{T}_{rA}+\Omega^{3}\partial_{r}\tilde{T}_{ur}-2\Omega^{2}\tilde{T}_{ur}+2F\Omega^{2}\tilde{T}_{rr}-\Omega^{3}F\partial_{r}\tilde{T}_{rr}-3\frac{\Omega^{4}}{\ell^{2}}\tilde{T}_{rr},$
(76)

where we have used the relation $\Upsilon_{r\mu}=\tilde{T}_{r\mu}$ to fulfill
the gauge conditions for $T_{\mu\nu}$. Thus the trace part of
$\Upsilon_{\mu\nu}$ is completely fixed by $\tilde{T}_{\mu\nu}$.

The transverse equations $\bar{\nabla}^{\mu}\Upsilon_{\mu A}=0$ yield

$\partial_{r}\Upsilon_{uA}=F\partial_{r}\tilde{T}_{rA}-\partial_{u}\tilde{T}_{rA}+\frac{2}{\Omega}\Upsilon_{uA}+(\frac{2\Omega}{\ell^{2}}-\frac{2F}{\Omega})\tilde{T}_{rA}+\frac{1}{\Omega^{2}}D^{B}\Upsilon_{AB}.$
(77)

Suppose that $\Upsilon_{uA}$ is given by the series

$\displaystyle\Upsilon_{uA}=\Upsilon^{0}_{uA}+\sum_{n=1}^{\infty}r^{n}\Upsilon_{uA}^{(n)}\,.$
(78)

All the order $\Upsilon_{uA}^{(n)}$ for $n\geq 1$ is fixed from the above
equation. While the leading order $\Upsilon^{0}_{uA}$ is free as an
integration constant. We continue with the $u$ component of the divergence-
free conditions, which gives

$\partial_{r}\Upsilon_{uu}=F\partial_{r}\tilde{T}_{ru}-\partial_{u}\tilde{T}_{ru}+(\frac{2}{\ell^{2}}\Omega-\frac{2}{\Omega}F)\tilde{T}_{ru}+\frac{1}{\Omega^{2}}D^{A}\Upsilon_{uA}+\frac{2}{\Omega}\Upsilon_{uu}.$
(79)

This equation controls $\Upsilon_{uu}$ up to an integration constant
$\Upsilon^{0}_{uu}$ at the leading order in the near horizon expansion.
Finally, we find that the traceless part of $\Upsilon_{AB}$ and the leading
order of $\Upsilon_{uu}$, $\Upsilon_{uA}$ of the auxiliary tensor are free as
initial data and the rest ingredients are fixed by the gauge conditions. The
$uu,uA$ components of the modified stress tensor at the leading orders are

$\displaystyle T_{uu}^{0}$
$\displaystyle=\partial_{u}\phi\partial_{u}\bar{\phi}-\Upsilon_{uu}^{0},$ (80)
$\displaystyle T_{uA}^{0}$
$\displaystyle=\frac{1}{2}\left(\partial_{u}\phi\partial_{A}\bar{\phi}+\partial_{A}\phi\partial_{u}\bar{\phi}\right)-\Upsilon_{uA}^{0}.$
(81)

The hard part of the near horizon supertranslation charge (42) is indeed
sensitive to the modification of the stress tensor. In this work, we follow
the previous choice in [41] to set $\Upsilon_{uu}^{0}=0$ and $T_{uA}^{0}=0$.

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