# Sensitivity to anisotropic stochastic gravitational-wave background with
space-borne networks

Zheng-Cheng Liang  MOE Key Laboratory of TianQin Mission, TianQin Research
Center for Gravitational Physics $\&$ School of Physics and Astronomy,
Frontiers Science Center for TianQin, CNSA Research Center for Gravitational
Waves, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, China  Zhi-Yuan
Li  MOE Key Laboratory of TianQin Mission, TianQin Research Center for
Gravitational Physics $\&$ School of Physics and Astronomy, Frontiers Science
Center for TianQin, CNSA Research Center for Gravitational Waves, Sun Yat-sen
University (Zhuhai Campus), Zhuhai 519082, China  En-Kun Li  MOE Key
Laboratory of TianQin Mission, TianQin Research Center for Gravitational
Physics $\&$ School of Physics and Astronomy, Frontiers Science Center for
TianQin, CNSA Research Center for Gravitational Waves, Sun Yat-sen University
(Zhuhai Campus), Zhuhai 519082, China  Jian-dong Zhang  Corresponding author:
<EMAIL_ADDRESS>MOE Key Laboratory of TianQin Mission, TianQin Research
Center for Gravitational Physics $\&$ School of Physics and Astronomy,
Frontiers Science Center for TianQin, CNSA Research Center for Gravitational
Waves, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, China  Yi-Ming
Hu  Corresponding author<EMAIL_ADDRESS>MOE Key Laboratory of TianQin
Mission, TianQin Research Center for Gravitational Physics $\&$ School of
Physics and Astronomy, Frontiers Science Center for TianQin, CNSA Research
Center for Gravitational Waves, Sun Yat-sen University (Zhuhai Campus), Zhuhai
519082, China

###### Abstract

This paper explores the detection capability of space-borne detectors to the
anisotropic stochastic gravitational-wave background (SGWB) without relying on
the low-frequency approximation. To assess the detection performance, we
calculate the power-law integrated sensitivity (PLIS) curve. Our results
demonstrate that a single detector has limited capabilities in detecting
multipole moments beyond the monopole ($l=0$), quadrupole ($l=2$), and
hexadecapole ($l=4$). However, when multiple detectors are combined, the
presence of multiple pointing directions and the separation between detectors
significantly enhance the detection capabilities for the other multipole
moments. For instance, when considering the dipole ($l=1$), combining TianQin
with TianQin II and LISA with TianQin significantly improves the detection
sensitivity by 2-3 orders of magnitude, compared with using a single TianQin
and a single LISA, respectively.

###### pacs:

04.25.dg, 04.40.Nr, 04.70.-s, 04.70.Bw

##  I Introduction

The detection of GWs has provided a new way to observe the Universe, with
almost a hundred GW events announced so far Abbott  _et al._ (2021). However,
not all GWs can be directly detected, leading to the existence of a large
number of independent and unresolved GWs that form a stochastic  background
(SGWB) Christensen (1992); Flanagan (1993); Allen and Romano (1999). In the
nano-Hertz ($\rm nHz$) band, a number of collaborations have announced the
successful detection of SGWB Agazie  _et al._ (2023); Xu  _et al._ (2023);
Antoniadis  _et al._ (2023); Reardon  _et al._ (2023). Furthermore, there is
great potential for future space-borne detectors Seto (2020); Liang  _et al._
(2022); Fan  _et al._ (2022) and ground-based detectors Abbott  _et al._
(2018); Chen  _et al._ (2019); Abbott  _et al._ (2019) to contribute to the
SGWB detection. An interesting aspect of the SGWB is its potential anisotropy,
where the intensity exhibits spatial variations Hils  _et al._ (1990); Geller
_et al._ (2018); Li  _et al._ (2022); Profumo and Yang (2023). Among the
various possible anisotropic SGWBs, the most promising candidate in the milli-
Hertz ($\rm mHz$) band is the foreground generated by the Galactic double
white dwarf (DWD) Robson  _et al._ (2019); Huang  _et al._ (2020).

Space-borne detectors are designed to detect GWs within the $\rm mHz$ band Luo
_et al._ (2016); Amaro-Seoane  _et al._ (2017); Hu and Wu (2017), where laser
noise is the dominant noise needs to be suppressed by orders of magnitude to
enable successful detection of GWs Giampieri  _et al._ (1996). To achieve this
goal, time delay interferometry (TDI) combination is proposed Tinto and
Armstrong (1999); Tinto and Dhurandhar (2021), and among all the TDI
combinations, the orthogonal channel group $\rm AET$ is commonly used Prince
_et al._ (2002). The $\rm A/E$ channels in this channel group are expected to
detect different polarization of the GW, through which the GW can be fully
characterized. The $\rm T$ channel, in contrast, is much less sensitive to the
GW at low frequencies than the $\rm A/E$ channels Hogan and Bender (2001).

After the effective suppression of laser noise, secondary noises, such as
optical-path noise and acceleration noise, become prominent in the detector
channel Krolak  _et al._ (2004). To extract the SGWB signal from these noises,
correlation analysis using either null-channel method Tinto  _et al._ (2001);
Hogan and Bender (2001); Cheng  _et al._ (2022) or cross-correlation method
Christensen (1992); Flanagan (1993) is required. The null-channel method
involves using a signal-insensitive channel to monitor the secondary noises of
other channels, while the cross-correlation method is applicable when there is
no correlation between the noise of the adopted channels. By performing auto-
or cross-correlation of the channel output, the correlation measurement of
SGWB signal can be extracted, which is related to the statistical property,
or, equivalently, the power spectral density (PSD) of SGWB Allen (1996).
Additionally, the correlation measurement also includes a geometrical factor,
which is determined by the channel geometry and the separation between
channels. Accumulating the geometric factor for all spatial orientations
allows the determination of the overlap reduction function (ORF). The ORF
represents the reduction of the correlation measurement relative to the PSD of
the SGWB.

To further investigate the anisotropy of the SGWB, it is possible to analyze
the geometrical factor by decomposing it into spherical harmonics with
multipole coefficients. These coefficients are directly related to the
multipole moments of the anisotropic SGWB Allen and Ottewill (1997); Cusin
_et al._ (2017). By calculating the ORF corresponding to each multipole
coefficient and taking into account the noise level of channel, the detection
sensitivity to the anisotropic SGWB can be determined. Previous studies have
highlighted the limitations of a single space-borne detector in detecting
multipole moments beyond the monopole ($l=0$), quadrupole ($l=2$), and
hexadecapole ($l=4$), which indicates that single detectors struggle to detect
other multipole moments Kudoh and Taruya (2005); Bartolo  _et al._ (2022).

This issue could be addressed by implementing a network of detectors with
diverse pointing directions, which enables better identification of the
intensity of the SGWB from various spatial orientations. Two space missions,
TianQin Luo  _et al._ (2016) and Laser interferometry Space Antenna (LISA)
Amaro-Seoane  _et al._ (2017), offer potential options for such a detector
network: the TianQin I+II network and the TianQin + LISA network. In the
TianQin I+II network, the first detector, TianQin, will have a fixed pointing
direction, while the second detector, TianQin II, will maintain a
perpendicular orientation to TianQin Ye  _et al._ (2019). On the other hand,
the TianQin + LISA network will involve LISA changing the pointing direction
over time. The main focus of this work is to explore the feasibility of
detecting additional multipole moments using the aforementioned detector
networks. To assess the detection capability, we utilize the power-law
integrated sensitivity (PLIS) curve Thrane and Romano (2013). This specific
sensitivity curve allows us to estimate the minimum detectable energy spectrum
density of the SGWB.

The paper is structured as follows. In Sec. II, we review the fundamental
properties of stochastic gravitational-wave background. We then discuss the
analysis for detecting the anisotropy of the SGWB in Sec. III. Our results are
presented in Sec. IV, based on the theoretical basis outlined in the previous
sections. Finally, we provide a brief summary in Sec. V.

##  II Theoretical Fundamental

In order to characterize the intensity of SGWB, we begin with the metric
perturbation in the transverse-traceless gauge. Since the SGWB is the
superposition of GWs, the metric perturbation $h(t,\vec{x})$ can be expressed
as follows:

$\displaystyle h(t,\vec{x})$ $\displaystyle=$
$\displaystyle\sum_{P=+,\times}\int_{-\infty}^{\infty}{\rm d}f\int_{S^{2}}{\rm
d}\hat{\Omega}_{\hat{k}}\,\widetilde{h}_{P}(f,\hat{k})\textbf{e}^{P}(\hat{k})$
(1) $\displaystyle\times e^{{\rm i}2\pi f[t-\hat{k}\cdot\vec{x}(t)/c]},$

where $\hat{k}$ is the wave vector of the GW, the polarization tensor
$\textbf{e}^{P}(\hat{k})$ refers to the polarization $P$ of the GW. In this
work, we focus on the Gaussian-stationary and unpolarized SGWB, which allows
us to characterize the statistical properties of SGWB through the mean value
and variance of the Fourier amplitude $\widetilde{h}_{P}(f,\hat{k})$. Based on
the fact that SGWB is composed of a large number of GWs, it is reasonable to
assume that the SGWB has a zero-mean Fourier amplitude
$\widetilde{h}_{P}(f,\hat{k})$:

$\langle\widetilde{h}_{P}(f,\hat{k})\rangle=0.$ (2)

On the other hand, the variance of $\widetilde{h}_{P}(f,\hat{k})$ is defined
as the PSD $\mathscr{P}_{\rm h}$ of SGWB:

$\langle\widetilde{h}_{P}(f,\hat{k})\widetilde{h}^{*}_{P^{\prime}}(f^{\prime},\hat{k}^{\prime})\rangle=\frac{1}{4}\delta(f-f^{\prime})\delta_{PP^{\prime}}\delta^{2}(\hat{k}-\hat{k}^{\prime})\mathscr{P}_{\rm
h}(|f|,\hat{k}),$ (3)

where the factor $1/4$ comes from the definition of one-side PSD and the
average of polarization.

In the case of an anisotropic SGWB, where the PSD exhibits a directional
preference, it is convenient to expand the PSD in terms of spherical
harmonics. By assuming that the intensity of the SGWB has no dependence
between direction and frequency, one can focus on characterizing the direction
dependence of the SGWB through Allen and Ottewill (1997)

$\mathscr{P}_{\rm
h}(f,\hat{k})=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}p_{lm}(f)Y_{lm}(\hat{k}),$ (4)

with the multipole moment

$p_{lm}(f)=\int_{S^{2}}{\rm d}\hat{\Omega}_{\hat{k}}\mathscr{P}_{\rm
h}(f,\hat{k})Y^{*}_{lm}(\hat{k}).$ (5)

To obtain the all-sky PSD of SGWB, the PSD, denoted as $\mathscr{P}_{\rm h}$,
needs to be integrated over all directions:

$\displaystyle S_{\rm h}(f)$ $\displaystyle=$ $\displaystyle\int_{S^{2}}{\rm
d}\hat{\Omega}_{\hat{k}}\mathscr{P}_{\rm h}(f,\hat{k})$ (6) $\displaystyle=$
$\displaystyle\sqrt{4\pi}\,p_{00}(f).$

Eq. (6) highlights that the all-sky PSD solely captures the monopole moment
$p_{00}$, while the higher-order multipole moments associated with the
anisotropy remain hidden. To study the anisotropy of SGWB, it is crucial to
establish an estimator that incorporates the quadratic form of
$\mathscr{P}_{\rm h}$:

$\displaystyle\int_{S^{2}}{\rm d}\hat{\Omega}_{\hat{k}}|\mathscr{P}_{\rm
h}(f,\hat{k})|^{2}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{l}p_{lm}(f)p^{*}_{lm}(f)$ (7)
$\displaystyle=$ $\displaystyle\sum_{l=0}^{\infty}(2l+1)\mathscr{A}_{l}(f),$

where the angular PSD plays a significant role in providing insights into the
anisotropy across various scales Thrane  _et al._ (2009):

$\mathscr{A}_{l}(f)=\frac{1}{2l+1}\sum_{m=-l}^{l}p_{lm}(f)p^{*}_{lm}(f).$ (8)

Especially, for an isotropic SGWB,

$\mathscr{A}_{0}(f)=\frac{S^{2}_{\rm h}(f)}{4\pi}.$ (9)

In addition, to characterize the distribution of energy across different
frequencies in SGWB, the dimensionless energy spectrum density $\Omega_{\rm
gw}$ is commonly used. The quantity allows us to specify the ratio of the GW
energy density ${\rm d}\rho_{\rm gw}$ within the frequency range [$f$,$f+{\rm
d}f$] to the critical energy density $\rho_{\rm gw}$ Allen (1996):

$\Omega_{\rm gw}(f)=\frac{1}{\rho_{\rm c}}\frac{{\rm d}\rho_{\rm gw}}{{\rm
d}(\ln{f})},$ (10)

where the critical energy density $\rho_{\rm c}=3H_{0}^{2}c^{2}/(8\pi G)$,
with the light speed $c$, the gravitational constant $G$, and the Hubble
constant $H_{0}$. The relationship between $\Omega_{\rm gw}$ and $S_{\rm h}$
can be established through Thrane and Romano (2013):

$\Omega_{\rm gw}(f)=\frac{2\pi^{2}}{3H_{0}^{2}}f^{3}S_{\rm h}(f).$ (11)

Figure 1: Schematic diagram of TianQin and LISA mission in the ecliptic
coordinate.

##  III Detection analysis

###  III.1 Detector design and channel

Currently, there are several proposed space missions for detecting GWs,
including LISA Amaro-Seoane  _et al._ (2017), TianQin Luo  _et al._ (2016),
Taiji Hu and Wu (2017), etc. In this paper, we constrain our focus on TianQin
and LISA.

As shown in Fig. 1, TianQin is a space-borne GW detector comprising three
identical satellites orbiting the Earth. These satellites have an orbital
period of $3.64$ days and are positioned at a radius of approximately
$10^{5}\,\,{\rm km}$. When in operation, the three satellites will form an
equilateral triangle with an armlength $L_{\rm tq}$ of about $1.7\times
10^{5}\,\,{\rm km}$. The working mode of TianQin follows a “three months on +
three months off” pattern. To ensure continuous detection, a second mission
called TianQin II is proposed to bridge the detection gap Ye  _et al._ (2019).
The orbital planes of TianQin and TianQin II are designed to be perpendicular
to each other, forming the TianQin I+II network. In fact, the initial working
mode does not allow for overlapping operating time between TianQin and TianQin
II. To address this issue, we adopt the improved working mode “four months on
+ two months off” Ye  _et al._ (2021), which allows for a four-month overlap
between TianQin and TianQin II within a year.

LISA is planned to be positioned in an orbit around the Sun, trailing
approximately $20\degree$ behind the Earth and maintaining a fixed angle of
$60\degree$ with the ecliptic plane. It will consist of three satellites
separated by a distance of $\sim 2\times 10^{6}\,\,{\rm km}$, which serves as
the armlength $L_{\rm lisa}$ of the detector. These three satellites will
complete one orbit in the detector plane over the course of one year. The
operating period and detection frequency band of LISA are similar to those of
TianQin, which opens up the possibility of establishing a network of TianQin +
LISA.

The triangular formation of the three satellites faces challenges in
maintaining an equilateral shape due to detector motion, making it difficult
to construct an equal-arm Michelson. This poses a challenge in canceling the
laser noise. An effective solution to address this issue is by utilizing TDI
combination Tinto and Armstrong (1999). The primary TDI channels, namely $\rm
X$, $\rm Y$, and $\rm Z$, can be established through each satellite and its
adjacent link. It is important to note that the angle between these three
channels is $60\degree$, resulting in a correlation. To address this
correlation and eliminate its impact, we can construct two orthogonal and
independent channels based on the three primary TDI channels. One possible
option is to incorporate the $\rm A/E$ channels, which are commonly combined
with the totally symmetrized Sagnac channel $\rm T$ Prince  _et al._ (2002).
Unlike the $\rm A/E$ channels, the Sagnac channel $\rm T$ is specifically
constructed to be insensitive to GWs in the low-frequency band, making it
useful for monitoring channel noise in GW detection. The channel group $\rm
AET$ can be constructed as follows Vallisneri  _et al._ (2008):

$\displaystyle{\rm A}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}({\rm
Z}-{\rm X}),$ $\displaystyle{\rm E}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}({\rm X}-2{\rm Y}+{\rm Z}),$
$\displaystyle{\rm T}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{3}}({\rm
X}+{\rm Y}+{\rm Z}).$ (12)

###  III.2 Detection measurement

The SGWB signal $h_{I}(t)$ in the detector channel $I$ is the convolution of
the metric perturbations $h(t,\vec{x})$ and the impulse response
$\mathbb{D}^{ab}(t,\vec{x})$ Romano and Cornish (2017). Although the channel
response will inevitably change due to the detector motion, we can simplify
the measurement process without significantly compromising its accuracy by
restricting our analysis to the short time scale of $[t_{0}-T/2,t_{0}+T/2]$:

$\displaystyle h_{I}(t,t_{0})$ $\displaystyle=$
$\displaystyle\mathbb{D}^{P}_{I}[t,\vec{x}(t_{0})]*h_{P}[t,\vec{x}(t_{0})]$
(13) $\displaystyle=$
$\displaystyle\sum_{P=+,\times}\int_{-\infty}^{\infty}{\rm d}f\int_{S^{2}}{\rm
d}\hat{\Omega}_{\hat{k}}F_{I}^{P}(f,\hat{k},t_{0})\widetilde{h}_{P}(f,\hat{k})$
$\displaystyle\times e^{{\rm i}2\pi f[t-\hat{k}\cdot\vec{x}(t_{0})/c]},$

where $\vec{x}$ denotes the position of the measurement at time $t$. The
response function of channel is the double contraction of the channel tensor
and the polarization tensor, i.e.,
$F_{I}^{P}={\textbf{D}}_{I}:{\textbf{e}}^{P}$ Cornish and Larson (2001). For
more details, readers can refer to Liang  _et al._ (2023). In this paper, we
will not extensively discuss these details but instead proceed to present the
response function of the $\rm AET$ channel group:

$\displaystyle F_{\rm A}^{P}(f,\hat{k},t_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{[}F_{\rm Z}^{P}(f,\hat{k},t_{0})e^{-{\rm
i}2\pi f\hat{k}\cdot\overrightarrow{A_{0}C_{0}}(t_{0})/c}-F_{\rm
X}^{P}(f,\hat{k},t_{0})\big{]},$ $\displaystyle F_{\rm
E}^{P}(f,\hat{k},t_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\big{[}F_{\rm X}^{P}(f,\hat{k},t_{0})-2F_{\rm
Y}^{P}(f,\hat{k},t_{0})e^{-{\rm i}2\pi
f\hat{k}\cdot\overrightarrow{A_{0}B_{0}}(t_{0})/c}+F_{\rm
Z}^{P}(f,\hat{k},t_{0})e^{-{\rm i}2\pi
f\hat{k}\cdot\overrightarrow{A_{0}C_{0}}(t_{0})/c}\big{]},$ $\displaystyle
F_{\rm T}^{P}(f,\hat{k},t_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\big{[}F_{\rm X}^{P}(f,\hat{k},t_{0})+F_{\rm
Y}^{P}(f,\hat{k},t_{0})e^{-{\rm i}2\pi
f\hat{k}\cdot\overrightarrow{A_{0}B_{0}}(t_{0})/c}+F_{\rm
Z}^{P}(f,\hat{k},t_{0})e^{-{\rm i}2\pi
f\hat{k}\cdot\overrightarrow{A_{0}C_{0}}(t_{0})/c}\big{]},$ (14)

where $A_{0}$, $B_{0}$, and $C_{0}$ denote the satellites of space-borne
detector, i.e., laser interference sites in the $\rm X$, $\rm Y$, and $\rm Z$
channels. Furthermore, the reference position of the channel is marked as
$A_{0}$.

In the frequency domain, the SGWB signal

$\displaystyle\widetilde{h}_{I}(f,t_{0})$ $\displaystyle=$
$\displaystyle\sum_{P=+,\times}\int_{S^{2}}\,{\rm{d}}\hat{\Omega}_{\hat{k}}F_{I}^{P}(f,\hat{k},t_{0})\widetilde{h}_{P}(f,\hat{k})$
(15) $\displaystyle\times e^{-{\rm i}2\pi f\hat{k}\cdot\vec{x}(t_{0})/c},$

and the PSD of the SGWB signal can be defined as

$\langle\widetilde{h}_{I}(f,t_{0})\widetilde{h}_{J}^{*}(f^{\prime},t_{0})\rangle=\frac{1}{2}\delta(f-f^{\prime})P_{{\rm
h}_{IJ}}(|f|,t_{0}).$ (16)

Here, $I=J$ refers to the auto PSD of one channel, while $I\neq J$ refers to
the cross PSD of two channels111To simplify, $P_{{\rm h}_{II}}$ reduces to
$P_{{\rm h}_{I}}$.. Combined with Eq. (15) and Eq. (16), we can establish a
following connection between $P_{{\rm h}_{IJ}}$ and $\mathscr{P}_{\rm h}$:

$\displaystyle P_{{\rm h}_{IJ}}(f,t_{0})$ $\displaystyle=$
$\displaystyle\int_{S^{2}}\,{\rm{d}}\hat{\Omega}_{\hat{k}}\mathcal{Y}_{IJ}(f,\hat{k},t_{0})\mathscr{P}_{\rm
h}(f,\hat{k}),$ (17)

where the geometrical factor involves both the channel response and the
separation between the channel:

$\displaystyle\mathcal{Y}_{IJ}(f,\hat{k},t_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{P=+,\times}F^{P}_{I}(f,\hat{k},t_{0})F^{P*}_{J}(f,\hat{k},t_{0})$
(18) $\displaystyle\times e^{-{\rm i}2\pi
f\hat{k}\cdot[\vec{x}_{I}(t_{0})-\vec{x}_{J}(t_{0})]/c}.$

To describe the directional dependence of the geometric factor, it is
necessary to further expand it in terms of spherical harmonics:

$\displaystyle\mathcal{Y}_{IJ}(f,\hat{k},t_{0})=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}a_{lm}(f,t_{0})Y_{lm}^{\ast}(\hat{k}),$
(19)

with the multipole coefficient

$\displaystyle a_{lm}(f,t_{0})=\int_{S^{2}}{\rm
d}\hat{\Omega}_{\hat{k}}\mathcal{Y}_{IJ}(f,\hat{k},t_{0})Y_{lm}(\hat{k}).$
(20)

Substituting Eq. (4) and Eq. (19) to Eq. (17), the PSD of the SGWB signal is
calculated as the sum of the product of the multipole coefficient and the
corresponding multipole moment Kudoh and Taruya (2005):

$P_{{\rm
h}_{IJ}}(f,t_{0})=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}a_{lm}(f,t_{0})p_{lm}(f),$
(21)

which allows for the determination of the angular PSD by taking into account
the contributions from various multipole components.

###  III.3 Signal-to-noise ratio

In addition to SGWB signal $h_{I}$, the output from channel $s_{I}$ is also
contributed by the channel noise, denoted as $n_{I}$. Hence, the complete
expression for the output is given by $s_{I}=h_{I}+n_{I}$. The PSD of the
output can be obtained by summing the signal PSD $P_{\rm h}$ and the noise PSD
$P_{\rm n}$:

$\displaystyle\langle\widetilde{s}_{I}(f,t_{0})\widetilde{s}^{\ast}_{J}(f^{\prime},t_{0})\rangle$
$\displaystyle=$
$\displaystyle\langle\widetilde{h}_{I}(f,t_{0})\widetilde{h}^{\ast}_{J}(f^{\prime},t_{0})\rangle+\langle\widetilde{n}_{I}(f)n^{\ast}_{J}(f^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2}\delta(f-f^{\prime})\big{[}P_{{\rm
h}_{IJ}}(|f|,t_{0})+P_{{\rm n}_{IJ}}(|f|)\big{]},$

where the channel noise is assumed to be stationary. Since the channel noise
and the SGWB signal are assumed to be uncorrelated, their intersection is
reduced to 0.

In the presence of noise, extracting the SGWB signal from the output becomes a
challenging task. It requires careful management and monitoring of the channel
noise. One effective approach to mitigate the impact of channel noise is to
employ cross-correlation between two channels with uncorrelated channel noise.
Ideally, the cross-correlation should result in a zero-value noise PSD
$P_{{\rm n}_{IJ}}$ Christensen (1992); Flanagan (1993). Alternatively, for a
single space-borne detector, the null channel could be utilized to monitor the
channel noise Tinto  _et al._ (2001); Hogan and Bender (2001). By utilizing
the above methods, the optimal signal-to-noise ratio (SNR) for SGWB detection

$\rho_{IJ}(t_{0})=\sqrt{\frac{2\,T}{1+\delta_{IJ}}\int_{f_{\rm min}}^{f_{\rm
max}}{\rm d}f\,\frac{|P_{{\rm h}_{IJ}}(f,t_{0})|^{2}}{P_{{\rm
n}_{I}}(f)P_{{\rm n}_{J}}(f)W_{IJ}(f,t_{0})}},$ (23)

where the correction function $W_{IJ}$ plays an important role when the SGWB
signal exceeds the channel noise Liang  _et al._ (2023). If one specifically
considers the contribution of different multipole moments to the SNR, where
each $m$ is derived from the same Gaussian statistics Bartolo  _et al._
(2022), i.e.,

$\mathscr{A}_{l}(f)=\delta_{ll^{\prime}}p_{lm}(f)p^{*}_{l^{\prime}m^{\prime}}(f),$
(24)

then the $l-$dependent SNR can be defined by substituting Eq. (21) into Eq.
(23):

$\displaystyle\rho^{(l)}_{IJ}(t_{0})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2\,T}{1+\delta_{IJ}}\int_{f_{\rm min}}^{f_{\rm
max}}{\rm
d}f\,\sum_{m=-l}^{l}\frac{a_{lm}(f,t_{0})a^{*}_{lm}(f,t_{0})p_{lm}(f)p^{*}_{lm}(f)}{P_{{\rm
n}_{I}}(f)P_{{\rm n}_{J}}(f)W^{lm}_{IJ}(f,t_{0})}}$ (25) $\displaystyle=$
$\displaystyle\sqrt{\frac{2\,T}{1+\delta_{IJ}}\int_{f_{\rm min}}^{f_{\rm
max}}{\rm
d}f\,\frac{|\Gamma^{(l)}_{IJ}(f,t_{0})|^{2}4\pi(2l+1)\mathscr{A}_{l}(f)}{P_{{\rm
n}_{I}}(f)P_{{\rm n}_{J}}(f)}},$

where the $l-$dependent ORF

$\displaystyle\Gamma_{IJ}^{(l)}(f,t_{0})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2l+1}\sum_{m=-l}^{l}\frac{|\Gamma_{IJ}^{lm}(f,t_{0})|^{2}}{W_{IJ}^{lm}(f,t_{0})}}$
(26) $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{4\pi(2l+1)}\sum_{m=-l}^{l}\frac{|a_{lm}(f,t_{0})|^{2}}{W_{IJ}^{lm}(f,t_{0})}},$

with the correction function

$\displaystyle W_{IJ}^{lm}(f,t_{0})$ $\displaystyle=$ $\displaystyle
1+\frac{P^{lm}_{{\rm h}_{I}}(f)P_{{\rm n}_{J}}(f)+P^{lm}_{{\rm
h}_{J}}(f)P_{{\rm n}_{I}}(f)}{P_{{\rm n}_{I}}(f)P_{{\rm n}_{J}}(f)}$
$\displaystyle+\frac{P^{lm}_{{\rm h}_{I}}(f)P^{lm}_{{\rm
h}_{J}}(f)+(1-\delta_{IJ})|P^{lm}_{{\rm h}_{IJ}}(f,t_{0})|^{2}}{P_{{\rm
n}_{I}}(f)P_{{\rm n}_{J}}(f)}.$

The SNR formula allows us to analyze and quantify the impact of each multipole
component on the SGWB detection, taking into account their statistical
properties.

Without loss of generality, the SNR corresponding to the total correlation
time $T_{\rm tot}$

$\rho^{(l)}_{IJ}=\sqrt{\frac{2\,T_{\rm tot}}{1+\delta_{IJ}}\int_{f_{\rm
min}}^{f_{\rm max}}{\rm
d}f\,\frac{|\bar{\Gamma}^{(l)}_{IJ}(f)|^{2}4\pi(2l+1)\mathscr{A}_{l}(f)}{P_{{\rm
n}_{I}}(f)P_{{\rm n}_{J}}(f)}},$ (28)

where the time-averaged ORF Liang  _et al._ (2022)

$\bar{\Gamma}^{(l)}_{IJ}(f)=\sqrt{\frac{1}{T_{\rm tot}}\int_{0}^{T_{\rm
tot}}{\rm d}t_{0}\,|\Gamma^{(l)}_{IJ}(f,t_{0})|^{2}}.$ (29)

In particular, when $l=0$, Eq. (28) and Eq. (29) correspond to an isotropic
SGWB.

###  III.4 Sensitivity curve

Detection SNR is commonly used to assess the detection capability of the
detector and detector network for the SGWB. However, in scenarios where one
aims to demonstrate the detection sensitivity, the PLIS curve serves as a more
suitable indicator Thrane and Romano (2013). This type of sensitivity curve,
specifically tailored for power-law SGWB, offers a robust method to assess the
detection sensitivity to SGWB, regardless of the intensity level of the SGWB.

To elaborate on the PLIS curve, let us begin with the power-law energy
spectrum density:

$\Omega^{(l)}_{\rm gw}(f)=\Omega^{(l)}_{0}(\epsilon)(f/f_{\rm
ref})^{\epsilon}|_{\epsilon=\epsilon_{0}},$ (30)

where $\Omega_{0}$ is normalized by the reference frequency $f_{\rm ref}$,
$\epsilon$ denotes the spectral index, and to be consistent with Eq. (11) at
$l=0$, we have

$\Omega^{(l)}_{\rm
gw}(f)=\frac{2\pi^{2}}{3H_{0}^{2}}f^{3}\sqrt{4\pi(2l+1)\mathscr{A}_{l}(f)}.$
(31)

Here, the angular power spectrum $\mathscr{A}_{l}$ of the SGWB at different
scales $l$ can be calculated using $\Omega^{(l)}_{\rm gw}$, which is, in
particular, numerically equal to the energy spectral density $\Omega_{\rm gw}$
of the SGWB when $l=0$.

Figure 2: ORF for a single TianQin and LISA, taking into account the channel
combinations $\rm AA$, $\rm AE$, $\rm AT$, and $\rm TT$. The solid line and
the dot-dashed line denote TianQin and LISA, respectively.

Substituting Eq. (31) into Eq. (28), and setting the SNR equal to the
threshold $\rho_{0}$ required for detection, one can determine the energy
spectrum density $\Omega^{(l)}_{\rm gw}$ for each index $\epsilon$ through
inverse solving. At each frequency, the maximum value of $\Omega^{(l)}_{\rm
gw}$ corresponding to a specific index $\epsilon$ can be selected, enabling
the construction of the PLIS curve:

$\Omega_{\rm PLIS}(f)={\rm max}_{\epsilon}[\Omega^{(l)}_{\rm gw}(f)].$ (32)

In particular, if we assume that the SGWB is significantly weaker than the
channel noise, resulting in the correction function being reduced to 1, the
process of reverse solving $\Omega^{(l)}_{\rm gw}$ can be simplified by
extracting $\Omega^{(l)}_{0}$ from the integral:

$\Omega^{(l)}_{0}=\rho_{0}\left[\frac{2\,T_{\rm
tot}}{1+\delta_{IJ}}\int_{f_{\rm min}}^{f_{\rm max}}{\rm d}f\frac{(f/f_{\rm
ref})^{2\epsilon}}{[\Omega_{{\rm n}_{IJ}}^{(l)}(f)]^{2}}\right]^{-1/2}.$ (33)

where the $l-$dependent effective sensitivity

$\Omega^{(l)}_{{\rm
n}_{IJ}}(f)=\frac{2\pi^{2}}{3H_{0}^{2}}f^{3}\frac{\sqrt{P_{{\rm
n}_{I}}(f)P_{{\rm n}_{J}}(f)}}{\bar{\Gamma}^{(l)}_{IJ}(f)}.$ (34)

If the energy spectrum density $\Omega^{(l)}_{\rm gw}$ of a multipole moment
is located above the PLIS curve, it indicates that the corresponding SNR is
expected to exceed the preset threshold. In such cases, the multipole moment
is likely to be detected successfully. Conversely, if $\Omega^{(l)}_{\rm gw}$
falls below the PLIS curve, it suggests that the detection is highly likely to
fail. Therefore, the lowest point on the PLIS curve denotes the minimum
detectable energy spectrum density, which is referred to as “sensitivity” when
discussing the detection capability.

##  IV Result

Table 1: The combination of channels for correlation in a single TianQin, LISA, and the TianQin I+II, TianQin +LISA network. The symbol ′ is employed to distinguish the channels of different detector in a detector network. detector/detector network | channel combination
---|---
TianQin or LISA |  $\rm AA$,$\rm AE$,$\rm AT$,$\rm TT$
TianQin I+II or TianQin + LISA |  $\\{\rm AA^{\prime}$,$\rm AE^{\prime}$,$\rm EA^{\prime}$,$\rm EE^{\prime}$}
$\\{\rm AT^{\prime}$,$\rm ET^{\prime}$,$\rm TA^{\prime}$,$\rm
TE^{\prime}\\}$,$\\{\rm TT^{\prime}\\}$

In this section, our focus is on demonstrating the detection sensitivity to an
anisotropic SGWB by presenting numerical results for the TianQin I+II and
TianQin + LISA networks. To ensure a comprehensive comparison, we also add the
result of single detectors, namely TianQin and LISA individually.

For a single detector, all possible correlations of the channels can be
utilized, including auto-correlation within the same channel and cross-
correlation between different channels. Since the ORF exhibits symmetry in the
channels, where $\Gamma_{\rm AA}^{l}=\Gamma_{\rm EE}^{l}$ and $\Gamma_{\rm
AT}^{l}=\Gamma_{\rm ET}^{l}$, we only need to consider the performance of four
pairs of channels. When analyzing detector networks and aiming to emphasize
the improvement achieved by multiple detectors in comparison to a single
detector, the focus shifts to solely considering the cross-correlation between
different detectors while disregarding the correlations within each individual
detector. The channel combinations are classified into three categories based
on the presence of null channel: those without null channels (0N), those with
one null channel (1N), and those with two null channels (2N). These categories
are consistently labeled throughout the paper. Table. 1 provides a list of the
correlation channel combinations used for our analysis. To simplify the
notation, we may use shorthand labels such as “tt” for TianQin I+II, and “tl”
for TianQin + LISA in the following figures and equations.

###  IV.1 Overlap reduction function

Figure 3: ORF of the TianQin I+II network within different initial phase angle
difference $\gamma_{0}$ normalized by the case $\gamma_{0}=0$. To comply with
the low-frequency approximation, we set the frequency $f$ to $0.01\,\,\rm Hz$.
It is important to note that for $l=0,2$, and 4, the ORFs are not affected by
$\gamma_{0}$. As a result, they coincide and appear as a single straight line
in the plot. Figure 4: Time-average ORF for the TianQin I+II network, where
“$\rm 0N$”, “$\rm 1N$”, and “$\rm 2N$” denote the channel combinations $\\{\rm
AA^{\prime}$,$\rm AE^{\prime}$,$\rm EA^{\prime}$,$\rm EE^{\prime}$}, $\\{\rm
AT^{\prime}$,$\rm ET^{\prime}$,$\rm TA^{\prime}$,$\rm TE^{\prime}\\}$, and
$\\{\rm TT^{\prime}\\}$, respectively.

####  IV.1.1 A singer detector

Fig. 2 illustrates the ORFs of a single TianQin and LISA. Due to the symmetry
of the spherical harmonic, which states that
$Y_{l-m}(-\hat{k})=(-1)^{l+m}Y^{*}_{lm}(\hat{k})$, the odd multipole moments
in the auto-correlation cancels, denoted as
$\Gamma_{II}=(-1)^{l}\,\,\Gamma_{II}$ Bartolo  _et al._ (2022). Consequently,
the auto-correlation of a single channel, represented by $\Gamma_{\rm AA}$ and
$\Gamma_{\rm TT}$, is unable to capture odd multipole moments. To overcome
this limitation, cross-correlation between different channels becomes
necessary. Even for the dipole ($l=1$), relying solely on the cross-
correlation of the $\rm A/E$ channels is insufficient. It is necessary to
incorporate the $\rm T$ channel, which serves as a null channel in the GW
detection Seto (2020). By combining the null channel and the signal-sensitive
channel, the presence of the dipole can be observed through the cross-
correlation between these two channels, represented by $\Gamma_{\rm AT}$.
Additionally, it is important to consider the limitations of the low-frequency
approximation for a single detector. This approximation fails when the
frequency scale reaches a characteristic frequency $f_{\ast}=c/(2\pi L)$,
where $L$ represents the armlength of the detector Cornish (2002). As LISA has
a longer armlength compared to TianQin, the ORF of a single LISA will start to
decline at lower frequencies than that of a single TianQin.

####  IV.1.2 TianQin I+II network

In contrast to a single TianQin, the TianQin I+II network requires only three
ORFs to be presented because the $\rm A$, $\rm E$, and $\rm T$ channels of
both TianQin and TianQin II share the same armlength and secondary noises:

$\displaystyle\Gamma_{\rm 0N}(f)$ $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{\rm AA^{\prime}}(f)|^{2}+|\Gamma_{\rm
AE^{\prime}}(f)|^{2}+|\Gamma_{\rm EA^{\prime}}(f)|^{2}+|\Gamma_{\rm
EE^{\prime}}(f)|^{2}},$ $\displaystyle\Gamma_{\rm 1N}(f)$ $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{\rm AT^{\prime}}(f)|^{2}+|\Gamma_{\rm
ET^{\prime}}(f)|^{2}+|\Gamma_{\rm TA^{\prime}}(f)|^{2}+|\Gamma_{\rm
TE^{\prime}}(f)|^{2}},$ $\displaystyle\Gamma_{\rm 2N}(f)$ $\displaystyle=$
$\displaystyle\Gamma_{\rm TT^{\prime}}(f),$ (35)

which simplifies the detection analysis for the TianQin I+II network.
Additionally, since detector networks are inherently more complex than
individual detectors, it becomes crucial to examine the impact of network
design on the ORF.

As mentioned in Ref. Liang  _et al._ (2022), the difference in initial angles
between TianQin and TianQin II, denoted as $\gamma_{0}=\alpha-\alpha^{\prime}$
(as illustrated in Fig. 1), affect the ORF of the TianQin I+II network.
However, for an isotropic SGWB, this impact is generally considered negligible
under the low-frequency approximation Seto (2020). In this study, we further
investigate the situation when dealing with an anisotropic SGWB. To narrow
down our analysis, we take advantage of the fact that the null channel $\rm T$
is insensitive to SGWB at frequencies much lower than the characteristic
frequency, which allows us to focus our initial analysis on the ORF
$\Gamma_{\rm 0N}$.

In Fig. 3, we present the normalized $\Gamma_{\rm 0N}$ for different
multipoles with $\gamma_{0}$ set to 0. To maintain the validity of the low-
frequency approximation, the frequency is set to $\rm 0.01\,\,Hz$. For $l=0$,
2, and 4, the ORF remains unchanged regardless of the value of $\gamma_{0}$.
However, for $l=1$, 3, 5, and 6, the ORF reaches the maximum and minimum when
$\gamma_{0}=\pi$ and $\gamma_{0}=0,2\pi$, which corresponds to the launch time
difference being set to a semi-integer and an integer multiple of the orbital
period, respectively. Particularly, for $l=6$, the selection of $\gamma_{0}$
can result in an almost threefold gap between the maximum and minimum of the
ORF. By setting the initial phase angle difference $\gamma_{0}$ to $\pi$ and
0, we illustrate the time-average ORF of the TianQin I+II network in Fig. 4.
The result further supports the conclusion that the effect of the initial
phase angle difference on the detection of an anisotropic SGWB cannot be
disregarded, even when applying the low-frequency approximation.

Figure 5: Effect of the channel separation $\Delta x$ on ORF for the TianQin
I+II network.

Additionally, we investigate the impact of channel separation $\Delta x$ on
the ORF (see Eq. (18)) by examining the $\rm A$ channel of TianQin and the
$\rm A^{\prime}$ channel of TianQin II in a specific case where
$\alpha=\alpha^{\prime}=0$. Fig. 5 illustrates the ORF for different
separations between the $\rm A$ and $\rm A^{\prime}$ channels. As the
separation increases, the characteristic frequency is continuously suppressed.
In frequency bands well below the characteristic frequency, the increase in
separation does not have a noticeable effect on the ORF for $l=0$, 2, and 4.
However, for the remaining multipole moments, increasing the separation leads
to a significant increase in the ORF, particularly for $l=6$. By increasing
the channel separation, the ORF of the detector network could be enhanced,
resulting in improved detection capability to the anisotropic SGWB.

####  IV.1.3 TianQin + LISA network

Figure 6: The ORF of the TianQin + LISA network undergoes time-dependent
changes, which is normalized by comparing them to the case when the two
detector planes are perpendicular for the first time. The dashed lines in the
plot correspond to two specific time points when the two detector planes are
perpendicular. To ensure the validity of the analysis, the frequency is set to
$1\times 10^{-5}\,\,\rm Hz$, which prevents the low-frequency approximation
from failing.

As for the TianQin + LISA network, it is important to consider the difference
in orbital periods between TianQin and LISA. The orbital period of LISA is
approximately 100 times longer than that of TianQin, allowing LISA to
effectively remain stationary during one orbital period of TianQin. This
averaging effect helps mitigate the impact of the initial phase angle
difference. However, it is important to note that the plane angle between
TianQin and LISA will change over time, necessitating closer attention to the
effect of the plane angle on the ORF for the TianQin + LISA network.

In Fig. 6, we present the normalized ORF of the TianQin + LISA network, with
respect to the case where TianQin is perpendicular to LISA. Due to the working
mode of TianQin, TianQin and LISA simultaneously detect SGWB for only half of
the year, while the dotted line in the graph corresponds to the period when
TianQin is off duty. After careful examination, it becomes evident that
setting the plane angle $\theta_{0}=\pi/2$ (perpendicular design) is not the
optimal choice for detecting the monopole ($l=0$) and hexadecapole ($l=4$).
However, the situation varies for other multipole moments. In particular, for
the quadrupole ($l=2$) moment, the best results are achieved when the two
detectors are positioned perpendicular to each other. It is crucial to
highlight that the channel separation $\Delta x$ of the TianQin + LISA network
varies over time. After six months, when the two detectors become
perpendicular again, the channel separation will no longer be the same as it
was half a year ago. Consequently, the ORFs at two perpendicular times are not
exactly equal, particularly for multipole moments with $l=1$, 3, 5, and 6.

Figure 7: Time-average ORF for the TianQin + LISA network, where “$\rm 0N$”,
“$\rm 1N_{I}$”, “$\rm 1N_{II}$”, and “$\rm 2N$” denote the channel
combinations $\\{\rm AA^{\prime}$,$\rm AE^{\prime}$,$\rm EA^{\prime}$,$\rm
EE^{\prime}$}, $\\{\rm AT^{\prime}$,$\rm ET^{\prime}\\}$,$\\{\rm
TA^{\prime}$,$\rm TE^{\prime}\\}$, and $\\{\rm TT^{\prime}\\}$, respectively.

Fig. 7 further illustrates the time-average ORF of the TianQin + LISA network.
Due to the differences in armlengths between TianQin and LISA, the channel
combination $\\{\rm AT^{\prime}$,$\rm ET^{\prime}$,$\rm TA^{\prime}$,$\rm
TE^{\prime}\\}$ should be divided into two groups $\\{\rm AT^{\prime}$,$\rm
ET^{\prime}\\}$ and $\\{\rm TA^{\prime}$,$\rm TE^{\prime}\\}$, denoted as
“$\rm 1N_{I}$” and “$\rm 1N_{II}$”, respectively. Note that, the separation
between TianQin and LISA, $\Delta x\approx 2\sin 10^{\degree}\,\rm AU$222AU
denotes the average distance between the Earth and the Sun, and $1\,\,\rm
AU\approx 1.496\times 10^{8}\,\,km$., is much longer than the armlengths of
the two detectors. Consequently, the characteristic frequency of the TianQin +
LISA network is further lowered to $\sim\rm 1\,\,mHz$ Liang  _et al._ (2022).
In the frequency band below $\rm 1\,\,mHz$, noticeable gaps exist between the
ORFs for different multipole moments. However, as the frequency increases
beyond $\rm 1\,\,mHz$, these gaps tend to significantly narrow. When the
frequency surpasses $\rm 10\,\,mHz$, the ORFs for different multipole moments
will converge to the similar magnitude, indicating a comparable level of
response across the multipole moments.

###  IV.2 Pow-law integrated sensitivity curve

The next step involves demonstrating the detection sensitivity to SGWB through
the PLIS curve. To evaluate the sensitivity, we consider all possible channel
combinations, including the null channel $\rm T$, which becomes a sensitive
channel at high frequencies Vallisneri  _et al._ (2008). Then the total
effective sensitivity

$\displaystyle\Omega^{(l)}_{{\rm n}_{\rm tot}}(f)$ $\displaystyle=$
$\displaystyle\left\\{\sum_{I,J}\left[\frac{1}{\Omega^{(l)}_{{\rm
n}_{IJ}}(f)}\right]^{2}\right\\}^{-1/2}$ (36) $\displaystyle=$
$\displaystyle\frac{2\pi^{2}}{3H_{0}^{2}}f^{3}\left\\{\sum_{I,J}\frac{|\bar{\Gamma}^{(l)}_{IJ}(f)|^{2}}{P_{{\rm
n}_{I}}(f)P_{{\rm n}_{J}}(f)}\right\\}^{-1/2},$

where the noise PSD of the channel group $\rm AET$ is given by Liang  _et al._
(2023)

$\displaystyle P_{\rm n_{\rm A/E}}(f)$ $\displaystyle=$
$\displaystyle\frac{2\sin^{2}\big{[}\frac{f}{f_{\ast}}\big{]}}{L^{2}}\bigg{[}\bigg{(}\cos\big{[}\frac{f}{f_{\ast}}\big{]}+2\bigg{)}S_{\rm
p}(f)+2\bigg{(}\cos\big{[}\frac{2f}{f_{\ast}}\big{]}+2\cos\big{[}\frac{f}{f_{\ast}}\big{]}+3\bigg{)}\frac{S_{\rm
a}(f)}{(2\pi f)^{4}}\bigg{]},$ $\displaystyle P_{\rm n_{T}}(f)$
$\displaystyle=$
$\displaystyle\frac{8\sin^{2}\big{[}\frac{f}{f_{\ast}}\big{]}\sin^{2}\big{[}\frac{f}{2f_{\ast}}\big{]}}{L^{2}}\bigg{(}S_{\rm
p}(f)+4\sin^{2}\big{[}\frac{f}{2f_{\ast}}\big{]}\frac{S_{\rm a}(f)}{(2\pi
f)^{4}}\bigg{)},$ (37)

which takes into account the optical-path noise $S_{\rm p}$ and the
acceleration noise $S_{\rm a}$. For more detailed information on the
parameters of TianQin and LISA, readers can refer to Ref. Mei  _et al._ (2021)
and Ref. Babak  _et al._ (2021), respectively. Furthermore, the operating time
for each scenario is set to one year. Specifically, for a single TianQin, a
single LISA, the TianQin I+II network, and the TianQin + LISA network, the
total correlation time is one year, half a year, four months, and half a year,
respectively Liang  _et al._ (2022).

In Fig. 8, we show the PLIS curve for both individual detectors and detector
networks using Eq. (32). To highlight the improved detection capability of a
detector network compared to a single detector in the anisotropic SGWB
detection, we compare the performance of a single TianQin with the TianQin
I+II network, as well as a single LISA with the TianQin + LISA network. The
detector network exhibits the comparable detection capability to a single
detector for multipole moments with $l=0$, 2, and 4. Furthermore, for the
remaining multipole moments, combining multiple detectors can significantly
improve the detection capability beyond that of a single detector. For
instance, in the case of detecting the dipole ($l=1$), when TianQin is
combined with TianQin II, the detection sensitivity improves from $1\times
10^{-8}$ to $1\times 10^{-11}$. Alternatively, when LISA is combined with
TianQin, the detection sensitivity improves from $5\times 10^{-11}$ to
$4\times 10^{-13}$. Additional cases are listed in Table. 2.

Table 2: The sensitivity to different multipole moments for four scenarios: a single TianQin, a single LISA, the TianQin I+II network, and the TianQin + LISA network. | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
TianQin | $1\times 10^{-13}$ | $1\times 10^{-8}$ | $1\times 10^{-12}$ | $4\times 10^{-10}$ | $7\times 10^{-12}$ | $1\times 10^{-8}$ | $1\times 10^{-6}$
LISA | $2\times 10^{-14}$ | $5\times 10^{-11}$ | $2\times 10^{-13}$ | $1\times 10^{-11}$ | $1\times 10^{-12}$ | $3\times 10^{-10}$ | $4\times 10^{-9}$
TianQin I+II | $3\times 10^{-13}$ | $1\times 10^{-11}$ | $6\times 10^{-13}$ | $3\times 10^{-11}$ | $6\times 10^{-12}$ | $2\times 10^{-10}$ | $9\times 10^{-9}$
TianQin + LISA | $3\times 10^{-13}$ | $4\times 10^{-13}$ | $6\times 10^{-13}$ | $8\times 10^{-13}$ | $1\times 10^{-12}$ | $2\times 10^{-12}$ | $2\times 10^{-12}$
Figure 8: PLIS sensitivity to different multipole moments of the SGWB for a
single TianQin, a single LISA, TianQin I+II, and TianQin + LISA. The SNR
threshold is set to 1, and the operating time is set to one year.

##  V Summary

This paper primarily focuses on the detection of the anisotropic SGWB using
detector networks. We conducted calculations to determine the PLIS curves for
both the TianQin I+II and TianQin + LISA networks, comparing them with single
detectors such as TianQin and LISA. Our analysis revealed that the detection
capabilities of the TianQin I+II network are similar to those of a single
TianQin for the monopole ($l=0$), quadrupole ($l=2$), and hexadecapole
($l=4$). However, for the remaining multipole moments, the TianQin I+II
network surpasses the detection capabilities of a single TianQin by several
orders of magnitude. Similar conclusions can be drawn when comparing a single
LISA with the TianQin + LISA network. For example, considering the dipole
($l=1$), which may arise from the motion of the solar system barycenter and
the Earth around it Chung  _et al._ (2022), the detection sensitivity of the
detector network is expected to improve by 2-3 orders of magnitude compared to
using a single detector. Furthermore, for the hexacontatetrapole, the
detection sensitivity can be enhanced by over 3 orders of magnitude.

We also investigated the influence of detector network design on the detection
of the anisotropic SGWB. In the TianQin I+II network, we found that carefully
considering the initial phase angle difference is crucial. Setting the launch
time difference between TianQin and TianQin II to a semi-integer multiple of
the orbital period yields better results compared to an integer multiple of
the period. The ORF corresponding to these two cases differs significantly,
even in frequency bands much lower than the characteristic frequency.
Additionally, we explored the impact of the detector plane angle using the
TianQin + LISA network. It was discovered that making the two detector planes
perpendicular to each other is not optimal for detecting the monopole ($l=0$)
and hexadecapole ($l=4$). However, this network design improves the detection
for the quadrupole ($l=2$). The specific requirements for the plane angle may
vary for different multipole moments, necessitating a case-by-case analysis.
Furthermore, we demonstrated how the channel separation can assist in
detecting the anisotropy of the SGWB. While it may not aid in the cases of
$l=0$, 2, and 4, increasing the channel separation greatly improves the ORF of
the detector network for $l=1$, 3, 5, and 6. This enhancement of the ORF
through increased channel separation contributes to improving the detection
capability, with a substantial magnitude of improvement.

###### Acknowledgements.

This work has been supported by the Guangdong Major Project of Basic and
Applied Basic Research (Grant No. 2019B030302001), the National Key Research
and Development Program of China (No. 2020YFC2201400), the Natural Science
Foundation of China (Grants No. 12173104), and the Natural Science Foundation
of Guangdong Province of China (Grant No. 2022A1515011862). We also thank Zu-
Cheng Chen, Jianwei Mei for helpful discussions.

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  *[GWs]: gravitational-wave
  *[GW]: gravitational-wave
  *[]:
  *[SGWB]: stochastic   () background
  *[SGWBs]: stochastic   () background
  *[DWD]: double white dwarf
  *[TDI]: time delay interferometry
  *[PSD]: power spectral density
  *[ORF]: overlap reduction function
  *[LISA]: Laser interferometry Space Antenna
  *[PLIS]: power-law integrated sensitivity
  *[SNR]: signal-to-noise ratio
  *[ORFs]: overlap reduction function


# Detection of Deepfake Environmental Audio

Hafsa Ouajdi  École Centrale Nantes
Nantes, France
<EMAIL_ADDRESS>Oussama Hadder  École Centrale Nantes
Nantes, France
<EMAIL_ADDRESS>Modan Tailleur, Mathieu Lagrange  Nantes Université,
École Centrale Nantes,
CNRS, LS2N, UMR 6004,
F-44000 Nantes, France
<EMAIL_ADDRESS>Laurie M. Heller  Dept. of Psychology,
Carnegie Mellon University,
Pittsburgh, PA, U.S.
<EMAIL_ADDRESS>
###### Abstract

With the ever-rising quality of deep generative models, it is increasingly
important to be able to discern whether the audio data at hand have been
recorded or synthesized. Although the detection of fake speech signals has
been studied extensively, this is not the case for the detection of fake
environmental audio.

We propose a simple and efficient pipeline for detecting fake environmental
sounds based on the CLAP audio embedding. We evaluate this detector using
audio data from the 2023 DCASE challenge task on Foley sound synthesis.

Our experiments show that fake sounds generated by 44 state-of-the-art
synthesizers can be detected on average with 98% accuracy. We show that using
an audio embedding learned on environmental audio is beneficial over a
standard VGGish one as it provides a 10 % increase in detection performance.
Informal listening to Incorrect Negative examples demonstrates audible
features of fake sounds missed by the detector such as distortion and
implausible background noise.

###### Index Terms:

Fake detection, Environmental sound, Deep learning, Classification, Deepfake
audio

##  I Introduction

The rapid evolution of generative models, such as those based on Diffusion
models, ushers in an era where the boundaries between reality and synthetic
content tend to blur more and more. In audio synthesis, there exists a sizable
literature dedicated to the detection of deep fakes in speech, aiming to
detect adversarial attacks ranging from misinformation dissemination to
identity theft [1].

As generative models become more sophisticated, particularly those rooted in
deep learning architectures, their capacity to produce eerily realistic audio
forgeries has grown exponentially. Innovations such as Variational
Autoencoders (VAEs) and Generative Adversarial Networks (GANs) have empowered
malicious actors to craft audio content that is virtually indistinguishable
from genuine recordings. Furthermore, deepfake audio is often used in
conjunction with deepfake video to create more realistic and convincing fake
movies. In light of these trends, the design of effective fake audio detection
systems is important.

Deepfake Audio refers to audio that has been generated or augmented using deep
learning techniques [2]. There are different types of Deepfake Audio: Text-to-
Speech [3], Voice conversion [4], emotion fakes[5], scene fakes[6], and
partial fakes [7].

While deepfake detection for speech is well studied, there appears to be
little research, to our knowledge, on the detection of fake content for
environmental sounds. This paper aims to fill this gap with respect to Foley
sound synthesis, i.e., sound categories such as hand clap, rain, etc.

We propose a simple and effective fake detection pipeline based on CLAP
embeddings [8]. Experiments are based on a publicly available audio dataset
developed during Task 7 of the IEEE AASP DCASE 2023 Challenge [9]. This is a
dataset of more than 6 hours of recorded audio and 28 hours of generated
audio.

The paper is organized as follows. Section II gives a brief review of the
relevant state of the art on deepfake detection in audio. Section III
introduces the proposed deepfake detector which is benchmarked using an
experimental protocol described in Section IV. Performance is discussed in
Section V and Section VI discusses the outcomes of informal listening of
detection mistakes of the proposed detector that suggest directions for
improvement. Code, supplementary material, and audio examples are available on
the companion page 111Companion page:
https://mathieulagrange.github.io/audioFoleyDeepFake.

##  II Related Work

AudioAudio (4s)Embedding(4,dim)Time Averaging(1,dim)Vectorization0 : Nonfake1
: Fake Figure 1: Overview of the pipeline used in the experiments for the
Deepfake detection, with a representation of the MLP’s network architecture.
The value of dim depends on the embedding method used.

This section discusses machine-learning techniques for Deepfake Audio
detection. In [10], the authors use linear regression to detect fake voices.
The features are extracted using a signal’s entropy calculated based on
Shannon’s equation for each second of the signal and for the whole signal. The
model reaches 98% accuracy, with all fake audio being correctly detected. In
[11], the authors adopt a combination of Bispectral analysis and MFCC (Mel
Frequency Cepstral Coefficients) for speech detection and feature extractions.
The former helps to identify components in generated audio that are not
present in human speech and the latter is used to detect the features in human
speech tied to the vocal tract that are absent in the AI-generated speech. The
features are then fed into seven different ML algorithms. The results show
that the Quadratic SVM model gives the highest accuracy of 96.1%. In [12], the
feature vector is extracted by filtering the signal under analysis and
extracting different statistics with short-term and long-term prediction
order. The features are then faded to a Linear SVM and Random Forest model.
The results show that the SVM outperforms the RF in different conditions.
Traditional machine-learning models require manual feature extraction.
Consequently, research was directed towards more sophisticated feature
extraction algorithms based on deep learning models.
In [13], they employed both CNNs (Convolutional Neural Net), using local
dependencies, and LSTMs (Long Short-Term Memory), using both local and
sequential dependencies, to classify audio based on MFCCs. The former model
helps achieve an accuracy of 80%. A CNN has been used with a similar strategy
in [14], which compares different approaches (MFCC, STFT, FTT of the signal)
to deduce that MFCC gives the best results and also that CNNs are beneficial
in such tasks because they can learn to detect important features. In [15], a
shallow CNN architecture, Deep2Net, was used. The authors transformed the
audio classification task into a computer vision problem by using a histogram
of the audio. The model reached a global accuracy of 0.985.

##  III Proposed approach

The detection of fake sounds is treated as a binary classification task. As
can be seen in Figure 1, the proposed architecture for solving this task
leverages the power of pre-trained audio embeddings. This allows us to
simplify the learning process by requiring less training data and using less
power as the learnable part of the detector consists of only a few Multilayer
Perceptron (MLP) or Convolutional Neural Network (CNN) layers.

The CNN architecture consists of two sequential convolutional layers with ReLU
activations and subsequent max pooling for feature abstraction, with a
flattening layer before a terminal linear transformation. However, due to the
poor performance of CNNs, we focus exclusively on reporting the performance of
the MLP model. This model is structured, as shown in 1, with three dense
layers. Each linear layer is followed by 1) ReLu activation functions to
reduce the backpropagation errors and accelerate the learning process, and 2)
dropout layers for regularization to avoid overfitting. A final dense layer
with a sigmoid function as the output activation function.

To provide input to the decision module, we tested 3 embeddings produced by
deep learning architectures, namely: vgg, clap, and pann.

The vgg embedding is produced by the VGGish network, a convolutional neural
network pre-trained for audio classification tasks, and is adapted from the
VGG [16] image classification architecture. This method adapts the VGG image
classification network to the audio domain by converting audio signals into
log-mel spectrograms, which are visual representations of the frequency and
amplitude of the sound. The spectrograms are then fed into the network, which
was pre-trained on a large-scale YouTube dataset with diverse audio
categories, such as music, speech, or animal sounds.
The clap embedding is produced by the Contrastive Language-Audio Pretraining
(Clap) model [17]. By leveraging a contrastive learning approach, CLAP enables
models to learn representations that encode the semantic content of language
and the acoustic characteristics of audio simultaneously. During pre-training,
these models are trained using contrastive learning objectives to project both
language and audio inputs into a shared embedding space. The model learns to
encode similar language-audio pairs closely together while pushing dissimilar
pairs apart. This process enables the extraction of rich embeddings that
capture semantic and acoustic similarities between inputs.
The pann embedding is produced by Pretrained Audio Neural Networks (PANNs)
[18], which are yet another class of models. These networks are trained on
spectrogram representations of audio signals that capture both the frequency
and temporal characteristics of sound. The pre-trained models are usually
initialized with weights from models pre-trained on large-scale audio
datasets, such as AudioSet or UrbanSound, which contain diverse audio
categories ranging from musical genres to environmental sounds.

##  IV Experiments

###  IV-A Dataset

The DCASE2023 Challenge [19] aimed to produce fake environmental sounds by
training on nonfake sounds. It consisted of two tracks: Track A, with 10
generation systems from participating teams using varied Foley sound synthesis
methods and permitting limited use of external resources; and Track B, with 28
generation systems based on provided code, prohibiting external resources.
Both tracks accepted rule-based and ensemble systems, provided they utilized
sounds only from the development set.

The resulting dataset consists of both nonfake and fake sounds across seven
distinct sound classes (dog_bark, footstep, gunshot, keyboard,
moving_motor_vehicle, rain, and sneeze_cough). Specifically, the dataset
comprises 5,550 nonfake sounds alongside 25,200 fake sounds, the latter of
which were generated by challenge participants. These sounds are analyzed
through the lens of four different embeddings, detailed in section II.B, each
with its unique dimensionality: VGGish embeddings at (4,128), Microsoft CLAP
(MS-Clap) embeddings at (4,1024), and two PANNs embeddings at (4,2048), where
’4’ refers to the four-second duration of the audio clips. For training and
evaluation purposes, we separate the dataset into a training set (70%), a
validation set (10%), and an evaluation set (20%) 222Dataset of recorded and
generated audio: https://zenodo.org/records/8091972.

| MLP VGGish | MLP MS-Clap | MLP PANN_Wavegram_logmel | MLP PANN_32k
---|---|---|---|---
Predicted | Nonfake (%) | Fake (%) | Nonfake (%) | Fake (%) | Nonfake (%) | Fake (%) | Nonfake (%) | Fake (%)
Nonfake | 15 | 3 | 17 | 1 | 14 | 3 | 14 | 5
Fake | 9 | 73 | 1 | 81 | 4 | 79 | 3 | 77
TABLE I: Confusion Matrix Evaluation for MLP models

###  IV-B Training Procedure

A simple preprocessing step is performed on the embeddings: we take the mean
(by axis 0) of each embedding to avoid the temporal sequence aspect (four-
second duration) affecting the deepfake detection. Figure 1 shows the pipeline
used in the experiment protocol. As we have four different input embeddings,
the procedure yielded four models to use and compare: MLP VGGish, MLP MS-Clap,
MLP PANN Wavegram, and MLP PANNcnn14-32k. These models were trained using the
Adam optimizer, configured with a learning rate of 7e-4. The training protocol
was standardized across models, with a batch size of 128 samples, a decision
threshold of 0.5 on both training and validation predictions, and training
over 100 epochs. Throughout this process, the Binary Cross-Entropy (BCE) 1
loss function is used to optimize the weights parameters in the
backpropagation process:

$J(y,\hat{y})=-\left(y\cdot\log(\hat{y})+(1-y)\cdot\log(1-\hat{y})\right)$ (1)

Where $y$ is the true label (0 or 1) and $\hat{y}$ is the predicted
probability.

During training, we implement a checkpointing mechanism that saves the state
of the model every 10 epochs of training. Following the best-accuracy
selection criterion [20], the final model is chosen based on the checkpoint
demonstrating the highest validation accuracy. For each implementation of the
detector, 10 training runs are performed and statistics are reported. We carry
out all the experiments on a machine with 4 physical cores and 8 threads, 16GB
of RAM, and an NVIDIA GeForce RTX 3060 graphic card with 6GB of RAM. To
address the imbalance between fake and nonfake samples, we employ a balancing
technique on the training set.

##  V Results

###  V-A Inference Time

To evaluate the computational efficiency of each model, we assessed the time
required for the embedding process and the MLP model’s inference for each
embedding on one random audio sample across 100 runs. The inference time (in
seconds) is then computed by averaging over the 100 inference times. In Table
II, the averaged inference time is expressed as a percentage of the real-time
duration of the audio sample. This metric provides a clear understanding of
the model’s speed relative to the actual audio duration.

###  V-B Overall Accuracy

We calculate both the relative overall validation and evaluation accuracy,
along with the time per epoch in seconds (s), and generate a Confusion Matrix
for the evaluation dataset. As presented in Table II, statistics such as
Overall Evaluation Accuracy and Training Time per Epoch are detailed for each
model. Among the experiments conducted under the same conditions, the MLP MS-
Clap model outperformed others with the highest Evaluation Accuracy of 98.02%,
surpassing both MLP PANN models by 5% and the MLP VGGish model by 10%.

TABLE II: Statistics on Overall Accuracy and the Inference Time expressed as a real-time percentage. Model x Embeddings | Accuracy (%) | Inference Time (%)
---|---|---
| Mean | Averaged percentage
MLP VGGish | 88.11 $\pm$ 0.73 | 0.423
MLP MS-Clap |  98.02 $\pm$ 0.18 | 1.82
MLP PANN-Wavegram | 93.15 $\pm$ 0.34 | 0.318
MLP PANNcnn14_32k | 93.04 $\pm$ 0.32 | 0.234

Among the 40 identical runs, we pick the best-performing run for each model
(i.e., the highest evaluation accuracy). The confusion matrices resulting from
the four chosen MLP models demonstrate a varied performance across the
different architectures. The MLP VGGish model shows difficulty with correctly
classifying nonfake sounds, as it produces 9% false positive judgements. The
MLP MS-Clap model shows the best proficiency in identifying the nonfake
category of sounds, with the lowest false prediction rate (2% of the dataset).
On the other hand, the MLP PANN-Wavegram and the MLP PANN_32k models behave
approximately the same by having a high evaluation accuracy and also a low
false prediction rate, yet they do not perform as well overall as the MLP Clap
model.

###  V-C Statistical Analysis

We test the four models in the same configurations as shown in Table II. We
train the models under each configuration for 10 iterations. We then perform
the Mann-Whitney U-test to assess the similarity of accuracy distribution. The
U-test is adopted instead of the t-test since it doesn’t impose the assumption
of normal distribution. In our case, the Null hypothesis assumes that run A is
similar to run B. If the p-value $<$ 0.05, we can confirm that the alternative
hypothesis is true (run A is similar to run B) with a confidence interval of
95%.
The accuracy of MLP MS-Clap is higher than that of MLP Pann-Wavegram with
5.02%, and that of MLP PANN 32k with 5.11%, and that of MLP VGGish with
10.23%. In all cases, the U-test confirms that those differences are
significant, as for all pairs the Null hypothesis is rejected.

Sound Class | dog_bark | footstep | gunshot | keyboard | moving_motor_vehicule | rain | sneeze_cough
---|---|---|---|---|---|---|---
Accuracy (%) | 99 | 98.1 | 98.3 | 97.7 | 98.1 | 98.5 | 97.7
TABLE III: Per Class Accuracy of the MLP MS-Clap deepfake predictor.

###  V-D Class-Wise Accuracy

As the MLP MS-Clap model is the best-performing model among the four trained
models, we chose to carry out our analysis with this model. To have a better
understanding of the behavior of the model’s performance, we look closely at
the model’s accuracy for each class (Table III). This closer look helps us
understand how the model performed with different types of data. We find that
the model was consistently good at identifying all classes (98% approximately
for each class), with dog_bark being the class where the model performs the
best, and it performs slightly less better than the others when it has to
predict on keyboard and sneeze_cough classes.333On the companion website a
confusion matrix is shown for each evaluated embedding. Entries of those
matrices are per system generator and sound class. This matrix shows that it
is challenging to detect some fake sounds that are generated by some specific
systems on specific classes.

###  V-E Accuracy vs. Generator Quality

We now compare our classifier’s performance against the outcomes of the
DCASE2023 challenge task 7. For this purpose, we conduct a comparison and
analysis between the scores of our predicted likelihoods by using the BCE as a
score function (the higher, the better), and the Fréchet Audio Distance (FAD)
scores [19] used for the official rankings of the challenge (the lower, the
better).

We observe a significant difference in correlation across the two tracks.
While Track A demonstrated a strong negative correlation of -0.86, indicating
a general agreement between the two scoring methods, Track B showed a notably
lower correlation of -0.27. This discrepancy may be due to two major
differences between the two tracks: Track B imposed restrictions on external
resources, whereas Track A had none, and Track B had more systems than Track A
(28 versus 10). Furthermore, the low correlation in Track B could be due to
the diversity, complexity, and originality of the features used in the
generating systems. For instance, feature extraction methods such as log-mel
spectrogram and log-magnitude spectrogram were used by Track B’s systems,
while in Track A the majority used Gaussian latent variables and a spectrogram
for feature extraction; additionally, some data augmentation strategies such
as time masking, tanh distortion, and sound wrapping were used in Track B
while being unused in Track A.

##  VI Discussion

We now discuss the outcome of informal listening tests that were performed to
qualitatively evaluate the acoustic properties of A) incorrect positives, and
B) incorrect negatives.

###  VI-A Incorrect Positives

Erroneously classifying a nonfake sound as a fake is an ”Incorrect Positive”
(or ”False alarm”) whereas classifying a fake sound as a fake is a ”Correct
Positive” (or ”Hit”). We examined 38 Incorrect Positive sounds, spanning all 7
categories, based on the erroneous high likelihood of the MLP MS-Clap model.
These sounds are from the DCASE development and evaluation dataset that were
presumed to be real recordings; however, although these sounds were aurally
screened by the organizers, the datasets may have nonetheless contained a few
Foley or processed sounds. To detect this possibility and also to detect other
reasons the system incorrectly classified a sound as fake, all authors
carefully listened to these sounds. We noted two recordings that had very
strong echoes. We suspected that two recordings were Foley-simulated footsteps
rather than recordings of real footsteps. Some recordings included non-target
sounds (e.g. footsteps during rain, or a snore after a sneeze). Many sounds
had significant background noise. We verbally characterized five sound
features that may have triggered an incorrect fake detection and used them to
create a menu of ”reasons” for a person suspecting that an audio is fake.
Next, one naive lab member blindly rated all 38 sounds as real or fake
(presented in random order along with the Incorrect Negative samples). A
response of ”fake” brought up a menu of five possible reasons for their
judgement: I hear extra artifacts that do not belong; The sound is
unrealistic; The sound does not fit the category; The sound is too repetitive;
The audio does not sound like it is from a recording; Other. The most common
reason given was unrealistic sound, followed by audible artifacts, followed by
not fitting the category. No other reasons were selected. The listener
correctly recognized 63.2% of these Incorrect Positives as nonfake, which was
marginally greater than a chance rate of 50% (p$<$.07, Binomial test) and
significantly greater than the 0% accuracy of our fake detector (p$<$.00001).

###  VI-B Incorrect Negatives

The 43 fake sounds that were erroneously classified as nonfake by the MLP MS-
Clap detector are referred to as ”Incorrect Negatives”. There were sounds from
all 7 categories. We noted a couple of recordings that contained very little
sound (perhaps a brief thud) and so there was not much available acoustic
information. Many sounds had significant background noise, and often the noise
sounded more like static than the typical noise heard in real recordings, such
as environmental noise or distortion from overloading a microphone
(clipping/saturation). Sometimes the onset of the target sound contained a
burst of noise. One sound had audible repetition; evidence of splicing/editing
was confirmed via a spectrogram. Next, as described above, one lab member
blindly rated all Incorrect Negatives as real or fake (presented in random
order along with the Incorrect Positives). The most common reason given was
audible artifacts, followed by unrealistic sound, followed by failure to fit
the category, followed by too repetitive. The overall pattern of reasons for a
fake judgement was similar for the Incorrect Negatives and Incorrect
Positives. The listener correctly recognized 81.4% of Incorrect Negatives as
fake, which was significantly greater than a chance rate of 50% (p$<$.00004,
Binomial test) and significantly greater than the 0% accuracy of our fake
detector (p$<$.00001). Their overall accuracy was 72.8%. Using the signal
detection theory framework, d’ is a measure of sensitivity to fake sounds
which is independent of the bias to decide real/fake. By design, our detector
had zero sensitivity to this set of 81 sounds, whereas our naive listener
displayed notable sensitivity with a d’=1.23 (derived from a hit rate = 0.814
and a false alarm rate = 0.368, i.e. the rate at which the listener made
errors on the set of Incorrect Positives.) In conclusion, we provide a ”proof
of concept” that there are audible acoustic cues that help distinguish fake
and nonfake sounds which have not yet been exploited by our deepfake detection
system. Human listening suggests opportunities for improvement in the
detection of distortion, realistic echoes and temporal structure, and patterns
of noise and repetition which may be noticeable to humans but not to a frame-
based detector such as the one considered in this study.

##  VII Acknowledgments

The authors thank Justine Sullivan for helping with the blind listening test
described in Section VI.

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