aainstitutetext: Chongqing University of Posts and Telecommunications,
400065, Chongqing, Chinabbinstitutetext: School of Electronic and Electrical
Engineering, Chongqing University of Arts and Science,
402160, Chongqing, Chinaccinstitutetext: Chongqing University,
400044, Chongqing, Chinaddinstitutetext: Departmentof Astronomy, Beijing
Normal University, Beijing,
100875,China

# High precision measurement of cosmic curvature: from gravitational waves and
cosmic chronometer

Y. He a,c,1 Y. Pan,111Corresponding author. b,1 D-P Shi,222Corresponding
author. c J. Li d S. Cao a,c W. Cheng<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Although the spatial curvature has been measured with very high precision, it
still suffers from the well known cosmic curvature tension. In this paper, we
propose an improved method to determine the cosmic curvature, by using the
simulated data of binary neutron star mergers observed by the second
generation space-based DECi-hertz Interferometer Gravitational-wave
Observatory (DECIGO). By applying the Hubble parameter observations of cosmic
chronometers to the DECIGO standard sirens, we explore different possibilities
of making measurements of the cosmic curvature referring to a distant past:
one is to reconstruct the Hubble parameters through the Gaussian process
without the influence of hypothetical models, and the other is deriving
constraints on $\Omega_{K}$ in the framework of non-flat $\Lambda$ cold dark
matter model. It is shown that in the improved method DECIGO could provide a
reliable and stringent constraint on the cosmic curvature
($\Omega_{K}=-0.007\pm 0.016$), while we could only expect the zero cosmic
curvature to be established at the precision of $\Delta\Omega_{K}=0.12$ in the
second model-dependent method. Therefore, our results indicate that in the
framework of methodology proposed in this paper, the increasing number of
well-measured standard sirens in DECIGO could significantly reduce the bias of
estimations for cosmic curvature. Such constraint is also comparable to the
precision of Planck 2018 results with the newest cosmic microwave background
(CMB) observations ($\Delta\Omega_{K}\approx 0.018$), based on the concordance
$\Lambda$CDM model.

###### Keywords:

Cosmological parameters — Gravitational waves — Cosmology:observations

##  1 Introduction

The determination of cosmic curvature is an important research in cosmology.
As we all known, satisfying the principles of cosmology, the space-time of our
universe can be described by the Friedmann-Lemaître-Robertson-Walker metric.
Cosmic curvature can help us understand whether the space of universe is open,
closed or flat. It is worth mentioning that in the research of some scholars,
the cosmic curvature is not completely independent, and there is a certain
dependence between dark energy and cosmic curvature(Clarkson2007, ; Wang2007,
; Hlozek2008, ).

In order to determine the cosmic curvature, many scholars have carried out
related research. Scholars mainly study curvature in two aspects. On the one
hand, based on the model assumption of the cold dark matter model, it is
feasible to fix the dark energy parameters in the model and introduce the
cosmic curvature density parameter. In these research, the constraint result
of the curvature density parameter by the Planck2018 cosmic microwave
background (CMB) is $-0.095<\Omega_{K}<-0.007$(Aghanim2020, ; DiValentino2020,
). The constraints of CMB combined with lens and baryon acoustic
oscillation(BAO) on the parameter of cosmic curvature density support the flat
universe, $\Omega_{K}=0.007~{}\pm 0.0019$(Aghanim2020, ). Besides, Gao et
al.Gao2020  studied the curvature density parameters and dark energy using the
latest supernova samples. The study of using full-shape and BAO to constrain
the model with curvature parameters has also been proposed(Chudaykin2021, ).
On the other hand, progress has been made in measuring curvature using model-
independent methods. Starting from the equation of curvature parameters, we
can construct the curvature parameters at different redshifts by providing
Hubble parameters($H(z)$) and luminosity distances($D_{L}$) at different
redshifts $\Omega_{K}=\frac{(H(z)D^{\prime}(z))^{2}-c^{2}}{H_{0}^{2}D(z)^{2}}$
(Clarkson2007, ). This method provides a good way to directly test the
curvature parameters by bypassing the hypothesis of the model. Li et al. used
$H(z)$ and angular diameter distance($D_{A}(z)$) given by BAO to test the
curvature parametersLi2014 . Supernova data and quasar data, which do not
depend on model assumptions, are also suitable for these works(Gao2020, ;
Yahya2014, ; Cai2016, ; Li2016, ; Wang2017, ; Liu2020b, ; Jesus2021, ;
Cao2019b, ; Wei2020, ). Some scholars use gravitational waves and $H(z)$ to
construct curvature at different redshiftsZheng2021 . It is worth noting that
this method needs to estimate the first and second derivatives of the
luminosity distance from the fitting function, which will lead to the increase
of uncertaintyYu2016 . Therefore, for the deficiency of this method, some
scholars have proposed proposed an improved model independent method to
measure curvature(Wei2017, ). In their research, the error caused by the
derivative of luminosity distance is overcome. Since then, some scholars have
used this method to test the curvature of the universeWei2018 ; Cao2019b ;
Wei2020 ; Yang2021 . Many scholars use model-independent methods to test the
curvature parameters. For example, based on the distance ratio contained in
the strong gravitational lensing, some scholars avoid assuming the model to
test the curvature(Liu2020a, ; Wang2020, ; Zhou2020, ; Qi2021, ). In addition,
some scholars have tested the curvature using strong lens time delays
dataLiao2017 . It should be noted that although cosmological models do not
need to be assumed in these methods, the Gaussian process used to reconstruct
data may have a model dependence tendencyColgain2021GP . It is worth
mentioning that recently, some scholars have used machine learning methods
combined with $H(z)$ and supernova samples to test the curvature parameters
$\Omega_{K}=0.028\pm 0.186$ Wang2021 . In their research, the method based on
artificial neural network can overcome the a priori problem of Hubble
constant($H_{0}$).

The successful detection of gravitational waves(GW) opens a new window for
cosmological research(Abbott2016, ; Abbott2017, ). Like supernovae, GW do not
rely on model assumptions in detection. Looking for merging events of binary
black holes or merging events of two neutron stars with electromagnetic
radiation can be used as standard sirens to study cosmology(Dalal2006, ;
Hlozek2008, ; Zhao2011, ; Liao2017NC, ; Cai2017, ; Du2022, ). Recently, black
hole-neutron star merging events have also been successfully detected
(Abbott2021, ). Moreover, the coupling coefficient between GW and matter is
small, which can carry more primordial wave source information, which benefits
of the cosmology. In this paper, the data from the space gravitational wave
detector DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) are
used to simulate the gravitational wave events that will be detected by DECIGO
in the future. And we use two methods to test curvature. In the first method,
we directly test the curvature density parameters in the luminosity distance,
but the cosmological model is not added to the luminosity distance. In the
second method, we add a cosmological model including curvature to the
luminosity distance, and constrain the curvature density parameters in the
model. First, we use the improved method proposed Wei et al.(Wei2017, ) to
test the curvature. And taking into account the impact of the number of data
samples on the constraints, we reconstruct the expansion rate, and the
curvature parameters are constrained. Then, we use gravitational wave data to
constrain $\Omega_{K}$ in the framework of non-flat $\Lambda$ cold dark matter
model($\Lambda CDM$+$\Omega_{K}$ model) to compare the curvature constraint
effect of the two methods. In addition, in the second method, we use
electromagnetic wave(EM) (supernovae and quasars) data to constrain the
curvature in $\Lambda CDM+\Omega_{K}$ model to compare with the constraints
from DECIGO GW data. Finally, we compare and analyze the results with the
research of other scholars.

The structure of this paper is as follows. In Sec. 2, we briefly introduce the
data and methods used. In Sec. 3, we will give the results and analysis. In
Sec. 4, we will make a summary of this article.

##  2 Data and Method

###  2.1 Gravitational Wave Detection From DECIGO

For GW events, we know that the luminosity distance of the merging event of
two stars can be obtained by detection, and the redshift information can be
obtained by directly detecting the merging event with neutron stars(Holz2005,
; Zhao2011, ; Abbott2017, ). So this kind of binary merging events can be used
as a standard siren to study cosmology.

DECIGO(Seto2001, ; Kawamura2006, ) is a GW detection project under
construction in Japan, which means DECi-hertz Interferometer Gravitational
wave Observatory. DECIGO’s detection frequency ranges from 0.1Hz to
10Hz(Kawamura2019, ). In this paper, we use DECIGO as the detector to simulate
the standard siren information provided by gravitational waves.

Consider two systems with masses $m_{1}$ and $m_{2}$, whose Fourier transform
can be expressed as

$\tilde{h}(f)=\frac{A}{d_{L}(z)}M_{z}^{5/6}f^{-7/6}e^{i\Psi(f)},$ (1)

Where, $\Psi(f)$ and $A=\left(\sqrt{6}\pi^{2/3}\right)^{-1}$ are the inspiral
phase term and geometrical average over the inclination angle of the system
respectively. The former parameter, $\Psi(f)$, is removed from the
calculation, which is a function of the coalescence time $t_{c}$, while the
latter is a constant. $d_{L}(z)$ is a function of luminosity distance with
respect to $z$ and $M_{z}=(1+z)\eta^{3/5}(m_{1}+m_{2})$ is the redshifted
chirp mass. $\eta=m_{1}m_{2}/(m_{1}+m_{2})^{2}$ is the symmetric mass
ratio.According to the studies of other scholars(Sathyaprakash2010, ;
Zhao2011, ), we assume that the neutron stars are uniformly distributed at
[1,2]$M_{\odot}$, and the coalescence time and initial phase of emission are
both zero(i.e. $\left(t_{c}=0,\phi_{c}\right)=0$). In this way, the unknown
parameters can be reduced to three:
$\theta=\left\\{M_{z},\eta,d_{L}\right\\}$.

We use the fisher matrix to estimate the uncertainty:

$\Gamma_{ab}=4Re\int_{f_{\min}}^{f_{\max}}\frac{\partial_{a}\tilde{h}_{i}^{*}(f)\partial_{b}\tilde{h}_{i}(f)}{S_{h}(f)}df,$
(2)

Where $\partial_{a}$ means to derive the parameter $\theta_{a}$. It can be
seen from the data released by DECIGO that it has eight equivalent
detectors(Seto2001, ; Kawamura2006, ). Therefore, if all detectors are taken
into account, the coefficient of $\Gamma_{ab}$ should be 8 times that of a
single detector.Noise spectrum Analysis from DECIGO(Kawamura2006, ;
Nishizawa2010, ; Kawamura2019, ):

$\displaystyle S_{h}(f)$ $\displaystyle=$ $\displaystyle 6.53\times
10^{-49}\left[1+\left(\frac{f}{7.36Hz}\right)^{2}\right]$ (3)
$\displaystyle+4.45\times
10^{-51}\times\left(\frac{f}{1Hz}\right)^{-4}\times\frac{1}{1+\left(\frac{f}{7.36Hz}\right)^{2}}$
$\displaystyle+4.94\times
10^{-52}\times\left(\frac{f}{1Hz}\right)^{-4}\mathrm{~{}Hz}^{-1},$

Where the first line on the right represents shot noise, the second line
represents radiation pressure noise, and the last line represents acceleration
noise.

In the process of simulating GW, we need to assume the parameters of the
cosmological model. In this paper, based on the data given by Bennett et
al.(Bennett2014, ), we choose the $\Lambda CDM$ model of
($H_{0}=69.6km~{}s^{-1}~{}Mpc^{-1},\Omega_{m}=0.286$). Luminosity distance
given by simulated GW

$d_{L,GW}(z)=\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{\sqrt{\Omega_{m}\left(1+z^{\prime}\right)^{3}+\left(1-\Omega_{m}\right)}},$
(4)

Where c is the speed of light and $H_{0}$ is the Hubble constant.

The distribution of wave sources that can be observed on Earth
is(Sathyaprakash2010, ; Cai2017, ):

$P(z)\propto\frac{4\pi D_{C}^{2}(z)R(z)}{H(z)(1+z)},$ (5)

Where, $H(z)$ represents the Hubble parameter, $D_{C}(z)$ represents the
comoving distance, and the representation of $R(z)$ has been used in many
articles(Schneider2001, ; Cutler2006, ; Cai2017, )

$R(z)=\left\\{\begin{array}[]{ll}1+2z,&z\leq 1\\\ \frac{3}{4}(5-z),&1<z<5\\\
0,&z\geq 5\end{array}\right.,$ (6)

The error expression of luminosity distance is:

$\displaystyle\sigma_{d_{L,GW}}$ $\displaystyle=$
$\displaystyle\sqrt{\left(\sigma_{d_{L,GW}^{\mathrm{inst}}}\right)^{2}+\left(\sigma_{D_{d,GW}^{\mathrm{lens}}}\right)^{2}}$
(7) $\displaystyle=$
$\displaystyle\sqrt{\left(\frac{2d_{L,GW}}{\rho}\right)^{2}+\left(0.05zd_{L,GW}\right)^{2}},$
(8)

Where, $d_{L,GW}^{\mathrm{inst}}$ represents the error caused by noise,
$d_{L,GW}^{\mathrm{lens}}$ represents the error caused by weak gravitational
lens effect. In the above equation, the extension of the second equation comes
from Zhao2011  and Sathyaprakash2010 , respectively. $\rho$ represents signal-
to-noise ratio.

Through the analysis of Kawamura2019 , DECIGO expects to detect more than
10000 binary merging events every year, and based on the analysis of
Cutler2009 , it is feasible to determine the redshift of these events through
their electromagnetic counterparts. Therefore, we simulate the luminosity
distance and the corresponding redshift of 10000 GW events, see fig. 1.

Figure 1:  Luminosity distance and redshift data simulated by 10000 GW events.

###  2.2 Gaussian Process

In this paper, in order to calculate the data at different redshifts at the
same redshift, we use the Gaussian process method to reconstruct the data.
This method was first used by Seikel et al.(Seikel2012, ), and has been
studied by many scholars(Zhang2018, ; Liao2019, ; Fu2013Yu, ; Li2016Yu, ;
Liu2020b, ; Zhou2020, ).

Gaussian process is a method of smoothing data. For the input data set,
multiple smooth data in a given range can be reconstructed, and the tasks of
data set expansion and redshift reconstruction can be well completed. Seikel
et al.(Seikel2012, ) used GP to reconstruct the dark energy state parameter
equation, which includes the use of GP, for which they have developed a third-
party library GaPP based on Python.

In this paper, we use Gaussian process to reconstruct Hubble parameter data.
The evolution of Hubble parameter with redshift represents the change of
cosmic expansion rate with the increase of distance. Jimenez et
al.(Jimenez2002, ) pointed out that we can use the relative galactic age to
constrain the cosmological parameters, in their study, the Hubble parameter is
expressed as

$H(z)=-\frac{1}{(1+z)}\frac{dz}{dt},$ (9)

In this work, we use 31 redshifts from 0.09 to 1.965 $H(z)$ data from cosmic-
chronometer approach(Jimenez2003, ; Simon2005, ; Stern2010, ; Moresco2012, ;
Moresco2015, ; Moresco2016, ; Ratsimbazafy2017, ).

We reconstruct 31 Hubble parameter data from the cosmic chronometer, and the
results are shown in fig. 2. In addition, we also integrate the reconstruction
results and get the $d_{p}(z)$ and its error within 1 $\sigma$, as shown in
fig. 3.

Figure 2:  We reconstruct 31 Hubble parameter data from the cosmic chronometer
and show its error within 1 $\sigma$. Figure 3:  We get $d_{p}(z)$ and its
error for the reconstructed Hz integral, and the red line represents the error
of $d_{p}(z)$ within 1 $\sigma$.

###  2.3 Luminosity Distance From The Cosmic Chronometer

As we all know, in the Friedmann-Lemaitre-Robertson-Walker(FLRW) metric, the
luminosity distance can be expressed as:

$\displaystyle
d_{L}(z)=\left\\{\begin{array}[]{ll}\frac{c(1+z)}{H_{0}\sqrt{\left|\Omega_{\mathrm{K}}\right|}}\sinh\left[\sqrt{\left|\Omega_{\mathrm{K}}\right|}\int_{0}^{z}\frac{dz^{\prime}}{E\left(z^{\prime}\right)}\right],&\Omega_{K}>0\\\
\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{E\left(z^{\prime}\right)},&\Omega_{K}=0.\\\
\frac{c(1+z)}{H_{0}\sqrt{\left|\Omega_{\mathrm{K}}\right|}}\sin\left[\sqrt{\left|\Omega_{\mathrm{K}}\right|}\int_{0}^{z}\frac{dz^{\prime}}{E\left(z^{\prime}\right)}\right],&\Omega_{K}<0\end{array}\right.$
(13)

$\Omega_{K}$ represents the curvature density parameter and $z$ represents the
redshift. $E(z)=H(z)/H_{0}$ is given by the ratio of $H(z)$ to $H_{0}$.
$H(z)$(Hubble parameter) denotes the expansion rate at $z$, and $H_{0}$ is the
Hubble parameter at $z=0$, which is called the Hubble constant. In order to
eliminate the influence of the model, we can introduce comoving distance:

$d_{p}(z)=\int_{0}^{z}\frac{dz^{\prime}}{H\left(z^{\prime}\right)}.$ (15)

If we express $d_{p}(z)$ in terms of the Hubble parameter, we can get a
luminosity distance given by the Hubble parameter,

$\displaystyle
d_{L}^{Hz}(z)=\left\\{\begin{array}[]{ll}\frac{c(1+z)}{H_{0}\sqrt{\left|\Omega_{\mathrm{K}}\right|}}\sinh\left[\sqrt{\left|\Omega_{\mathrm{K}}\right|}H_{0}d_{p}(z)\right],&\Omega_{K}>0\\\
\frac{c(1+z)}{H_{0}}H_{0}d_{p}(z),&\Omega_{K}=0.\\\
\frac{c(1+z)}{H_{0}\sqrt{\left|\Omega_{\mathrm{K}}\right|}}\sin\left[\sqrt{\left|\Omega_{\mathrm{K}}\right|}H_{0}d_{p}(z)\right],&\Omega_{K}<0\end{array}\right.$
(19)

Accordingly, the error of $d_{L}^{Hz}(z)$ is given by error transmission,

$\displaystyle\sigma_{d_{L}^{Hz}(z)}=\left\\{\begin{array}[]{ll}\frac{c(1+z)}{H_{0}}\cosh\left[\sqrt{\left|\Omega_{K}\right|}H_{0}d_{p}(z)\right]\sigma_{d_{p}(z)},&\Omega_{K}>0\\\
c(1+z)\sigma_{d_{p}(z)},&\Omega_{K}=0\\\
\frac{c(1+z)}{H_{0}}\cos\left[\sqrt{\left|\Omega_{K}\right|}H_{0}d_{p}(z)\right]\sigma_{d_{p}(z)},&\Omega_{K}<0\end{array}\right.$
(24)

In this paper, we set the Hubble constant to $H_{0}=69.6\pm
0.7~{}km~{}s^{-1}Mpc^{-1}$ Bennett2014 , so the only free parameter contained
in the $d_{L}^{Hz}(z)$ is $\Omega_{K}$. We will use GW data to constrain the
parameter $\Omega_{K}$ through Markov chain Monte Carlo method in Sec. 3.

##  3 Result and Discussion

###  3.1 Improved Curvature Test Method

In the 2.2, we use the Gaussian process to reconstruct the Hubble parameter
data and obtain the luminosity distance($d_{L}^{Hz}(z)$) given by $H(z)$. It
is worth noting that the luminosity distance obtained by this method is not
exactly the same as that given by GW, so we smooth the $d_{L}^{Hz}(z)$.
Considering that there may be unknown errors in gravitational wave detection,
we add 10% systematic error to $d_{L}^{GW}(z)$. Therefore, we can constrain
the curvature density parameter through $\chi^{2}$ method:

$\chi^{2}=\sum_{i}\frac{\left[d_{L}^{Hz}\left(z_{i};\Omega_{K}\right)-d_{L}^{\mathrm{GW}}\left(z_{i}\right)\right]^{2}}{\sigma_{d_{L}^{Hz},i}^{2}+\sigma_{d_{L}^{\mathrm{GW}},i}^{2}},$
(26)

The curvature parameter result is shown in Fig.4, and in order to show the
constraint effect more intuitively, we compare the constraint results of
curvature parameters given by other observation data using the same method
(see Tab.1).

Figure 4:  The constraint results of the reconstructed cosmic chronometer and
DECIGO on the curvature parameter.

As can be seen from the Tab. 1, the cosmic curvature density parameter which
is constrained by the luminosity distance obtained from the reconstructed
$H(z)$ and the simulated GW data from DECIGO is $\Omega_{K}=-0.007\pm 0.016$.
In order to better compare the constraint ability of DECIGO to curvature
parameters, we add the research results of other scholars to the table. The
constraint result given by the third generation gravitational wave detector ET
is $\Omega_{K}=0.035\pm 0.039$, the curvature constraint ability of DECIGO is
better than that of ET. We also show the curvature constraint results of
supernovae and quasars. It is easy to see that the curvature constraint
accuracy given by the GW data is more than 90% higher than that given by
quasars ($\Omega_{K}=0.0\pm 0.3$) and supernovae data ($\Omega_{K}=0.09\pm
0.25$) respectively. In addition, we also noticed that Zheng et al.(Zheng2021,
) used the GW simulation data of DECIGO and ET to test the cosmic curvature,
and they used the third-order logarithm polynomial approximation of $D_{L}(z)$
with undetermined coefficients in their research, and directly constrained the
cosmic curvature through the research results of Clarkson et al.(Clarkson2007,
). They use DECIGO(0<z<2) to simulate data to constrain curvature to
$\Omega_{K}=0.004\pm 0.09$, and the result of using ET data in the same way is
$\Omega_{K}=0.01\pm 0.10$. Compared with the results of Zheng et
al.(Zheng2021, ), the research method used in this paper gives higher
constraint accuracy. As the space gravitational wave detector, the accuracy of
DECIGO is more than 50% higher than that of ET detector when using the same
method to measure curvature density parameters. In order to more intuitively
compare the results given by different data, we visualize the comparison of
the data (see Fig. 5).

Data | $\Omega_{K}$ |  | Source
---|---|---|---
DECIGO | $-0.007\pm 0.016$ |  | This Work
ET | $0.035\pm 0.039$ |  | (Wei2018, )
Quasar | $0.0\pm 0.3$ |  | (Cao2019b, )
supernove | $0.09\pm 0.25$ |  | (Wei2017, )
Table 1:  The results obtained by using the model-independent method to
constrain curvature. Figure 5:  Comparison of the results of four kinds of
data using model-independent method.

###  3.2 Curvature Test from Cosmological Model

In addition, we also investigate the constraint effect of DECIGO on curvature
density parameters under the assumption of cosmological model. The $\Lambda
CDM+\Omega_{K}$ model is,

$E(z)=\Omega_{m}(1+z)^{3}+\Omega_{K}(1+z)^{2}+(1-\Omega_{m}-\Omega_{K}),$ (27)

At the same time, we also combine the supernovaScolnic2018  data with
quasarCao2017a ; Cao2017b  data (EM) to constrain curvature parameter to
discuss the constraint effect of GW data and EM data on the curvature density
parameters in the $\Lambda CDM+\Omega_{K}$ model. Meanwhile, we divide the
sub-samples of simulated GW with redshifts $0<z<2$ from the full samples with
redshifts $0<z<5$ and assess the accuracy of $\Omega_{K}$ given by the model-
dependent($\Lambda CDM+\Omega_{K}$ model) method and Improved curvature test
method.

Figure 6:  The result of constraining the $\Lambda CDM+\Omega_{K}$ model using
DECIGO and EM, respectively.

As shown in the Tab. 2, The results of constraining $\Lambda CDM+\Omega_{K}$
model in the table show that the curvature best values and their $68\%$
confidence errors are $\Omega_{K}=-0.012\pm 0.11$ and $\Omega_{K}=-0.031\pm
0.05$ constrained by sub-sample DECIGO data at the redshift $0$ to $2$ and
full sample at $z=0$ to $5$ respectively. In addition, when two kinds of EM
(supernova + quasar) data are used, the constraint of curvature density
parameter is $\Omega_{K}=-0.034\pm 0.15$. For intuitive comparison, we also
show the model-based constraint results of three kinds of data in Fig. 7.

Data | $\Omega_{K}$ | $\Omega_{m}$ | $H_{0}$
---|---|---|---
EM | $-0.034\pm 0.15$ | $0.292\pm 0.071$ | $71.14\pm 0.25$
DECIGO (0<z<2) | $-0.012\pm 0.11$ | $0.284\pm 0.046$ | $70.16\pm 0.43$
DECIGO (0<z<5) | $-0.031\pm 0.05$ | $0.292\pm 0.016$ | $70.24\pm 0.40$
Table 2:  Results of $\Lambda CDM+\Omega_{K}$ models with different data
constraints. Figure 7:  The $\Lambda CDM+\Omega_{K}$ model-dependent curvature
density parameter $\Omega_{K}$

Our results show that in the method of directly assuming cosmological
models($\Lambda CDM+\Omega_{K}$), the accuracy of curvature parameter error
($\Delta\Omega_{K}=0.05$) given by the full sample with red shift
($z\in[0,5]$) is higher, which is only $0.5$ times of the curvature
error($\Delta\Omega_{K}=0.11$) given by the sub-sample ($z\in[0,2]$), and only
$0.3$ times of the error ($\Delta\Omega_{K}=0.15$) given by the EM data
sample. It can be seen that the error given by the full sample ($z\in[0,5]$)
is more than twice the result of Planck data constraint
$\Delta\Omega_{K}=0.018$ (Aghanim2020, ). In addition, it is worth noting that
the accuracy of the curvature parameter ($\Delta\Omega_{K}=0.11$) is lower
than that of the improved curvature test method ($\Delta\Omega_{K}=0.016$)
error given under the same redshift range ($z\in[0,2]$), as shown in Tab. 1.
However, the curvature constraint results obtained by improved curvature test
method are similar to those obtained by Planck data. For EM data, compared
with the constraint result of using supernova data alone on the model is
$\Omega_{K}=-0.062^{+0.189}_{-0.169}$ (Gao2020, ), the accuracy of curvature
is improved after adding quasars, but it is still lower than those from GW
data. Therefore, for gravitational waves, a improved curvature test method to
constrain curvature can be able to obtain higher accuracy results. In
addition, for the constraint results of matter density parameters
($\Omega_{m}$) in the $\Lambda CDM+\Omega_{K}$ model, the constraint accuracy
of GW data is also higher than that of EM data, and this constraint ability
becomes stronger with the increase of the number of events with higher
redshift of GW. For the Hubble constant($H_{0}$), the error results
constrained by the two kinds of GW data are slightly larger than those given
by EM data, but they are consistent with each other within the $1\sigma$ error
range.

##  4 Conclusion

Gravitational waves as a standard siren may open a new window for the study of
cosmology, and more interesting results are expected through gravitational
wave detection. In this paper, we use a improved curvature test method to
study curvature. Firstly, 31 sets of Hubble parameter data from the cosmic
chronometer are reconstructed using Gaussian process. After integrating the
reconstructed data, a set of luminosity distance $d_{L}^{Hz}(z)$ given by
$H(z)$ is obtained. Then, we use the data from DECIGO to simulate 10000 GW
events, and obtain their redshift, luminosity distance and corresponding
error. By comparing the luminosity distance given by the two kinds of data,
the curvature parameters are constrained. In our work, 1. we simulated 10000
GW events based on the estimation of future detection events by the DECIGO
project. 2. The curvature constraint results of the third generation GW
detectors (ET) and the space gravitational wave detectors (DECIGO) are
compared in our study. When the cosmic curvature is constrained by the
improved curvature test method, ET and DECIGO get the constraint results of
$\Omega_{K}=0.035\pm 0.039$,$\Omega_{K}=-0.007\pm 0.016$, respectively. The
results show that the space gravitational wave detectors can provide stronger
constraint effect. 3. We also compare the results of using improved curvature
test method and the method of using $\Lambda CDM+\Omega_{K}$ model to
constrain curvature based on DECIGO data, which are $\Omega_{K}=-0.007\pm
0.016$, $\Omega_{K}=-0.031\pm 0.05$. It shows that for DECIGO, improved
curvature test method can get stronger constraint effect, and the curvature
constraint accuracy obtained by improved curvature test method can be similar
to that of curvature constraint given by Planck2018 microwave background
($\Delta\Omega_{K}=0.018$) (Aghanim2020, ).

At the same time, we also compare the constraint results given by other
observation data. Compared with the third-generation gravitational wave
detector ET(Wei2018, ) and supernovae(Gao2020, ) and quasars(Cao2019b, ), the
error given by DECIGO is half that of ET, and the accuracy is one order of
magnitude higher than that of supernovae and quasars.

In addition, we also use DECIGO to constrain the $\Lambda CDM+\Omega_{K}$
model, and take the EM data as the control group. The results show that in the
case of constraint model, the curvature constraint effect of DECIGO is
slightly higher than that of EM. Meanwhile, GW have great potential for
curvature constraints under improved curvature test method conditions, and by
comparing the current constraint results, the gravitational wave detector
DECIGO has higher constraint ability of curvature constraints than ET and some
other current research results. We expect future gravitational wave detection
projects to provide a more accurate constraint on cosmic curvature. At the
same time, we note that some scholars have proposed a framework including
peculiar velocity correction for GW sourcesMukherjee2021 , which will provide
guidance for our next work direction.

###### Acknowledgements.

This work was supported in part by the National Natural Science Foundation of
China under Grant Nos. 11873001, 12047564 and by the Fundamental Research
Funds for the Central Universities under Grant No. 2020CDJQY-Z003. This
project is sponsored by the Scientific Research and Innovation Project of
Graduate Students in Chongqing No. CYS20272.

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The neutron skin-thickness of 208Pb determined by electron and proton
scattering

Toshio Suzuki, Rika Danjo and Toshimi Suda

Research Center for Accelerator and Radioisotope Science, Tohoku University

Sendai 982-0826, Japan

email<EMAIL_ADDRESS>
Abstract: Electron as well as proton elastic scattering is not able to
determine the point proton and point neutron densities,
$\rho_{\tau}(r),(\tau=p,n)$, separately. If both scatterings are analyzed
consistently, those densities would be determined uniquely, since the two
densities are observed by different combinations from each other. The previous
experiments did not provide $\rho_{\tau}(r)$ uniquely, but the values of the
mean square radii of $\rho_{p}(r)$, $\langle\,r^{2}\,\rangle_{p}$, and of
$\rho_{n}(r)$, $\langle\,r^{2}\,\rangle_{n}$, are shown to be determined
consistently through the fourth moment of the observed charge density,
$\langle\,r^{4}\,\rangle_{c}$, in 208Pb. The previous analyses of
$(\gamma,\pi^{0})$ and $\bar{p}$-nucleus obtained a similar value of
$\langle\,r^{2}\,\rangle_{n}$, but they do not yield the experimental value of
$\langle\,r^{4}\,\rangle_{c}$ observed in electron scattering.

One of the long-standing problems in the nuclear physics is if the mean square
radius of the point-neutron distribution(n-msr) 11endnote: 1The abbreviation
of the ‘rms’(root mean square)-radius is frequently used in the literature,
but it is convenient for the present purpose to employ ‘msr’ for the mean
square radius, because electron scattering observes the value of the msr
($\langle\,r^{2}\,\rangle_{c}$), together with the higher moments,
$\langle\,r^{2n}\,\rangle_{c},(n=2,3,\cdots)$, rather than the square root of
the msr. in nuclei is determined experimentally. On the one hand, the msr of
the point proton distribution(p-msr) has been determined almost model-
independently through the probe with well-known electromagnetic interaction
like electron scattering[1, 2], and has played an essential role for
understanding nuclear structure from the beginning of nuclear physics
history[3]. On the other hand, its counter-part, the n-msr, has been still
under discussions experimentally and theoretically, because its experimental
study has to employ probes with strong interaction. Compared with
electromagnetic interaction, the understanding of the strong interaction in
the many-nucleon system is not known without ambiguity and, moreover, the
reaction mechanism with strong interaction is not yet well established, as in
proton scattering[4, 5, 7, 8, 10, 9, 11]. As a result, there is no data table
of the n-msr yet, unlike those of the p-msr throughout the periodic table[12,
13].

Both electron and proton as probes interact with the protons and neutrons in
the nucleus, but each scattering itself is not able to observe the point-
proton distribution, $\rho_{p}(r)$ and the point-neutron distribution,
$\rho_{n}(r)$, separately. Electron scattering observes the nuclear charge
density, $\rho_{c}(r)$, as[2]

$\rho_{c}(r)=\rho_{cp}(r)+\rho_{cn}(r),$ (1)

where $\rho_{c\tau}(r),(\tau=p,\,n)$ denote the $\rho_{\tau}(r)$ folded by a
single proton and neutron charge distributions, respectively, which are
supposed to be known in other experiments. Proton scattering is analyzed by
assuming, for example, the potential as[8]

$U(r)=\sum_{\tau}\int
d^{3}r^{\prime}\rho_{\tau}(r)t_{\tau}(|\bm{r}-\bm{r}^{\prime}),$ (2)

where $t_{\tau}$ indicate the free nucleon-nucleon $t-$matrix. Thus, the
electron and proton are scattered by $\rho_{\tau}(r)$ in different ways. In
principle, therefore, if proton scattering were described transparently as
electron scattering, the combined data would provide almost uniquely
$\rho_{\tau}(r)$.

Ref.[4, 5, 6, 7] aimed to analyze electron and proton scattering consistently.
In the first step, the author of Ref.[5] obtained $\rho_{\tau}$ by using
experimental values of the right-hand side in Eq.(1), but assuming each
contribution of $\rho_{\tau}(r)$ to it model-dependently[14], because electron
scattering can not observe them separately as mentioned before. Next, proton
scattering is analyzed with the use of the obtained $\rho_{\tau}(r)$, and he
determines the best $\rho_{\tau}(r)$ to reproduce the proton-scattering cross
sections. Third, the obtained new $\rho_{\tau}(r)$ is examined if the original
electron scattering data are reproduced. According to such iterations, it is
found that a few repetitions are enough for the convergence, if the first
trial function of $\rho_{\tau}(r)$ are well prepared[5]. The model-dependence
in the first step is expected to disappear in the iterations.

Nevertheless, various experiments and their analyses[15, 16, 17, 18] together
with proton scattering[4, 5, 6, 7, 8, 10, 9, 11] were repeated for the last
forty years. The reason may be because there is no unique understanding of the
reaction mechanism and used parameters, so that there is not enough confidence
for $\rho_{n}(r)$ to be derived in the analyses of the proton scattering.
Indeed, all the analyses in Refs.[4, 5, 6, 7, 8, 9, 10, 11] used different
descriptions for the strong interaction with $\rho_{\tau}(r)$, sometimes with
physically different parameters like assumed effective masses of nucleons and
mesons in the nuclear medium as in Refs.[7, 8, 9] and [10, 11].

Among experiments accumulated until now, however, the noteworthy fact is that,
when the $\rho_{p}(r)$ are inferred from the electron-scattering data, all the
analyses of the proton scattering[4, 5, 6, 7, 8, 9, 10, 11] predict the value
of the neutron skin thickness, $\delta R$, to be about $0.14\sim 0.2$, where
$\delta R$ is defined by $\delta
R=\sqrt{\langle\,r^{2}\,\rangle_{n}}-\sqrt{\langle\,r^{2}\,\rangle_{p}}$ with
the n-msr and p-msrh denoted as $\langle\,r^{2}\,\rangle_{n}$ and
$\langle\,r^{2}\,\rangle_{p}$, respectively. This fact implies that the value
of $\langle\,r^{2}\,\rangle_{n}$ proportional to $\int r^{4}\rho_{n}(r)$dr
depends weakly on the various reaction mechanisms assumed and different
parameters used in the proton-scattering analyses, as far as the unfolded
charge densities inferred from electron scattering are employed. In order to
confirm the reliability of the value of $\langle\,r^{2}\,\rangle_{n}$,
therefore, it may not be appropriate to compare the $\rho_{n}(r)$-profiles
with one another, within proton-scattering analyses. Instead, it is better to
explore a bridge between electron- and proton-scattering analyses, to confirm
the validity of $\langle\,r^{2}\,\rangle_{n}$. It will be shown below that the
moments of the charge distribution observed in electron scattering play a role
of such a bridge.

Fortunately, the second moment(msr) of $\rho_{c}(r)$,
$\langle\,r^{2}\,\rangle_{c}$, does not depend on $\rho_{cn}(r)$[19]. The
p-msr is given by the second moment as[20]

$\langle\,r^{2}\,\rangle_{c}=\frac{1}{Z}\langle\,0\,|\,\sum_{k=1}^{Z}r_{k}^{2}\,|\,0\,\rangle+r^{2}_{p}+r^{2}_{n}\frac{N}{Z}+C_{\rm
rel},$ (3)

where the first term of the right-hand side stands for the p-msr,
$\langle\,r^{2}\,\rangle_{p}$, while $r^{2}_{p}$ and $r^{2}_{n}$ indicate the
mean square radii of the proton and neutron themselves, respectively[20].
Ref.[20] used their values as $r_{p}=0.877$ fm and $r^{2}_{n}=-0.116$ fm2. The
last term of Eq.(3) represents the relativistic correction up to order of
$1/M^{2}$ which is written as[20]

$C_{\rm
rel}=\frac{1}{M^{2}}\left(\frac{1}{Z}\sum_{k=1}^{A}\mu_{k}\langle\,0\,|\,\bm{\ell}_{k}\\!\cdot\\!\bm{\sigma}_{k}\,|\,0\,\rangle+\frac{3}{4}+\frac{1}{2Z}\sum_{k=1}^{Z}\langle\,0\,|\,\bm{\ell}_{k}\\!\cdot\\!\bm{\sigma}_{k}\,|\,0\,\rangle\right),$
(4)

with $M$ denoting the nucleon mass to be $939$ MeV, and $\mu_{k}$ the
anomalous magnetic moment, $\mu_{p}=1.793$ for proton and $\mu_{n}=-1.913$ for
neutron. Estimation of the relativistic correction is almost model-independent
in 208Pb as a double closed shell nucleus. For example, Ref.[20] estimated,
against the experimental value of $\langle\,r^{2}\,\rangle_{c}=30.283(0.154)$
fm2[12, 13], the value of $C_{\rm rel}$ to be about $0.021$ fm2 with the use
of the Skyrme mean-field models, yielding $\langle\,r^{2}\,\rangle_{p}=29.671$
fm2. We note that the last $\mu_{p}-$independent term in the parenthesis of
the right-hand side in Eq.(4) stems from the the Foldy-Wouthuysen
transformation[1] of the four-component wave function to the two-component
one, together with the first two terms[20].

| $\langle\,r^{4}\,\rangle_{c}$ | $\langle\,r^{2}\,\rangle_{c}$ | $\langle\,r^{4}\,\rangle_{p}$ | $\langle\,r^{2}\,\rangle_{p}$ | $\langle\,r^{2}\,\rangle_{n}$ | $\Delta_{2}/\Delta_{4}$ | $\delta R$
---|---|---|---|---|---|---|---
$(e,N)$ | $1171.981$ | $30.283$ |  | $(29.671)$ |  | $(\Delta_{2}=0.612)$ |
$(p,N)$ | $[1173.3]$ | $[30.265]$ | $1119.6$ | $29.790$ | $31.900$ | $(0.475/3.416)$ | $0.198$
LSA | $[1171.981]$ | $[30.283]$ | $1111.855$ | $29.738$ | $31.507$ | $\Delta_{4}=2.605$ | $0.160$
$(\bar{p},N)$ | $(1156.047)$ |  | $(1098.016)$ | $(29.554)$ | $31.311$ |  | $0.159$
$(\gamma,\pi^{0})$ | $(1155.040)$ |  | $(1096.854)$ | $(29.569)$ | $31.114$ |  | $0.140$
Table 1:  The $n$th moments of the charge, proton, and neutron distribution in
208Pb obtained in various analyses. The values in the square brackets indicate
the experimental values used in the analyses, while those in the parentheses
are calculated with the assumed distributions. The values of LSA are obtained
by the least squares analysis of the non-relativistic mean-field models in
Ref.[20]. The neutron skin-thickness, $\delta R$, is defined by
$(\sqrt{\langle\,r^{2}\,\rangle_{n}}-\sqrt{\langle\,r^{2}\,\rangle_{p}})$. All
the values are given in units of fmn. For the details, see the text.

The $n(\geq 4)$th moment of the charge distribution depends not only on the
$m(\leq n)$th moments of $\rho_{p}(r)$[19], but also on the $m(\leq(n-2))$th
ones of $\rho_{n}(r)$. For example, $\langle\,r^{4}\,\rangle_{c}$ is given
by[20]

$\langle\,r^{4}\,\rangle_{c}=\langle\,r^{4}\,\rangle_{p}+\frac{10}{3}r_{p}^{2}\langle\,r^{2}\,\rangle_{p}+\frac{10}{3}r_{n}^{2}\langle\,r^{2}\,\rangle_{n}\frac{N}{Z}+\Delta_{4},$
(5)

where $\Delta_{4}$ represents the fourth moment of a single proton and neutron
charge distribution and relativistic corrections. Ref.[19, 20] shows the
explicit expression of $\Delta_{4}$ and its value is estimated model-
dependently in Ref.[20]. The last three terms of Eq.(3) will be expressed as
$\Delta_{2}$ from now on in the same way.

On the one hand, in electron scattering, the experimental value of
$\langle\,r^{4}\,\rangle_{c}$ is known[20, 21] and the second term of the
right-hand side of the above equation is determined by Eq.(3), taking into
account a small correction due to $\Delta_{2}$. On the other hand, in proton
scattering, Ref.[8] summarized the values of the moments in the first three
values of the right-hand side, which were obtained from $\rho_{\tau}(r)$
derived in their analyses of the experimental data in Ref.[4, 5, 6, 7].
Ref.[8] listed also the value of the moments of the charge distribution
obtained in the analysis of electron-scattering data[22]. These values are
listed in the $(e,N)-$ and $(p,N)-$rows of Table 1 in electron and proton
scatterings, respectively.

The experimental values of $\langle\,r^{4}\,\rangle_{c}=1171.981$ fm4 and
$\langle\,r^{2}\,\rangle_{c}=30.283$ fm2 in the $(e,N)$-row are obtained in
the sum-of-Gaussians analysis of electron-scattering data[12, 21], which are
almost equal to those by the Fourier-Bessel one[20, 24], while Ref.[8] employs
the values taken from Ref.[22] which assumed the three-point Gaussian
distribution for the charge density. The values of $r_{p}$ and $r_{n}^{2}$ in
Ref.[5] cited in Ref.[8], are taken from Ref.[25] which provides $0.836$ fm
and $-0.117$ fm2, respectively. By employing these values, in order to
reproduce the experimental value of $\langle\,r^{4}\,\rangle_{c}=1173.3$ fm4,
we need the value of $\Delta_{2}=0.475$ fm2 and $\Delta_{4}=3.416$ fm4, which
are comparable to $\Delta_{2}=0.612$ fm2 and $\Delta_{4}=2.605$ fm4 listed in
the $(e,N)$\- and LSA-rows in Table 1, respectively. These values of LSA have
been obtained by the least squares analysis(LSA) of the non-relativistic mean-
field(NMF)-models in Ref.[20].

The least squares method is as follows. First, a number of the NMF-models are
chosen arbitrarily in the literature[23], where there are more than $100$
versions accumulated for the last $50$ years. By using them, the values of
$\langle\,r^{4}\,\rangle_{c}$ and $\langle\,r^{2}\,\rangle_{p}$ are calculated
according to Eqs.(3) and (5). Next, those values are plotted in the
$\langle\,r^{2}\,\rangle_{p}-\langle\,r^{4}\,\rangle_{c}$ plane, to obtain the
least squares regression line(LSL) between $\langle\,r^{4}\,\rangle_{c}$ and
$\langle\,r^{2}\,\rangle_{p}$. Finally, The value of
$\langle\,r^{2}\,\rangle_{p}$ accepted for the used model-framework is
determined by the cross point(LSL-value) of the LSL and the line of the
experimental value of $\langle\,r^{4}\,\rangle_{c}$. Ref.[24] proved that the
LSL-values provides uniquely the value of each component of the reference
formula like Eqs.(3) and (5), as listed in Table 1. It should be noted that
these values of the components are not experimental values, but are allowed in
the model-framework as for the NMF-models[24]. The relativistic mean-
field(RMF)-models provide different LSL-values[20] from those in Table 1,
because the definitions of $\langle\,r^{n}\,\rangle_{c}$ and
$\langle\,r^{n}\,\rangle_{\tau}$ themselves are different from Eqs.(3) and
(5). The LSL-values obtained for the RMF-models which should be compared with
the values obtained with the relativistic analysis of proton scattering,
including the relativistic spin-orbit charge density[27] which is more
important for $\Delta_{4}$ than that in Table 1[20]. As far as the authors
know, however, there is not such a analysis which is performed consistently
with electron-scattering data by iteration-method[26]. .

Now, since proton[4, 5, 6, 7, 8]\- and electron[12, 22]-scatterings were
analyzed by using the same $\rho_{\tau}$, the experimental values of
$\langle\,r^{4}\,\rangle_{c}$ and $\langle\,r^{2}\,\rangle_{c}$ observed in
electron scattering should be reproduced by the values of their components
determined by proton-scattering analyses, according to Eqs.(3) and (5). Table
1 shows that the values of the components in the $(p,N)$-row reproduce
remarkably well the experimental values of $\langle\,r^{4}\,\rangle_{c}$ and
$\langle\,r^{2}\,\rangle_{c}$. We note that in order to obtain
$\langle\,r^{4}\,\rangle_{c}=1173.3$, Ref.[8] published in 1995 did not use
the right-hand side of Eq.(5) in Ref.[19] which appeared in 2019.

There are a few comments on Table 1. First, the values of the components
listed in the $(p,N)$-row are in good agreement with the LSL-values. The only
exception is between the values of $\langle\,r^{4}\,\rangle_{p}$. The value
given by proton scattering is $1119.6$ fm4, while that by the LSA $1111.855$
fm4. In electron scattering, however, the observed
$\langle\,r^{4}\,\rangle_{c}$ depends on the values of
$\langle\,r^{4}\,\rangle_{p}$, $\langle\,r^{2}\,\rangle_{p}$ and $r^{2}_{p}$
as
$\langle\,r^{4}\,\rangle_{p}+\frac{10}{3}r_{p}^{2}\langle\,r^{2}\,\rangle_{p}$
in Eq.(5). In taking into account the value of $r_{p}=0.836$ fm used in the
proton-scattering analyses and $0.877$ fm in the LSA, the above sums become
$1188.096$ fm4 and $1189.000$ fm4 in the proton scattering analysis and the
LSA, respectively. Thus, it is seen that electron scattering provides a strong
constraint on the values of the moments of $\rho_{p}(r)$.

Second, Eq.(5) is written as

$\langle\,r^{4}\,\rangle_{c}=\langle\,r^{4}\,\rangle_{p}+\frac{10}{3}(r_{p}^{2}+r^{2}_{n}\frac{n}{Z})\langle\,r^{2}\,\rangle_{p}+\frac{10}{3}r_{n}^{2}\frac{N}{Z}(\langle\,r^{2}\,\rangle_{n}-\langle\,r^{2}\,\rangle_{p})+\Delta_{4}$
(6)

The Table 1 provides for (p-N) and LSA, respectively, as

$\displaystyle 1173.3$ $\displaystyle=1119.6+51.548-1.264+3.416,$ (7)
$\displaystyle 1171.981$ $\displaystyle=1111.855+58.572-1.051+2.605,$ (8)

where each value in the right-hand side corresponds to those in Eq.(6). As
mentioned in the previous comment, the sum of the first two terms of the above
equations are constrained by $\rho_{p}(r)$. The last term denotes the value of
$\Delta_{4}$ which depends on the fourth moment of the nucleon and the spin-
orbit density[19, 20]. Thus, the contribution of
$(\langle\,r^{2}\,\rangle_{n}-\langle\,r^{2}\,\rangle_{p})$-term to
$\langle\,r^{4}\,\rangle_{c}$ is about $0.1\%$, reflecting
$|r^{2}_{n}|<r^{2}_{p}$. Hence, in electron scattering only, it seems to be
impossible to determine the value of $\delta R$. In contrast to Eq.(6) in
electron scattering, Eq.(2) indicates

$t_{p}\rho_{p}+t_{n}\rho_{n}=(t_{p}+t_{n})\rho_{p}+t_{n}(\rho_{n}-\rho_{p}),$
(9)

implying that the last term contributes much to proton-scattering cross
section. Indeed, for example, Ref. [6] pointed out that $\rho_{n}(r)$, which
provides $\delta R>0$, is sensitive to the maximum- and minimum-position and
the slope of the angular distribution in proton-scattering cross section.

Third, Table 1 lists the values of $\langle\,r^{2}\,\rangle_{n}$ observed in
$\bar{p}-$atom as $(\bar{p},N)$[16], and in coherent $\pi$ production as
$(\gamma,\pi^{0})$[17] for reference. Their values were obtained by using the
two-point Fermi distributions of $\rho_{\tau}(r)$. When using the given two
parameters, and $r_{p}=0.877$ fm and $r^{2}_{n}=-0.116$ fm2, the values of
$\langle\,r^{4}\,\rangle_{c}$ are obtained as in Table 1 22endnote: 2Refs.[16]
and [17] provided the values of $\delta R$ to be $0.16$ and$0.15$ fm,
respectively, obtained approximately by the two-point Fermi distributions.
Table 1 lists the values by the exact formulae with the same two parameters
for the Fermi distribution, because the values of the fourth moments are
required for comparison.. They are much smaller than others, implying that
consistent analyses are necessary for discussions of $\delta R$.

Finally, all the arguments in the present paper have not taken into account
the experimental errors, because our purpose is to understand a qualitative
relationship between electron and proton scatterings. Precise determination of
not only the values of $r_{p}$ and $r^{2}_{n}$, but also those of the higher
moments$(n>4)$ of the charge distribution would provide more information on
$\rho_{n}(r)$ and $\delta R$[24]. For example, Ref.[8] lists the value of
$\langle\,r^{4}\,\rangle_{n}$ to be $1317.3$ fm4, while the LSA in Ref.[24]
$1282.926$ fm 4. In comparing $\delta R$s of various experiments with one
another, differences between the estimated values of
$\langle\,r^{2}\,\rangle_{p}$ should be also examined.

In conclusion, both electron- and proton-scattering experiments are necessary
for determination of the moments of $\rho_{n}(r)$. The fourth moment,
$\langle\,r^{4}\,\rangle_{c}$, of the charge distribution plays a role to
verify the consistency between the moments of $\rho_{\tau}(r)$ estimated by
those experiments. Such a role is expected to become more important in future
in studying neutron-rich unstable nuclei[28], where contributions from the
moments of the $\rho_{n}(r)$ to $\langle\,r^{n}\,\rangle_{c}$ would more
increase than in stable nuclei[29].

## Acknowledgments

This work was supported by JSPS KAKENHI Grant Numbers JP22K18706.

###### Notes

  1. 1 The abbreviation of the ‘rms’(root mean square)-radius is frequently used in the literature, but it is convenient for the present purpose to employ ‘msr’ for the mean square radius, because electron scattering observes the value of the msr ($\langle\,r^{2}\,\rangle_{c}$), together with the higher moments, $\langle\,r^{2n}\,\rangle_{c},(n=2,3,\cdots)$, rather than the square root of the msr.
  2. 2 Refs.[16] and [17] provided the values of $\delta R$ to be $0.16$ and$0.15$ fm, respectively, obtained approximately by the two-point Fermi distributions. Table 1 lists the values by the exact formulae with the same two parameters for the Fermi distribution, because the values of the fourth moments are required for comparison.

## References

  * [1] J. D. Bjorken and S. D. Drell, Relativistic quantum mechanics (McGraw Hill Book Company, 1964).
  * [2] T. deForest and J. D. Walecka, Adv. Phys. 15, 1 (1966).
  * [3] A. Bohr and B. R. Mottelson, Nuclear structure, vol.1 (World Scientific Publishing Co. Pte. Ltd., 1998).
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  * [18] A. Tamii, et al., Phys. ReV. Lett. 107, 062502 (2011).
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  * [21] H. J. Emrich, PhD thesis, Johannes-Gutenberg-Universit$\ddot{a}$t, Mainz,1983.
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# Angle structures on $3$-manifolds

Anton Mellit<EMAIL_ADDRESS>Faculty of Mathematics, University of
Vienna,
Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

###### Abstract.

Given a compact oriented triangulated $3$-manifold we find a non-trivial
condition satisfied by certain labelings of the tetrahedra by elements of an
arbitrary abelian group which we call angle structures. Smoothness of the
manifold is used in an essential way. This is inspired by the notion of the
volume of hyperbolic manifolds, which would correspond to the case when the
abelian group is the multiplicative group of $\mathbb{C}$, but the
construction here seems to be more general, in particular it only uses the
abelian group structure.

Partially supported by the projects Y963-N35 and P-31705 of the Austrian
Science Fund. This note was written in 2007 during the author’s stay at MPIfM.
It is uploaded to arXiv at the request of Herbert Gangl.

Let $A$ be an abelian group. Let $X$ be a trangulated oriented $3$-dimensional
manifold. Let $\Delta=(a,b,c,d)$ be an oriented $3$-dimensional simplex of the
triangulation.

######  Definition 1.

An angle structure on $\Delta$ a collection of $12$ elements of $A$, denoted
$k_{xy}$, one for each oriented edge $xy$ satisfying relations:

$k_{xy}=k_{yx},\;\sum_{y}k_{xy}=0\;\text{for any vertex $x$.}$

We have

$k_{ab}+k_{ac}+k_{ad}=0,\;k_{ab}+k_{bc}+k_{bd}=0,\;k_{ac}+k_{bc}+k_{cd}=0,\;k_{ad}+k_{bd}+k_{cd}=0.$

This implies

$2(k_{ab}-k_{cd})=2(k_{ac}-k_{bd})=2(k_{ad}-k_{bc})=0.$

Suppose each tetrahedron of the triangulation has an angle structure. The
corresponding elements will be denoted $k_{ab}^{cd}\in A$, i.e. for a simplex
$\Delta=(a,b,c,d)$ we have $12$ elements listed below:

$k_{ab}^{cd}=k_{ba}^{dc},k_{ac}^{db}=k_{ca}^{bd},k_{ad}^{bc}=k_{da}^{cb},k_{bc}^{ad}=k_{cb}^{da},k_{bd}^{ca}=k_{db}^{ac},k_{cd}^{ab}=k_{dc}^{ba}.$

Therefore the notation is invariant with respect to even permutations of
vertices of $\Delta$.

######  Definition 2.

For each edge $ab$ of $X$ let $(abx_{i}x_{i+1})$ be the $3$-simplices adjacent
to $ab$ with $i=0,1,\ldots,n-1$, and $x_{n}=x_{0}$. Suppose the following
condition is satisfied:

$\sum_{i}k_{ab}^{x_{i}x_{i+1}}=0.$

Then we call our manifold _angled_.

Suppose we have a finite set of tetrahedra with angle structure. Is it
possible to glue them together and obtain an angled oriented manifold? We
provide a necessary condition.

######  Theorem 3.

In an angled oriented manifold the following condition is satisfied in the
group $\Lambda^{2}A$:

$\sum_{abcd}4k_{ab}^{cd}\wedge k_{ac}^{db}+k_{ad}^{bc}\wedge
k_{ad}^{bc}+k_{bc}^{ad}\wedge k_{bc}^{ad}=0,$

the sum is over $3$-simplices which compose the fundamental class of the
manifold.

######  Remark 4.

Let $A=\mathbb{C}^{\times}$ and each tetrahedron is realized as an ideal
tetrahedron in $\mathbb{H}^{3}$ with cross-ratio $z$. Then its angles are $z$,
$\frac{1}{1-z}$, $1-\frac{1}{z}$. To make the product $1$ one can change
angles to $-z$, $\frac{1}{z-1}$, $\frac{1}{z}-1$. Then we see that up to
$2$-torsion the sum of tensors $z\wedge(1-z)$ is $0$, which is well known. In
other words, hyperbolic $3$-manifolds provide elements in the Bloch group, see
[NZ85], [DS82]. However our approach seems to be more general.

The rest of this paper provides a proof of the theorem.

Suppose $X$ is angled. Then we can construct elements $h_{ab}^{c}\in A$ with
the property

$k_{ab}^{cd}=h_{ab}^{d}-h_{ab}^{c}$

for each oriented simplex $(a,b,c,d)$. If $h^{\prime c}_{ab}$ is another such
family then there is a family $q_{ab}$ with

$h^{\prime c}_{ab}-h_{ab}^{c}=q_{ab}.$

Fix a vertex $a$. Let $ab$ and $ab^{\prime}$ be edges. Join $ab$ and
$ab^{\prime}$ by a sequence of triangles $(ab_{i}b_{i+1})$ for
$i=0,1,\ldots,n-1$, $b_{0}=b$, $b_{n}=b^{\prime}$. Put

$m_{a}(b,b^{\prime})=\sum_{i}(h_{ab_{i}}^{b_{i+1}}-h_{ab_{i+1}}^{b_{i}}).$

This does not depend on the choice of the sequence since for any oriented
$3$-simplex $(abcd)$ we have

$(h_{ab}^{c}-h_{ac}^{b})+(h_{ac}^{d}-h_{ad}^{c})+(h_{ad}^{b}-h_{ab}^{d})=-k_{ab}^{cd}-k_{ac}^{db}-k_{ad}^{bc}=0.$

Therefore there is a family $q_{ab}\in A$ with the property

$m_{a}(b,b^{\prime})=q_{ab^{\prime}}-q_{ab}.$

If $q^{\prime}_{ab}$ is another such family there exists a family $p_{a}\in A$
with

$q^{\prime}_{ab}-q_{ab}=p_{a}.$

In particular for any $2$-simplex $(abc)$ we have

$h_{ab}^{c}-h_{ac}^{b}=q_{ac}-q_{ab}.$

We see that we can replace $h_{ab}$ with $h_{ab}^{c}+q_{ab}$ to make $h$
satisfying

$h_{ab}^{c}=h_{ac}^{b}\;\text{for any $2$-simplex $(abc)$.}$

If $h^{\prime c}_{ab}$ is another family with this condition then there is a
family $p_{a}\in A$ with the property

$h^{\prime c}_{ab}-h^{c}_{ab}=p_{a}.$

Put $h^{\prime c}_{ab}=-h^{c}_{ba}$. Then for any oriented $3$-simplex
$(abcd)$

$k_{ab}^{cd}=k_{ba}^{dc}=h_{ba}^{c}-h_{ba}^{d}=h^{\prime d}_{ab}-h^{\prime
c}_{ab}.$

Therefore there exists a family $q_{ab}$ (this $q$ is different from the one
used before) with property

$h^{c}_{ab}-h^{\prime c}_{ab}=q_{ab}.$

This means that for any $2$-simplex $(abc)$ we have

$h^{c}_{ab}+h^{c}_{ba}=q_{ab}.$

Let us summarize the properties of $h^{c}_{ab}$:

$h_{ab}^{c}=h_{ac}^{b},\;h^{c}_{ab}+h^{c}_{ba}=q_{ab},\;k_{ab}^{cd}=h_{ab}^{d}-h_{ab}^{c}.$

Let $(abc)$ be a $2$-simplex. Then

$q_{ab}+q_{ac}-q_{bc}=h_{ab}^{c}+h_{ba}^{c}+h_{ac}^{b}+h_{ca}^{b}-h_{bc}^{a}-h_{cb}^{a}=2h_{ab}^{c}.$

Consider the following element in $\Lambda^{2}A$:

$\phi_{abc}=h_{ab}^{c}\wedge h_{bc}^{a}+h_{bc}^{a}\wedge
h_{ca}^{b}+h_{ca}^{b}\wedge h_{ab}^{c}.$

This element is invariant under cyclic permutations:

$\phi_{abc}=\phi_{bca}=\phi_{cab}.$

Moreover,

$\phi_{acb}=h_{ac}^{b}\wedge h_{cb}^{a}+h_{cb}^{a}\wedge
h_{ba}^{c}+h_{ba}^{c}\wedge h_{ac}^{b}=h_{ab}^{c}\wedge
h_{ca}^{b}+h_{ca}^{b}\wedge h_{bc}^{a}+h_{bc}^{a}\wedge
h_{ab}^{c}=-\phi_{abc}.$

Since $h_{bc}^{a}=q_{ab}-h_{ab}^{c}$ and $h_{ca}^{b}=q_{ac}-h_{ab}^{c}$, we
can also write $\phi_{abc}$ as

$\phi_{abc}=2h_{ab}^{c}\wedge(q_{ab}-q_{ac})+q_{ab}\wedge
q_{ac}-h_{ab}^{c}\wedge h_{ab}^{c}\\\ =q_{ac}\wedge q_{ab}+q_{ab}\wedge
q_{bc}+q_{bc}\wedge q_{ac}+q_{ab}\wedge q_{ab}+q_{ac}\wedge
q_{ac}+h_{ab}^{c}\wedge h_{ab}^{c}\\\ =q_{ac}\wedge q_{ab}+q_{ab}\wedge
q_{bc}+q_{bc}\wedge q_{ac}+q_{bc}\wedge q_{bc}+h_{ab}^{c}\wedge h_{ab}^{c}.$

Let

$\phi_{abc}^{0}=q_{ac}\wedge q_{ab}+q_{ab}\wedge q_{bc}+q_{bc}\wedge
q_{ac},\;\phi_{abc}^{1}=q_{bc}\wedge q_{bc}+h_{ab}^{c}\wedge h_{ab}^{c}.$

For any oriented $3$-simplex $(abcd)$ put

$\psi^{*}_{abcd}=\phi^{*}_{bcd}-\phi^{*}_{acd}+\phi^{*}_{abd}-\phi^{*}_{abc}.$

Then

$\psi^{0}_{abcd}=q_{bc}\wedge q_{cd}+q_{cd}\wedge q_{bd}+q_{bd}\wedge
q_{bc}-q_{ac}\wedge q_{cd}-q_{cd}\wedge q_{ad}-q_{ad}\wedge q_{ac}\\\
+q_{ab}\wedge q_{bd}+q_{bd}\wedge q_{ad}+q_{ad}\wedge q_{ab}-q_{ab}\wedge
q_{bc}-q_{bc}\wedge q_{ac}-q_{ac}\wedge q_{ab}\\\
=q_{ab}\wedge(q_{bd}-q_{ad}-q_{bc}+q_{ac})+q_{cd}\wedge(-q_{bc}+q_{bd}+q_{ac}-q_{ad})+(q_{bd}+q_{ac})\wedge(q_{bc}+q_{ad})\\\
=(q_{bc}+q_{ad})\wedge(q_{ab}+q_{cd})+(q_{ab}+q_{cd})\wedge(q_{bd}+q_{ac})+(q_{bd}+q_{ac})\wedge(q_{bc}+q_{ad}).$

We may rewrite

$\displaystyle q_{bc}+q_{ad}$
$\displaystyle=h_{bc}^{a}+h_{cb}^{a}+h_{ad}^{c}+h_{da}^{c}=h_{bc}^{a}+h_{dc}^{a}+q_{ac}-k_{ca}^{bd},$
$\displaystyle q_{ab}+q_{cd}$
$\displaystyle=h_{ab}^{c}+h_{ba}^{c}+h_{cd}^{a}+h_{dc}^{a}=h_{bc}^{a}+h_{dc}^{a}+q_{ac}+k_{ca}^{bd}.$

Therefore

$\psi^{0}_{abcd}=2k_{ca}^{bd}\wedge(q_{bd}+q_{ac})-2k_{ca}^{bd}\wedge(h_{bc}^{a}+h_{dc}^{a}+q_{ac})+k_{ca}^{bd}\wedge
k_{ca}^{bd}+h_{bc}^{a}\wedge h_{bc}^{a}+h_{dc}^{a}\wedge
h_{dc}^{a}+q_{ac}\wedge q_{ac}\\\
=2k_{ca}^{bd}\wedge(q_{bd}-h_{bc}^{a}-h_{dc}^{a})+q_{bd}\wedge
q_{bd}+q_{ac}\wedge q_{ac}.$

Take into account that

$q_{bd}-h_{bc}^{a}-h_{dc}^{a}=h_{bd}^{a}+h_{db}^{a}-h_{bc}^{a}-h_{dc}^{a}=h_{ba}^{d}+h_{da}^{b}-h_{ba}^{c}-h_{da}^{c}=-k_{ba}^{dc}+k_{da}^{cb},$

we obtain

$\psi^{0}_{abcd}=2k_{ca}^{bd}\wedge(k_{da}^{cb}-k_{ba}^{dc})+q_{bd}\wedge
q_{bd}+q_{ac}\wedge q_{ac}=4k_{ab}^{cd}\wedge k_{ac}^{db}+q_{bd}\wedge
q_{bd}+q_{ac}\wedge q_{ac}.$

Now we turn to $\psi^{1}_{abcd}$.

$\psi^{1}_{abcd}=q_{cd}\wedge q_{cd}+h_{bc}^{d}\wedge h_{bc}^{d}+q_{cd}\wedge
q_{cd}+h_{ac}^{d}\wedge h_{ac}^{d}+q_{ab}\wedge q_{ab}+h_{da}^{b}\wedge
h_{da}^{b}+q_{ab}\wedge q_{ab}+h_{ca}^{b}\wedge h_{ca}^{b}\\\ =q_{ad}\wedge
q_{ad}+q_{bc}\wedge q_{bc}+k_{da}^{cb}\wedge k_{da}^{cb}+k_{cb}^{da}\wedge
k_{cb}^{da}.$

Therefore

$\psi_{abcd}=4k_{ab}^{cd}\wedge k_{ac}^{db}+k_{ad}^{bc}\wedge
k_{ad}^{bc}+k_{bc}^{ad}\wedge k_{bc}^{ad}.$

We see that $\psi_{abcd}$ depends only on the angles and its sum over the
manifold is zero. If $2$ is invertible in $A$ then

$\frac{1}{4}\psi_{abcd}=k_{ab}\wedge k_{ac}.$

This specializes to $z\wedge(1-z)$ in the case of an ideal hyperbolic
tertrahedron from Remark 4.

## References

  * [DS82] Johan L. Dupont and Chih Han Sah.  Scissors congruences. II.  J. Pure Appl. Algebra, 25(2):159–195, 1982.
  * [NZ85] Walter D. Neumann and Don Zagier.  Volumes of hyperbolic three-manifolds.  Topology, 24(3):307–332, 1985.


Xin Jing1,3, Luyang Zhang2, Jiangjian Xie2 Corresponding author:
<EMAIL_ADDRESS>Alexander Gebhard3, Alice Baird5, Björn Schuller1,3,6

# DB3V: A Dialect Dominated Dataset of Bird Vocalisation
for Cross-corpus Bird Species Recognition

###### Abstract

In ornithology, bird species are known to have varied dialects in their calls
across different regions. Consequently, computational methods to identify bird
species only through their calls face critical challenges. There is growing
interest in understanding the impact of dialects on the effectiveness of bird
species recognition. Despite potential mitigation through the expansion of
dialect datasets, the absence of publicly available testing data currently
impedes robust benchmarking efforts. This paper presents the Dialect Dominated
Dataset of Bird Vocalisation (D3BV ), the first cross-corpus dataset that
focuses on dialects in bird vocalisations. The D3BV comprises more than 25
hours of audio recordings from 10 bird species distributed across three
distinct regions in the contiguous United States (CONUS). In addition to
presenting the dataset, we conduct analyses and establish baseline models for
cross-corpus bird recognition. The data and code are publicly available
online: BLINDED

Index Terms: bioacoustic, computer audition, audio dataset, species
identification

##  1 Introduction

Bird monitoring is pivotal for biodiversity assessment and environmental
conservation [r2, r3]. Passive Acoustic Monitoring (PAM) technology, as
outlined in prior studies [r1], has emerged as a key method for this purpose.
In recent years, PAM devices have been extensively employed to capture bird
vocalisations automatically. Subsequently, PAM technology facilitates species
identification and diversity assessment through the analysis of these captured
vocalisations [xie2023review].

Long-term passive monitoring, however, will result in huge amounts of audio
data, and with a scarcity in the skilled human resources required to
effectively label this data automated workflows are of high importance
[stowell2022computational]. Although automatic bird recognition methods
employing deep learning (DL) have become state-of-the-art for boosting data
processing efficiency [xie2023review], bird recognition in the wild remains an
open challenge.

The diversity observed in bird vocalisations underscores the existence of
widespread dialects across different geographical regions [aplin2019culture,
smeele2024multilevel]. Research has identified microgeographical dialects,
spanning less than 2 km, which significantly impact the efficacy of automatic
bird recognition methods [r6, morgan2022open]. These regional variations in
vocalisations present a formidable challenge for improving the accuracy of
recognition methods [morgan2022open, r5, xie2023cross], necessitating further
investigation into cross-dialect recognition strategies. The presence of
dialects within a database parallels the challenge of dealing with disparate
training and test sets stemming from distinct corpora or domains, thereby
introducing a cross-corpus recognition hurdle. While expanding the collection
of diverse dialect datasets may offer some relief, tackling the variability
present in the testing set demands a more robust approach [tan2024acoustic].
Numerous strategies have been explored to bolster the model's generalisation
across various datasets [conde2021weakly, li2023multimodal, zhang2023optimal,
gebhard2023exploring, jing2022temporal].

In scenarios like bird monitoring, where the specifics of the target domain
are often uncertain, normalisation techniques can play a pivotal role in
enhancing the model's generalisability. Instance normalisation (IN)
[Ulyanov2016Ins] stands as one such method, effectively mitigating instance-
specific domain disparities and enhancing recognition performance across
diverse tasks. To compensate for the loss of classification information
resulting from normalisation while simultaneously eliminating irrelevant
device-specific nuances, approaches like audio residual normalisation
[kim2021domain] and relaxed instance frequency normalisation [kim2022domain]
have been employed, thereby bolstering the recognition model's
generalisability across different audio sample devices. Tang et al.
[tang2022acoustic] employed hybrid augmentation across diverse datasets to
broaden the domain coverage of the training dataset, coupling it with per-
channel energy normalisation (PCEN) to alleviate acoustic distortion and
consequently improve cross-domain bird audio detection accuracy. In a similar
vein, Xie et al. [xie2023cross] devised an enhanced instance frequency
normalisation strategy aimed at enhancing feature-specific expression, proving
to be advantageous in cross-corpus recognition. These normalisation
methodologies hold promise in mitigating dialect effects within bird
vocalisation recognition.

However, before processing methodologies can be explored high-quality data
remains fundamental to the development of any deep learning model. To the best
of the authors knowledge, there is no bird vocalisation dataset that focuses
on dialects111Datasets for audio-driven ornithological machine learning tasks
submitted as supplementary material. In this study, we establish a Dialect
Dominated Dataset of Bird Vocalisations (D3BV ), which contains 3 sub-datasets
from different geographic regions of the CONUS. The reason for this is that
the CONUS has a different climatic distribution and a rather homogeneous
biomes system, with Desert shrubs, Savanna, and Temperate broadleaved forests
stretching from west to east (shown in Figure 1). Considering that vegetation
is a vital factor influencing bird activity, we separated the data into three
distinct sections based on various plant types [NAGDC]. In addition, ten
common bird species in the CONUS are selected to ensure their presence in all
three regions.

We present a comprehensive overview of our dataset, and conduct a series of
experiments aimed at discerning differences within bird vocalisations of
identical species. Our experimental findings unveil the existence of dialects
within the same bird species across different regions. Furthermore, the
outcomes of our experiments, spanning several deep learning models,
demonstrate the influence of regional dialects on model generalisation and
robustness.

Figure 1: A map of the contiguous United States, illustrating three distinct
geographical regions (D1-D3) delineated by variations in climate and
vegetation patterns. Markers indicate co ordinates for the audio samples of
D3BV , with different marker colours representing different regions. The map
was generated using ArcGIS. Figure 2: The spectrograms of three typical bird
species in different regions across the contiguous United States in the D3BV .
The variations in frequency range, intervals, and other features of bird
vocalisations indicate the existence of regional dialects within the same bird
species. Figure 3: Amount of segments of different bird species across regions
with Corresponding Codes (#0 – #9) in Table 2

Our main contributions can be summarised as follows:

  * •

we collected a dialect-dominated, preprocessed wild bird vocalisation dataset,
enriched with detailed annotations.

  * •

we provide baseline models to benchmark the bird species recognition based on
the cross dialects vocalisation dataset.

To the best of our knowledge, D3BV is the first open bird vocalisation dataset
aimed at dialect effect study. We want to contribute to the field of bird
vocalisation recognition by releasing our data set open-source and free for
research use.

##  2 D3BV Dataset

###  2.1 Data Collection

The D3BV dataset encompasses audio clips from 10 bird species in three regions
of the CONUS, amounting to a total duration of 91 752 seconds. The bird
vocalisation audio was gathered from "Xeno-canto"222https://xeno-canto.org/ ,
a platform dedicated to sharing wildlife sound recordings globally. The shared
recordings are licensed under $\copyright$ Creative Commons Attribution-
NonCommercial-ShareAlike 4.0. Our selection criteria for bird species are
based on their extensive distribution and the availability of high-quality
audio data (labeled 'A' for sound quality by recordists) across all three
regions, as indicated in the associated metadata.

When sourcing the data we chose to focus on the CONUS (the latitude range of
24°N to 60°N) as the target area for the following reasons:

1\. A wide range of bird species are found at all latitudes and longitudes,
making it possible to investigate the dialects of the same species in
different geographic regions.

2\. The obvious pattern of regional variations in climate and vegetation can
be broadly divided into three main geographical regions:

  * •

Western Cordillera (Region D1) ($\approx$100°W-130°W): This region is
characterised by mountain ranges, including the Rocky Mountains and the
Coastal Ranges. It extends along the western part of the contiguous United
States.

  * •

Interior Plains (Region D2) ($\approx$90°W-100°W): The vast and flat area
known as the Great Plains lies between the Western Cordillera and the
Appalachian Mountains. It is a region of extensive grasslands.

  * •

Eastern Highlands (Region D3) ($\approx$70°W-90°W): The Appalachian Mountains,
running along the eastern part of the continent, form the Eastern Highlands.
This region consists of forested mountains and valleys.

We collected data corresponding to the regions outlined in Table 1.
Additionally, in Table 2 we present comprehensive details of the 10 selected
bird species, accompanied by the corresponding sample number across the three
distinct regions.

Figure 1 provides a visual representation of the recording locations,
employing different marker colors—white, red, and blue—to denote D1, D2, and
D3, respectively. Furthermore, Figure 2 offers a detailed examination of
spectrogram disparities among three representative bird species across the
three regions, revealing the presence of dialectical variations within the
selected species. Variations observed among regions exhibit distinct
characteristics across species. Specifically, in the context of Agelaius
phoeniceus, there are varying frequencies of repetition in vocalisations. In
the case of Cardinalis Cardinalis, differences arise in the overall call
durations, and for Turdus Migratorius, variations are noted in the presence or
absence of significant high-frequency harmonics.

Table 1: Distribution of audio clips across the regions. Region | # samples | Longitude ranges
---|---|---
D1 | 5781 | (100.2167°W, 129.9241°W)
D2 | 831 | (90.3694°W, 99.9995°W)
D3 | 4857 | (62.2240°W, 89.9656°W)
Table 2: Overview of the D3BV dataset in terms of duration (seconds), frequency range (kHz), and number of audio files per species. # Bird Species | Common name | Sound Type | Total Time (sec.) | Number of Segments | Range (kHz)
---|---|---|---|---|---
0 Agelaius phoeniceus | Red-winged Blackbird | Songs | 17 504 | 2188 | 2.8-5.7
1 Cardinalis cardinalis | Northern Cardinal | Songs | 17 944 | 2243 | 3.5-4.0
2 Certhia americana | Brown Creeper | Calls | 3912 | 489 | 3.7-8.0
3 Corvus brachyrhynchos | American Crow | Calls | 9624 | 1203 | 0.5-1.8
4 Molothrus ater | Brown-headed Cowbird | Calls | 4304 | 538 | 0.5-12.0
5 Setophaga aestiva | American Yellow Warbler | Songs | 8288 | 1036 | 3.0-8.0
6 Setophaga ruticilla | American Redstart | Songs | 7080 | 885 | 3.0-8.0
7 Spinus tristis | American Goldfinch | Songs | 4784 | 598 | 1.6-6.7
8 Tringa semipalmata | Willet | Calls | 2184 | 273 | 1.5-2.5
9 Turdus migratorius | American Robin | Songs | 16 128 | 2016 | 1.8-3.7
Total | - | - | 91 752 | 11 469 | -
Table 3: Cross-corpus baseline results, from four modeling strategies (see Section 3. The evaluation is conducted by cross-corpus training and testing sets from the three distinct regions (D1-D3) and assessing their performance. We report accuracy (ACC), Unweight Average Recall (UAR), and F1-score (F1). Bold indicates the best performance on the cross-region vocalisation recognition Model | Train Dataset | Test Dataset
---|---|---
D1 | D2 | D3
ACC | UAR | F1 |  | ACC | UAR | F1 |  | ACC | UAR | F1
TDNN+TN | D1 | 0.9707 | 0.9537 | 0.9664 |  | 0.5012 | 0.4018 | 0.3624 |  | 0.4633 | 0.4296 | 0.3939
D2 | 0.2275 | 0.2308 | 0.1752 |  | 0.9762 | 0.9895 | 0.9889 |  | 0.3799 | 0.2545 | 0.2480
D3 | 0.4978 | 0.4398 | 0.4226 |  | 0.6486 | 0.5397 | 0.4855 |  | 0.9631 | 0.946 | 0.9493
TDNN+BN | D1 | 0.9621 | 0.9556 | 0.9550 |  | 0.6027 | 0.4907 | 0.4295 |  | 0.5768 | 0.5206 | 0.4900
D2 | 0.2595 | 0.2864 | 0.2127 |  | 0.9048 | 0.8724 | 0.8801 |  | 0.4500 | 0.3367 | 0.3315
D3 | 0.5704 | 0.5125 | 0.4930 |  | 0.7500 | 0.6209 | 0.5958 |  | 0.9693 | 0.9544 | 0.9479
TDNN+GW | D1 | 0.9552 | 0.9518 | 0.9512 |  | 0.5906 | 0.4683 | 0.4006 |  | 0.5949 | 0.5366 | 0.5097
D2 | 0.3045 | 0.3078 | 0.2269 |  | 0.9167 | 0.8509 | 0.8780 |  | 0.4885 | 0.3580 | 0.3357
D3 | 0.5948 | 0.5363 | 0.5126 |  | 0.7488 | 0.6294 | 0.5917 |  | 0.9570 | 0.9287 | 0.9369
TDNN+IFN | D1 | 0.9052 | 0.8721 | 0.8734 |  | 0.7246 | 0.6262 | 0.5837 |  | 0.6979 | 0.6313 | 0.6073
D2 | 0.3628 | 0.3651 | 0.2641 |  | 0.8214 | 0.6664 | 0.6565 |  | 0.5086 | 0.4085 | 0.3521
D3 | 0.6439 | 0.5740 | 0.5540 |  | 0.7766 | 0.5838 | 0.5918 |  | 0.8852 | 0.8217 | 0.8489

###  2.2 Data preprocessing

All audio files are converted to the Waveform Audio File Format (WAV) with a
mono channel. As depicted in Table 2, the majority of bird vocalisations
exhibit a frequency range within [0-8 kHz]. Although Molothrus ater has a
maximum frequency of 12 kHz, our analysis reveals that all instances of
Molothrus ater in our dataset had their highest frequencies below 8 kHz.
Consequently, we uniformly set the sampling frequency to 16 kHz.

We adopt a method inspired by [morgan2022open], clipping the audio into
8-second segments. For segments shorter than 8 seconds, zero-padding is
applied. The segments are then thoroughly manually checked to validate the
presence of bird vocalisation lasting more than one second in each segment.
This filtering strategy improves the dataset's robustness and reliability
while also aligning with the suitable preprocessing procedures for subsequent
analysis. Figure 3 lists the number of segments of different bird species
across regions.

##  3 Baseline Experiment

###  3.1 Baseline and Compared Model

Prior research has demonstrated that Time-Delay Neural Networks (TDNNs) and
their variations [xie2023cross], exhibit superior performance compared to
alternative methods in the realm of cross-corpus recognition of bird
vocalisations. Therefore, to offer a baseline for other to benchmark the D3BV
dataset against, we apply four TDNN-based models with different normalisation
methods in this work.

  * •

TDNN + TN [xie2023cross]: We apply Time-Norm normalisation, which normalises
each log-Mel spectrogram on the temporal axis, to replace the BN layer in the
TDNN + BN model.

  * •

TDNN + BN [xie2023cross]: Compared to the original TDNN [fan2021deep], Batch
Normalisation (BN) and a Rectified Linear Unit (ReLU) layer are appended after
each Conv1D layer before the pooling layer.

  * •

TDNN + GW [xie2023cross]: GW [Huang2021Group] is more efficient on exploiting
the advantages of the whitening operation while keeping a balance between
learning efficiency and representational capacity.

  * •

TDNN + IFN [xie2023cross]: In [xie2023cross], it is demonstrated that the
frequency of a vocalisation varies more significantly between different
domains in the context of bird species recognition. Therefore, the Instance
Frequency normalisation (IFN) was proposed to eliminate the instance-specific
contrast information on the frequency dimension for a more efficient
extraction of domain-invariant features. The formula of IFN is:

$IFN(x)=\frac{x-\mu_{in}}{\sqrt{\sigma_{in}+\delta}},$ (1)

where, $\mu_{in},\sigma_{in}\in\mathbb{R}^{NxF}$ are the mean and standard
deviation of the input feature $x\in\mathbb{R}^{N\times F\times T}$, in which
$N,F$, and $T$ are batch size, frequency dimension, and time dimension
respectively. $\delta$ is a small number to improve numerical stability.

###  3.2 Experiment Setups

From each audio sample, we extract Log Mel spectrograms with an FFT size of
2048, a hop length of 512, and 128 Mel bins. As shown in Table 3, we conduct
the experiments across regions by setting D$n$D$m$ ($n,m\in(1,2,3)$). When
$n=m$ (in-region), training and testing occur on the same region data, while
$n\neq m$ (cross-region) indicates different regions involved in the training
and testing. For in-region recognition, the sub-dataset is partitioned into an
8:1:1 ratio of training, validation, and test sets. When referring to cross-
region recognition, the data within a single regional dataset will not be
split like the in-region dataset setup. The bird data from the current region
(D$n$) will be utilised as training data, while all data from the other two
regions (D$m,n\neq m$) will serve as test data respectively. The loss function
is Cross Entropy function. Training is conducted with a batch size 32 and the
learning rate is 1e-4, for 40 epochs. The Adam optimiser with 0.8 decay every
3 steps is applied for all the models.

###  3.3 Result and Analysis

The results, as summarised in Table 3, consistently demonstrate that
performances within the same region (D$n$D$n$) generally surpass those across
different regions (D$n$D$m$) across all models. This observation strongly
suggests the presence of dialectal variations among regions.

Furthermore, the performance disparities observed in models trained on D1D2
and D1D3 datasets vividly underscore the influence of geographical distance on
bird dialects. Notably, as the distance from the training data collection area
increases, there is a noticeable degradation in model performance. These
findings align with previous research [lewis2021uses], which suggests that
dialectal variations often correlate with the geographical separation of bird
populations.

It's worth mentioning that models trained on D2 data exhibit the least
effective performance when tested on cross-region datasets. This is primarily
attributed to D2 being situated in a region characterized by grasslands and
relatively lower bird population density than the other two regions, resulting
in a smaller number of audio samples in the D2 dataset.

Moreover, comparing normalisation methods, it's evident that Batch
Normalisation (BN) normalises samples within a single batch, thereby
potentially enhancing the model's generalisation ability. Conversely, Group
Whitening (GW) normalises samples within constructed groups, offering
improvements over BN by addressing within-batch normalisation limitations,
ultimately enhancing generalisation

Both BN and GW normalise the time and frequency dimensions simultaneously,
whereas Temporal Normalisation (TN) and Instance Frequency Normalisation (IFN)
independently normalise time and frequency dimensions. With this in mind, we
observe that TN exhibited the poorest performance, while IFN demonstrated the
highest. This suggesting that frequency normalisation plays a stronger role in
mitigating the influence of dialect and enhances the cross-corpus recognition
capability of models. These findings align with existing literature
[Stowell2014large], which suggests that birds adapt to diverse acoustic
environments through Frequency Modulation (FM), underscoring the significance
of frequency variations as a fundamental characteristic of dialectal
differences. Nevertheless, the performance of IFN requires further
improvement, and it is possible that taking the time dimension of variation
into consideration can aid this. Drawing on literature [kim2022domain],
relaxed instance frequency normalisation may be a potential approach to
implement next.

##  4 Conclusions

This paper introduces D3BV, a novel dataset of bird vocalisations designed
primarily for investigating the influence of dialects on bird species
recognition tasks. In addition, baseline models have been established to
provide a benchmark for the challenge of cross-corpus bird species recognition
based on vocalisations. We anticipate that both the D3BV dataset and our
experimental findings will contribute to the bird vocalisation recognition
research community, serving as a foundation and reference for future
endeavors. We foresee opportunities for further exploration in the development
of appropriate normalisation techniques. Moreover, we advocate for a
comprehensive examination of bird dialects, emphasizing the importance of
approaching this subject from an ornithological perspective. We believe that
the dataset proposed herein marks the initial step towards a thorough
quantitative investigation of bird dialects.

##  5 Acknowledgements

The authors appreciate the valuable contributions of discussion participants
and the support that facilitated the development of this research.