# White dwarf binary modulation can help stochastic gravitational wave
background search

Shijie Lin  Institute for Frontier in Astronomy and Astrophysics, Beijing
Normal University, Beijing, 102206, People’s Republic of China  Department of
Astronomy, Beijing Normal University, Beijing 100875, People’s Republic of
China.  Bin Hu<EMAIL_ADDRESS>Institute for Frontier in Astronomy and
Astrophysics, Beijing Normal University, Beijing, 102206, People’s Republic of
China  Department of Astronomy, Beijing Normal University, Beijing 100875,
People’s Republic of China.  Xue-Hao Zhang  Lanzhou Center for Theoretical
Physics, Key Laboratory of Theoretical Physics of Gansu Province, School of
Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, China  Yu-Xiao Liu  Lanzhou Center for Theoretical
Physics, Key Laboratory of Theoretical Physics of Gansu Province, School of
Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, China

###### Abstract

For the stochastic gravitational wave backgrounds (SGWBs) search centred at
the milli-Hz band, the galactic foreground produced by white dwarf binaries
(WDBs) within the Milky Way contaminates the extra-galactic signal severely.
Because of the anisotropic distribution pattern of the WDBs and the motion of
the spaceborne gravitational wave interferometer constellation, the time-
domain data stream will show an annual modulation. This property is
fundamentally different from those of the SGWBs. In this Letter, we propose a
new filtering method for the data vector based on the annual modulation
phenomenon. We apply the resulted inverse variance filter to the LISA data
challenge. The result shows that for the weaker SGWB signal, such as energy
density $\Omega_{\rm astro}=5\times 10^{-12}$, the filtering method can
enhance the posterior distribution peak prominently. For the stronger signal,
such as $\Omega_{\rm astro}=15\times 10^{-12}$, the method can improve the
Bayesian evidence from ‘substantial’ to ‘strong’ against null hypotheses. This
method is model-independent and self-contained. It does not ask for other
types of information besides the gravitational wave data.

††preprint: APS/123-QED

_Introduction.–_ The stochastic gravitational wave background (SGWB) [1, 2, 3]
is one of the primary targets of all types of gravitational detectors. There
are two kinds of SGWB signals, namely those from astrophysical sources and
from cosmological origins. The former is produced by many independent and
unresolved continuous gravitational wave sources, such as black hole binaries
[4] and neutron star binaries [5]. The incoherent superposition from each of
the single sources leads to a stochastic nature of the gravitational wave
background [6, 7]. The cosmological SGWB is generated from the exotic
mechanisms in the early universe, such as cosmic inflation, first-order phase
transition and cosmic strings [8, 9], etc.

The search for SGWB signal is performed in multi-frequency bands. The LIGO-
Virgo-KAGRA (LVK) collaboration looks for this signal in the frequency range
between 10 Hz and 10 $k$Hz, where the optimal sensitivity is centred at 100
Hz. Unfortunately, up to the third observing run (O3), there is no detection
evidence [10, 11, 12]. For instance, the upper limit of the dimensionless
energy density $\Omega_{\rm GW}\leq 5.8\times 10^{-9}$ at the $95\%$ credible
level for a flat spectrum in the most sensitive part of the LIGO band ($20-86$
Hz). Furthermore, the pulsar timing array (PTA) can be utilised to search SGWB
in the nano-Hz band. The recent NANOGrav (the North American Nanohertz
Observatory for Gravitational Waves) 12.5-year data reported a strong evidence
of a stochastic process. Unfortunately, this process has no statistically
significant quadrupolar spatial correlations, which would consider necessary
to claim an SGWB detection predicted by the theory of general relativity [13].

The spaceborne GW interferometers like LISA [14], Taiji [15] and TianQin [16]
are able to detect SGWB centred at the milli-Hz band ($10^{-4}-10^{-1}$Hz).
However, in this frequency range, there is a significant galactic foreground
contamination produced by tens of millions of white dwarf binaries (WDBs) [17,
18, 19, 20]. In the LISA Data Challenge (LDC) [21], methods to separate the
galactic foreground from the SGWB signal are in demand, and some solutions
have been proposed [22, 23]. To separate and remove the galactic foreground,
there are two basic directions [22]. First, the foreground of the Milky Way
has different energy spectrum shapes from instrument noise and the typical
SGWB models, which can help us effectively distinguish noise and multiple
different SGWB components. For the galactic foreground, some high-frequency
WDBs can even be identified and eliminated from the data, thus reducing the
impact of the galactic foreground at high frequencies [24, 25]. Second, since
most of the WDBs in the Milky Way are distributed in the disk [19, 26], the
galactic foreground spatial distribution is anisotropic, which is obviously
different from the isotropic SGWB. The noise intensity generated by the
galactic foreground will strongly depend on the direction of the detector
constellation. Since the angle between the LISA detector plane and the
equatorial coordinate system will not change a lot within the total
observation time [14], its time-domain signal will show annual modulation in
the one-year observation data [27, 22, 23]. We can use this property to
separate the galactic foreground from the extra-galactic SGWB. In this Letter,
we utilise the annual modulation to reduce the galactic foreground
contamination. The resulting method is nothing but the inverse variance filter
(IVF), which is widely used in cosmic microwave background (CMB) data
analysis, such as foreground component separation [28] and gravitational
lensing studies [29].

_Methodology.–_ For a time-domain data stream, $d(t)$, whose campaign length
is few years, one can divide it into $N$ slots with a typical width of about
one week,

$d_{i}(t)=g_{i}(t)+n(t)+s(t)\;,$ (1)

where $g,n,s$ denote for the galactic foreground, gaussian stationary noise
and isotropic SGWB signal, respectively. The sub-index labels the slot sort.
Now, we weight the original data stream to reduce the foreground

$\tilde{d}(t)=\sum_{i}\omega_{i}d_{i}(t)\;,\quad\sum_{i}\omega_{i}=N\;,$ (2)

where the second equation ensures that the weighting will not damage SGWB
strength. We ask the weight to give the minimum variance of the periodic
diagram. To do so, we give the power spectrum density (PSD) of the data stream
$\tilde{d}(t)$

$\displaystyle\tilde{S}(f)$ $\displaystyle=\langle|\tilde{d}(f)|^{2}\rangle\;$
$\displaystyle=\langle\frac{|\sum_{i}\omega_{i}g_{i}(f)|^{2}}{N}\rangle+S_{n}(f)+S_{s}(f),\;$
(3)

where $\tilde{d}(f)$ is the fourier transform of the $\tilde{d}(t)$. One can
easily see that the weight will not influence the PSD of instrument noise and
the SGWB signal, which are stationary and gaussian during the observing
campaign.

Using the precondition $\sum_{i}\omega_{i}=N$ and Lagrangian multiplier, one
can derive the optimal weight as

$\omega_{i}=N\frac{\Big{[}\int g^{2}_{i}(f)df\Big{]}^{-1}}{\sum_{j}\Big{[}\int
g^{2}_{j}(f)df\Big{]}^{-1}}\;.$ (4)

One can see that this is simply the inverse variance filter. The behind logic
is the following. On the one hand, the galactic foreground is anisotropic and
follows the spatial distribution of the galactic plane. With the motion of the
detector constellation, the galactic foreground is loud when the constellation
points to the galactic plane; when the constellation points to the high
galactic latitude, the galactic foreground is relatively low. On the other
hand, the SGWB is isotropic. Hence, one can give a low weight to the former
and a high weight to the latter.

Figure 1:  PSD of the simulated components and the fitting results.

_Results.–_ In the rest, we will use the LISA Data Challenge simulation to
demonstrate the robustness of this method. LISA constellation consists of
three spacecraft with a triangular configuration, separated from one another
at a distance of $2.5\times 10^{6}\;km$. To overcome the laser frequency
noise, the LISA experiment adopts the time-delay interferometry (TDI)
technology [30]. In this Letter, we use the Michelson TDI configuration
(X,Y,Z) to compose the optimal TDI configuration (A,E,T) [31]. For simplicity,
we assume the equal arm configuration. Under these circumstances, the
T-channel is a nearly null channel for GW signal search, but it can be used to
calibrate the instrumental noise [32]. Following the same methodology
presented in [33], we use the analytic expressions for the response functions
and instrumental noise PSD. We consider two major components of the
instrumental noise, namely the acceleration noise ($N_{\rm acc}$) and optical
path-length fluctuation ($N_{\rm op}$).

Furthermore, we adopt the galactic foreground model given by LDC1-4 data [34],
which consists of about 30 million WDBs. For each binary, LDC1-4 contains the
amplitude, the frequency, the frequency derivative, the ecliptic latitude and
longitude, the inclination, the initial phase and the polarization angle [25].
Among the 30 million WDBs, about ten thousand systems can be resolved
individually. These strong point sources must be identified and removed
accurately. To do so, we use the method from galactic binary separation by
iterative extraction and validation using extended range (GBSIEVER) [25]. The
principal novel features of GBSIEVER are using particle swarm optimization to
maximize the F-statistic, fast template generation using under-sampling, and
mitigation of spurious sources using a cross-validation scheme. By this means,
we remove about ten thousand strong point sources. The initial PSD and the
residual of the galactic foreground are shown as light-grey and light-purple
curves in Fig.1, respectively. One can clearly see that our foreground
cleaning method significantly improves the results in the frequency range of
($10^{-3}-10^{-2}$) Hz.

Finally, we consider the SGWB from the astrophysical origin, sourced by many
compact binaries, mostly stellar-origin black holes and neutron star binaries
(BBH+BNS). The gravitational wave emission of them is incoherently superposed.
According to previous work [3], this component can be well approximated by a
power law function with a slope $\alpha_{\rm astro}=2/3$. In this Letter, we
inject two amplitude values, namely $\Omega_{\rm astro}=5\times 10^{-12}$ and
$15\times 10^{-12}$, respectively.

Armed with these preparations, we simulate the time domain stream, $d(t)$, for
two years length and with 15 seconds sampling rate. Then, we use the following
three steps to process the IVF operation.

  * •

First, we divide the time domain data into 100 slots, $\\{d_{i}(t)\\}$, in
which each slot has about 1 week time length.

  * •

Second, we calculate the IVF for each slot with Eq.(4). The frequency
integration range covers $[1\times 10^{-4}-7\times 10^{-3}]$Hz, where the
galactic foregrounds are prominent.

  * •

At last, we obtain the filtered data, $\tilde{d}(t)$, by multiplying
$\omega_{i}$ to the original data. Then, we calculate the PSD of the filtered
data and estimate different components in the frequency domain.

Figure 2:  Posterior distribution function of SGWB energy density,
$\Omega_{\rm astro}$, with (blue) and without (orange) the IVF. The injected
value is $\Omega_{\rm astro}=5\times 10^{-12}$. The solid and dashed vertical
lines with the corresponding colors denote for the 1$\sigma$ and 2$\sigma$
ranges, respectively. Figure 3:  The probability distribution function of the
Bayes factor with and without the IVF. The injected SGWB energy density value
is $\Omega_{\rm astro}=15\times 10^{-12}$. The number of the simulation is 200
for each case.

Next, we estimate the SGWB energy density, foregrounds as well as instrumental
noise parameters via the Monte Carlo method. We write the PSD model as,
$S_{n,I}+S_{{\rm gw},I}$, where $S_{n,I}$ is the instrumental noise PSD; and
$S_{{\rm gw},I}$ denotes for the total GW energy density $\Omega_{\rm
gw}=\Omega_{\rm fg}+\Omega_{\rm astro}$. To fit the galactic foreground, we
use a broken power-law model [35]. The number of the parameters in our
analysis is seven ($N_{\rm pos},N_{\rm
acc},A_{1},\alpha_{1},A_{2},\alpha_{2},\Omega_{\rm astro}$), two for
instrument noise, four for galactic foregrounds and one for the SGWB
amplitude.

We use both the Markov chain Monte Carlo (MCMC) method to estimate the model
parameters and dynamic nested sampling (DNS) to estimate Bayesian posteriors
and evidences. The packages for MCMC and DNS are emcee [36] and dynesty [37],
respectively. We adopt the Whittle likelihood [2]

$\displaystyle L(\mathbf{d}|\theta)=$
$\displaystyle-\dfrac{1}{2}\sum_{k=0}^{N}\Bigg{[}\dfrac{PSD_{A}}{S_{{\rm
gw},A}+S_{n,A}}+\dfrac{PSD_{E}}{S_{{\rm
gw},E}+S_{n,E}}+\dfrac{PSD_{T}}{S_{n,T}}$ (5)
$\displaystyle+\log\Big{\\{}8\pi^{3}(S_{{\rm gw},A}+S_{n,A})(S_{{\rm
gw},E}+S_{n,E})S_{n,T}\Big{\\}}\Bigg{]}.$

Since the GW signal in the T-channel is much lower than the instrumental
noise, we do not give the GW model in the T-channel but add it in our analysis
to estimate the instrumental noise parameters.

In Fig. 1, we show the fitting results from sampling points as the thin
smoothing lines. Dark purple and orange transparent lines are for the
foreground and SGWBs, respectively. The bold dashed curves denote for the best
fit. In Fig. 2, we plot the posterior of $\Omega_{\rm astro}$. The blue and
orange curves denote for the results with and without the filtered method. The
injected SGWB energy density is $\Omega_{\rm astro}=5\times 10^{-12}$, marked
as the red vertical line. Obviously, the filtered method makes the posterior
peak more prominent. As demonstrated in the sub-panel, the posterior
probability of $\Omega_{\rm astro}$ has a long tail extending to the lower
values. After applying the IVF method, the left-hand tail gets depressed, and
the corresponding masses will move to the right and contribute to the main
peak. This benefit will also appear when we increase the amplitude of the
SGWB.

For a higher amplitude ($\Omega_{\rm astro}=15\times 10^{-12}$), instead of
the parameter estimation, we turn to the Bayesian model selection. We consider
two models: Model $M_{0}$ only contains the instrument noise and galactic
foregrounds; model $M_{1}$ has the instrumental noise, galactic foregrounds
and SGWB. We simulated 200 realizations for each case with/without IVF to see
the distribution of the Bayesian factor as shown in Fig. 3. The blue and
orange horizontal hinges denote for the quartiles, namely the $25\%-75\%$
confidence level. The tiny vertical line in the middle of the hinges denotes
for the median value. The horizontal error bar ranges lie 1.5 times the inter-
fourth range from the median. The empty circles denote for the outliers.
Without applying IVF, we obtain a Bayesian factor $B_{10,{\rm
nof}}=9.6^{+3.8}_{-2.3}$; after applying IVF, we obtain a Bayesian factor
$B_{10,{\rm f}}=16.1^{+4.5}_{-5.0}$. The corresponding difference statistics,
$\Delta B_{10}=B_{10,{\rm f}}-B_{10,{\rm nof}}$, reads $\Delta
B_{10}=5.8_{-4.6}^{+4.1}$. One can see that the IVF operation can improve the
Bayesian evidence from ‘substantial’, $B_{10}\in(3.2,10)$, to ‘strong’,
$B_{10}\in(10,100)$, against null hypotheses [38]. These results show that WDB
modulation will significantly improve confidence in model selection.

_Conclusion.–_ The search for the SGWB signal from astrophysical and
cosmological origins is one of the major science cases for the spaceborne
gravitational wave observatory. However, the presence of the galactic
foreground will obstruct our road. Few works have discussed selecting tens of
thousands of the resolved WDBs within the galactic foreground [24, 34, 39, 40,
41, 25, 42, 43]. Some works have suggested that there would be confusion noise
after removing each of the resolved WDBs and developed models of the ideal
confusion noise [44, 41, 45]. Some algorithms to separate different components
have been proposed [22, 46, 35, 23, 47]. The annual modulation produced by the
galactic WDBs can provide useful information to distinguish the galactic
foreground from SGWBs.

In this Letter, we propose a filtering method based on the annual modulation
information. The advantage of this method is its model independency. We do not
need to know how the WDBs are distributed in the galaxy, and we do not need to
know the accurate position of the spaceborne GW interferometer. Every process
is made on the observational gravitational wave data alone (simulated in this
Letter) and no external data is needed. Besides, this method is independent of
how well we select the resolved source. In this Letter, we only discuss the
astrophysical originated SGWBs, described by a power-law function with the
slope $\alpha=2/3$. This method will also be suitable for other forms of the
SGWBs, like the flat spectrum for inflation [3], the gaussian-bump spectrum
for the first-order phase transition [48, 49] and the spectrum for the cosmic
string [50]. This idea does not depend on the amplitude of the SGWB signal. We
show that it is well-behaved for both lower and higher amplitude. In
conclusion, the annual modulation originated from WDBs can help stochastic
gravitational wave background search.

_Acknowledgements.–_ This work is supported by the National Key R&D Program of
China No. 2021YFC2203001 and No. 2021YFC2203003, and the National Natural
Science Foundation of China through Grant No. 12247101.

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