# Large Language Models Show Human-like Social Desirability Biases in Survey
Responses

Aadesh Salecha  Stanford University  Molly E. Ireland  Receptiviti  Shashanka
Subrahmanya  Stanford University  João Sedoc  New York University  Lyle H.
Ungar  University of Pennsylvania  Johannes C. Eichstaedt  Stanford University

## Abstract

As Large Language Models (LLMs) become widely used to model and simulate human
behavior, understanding their biases becomes critical. We developed an
experimental framework using Big Five personality surveys and uncovered a
previously undetected social desirability bias in a wide range of LLMs. By
systematically varying the number of questions LLMs were exposed to, we
demonstrate their ability to infer when they are being evaluated. When
personality evaluation is inferred, LLMs skew their scores towards the
desirable ends of trait dimensions (i.e., increased extraversion, decreased
neuroticism, etc). This bias exists in all tested models, including GPT-4/3.5,
Claude 3, Llama 3, and PaLM-2. Bias levels appear to increase in more recent
models, with GPT-4’s survey responses changing by 1.20 (human) standard
deviations and Llama 3’s by 0.98 standard deviations—very large effects. This
bias is robust to randomization of question order and paraphrasing. Reverse-
coding all the questions decreases bias levels but does not eliminate them,
suggesting that this effect cannot be attributed to acquiescence bias. Our
findings reveal an emergent social desirability bias and suggest constraints
on profiling LLMs with psychometric tests and on using LLMs as proxies for
human participants.

Keywords $\cdot$ large language models $\cdot$ cognitive bias $\cdot$ AI
$\cdot$ psychometrics $\cdot$ emergent behavior

Large Language Models have demonstrated remarkable proficiency in a wide array
of tasks, ranging from language translation and creative writing to code
generation problem-solving. As these models are trained on vast amounts of
human-generated text data, they can emulate human textual behavior and exhibit
emergent capabilities that were not anticipated during their development [1].
Researchers are using LLMs’ emulation capabilities to simulate data from human
participants across diverse psychological and behavioral experiments [2, 3,
4].

Personality traits like the Big Five are among the most studied individual
differences among human subjects. The Big Five aims to capture everything that
can be known about a person in five mostly orthogonal trait domains:
Extraversion, Openness to Experience, Conscientiousness, Agreeableness, and
Neuroticism. Though originally intended to be value-neutral, most Big Five
assessments are acknowledged to have evaluative content [5], with people
preferring lower Neuroticism (i.e., negativity and vulnerability to stress)
and higher scores on the remaining four traits. Big Five traits are also
robust predictors of various human behavioral tendencies and life outcomes.
Researchers have therefore subjected LLMs to Big Five surveys and correlated
traits with downstream task performance [6], in addition to comparing trait
distribution to human norms [7]. But, the assessment of these personality
traits via self-report questionnaires is vulnerable to response biases [8]
such as acquiescence (tending to agree with questions regardless of their
content) and social desirability biases (skewing responses toward perceived
societal ideals). Prior research has used cognitive psychology experiments to
uncover emergent biases in LLMs [9] including an acquiescence bias [10].
Social desirability biases are some of the most persistent sources of error
variance on surveys that include items relevant to social norms, such as most
personality traits [11].

We designed a novel experiment to assess social desirability tendencies in
various LLMs. By systematically varying the number of questions an LLM is
exposed we both uncover a persistent social desirability bias as well as the
mechanism driving it.

## Results

To evaluate response biases in LLMs, we conducted a series of experiments
using a standardized 100-item Big Five personality questionnaire. We
administered the questionnaire in batches, systematically varying the number
of questions per batch (denoted as $Q_{n}$). To ensure the LLM had no access
to previous items, we started a new chat session for each batch. We provided
standardized instructions asking the LLMs to respond on a 5-point Likert
scale. Our analysis encompassed models from OpenAI, Anthropic, Google, and
Meta to ensure broad generalizability. See Supplementary Information (SI) for
full methods.

### Manifestation of Social Desirability Bias

Our experiments revealed that LLMs consistently skew their Big Five factor
scores towards the more socially desirable ends of the trait dimensions. This
propensity was most notable in GPT-4 (Fig. 1A, B); as we increased $Q_{n}$
(question batch size) from 1 to 20, scores for positively-perceived
traits–Extraversion, Conscientiousness, Openness, and Agreeableness–increased
by about 0.75 points (1.22 human standard deviations). Conversely, for
Neuroticism, a culturally devalued trait, the score decreased from 2.87 to
2.02 (1.10 human SDs; Fig. 1B).

### Generalizability Across LLMs

An analysis across different proprietary and open-source models, including:

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B

Trait |  $Q_{n}$ = 1 |  $Q_{n}$ = 20 |  GPT-4 Diff. |  Diff. in units of human SD [12]
---|---|---|---|---
Openness (+) |  3.31 |  4.09 |  +0.77 |  1.40
Conscientiousness (+) |  3.25 |  4.20 |  +0.96 |  1.42
Extroversion (+) |  3.00 |  3.74 |  +0.73 |  0.89
Agreeableness (+) |  3.64 |  4.41 |  +0.77 |  1.20
Neuroticism (+) |  2.87 |  2.02 |  -0.85 |  1.11
Average |  |  |  0.82 |  1.20

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Figure 1: (A) As the number of questions asked in a prompt ($Q_{n}$), GPT-4’s
responses to Big Five survey questions skewed closer to the socially desirable
ends of the scale (N = 30 trials, CI = 95%, *p $<$ 0.001). The general
positive and negative perceptions associated with traits are represented by
(+) and (-). (B) Summary of GPT-4’s social desirability bias. We calculated
the difference between administering surveys, one question per prompt and 20
questions per prompt, and showed the equivalent difference in terms of human
SDs based on population norms for the Big Five [12] Diff. = difference. (C) To
compare this bias across LLMs, we compute the difference in Personality Factor
Scores when administering the survey 1 vs. 10 questions per prompt (averaged
across N = 30 trials per model). (D) When comparing the average absolute
difference (Avg. Diff.) between $Q_{10}$ and $Q_{1}$ and the equivalent in
human SDs, we find that across LLM families, the larger and more recent models
have more bias.

GPT-4, Claude 3, Llama 3, and PaLM 2, revealed the prevalence of social
desirability bias across LLM families(Fig. 1C). All tested LLMs displayed this
bias with larger and more recent models exhibiting more bias.

### Driving Mechanism for Bias

LLMs seem to modulate their responses based on the perceived evaluation
context, whether automatically inferred or explicitly provided.

Implicit Inference of Evaluation Context When exposed to as few as five
randomly selected questions from the Big Five questionnaire, GPT-4, Claude 3,
and Llama 3 were able to identify that these questions belong to a personality
survey with over 90% accuracy. PaLM 2 and GPT-3.5 were less perceptive at 55%
and 45% accuracy, respectively (Fig. 2A). LLMs’ ability to correctly identify
the evaluation context seems to be correlated with their bias levels.

Explicit Prompting of Evaluation Context  Similar to humans, when the LLMs
were explicitly told that they were completing a Big Five personality survey
in the prompt, responses skewed towards the socially desirable end of the
spectrum, even when only presented with a single question (Fig. 2B). Explicit
prompting had an effect comparable to asking five questions at a time,
corroborating that LLMs seem to modulate their scores when under the
impression that they are being evaluated.

### Differential Impact of Positive and Reverse Coding

We tested two variations of the Big Five survey: one fully reverse-coded (with
negations in the questions) and another with only positively-coded items. The
reverse-coded version reduced the bias, with the average score changing only
by 0.37 points (0.54 human SD). Positively coding all questions did not
significantly affect the bias, which remained at an average change of 0.76
points (1.15 human SD) (Fig. 2C), suggesting that the effects cannot be
entirely attributed to acquiescence bias. Reverse-coded items load on separate
factors than positively-coded items [13] and likely occupy a different region
of semantic space.

### Robustness Across Paraphrasing, Question Randomization, and Temperatures

To address concerns that LLMs may be relying on the memorized versions of the
Big Five items, we also administered paraphrased variants of the survey and
found similar levels of bias. Additionally, we used three distinct
randomization strategies (as described in SI. Section D.) to assemble the
question sets. We found that such randomization had a minimal effect on the
bias, pointing to the absence of significant question-order effects. Finally,
we underscore the robustness of this bias by demonstrating it across various
LLM temperatures, which control the degree of randomness of their output. We
used values ranging from 0.0 (deterministic) to 1.2.

### Consistency and Reliability of LLM responses

Similarly to previous studies [6], we found that LLMs maintained a high degree
of internal consistency in their responses (Cronbach’s $\alpha>$ 0.8 for
individual subscales and $\alpha>$ 0.93 for all items), where 0.7 is
considered adequately reliable. We also found high split-test reliability
scores (0.79 after Spearman-Brown correction).

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Figure 2: (A) Comparing LLMs’ ability to identify the source of questions as a
personality survey as a function of number of questions. (B) Big Five scores
for GPT-4 with and without explicitly prompting that the LLM is completing a
Big Five personality survey. The no-explicit-prompting condition is identical
to that described in Fig. 1A. The information gained from explicit prompting
roughly has the same effect as asking five questions at once. (C) The coding
scheme of the questions had a substantive effect on the bias levels. GPT-4’s
average difference decreased from 0.81 points (1.22 human SD) when using the
standard International Personality Item Pool (IPIP) coding to 0.38 points
(0.54 human SD) with all items reverse-coded. Positively coding all items did
not have a significant effect on the bias.

## Discussion

We observe a persistent social desirability bias across various proprietary
and open-source models. The driving mechanism for this behavior seems to be
LLMs’ (implicit) awareness of the evaluation context. The evidence for bias
was consistent across randomization paradigms, paraphrasing of questions, and
temperature ranges. Reverse-coding items weakened but did not eliminate bias,
highlighting the need for further research into the development of mitigation
strategies. Our findings demonstrate potential subtleties with using
psychometric tests to evaluate LLM behavior and raise concerns around using
LLMs to simulate human participants.

Prior research has used survey-based personality assessments to study LLM
behavior. LLMs’ personality traits have been correlated with performance in
downstream text generation tasks [6] and also used as a part of a Turing test
to study behavioral similarity to humans [7]; however, our findings suggest
that these results may have been influenced by implicit social desirability
biases.

We suspect our findings will generalize to measures that parallel the Big Five
in popularity–and thus representation in LLMs’ training data. The same
response biases may not occur with constructs that are less represented in
models’ training data or less socially evaluative.

There are a few tactics that may mitigate response biases in simulated human
data derived from LLMs. Reverse-coding items may be the most straightforward
remedy; in our experiments, it reduced the bias by roughly half. Adapting
surveys to be less explicitly evaluative (or using existing less-evaluative
scales such as 5) may further reduce such biases. The most effortful but also
most effective safeguard would be to follow classic psychometric advice [14]
to replicate results across multiple measures (e.g., self-report and
behavioral data), paradigms (e.g., different experiments or study designs),
and data sources (including the replication of effects in human samples).

There is no doubt that LLMs are skilled at imitating humans in manifold tasks.
Whether they can also generate distributions of data that accurately reflect
human psychology within and across cultures remains in doubt [15]. We show
evidence of response biases to a common personality assessment and provide
recommendations for remediating these biases. Simulated human data from LLMs
have the potential to profoundly expand our ability to carry out psychological
experiments—but that will only be possible once systematic influences
impacting LLM responses are well understood.

## Methods

### LLM Configuration

We used a suite of widely available models: OpenAI’s text-davinci-002,
GPT-3.5, and GPT-4 (the 0613 fixed versions), Anthropic’s Claude 3 Opus
(claude-3-opus-20240229), Claude 3 Haiku (claude-3-haiku-20240229), Google’s
PaLM 2 (chat-bison-001), and Meta’s Llama 3 70B Instruct, Llama 2 70B Chat.
All experiments were carried out August 1-December 1, 2023. Default settings
were maintained for all models, except for the temperature parameter, which
was varied systematically.

### Variation of Temperature Parameter

We assessed the impact of the LLMs’ stochastic output on response biases by
varying the temperature parameter at 0.0, 0.4, 0.8, and 1.2. This range
allowed us to demonstrate the robustness of the bias across different levels
of response randomness.

### Paraphrasing and Randomization

Paraphrased variants of the survey were manually created to maintain semantic
integrity while altering phrasing. We employed three randomization approaches
– complete randomization of all items, randomization of questions within each
factor, and no randomization (i.e., administer the items in their original
sequence from the International Personality Item Pool).

### Questionnaire Administration to LLMs

The Big Five personality assessment was Goldberg’s (1999) IPIP representation
of Costa and McCrae’s NEO-PI-R Domains [16], comprising 100 items scored on a
5-point Likert scale (Supplementary Information section SI.3.). LLMs received
instructions akin to those provided to human participants. The only adaptation
to the instructions for LLMs was to explicitly ask it to respond with exactly
$Q_{n}$ numbers corresponding to the batch size. The Supplementary Information
(section SI.1. and SI.2.) detail the exact prompts verbatim. We dynamically
replaced {{Qn}} and {{Survey_Items}} with the batch size and the corresponding
survey items respectively.

### Data and Code Availability

All study replication materials, including verbatim LLM conversations and
codebooks to replicate and analyze our findings, are available on the Open
Science Framework (OSF) at osf.io/3fq2n/.

## References

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# Sky location of Massive Black Hole Binaries in the foreground of Galactic
white dwarf binaries

Pan Guo1,4 Hong-Bo Jin1,2,3 Corresponding author Email<EMAIL_ADDRESS>Cong-
Feng Qiao1,4 Yue-Liang Wu1,4,5 1The International Centre for Theoretical
Physics Asia-Pacific, University of Chinese Academy of Sciences, Beijing
100190, China
2Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal
University, Beijing, China
3National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China
4The School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
5Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing
100190, China

###### Abstract

Sky locating massive black hole binaries (MBHBs) from the foreground of double
white dwarf (DWD) is essential for space-based gravitational wave (GW)
detection. In an orbit period of the space crafts, there is an optimal orbital
position of the GW detectors to observe GW sources, where the signal intensity
is at its peak. From the model Q3-d, five MBHB sources are selected based on
the optimal observation orbital positions of the GW detectors, which are
associated with the orientation of the MBHB perpendicular to the detection
arms. For two MBHB sources of lower intensity, luminosity distance
uncertainties, $\Delta D_{L}$/$D_{L}$ at the 95$\%$ confidence level from the
overlapping GW signals of MBHB and DWD sources, when employing wavelet
decomposition and reconstruction methods, are improved by $\sim$ 2 times.
Besides, the angular resolutions $\Delta\Omega_{s}$ are also improved by a
factor of $\sim$ 20\. These results imply that we can obtain relatively high
accuracy of quickly locating MBHB from the overlapped GW signals with DWDs at
the best observation orbit position. The luminosity distance uncertainties at
the 95$\%$ confidence level for MBHB sources with the higher sign-noise ratio,
have constraints on the precision of the Hubble constant.

##  I Introduction

During the O1-O3 observing runs, ground-based observatories from the LIGO,
Virgo, and KAGRA collaborationsAbbott et al. (2023a, b) have detected nearly
one hundred gravitational wave events with frequency above 10 Hz.
Gravitational wave detection from space-based observatories, such as
LISA(Amaro-Seoane et al., 2017) and Taiji (Hu and Wu, 2017), operates in the
low-frequency range, spanning from $10^{-4}$ to $10^{-1}Hz$. Gravitational
wave (GW) signals emitted by the merger of massive black holes persist for an
extended duration within the sensitive frequency band of space-based
detectors, often overlapping with the continuous, nearly monochromatic GW
signals produced by Galactic white dwarf binaries. The number of Galactic
white dwarf binaries is in the order of $10^{7}$, and this type of source is
also called the foreground signal(Farmer and Phinney, 2003; Nelemans, 2009).
The number of resolvable foreground signals are about $10^{4}$ (Littenberg et
al., 2020; Zhang et al., 2021a).

The identification of foreground signals remains a challenge in the detection
of space-based gravitational waves. Overlapping gravitational wave (GW)
sources can interfere with the detection of other signals, including those
from merging massive black hole binaries(Klein et al., 2016). The data
analysis of those overlapping signals is one of the LISA Data Challenges(LDC,
) and the Taiji Data Challenge (TDC, ). The data analysis of overlapped GW
signals become already a general issue to be solved.

locating the sky position of the gravitational source is a key scientific goal
for gravitational wave observations(Błaut, 2011). Quickly locating the
identified binaries is the basic requirement for the space-based gravitational
wave detection, that is used to calibrate the detection data and reduce the
noises of detectors. Moreover, accurately determining the sky positions of
these gravitational sources facilitates multi-messenger observations when
combined with other gravitational wave detectors and astronomical data.
Additionally, the uncertainty in the sky location parameter, specifically the
luminosity distance, derived from these sources can contribute to refining the
estimation of the Hubble constant within cosmological models through the
distance-redshift relationshipSong et al. (2024).

For the sky location methods of gravitational wave signals, there are mainly
two types: One method is to use FIM (Fisher information matrix) to estimate
the source parameter error, which can give the estimation of the spatial
location of the gravitational source with high signal-to-noise ratio (SNR),
and the inverse FIM gives the covariance matrix of the parameters. The angular
resollocationution of the detector and the estimated signal parameters can
also be calculated as a function of the spatial position and frequency of the
source(Wen and Chen, 2010). Different detectors of joint detection can
effectively achieve better waves space positioning accuracy (Ruan et al.,
2021; Wen-Hong et al., 2020; Wang et al., 2020; Zhang et al., 2021b). The
parameter error estimation method of FIM is limited by the signal-to-noise
ratio, and it is only a general error estimation of the detector’s ability to
locate the gravitational source.

The second kind of method is based on Bayesian method, the posterior
distribution describes the uncertainty of the gravitational source parameters.
The most successful of the bayesian approach is based on markov chain monte
carlo (MCMC) methods (Crowder and Cornish, 2007a, b; Littenberg, 2011), this
kind of method provides richer information about the source parameters, but
compared with the second kind, the calculation cost is higher. However, the
MCMC method is a classical algorithm for parameter estimation of gravitational
sources. This method can realize the parameter estimation of space
gravitational source completely.

In our previous work, see ReferenceGuo et al. (2024) for details, the Taiji
response function is expressed in the transverse-traceless gauge and
calculated numerically in the time domain. We find that the amplitude of GW
signals are modulated by the periodic motion of GW detectors on the solar
orbit. The searching of GW signals and parameter estimation of GW sources from
the experimental data of the space-based mission do not ignore the orbit
positions relevant to GW sources.

In this paper, we focus on the sky location of massive black hole binaries
from foreground of Galactic white dwarf binaries, considering the amplitude
modulation effect of different detector positions on the detector response.The
best observation based on short time of massive black hole binaries is
achieved in this paper. Three models for the population of massive black hole
binary, i.e., pop III, Q3-nod, and Q3-d(Klein et al., 2016), are considered to
predict the events of massive black hole binary mergers. For some specific
Q3-d sources, the best location the sky position of the sources at the best
detector position is achieved in this paper. For the relatively strong Q3-d
sources, the accuracy of the parameters obtained from overlapped signals is
close to that without overlapped signals. However, for MBHB sources of lower
intensity, the accuracy of the parameters obtained from overlapped signals is
much worse than that without overlapped signals. The method of wavelet
decomposition and reconstruction can significantly weaken the impact of
overlapping with DWDs for MBHB sources of lower intensity.

The structure of this paper is as follows: in Sec.II , we mainly introduce the
signal model of the MBHB and the DWDs sources; the detector response and the
method used in this paper are shown in Sec.III, the detector response is
presented and best orbit position for observation is obtained ; in Sec.IV, Sky
location of MBHB at the best detector position is achieved; in Sec.V , the
full text is summarized.

##  II Signal Model

###  II.1 Massive Black Hole Binary

Three models for the population of MBHB, i.e., pop III, Q3-nod, and Q3-d(Klein
et al., 2016), are considered to predict the events of MBHB mergers. The
parameters are obtained from the website $https://www.gw-universe.org/obs-
space.html$. The detector parameters are set as follows, see the table 1 for
details.

parameter | value | parameter | value
---|---|---|---
the Arm length[m] | 3.0e9 | the Laser pow[W] | 2.2
Telescope diameter[m] | 0.35 |  Accelaration Noise[Hz-1] | 3.9e-44
Laser Shot Noise factor[Hz-1] | 5.3e-38 |  Other Optical Metrology system noise [Hz-1] | 2.81e-38
The Time [days] | 365 | SNR-threshol | 8
Cosmology |  Flat$\Lambda$CDM-Planck15 |  |
Table 1: Parameter list of Detectors.

We choose the model Q3-d that is closest to the reality among the three
models, and 5 MBHB sources data is listed according to the time length of one
year.

There are eight parameters: intrinsic mass of primary $/$ secondary BH (M⊙)
$M1/M2$, cosmological redshift $z$, spin of primary$/$secondary BH $s1/2$,
luminosity distance (Gpc) $D_{L}$, Ecliptic latitude (rad) $\beta$, Ecliptical
longitude (rad) $\lambda$, and signal to noise ratio SNR. We select a higher-
multipole gravitational waveform model for an eccentric binary black holes
based on the effective-one-body-numerical-relativity formalism, waveform model
SEOBNRE (Cao and Han, 2017; Liu et al., 2022). In the simulation experiment in
this paper, the deflection Angle $\iota$ is set to 60 degrees, and the
polarization Angle $\psi$ is set to 0. Specifically, 8 parameters are
required, see the table 2 for details.

Q3-d |  $M1$ [M⊙] |  $M2$ [M⊙] | $z$ |  $D_{L}$ [Gpc] |  $\beta$ [rad] |  $\lambda$ [rad] | SNR
---|---|---|---|---|---|---|---
Q3-delay-1 | $209156$ | $27895$ | $7.25$ | $73.581$ | $-0.06470$ | $3.180$ | $82.54$
Q3-delay-2 | $1361322$ | $770150$ | $1.12$ | $7.870$ | $0.2066$ | $4.536$ | $1818$
Q3-delay-3 | $1272097$ | $1127338$ | $1.69$ | $12.986$ | $-0.7767$ | $3.965$ | $3053$
Q3-delay-4 | $389952$ | $103676$ | $5.54$ | $53.903$ | $-0.1365$ | $0.7692$ | $180.5$
Q3-delay-5 | $103676$ | $275282$ | $0.749$ | $4.758$ | $0.1067$ | $4.469$ | $3661$
Table 2: Parameter list of model Q3-d.

###  II.2 Galactic white dwarf binaries

For Taiji, since the band of space gravitational wave detection is ranging
from $10^{-4}$ to $10^{-1}Hz$, a large number of Galactic white dwarf binaries
are overlapped in this band, that is, Galactic foreground is formed. DWDs are
described by a set of eight parameters: frequency $f$, frequency derivative
$\dot{f}$, amplitude ${\cal A}$, sky position in ecliptic coordinates
$(\lambda,\beta)$, orbital inclination $\iota$, polarisation angle $\psi$, and
initial phase $\phi_{0}$ (Cutler, 1998; Roebber et al., 2020; Karnesis et al.,
2021). GWs emitted by a monochromatic source are characterized and quantified
using the quadrupole approximationLandau and Lifshitz (1962); Peters and
Mathews (1963). In this approximation, the GW signal is described as a
combination of the two polarizations ($+$ and $\times$):

$h_{+}(t)=\mathcal{A}(1+\cos\iota^{2})\cos(\Phi(t))$ (1)
$h_{\times}(t)=2\mathcal{A}\cos\iota\sin(\Phi(t))$ (2)

with

$\Phi(t)=2\pi ft+\pi\dot{f}t^{2}+\phi_{0}$ (3)
$\dot{f}=\frac{96}{5}\pi^{8/3}\left(\frac{G{\cal
M}}{c^{3}}\right)^{5/3}f^{11/3},$ (4)
$\mathcal{A}=\frac{2(G\mathcal{M})^{5/3}}{c^{4}D_{L}}(\pi f)^{2/3},$ (5)

where $\mathcal{M}\equiv(m_{1}m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}$ is the chirp
mass, and $D_{L}$ is luminosity distance, and $G$ and $c$ are the
gravitational constant and the speed of light, respectively.

The parameters of DWDs sources are obtained from the parameter distribution,
which is detailed in the paperGuo et al. (2024).

The coordinates under the $SSB$-frame are also composed of the spatial
position coordinates ($\beta,\lambda$). So we can calculate the amplitude of
each DWDs signal source at each the spatial position coordinates
($\beta,\lambda$) from the DWDs in the Synthetic Galactic Population. The
formula for calculating the amplitude is the Eq. 5. The latitude
($\beta\in\left[-\pi/2,\pi/2\right]$) is divided into $100$ equal parts and
the longitude ($\lambda\in\left[0,2\pi\right]$) is divided into $100$ equal
parts. So we can get the grids of $100\times 100$ on the sky map. Because
there are so many of DWDs, there are many DWDs signals in each spatial grid.
Here, we choose only the maximum value and keep its corresponding DWDs source
as the source at that spatial location. So we can obtain ten thousand DWDs
sources with larger amplitudes. Ten thousand DWDs sources are projected onto
the detector arms ($L_{12}$, $L_{23}$, and $L_{31}$), the observation time is
set to one year, and the sampling frequency is set to 0.1Hz. The polarisation
angle is set $\psi=0$, and initial orbital phase is set $\phi_{0}=0$. Finally,
all the projection signals are added together to get the DWDs sources
confusion signal. The time domain and frequency domain of the DWDs sources
confusion signal on the detector arm $L_{12}$ are as seen in Fig. 1.

(a) Time domain

(b) Frequency domain

Figure 1: The projection results on the detector $L_{12}$ of DWDs confusion.

###  II.3 Gravitational Wave Model

This is a source frame, but for detectors, a Solar System Barycenter (SSB)
frame, based on the ecliptic plane is selected111https://lisa-
ldc.lal.in2p3.fr/static/data/pdf/LDC-manual-002.pdf.In the $SSB$-frame, the
standard spherical coordinates $\left(\theta,\phi\right)$, and the associated
spherical orthonormal basis vectors
$\left(\bf{e}r,\bf{e}\theta,\bf{e}\phi\right)$ are confirmed. The position of
the source in the sky will be parametrized by the ecliptice latitude
$\beta=\pi/2-\theta$ and the ecliptic longitude $\lambda=\phi$. The GW
propagation vector $\hat{k}$ in spherical coordinates is now

$\hat{k}=-\bf{e}r=-\cos\beta\cos\lambda\,\hat{x}-\cos\beta\sin\lambda\,\hat{y}-\sin\beta\,\hat{z}\,.$
(6)

Introduce reference polarization vectors as

$\displaystyle\hat{u}$ $\displaystyle=$
$\displaystyle-\bf{e}\theta=-\sin\beta\cos\lambda\,\hat{x}-\sin\beta\sin\lambda\,\hat{y}+\cos\beta\,\hat{z}$
$\displaystyle\hat{v}$ $\displaystyle=$
$\displaystyle-\bf{e}\phi=\sin\lambda\,\hat{x}-\cos\lambda\,\hat{y}$ (7)

The last degree of freedom between the frames corresponds to a rotation around
the line of sight, and is represented by the polarization angle $\psi$. The
polarization tensors are given by

$\displaystyle{\mbox{\boldmath$\epsilon$}}^{+}$ $\displaystyle=$
$\displaystyle\cos(2\psi){\bf e}^{+}-\sin(2\psi){\bf e}^{\times}$
$\displaystyle{\mbox{\boldmath$\epsilon$}}^{\times}$ $\displaystyle=$
$\displaystyle\sin(2\psi){\bf e}^{+}+\cos(2\psi){\bf e}^{\times}\,,$ (8)

where the basis tensors ${\bf e}^{+}$ and ${\bf e}^{\times}$ are expressed in
terms of two orthogonal unit vectors,

$\displaystyle{\bf e}^{+}$ $\displaystyle=$
$\displaystyle\hat{u}\otimes\hat{u}-\hat{v}\otimes\hat{v}$ $\displaystyle{\bf
e}^{\times}$ $\displaystyle=$
$\displaystyle\hat{u}\otimes\hat{v}+\hat{v}\otimes\hat{u}\,.$ (9)

GW traveling in the $\hat{k}$ direction are written as the linear combination
of two independent polarization states,

${\bf
h}(\xi)=h_{+}(\xi){\mbox{\boldmath$\epsilon$}}^{+}+h_{\times}(\xi){\mbox{\boldmath$\epsilon$}}^{\times}\,,$
(10)

where the wave variable $\xi=t-\hat{k}\cdot{\bf x}$ gives the surfaces of
constant phase.

##  III Short-time Observation from Taiji

###  III.1 Detector Response

Like LISA, the individual Taiji spacecraft will follow independent Keplerian
orbitsRubbo et al. (2004). The spacecraft positions are expressed as a
function of time. The parameter $\kappa$ gives the initial ecliptic longitude.
By setting the mean arm-length equal to those of the Taiji baseline,
$L=3\times 10^{9}$ m.

The laser link signal is emitted from the spacecraft $i$ at $t_{i}$ moment and
received by the spacecraft $j$ at $t_{j}$ moment. Assume that $c=1$. The
gravitational wave travels in the $\hat{k}$ direction and ${\mathbf{x}(t)}$ is
the position on the laser link at time t. The Eq. (11) can calculate the
$\delta\ell_{ij}$ in a numerical way in the time domainGuo et al. (2024).

$\delta\ell_{ij}=\frac{1}{2}\hat{r}_{ij}(t)\otimes\hat{r}_{ij}(t):\int_{t_{i}}^{t_{j}}{\bf
h}(t-\hat{k}\cdot{\mathbf{x}(t)})dt$ (11)

where, $\hat{r}_{ij}(t)$ denotes the unit vector

$\hat{r}_{ij}(t_{i})=\frac{{\bf x}_{j}(t_{j})-{\bf
x}_{i}(t_{i})}{\ell_{ij}}\,.$ (12)

and ${\bf h}(\xi)$ is the gravitational wave tensor in the transverse-
traceless gauge. The colon here denotes a double contraction, ${\bf a}:{\bf
b}=a^{ij}b_{ij}$.

###  III.2 Best orbit position for observation

The annual modulation effect of space gravitational wave detector on the
source of gravitational wave is mainly the result of modulating the amplitude
of the source signal. The annual modulation effect is mainly caused by the
change of the Angle between the propagation direction of the gravitational
wave (GW) source and the detector arm during the one-year orbit. In this case,
we’re defining the Angle $\theta$ Guo et al. (2024)

$\theta=\arccos{\left|\frac{\hat{k}\cdot\hat{r}_{ij}}{|\hat{k}|\left|\hat{r}_{ij}\right|}\right|}$
(13)

where $\hat{k}$ is the propagation direction of the GW source and
$\hat{r}_{ij}(t)$ denotes the unit vector in the Eq. (12). Here, we set
$t_{i}=0s$ and $t_{j}=10s$ . When the laser has the same transmitting time and
receiving time, the Angle $\theta$ is different when the detectors are at
different orbital positions.

In the extreme case, when the Angle $\theta=0^{\circ}$, the projection of the
GW source on the detector arms ($L_{12}$, $L_{23}$, and $L_{31}$) is the
smallest; when the Angle $\theta=90^{\circ}$, the GW source is projected on
the detector arms ($L_{12}$, $L_{23}$, and $L_{31}$) to the maximum. In this
case, the variable, the initial ecliptic longitude $\kappa$ in detector orbit,
is selected to present the detector position.

However, the Angle $\theta$ can not vary from $0^{\circ}$ to $90^{\circ}$ for
GW sources in the whole space for all the variable $\kappa$ value
($\kappa\in\left[0,2\pi\right]$). Due to the constraints of the detector
configuration and orbit position, the modulation effect of the source signal
in the all-sky observation will be different. See more details in Guo et al.
(2024).

Here We select 5 MBHB sources from Q3-delay model, Q3-delay-1$\sim$Q3-delay-5
to observe the annual modulation effect of the detector on them. Here, take
365 different $\kappa$ values uniformly in the interval
$\kappa\in\left[0,2\pi\right]$. The change of the Angle $\theta$ on
$L_{12},L_{23}$ and $L_{31}$ and the mean of Angle $\theta$ on $L_{12},L_{23}$
and $L_{31}$ along with orbit position parameter $\kappa$ for MBHB:
Q3-delay-1$\sim$MBHB: Q3-delay-5, respectively are shown in Fig. 2.

For MBHB: Q3-delay-1, the Angle $\theta$ on $L_{23}$ can vary from about
$70^{\circ}$ to $90^{\circ}$. However, the Angle $\theta$ on $L_{12}$ and
$L_{31}$ can vary from about $20^{\circ}$ to about $70^{\circ}$. In other
words, the projection signal amplitude of MBHB: Q3-delay-1 on the $L_{23}$
always maintains a relatively large, while the projection signal amplitude of
MBHB: Q3-delay-1 on $L_{12}$ or $L_{31}$ decreases due to the influence of
different detectors positions.

When processing the one-year observation data, we must pay more attention to
the data segment with the largest projection amplitude of the GW source to be
solved, which makes the estimation results of the source parameters more
accurate. The mean of the Angle $\theta$ on $L_{12}$, $L_{31}$ and $L_{23}$
can synthesize the size change of the GW source on the three detector arms.
When the mean value reaches the maximum, the corresponding orbit position
parameter $\kappa$ is the best observation position. Therefore, the best
observation position of a MBHB source is obtained according to the variation
of Angle $\theta$ for all the variable $\kappa$ value
($\kappa\in\left[0,2\pi\right]$). So we can get the best observation position
$\kappa$ of 5 MBHB sources.

According to the best observation position of MBHB sources, by projecting the
5 signals of the model Q3-d, three observation data on the detector arms for
each MBHB are obtained. By combining the responses of 5 MBHB sources of model
Q3-d with the 10,000 DWDs confusion signal responses, 5 segments of overlapped
GW signals data containing MBHB are obtained.

(a) MBHB:Q3-delay-1 (b) MBHB:Q3-delay-2 (c) MBHB:Q3-delay-3

(d) MBHB:Q3-delay-4 (e) MBHB:Q3-delay-5

Figure 2: The change of the Angle $\theta$ on $L_{12},L_{23}$ and $L_{31}$ and
the mean of Angle $\theta$ on $L_{12},L_{23}$ and $L_{31}$ along with orbit
position parameter $\kappa$ for MBHB sources: Q3-delay-1$\sim$Q3-delay-5,
respectively.

##  IV Sky location of MBHB

###  IV.1 Bayesian inference

The Bayesian inference is based on calculating the posterior probability
distribution function (PDF) of the unknown parameter set
$\bm{\theta}=\\{\theta_{1},\dots,\theta_{m}\\}$ in a given model, which
actually updates our state of belief from the prior PDF of $\bm{\theta}$ after
taking into account the information provided by the experimental data set $D$.
The posterior PDF is related to the prior PDF by the Bayes’s therom

$\displaystyle
p(\bm{\theta}|D)=\frac{\mathcal{L}(D|\bm{\theta})\pi(\bm{\theta})}{p(D)},$
(14)

where $\mathcal{L}(D|\bm{\theta})$ is the likelihood function, and
$\pi(\bm{\theta})$ is the prior PDF which encompasses our state of knowledge
on the values of the parameters before the observation of the data. The
quantity $p(D)$ is the Bayesian evidence which is obtained by integrating the
product of the likelihood and the prior over the whole volume of the parameter
space

$p(D)=\int_{V}\mathcal{L}(D|\theta)\pi(\bm{\theta})d\theta.$ (15)

The evidence is an important quantity for Bayesian model comparison.It is
straight forward to obtain the marginal PDFs of interested parameters
$\\{\theta_{1},\dots,\theta_{n}\\}(n<m)$ by integrating out other nuisance
parameters $\\{\theta_{n+1},\dots,\theta_{m}\\}$

$p(\theta_{1},\dots,\theta_{n})_{\text{marg}}=\int
p(\bm{\theta}|D)\prod_{i=n+1}^{m}d\theta_{i}.$ (16)

The marginal PDF is often used in visual presentation. If there is no
preferred value of $\theta_{i}$ in the allowed range ($\theta_{i,\text{min}}$,
$\theta_{i,\text{max}}$), the priors are taken as a flat distribution

$\displaystyle\pi(\theta_{i})\propto\left\\{\begin{tabular}[]{ll}1,&\text{for
} $\theta_{i,\text{min}}<\theta_{i}<\theta_{i,\text{max}}$\\\
0,&\text{otherwise}\end{tabular}\right..$ (19)

The likelihood function is often assumed to be Gaussian

$\displaystyle\mathcal{L}(D|\bm{\theta})=\prod_{i}\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\left[-\frac{(f_{\text{th},i}(\bm{\theta})-f_{\text{exp},i})^{2}}{2\sigma_{i}^{2}}\right],$
(20)

where $f_{\text{th},i}(\bm{\theta})$ are the predicted $i$-th observable from
the model which depends on the parameter set $\bm{\theta}$, and
$f_{\text{exp},i}$ are the ones measured by the experiment with uncertainty
$\sigma_{i}$.

When specifying the form of the likelihood function, a posterior PDF is
determined using the Markov Monte Carlo (MCMC) method based on the prior PDF
and likelihood function. Specifically, this article uses the Metropolis-
Hastings sampling method to obtain a posterior PDF.

###  IV.2 Wavelet decomposition and reconstruction

Wavelet transform (WT) (Chui, 1992; Meyer et al., 1992; Benedetto, 1993;
Misiti et al., 2013) is a very important and classic method in signal
processing. This article mainly studies the wavelet decomposition and
reconstruction methods, and chooses the discrete wavelet transform (DWT) for
calculation. First, downsample the object to be decomposed, and then perform
wavelet decomposition of the signal. Through two different filters, namely
high-pass and low-pass filters, convolved with the signal respectively, the
coefficients after wavelet decomposition are obtained. It is divided into two
types of coefficients, one is the detail coefficients (cD) obtained by high-
pass filtering and the approximation coefficients (cA) obtained by the low-
pass filter. The specific wavelet decomposition process is shown in the figure
3. Then select the detail coefficients (cD) and the approximation coefficients
(cA) to obtain the reconstructed signal through wavelet reconstruction which
is viewed as the inverse process of wavelet decomposition.

Figure 3: Wavelet decomposition process of 3 level(Wodarski et al., 2020;
Singh et al., 2011).

###  IV.3 Posterior

The MBHB source position parameters are set by
$\bm{\theta}=\\{sin\beta,\lambda,D_{L}\\}$. The prior is selected as the area
near the true parameter values of 5 sources to be observed. Considering the
program running time, for this experiment we selected 1 MCMC chain with a
sample number of $136000$, and calculated the log likelihood value for each
group of samples. Then the posterior distribution of the parameters is
obtained through the Metropolis-Hastings sampling method.

In this paper, three sets of comparisons are carried out. The first set is to
solve the independent sources Q3-delay-1$\sim$Q3-delay-5 directly by FFT of
their response data, abbreviated as MBHB(FFT); The second set is to solve the
overlapped GW signals of Q3-delay-1$\sim$Q3-delay-5 and DWDs by FFT directly
on their response data, abbreviated as MBHB+DWDs(FFT). The third set is to
perform wavelet decomposition on the response data of the overlapped GW
signals of Q3-delay-1, Q3-delay-4 and DWDs, and then perform FFT solution on
the new data reconstructed after filtering the approximate coefficients,
abbreviated as MBHB+DWDs(DWT+FFT). Here, $level=6$ and $db6$ is selected for
wavelet type of DWT.

Here, taking the $\chi^{2}$ logarithm of the likelihood function gives the
following form:

$\displaystyle\chi^{2}=\frac{(h_{\text{th},i}(\bm{\theta})-h_{\text{exp},i})^{2}}{\sigma_{i}^{2}},$
(21)

This paper mainly studies the observation results of the detector’s three arms
$L_{12}$, $L_{23}$ and $L_{31}$. Therefore, in the Eq. (21), $i$ values are
$1,2and3$, corresponding to the projection of the GW source on the three
detector arms respectively. The projected data obtained from the GW source to
be solved at the best observation position is selected as the experimental
data $h_{\text{exp},i}$=$h_{MBHB,i}$ for MBHB(FFT) and
$h_{\text{exp},i}$=$h_{MBHB,i}+h_{DWDs}$ for MBHB+DWDs(FFT). The uncertainty
of the experimental data $\sigma_{i}$=$1/\rm{SNR_{1}}\times h_{MBHB,i}$ for
MBHB(FFT) and $\sigma_{i}$=$1/\rm{SNR_{1}}\times h_{MBHB,i}$\+
$1/\rm{SNR_{2}}\times h_{DWDs,i}$ for MBHB+DWDs(FFT), where $\rm SNR_{1}$ and
$\rm SNR_{2}$ are calculated by the Eq. (22) through the projected data
obtained $h_{MBHB,i}$ and $h_{DWDs,i}$ and the analytic model for the
sensitivity curve $S_{n}$, specifically formula (13)(Robson et al., 2019) for
MBHB(FFT), adding formula (14)(Robson et al., 2019) to formula (13)(Robson et
al., 2019) for MBHB+DWDs(FFT). The observation time $T_{OB}=5\times 10^{5}s$.
See details in the Table 3.

$\displaystyle\rm
SNR_{1,2}=\sqrt{\int\frac{|h_{MBHB,DWDs}^{2}(f)|}{S_{n}(f)}df},$ (22)

SNR | i | Q3-delay-1 | Q3-delay-2 | Q3-delay-3 | Q3-delay-4 | Q3-delay-5
---|---|---|---|---|---|---
SNR1(MBHB) | 1 | 4.9 | 326 | 362 | 22 | 366
2 | 7.0 | 285 | 196 | 19 | 334
3 | 5.2 | 248 | 258 | 22 | 282
SNR1(MBHB+DWDs) | 1 | 4.9 | 220 | 315 | 22 | 252
2 | 4.9 | 155 | 141 | 18 | 233
3 | 4.9 | 148 | 228 | 21 | 187
SNR2(MBHB+DWDs) | 1 | 1.0 | 2.4 | 1.9 | 4.6 | 2.4
2 | 3.6 | 1.1 | 0.92 | 3.7 | 1.2
3 | 0.76 | 2.8 | 1.6 | 1.3 | 3.0
Table 3: SNR1 and SNR2 for MCMC of Q3-d sources. $i$ are $1,2,3$ corresponding
to the detector’s three arms $L_{12}$, $L_{23}$ and $L_{31}$, respectively.

(a) MBHB(FFT):Q3-delay-2

(b) MBHB(FFT):Q3-delay-3

(c) MBHB(FFT):Q3-delay-5

(d) MBHB+DWDs(FFT):Q3-delay-2

(e) MBHB+DWDs(FFT):Q3-delay-3

(f) MBHB+DWDs(FFT):Q3-delay-5

Figure 4: The contract of posterior distribution of the sky position
parameters ($sin\beta,\lambda,D_{L}$) of MBHB(FFT) and MBHB+DWDs(FFT) for
Q3-delay-2, Q3-delay-3 and Q3-delay-5. The black (blue) vertical line in the
post-pdf of each one-dimensional marginal posterior PDF represents the true
value (statistical mean). The black star (blue cross) in the pdf after each
one-dimensional marginal posterior PDF represents the true value (statistical
mean). The red plus sign in each one-dimensional and one-dimensional marginal
posterior PDF indicates the best fit value. The contours contain the $68\%$
and $95\%$ probability regions of the parameter estimates.

The specific posterior distributions of the first set and the second set of
Q3-delay-2, Q3-delay-3 and Q3-delay-5 are shown in Figure 4. The specific
posterior distributions of the first set, the second set and the third set of
Q3-delay-1 and Q3-delay-4 are shown in the figure 5. The figure shows the one-
dimensional and two-dimensional marginal posterior probability density
distributions of the sky position parameters ($sin\beta,\lambda,D_{L}$) of the
source, and the best fitting values and statistical averages are obtained. The
contour lines enclose the $68\%$ and $95\%$ probability regions of the
parameter estimates. The black (blue) vertical line in the post-pdf of each
one-dimensional marginal posterior PDF represents the true value (statistical
mean). The black star (blue cross) in the pdf after each one-dimensional
marginal posterior PDF represents the true value (statistical mean). The red
plus sign in each one-dimensional and one-dimensional marginal posterior PDF
indicates the best fit value. The contours contain the $68\%$ and $95\%$
probability regions of the parameter estimates.

(a) MBHB(FFT):Q3-delay-1

(b) MBHB+DWDs(FFT):Q3-delay-1

(c) MBHB+DWDs(DWT+FFT):Q3-delay-1

(d) MBHB(FFT):Q3-delay-4

(e) MBHB+DWDs(FFT):Q3-delay-4

(f) MBHB+DWDs(DWT+FFT):Q3-delay-4

Figure 5: The contract of posterior distribution of the sky position
parameters of MBHB(FFT), MBHB+DWDs(FFT) and MBHB+DWDs(DWT+FFT) for Q3-delay-1
and Q3-delay-4.The black (blue) vertical line in the post-pdf of each one-
dimensional marginal posterior PDF represents the true value (statistical
mean). The black star (blue cross) in the pdf after each one-dimensional
marginal posterior PDF represents the true value (statistical mean). The red
plus sign in each one-dimensional and one-dimensional marginal posterior PDF
indicates the best fit value. The contours contain the $68\%$ and $95\%$
probability regions of the parameter estimates.

The error in the solid angleBłaut (2011) is given by,

$\Delta\Omega_{s}=2\pi\cos{\beta}\sqrt{C_{\beta\beta}C_{\lambda\lambda}-C_{\beta\lambda}^{2}}$
(23)

The covariance matrix C is calculated from the posterior distribution of the
parameters $\beta$ and $\lambda$.

Table 4 shows the results of parameter estimation of
Q3-delay-1$\sim$Q3-delay-5. The sky location parameters and corresponding true
values, the confidence interval at 68$\%$ and 95$\%$ confidence level,
luminosity distance uncertainty, $\Delta D_{L}$/$D_{L}$ based on 95$\%$
confidence level and angular resolution $\Delta\Omega_{s}$ are shown in this
table.

Q3-d | three sets of comparisons | parameter | real value |  95$\%$ CL |  lg($\Delta D_{L}$/$D_{L}$) |  lg($\Delta\Omega_{s}[deg^{2}]$)
---|---|---|---|---|---|---
Q3-delay-1 | MBHB(FFT) |  DL [Gpc] | 73.58 | [66.81, 80.73] | -0.7231 | 1.996
sin$\beta$ | -0.06466 | [-0.1669, 0.03313]]
$\lambda$[deg] | 182.2 | [169.7, 195.8]
MBHB+DWDs(FFT) |  DL [Gpc] | 73.58 | [59.7, 89.16] | -0.3975 | 3.352
sin$\beta$ | -0.06466 | [-0.5422, 0.4818]
$\lambda$[deg] | 182.2 | [129.2, 230.3]
MBHB+DWDs(DWT+FFT) |  DL [Gpc] | 73.58 | [65.64, 79.96] | -0.7108 | 2.049
sin$\beta$ | -0.06466 | [-0.1629, 0.04774]
$\lambda$[deg] | 182.2 | [169.1, 196.6]
Q3-delay-2 | MBHB(FFT) |  DL [Gpc] | 7.870 | [7.780, 7.960] | -1.640 | -0.7395
sin$\beta$ | 0.2051 | [0.1955, 0.2147]
$\lambda$[deg] | 259.9 | [259.3, 260.5]
MBHB+DWDs(FFT) |  DL [Gpc] | 7.870 | [7.767, 7.987] | -1.553 | -0.3367
sin$\beta$ | 0.2051 | [-0.2358, -0.04041]
$\lambda$[deg] | 259.9 | [259.1, 260.7]
Q3-delay-3 | MBHB(FFT) |  DL [Gpc] | 12.98 | [12.97, 12.99] | -2.812 | -0.4672
sin$\beta$ | -0.7009 | [-0.7122, -0.6886]
$\lambda$[deg] | 227.2 | [223.4, 231.2]
MBHB+DWDs(FFT) |  DL [Gpc] | 12.98 | [12.97, 12.99] | -2.812 | 0.009341
sin$\beta$ | -0.7009 | [-0.7177, -0.6837]
$\lambda$[deg] | 227.2 | [222.4, 231.9]
Q3-delay-4 | MBHB(FFT) |  DL [Gpc] | 53.90 | [49.61, 59.63] | -0.7307 | 1.044
sin$\beta$ | -0.1361 | [-0.2527, -0.05645]
$\lambda$[deg] | 44.07 | [41.13, 44.87]
MBHB+DWDs(FFT) |  DL [Gpc] | 53.90 | [43.7, 63.68] | -0.431 | 2.464
sin$\beta$ | -0.1361 | [-0.2619, 0.1651]
$\lambda$[deg] | 44.07 | [19.84, 49.42]
MBHB+DWDs(DWT+FFT) |  DL [Gpc] | 53.90 | [47.57, 62.95] | -0.5447 | 1.051
sin$\beta$ | -0.1361 | [-0.2608, -0.08463]
$\lambda$[deg] | 44.07 | [41.86, 45.71]
Q3-delay-5 | MBHB(FFT) |  DL [Gpc] | 4.758 | [4.748, 4.768] | -2.376 | -1.125
sin$\beta$ | 0.1065 | [0.0923, 0.1207]
$\lambda$[deg] | 256 | [255.4, 256.7]
MBHB+DWDs(FFT) |  DL [Gpc] | 4.758 | [4.748, 4.768] | -2.376 | -0.1817
sin$\beta$ | 0.1065 | [0.08503, 0.125]
$\lambda$[deg] | 256 | [255.3, 256.9]
Table 4: The results of three sets of comparisons of sky location parameter
estimation of Q3-delay-1$\sim$Q3-delay-5. Three sets of comparisons are given
in column 2. The spatial position parameters and their true values are given
in column 3. The confidence interval and luminosity distance uncertainty,
$\Delta D_{L}$/$D_{L}$ based on 95$\%$ confidence level are given in column 4.
The angular resolution $\Delta\Omega_{s}$ is shown in column 5.

From the Figure 4, Figure 5 and Table 4, we can find that, for the relatively
stronger MBHB sources, such as Q3-delay-2, Q3-delay-3, and Q3-delay-5,
comparing the first set MBHB(FFT) and the second set MBHB+DWDs(FFT), we find
that at a 95$\%$ confidence level, the parameters derived from the second set
MBHB+DWDs(FFT) are comparable to those obtained from the first set MBHB(FFT).
However, for MBHB sources of lower intensity, such as Q3-delay-1 and
Q3-delay-4, comparing the first set MBHB(FFT) and the second set
MBHB+DWDs(FFT), we find that at a 95$\%$ confidence level, the parameters
derived from the second set MBHB+DWDs(FFT) much worse than that the first set
MBHB(FFT). For MBHB sources of lower intensity, luminosity distance
uncertainties, $\Delta D_{L}$/$D_{L}$ derived at a 95$\%$ confidence level
from the third set MBHB+DWDs(DWT+FFT), when employing wavelet decomposition
and reconstruction methods, are improved by $\sim$ 2 times compared to that
from the second set MBHB+DWDs(FFT), corresponding to Q3-delay-1 and Q3-delay-4
respectively. The angular resolutions $\Delta\Omega_{s}$ derived from from the
third set MBHB+DWDs(DWT+FFT), when employing wavelet decomposition and
reconstruction methods, are improved by $\sim$ 20 times compared to that from
the second set MBHB+DWDs(FFT), corresponding to Q3-delay-1 and Q3-delay-4
respectively.The method of wavelet decomposition and reconstruction can
significantly weaken the impact of overlapping with DWDs for MBHB sources of
lower intensity.

##  V Conclusions

The GW signals from MBHB merging linger for much longer in the detector
sensitive band, which overlap with the continuous and nearly monochromatic GW
signals of DWDs sources. The data analysis of overlapped GW signals become
already a general issue to be solved. In this paper, we focus on locating the
sky position of MBHB from overlapped GW signals with DWDs sources. In fact,
the amplitude of GW signals are modulated by the periodic motion of GW
detectors on the solar orbit. For the first time, the effects of the periodic
orbit position parameter of space-based laser interferometer detectors on the
locating the sky position of MBHB are considered.

The model, Q3-d, for the population of massive black hole binary is selected.
At the best orbit position, the observations of Q3-d sources are performed.
Location parameter estimation of the five MBHB sources are performed using the
Metropolis-Hastings MCMC method. In this paper, three sets of comparisons are
carried out. The first set is to solve the independent sources
Q3-delay-1$\sim$Q3-delay-5 directly by FFT of their response data, abbreviated
as MBHB(FFT); The second set is to solve overlapped GW signals of
Q3-delay-1$\sim$Q3-delay-5 and DWDs sources by FFT directly on their response
data, abbreviated as MBHB+DWDs(FFT). The third set is to perform wavelet
decomposition on the response data of overlapped GW signals of Q3-delay-1,
Q3-delay-4 and DWDs sources, and then perform FFT solution on the new data
reconstructed after filtering the approximate coefficients, abbreviated as
MBHB+DWDs(DWT+FFT). Here, $level=6$ and $db6$ is selected for wavelet type of
DWT.

For the relatively stronger MBHB sources, such as Q3-delay-2, Q3-delay-3, and
Q3-delay-5, comparing the first set MBHB(FFT) and the second set
MBHB+DWDs(FFT), we find that at a 95$\%$ confidence level, the parameters
derived from the second set MBHB+DWDs(FFT) are comparable to those obtained
from the first set MBHB(FFT). However, for MBHB sources of lower intensity,
such as Q3-delay-1 and Q3-delay-4, comparing the first set MBHB(FFT) and the
second set MBHB+DWDs(FFT), we find that at a 95$\%$ confidence level, the
parameters derived from the second set MBHB+DWDs(FFT) much worse than that the
first set MBHB(FFT). The method of wavelet decomposition and reconstruction is
used in the paper.

For MBHB sources of lower intensity, luminosity distance uncertainties,
$\Delta D_{L}$/$D_{L}$ derived at a 95$\%$ confidence level from the
overlapping GW signals of MBHB and DWD sources, when employing wavelet
decomposition and reconstruction methods, are improved by a factor of $\sim$
2, corresponding to Q3-delay-1 and Q3-delay-4 respectively. The angular
resolutions $\Delta\Omega_{s}$ derived from from the third set
MBHB+DWDs(DWT+FFT), when employing wavelet decomposition and reconstruction
methods, are improved by $\sim$ 20 times compared to that from the second set
MBHB+DWDs(FFT), corresponding to Q3-delay-1 and Q3-delay-4 respectively. The
method of wavelet decomposition and reconstruction can significantly weaken
the impact of overlapping with DWDs for MBHB sources of lower intensity.

The angular resolution $\Delta\Omega_{s}$ and luminosity distance uncertainty,
$\Delta D_{L}$/$D_{L}$ of the method based on a short observation time are
smaller than the ones in Ruan et al. (2021) based on observation time of about
10 days. Through the distance-redshift relation, the luminosity distance
uncertainty obtained by gravitational source can help constrain the accuracy
of the Hubble constant in cosmologySong et al. (2024). lg($\Delta
D_{L}$/$D_{L}$) at 95$\%$ confidence level of Q3-delay-3 and Q3-delay-5 for
Taiji is lower than that for ET2CESong et al. (2024) which is about $-2$ and
lg($\Delta D_{L}$/$D_{L}$) at 95$\%$ confidence level of Q3-delay-2 for Taiji
is lower than that for ETSong et al. (2024) which is about $-1.5$. The
luminosity distance uncertainty at 95$\%$ confidence level obtained by MBHB
based on the method in this paper can enhance the constraints on the accuracy
of the Hubble constant in cosmology.

## Acknowledgements

This work has been supported by the Fundamental Research Funds for the Central
Universities. This work has been supported in part by the National Key
Research and Development Program of China under Grant No.2020YFC2201500, the
National Science Foundation of China (NSFC) under Grants Nos. 12147103,
11821505, 11975236, and 12235008, the Strategic Priority Research Program of
the Chinese Academy of Sciences under Grant No. XDB23030100. In this paper,
The numerical computation of this work was completed at TAIJI Cluster of
University of Chinese Academy of Sciences.

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Ginwidth=,height=,keepaspectratio mm[#1]

# 21cmSense v2: A modular, open-source 21 cm sensitivity calculator

Steven G. Murray  0000-0003-3059-3823 Jonathan Pober  0000-0002-3492-0433
Matthew Kolopanis  School of Earth and Space Exploration, Arizona State
University, Tempe, AZ, USA  0000-0002-2950-2974

(18 January 2024)

## Summary

The 21cm line of neutral hydrogen is a powerful probe of the high-redshift
universe (Cosmic Dawn and the Epoch of Reionization), with an unprecedented
potential to inform us about key processes of early galaxy formation, the
first stars and even cosmology and structure formation (ref-Liu2020Liu & Shaw,
2020), via intensity mapping. It is the subject of a number of current and
upcoming low-frequency radio experiments, including the MWA (ref-mwaTingay et
al., 2013), LOFAR (ref-lofarvan Haarlem et al., 2013), HERA (ref-heraDeBoer et
al., 2017) and the SKA (ref-Pritchard2015Pritchard et al., 2015), which
complement the detailed information concerning the brightest sources in these
early epochs from powerful optical and near-infrared telescopes such as the
JWST (ref-jwstCastellano et al., 2022).

21cmSense is a Python package that provides a modular framework for
calculating the sensitivity of these experiments, in order to enhance the
process of their design. This paper presents version v2.0.0 of 21cmSense,
which has been re-written from the ground up to be more modular and
extensible, and to provide a more user-friendly interface – as well as
converting the well-used legacy package, presented in (ref-Pober2013Pober et
al., 2013, ref-Pober20142014) from Python 2 to 3.

21cmSense can compute sensitivity estimates for both map-making (ref-fhdBarry
et al., 2019) and delay-spectrum (ref-Parsons2012Parsons et al., 2012)
approaches to power-spectrum estimation. The full sensitivity calculation is
rather involved and computationally expensive in its most general form,
however 21cmSense uses a few key assumptions to accelerate the calculation:

  1. 1.

Each baseline (pair of antennas) in the interferometer intrinsically measures
a dense blob of 2D spatial Fourier modes of the sky intensity distribution,
centred at a particular Fourier coordinate $(u,v)$ given by the displacement
vector between the antennas forming the baseline, and covering an area in this
$(u,v)$-space that is given by the Fourier-transform of the primary beam of
the instrument. The Fourier-space representation of the sky is thus built up
by collecting many baselines that cover the so-called “$(u,v)$-plane”.
21cmSense approximates this process of synthesising many baselines by nearest-
grid-point interpolation onto a regular grid in the $(u,v)$-plane.
Furthermore, importantly the $(u,v)$-grid is chosen to have cells that are
comparable to the instrument’s Fourier-space beam size, so that a particular
baseline essentially measures a single cell in the grid, and no more. This
maximizes resolution while keeping the covariance between cells small. This
removes the need for tracking the full covariance between cells, and also
removes the need to perform a beam convolution, which can be expensive.

  2. 2.

We do not consider flagging of visibilities due to RFI and other systematics,
which can complicate the propagation of uncertainties.

Some of the key new features introduced in this version of 21cmSense include:

  1. 1.

Simplified, modular library API: the calculation has been split into modules
that can be used independently (for example, a class defining the Observatory,
the Observation and the Sensitivity). These can be used interactively via
Jupyter (ref-jupyterKluyver et al., 2016) or other interactive interfaces for
Python, or called as library functions in other code.

  2. 2.

Command-line interface: the library can be called from the command-line,
allowing for easy scripting and automation of sensitivity calculations.

  3. 3.

More accurate cosmological calculations using astropy (ref-astropyAstropy
Collaboration et al., 2018; ref-Robitaille2013Robitaille et al., 2013)

  4. 4.

Improved documentation and examples, including a Jupyter notebook that walks
through the calculation step-by-step.

  5. 5.

Generalization of the sensitivity calculation. The Sensitivity class is an
abstract class from which the sensitivity of differing summary statistics can
be defined. Currently, its only implementation is the PowerSpectrum class,
which computes the classic sensitivity of the power spectrum. However, the
framework can be extended to other summaries, for example wavelets (ref-
Trott2016aTrott, 2016).

  6. 6.

Improved speed: the new version of 21cmSense is significantly faster than the
legacy version, due to a number of vectorization improvements in the code.

  7. 7.

Built-in profiles for several major experiments: MWA, HERA and SKA-1. These
can be used as-is, or as a starting point for defining a custom instrument.

An example of the predicted sensitivity of the HERA experiment after a year’s
observation at $z=8.5$ is shown in Figure 2, corresponding to the sampling of
the $(u,v)$-grid shown in Figure 1. The sensivity here is represented as a
“noise power” (i.e. the contribution to the power spectrum from thermal
noise). This figure also demonstrates that the new 21cmSense is capable of
producing sensitivity predictions in the cylindrically-averaged 2D power
spectrum space, which is helpful for upcoming experiments.

Figure 1: Sampling of the $(u,v)$-plane for the HERA experiment during a full
year of observations. Figure 2: Predicted noise-power of 1000 hours (one year)
of HERA observations, as a function of perpendicular and line-of-sight fourier
scale. The noise-power is represented for each $k$-model.

## Statement of need

21cmSense provides a simple interface for computing the expected sensitivity
of radio interferometers that aim to measure the 21cm line of neutral
hydrogen. This field is growing rapidly, with a number of experiments
currently underway or in the planning stages. Historically, 21cmSense has been
a trusted tool for the design of these experiments (ref-Greig2020Greig et al.,
2020; ref-Pober2013Pober et al., 2013, ref-Pober20142014) and for forecasting
parameter constraints (ref-Greig2015Greig & Mesinger, 2015, ref-Greig20172017,
ref-Greig20182018). This overhauled, modularized version of 21cmSense provides
a more user-friendly interface, improved performance, and the extensibility
required for the next generation, as evidenced by its usage in the literature
(ref-Breitman2024Breitman et al., 2024; ref-Schosser2024Schosser et al.,
2024).

## Acknowledgements

We acknowledge helpful conversations with Danny Jacobs.

## References

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