11institutetext: Xuefei Gong 22institutetext: Institute of Applied
Mathematics, Academy of Mathematics and System Science, Chinese Academy of
Science, Beijing, China, 100190. 22email<EMAIL_ADDRESS>33institutetext:
Shengnian Xu 44institutetext: Institute of Applied Mathematics, Academy of
Mathematics and System Science, Chinese Academy of Science, Beijing, China,
100190. 44email<EMAIL_ADDRESS>55institutetext: Shanquan Gui
66institutetext: School of Physical Science and Technology, Lanzhou
University, Lanzhou, 730000, China; Department of Astronomy, School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China.
66email<EMAIL_ADDRESS>77institutetext: Shuanglin Huang
88institutetext: Institute of Applied Mathematics, Academy of Mathematics and
System Science, Chinese Academy of Science, Beijing, China, 100190; University
of Chinese Academy of Sciences, Beijing 100049, China. 88email:
<EMAIL_ADDRESS>99institutetext: Yun-Kau Lau 1010institutetext: Morningside
Center of Mathematics and Institute of Applied Mathematics, Academy of
Mathematics and System Science, Chinese Academy of Science, Beijing, China,
100190.
1010email<EMAIL_ADDRESS>
# Mission Design for the TAIJI misson and Structure Formation in Early
Universe

Xuefei Gong  Shengnian Xu  Shanquan Gui  Shuanglin Huang and Yun-Kau Lau
Corresponding author

###### Abstract

Gravitational wave detection in space promises to open a new window in
astronomy to study the strong field dynamics of gravitational physics in
astrophysics and cosmology. The present article is an extract of a report on a
feasibility study of gravitational wave detection in space, commissioned by
the National Space Science Center, Chinese Academy of Sciences almost a decade
ago. The objective of the study was to explore various possible mission
options to detect gravitational waves in space alternative to that of the
(e)LISA mission concept and look into the requirements on the technological
fronts. On the basis of relative merits and balance between science and
technological feasibility, a set of representative mission options were
studied and in the end a mission design was recommended as the starting point
for research and development in the Chinese Academy of Sciences. The mission
design was eventually adopted by the current TAIJI mission as the baseline
parameters for the project. Subject to technological constraints, the baseline
parameters of the TAIJI mission were designed in such a way to optimise the
capability of a spaceborne gravitational wave detector to probe high redshift
light seed, intermediate mass black holes and thereby shed important light on
the structure formation in early Universe.

##  1 Keywords

Gravitational wave detection in space; Mission design; Intermediate mass black
hole binaries at high redshift Universe.

##  2 1\. Introduction

In the year 2011, the general relativity group of the Morningside Center of
Mathematics was commissioned by the National Space Science Center, Chinese
Academy of Sciences to conduct a feasibilty study on the detection of
gravitational wave in space, as a project in the second phase of the
pioneering (Xiandao) program in space. At then, the objective of the study was
to explore various possible mission options to detect gravitational waves in
space alternative to that of the (e)LISA mission concept and look into the
requirements on the technological fronts. Though some preliminary study was
made in the first phase of the pioneering program 01 , a more complete and
systematic study was needed to assess the feasibility for China to undertake
further research and development in this direction, in tandem with Europe and
US at that time.

A report was submitted at the end of the two year study to the National Space
Center in which certain mission design was recommended as a starting point of
the development. This mission design was later adopted by the current TAIJI
mission as the baseline parameters for the project. In the intervening years,
certain parts of the report were made public both in Chinese and English 02 ;
020 . The core of the report is concerned with science objectives and it was
translated and published in English 03 . However the English translation left
a lot to be desired and obscured the real content of the report. It is the aim
of this article to put together the mission design and the primary science
drivers of the TAIJI mission design recommended in the report in a more
concise way in English, in the hope that the content of the report will be
accessible to a boarder spectrum of readership. At the same time it serves to
put on record the early phase of development of gravitational wave detection
in space in China.

The outline of the article may be described as follows. Section 2 will be
concerned with a catalogue of the gravitational wave sources at the frequency
window $10^{-4}-0.1Hz$. In section 3 we will retrace the line of the thoughts
leading eventually to the mission design of the TAIJI mission. In section 4,
we will present the simulation of the cosmic growth history of supermassive
black holes and give event rate estimate for the detection of intermediate
mass black holes at high redshift Universe. Section 5 is concerned with some
simple event rate estimate on the detection of intermediate mass ration
inspirals in dense star clusters. Some concluding remarks will be made in
section 6 to conclude this article.

##  3 2\. A survey of gravitational wave sources.

In comparison with ground-based detection, the characteristic mass and energy
scale of gravitational wave sources in the detection of gravitational wave in
space are generally much larger. Prospective gravitational wave sources in the
frequency window $10^{-4}Hz$ to $0.1Hz$ are displayed in Table 1.

Table 1: Expected gravitational wave sources within the frequency window $10^{-4}-0.1Hz$. Gravitational wave sources |  Brief description of source and the significance of its detection in astronomy and fundamental physics
---|---
Compact binaries (binary white dwarf and neutron star systems). |  Tens of millions of compact binaries in the Milky Way and beyond. Binaries with precisely measured parameters will be used to calibrate the instruments of a gravitational wave detector and stellar dynamics. The unresolved signals generate a foreground that hampers the detection of other gravitational wave sources. At the same time, the foreground provides important information on the galactic structure of the Milky Way, stellar evolution history and formation mechanism.
Intermediate mass black hole binaries at high redshifts. |  Intermediate mass black holes descended from the gravitational collapse of heavy Pop III stars, believed to be the seed black holes of the supermassive black holes in galactic centers at the present epoch. The detection will give clues on the galaxy-black hole coevolution and distinguish various models and mechanisms for the cosmic growth of supermassive black holes and galaxies.
Supermassive black holes merger |  The coalsecence of supermassive black holes that accompany galaxy mergers. The detection provides new way to understand the dynamics of galaxy mergers. The high signal to noise ratios also means that it is a good candidate for multi-messenger astronomy and give precision measurement on the redshift-luminosity relation for cosmic evolution.
Extreme mass ratio inspirals. |  Compact celestial bodies in the galactic centers captured by the supermassive black holes or stars generated by the formation and fast evolution of massive stars in the accretion disk surrounding the supermassive black hole. The detection presents a new way to probe galactic dynamics. The precision measurement of black hole parameters enables us to study spacetime structure in strong gravity field regime, check on the no-hair theorem of black holes and distinguish alternative gravity theories.
Intermediate mass ratio inspiral in dense star clusters |  Inspiral systems formed by compact celestial bodies and intermediate-mass black holes in the center of dense star cluster. Provide unequivocal evidence for the existence of intermediate-mass black holes and reveal the dynamics of star cluster and the formation mechanism for intermediate-mass black holes.
Gravitational waves in early Universe |  Gravitational waves generated by the big bang, inflation and possibly other cosmological processes like for instance the first order electroweak phase transition.
Burst signals. |  Hyperbolic encounters of black holes or celestial bodies and possibly from unknown sources.
Unmodelled sources. |  New physics and new astronomy

###  3.1 2.1 Compact Binary Star Systems.

Through optical and X-ray observations, a number of compact binary star
systems with inspiral period shorter than one hour in the Milky Way were
identified, including some binary systems with mass-transfer taking place.
These binaries are confirmed gravitational wave sources at the millihertz
band. It is expected that, with the increase in precision in our observations,
more binaries of this kind 26 ; 27 ; 28 ; 49  will be discovered. As the
astrophysical parameters and positions of these systems may be measured
precisely, the gravitational waves generated by these sources may be predicted
in a very accurate way bender-hils ; farmer-phinney . These sources, termed
verification binaries, may be used in a reverse way as a standard candle to
calibrate a gravitational wave detector in space.

Though weak in strength in the gravitational wave amplitude in comparsion with
those generated by binary black hole mergers, due to the abundance in numbers
(up to millions) and much shorter distance from a spaceborne detector,
currently planned spaceborne detector will detect gravitational wave signals
generated by millions of compact binaries in and outside of the Milky Way.
Data analysis will enable us to identify signals from around ten thousand
individual compact binaries and determine the astrophysical parameters of
these wave sources, such as the mass, luminosity distance, position, and the
orbital inclination of the binary stars. The rest unresolved signals will
become a foreground (confusion) noise around the $mHz$ band and hamper the
detection of other gravitational wave sources. At the same time, the
foreground signal will be modulated by the annual motion of a spaceborne
detector with respect to the orientation of the Galactic disk. The strength,
spectral shape and annual variation of the foreground may then also be used to
constrain and probe galactic structure, stellar evolution history and
formation mechanism which are otherwise difficult to probe by conventional
electromagnetic astronomy.

###  3.2 2.2 Binary Black Hole Mergers

These are the majority of gravitational wave sources for detection in space
and holds the promise of unravelling important and significant information
concerning astrophysical and cosmological physics. Black hole binaries may be
classified in accordance with the mass ratios between the two black holes
undergoing inspiral and coalescence processes.

#### 2.2.1 Intermediate mass black holes binaries at high redshift Universe

Astronomical Observations indicate that supermassive black holes are present
in the centers of both normal and active galaxies, and that the velocity
dispersion of stars in the galactic bulge is closely correlated with the mass
of the supermassive black hole in the galactic center, i.e, the $M-\sigma$
relation 2 ; 3 ; 4  . The $M-\sigma$ relation reveals that a co-evolution
history possibly exists between the galaxy and its central black hole. Quasars
observed by the SDSS at redshifts larger than 6 powered by the accretion of
supermassive black holes with $\sim 10^{9}M_{\odot}$ in mass require that the
seeds of supermassive black holes should form in the earlier period of the
Universe 5 ; 6 ; 7 . This gives important constraints to the formation of
supermassive black holes and the occurrence of the high-contrast density
fluctuations in early universe.

Observational results of the cosmic background radiation indicate that energy
fluctuations of the fractional order of $10^{-5}$ exist on the isotropic
background after the recombination era at $z\sim 1000$. The gravitational
(Jeans) instability induced collapse and aggregation in these high-density
places of the early Universe, and the first generation dark matter halos
formed at around $z=20\sim 30$, in which the gas finally experienced cooling,
fragmentation, and agglomerating of $H_{2}$ molecules and the Pop III ( first
generation) stars as a result 8 ; 9 . A lack of metal abundance nor efficient
cooling mechanism mean that the Pop III stars are very massive. The first
light from these stars ended the dark age era of the Universe, while its
energetic UV component ionised the fragile neutral $H_{2}$ molecules and
heralded the dawn of reionization. With zero metal abundance, the stellar wind
loss caused by radiation may be neglected, the lost mass in the final stage of
collapse is very small. Numerical simulations indicate that the Pop III stars
of $100\sim 260M_{\odot}$ disappear directly through the supernova explosions
due to electron-positron pair instability, while the stellar bodies with
heavier mass collapsed to form the black holes with mass greater than $\sim
150M_{\odot}$ 10 . This falls in the range of intermediate mass black holes
which is one possible origin of the seed black holes for supermassive black
holes observed at the present epoch. Alternatively, gas clouds may directly
collapse to form the seed black holes with mass heavier than
$10^{4}M_{\odot}$. Through an effective angular momentum transfer, the
collapse of stellar cluster nuclei may also form seed black holes of
$10^{3}\sim 10^{4}M_{\odot}$ 17 ; 18 .

According to the hierarchical growth model of cold dark matter, the
supermassive black holes located at center of a galaxy at the present epoch
evolved from density fluctuations before re-ionization: the small-mass sub-
galactic structures formed first, then merged with each other continuously to
form bigger and bigger structures 10 ; 11 ; 12  In this process, if black
holes existed in the galactic centers, these black holes would experience the
continuous accretion and merger processes of the dark matter halos and host
galaxies, and formed the currently observed massive black holes at different
redshifts 11 -19 .

A detailed understanding of the formation and growth of supermassive black
holes is of clear cosmological as well as astrophysical significance 190 . It
represents in part of our endeavour to understand galaxy evolution and
structure formation of the Universe. Electromagnetic astronomy enables us to
probe the dynamics of a galaxy at redshifts around 6 or a little higher. It is
expected that the prospective launch of the JWST telescope will extend our
horizon to higher redshifts in the infrared red regime of electromagnetic
waves. At high redshifts beyond the reach of electromagnetic astronomy,
gravitational waves detection provides a unique way to probe the growth of
black holes and galaxies through probing the merger of intermediate mass black
holes binaries descended from the Pop III stars. Gravitational waves may be
considered as an entirely new window to observe the galaxy-black hole co-
evolution process and obtain important information regarding the history of
structure evolution beginning from the end of the dark age era 20 ; 21 .

#### 2.2.2 Supermassive black hole mergers.

In the merger of two galaxies, provided there are supermassive black holes
residing in the the centers of both galaxies, the two supermassive black holes
will be dragged into the newly formed galactic center due to the dynamic
friction with the surrounding stars, and gradually a distance is reached
within which gravitational interaction becomes dominant and a bound state is
formed. Through the mutual three-body interaction with the surrounding small
celestial bodies or the friction with gas, the potential energy is reduced and
the orbit is further shrunk. When the binary black hole system transfers their
gravitational energy and angular momentum to the surrounding stars through the
interactions with them, the surrounding stars acquire higher energy and
angular momentum and become capable of escaping from the galactic center. When
the stellar environment is deficient in gas and the galactic structure is
spherical symmetric, there is then a lack of dynamical drag that drives
further the merger of two black holes (i.e., the final parsec problem).
Current study indicates that after merger the structures of galaxies
themselves are not spherical symmetric and the the stellar orbits are
aspherical. There are enough stars capable of getting into the vicinity of the
binary black holes, so that the two black holes can keep their closeness by
ejecting stars, and the distance between the black holes further reduces;
finally, gravitational waves generated by the gravitational interaction of the
binary black holes begins to play a dominant role and brings away the energy
and angular momentum, ending up finally coalescence of the binary black holes.

In the case when the stellar environment is gas rich, after the formation of
binary black holes, a common accretion disk for the binary black holes forms
from the surrounding gas. The binary black holes further accrete matter,
transfer the angular momentum to surrounding stars and finally forms a single
black hole through the emission of gravitational waves 13 ; 14 ; 15 ; 17 ; 56
. Many complicated astrophysical factors and processes are involved, such as
gas accretion, dynamic interaction with the small stars, structure of the
accretion disk, accretion efficiency, and gravitational recoil of the final
merger, etc. Detection of gravitational waves provides an entirely new way to
the understanding of these physical processes 21 ; 57 ; 58

The signal-noise ratio for the detection of the merger of two supermassive
black holes is in general very large. The gravitational wave signals generated
in the final merging process is detectable even in the time domain. Through
the accumulation in time sufficient signal to noise ratio (for example, one
month or one year before merger), certain astrophysical parameters like for
instance the luminosity distance, orbital plane and possibly other parameters
of the binary supermassive black hole merging system may be precisely measured
and these otherwise will be difficult to obtain by electromagnetic means.
Combined with the information of redshifts from measurements in
electromagnetic wave astronomy, the binary supermassive black hole merger wave
sources will become a standard siren with large signal-noise ratio and
redshift range (in analogy to the standard candle of electromagnetic wave
astronomy). This will enable us to determine precisely the redshift-luminosity
relation and other cosmic evolution information 20 ; 21 .

#### 2.2.3 Extreme mass ratio inspirals.

In the center of a galaxy, through dynamical capture of a compact celestial
body (stellar mass black hole, neutron star or a white dwarf) by the
supermassive black hole located at the galactic center or possibly certain
companion stars generated by the formation and fast evolution of massive stars
in the accretion disk surrounding the supermassive black hole, a binary
system, termed extreme mass ratio inspiral (EMRI), is formed with mass ratios
$\sim 10^{6}$ or bigger between the supermassive black hole and the compact
celestial body. EMRIs are an important gravitational wave source for detection
in space around the $mHz$ regime 62 ; 63 ; 64 . Though the event rate is
highly uncertain, the general expectation is that EMRIs should be present in a
galactic center. A long duration of observation is needed to accumulate the
EMRI signal to noise ratio to a level above the threshold for detection.

The detection of gravitational waves from EMRIs provides a new window to study
and explore the galactic center region which otherwise would be difficult to
resolve by means of electromagnetic observations. Through probing the dynamics
of EMRIs near the galactic center, we will be able to measure the mass, spin
and higher order multipole moments of a supermassive black hole and the
detectable range extends far beyond the local Universe. The detection is also
a way to confirm the existence of a supermassive black hole in the center of
an inactive galaxy in the $(10^{5}\sim 10^{7}M_{\odot})$ regime and enable us
to do precision measurements on black hole parameters.

The power of gravitational radiation generated by an EMRI system is rather
weak and the orbital evolution of the compact celestial body is very slow. The
adiabatic evolution of Kerr black hole geodesics may be used to describe the
orbital dynamics generated by gravitational waves. Though the periodically
averaged frequency variation is generally unremarkable, due to the existence
of eccentricity and a number of precession effects, the gravitational waves
emitted by an EMRI system has rich short-period frequency variations and the
phases with complex structures 68 -72 . As a gravitational wave source, an
EMRI system is characterised by 17 parameters. It is a great challenge to
handle the adiabatic and multiscale evolution of the spacetime structure and
the corresponding geodesics. The data processing in a large and complex
parametric space together with a weak and long time wave train will also be a
problem we have to contend with. The mass, angular momentum, mass quadrupole
moment of a supermassive black hole retrieved from the gravitational wave
signals of EMRIs 73 ; 74  also presents a way to check on the no-hair theorem
of black holes 68 ; 76  and to distinguish alternative gravity theories.

#### 2.2.4 Intermediate mass ratio inspirals.

The observations of ultraluminous X-ray sources (ULXs) and stellar dynamics in
star clusters suggest the possible existence of intermediate-mass black holes
of $10^{2}\sim 10^{4}M_{\odot}$ in a dense star clusters. Observations
indicate that ultraluminous compact sources are very common in our local
Universe, the well known ones are M82-X1, M15, NGC3628, and the Centaurus
$\omega$ etc. 70 ; 78 ; 79 ; 80 . Generally, the ULX sources locate at a
certain distance from the centers of galaxies, which means that their mass is
not very large (does not exceed $10^{5}M_{\odot}$). It is found that some ULX
sources are surrounded by gaseous loops illuminated by X-rays, this means that
at least the X-rays emitted by a part of ULX sources are not highly
concentrated in the line of sight connecting the source to the Earth. If the
energy released by the ULX sources is isotropic and the luminosity does not
exceed the Eddington limit, it may be deduced that the mass for these X-ray
sources is in the range of $10^{2}\sim 10^{4}M_{\odot}$. A strong quasi-
periodic light variability has been detected for some ULX sources. The
accretion by an intermediate-mass black holes at the center of a star cluster
is one possible explanation for these ULX sources.

N body simulations of compact star clusters also suggests that intermediate-
mass black holes can be produced by the fast collision and merging of stellar
mass black holes in the central regions of star clusters, or by the repeated
aggregation of massive stars. Some simulations even suggest that multi
intermediate mass black holes may form in young star clusters 81 -85 . In the
star clusters containing intermediate-mass black holes, the compact celestial
bodies, such as the stellar mass black holes, neutron stars, etc., can be
captured by the intermediate-mass black hole through the dynamic processes of
two-body exchange or hierachical three-body interaction to form a binary
system with mass ratios of a few tens to one thousand 83 -88 , which is termed
the intermediate mass ratio inspiral (IMRI). Further, when a star cluster with
an intermediate-mass black hole evolves toward the galactic nucleus, it will
gradually be torn up by the tidal force, the remnant intermediate-mass black
hole and the supermassive black hole in the galactic center will form an IMRI
on a bigger mass scale 87 .

At present, the theory that an intermediate-mass black hole in the center of a
dense star cluster provides a plausible explanation of the ULX observations.
on, very strong There are however many other possible explanations for the
observations. Conventional electromagnetic astronomy often is not able to
provide good resolution of the center of a cluster. However, gravitational
waves can be violently emitted in the final inspiral stage of IMRI systems.
The gravitational wave detection of IMRIs in dense star clusters will give
equivocal evidence for the existence of intermediate-mass black holes in dense
stellar environment. At the same time, the detection provides an entirely new
avenue to probe the dynamics of star clusters and addresses important
astrophysical problems like the formation mechanism of intermediate-mass black
holes, the initial mass-function and dynamical evolution history of star
clusters etc.

The characteristic frequencies of gravitational waves emitted by IMRIs in a
dense star clusters fall into the $0.01\sim 1Hz$ regime. To optimise detection
of these sources, baseline parameters of a spaceborne detector is required to
tune to this frequency range as the most sensitive regime of a detector. At
the same time, better sensitivity in laser metrology is also needed to
compensate for the weak signal strength due to smaller intermedate mass scale
87 -90 .

#### 2.2.5 Gravitational waves from early Universe

Gravitational waves generated during the big bang and subsequent inflation era
provide the earliest clue in our quest for understanding of the origin and
creation of our Universe. The primary science objectives of The BBO (Big Bang
Observer) 100 ; 101  and the DECIGO (Deci-Hertz Gravitational wave
Observatory) 102 ; 103 ; 104  project are to detect directly the primordial
gravitational waves from the early universe. However these projects pose
extremely demanding requirements on the technological fronts. With the
foreseeable development in technologies in the next two decades, it is more
sensible to be content with giving an upper bound on this gravitational wave
background. There are many competing theories and models which seek to
describe the dynamics soon after the big bang, among these are the Big Bang-
inflation process, the first-order electroweak phase transition, the extra-
dimensional dynamics predicted by the superstring theory, the cosmic string
network, 20 ; 99  etc. The current mission design will be able to impose
meaningful constraints on the parameter space of these models.

#### 2.2.6 Burst signals.

Burst signals are difficult to forecast their waveforms and foresee their
detection at this stage. These signals may be originated from the short-
distance capture/grazing among celestial bodies, or from cosmological process
like the breaking up of a cosmic string and possibly some new and unmodelled
astrophysical or even quantum gravitational processes.

##  4 3\. Mission design

The starting point of the study is to explore the feasibility of detection in
the band gap from $0.1Hz$ to $10Hz$ between LISA and LIGO. To this end, it is
natural to consider in the mission design to shorten the armlength of the
interferometer and increase the precision of the laser metrology by
suppressing shot noise and enlarge the diameter of the telescopes. This
happens to overlap with the ALIA mission concept proposed by Peter Bender
bender1 ; bender2  which is conceivably in many ways the simplest adaptation
of the LISA mission to a frequency band one order of magnitude higher than
that of LISA, with the promise of mapping out mass and spin distribution of
light seed black holes at high redshifts.

However, when the key technologies of the ALIA mission was further looked at,
the sub-picometer interferometry requirement in the laser metrology part poses
a major obstacle on the technological side of the mission. With a view that
China will have a reasonable chance to realise the mission in the next few
decades and to minimise possible risks in future research and development of
the key technologies, further relaxation of the baseline parameters to a more
realistic level seems to be a natural step to take.

Based on the relative merits between science and technological viability, a
number of mission options from both scientific and technological perspective
are carefully studied. Mainly due to foreseeable technological limitations in
laser interferometry in the next two decades, in the end a compromise between
scientific significance and technological feasibility is reached and the
following baseline parameters were chosen for further studies given in Table
2. Apart from the detection of gravitational wave sources in the
$10^{-4}-0.1Hz$ frequency window, the primary science driver is set to probe
high redshifts intermedate mass black hole binaries, with a view to understand
structure formation in early Universe and galaxy-black hole co-evolution. In
doing so, the mission will become a part of an astronomy program, working
closely with future infrared red astronomy and radio astronomy programs in
China to explore high redshift Universe after the dark age era. The baseline
parameters are subject to minor variations, in particular the position noise
may be relaxed further to 10pm or more, yet the science of the mission is
still worth pursuing.

Table 2:  arm length | telescope | laser | 1-way position | acceleration noise
---|---|---|---|---
($m$) | diameter($m$) | power($W$) | noise($pm\cdot Hz^{-1/2}$) | ($m\cdot s^{-2}\cdot Hz^{-1/2}$)
$3\times 10^{9}$ | 0.45$\sim$0.6 | 2 | 5$\sim$8 | $3\times 10^{-15}(>0.1mHz)$
$5\times 10^{8}$(ALIA) | 1.0 | 30 | 0.1 | $3\times 10^{-16}(>1mHz)$
$5\times 10^{9}$(LISA) | 0.4 | 2 | 18 | $3\times 10^{-15}(>0.1mHz)$
$1\times 10^{9}$(eLISA) | 0.2 | 2 | 11 | $3\times 10^{-15}(>0.1mHz)$

For reference purpose, the baseline design parameters of ALIA, LISA/eLISA are
also given in Table 2. The relevant sensitivity curves are displayed in Figure
1. Apart from the instrumental noises, confusion noise generated by both
galactic and extra-galactic compact binaries are also taken into
consideration. Relevant confusion levels are converted from estimations by
Hils and Bender bender-hils  and Farmer and Phinney farmer-phinney .

Figure 1: Sensitivity curves of mission designs with different choice of
baseline parameters, with ALIA, LISA and eLISA included for the purpose of
comparsion.

For black hole binaries with mass ratio $1:4$, typical of what one would
expect from hierarchical black hole growth at high redshift, the all angle
averaged detection range are plotted in Figure 2. Apart from galactic
confusion noise, upper level (dashed curve) and lower level (dotted dashed
curve) of confusion noise generated by extragalactic compact binaries as those
estimated by farmer-phinney  are also taken into account.

In calculating the averaged SNR, we have used hybrid waveforms in the
frequency domain with black hole spin not taken into account ajith1 ; ajith2 .
For one year of observation before merger, the contributions in SNR due to
large spin is indeed negligible according to our calculations. Spin is
relevant only in the parameter estimation stage, which will not be discussed
in the present work. As may be seen from Figure 2, for a given redshift, the
proposed mission concept is capable of detecting lighter black hole binaries
in comparsion with eLISA/LISA and thereby provides better understanding of the
hierarchical assembling process in early Universe.

Figure 2: All-angle averaged detection range of a single Michelson channel
with threshold SNR of 7 for $1:4$ mass ratio intermediate black hole binaries,
one year observation prior to merger. For each mission option, both upper and
lower confusion noise levels (represented by the dashed curve and dotted
dashed curve respectively) due to extragalactic compact binaries are
considered. Figure 3: The signal-noise ratio contours for detecting IMRIs at
different redshifts by a single Michelson channel with $3\times 10^{6}km$
armlength and one year observation before merger. (a) $5pm\cdot Hz^{-1/2}$,
$z=2$, (b) $5pm\cdot Hz^{-1/2}$, $z=6$, (c) $5pm\cdot Hz^{-1/2}$, $z=10$, (d)
$8pm\cdot Hz^{-1/2}$, $z=2$, (e) $8pm\cdot Hz^{-1/2}$, $z=6$, and (f)
$8pm\cdot Hz^{-1/2}$ , $z=10$. Confusion noise considered is at an
intermediate level between the upper and lower limit

Apart from intermediate mass black hole binaries at high redshift, the
designed sensitivity at around $0.01Hz$ measurement band means that the
instrument is also capable of detecting IMRIs harboured at globular clusters
or dense young star clusters at low redshift ($z<0.6$). See 88  for a further
discussion of the capture dynamics of an IMRI in dense star clusters.
Displayed in Fig 3 are the detection ranges of IMRIs with different mass
ratios one year prior to merger. The stellar black hole is fixed to be
$10M_{\odot}$, while the mass of an intermediate mass black hole is subject to
variation in order to generate different mass ratios in the figure.

Figure 4: All-angle averaged detection range under a single Michelson
threshold SNR of 7 for IMRIs with reduced masses of $10M_{\odot}$, one year
observation prior to merger. For each mission option, both upper and lower
confusion noise levels (represented by the dashed curve and dotted dashed
curve respectively) due to extragalactic compact binaries are considered.
Figure 5: The signal-noise ratio contours for detecting IMRIs with the reduced
mass of $10M_{\odot}$) by a single Michelson channel with armlength of
$3\times 10^{6}km$, one year of observation before merger. (a) $5pm\cdot
Hz^{-1/2}$, and (b) $8pm\cdot Hz^{-1/2}$. The confusion noise considered is at
an intermediate level between the upper and lower limits.

##  5 4\. Simulations of cosmic growth and merger of black holes and event
rate estimates

To understand the detection capability in high redshift Universe of the
mission options given in Table 2, we carry out a Monte Carlo simulation of
black hole merger histories based on the EPS formalism and semi-analytical
dynamics.

Pop III remnant black holes of $150M_{\odot}$ are placed in $3.5\sigma$ biased
halos at $z=20$ with initial spins of the seeds generated randomly. By
prescribing VHM-type dynamics 13 ; 14 , we trace downwards the black hole
merging history. The halo mass ratio criteria for major merger is set to be
greater than $0.1$. Both the prolonged accretion and the chaotic accretion
scenario are considered. Black hole spins coherently evolve through both
mergers and accretions processes and their magnitudes influence strongly the
mass-to-energy conversion efficiency. We assume efficient gaseous alignment of
the black holes so that the hardening time is short and only moderate
gravitational radiation recoils take place. Numerical simulations khan
suggest that the hardening and merger time scales remain short even in gas
free environment. In calculations relevant to GW observations, we assume a
threshold SNR of 7 for detection in the sense of single Michelson
interferometer and one year observation prior to merger.

The results are schematically given in Figure 6 and Figure 7.

Figure 6: Coalescence rate predicted by the Monte Carlo simulations. Figure 7:
Event rates estimates for the mission designs.

We assess our simulations by fitting the black hole mass functions and
luminosity functions at six almost equally divided successive redshift
intervals ranging from $z=0.4$ to $2.1$ (see Figure 8). In the prolonged
accretion scenario, the results deviate from the observational constrains
given by Soltan type argument when going up to redshift $z>1.5$. It may
therefore underestimate the black holes growth rate and perhaps also the
coalescence rate. Observationally the existence of very high redshift ($z>6$)
AGNs implies that feed back mechanisms may be very different at early epoch so
that fast growth of the seed black hole could be possible.

Figure 8: The degree of coincidence of the luminosity function of quasars
given by the Monte Carlo simulation with observed data

In terms of coalescence rate, our result displayed in Figure 4 is in overall
agreement with the results given by Sesana et al 21 ; 57  and Arun et al arun
, though the coalescence counts given by their simulations are about two or
three times higher. It is likely due to various numerical discrepancies in the
simulations. Overall, our black hole mass growth is slower, particularly in
the prolonged accretion scenario. At $z=15$, the total mass of the black hole
binaries typically are still less than $600M_{\odot}$ in the prolonged model
and this may lead to a smaller counts in detectable sources. Our results are
expected to give a very conservative (pessimistic) estimate of black hole
binaries merger event rate.

The astrophysics encapsulated in our simulation represents the state of art
understanding of structural formation after the dark age. Due to our poor
understanding of the evolution of the Universe at this epoch, it is likely
that the simulation overlooks many details of the physical processes involved.
The event rate count should be looked upon in a cautious way. Instead of
reading into the precise numbers, it serves as an indication what spaceborne
gravitational wave detector is capable of and in our case, the advantage of
setting the most sensitive regime of the measurement band from a few $mHz$ to
$0.01Hz$.

##  6 5\. Event rate estimate for the detection of IMRIs in dense star
clusters

Consider the following scenario: 84 ; 85 ; 86 :

(1) In the accessible range of the universe for the mission design, the
spatial number density of star clusters is a constant in respect to the volume
calculated by the luminosity distance. It has the same value as that of the
local universe. For globular and young clusters: $n_{GC}\approx 8h^{3}\cdot
Mpc^{-3}$, $n_{YC}\approx 3h^{3}\cdot Mpc^{-3}$, $h=H_{0}/(100km\cdot
s^{-1}\cdot Mpc^{-1})$ and we take $h=0.73$. In the event rate estimate in
what follows, we consider only the globular clusters and assume that the total
number density of dense star clusters is given by $n_{C}\approx 8h^{3}\cdot
Mpc^{-3}$, which will give a conservative estimation of the event rate.

(2) The probability of a small compact celestial body being captured by an
intermediate-mass black hole at the center of a cluster is given by

$\nu(M,\mu,z)\approx 10^{-10}\frac{M}{\mu}a^{-1},$

i.e., it is directly proportional to the mass of the intermediate-mass black
hole, and inversely proportional to the reduced mass. This assumption is
motivated by the dynamical analysis on globular clusters 82 -85 .

In the event rate calculations, for wave sources with non-negligible redshift
effect, a scaling factor $1+z$ is required in principle, but the coalescence
rate itself is just an of order of magnitude estimate. Further, the accessible
range of redshifts considered by the current mission options is quite small.
This scaling factor will be neglected here.

(3) In dense star clusters where the intermediate-mass black holes is located,
the mass distribution function of the intermediate-mass black holes is assumed
to be

$f(M)=\frac{f_{tot}}{ln\frac{M_{max}}{M_{min}}}\frac{1}{M}.$

$M_{{min}}$ and $M_{{max}}$ are respectively the lower and upper limits of the
mass-distribution range of intermediate mass black holes, they are taken
respectively as $10^{2}M_{\odot}$ and $10^{4}M_{\odot}$, and beyond this range
$f(M)=0$. $f_{tot}$ is the fraction of dense star clusters containing
intermediate-mass black holes. Its value is highly uncertain and we shall take
a conservative estimate that $f_{tot}=0.1$ 84 ; 86 .

Take the observational time of one year before merger, the reduced mass of an
IMRI system is taken to be $10M_{\odot}$ and the threshold value of signal-
noise ratio of a single Michelson detection is taken as 7. The event rate may
then be calculated using the following formula 84 ; 87 :

$R=\frac{4\pi}{3}\int^{M_{max}}_{M_{min}}[D_{L}(M,\mu)]^{3}\nu(M,\mu,z)n_{c}f(M)dM.$

and the result is given in Table 3.

Table 3: Prospective detection rate for IMRIs in globular clusters. Mission option | Upper level of confusion | Lower level of confusion
---|---|---
ALIA | $z_{c}=5:\sim 8000$ | $\sim 12000$
| $z_{c}=3:\sim 6000$ | $\sim 7000$
5pm(D=0.6m) | $\sim 90$ | $\sim 130$
8pm(D=0.45m) | $\sim 26$ | $\sim 32$
LISA ($5\times 10^{6}$ km in armlength) | $\sim 3$

The above event rate estimate is subject to many uncertainties and perhaps we
should not attach too much importance to the precise numbers. Instead, the
calculations serves as an indication of the detection potential of the mission
concept as far as IMRIs at low redshifts are concerned. Further, as event rate
goes up as the cubic of the improvement in sensitivity, it also brings out the
advantage of shifting slightly the most sensitive region of the measurement
band to a few hundredth $Hz$, as far as detection of IMRIs is concerned. It
should also be remarked that collision of dense star clusters 88  constitutes
a possible intermediate mass black hole binaries gravitational wave sources,
while the inspiral of massive black holes ($\sim 10^{3}M_{\odot}$ to $\sim
10^{4}M_{\odot}$) into the supermassive black hole at the center of a galaxy
is also a promising IMRI source 87 ; liu . However, the corresponding event
rates would be difficult to estimate.

##  7 Concluding remarks

Gravitational wave detection in space promises to open a new window in the
quest for understanding of our Universe, in particular as a new way to probe
the formation of galactic structure at early Universe discussed here. With the
recent revised definition of the LISA mission in which the armlength of the
laser interferometer is shortened to 2.5 million kilometers, modulo some minor
variations in the baseline parameters, there is basically no longer any
difference between LISA and TAIJI in terms of the mission definition. A global
effort to realise such a mission seems to be the next natural step forward,
though it does not seem possible in view of the current political climate.
Still it is an option we should keep in mind, in the hope that when the
occasion is right this becomes a realistic step to be taken.

## Acknowledgements

We were very grateful to Prof. Shing-Tung Yau, director of the Morningside
Center of Mathematics, for very strong financial backing that enabled us to
conduct the study in a very comprehensive way and thereby laid a solid
foundation for the current development in China. His encouragement also served
as a source of inspiration for the study. The feasibility study was conducted
with some very generous technical assistance from the Albert Einstein
Institute, Hannover and Peter Bender. Our thanks also go to Profs. Wenrui Hu
and Shuangnan Zhang for support.

## References

  * (1) Gong XF, Xu SN, Bai S, et al. (2011) A scientific case study of an advanced LISA mission. Classical and Quantum Gravity 28(9):094012.
  * (2) Gong XF, Lau YK, Xu SN, et al (2015) Descope of the ALIA mission. In: Journal of Physics: Conference Series (Vol. 610, No. 1, p. 012011). IOP Publishing.
  * (3) Gong XF, Xu SN, Yuan YF, et al. (2015) Laser interferometric gravitational wave detection in space and structure formation in the early universe(in Chinese). Progress in Astronomy, 33(1):59-83.
  * (4) Gong XF, Xu SN, Yuan YF, et al. (2015) Laser interferometric gravitational wave detection in space and structure formation in the early universe. Chinese Astronomy and Astrophysics, 39(4):411-446.
  * (5) Burgay M, D’Amico N, Possenti A, et al. (2003) An increased estimate of the merger rate of double neutron stars from observations of a highly relativistic system. Nature 426(6966):531-533.
  * (6) Lyne AG, Burgay M, Kramer M, et al. (2004) A double-pulsar system: a rare laboratory for relativistic gravity and plasma physics. Science 303(5661):1153-1157.
  * (7) Antoniadis J, Freire PC, Wex N, et al. (2013) A massive pulsar in a compact relativistic binary. Science 340(6131).
  * (8) Marsh T (2011) Double white dwarfs and LISA. Classical and Quantum Gravity 28(9):094019.
  * (9) Nelemans G, Zwart SP, Verbunt F, Yungelson L (2001) Population synthesis for double white dwarfs-II. Semi-detached systems: AM CVn stars. Astronomy & Astrophysics 368(3):939-949.
  * (10) Bender PL, Hils D (1997) Confusion noise level due to galactic and extragalactic binaries. Classical and Quantum Gravity 14(6):1439.
  * (11) Farmer AJ, Phinney ES (2003) The gravitational wave background from cosmological compact binaries. Monthly Notices of the Royal Astronomical Society 346(4):1197-1214.
  * (12) Ferrarese L, Merritt D (2000) A fundamental relation between supermassive black holes and their host galaxies. The Astrophysical Journal Letters 539(1):L9.
  * (13) Gebhardt K, Bender R, Bower G, et al. (2000) A relationship between nuclear black hole mass and galaxy velocity dispersion. The Astrophysical Journal Letters 539(1):L13.
  * (14) Ferrarese L (2002) Beyond the bulge: a fundamental relation between supermassive black holes and dark matter halos. The Astrophysical Journal 578(1):90.
  * (15) Fan X, Narayanan VK, Lupton RH, et al. (2001) A survey of $z>5.8$ quasars in the Sloan Digital Sky Survey. I. Discovery of three new quasars and the spatial density of luminous quasars at $z\sim 6$. The Astronomical Journal 122(6):2833.
  * (16) Fan X, Strauss MA, Schneider DP, et al. (2001) High-redshift quasars found in Sloan Digital Sky Survey commissioning data. IV. Luminosity function from the fall equatorial stripe sample. The Astronomical Journal 121(1):54.
  * (17) Fan X, Hennawi JF, Richards GT, et al. (2004) A survey of $z>5.7$ quasars in the Sloan Digital Sky Survey. III. Discovery of five additional quasars. The Astronomical Journal 128(2):515.
  * (18) Madau P, Rees MJ (2001) Massive black holes as population III rem- nants. The Astrophysical Journal Letters 551(1):L27.
  * (19) Heger A, Woosley SE (2002) The nucleosynthetic signature of population III. The Astrophysical Journal 567(1):532.
  * (20) Press WH, Schechter P (1974) Formation of galaxies and clusters of galaxies by self-similar gravitational condensation. The Astrophysical Journal 187:425-438.
  * (21) Koushiappas SM, Zentner AR (2006) Testing models of supermassive black hole seed formation through gravity waves. The Astrophysical Journal 639(1):7.
  * (22) Koushiappas SM, Bullock JS, Dekel A (2004) Massive black hole seeds from low angular momentum material. Monthly Notices of the Royal Astronomical Society 354(1):292-304.
  * (23) Lacey C, Cole S (1993) Merger rates in hierarchical models of galaxy formation. Monthly Notices of the Royal Astronomical Society 262(3):627-649.
  * (24) Cole S, Lacey CG, Baugh CM, Frenk CS (2000) Hierarchical galaxy formation. Monthly Notices of the Royal Astronomical Society 319(1):168-204.
  * (25) Volonteri M, Haardt F, Madau P (2003) The assembly and merging history of supermassive black holes in hierarchical models of galaxy formation. The Astrophysical Journal 582(2):559.
  * (26) Volonteri M, Madau P, Haardt F (2003) The formation of galaxy stel- lar cores by the hierarchical merging of supermassive black holes. The Astrophysical Journal 593(2):661.
  * (27) Volonteri M, Madau P, Quataert E, Rees MJ (2005) The distribution and cosmic evolution of massive black hole spins. The Astrophysical Journal 620(1):69.
  * (28) Volonteri M, Rees MJ (2005) Rapid growth of high-redshift black holes. The Astrophysical Journal 633(2):624.
  * (29) Begelman MC, Volonteri M, Rees MJ (2006) Formation of supermassive black holes by direct collapse in pre-galactic haloes. Monthly Notices of the Royal Astronomical Society 370(1):289-298.
  * (30) National Research Council (2010) New Worlds, New Horizons in Astronomy and Astrophysics. The National Academies Press, Washington, DC.
  * (31) Sathyaprakash BS, Schutz BF (2009) Physics, astrophysics and cosmology with gravitational waves. Living reviews in relativity, 12(1):2.
  * (32) Sesana A, Volonteri M, Haardt F (2007) The imprint of massive black hole formation models on the LISA data stream. Monthly Notices of the Royal Astronomical Society 377(4):1711-1716.
  * (33) Berti E, Volonteri M (2008) Cosmological black hole spin evolution by mergers and accretion. The Astrophysical Journal 684(2):822.
  * (34) Sesana A, Gair J, Berti E, Volonteri M (2011) Reconstructing the massive black hole cosmic history through gravitational waves. Physical Review D 83(4):044036.
  * (35) Gair JR, Sesana A, Berti E, Volonteri M (2011) Constraining properties of the black hole population using LISA. Classical and Quantum Gravity 28(9):094018.
  * (36) Amaro-Seoane P, Gair JR, Freitag M, Miller MC, Mandel I, Cutler CJ, Babak S (2007) Intermediate and extreme mass-ratio inspirals- astrophysics, science applications and detection using LISA. Classical and Quantum Gravity 24(17):R113.
  * (37) Miller MC, Freitag M, Hamilton DP, Lauburg VM (2005) Binary encounters with supermassive black holes: zero-eccentricity LISA events. The Astrophysical Journal Letters 631(2):L117.
  * (38) Levin Y (2007) Starbursts near supermassive black holes: young stars in the galactic centre, and gravitational waves in LISA band. Monthly Notices of the Royal Astronomical Society 374(2):515-524.
  * (39) Ryan FD (1995) Gravitational waves from the inspiral of a compact object into a massive, axisymmetric body with arbitrary multipole moments. Physical Review D 52(10):5707.
  * (40) Ryan FD (1997) Accuracy of estimating the multipole moments of a massive body from the gravitational waves of a binary inspiral. Physical Review D 56(4):1845.
  * (41) Barack L, Cutler C (2007) Using LISA extreme-mass-ratio inspiral sources to test off-kerr deviations in the geometry of massive black holes. Physical Review D 75(4):042003.
  * (42) Finn LS, Thorne KS (2000) Gravitational waves from a compact star in a circular, inspiral orbit, in the equatorial plane of a massive, spinning black hole, as observed by LISA. Physical Review D 62(12):124021.
  * (43) Drasco S (2006) Strategies for observing extreme mass ratio inspirals. Classical and Quantum Gravity 23(19):S769.
  * (44) Cornish NJ (2011) Detection strategies for extreme mass ratio inspirals. Classical and Quantum Gravity 28(9):094016.
  * (45) Gair JR, Porter E, Babak S, Barack L (2008) A constrained Metropolis-Hastings search for EMRIs in the mock LISA data challenge 1b. Classical and Quantum Gravity 25(18):184030.
  * (46) Drasco S, Hughes SA (2004) Rotating black hole orbit functionals in the frequency domain. Physical Review D 69(4):044015.
  * (47) Matsushita S, Kawabe R, Matsumoto H, Tsuru TG, Kohno K, Morita KI, Okumura SK, Vila-Vilaró B (2000) Formation of a massive black hole at the center of the superbubble in M82. The Astrophysical Journal Letters 545(2):L107.
  * (48) Van der Marel RP (2003) Intermediate-mass black holes in the universe: A review of formation theories and observational constraints. arXiv preprint astro-ph/0302101.
  * (49) Fabbiano G (2005) The hunt for intermediate-mass black holes. Science 307(5709):533-534.
  * (50) Maccarone TJ, Kundu A, Zepf SE, Rhode KL (2007) A black hole in a globular cluster. Nature 445(7124):183-185.
  * (51) Zwart SFP, McMillan SL (2002) The runaway growth of intermediate-mass black holes in dense star clusters. The Astrophysical Journal 576(2):899.
  * (52) Miller MC, Hamilton DP (2002) Production of intermediate-mass black holes in globular clusters. Monthly Notices of the Royal Astronomical Society 330(1):232-240.
  * (53) Miller MC (2002) Gravitational radiation from intermediate-mass black holes. The Astrophysical Journal 581(1):438.
  * (54) Miller MC, Colbert EJM (2004) Intermediate-mass black holes. International Journal of Modern Physics D 13(01):1-64.
  * (55) Miller MC (2005) Probing general relativity with mergers of super- massive and intermediate-mass black holes. The Astrophysical Journal 618(1):426.
  * (56) Will CM (2004) On the rate of detectability of intermediate-mass black hole binaries using LISA. The Astrophysical Journal 611(2):1080.
  * (57) Konstantinidis S, Amaro-Seoane P, Kokkotas, KD (2013) Investigating the retention of intermediate-mass black holes in star clusters using N-body simulations. Astronomy & Astrophysics, 557:A135.
  * (58) Mandel I, Gair JR (2009) Can we detect intermediate mass ratio inspirals? Classical and Quantum Gravity 26(9):094036.
  * (59) Gair JR, Mandel I, Miller MC, Volonteri M (2011) Exploring intermediate and massive black-hole binaries with the einstein telescope. General Relativity and Gravitation 43(2):485-518.
  * (60) Binetruy P, Bohe A, Caprini C, Dufaux JF (2012) Cosmological backgrounds of gravitational waves and eLISA/NGO: phase transitions, cos- mic strings and other sources. Journal of Cosmology and Astroparticle Physics 2012(06):027.
  * (61) Phinney S, Bender P, Buchman R, et al. (2004) The Big Bang Observer: Direct detection of gravitational waves from the birth of the universe to the present. NASA Mission Concept Study.
  * (62) Crowder J, Cornish NJ (2005) Beyond LISA: Exploring future gravitational wave missions. Physical Review D 72(8):083005.
  * (63) Seto N, Kawamura S, Nakamura T (2001) Possibility of direct measurement of the acceleration of the universe using 0.1 hz band laser interferometer gravitational wave antenna in space. Physical Review Letters 87(22):221103.
  * (64) Kawamura S, Ando M, Nakamura T, et al. (2008) The Japanese space gravitational wave antenna; DECIGO. Journal of Physics: Conference Series 120(3):032004.
  * (65) Ando M, Kawamura S, Seto N, et al. (2010) DECIGO and DECIGO pathfinder. Classical and Quantum Gravity 27(8):084010.
  * (66) Bender PL (2004) Additional astrophysical objectives for LISA follow- on missions. Classical and Quantum Gravity 21(5):S1203.
  * (67) Bender PL, Begelman MC (2005) Massive black hole formation and growth. In: Favata F, et al(eds) ESA SP-588, European Space Agency, p33.
  * (68) Ajith P, Babak S, Chen Y, et al. (2008) Template bank for gravitational waveforms from coalescing binary black holes: Nonspinning binaries. Physical Review D 77(10):104017.
  * (69) Ajith P, Hannam M, Husa S, et al. (2011) Inspiral-merger-ringdown waveforms for black-hole binaries with nonprecessing spins. Physical Review Letters 106(24):241101.
  * (70) Arun KG, Babak S, Berti E, et al. (2009). Massive black-hole binary inspirals: results from the LISA parameter estimation taskforce. Classical and Quantum Gravity, 26(9):094027.
  * (71) Khan FM, Holley-Bockelmann K, Berczik P, Just A (2013) Supermassive black hole binary evolution in axisymmetric galaxies: the final parsec problem is not a problem. The Astrophysical Journal, 773(2):100.
  * (72) Liu FK, Li S, Komossa S (2014) A milliparsec supermassive black hole binary candidate in the galaxy SDSS J120136.02+300305.5. The Astrophysical Journal, 786(2):103.


Virial Expansions for Correlation Functions in Canonical Ensemble

A. L. Rebenko

Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv,
Ukraine.
---

Abstract

The infinite set of coupled integral nonlinear equations for correlation
functions in the case of classical canonical ensemble is considered. Some kind
of graph expansions of correlation functions in the density parameter are
constructed. Existing of unique solutions for small value of density and high
temperature is discussed.

Keywords: 2-connected graphs, Kirkwood-Salzburg equation, virial expansion.
Mathematics Subject Classification 2020: 82B31.

##  Introduction.

The first mathematically rigorous studies of infinite statistical systems were
the works of Bogolyubov [1] in the canonical ensemble and Ruelle [22, 23] in
the grand canonical ensemble. In the monograph [1] Bogolyubov developed a
theory of expansions in degrees of density, which, although not yet rigorously
mathematically justified, but was based on the fundamental principles of
statistical mechanics and included all known at that time expressions of
Ursel-Mayer theory on the basis of his derived integro-differential equations
for $m$ -particle distribution functions. Really rigorous result was
formulated in [2] (see details in [6, 7]), but his description covered only
non-negative interaction potentials of the particles. It was only in 1969 that
Bogolyubov, Petrina, and Khatset [3] succeeded in generalizing these results
to the case of stable regular potentials. Following Ruelle’s operator method,
they derived a recurrent relation, which, in the thermodynamic limit, turned
into the Kirkwood-Salzburg equation. But the role of activity $z$ is played
some function of the density $a(\varrho)$, which is determined by series in
the set of all distribution functions. This essentially means that the
equation is an infinite system of nonlinear equations, and its iterations are
not a series expansion in the density. A little later, these equations were
derived by Pogorelov [15] directly from Bogolyubov’s integro-differential
equations (see [1]) and a theorem on the existence of a unique solution (as of
nonlinear operator equation) for small values of density was proved.

The virial expansion is most often understood as the expansion of pressure in
terms of the density [11] and obtained, by formally inverting the density as a
power series of the fugacity and composing the latter with the expansion for
the pressure (Mayer’s power series in the fugacity). In addition, we work with
expansions of correlation functions in the thermodynamic limit. Existence of a
thermodynamic limit for correlation functions in a canonical ensemble ”is more
subtle than for thermodynamic quantities like pressure and free energy which
are on a logarithmic scale” [9]. One more thing, the terms of the Mayer’s
cluster expansions can be expressed in terms of forest-graphs (see, i.e. [23,
12, 21], while those of the virial expansion are usually written in terms of
2-connected diagrams (see, for instance, the classical reference [25]).

In this article we construct first an expansion for correlation functions by
some special type of forest-graphs, but with huge number of terms in every
order of density. Due to the fact that this series consists of two parts that
have opposite signs, it is possible to reduce the contributions and, as a
result, obtain its representation by some set of graphs, connected components
of which are 2-connected graphs. Such set of graphs was also defined in the
article of Jansen [5] (see sect. 2).

The well-known proofs of convergence for the virial expansion are based on the
above mentioned inversion. Obviously the first result goes back to the work of
Lebowitz and Penrose [10] and some alternative approaches were given by
Groeneveld [4] (see also [16, 17, 13], which was continued in a more general
and modern presentation in the work [5]. The present work apparently has quite
a few points of contact with the work [5].

The article is organized as follows. In Section 2 we give a short mathematical
introduction of concepts and formulas in which the presentation of the article
is carried out. In Section 3, we briefly show the derivation of equations for
correlation functions in the canonical ensemble. The fourth section is devoted
to the representation of the solution in the form of the Lebesgue-Poisson
integral of some kernel, for which we write the recurrent nonlinear equation.
And finally, in the last section, the question of its solution in the form of
an expansion in terms of powers of density and the convergence of the integral
representation for correlation functions are discussed.

##  On some mathematical concepts

###  Configuration spaces and Lebesgue-Poisson measure.

Let $\sigma$ be a Lebesgue measure in $\mathbb{R}^{d}$. Configuration space
$\Gamma\;:=\;\Gamma_{\mathbb{R}^{d}}$ consists of all locally finite subsets
of the space $\mathbb{R}^{d}$. This definition is quite natural from the point
of view of physics, because an infinite number of particles cannot be in a
bounded volume.

Denote the set of all finite configurations of the space $\Gamma$ by
$\Gamma_{0}$, and by $\Gamma^{(n)}$ configuration space with a fixed number of
points.

If all such configurations are in some bounded set
$\Lambda\subset\mathbb{R}^{d}$, then the corresponding space will be
$\Gamma^{(n)}_{\Lambda}$.

State of an ideal gas in equilibrium statistical mechanics is described by the
Poisson measure $\pi_{z\sigma}$ on configuration space $\Gamma$, where $z>0$
is the activity (physical parameter that associated with the density of
particles in the system).

But we need so-called Lebesgue-Poisson measure, i.e. the Poisson measure on
the space of finite configurations, which in some sense is absolutely
continuous with respect to the Lebesgue measure in contrast to the
$\pi_{\sigma}$ measure for $\Gamma$. Integral for Lebesgue-Poisson measure in
space $\Gamma_{0}$ (or $\Gamma_{\Lambda}$) is determined by the formula:

$\displaystyle\int_{\Gamma_{X}}F(\gamma)\lambda_{z\sigma}(d\gamma)\;$
$\displaystyle:=\;\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\int_{X}\cdots\int_{X}F(\\{x_{1},...,x_{n}\\})\sigma(dx_{1})\cdots\sigma(dx_{n})=$
$\displaystyle=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\int_{X}\cdots\int_{X}F_{n}(x_{1},...,x_{n})dx_{1}\cdots
dx_{n},$ (2.1)

for all measurable functions $F=\\{F_{n}\\}_{n\geq 0}$, $F_{n}\in
L^{1}(X^{n})$, and $\Gamma_{X}\in\\{\Gamma_{0},\Gamma_{\Lambda}$). Such
measure not only simplifies the recording of cumbersome expressions and
combinatorial proofs, but also allows to get some new results based on
properties infinite divisibility (see, for example, [18, 20, 19, 14].

In the language of integrals with the measure $\lambda_{z\sigma}$ we give one
well-known and important identity, which will be used very often below (see,
for example, [8]).

###### Lemma 2.1

For all measurable functions $G:\Gamma_{0}\mapsto\mathbb{R}$ and
$H:\Gamma_{0}\times\Gamma_{0}\mapsto\mathbb{R}$, for which
$G(\xi\cup\gamma)H(\xi,\gamma)\in
L^{1}(\Gamma_{0}\times\Gamma_{0},\lambda_{\sigma}\otimes\lambda_{\sigma})$,
the following equality is true:

$\int_{\Gamma_{0}}G(\gamma)\sum_{\xi\subseteq\gamma}H(\xi,\gamma\setminus\xi)\lambda_{\sigma}(d\gamma)=\int_{\Gamma_{0}}\int_{\Gamma_{0}}G(\xi\cup\gamma)H(\xi,\gamma)\lambda_{\sigma}(d\gamma)\lambda_{\sigma}(d\xi).$
(2.2)

Proof. Let
$\xi\restriction\Gamma_{0}^{(k)}=\\{x_{1},...,x_{k}\\}:=\\{x\\}_{1}^{k}$,
$\gamma\restriction\Gamma_{0}^{(m)}=\\{x_{k+1},...,x_{k+m}\\}:=\\{x\\}_{k+1}^{k+m}$.
Then

$\displaystyle\int_{\Gamma_{0}}\int_{\Gamma_{0}}G(\xi\cup\gamma)H(\xi,\gamma)\lambda_{\sigma}(d\gamma)\lambda_{\sigma}(d\xi)=$
$\displaystyle=\sum_{k,m=0}^{\infty}\frac{1}{k!m!}\int_{\mathbb{R}^{dk}}\int_{\mathbb{R}^{dm}}G(\\{x\\}_{1}^{k+m})H(\\{x\\}_{1}^{k},\\{x\\}_{k+1}^{k+m})\sigma(dx)^{k+m}=$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{k=0}^{n}{n\choose
k}\int_{\mathbb{R}^{dn}}G(\\{x\\}_{1}^{n})H(\\{x\\}_{1}^{k},\\{x\\}_{k+1}^{n})\sigma(dx)^{n}=$
$\displaystyle=\int_{\Gamma_{0}}G(\gamma)\sum_{\xi\subseteq\gamma}H(\xi,\gamma\setminus\eta)\lambda_{\sigma}(d\gamma).$

$\blacksquare$

Note that the introduction of the measure (2.1) and the application of the
lemma 2.1 greatly simplifies the calculations of many combinatorial
statements.

###  Algebra of generating functionals.

For any function $\psi\in C_{0}(\Gamma_{0})$ define generating functional :

$\displaystyle\widetilde{F}_{\psi}(j)$
$\displaystyle\;=\;\sum_{N=0}^{\infty}\frac{1}{N!}\int_{\mathbb{R}^{dN}}(dx)^{N}j(x_{1})\cdots
j(x_{N})\psi(x_{1},\ldots,x_{N})=$
$\displaystyle\;=\;\int_{\Gamma_{0}}e(j;\gamma)\psi(\gamma)\lambda_{\sigma}(d\gamma),$
(2.3)

where functional $e(j;\gamma)$ is defined by the formula

$\displaystyle e(j;\gamma):=\;$
$\displaystyle\begin{cases}1,\;&\gamma=\emptyset,\\\
\prod_{x\in\gamma}j(x),\;&\gamma\in\Gamma_{0}\setminus\\{\emptyset\\},\end{cases}$
(2.4)

and $j\in C_{0}(\mathbb{R}^{d})$ is bounded (take for simplicity $j\leq 1$),
continuum, nonnegative function.

It is easy to calculate using the lemma 2.1 that

$\widetilde{F}_{\psi_{1}}(j)\widetilde{F}_{\psi_{2}}(j)\;=\;\widetilde{F}_{\psi_{1}*\psi_{2}}(j)=\int_{\Gamma_{0}}e(j;\gamma)(\psi_{1}*\psi_{2})(\gamma)\lambda_{\sigma}(d\gamma),$
(2.5)

where $\psi_{1}*\psi_{2}$ is product $\psi_{1}$ and $\psi_{2}$ in commutative
algebra $\mathcal{A}$, which was introduced by Ruelle (see [23], Ch. 4).
Formula (2.5) establishes an unambiguous correspondence between the algebra of
generating functionals with operation of ordinary multiplication and the
algebra $\mathcal{A}$. Recall that for $\gamma=\\{x_{1},x_{2},...x_{n}\\}$

$(\psi_{1}*\psi_{2})(\gamma)=\sum_{\xi\subseteq\gamma}\psi_{1}(\xi)\psi_{2}(\gamma\setminus\xi).$
(2.6)

##  Description of the system of interacting particles in the canonical
ensemble.

###  Interaction between particles

We consider the 2-particle interaction, which is described by the potential
$V_{2}(x,y)=\phi(|x-y|),\;\phi(0)=+\infty$. For any configuration
$\gamma\in\Gamma_{0}$ an interaction energy is

$U(\gamma)=U_{\phi}(\gamma):=\sum_{\\{x,y\\}\subset\gamma}\phi(|x-y|),$ (3.7)

and interaction $W(\eta;\gamma)$ between particles of configuration
$\eta\in\Gamma_{0}$ with particles of configuration $\gamma\in\Gamma_{0}$ is

$W(\eta;\gamma):=\sum_{\begin{subarray}{c}x\in\eta\\\
y\in\gamma\end{subarray}}\phi(|x-y|).$ (3.8)

We impose the following conditions on the potential of interaction:

(A): 1. Stability:

$U(\gamma)\;\geq\;-B|\gamma|,\;B\,\geq 0,\;\gamma\in\Gamma_{0},$ (3.9)

2\. Regularity:

$C(\beta)\;=\;\int_{\mathbb{R}^{d}}dx|e^{-\beta\phi(|x|)}\;-\;1|\;<\;+\infty,\beta=\frac{1}{kT}.$
(3.10)

###  Derivation of equations for correlation functions.

The correlation functions of the canonical ensemble of the system $N$ of
particles in some limited volume $\Lambda$ in configuration
$\eta=\\{x_{1},\ldots,x_{m}\\}$ are determined by the sequence

$\displaystyle\rho^{(N)}_{\Lambda}(\eta)\;:=\;\begin{cases}&1,\;\;\text{at}\;\eta=\emptyset,\\\
&\varrho=\frac{N}{V}\;\;\text{at}\;\;|\eta|=1,V=\sigma(\Lambda),\\\
&0,\;\text{at};|\eta|=m\;>\;N,\end{cases}$ (3.11)

and for $2\leq m\leq N$,

$\rho^{(N)}_{\Lambda}(\eta)\;:=\;\frac{1}{Z_{\Lambda}(\beta,N)}\int_{\Gamma^{(N-m)}_{\Lambda}}\frac{d\gamma}{(N-m)!}e^{-\beta
U(\eta\cup\gamma)},$ (3.12)

where

$Z_{\Lambda}(\beta,N)\;=\;\frac{1}{N!}\int_{\Gamma^{(N)}_{\Lambda}}e^{-\beta
U(\gamma)}d\gamma,$ (3.13)

and $\varrho$ is the density of the particles in the system.

To derive equation for correlation functions using lemma 2.1 we write down the
right hand part of (3.12) by the integral (2.1):

$\rho^{(N)}_{\Lambda}(\eta)\;=\;\frac{1}{Z_{\Lambda}(\beta,N)}\int_{\Gamma_{\Lambda}}e^{-\beta
U(\eta\cup\gamma)}1\\!\\!1_{N-m}(\gamma)\lambda_{\sigma}(d\gamma),\;|\eta|\leq
N,$ (3.14)

where

$\displaystyle 1\\!\\!1_{N-m}(\gamma)\;:=\;\begin{cases}&1,\;|\gamma|=N-m,\\\
&0,\;|\gamma|\neq N-m.\end{cases}$ (3.15)

Introduce a notation $\eta^{(\hat{x})}:=\eta\setminus\\{x\\}$. Let in
configuration $\eta=\\{x_{1},...,x_{m}\\}$ the coordinate $x_{1}$ such that,
for which the following inequality is true:

$W(x_{1};\eta^{(\hat{x}_{1})})=\sum_{k=2}^{m}\phi(|x_{1}-x_{k}|)\;\geq\;-B,\;B\geq
0.$ (3.16)

Then

$U(\eta\cup\gamma)=W(x_{1};\eta^{(\hat{x}_{1})})+W(x_{1};\gamma)+U(\eta^{(\hat{x}_{1})}\cup\gamma)$
(3.17)

and

$e^{-\beta
W(x_{1};\gamma)}=\prod_{y\in\gamma}\left[(e^{-\beta\phi(|x_{1}-y|)}-1)+1\right]=\sum_{\xi\subseteq\gamma}K(x_{1};\xi),$
(3.18)

where

$\displaystyle
K(x_{1};\xi)\;:=\;\begin{cases}&\prod_{y\in\xi}(e^{-\beta\phi(|x_{1}-y|)}-1),|\xi|\geq
1,\\\ &1,\;\xi=\emptyset.\end{cases}$ (3.19)

Then

$\displaystyle\rho^{(N)}_{\Lambda}(\eta)\;=\;$ $\displaystyle\frac{e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}}{Z_{\Lambda}(\beta,N)}\int_{\Gamma_{\Lambda}}e^{-\beta
U(\eta^{(\hat{x}_{1})}\cup\gamma)}\times$ $\displaystyle\times
1\\!\\!1_{N-m}(\gamma)\sum_{\xi\subseteq\gamma}K(x_{1};\xi)1\\!\\!1_{\Lambda}(\gamma\setminus\xi)\lambda_{\sigma}(d\gamma).$
(3.20)

Apply to the integral in (3.2) the lemma 2.1 with

$G(\gamma)=e^{-\beta
U(\eta^{(\hat{x}_{1})}\cup\gamma)}1\\!\\!1_{N-m}(\gamma)\;\;\text{and}\;\;H(\xi;\gamma\setminus\xi)=K(x_{1};\xi)1\\!\\!1_{\Lambda}(\gamma\setminus\xi).$
(3.21)

Then

$\displaystyle\rho^{(N)}_{\Lambda}(\eta)\;$ $\displaystyle=\;\frac{e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}}{Z_{\Lambda}(\beta,N)}\int_{\Gamma_{\Lambda}}\int_{\Gamma_{\Lambda}}e^{-\beta
U(\eta^{(\hat{x}_{1})}\cup\xi\cup\gamma)}\times$ $\displaystyle\times
1\\!\\!1_{N-m}(\xi\cup\gamma)K(x_{1};\xi)1\\!\\!1_{\Lambda}(\gamma)\lambda_{\sigma}(d\gamma)\lambda_{\sigma}(d\xi).$
(3.22)

Rewrite the right hand side in the following way

$\displaystyle\rho^{(N)}_{\Lambda}(\eta)=e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}\frac{Z_{\Lambda}(\beta,N-1)}{Z_{\Lambda}(\beta,N)}\int_{\Gamma_{\Lambda}}K(x_{1};\xi)\times$
(3.23)
$\displaystyle\times\left[\frac{1}{Z_{\Lambda}(\beta,N-1)}\int_{\Gamma_{\Lambda}}e^{-\beta
U(\eta^{(\hat{x}_{1})}\cup\xi\cup\gamma)}1\\!\\!1_{N-m-|\xi|}(\gamma)\lambda_{\sigma}(d\gamma)\right]\lambda_{\sigma}(d\xi).$

Taking into account (3.14)($N-m-|\xi|=N-1-(m-1+|\xi|)$) we obtain:

$\rho^{(N)}_{\Lambda}(\eta)=e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}a_{N}^{\Lambda}\int_{\Gamma_{\Lambda}}K(x_{1};\xi)\rho^{(N-1)}_{\Lambda}(\eta^{(\hat{x}_{1})}\cup\xi)\lambda_{\sigma}(d\xi),$
(3.24)

where $|\eta|=m$ $\eta=\\{x_{1},...,x_{m}\\}$, $1<m<N$, and

$a_{N}^{\Lambda}\;=\;\frac{Z_{\Lambda}(\beta,N-1)}{Z_{\Lambda}(\beta,N)}.$
(3.25)

For $m=1$

$\rho^{(N)}_{\Lambda}(\\{x_{1}\\})=\frac{N}{V}=a_{N}^{\Lambda}\int_{\Gamma_{\Lambda}}K(x_{1};\xi)\rho^{(N-1)}_{\Lambda}(\xi)\lambda_{\sigma}(d\xi),\;\;V=\sigma(\Lambda).$
(3.26)

Note, also, that

$\rho^{(N)}_{\Lambda}(\eta)=0,\;\;\text{if}\;\;|\eta|\;\geq\;N.$ (3.27)

In the thermodynamic limit

$\Lambda\uparrow\mathbb{R}^{d},\;\;\lim_{\begin{subarray}{c}V\rightarrow\infty\\\
N\rightarrow\infty\end{subarray}}\frac{V}{N}=v,\;\;\frac{1}{v}=\varrho,$
(3.28)

existing of which was proved in [3] we have the following equation

$\rho(\eta)\;=\;\varrho a(\varrho)e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}\int_{\Gamma_{0}}K(x_{1};\xi)\rho(\eta^{(\hat{x}_{1})}\cup\xi)\lambda_{\sigma}(d\xi),$
(3.29)

where

$\lim_{\begin{subarray}{c}V=N/\varrho\\\
N\rightarrow\infty\end{subarray}}\frac{Z_{\Lambda}(\beta,N-1)}{Z_{\Lambda}(\beta,N)}\;=\;\varrho
a(\varrho),\;\;V=\sigma(\Lambda)$ (3.30)

and

$a(\varrho)=\frac{1}{\widetilde{Q}(\rho)},\;\widetilde{Q}(\rho)\;=\;\int_{\Gamma_{0}}K(x_{1};\xi)\rho(\xi)\lambda_{\sigma}(d\xi).$
(3.31)

##  Construction of expansion for correlation functions in the density
parameter $\varrho$.

Solution of the equation (3.29) by the iteration method (Ruelle operator
method [23]) or by the Minlos-Poghosyan method [12] (about the connection
between these methods see [21]) leads to the series in the parameter $\varrho
a(\varrho)$. So, in fact, these equations are nonlinear, because $a(\varrho)$
depends on all correlation functions (see (3.29) and (3.31)). In this section,
we construct expansion in the parameter density $\varrho$, considering the
equation (3.29) as nonlinear equation.

###  Solution of equations for correlation functions.

Rewrite the equation (3.29), (3.31) as follows:

$\widetilde{Q}(\rho_{j})\rho_{j}(\eta)\;=\;\varrho e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}j(x_{1})\int_{\Gamma_{0}}K(x_{1};\xi)\rho_{j}(\eta^{(\hat{x}_{1})}\cup\xi)\lambda_{\sigma}(d\xi),$
(4.32)
$\widetilde{Q}(\rho_{j})\;=\;\int_{\Gamma_{0}}K(x_{1};\xi)\rho_{j}(\xi)\lambda_{\sigma}(d\xi),$
(4.33)

where, following [12], a continuous positive function
$j:\mathbb{R}^{d}\rightarrow\mathbb{R}_{+}$ is introduced and $\rho(\eta)$
becomes functional such as $\rho(\eta)\;=\;\rho_{j}(\eta)|_{j=1}.$

We will look for the solution of these equations in the form:

$\rho_{j}(\eta)\;=\;e(j;\eta)\int_{\Gamma^{0}}T(\eta|\gamma)e(j;\gamma)\lambda_{\sigma}(d\gamma),$
(4.34)

where $e(j;\eta)$ is given by (2.4).

The idea of finding a solution is to consider the left side of the equation
(4.32) as the product of two generating functionals (see (2.5)) and compare it
with the right part, which is also written in the form of a generating
functional of the form (2.2). So, substituting in (4.33) the corresponding
expression for $\rho_{j}(\xi)$ we have:

$\widetilde{Q}(\rho_{j})\;=\;\int_{\Gamma_{0}}\int_{\Gamma_{0}}e(j;\xi\cup\gamma)K(x_{1};\xi)T(\xi|\gamma)\lambda_{\sigma}(d\gamma)\lambda_{\sigma}(d\xi).$
(4.35)

Apply to the integral on the right side of the (4.35) lemma 2.1 with

$G(\xi\cup\gamma)=e(j;\xi\cup\gamma)\;\;\text{and}\;\;H(\xi;\gamma)=K(x_{1};\xi)T(\xi|\gamma).$
(4.36)

Then

$\widetilde{Q}(\rho_{j})\;=\;\widetilde{F}_{\psi_{1}}(j)\;=\;\int_{\Gamma_{0}}e(j;\gamma)\psi_{1}(\gamma)\lambda_{\sigma}(d\gamma),$
(4.37)

with

$\psi_{1}(\gamma)\;=\;\sum_{\xi\subseteq\gamma}K(x_{1};\xi)T(\xi|\gamma\setminus\xi).$
(4.38)

Write down (4.34) in the form of generating functional:

$\rho_{j}(\eta)\;=\;\widetilde{F}_{\psi_{2}}(j)\;=\;\int_{\Gamma_{0}}e(j;\gamma)\psi_{2}(\gamma)\lambda_{\sigma}(d\gamma),$
(4.39)

with

$\psi_{2}(\gamma)\;=\;e(j;\eta)T(\eta|\gamma).$ (4.40)

Left hand side (4.32) is:

$\widetilde{Q}(\rho_{j})\rho_{j}(\eta)\;=\;\widetilde{F}_{\psi_{1}}(j)\widetilde{F}_{\psi_{2}}(j)\;=\;\widetilde{F}_{\psi_{1}*\psi_{2}}(j)\;=\;\int_{\Gamma_{0}}e(j;\gamma)(\psi_{1}*\psi_{2})(\gamma)\lambda_{\sigma}(d\gamma),$
(4.41)

where

$\displaystyle(\psi_{1}*\psi_{2})(\gamma)$
$\displaystyle=\sum_{\xi\subseteq\gamma}\psi_{1}(\xi)\psi_{2}(\gamma\setminus\xi)=\sum_{\xi\subseteq\gamma}\psi_{2}(\gamma\setminus\xi)\psi_{1}(\xi)=$
(4.42)
$\displaystyle=e(j;\eta)\sum_{\xi\subseteq\gamma}T(\eta|\gamma\setminus\xi)\sum_{\zeta\subseteq\xi}K(x_{1};\zeta)T(\zeta|\xi\setminus\zeta)$

Right hand side (4.32) after substitution of
$\rho_{j}(\eta^{(\hat{x}_{1})}\cup\xi)$ in the form (4.34) will look as:

$\varrho e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}e(j;\eta)\int_{\Gamma_{0}}\int_{\Gamma_{0}}e(j;\xi\cup\gamma)K(x_{1};\xi)T(\eta^{(\hat{x}_{1})}\cup\xi|\gamma)\lambda_{\sigma}(d\gamma)\lambda_{\sigma}(d\xi).$
(4.43)

Let us use lemma 2.1 again with

$G(\xi\cup\gamma)=e(j;\xi\cup\gamma)\;\;\text{and}\;\;H(\xi;\gamma)=K(x_{1};\xi)T(\eta^{(\hat{x}_{1})}\cup\xi|\gamma).$
(4.44)

Then right hand side of (4.32) will be as

$\varrho e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}e(j;\eta)\int_{\Gamma_{0}}e(j;\gamma)\sum_{\xi\subseteq\gamma}K(x_{1};\xi)T(\eta^{(\hat{x}_{1})}\cup\xi|\gamma\setminus\xi)\lambda_{\sigma}(d\gamma).$
(4.45)

We equate the functions near $e(j;\gamma)$ under the integrals of the
expressions (4.41) and (4.45) and, reducing the multiplier $e(j;\eta)$ we get
the relation:

$\displaystyle\sum_{\xi\subseteq\gamma}T(\eta|\gamma\setminus\xi)\sum_{\zeta\subseteq\xi}K(x_{1};\zeta)T(\zeta|\xi\setminus\zeta)\;=$
(4.46) $\displaystyle=\;\varrho e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}\sum_{\xi\subseteq\gamma}K(x_{1};\xi)T(\eta^{(\hat{x}_{1})}\cup\xi|\gamma\setminus\xi).$

Select the term with $\xi=\emptyset$ in the left side of this relation and
take into account that $T(\emptyset|\emptyset)=1$ and $T(\emptyset|\gamma)=0$,
if $\gamma\neq\emptyset)$, we obtain the following nonlinear recurrent
relation:

$\displaystyle T(\eta|\gamma)$ $\displaystyle=\varrho e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}\sum_{\xi\subseteq\gamma}K(x_{1};\xi)T(\eta^{(\hat{x}_{1})}\cup\xi|\gamma\setminus\xi)-$
$\displaystyle-\sum_{\emptyset\neq\xi\subseteq\gamma}T(\eta|\gamma\setminus\xi)\sum_{\emptyset\neq\zeta\subseteq\xi}K(x_{1};\zeta)T(\zeta|\xi\setminus\zeta).$
(4.47)

Initial conditions:

$T(\emptyset|\emptyset)\;=\;1,\;\;\;T(\emptyset|\gamma)\;=\;0,\text{if}\;\;\gamma\neq\emptyset,$
(4.48) $T(\\{\eta|\gamma)\;=\;0,\text{if}\;\;\eta\cap\gamma\neq\;\emptyset.$
(4.49)

###  Graphic interpretation of solutions.

As in the case of the Kirkwood-Salzburg equations for a large canonical
ensemble (see details in [21]) the analytical expression of the kernel
$T(\eta|\gamma)$ can be given as contributions of forest-graphs. The number
$N(m|n),m=|\eta|,n=|\gamma|$ all forest graphs can be calculated by solving
the following recurrent equation:

$N(m|n)=\sum_{k=0}^{n}{n\choose k}N(m+k-1|n-k)+\sum_{k=1}^{n}{n\choose
k}N(m|n-k)\sum_{l=1}^{k}{k\choose l}N(l|k-l).$ (4.50)

If we reject the nonlinear term, the recurrent equation can be solved exactly:
$N(m|n)=m(m+n)^{n-1}$ (see [21]). This formula makes it possible to prove the
convergence of the integral (4.34) at $j=1$ and small density values. But the
nonlinear term apparently significantly increases the number of forest graphs.
But according to the formula (4.1), each contribution of the forest graph,
which contains a tree with the contribution of the factor
$K(x_{1};\xi),\xi\in\gamma$, which appears from the second sum in (4.1) is
included with a minus sign. So, for example calculating

$T(x_{1}|y)\;=\;\varrho^{2}\left[K(x_{1};y)\;-\;K(x_{1};y)\right]=0.$ (4.51)

Therefore, it is necessary to analyze this recurrent relationship in more
detail. It is easy to get the following proposition.

###### Proposition 4.1

From initial conditions (4.48)-(4.49) it follows that

$T(x_{1}|\emptyset)\;=\;\varrho,\;\;T(x_{1}|\gamma)\;=\;0\;\text{for}\;\gamma\neq\emptyset.$
(4.52)

It gives the needed equality for density $\rho(x_{1})\;=\;\varrho$.
In addition $T(\eta|\emptyset)\;=\;\varrho^{|\eta|}\exp[-\beta U(\eta)]$.

The proof follows from (4.1) by induction procedure. To obtain the solution of
(4.1) in the language of graph theory [24, 4] define the set graphs
$\mathcal{D}(\eta;\gamma)$.

###### Remark 4.1

The set $\mathcal{D}(\eta;\gamma)$ is the set of rooted graph, connected
components of which are 2-connected graphs with respect to configuration
$\gamma$. Any graph $G\in\mathcal{D}(\eta;\gamma)$ has $m=|\eta|$ vertices of
configuration $\eta$ and $n=|\gamma|$ vertices of configuration $\gamma$.
Every vertex of $\eta$ can be free of lines or can connect only with vertices
$y$ of configuration $\gamma$. Every vertex $y\in\gamma$ connects with
vertices $x$ or $y$ by at least two lines. This definitions is very close to
definitions of the set of white and black vertices $\mathcal{D}(W;B)$ in [5],
Section 2. The role of the roots of the graph is performed by the
configuration $\eta$. A singleton $G\in\mathcal{D}(x_{1};\emptyset)$. The
contribution of any vertex is $\varrho$. The contribution of line $w(l_{xy})$,
which connected two vertices $x$ and $y$ or $y_{i}$ and $y_{j}$ is $K(x;y)$.

###### Remark 4.2

According to (4.34), integration is performed at the points of configuration
$\gamma$, and according to equation (4.1), in the kernels $T(\theta;\xi)$, the
configuration $\xi\subseteq\gamma$, and the configuration $\theta$ can contain
both variables from $\eta$ and $\gamma$. Therefore, the corresponding graphs
of sets $\mathcal{D}(\theta;\xi)$ can contain only black vertices.

The main statement of the section is

###### Proposition 4.2

The solution of the recurrence relation (4.1) can be rewritten by
contributions from the graphs of the set $\mathcal{D}(\eta;\gamma)$:

$T(\eta|\gamma)\;=\;\varrho^{|\eta|+|\gamma|}\exp[-\beta
U(\eta)]\sum_{G\in\mathcal{D}(\eta;\gamma)}w_{G}(L;\eta,\gamma),$ (4.53)

where $L$ is a configuration of lines(edges).

The main points of proof. The proof is based on the induction procedure. It is
very easy to see the cancellation of the contributions of the corresponding
graphs, for example, for kernels $T(x_{1},x_{2}|y_{1})$ and
$T(x_{1},x_{2}|y_{1},y_{2})$. To facilitate these calculations, it is better
to consider instead of the equation (4.1) the equation for kernels that are
integrated in variables $y_{1},...,y_{n}$:

$\widehat{T}(\eta|n):=\int_{\Gamma^{(n)}}T(\eta|\gamma)d\gamma=\int_{\mathbb{R}^{dn}}(dy)^{n}T(\eta|\\{y\\}_{1}^{n}),$
(4.54)

especially since this is the value that must be substituted into the formula
for the solution (4.34).

$\displaystyle\widehat{T}(\eta|n)$ $\displaystyle=\varrho e^{-\beta
W(x_{1};\eta^{(\hat{x}_{1})})}\sum_{k=0}^{n}{n\choose
k}\int(dy)^{k}K(x_{1};\\{y\\}_{1}^{k})\widehat{T}(\eta^{(\hat{x}_{1})}\cup\\{y\\}_{1}^{k})|n-k)-$
$\displaystyle-\sum_{k=1}^{n}{n\choose k}\sum_{l=1}^{k}{k\choose
l}\int(dy)^{l}K(x_{1};\\{y\\}_{1}^{l})\widehat{T}(\eta|n-k)\widehat{T}(\\{y\\}_{1}^{l}|k-l).$
(4.55)

For $k=0$ in the first line of formula (4.2), the term
$\widehat{T}(\eta^{(\hat{x}_{1})}|n)$ corresponds to the analytical
contribution of all graphs $\mathcal{D}(\eta;\gamma)$, in which the vertex
$x_{1}$ has no edges. For $k\geq 1$ the set of graphs for
$\widehat{T}(\eta^{(\hat{x}_{1})}\cup\\{y\\}_{1}^{k})|n-k)$ is arranged in
such a way that $k$ vertices from $n$ vertices of the configuration $\gamma$
are transformed into vertices, which are connected to the graphs of
$\widehat{T}(\eta^{(\hat{x}_{1})}|n-k)$ either by means of lines or as free
vertices. If a vertex $y_{i}$ is already connected by at least one line, then
joining a line with vertex $x_{1}$ leaves this graph in the set
$\mathcal{D}(\eta;\gamma)$. If the vertex did not have edges, then operation
$\int(dy)^{l}K(x_{1};\\{y\\}_{1}^{l})$ does not ensure that it belongs to the
set $\mathcal{D}(\eta;\gamma)$, and such a diagram should be reduced.This is
ensured by the presence of a similar contribution in the second line of the
formula (4.2). This variant occurs in $l$ of $k$ cases for each of the graphs
in $\widehat{T}(\eta|n-k)$. $\blacksquare$

###  Conclusion on the convergence of the expansion.

Existence of a unique solution of the nonlinear equation (3.29) as an operator
equation in the Banach space $E_{\xi}$ was proved in [15]. In a recent work of
Jansen [5] a generalized version of Groeneveld’s convergence criterion for the
virial expansion and generating functionals for weighted 2-connected graphs
was proven. In fact, the present paper turned out to be very close to article
[5], and here one can apply the criteria for the convergence of expansions.
Therefore, we do not yet discuss this issue in this preliminary publication.
Although perhaps a more thorough study of the recurrence relation (4.2) may
introduce some adjustments.

AVAILABILITY OF DATA

Data available on request from the author

## References

  * [1] Bogolyubov N. N. _Problems of Dynamical Theory in Statistical Physics [in Russian]_. M.: GITTL (1946). Translation: Bogoliubov, N. N. _Problems of a dynamical theory in statistical physics._ In: de Boer, J., Uhlenbeck, G. E. (eds.) Studies in Statistical Mechanics, 1, pp. 1–118. North-Holland, Amsterdam (1962)
  * [2] Bogolyubov, N. N., Khatset, B. I., On some mathematical problems of the theory of statistical equilibrium. _Doklady Akad. Nauk SSSR (N.S.)_ 66, (1949), 321–324.
  * [3] Bogolyubov, N. N., Petrina, D. Ya., Khatset, B. I. Mathematical description of the equilibrium state of classical systems on the basis of the canonical ensemble formalism, _Teoret. Mat. Fiz._ 1(2), 251–274 (1969).
  * [4] Groeneveld, J. Estimation methods for Mayer’s graphical expansions. IV. Estimation methods of degree 2, _Nederl. Akad. Wetensch. Proc. Ser._ B 70, 490–507(1967).
  * [5] Jansen, S. Revisiting Groeneveld’s approach to the virial expansion, _Jour. of Math. Phys._ ,62, (2), 023302 (2021); https://doi.org/10.1063/5.0030148
  * [6] Khatset, B. I. Asymptotic expansions by degrees of density of the distribution function of systems in the state of statistical equilibrium [in Ukraine], _Nauk. Zapiski Zhitom. Pedag. Inst., Fizmat. Seriya_ , 3, p. 113–138 (1956).
  * [7] Khatset, B. I. Asymptotic expansions by degrees of density of the distribution function of systems in the state of statistical equilibrium [in Ukraine], _Nauk. Zapiski Zhitom. Pedag. Inst., Fizmat. Seriya_ , 3, p.139–157 (1956).
  * [8] Kuna, T. Studies in configuration space analysis and applications , PhD thesis. Bonner Mathematische Schriften Nr. 324, University of Bonn, 1999.
  * [9] Kuna, T. and Tsagkarogiannis D. Convergence of density expansions of correlation functions and the Ornstein-Zernike equation, Ann. Henri Poincare 19, no. 4, 1115–1150 (2018).
  * [10] Lebowitz, J. L. and Penrose, O. Convergence of Virial Expansions, _J. Math. Phys._ , 5, (1964), 841; https://doi.org/10.1063/1.1704186
  * [11] Mayer, J. E., Mayer, M. G. _Statistical Mechanics._ Wiley, New York (1940)
  * [12] Minlos R. A. and Poghosyan S. K., Estimates of Ursell functions, group functions, and their derivatives, _Teoret. Mat. Fiz._ 31(2), 199–213 (1977) (in Russian); _Theor. Math. Phys._ 31(2), 408–418 (1977).
  * [13] Nguyen, T. X. and Fern´andez, R. Convergence of cluster and virial expansions for repulsive classical gases, _J. Stat. Phys._ 179, no. 2, 448–484 (2020).
  * [14] Petrenko, S. N.Rebenko, A. L. Superstable criterion and superstable bounds for infinite range interaction I: two-body potentials,  Meth. Funct. Anal. and Topology, 13, 50–61 (2007).
  * [15] Pogorelov, Yu. G. Convergence of virial expansions for a classical canonical ensemble, _Theoret. and Math. Phys.,_ 24,2 , 808–812 (1975). https://doi.org/10.1007/BF01029066
  * [16] Pulvirenti, E. and Tsagkarogiannis, D. Cluster expansion in the canonical ensemble, _Comm. Math. Phys._ 316, no. 2, 289–306 (2012).
  * [17] Ramawadh, S. and Tate, S. J. Virial expansion bounds through tree partition schemes, _Online preprint, arXiv:1501.00509_ [math-ph], (2015).
  * [18] Rebenko, A. L. Poisson measure representations and cluster expansion in classical statistical mechanics, _Commun. Math. Phys._ , 151, p. 427–435 (1993).
  * [19] Rebenko, A. L. New proof of Ruelle’s superstability bounds _J. Stat. Phys._ , 91, № 3/4, p. 815–826 (1998).
  * [20] Rebenko, A. L., Shchepan’uk, G. V. The convergence of cluster expansions for continuous systems with many-body interactions. _J. Stat. Phys._ , 88, No.3/4, p. 665–689 (1997).
  * [21] Rebenko, O. L. On the connection of some approaches to solving the Kirkwood-Salzburg equations,  Ukr. Math. Journ. 73, No 3, p. 93–106 (2021).
  * [22] Ruelle, D. Correlation functions of classical gases, _Ann. Phys._ , 25, №1, p. 109–120 (1963).
  * [23] Ruelle, D. Statistical Mechanics, (Rigorous results),  W.A. Benjamin, inc. N.Y.–Amsterdam, (1969).
  * [24] Stell, G. Cluster expansions for classical systems in equilibrium, The Equilibrium Theory of Classical Fluids (H. L. Frisch and J. L. Lebowitz, eds.), Benjamin, New York, pp. 171–261 ( 1964).
  * [25] Uhlenbeck, G.E., Ford, G. W. Lectures in Statistical Mechanics. Providence, RI: Amer. math. Soc. ( 1963)