# Stability of the Potential Super Jupiter in Alpha Centauri System

Tinglong Feng<EMAIL_ADDRESS>Xi’an Jiaotong University

###### Abstract

The stability of potential exoplanets in binary star systems remains a
critical challenge in celestial mechanics. This study investigates the
stability of a hypothetical Jupiter-mass planet in the Alpha Centauri AB
system by drawing parallels with the recently detected Neptune-mass planet in
the GJ65AB system, which shares similar mass ratios and orbital
eccentricities. Utilizing the Kolmogorov-Arnold-Moser (KAM) theorem and the
Mean Exponential Growth of Nearby Orbits (MEGNO) parameter within the REBOUND
software package, we perform numerical simulations to identify stable orbital
configurations. Our results indicate that a Jupiter-mass planet could
theoretically maintain a stable orbit in Alpha Centauri AB, with orbital
parameters derived from the stable zones observed in GJ65AB. Despite the
absence of observational evidence for such a planet in Alpha Centauri AB, this
study suggests that similar binary systems could exhibit analogous stability
characteristics. Future observational efforts are recommended to explore this
potential and refine our understanding of planetary formation in binary star
systems.

##  1 Introduction

The three-body problem, which seeks stable orbit configurations among
gravitating bodies, is a longstanding challenge in celestial mechanics. Recent
research has prioritized specific astronomical systems, employing a range of
methods including numerical simulations(Doolin & Blundell (2011),
Georgakarakos et al. (2024)), analytical techniques (Shevchenko (2015)), and
machine learning (Lam & Kipping (2018)).

As the closest triple stellar system to Earth, Alpha Centauri system has
attracted diverse studies in astronomy, including exoplanet
stability(Giuppone, C. A. & Correia, A. C. M. (2017), Cuello & Sucerquia
(2024)). Latest observations hinted at the existence of a low mass planet in
Alpha Centauri (Wagner et al. (2021)), but no Jupiter-mass planet has ever
been detected up to now. Given the significant distance from star C to stars
A/B, the stability of planets around stars A/B can be regarded as a stability
problem in a binary system(Cuello & Sucerquia (2024)).

Recently, a Neptune-mass candidate planet has been detected in M-dwarf binary
system GJ65(GRAVITY Collaboration et al. (2024)). GJ65AB and Alpha Centauri AB
have almost the same mass ratio and eccentricity, from which we may suppose
the planet stability for these two system may share similar property.

In this work, we study the similarity between GJ65AB and Alpha Centauri AB,
and explore the stability of a potential super Jupiter in the latter. In
section 2 we review the exoplanet stability theorem based on KAM theorem, from
which we assume the similarity method. In section 3 we compare the parameters
between the two systems, and use the similarity method to derive the mass and
semimajor of the potential stable planet in Alpha Centauri AB, according to
parameters of the newly detected planet in GJ65AB. In section 4 we present
numerical simulations to verify the stability of the newly detected super
Neptune in GJ65AB and the potential super Jupiter derived from similarity
method in Alpha Centauri AB. Finally, we conclude our results in section 5.

##  2 Stability of Exoplanet System

In order to discuss the stability of exoplanet system, we first need to review
some point of KAM theorem(Shikita et al. (2010)). Consider $N$-body problem
where the Hamiltonian is given by

$H=\sum_{j=1}^{N}\frac{p_{j}^{2}}{2m_{j}}-\sum_{1\leq j\leq k\leq
N}G\frac{m_{j}m_{k}}{|x_{j}-x_{k}|}.$ (1)

It is well known that for $N\geq 3$ the Hamiltonian system is not integrable.
Nevertheless, it can be regarded as a small perturbation of an integrable
system. If the total Hamiltonian $H(p,q)$ satisfies

$H(p,q)=H_{0}(p,q)+\epsilon H_{1}(p,q)$ (2)

where $H_{0}$ is completely integrable, then for a sufficiently small
$\epsilon$ the quasi-periodic solutions (corresponding to a stable motion in
planet systems) survive under this small perturbation. That is the result of
KAM theorem.

Now we turn to the stability problem for the planets in binary star system.
For $N=3$ case, if we set $M_{1}\gg m,M_{2}\gg m$ and recall that Kepler
problem is completely integrable, then (2) becomes

$H(p,q)=H_{0}(p,q)-G\frac{mM_{1}}{|x_{3}-x_{1}|}-G\frac{mM_{2}}{|x_{3}-x_{2}|}$
(3)

where $H_{0}$ is the Hamiltonian of binary stars $M_{1},M_{2}$ and the kinetic
energy of planet $m$ is ignored. Compare (2) and (3) we could find that the
$\epsilon$ only depends on the mass ratio of these three bodies
$m:M_{1}:M_{2}$ and the relative distance of them
$|x_{1}-x_{2}|:|x_{2}-x_{3}|:|x_{1}-x_{3}|$, while the latter can be described
by eccentricity of orbits. Hence we could guess that for two binary star
systems with nearly identical mass ratio and eccentricity, they will have
similar planet stability property. For instance, consider two binary systems
A$(M_{1},M_{2})$ and B$(M_{1}^{\prime},M_{2}^{\prime})$ where
$M_{1}/M_{1}^{\prime}=M_{2}/M_{2}^{\prime}=\alpha$ and
$e=e^{\prime},a/a^{\prime}=\beta$ (with ’$a$’ representing the semimajor
axis). If a planet in system A with mass $m$, semimajor axis $a_{3}$, and
eccentricity $e_{3}$ exhibits stable motion, then a planet in system B with
mass $m/\alpha$, semimajor axis $a_{3}/\beta$ and eccentricity $e_{3}$ should
also have stable motion.

##  3 Methods

###  3.1 Similarities between GJ65AB and Alpha Centauri AB

According to the latest astrometry results(Akeson et al. (2021), Kervella,
Pierre et al. (2022)), GJ65AB and Alpha Centauri AB have nearly identical mass
ratio and eccentricity (Table 1). If the former is denoted as system A and the
latter as system B, as in section 2, then we have corresponding $\alpha\simeq
1/9$ and $\beta\simeq 1/4.2$. A novel astrometric detection result GRAVITY
Collaboration et al. (2024) shows the existence of a stable Neptune-mass
planet in GJ65 system, based on which and the hypothesis in section 2 we could
suppose the possibility of potential Jupiter-mass planet in Alpha Centauri AB
system, at least from the standpoint of stability. The parameters of both
planets are listed in Table 1.

Table 1: The comparison of GJ65 AB and Alpha Centauri AB, parameters are from
recent astrometric measurements Kervella, Pierre et al. (2022) and Akeson et
al. (2021). The planet’s parameters in GJ65 AB are from GRAVITY Collaboration
et al. (2024), and the values in parentheses are derived using similarity
method.

Parameter | GJ65AB | Alpha Centauri AB
---|---|---
$a$ (AU) | $5.459\pm 0.002$ | $23.336\pm 0.013$
$e$ | $0.6172\pm 0.0001$ | $0.51947\pm 0.00015$
$\omega$ (deg) | $103.2\pm 0.1$ | $231.519\pm 0.027$
$m_{A}(M_{\odot})$ | $0.122\pm 0.002$ | $1.0788\pm 0.0029$
$m_{B}(M_{\odot})$ | $0.116\pm 0.002$ | $0.9092\pm 0.0025$
Potential Planet Parameter: |  |
$a$ (AU) | $0.283\pm 0.002$ | $(1.189)$
$e$ | $0.33\pm 0.30$ | $(0.33)$
$m(M_{\oplus})$ | $40$ | $(350)$

###  3.2 Numerical Simulation

The similarities between GJ65AB and Alpha Centauri AB, together with the newly
detected stable super Neptune in GJ65 system, suggest the stability of the
corresponding potential super Jupiter in Alpha Centauri AB. In order to verify
this, we employ the Mean Exponential Growth of Nearby Orbits (MEGNO) stability
parameter(Cincotta, P. M. & Simó, C. (2000)).

The MEGNO parameter evaluate the stability of N-body gravitational system by
measuring how orbits in phase space diverge over time. In MEGNO, values near
two indicate stable orbits, while significantly higher values suggest chaos
and instability. For this study, we used the MEGNO tool from the REBOUND
software package(Rein, H. & Liu, S.-F. (2012)).

Here is our experimental approach: first, in GJ65 system, we placed the planet
with mass and orbit parameters following the newly astrometric detection
result so that we can verify the stability of this exoplanet. Then, in Alpha
Centauri AB we placed the potential planet with corresponding parameters in
Table 1, derived from similarity method, to verify whether it is in stable
region.

##  4 Results

Figure 1: Stability maps based on MEGNO values for Neptune-mass planet in
GJ65AB(top panel) and for Jupiter-mass planet in Alpha Centauri AB(below
panel). Dynamically stable regions are coloured in green.

First we perform the simulation of planet in GJ65 system. Using the most
recent orbital solution based on astrometric measurements, we initialized the
binary orbital parameters with the following values: semimajor axis $a=5.459$
AU, eccentricity $e=0.6172$, and argument of periastron $\omega=103.2$ deg.
The masses were set to $m_{A}=0.122M_{\odot}$ and $m_{B}=0.116M_{\odot}$,
respectively. For simplicity, we supposed the host star of the planet is star
A, and set the mass of the planet $40M_{\oplus}$, according to recent
detection. For this simulation, we restricted the semimajor axis of the planet
to range from $0.1$ to $1.0$ au, and eccentricities less than $0.5$.

The top panel in Figure 1 shows the result of the simulation. The green
regions indicate the stable zones in the $a-e$ parameter space, while the red
regions are chaotic zones. We identified the stable zone spanning from $0.1$
to $\sim 0.35$ au, which contains all the stable orbits for $e$ ranging from
$0$ to $0.5$. The astrometric detection suggests the planet has the following
parameters: $a\approx 0.283$ au and $e\approx 0.33$, which are indeed located
in the stable zones of our simulation, and the stability of this planet has
been verified.

Next we turn to Alpha Centauri AB. According to highly precise astrometric
results, we set the parameters: $a=23.336$ AU, $e=0.5195$, $\omega=231.5$ deg,
$m_{A}=1.0788M_{\odot}$ and $m_{B}=0.9092M_{\odot}$. The potential planet was
also supposed to be around star A, and the planet’s mass was set to be about
$350M_{\oplus}$ based on similarity method, i. e., a Jupiter mass exoplanet.
For this simulation, we restricted the semimajor axis of the planet to range
from $0.1$ to $5.0$ au, and eccentricities less than $0.5$.

The panel below in Figure 1 displays the result of this simulation, where
green zones indicate stability and red ones represent chaos. We figured out
the stable zone, with $a$ spanning from $0.1$ to $\sim 2.2$ au and $e$ ranges
from $0$ to $0.5$. According to the similarity discussed at the beginning of
this section, the potential stable planet should have parameters of $a\approx
1.189$ and $e\approx 0.33$, exactly contained in the stable zone of our
simulation. This result verified our postulate that, if there exists a
Jupiter-mass planet in Alpha Centauri AB system with corresponding orbital
parameters, it should be stable.

##  5 Conclusion

In this work, we propose that exoplanets in similar binary systems, i.e.,
systems with nearly identical mass ratios and eccentricities, may exhibit
similar stability properties according to the KAM theorem. From this
hypothesis, together with the newly detected Neptune-mass planet in GJ65
system which is similar to Alpha Centauri AB, we assume the existence of a
potential Jupiter-mass planet with corresponding orbital parameters in Alpha
Centauri AB should also be possible, at least from the standpoint of dynamical
stability. We have used the numerical tool MEGNO to successfully verify our
assumption.

Nevertheless, we have not detected any Jupiter-mass planet candidates in Alpha
Centauri AB system up to now(Wagner et al. (2021)). To explain this we may
come up with two possible cases:

  1. 1.

There exists Jupiter-mass planet in Alpha Centauri AB, but we have not
detected it yet.

  2. 2.

There does not exist a Jupiter-mass planet in Alpha Centauri AB. This case
suggests the formation of giant planets in binary systems depends not only on
the mass ratio and eccentricity but also on the absolute value of the host
star’s mass.

These two cases may indicate further studies and detection.

## References

  * Akeson et al. (2021) Akeson, R., Beichman, C., Kervella, P., Fomalont, E., & Benedict, G. F. 2021, The Astronomical Journal, 162, 14, doi: 10.3847/1538-3881/abfaff
  * Cincotta, P. M. & Simó, C. (2000) Cincotta, P. M., & Simó, C. 2000, Astron. Astrophys. Suppl. Ser., 147, 205, doi: 10.1051/aas:2000108
  * Cuello & Sucerquia (2024) Cuello, N., & Sucerquia, M. 2024, Universe, 10, doi: 10.3390/universe10030120
  * Doolin & Blundell (2011) Doolin, S., & Blundell, K. M. 2011, Monthly Notices of the Royal Astronomical Society, 418, 2656, doi: 10.1111/j.1365-2966.2011.19657.x
  * Georgakarakos et al. (2024) Georgakarakos, N., Eggl, S., Ali-Dib, M., & Dobbs-Dixon, I. 2024, Empirical stability criteria for 3D hierarchical triple systems I: Circumbinary planets.  https://arxiv.org/abs/2404.13746
  * Giuppone, C. A. & Correia, A. C. M. (2017) Giuppone, C. A., & Correia, A. C. M. 2017, A&A, 605, A124, doi: 10.1051/0004-6361/201730491
  * GRAVITY Collaboration et al. (2024) GRAVITY Collaboration, Abuter, R., Amorim, A., et al. 2024, A&A, 685, L9, doi: 10.1051/0004-6361/202449547
  * Kervella, Pierre et al. (2022) Kervella, Pierre, Arenou, Frédéric, & Thévenin, Frédéric. 2022, A&A, 657, A7, doi: 10.1051/0004-6361/202142146
  * Lam & Kipping (2018) Lam, C., & Kipping, D. 2018, Monthly Notices of the Royal Astronomical Society, 476, 5692, doi: 10.1093/mnras/sty022
  * Rein, H. & Liu, S.-F. (2012) Rein, H., & Liu, S.-F. 2012, A&A, 537, A128, doi: 10.1051/0004-6361/201118085
  * Shevchenko (2015) Shevchenko, I. I. 2015, The Astrophysical Journal, 799, 8, doi: 10.1088/0004-637X/799/1/8
  * Shikita et al. (2010) Shikita, B., Koyama, H., & Yamada, S. 2010, The Astrophysical Journal, 712, 819, doi: 10.1088/0004-637X/712/2/819
  * Wagner et al. (2021) Wagner, K., Boehle, A., Pathak, P., et al. 2021, Nature Communications, 12, 922, doi: 10.1038/s41467-021-21176-6


††institutetext: School of Physics, Sun Yat-Sen University, Guangzhou 510275,
China

# Correlation length of the angular mode for an approximate $U(1)$ symmetry
during inflation

Chengcheng Han<EMAIL_ADDRESS>
###### Abstract

It is known that a light scalar field obtains fluctuations in the de Sitter
inflationary background. Such fluctuations could provide an initial condition
for baryogenesis through the Affleck-Dine mechanism, where an approximate
$U(1)_{B}$ symmetry is usually assumed. However, an interpretation of the
baryon number generation in this way is strongly related to the correlation
length of the angular mode. In this work, we calculate the correlation length
of the angular mode for a model exhibiting an approximate $U(1)$ symmetry. We
find that for a massive nearly non-interacting field, the correlation length
of the angular mode is determined by the mass parameter of the model and it is
similar to $H^{-1}\exp(H^{2}/m^{2})$. Applying this result to baryogenesis via
the Affleck-Dine mechanism with a stochastic origin, we find that only for
$m\ll\mathcal{O}(0.1)H$(assuming $N_{*}=60$) can the correlation length of the
baryon number density be much larger than our current horizon size, such that
we live in the baryon-rich region. If this is not true, at early times our
universe would consist of numerous patches of baryon-rich and anti-baryon-rich
regions with the average baryon number being nearly zero.

††preprint:

##  1 Introduction

It is widely believed the universe undergoes fast expansion in its early stage
(inflation), characterized by a quasi-de Sitter space-time Brout:1977ix ;
Sato:1980yn ; Guth:1980zm ; Linde:1981mu ; Albrecht:1982mp . The period of
inflation not only solves the flatness problem, the horizon problem, but also
seeds the primordial fluctuations of the Cosmic Microwave Background (CMB)
temperature, which finally resulted in the large structure of our universe.
During this stage, all light scalar fields obtain fluctuations with an
amplitude around $\frac{H}{2\pi}$ in each e-fold, and the distribution
function of the field values can be described by the Fokker-Planck equations.
For sufficient long inflationary time, the fluctuations continue to
accumulate, and finally, the distribution function reaches an equilibrium
state Bunch:1978yq ; Starobinsky:1994bd . For a massive nearly non-interacting
field, the equilibrium solution is characterised by a correlation length of
around $H^{-1}\exp(H^{2}/m^{2})$ within which we can take the field value as a
constant Bunch:1978yq ; Starobinsky:1994bd . This equilibrium distribution
provides an initial condition for the vacuum displacement of the scalar field.
If this field carries a $U(1)_{B}$ or $U(1)_{L}$ charge, it may generate the
baryon asymmetry of our universe through the Affleck-Dine mechanism
Affleck:1984fy ; Dine:1995uk ; Dine:1995kz . However, due to the stochastic
behavior of the field, the average baryon number in the whole universe
vanishes. But it has been argued that if $m\ll H$ Dine:1995kz ; Linde:1985gh ,
the correlation length of the field is much larger than our current horizon
size, such that we could live in a baryon-rich patch.

Recently, some studies have tried to use similar ideas to generate the baryon
asymmetry via the Affleck-Dine mechanism Wu:2020pej ; Wu:2021mwy ; Wu:2021gtd
. However, to avoid the baryonic isocurvature limit from CMB observation, they
require the mass of the scalar field to be close to the inflationary Hubble
parameter. Since the correlation length of the field is around
$H^{-1}\exp(H^{2}/m^{2})$, in the early time our current universe should
consist of numerous Hubble patches with different baryon numbers.
Nevertheless, the observed baryon number of our universe should be the average
of all patches. Now we come up with a question: whether the average of the
baryon number in our universe (close to) zero or not?

This question is closely related to the correlation length of the angular mode
$\theta$. In the Affleck-Dine mechanism, the baryon number asymmetry is
usually proportional to $\sin n\theta$, where the value of $n$ depends on the
models. Due to the stochastic behavior of the field, $\theta$ is randomly
distributed and the average of the baryon number in the whole universe should
be zero. However, if the correlation length of $\theta$ is much larger than
our current horizon size, we can still live in a baryon-rich patch, even if
the average of the baryon number in the whole universe is zero. On the other
hand, if the correlation length of $\theta$ is much smaller than our current
horizon size, then at early times our universe consists of many baryon-rich
patches and anti-baryon-rich patches with the same probabilities, then the
average of the baryon number in our universe is nearly zero111It can not be
exactly zero. According to the central limit theorem, the average of the
baryon number in our universe should be around $|Y_{B}|e^{-3/2N}$ where
$Y_{B}$ is the baryon to entropy ratio defined as $\sqrt{\langle
Y_{B}^{2}\rangle}$ and $N\sim N_{*}/\log(HR_{c})$ is the number of the patches
contained in our current horizon. Here $N_{*}\approx 60$ is the e-fold number
of our observed universe..

Since the correlation length for a weakly coupled scalar field depends on its
mass, one may argue that for a small $U(1)_{B}$ breaking term, the angular
mode would have an effective “mass” much smaller than the Hubble parameter. In
this case, one may conclude that the correlation length of the $\theta$ could
be much larger than our current horizon size. A similar argument has been used
in the works Wu:2020pej ; Wu:2021mwy ; Wu:2021gtd .

On the other hand, since the Affleck-Dine field contains two real fields
$\phi_{R}$ and $\phi_{I}$ with similar correlation length $R_{c}\approx
H^{-1}\exp(H^{2}/m^{2})$, one can imagine that the whole universe is divided
by many cells with a scale of $R_{c}$. Within each cell, we can take
$\phi_{I}$ and $\phi_{R}$ as a constant, but for different cells, the field
values of $\phi_{R,I}$ are not strongly correlated and they can be taken as
random distributed222The distribution functions are decided by the equilibrium
solution of the Fokker-Planck equations.. Due to the approximate $U(1)$
symmetry, the sign of $\phi_{R}$ and $\phi_{I}$ are also random with equal
probabilities to be positive or negative. Note that
$\tan\theta=\frac{\phi_{I}}{\phi_{R}}$, so the sign of $\tan\theta$ is also
randomly distributed. Therefore, we expect that the correlation length of
$\theta$ should not be too much larger than $R_{c}$, as illustrated in Fig. 1.

To resolve this contradiction, in this work we implicitly calculate the
correlation length of $\theta$. We find that if the model presents an
approximate $U(1)$ symmetry, the correlation length of the angular mode is
indeed not much larger than $R_{c}$. The paper is organized as follows, in
Sec. 2 we show the general formalism for the calculation, in Sec. 3 we take
the single field as one typical example, and we calculate the correlation
length of angular mode for the exact $U(1)$ case in Sec. 4 and approximate
$U(1)$ case in Sec. 5. We also calculate the correlation length of the baryon
number density in Sec. 6 and we discuss the results in Sec. 7.

Figure 1: Illustration of the correlation length for the angular mode. Within
each cell, the $\phi_{R},\phi_{I}$ can be taken as a constant, with the size
of each cell is around $R_{c}$. Here $\phi_{R,I}^{+}$($\phi_{R,I}^{-}$)
denotes the region with $\phi_{R,I}>0$($\phi_{R,I}<0$) and
$\tan\theta=\frac{\phi_{I}}{\phi_{R}}$.

##  2 General formalism

In this section, we present the general formalism for our calculation. This
part is based on the work Starobinsky:1994bd . During inflation, the scalar
fields follow a stochastic process, which can be described by the Fokker-
Planck equation,

$\displaystyle\frac{\partial\rho(\varphi,t)}{\partial
t}=\frac{1}{3H}\left[\rho(\varphi,t)\nabla^{2}V(\varphi)+\nabla
V(\varphi)\cdot\nabla\rho(\varphi,t)\right]+\frac{H^{3}}{8\pi^{2}}\nabla^{2}\rho(\varphi,t)~{},$
(1)

where $V(\varphi)$ is the potential for $\varphi$ and $\rho(\varphi,t)$ is the
probability distribution function. Here $\nabla$ only acts on the field space.

To solve the Fokker-Planck equation, we can assume,

$\displaystyle\rho(\varphi,t)=\exp\left(-\frac{4\pi^{2}V(\varphi)}{3H^{4}}\right)\sum_{n=0}^{\infty}a_{n}\Phi_{n}(\varphi)e^{-{\Lambda_{n}}(t-t_{0})}~{}.$
(2)

Define

$\displaystyle v(\varphi)\equiv\frac{4\pi^{2}}{3H^{4}}V(\varphi)~{}.$ (3)

Then $\Phi_{n}$ satisfies the following eigenvalue equation,

$\displaystyle\frac{1}{2}\left[-\nabla^{2}+\left({\nabla
v(\varphi)}^{2}-\nabla^{2}v(\varphi)\right)\right]\Phi_{n}(\varphi)=\frac{4\pi^{2}}{H^{3}}\Lambda_{n}\Phi_{n}(\varphi)~{}.$
(4)

The coefficients $a_{n}$ are given by the initial condition of $\rho$ at
$t=t_{0}$ as

$\displaystyle a_{n}=\int
d\varphi\rho(\varphi,t_{0})e^{v(\varphi)}\Phi_{n}(\varphi)~{}.$ (5)

Note that there is always a solution which is called the static equilibrium
solution with $\Lambda_{0}=0$.

$\displaystyle\Phi_{0}(\varphi)=\mathcal{N}^{-1/2}e^{-v(\varphi)}~{},$ (6)

where $\mathcal{N}$ is calculated by normalization condition

$\displaystyle\mathcal{N}\equiv\int e^{-2v(\varphi)}d\varphi~{}.$ (7)

After enough time has passed, any solution will finally approach the static
equilibrium solution,

$\displaystyle\rho_{q}(\varphi)\equiv\mathcal{N}^{-1}\exp\left(-\frac{8\pi^{2}}{3H^{4}}V(\varphi)\right)=\mathcal{N}^{-1}e^{-2v(\varphi)}~{}.$
(8)

We can calculate the average of any variable $F(\varphi)$ and $F^{2}(\varphi)$
in terms of the equilibrium solution,

$\displaystyle\langle F\rangle=\int d\varphi F(\varphi)\rho_{q}(\varphi)~{},$
(9)

and

$\displaystyle\langle F^{2}\rangle=\int d\varphi
F^{2}(\varphi)\rho_{q}(\varphi)~{}.$ (10)

On the other hand, in the equilibrium state, we can derive the correlation
function for any variable $F(\varphi)$,

$\displaystyle G_{F}(|t_{1}-t_{2}|)\equiv\langle F(\varphi({\bf
x},t_{1}))F(\varphi({\bf x},t_{2}))\rangle~{}.$ (11)

We have

$\displaystyle G_{F}(t)=\int F(\varphi)\Xi(\varphi,t)d\varphi~{},$ (12)

where

$\displaystyle\Xi(\varphi,t)\equiv\int
F(\varphi_{1})\rho_{q}(\varphi_{1})\Pi[\varphi(t)|\varphi_{1}(0)]d\varphi_{1}~{},$
(13)

satisfying the Fokker-Planck equation with the initial condition,

$\displaystyle\Xi(\varphi,0)=F(\varphi)\rho_{q}(\varphi)~{}.$ (14)

Since $\Xi(\varphi,t)$ satisfies the Fokker-Planck equation, it can be written
as,

$\displaystyle\Xi(\varphi,t)=\exp\left(-\frac{4\pi^{2}V(\varphi)}{3H^{4}}\right)\sum_{n=0}^{\infty}A_{n}\Phi_{n}(\varphi)e^{-{\Lambda_{n}}t}~{},$
(15)

where

$\displaystyle A_{n}$ $\displaystyle=$ $\displaystyle\int
F(\varphi)\rho_{q}(\varphi)e^{v}\Phi_{n}(\varphi)d\varphi=\mathcal{N}^{-1}\int
F(\varphi)e^{-v}\Phi_{n}(\varphi)d\varphi$ (16) $\displaystyle=$
$\displaystyle\mathcal{N}^{-1/2}\int
F(\varphi)\Phi_{0}\Phi_{n}(\varphi)d\varphi~{}.$

Then we have

$\displaystyle G_{F}(|t|)$ $\displaystyle=$
$\displaystyle\sum_{n=0}A_{n}e^{-\Lambda_{n}t}\int
F(\varphi)e^{-v}\Phi_{n}(\varphi)d\varphi=\mathcal{N}\sum_{n=0}A_{n}^{2}e^{-\Lambda_{n}t}=\sum_{n=0}\tilde{A}_{n}^{2}e^{-\Lambda_{n}t}~{},$
(17)

with

$\displaystyle\tilde{A}_{n}$ $\displaystyle=$ $\displaystyle\int
F(\varphi)\Phi_{0}(\varphi)\Phi_{n}(\varphi)d\varphi~{}.$ (18)

For a long enough period of time, the dominate contribution of $G_{F}(t)$
comes from the lowest state $\Lambda_{\rm min}$ with $\tilde{A}_{\rm min}\neq
0$. We can use the above formula to calculate the average of the variable $F$
and $F^{2}$,

$\displaystyle\langle F\rangle$ $\displaystyle=$ $\displaystyle\left(\int
F(\varphi)\rho_{q}d\varphi\right)=\left(\int
F(\varphi)\Phi_{0}(\varphi)\Phi_{0}(\varphi)d\varphi\right)=\tilde{A}_{0}~{},$
$\displaystyle{\langle F^{2}\rangle}$ $\displaystyle=$ $\displaystyle
G_{F}(0)=\sum_{n=0}\tilde{A}_{n}^{2}~{}.$ (19)

If we define the fluctuations of $F$ as $\delta F=F-\langle F\rangle$, then

$\displaystyle{\langle(\delta F)^{2}\rangle}=\langle F^{2}\rangle-\langle
F\rangle^{2}=\sum_{n=1}\tilde{A}_{n}^{2}~{}.$ (20)

For the spatial correlation function $\xi_{F}({\bf x})=\langle F(\varphi({\bf
x_{1}},t))F(\varphi({\bf x_{2}},t))\rangle$(${\bf x=x_{1}-x_{2}}$), in the
large ${\bf|x|}$ limit, we just need to take $t\rightarrow
2H^{-1}\log(aH{\bf|x|})$,

$\displaystyle\xi_{F}({\bf x})$ $\displaystyle=$
$\displaystyle\sum_{n=0}\tilde{A}_{n}^{2}(aH{\bf|x|})^{-2\Lambda_{n}H^{-1}}~{}.$
(21)

For the case of $\tilde{A}_{0}=0$, we can define the correlation length that
the variable $F(\varphi)$ satisfies,

$\displaystyle\xi_{F}(R_{c})$ $\displaystyle=$ $\displaystyle G_{F}(0)/2~{}.$
(22)

Then within the length scale $R_{c}$ we can take $F$ approximately as a
constant. Usually this scale is decided by the lowest eigenvalue
$\Lambda_{\min}$ with $\tilde{A}_{\rm min}\neq 0$.

##  3 Example for a single field

In this section, we will use the above formulas to calculate the example of a
single scalar field where only a mass term is included. These results are
already known and there is nothing new here. But this result is important to
allow us to extend to the case of multi-dimensions, particularly into
2-dimensions which we are interested in. Now the potential is

$\displaystyle
V(\varphi)=\frac{1}{2}m^{2}\varphi^{2},~{}~{}~{}v(\varphi)\equiv\frac{4\pi^{2}}{3H^{4}}V(\varphi)~{}.$
(23)

The eigenvalue equation is,

$\displaystyle\frac{1}{2}\left[-\frac{\partial^{2}}{\partial\varphi^{2}}+\left({v^{\prime}(\varphi)}^{2}-v^{\prime\prime}(\varphi)\right)\right]\psi_{n}(\varphi)=\frac{4\pi^{2}}{H^{3}}\Lambda_{n}\psi_{n}(\varphi)~{}.$
(24)

We can define an effective potential as,

$\displaystyle\mathcal{V}=\frac{1}{2}\left(v^{\prime}(\varphi)^{2}-v^{\prime\prime}(\varphi)\right)=\frac{1}{2}\omega^{2}\varphi^{2}-\frac{1}{2}\omega~{},$
(25)

where $\omega=\frac{4\pi^{2}m^{2}}{3H^{4}}$. Then the eigenvalue equation
becomes,

$\displaystyle\frac{1}{2}\left[-\frac{\partial^{2}}{\partial\varphi^{2}}+\omega^{2}\varphi^{2}\right]\psi_{n}(\varphi)-\frac{1}{2}\omega\psi_{n}(\varphi)=\frac{4\pi^{2}}{H^{3}}\Lambda_{n}\psi_{n}(\varphi)~{}.$
(26)

This is nothing but the eigenvalue equation for an oscillator with a frequency
$\omega$. The difference is that the zero energy has been removed by the terms
$-\frac{1}{2}\omega\psi_{n}(\varphi)$. One can easily find that the
eigenvalues are,

$\displaystyle\Lambda_{n}=\frac{H^{3}}{4\pi^{2}}n\omega=\frac{nm^{2}}{3H},~{}n=0,1,2...~{}.$
(27)

The eigenfunctions can be written as,

$\displaystyle\psi_{n}=N_{n}\exp\left(-\frac{1}{2}\alpha^{2}\varphi^{2}\right)H_{n}(\alpha\varphi),~{}~{}N_{n}=(\alpha/\sqrt{\pi}2^{n}n!)^{1/2}~{},$
(28)

with $\alpha=\sqrt{\omega}$ and $H_{n}$ is the Hermite polynomials. For
simplicity, we can use the Dirac Bra-ket notation to describe the
eigenfunction $\psi_{n}$ as $|n\rangle$ with the eigenvalue $\Lambda_{n}$. We
also have the following relation,

$\displaystyle\varphi|n\rangle=\frac{1}{\alpha}\left[\sqrt{\frac{n}{2}}|n-1\rangle+\sqrt{\frac{n+1}{2}}|n+1\rangle\right]~{}.$
(29)

It is then not difficult to find that for $F(\varphi)=\varphi^{2}$, we have

$\displaystyle\langle F\rangle=\langle\varphi^{2}\rangle=\langle
0|\varphi^{2}|0\rangle=\frac{1}{2\alpha^{2}}=\frac{3H^{4}}{8\pi^{2}m^{2}}~{}.$
(30)

For $F(\varphi)=\varphi$, we have

$\displaystyle\langle 0|\varphi|0\rangle=0,~{}~{}\langle
0|\varphi|1\rangle=\frac{1}{\sqrt{2}\alpha}~{}=\tilde{A}_{1}$ (31)

The spatial correlation function of $F(\varphi)=\varphi$ is

$\displaystyle\xi({\bf
x})=|\tilde{A}_{1}|^{2}(aH{\bf|x|})^{-2\Lambda_{1}H^{-1}}=\langle\varphi^{2}\rangle(aH{\bf|x|})^{-2\Lambda_{1}H^{-1}}~{}.$
(32)

Then we have

$\displaystyle
R_{c}=H^{-1}\exp{\left(\frac{3{\rm\log}2}{2}\frac{H^{2}}{m^{2}}\right)}\approx
H^{-1}\exp{\left(\frac{H^{2}}{m^{2}}\right)}~{},$ (33)

therefore, the correlation length is totally decided by the mass of the
field(we set the $a=1$ at the end of inflation). For $N_{*}=60$, if the mass
is $m\lesssim 0.1H$, the correlation length $\varphi$ can be much larger than
our observed universe. If $m\gtrsim 0.13H$, in the early time our current
universe is composed of different patches with different $\varphi$ following a
distribution described by $\rho_{q}(\varphi)$.

##  4 Correlation length of the angular mode for an exact $U(1)$ symmetry

In this section, we will calculate the correlation length of the angular mode
for an exact $U(1)$ symmetry. In this case, the $\theta$ is nearly massless
and the correlation length may appear to be very large. We will show that this
naive estimation is wrong. For multi-dimension Fokker-Planck equation,

$\displaystyle-\frac{1}{2}\nabla^{2}\Phi_{n}+\frac{1}{2}\left[(\nabla
v)^{2}-\nabla^{2}v\right]\Phi_{n}=\frac{4\pi^{2}\Lambda_{n}}{H^{3}}\Phi_{n}~{}.$
(34)

Let us look at a simple case,

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}|\phi|^{2}=\frac{1}{2}m^{2}\phi_{1}^{2}+\frac{1}{2}m^{2}\phi_{2}^{2}~{},$
(35)

where $\phi=\frac{1}{\sqrt{2}}(\phi_{1}+i\phi_{2})$. From our previous
calculation, it shows that the universe is divided into different patches with
a correlation length of $R_{c}\approx
H^{-1}\exp\left(\frac{H^{2}}{m^{2}}\right)$. However, if we use the radial-
angular coordinates $\phi=\frac{r}{\sqrt{2}}e^{i\theta}$, we have

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}|\phi|^{2}=\frac{1}{2}m^{2}r^{2}~{}.$ (36)

Then we have one massive mode and one massless mode. One may argue that the
correlation length for $\theta$ should be much larger than $R_{c}$. We will
show that it is not true considering two different coordinate representations.

###  4.1 $\phi_{1}$, $\phi_{2}$ representation

Since the $\phi_{1},\phi_{2}$ do not couple to each other, the eigenvalues are
easy to solve. The total eigenfunctions are just the product of each separate
eigenfunction. Let us define the eigenfunction with eigenvalue $\Lambda_{n,l}$
as

$\displaystyle\Psi_{n,l}(\phi_{1},\phi_{2})$ $\displaystyle=$
$\displaystyle\psi_{n}(\phi_{1})\tilde{\psi}_{l}(\phi_{2}),~{}\Lambda_{n,l}=\frac{(n+l)m^{2}}{3H},~{}n,l=0,1,2...~{}.$
(37)

There is a degeneracy for the eigenvalue $\Lambda_{n,l}$ with
$N_{\Lambda_{n,l}}=n+l+1$. Here we only list some of the eigenfunctions with
the lowest eigenvalues,

$\displaystyle\Psi_{0,0}(\phi_{1},\phi_{2})$ $\displaystyle=$
$\displaystyle\psi_{0}(\phi_{1})\tilde{\psi}_{0}(\phi_{2}),~{}\Lambda_{0}=0$
(38) $\displaystyle\Psi_{0,1}(\phi_{1},\phi_{2})$ $\displaystyle=$
$\displaystyle\psi_{0}(\phi_{1})\tilde{\psi}_{1}(\phi_{2}),~{}\Lambda_{0,1}=\frac{m^{2}}{3H}$
(39) $\displaystyle\Psi_{1,0}(\phi_{1},\phi_{2})$ $\displaystyle=$
$\displaystyle\psi_{1}(\phi_{1})\tilde{\psi}_{0}(\phi_{2}),~{}\Lambda_{1,0}=\frac{m^{2}}{3H}~{}.$
(40)

We can also use the Dirac Bra-ket notation $|n,l\rangle$ to denote the state
with eigenfunction $\psi_{n}(\phi_{1})\tilde{\psi}_{l}(\phi_{2})$.

Let us calculate the correlation length of the angular mode
$F(\phi_{1},\phi_{2})=\theta={\rm
Arcsin}\frac{\phi_{2}}{\sqrt{\phi_{1}^{2}+\phi_{2}^{2}}}$. Due to the symmetry
of the function, the first non-vanishing $\tilde{A}$ appears at

$\displaystyle\tilde{A}_{0,1}$ $\displaystyle=$ $\displaystyle\int{\rm
Arcsin}\frac{\phi_{2}}{\sqrt{\phi_{1}^{2}+\phi_{2}^{2}}}\Psi_{0,0}(\phi_{1},\phi_{2})\Psi_{0,1}(\phi_{1},\phi_{2})d\phi_{1}d\phi_{2}~{},$
(41)

with the corresponding eigenvalue $\frac{m^{2}}{3H^{2}}$. Numerically we find
$A_{0,1}=0.8\neq 0$, then $R_{\theta}\approx R_{c}$.

However, it is more intriguing to calculate the correlation length of
$\sin\theta=\frac{\phi_{2}}{\sqrt{\phi_{1}^{2}+\phi_{2}^{2}}}$. Within the
correlation length $R_{\theta}$, we can take $\theta$ as a constant, then the
same is true for $\sin\theta$. Therefore, we expect the correlation length of
these two variables might be similar (generally $f(\theta)$ and $\theta$ could
have different correlation lengths). More importantly, it is much easier to
apply this variable to the $r-\theta$ representation. As usual,

$\displaystyle\tilde{A}_{0,1}$ $\displaystyle=$
$\displaystyle\int\frac{\phi_{2}}{\sqrt{\phi_{1}^{2}+\phi_{2}^{2}}}\Psi_{0,0}(\phi_{1},\phi_{2})\Psi_{0,1}(\phi_{1},\phi_{2})d\phi_{1}d\phi_{2}~{}.$
(42)

Numerically, we find $\tilde{A}_{0,1}\approx 0.63$, and thus, the spatial
correlation function is

$\displaystyle\xi_{\sin\theta}({\bf x})$ $\displaystyle=$
$\displaystyle\tilde{A}_{1}^{2}(aH{\bf|x|})^{-2\Lambda_{0,1}H^{-1}}~{},$ (43)

where $\Lambda_{0,1}=\frac{m^{2}}{3H}$. Then we have $R_{\sin\theta}\approx
H^{-1}\exp\left(\frac{H^{2}}{m^{2}}\right)$. Indeed the correlation length is
similar to $R_{c}$.

###  4.2 $r$, $\theta$ representation

For the eigenvalue equation,

$\displaystyle-\frac{1}{2}\nabla^{2}\Phi_{n}+\frac{1}{2}\left[(\nabla
v)^{2}-\nabla^{2}v\right]\Phi_{n}=\frac{4\pi^{2}\Lambda_{n}}{H^{3}}\Phi_{n}~{},$
(44)

we have

$\displaystyle\nabla^{2}=\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}~{}.$ (45)

in the $r-\theta$ representation. The effective potential $\mathcal{V}$ is
only a function of $r$.

$\displaystyle\mathcal{V}=\frac{1}{2}\left[(\nabla
v)^{2}-\nabla^{2}v\right]~{}.$ (46)

Due to the rotation symmetry, the wave function can be written as,

$\displaystyle\Psi_{n,l}(r,\theta)=R_{n,l}(r)Y_{l}(\theta),~{}l=0,\pm 1,\pm
2...~{},$ (47)

where

$\displaystyle Y_{l}(\theta)=\frac{1}{\sqrt{2\pi}}e^{il\theta},~{}l=0,\pm
1,\pm 2...~{},$ (48)

and we have

$\displaystyle\int^{2\pi}_{0}d\theta
Y_{l}(\theta)Y^{*}_{l^{\prime}}(\theta)=\delta_{l,l^{\prime}}~{}.$ (49)

Since $v$ is only a function of $r$, the eigenvalue equation becomes

$\displaystyle-\left[R^{\prime\prime}_{n,l}+\frac{R^{\prime}_{n,l}}{r}-\frac{l^{2}}{r^{2}}R_{n,l}\right]+\left[(v^{\prime})^{2}-v^{\prime\prime}-\frac{v^{\prime}}{r}\right]R_{n,l}=\frac{8\pi^{2}\lambda_{n,l}}{H^{3}}R_{n,l}~{},$
(50)

where we have used the following relation

$\displaystyle\frac{\partial^{2}}{\partial\theta^{2}}Y_{l}$ $\displaystyle=$
$\displaystyle-l^{2}~{}.$ (51)

The eigensystem is similar to the 2 dimension oscillator

$\displaystyle\lambda_{n,l}$ $\displaystyle=$
$\displaystyle(2n+|l|)\frac{m^{2}}{3H},~{}n=1,2,3...,~{}l=0,\pm 1,\pm 2...$
$\displaystyle N_{\lambda_{n,l}}$ $\displaystyle=$ $\displaystyle
2n+|l|+1~{}.$ (52)

This result is totally consistent with that found for the
$\phi_{1}$,$\phi_{2}$ coordinates. Here we list some of the wave-functions

$\displaystyle\Phi_{0,0}(r,\theta)$ $\displaystyle=$ $\displaystyle
R_{0,0}(r)\frac{1}{\sqrt{2\pi}}$ $\displaystyle\Phi_{0,1}(r,\theta)$
$\displaystyle=$ $\displaystyle R_{0,1}(r)\frac{1}{\sqrt{2\pi}}e^{i\theta}$
$\displaystyle\Phi_{0,-1}(r,\theta)$ $\displaystyle=$ $\displaystyle
R_{0,1}(r)\frac{1}{\sqrt{2\pi}}e^{-i\theta}~{}.$ (53)

Actually it is more convenient to use another set of eigenfunctions,

$\displaystyle\Phi_{n,0}(r,\theta)$ $\displaystyle=$ $\displaystyle
R_{n,0}(r)\frac{1}{\sqrt{2\pi}},~{}n=0,1,2,3...,$
$\displaystyle\Phi_{n,l}(r,\theta)$ $\displaystyle=$ $\displaystyle
R_{n,l}(r)\frac{1}{\sqrt{\pi}}\cos l\theta,~{}n=0,1,2,3...,~{}l=1,2,3...$
$\displaystyle\Phi^{\prime}_{n,l}(r,\theta)$ $\displaystyle=$ $\displaystyle
R_{n,l}(r)\frac{1}{\sqrt{\pi}}\sin l\theta,~{}n=0,1,2,3...,~{}l=1,2,3...~{}.$
(54)

The lowest eigenvalue which has non-vanishing $\tilde{\mathcal{A}}$ is
$\lambda_{0,1}=\frac{m^{2}}{3H}$ with

$\displaystyle\tilde{\mathcal{A}}_{0,1}^{\prime}$ $\displaystyle=$
$\displaystyle\int\sin\theta\Phi_{0,0}(r,\theta)\Phi^{\prime}_{0,1}(r,\theta)rdrd\theta$
(55) $\displaystyle=$ $\displaystyle\int
R_{0,0}(r)R_{0,1}(r)rdr\frac{1}{\sqrt{2}}~{}.$

Numerically, we find $\tilde{\mathcal{A}}^{\prime}_{0,1}\approx 0.63$, which
is totally consistent with the previous calculation in which we used the
$\phi_{1},\phi_{2}$ coordinates.

##  5 Correlation length of the angular mode for an approximate $U(1)$
symmetry

For the case of an approximate $U(1)$ symmetry, we add a small breaking term
for the $U(1)$ symmetry, which is a necessary condition for the baryogenesis
by the Affleck-Dine mechanism. In this scenario, one may argue that for the
approximate $U(1)$ symmetry, $\theta$ is not exactly massless and obtains a
finite (small) mass, such that the correlation length could be much longer. In
the following, we will show, that even if $U(1)$ is an approximate symmetry,
the correlation length of $\theta$ is still close to $R_{c}$. Let us consider
an example,

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}|\phi|^{2}+\left[\frac{1}{2}\mu^{2}\phi^{2}+h.c\right]$ (56)
$\displaystyle=$
$\displaystyle\frac{1}{2}(m^{2}+\mu^{2})\phi_{1}^{2}+\frac{1}{2}(m^{2}-\mu^{2})\phi_{2}^{2}$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2}m_{1}^{2}\phi_{1}^{2}+\frac{1}{2}m_{2}^{2}\phi_{2}^{2}~{},$

where the $\mu^{2}$ term breaks the $U(1)$ symmetry and we assume that such a
term is small compared to the $m^{2}$ term. In this case, we still have an
exact solution.

$\displaystyle\Psi_{n,l}(\phi_{1},\phi_{2})$ $\displaystyle=$
$\displaystyle\psi_{n}(\phi_{1})\tilde{\psi}_{l}(\phi_{2}),~{}\Lambda_{n,l}=\frac{(nm_{1}^{2}+lm_{2}^{2})}{3H},~{}n,l=0,1,2...~{}.$
(57)

Here we list some of the wave functions with lowest eigenvalues,

$\displaystyle\Psi_{0,0}(\phi_{1},\phi_{2})$ $\displaystyle=$
$\displaystyle\psi_{0}(\phi_{1})\tilde{\psi}_{0}(\phi_{2}),~{}\Lambda_{0}=0$
(58) $\displaystyle\Psi_{0,1}(\phi_{1},\phi_{2})$ $\displaystyle=$
$\displaystyle\psi_{0}(\phi_{1})\tilde{\psi}_{1}(\phi_{2}),~{}\Lambda_{1}=\frac{m^{2}_{2}}{3H^{2}}$
(59) $\displaystyle\Psi_{1,0}(\phi_{1},\phi_{2})$ $\displaystyle=$
$\displaystyle\psi_{1}(\phi_{1})\tilde{\psi}_{0}(\phi_{2}),~{}\Lambda_{2}=\frac{m^{2}_{1}}{3H^{2}}~{}.$
(60)

The degeneracy of the spectrum has been broken because $m_{1}>m_{2}$. Now
again, the lowest eigenvalue with non-vanishing $\tilde{A}$ is
$\frac{m^{2}_{2}}{3H}$ with

$\displaystyle\tilde{A}_{0,1}$ $\displaystyle=$
$\displaystyle\int\frac{\phi_{2}}{\sqrt{\phi_{1}^{2}+\phi_{2}^{2}}}\Psi_{0,0}(\phi_{1},\phi_{2})\Psi_{0,1}(\phi_{1},\phi_{2})d\phi_{1}d\phi_{2}~{}.$
(61)

For $\mu^{2}\ll m^{2}$, the value of $\tilde{A}_{0,1}$ is essentially not
changed. But the eigenvalue has been changed from $\frac{m^{2}}{3H^{2}}$ into
$\frac{m^{2}_{2}}{3H^{2}}$ and the correlation length indeed becomes larger

$\displaystyle R_{\sin\theta}\approx
H^{-1}\exp\left({H^{2}}/{(m^{2}-\mu^{2})}\right)~{}.$ (62)

Since we are considering the case of $\mu^{2}\ll m^{2}$, this value is still
close to $R_{c}$. More importantly, in the limit of $\mu^{2}\rightarrow 0$, we
get the exact $U(1)$ symmetry result.

To be complete, we also calculate the result in the $r,\theta$ coordinates. In
this case we can use the formulas used in perturbation theory for quantum
mechanics. We have

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}|\phi|^{2}+\left[\frac{1}{2}\mu^{2}\phi^{2}+h.c\right]$ (63)
$\displaystyle=$
$\displaystyle\frac{1}{2}m^{2}r^{2}+\frac{1}{2}\mu^{2}r^{2}\cos 2\theta~{}.$

Since

$\displaystyle\Phi_{0}$ $\displaystyle=$
$\displaystyle\mathcal{N}^{-1/2}e^{-v}~{},$ (64)

obviously the

$\displaystyle\tilde{A}_{0}=\int\Phi_{0}^{2}\sin\theta rdrd\theta=0~{}.$ (65)

Note that $\Phi_{0}\neq\Phi^{(0)}_{0,0}$ since $\Phi_{0}$ is the exact
eigenfunction with the breaking term included. Now we need to find out the
lowest eigenvalue with $\tilde{A}\neq 0$. The effective potential can be
written as

$\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(\nabla
v)^{2}-\nabla^{2}v\right]=\left[\frac{1}{2}\left(\frac{4\pi^{2}}{3H^{4}}\right)^{2}m^{4}r^{2}\frac{4\pi^{2}m^{2}}{3H^{4}}\right]+\left(\frac{4\pi^{2}}{3H^{4}}\right)^{2}\mu^{2}m^{2}r^{2}\cos
2\theta+\mathcal{O}(\mu^{4})$ (66) $\displaystyle=$
$\displaystyle\left({1}/{2}\omega_{0}^{2}r^{2}-\omega_{0}\right)+\left(\omega_{0}\omega_{1}r^{2}\right)\equiv\mathcal{V}_{0}+\mathcal{V}_{b}~{},$

where we define $\omega_{0}=\frac{4\pi^{2}m^{2}}{3H^{4}}$ and
$\omega_{1}=\frac{4\pi^{2}\mu^{2}}{3H^{4}}$. Clearly $\mathcal{V}_{b}$ breaks
the degeneracy of the eigenvalues for states with

$\displaystyle\Phi^{(0)}_{0,1}(r,\theta)$ $\displaystyle=$ $\displaystyle
R^{(0)}_{0,1}(r)\frac{1}{\sqrt{\pi}}\cos\theta,$
$\displaystyle{\Phi^{\prime}}^{(0)}_{0,1}(r,\theta)$ $\displaystyle=$
$\displaystyle R^{(0)}_{0,1}(r)\frac{1}{\sqrt{\pi}}\sin\theta~{}.$ (67)

However, it does not mix these two states, and the eigenvalues of each state
become,

$\displaystyle\lambda^{b}_{0,1}$ $\displaystyle=$
$\displaystyle\lambda_{0,1}^{(0)}+\frac{H^{3}}{4\pi^{2}}\int{{\Phi_{0,1}^{(0)}}}^{2}\mathcal{V}_{b}rdrd\theta$
$\displaystyle{\lambda^{\prime}}^{b}_{0,1}$ $\displaystyle=$
$\displaystyle\lambda_{0,1}^{(0)}+\frac{H^{3}}{4\pi^{2}}\int{{{{\Phi^{\prime}}^{(0)}_{0,1}}}}^{2}\mathcal{V}_{b}rdrd\theta~{}.$
(68)

We have

$\displaystyle\frac{H^{3}}{4\pi^{2}}\int{{\Phi_{0,1}^{(0)}}}^{2}\mathcal{V}_{b}rdrd\theta$
$\displaystyle=$
$\displaystyle\frac{\omega_{0}}{3H}\frac{\mu^{2}}{2}\int{R^{(0)}_{0,1}}^{2}{r^{3}}dr$
$\displaystyle\frac{H^{3}}{4\pi^{2}}\int{{{{\Phi^{\prime}}^{(0)}_{0,1}}}}^{2}\mathcal{V}_{b}rdrd\theta$
$\displaystyle=$
$\displaystyle-\frac{\omega_{0}}{3H}\frac{\mu^{2}}{2}\int{R^{(0)}_{0,1}}^{2}{r^{3}}dr~{}.$
(69)

The last integration seems to be not easily determined. Notice that it is the
average of $r^{2}=\phi_{1}^{2}+\phi_{2}^{2}$ for the state with
$\Lambda=\frac{m^{2}}{3H}$. We can use the $\phi_{1}$ $\phi_{2}$ coordinates
to estimate this integration, since we have

$\displaystyle\langle\Psi^{(0)}_{0,1}|\phi_{1}^{2}+\phi_{2}^{2}|\Psi^{(0)}_{0,1}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi^{(0)}_{1,0}|\phi_{1}^{2}+\phi_{2}^{2}|\Psi^{(0)}_{1,0}\rangle=\frac{2}{\omega_{0}}~{}.$
(70)

then we get

$\displaystyle\int{R^{(0)}_{0,1}}^{2}{r^{3}}dr=\frac{2}{\omega_{0}}~{}.$ (71)

In the end we find

$\displaystyle\lambda^{b}_{0,1}$ $\displaystyle=$
$\displaystyle(m^{2}+\mu^{2})\frac{1}{3H}$
$\displaystyle{\lambda^{\prime}}^{b}_{0,1}$ $\displaystyle=$
$\displaystyle(m^{2}-\mu^{2})\frac{1}{3H}~{}.$ (72)

Since we have $\mu^{2}\ll m^{2}$, we do not expect $\tilde{\mathcal{A}}_{0,1}$
to change too much. The essential thing here is that the $\lambda$ has matched
the value of the exact solutions in the $\phi_{1},\phi_{2}$ representation and
the correlation length is determined by the eigenvalue. Since the correlation
length is mainly determined by the lowest eigenvalue with
$\tilde{\mathcal{A}}\neq 0$, we arrive at the same conclusion, that the
correlation length of the angular mode should not be much larger than $R_{c}$.

One can also extend this result to other small $U(1)$ breaking cases, from
which we can then conclude that even if a small breaking term is included, the
correlation length $\theta$ can not be much larger than $R_{c}$.

##  6 Correlation length for the baryon number density

It is known that a scalar field which carries baryon charge can generate a
baryon asymmetry through the Affleck-Dine mechanism. Tn the following, we
analyse the correlation length of the baryon number density. The displacement
of the field originates from quantum fluctuations. For the potential, we have

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}|\phi|^{2}+\left[\kappa_{n}\phi^{n}+h.c\right]~{},$ (73)

and

$\displaystyle\dot{n}_{B}+3Hn_{B}={\rm Im}(\phi\frac{\partial
V}{\partial\phi})~{}.$ (74)

The baryon number density at $H=m$ can be estimated as,

$\displaystyle n_{B}\sim\frac{n\kappa_{n}{\rm
Im}(\phi^{n})}{m}=\frac{n\kappa_{n}r^{n}\sin n\theta_{0}}{2^{n/2}m}~{}.$ (75)

Since $\sin n\theta_{0}$ is randomly distributed, the average is given by
$\langle n_{B}\rangle=0$. But this just means that the whole universe has an
average baryon number of zero, but not necessarily in our observed universe.
If the correlation length of $n_{B}$ is much larger than our current universe,
we may still live in a region with a positive baryon number. To calculate the
correlation length, we set $n=4$ as an example,

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}|\phi|^{2}+\left[\kappa\phi^{4}+h.c\right]$ (76) $\displaystyle=$
$\displaystyle\frac{1}{2}m^{2}r^{2}+\frac{1}{2}\kappa r^{4}\cos 4\theta~{}.$

The baryon number density is

$\displaystyle n_{B}=\frac{\kappa r^{4}\sin
4\theta}{2m}=\kappa\frac{4\phi_{1}\phi_{2}(\phi_{1}^{2}-\phi_{2}^{2})}{2m}~{}.$
(77)

The correlation function is

$\displaystyle\langle
n_{B}(0)n_{B}(t)\rangle=\sum\tilde{A}_{n}^{2}e^{-\Lambda_{n}t}~{},$ (78)

where

$\displaystyle\tilde{A}_{n}$ $\displaystyle=$ $\displaystyle\int
n_{B}\Phi_{0}(\varphi)\Phi_{n}(\varphi)d\varphi~{}.$ (79)

Due to the $\sin 4\theta$ term, the major contribution is from $\Phi_{0,4}$
with eigenvalue $\lambda_{0,4}=\frac{4m^{2}}{3H}$. Numerically, we find that

$\displaystyle\tilde{A}_{0,4}$ $\displaystyle=$ $\displaystyle\int\frac{\kappa
r^{4}}{2m}R_{0,0}(r)R_{0,4}rdr\frac{1}{\sqrt{2}}=\frac{1.8\kappa}{m\omega^{2}}~{}.$
(80)

Then

$\displaystyle\langle
n_{B}(0)n_{B}(t)\rangle\approx\frac{3\kappa^{2}}{m^{2}\omega^{4}}e^{-\frac{4m^{2}}{3H}t}~{}.$
(81)

It is also useful to consider the $\phi_{1},\phi_{2}$ coordinates. For
$\Lambda=\frac{4m^{2}}{3H}$, there are five states, $|0,4\rangle$,
$|4,0\rangle$, $|1,3\rangle$, $|3,1\rangle$, $|2,2\rangle$. By defining
$f(\phi_{1},\phi_{2})=4\phi_{1}\phi_{2}(\phi_{1}^{2}-\phi_{2}^{2})$, it is not
difficult to find that

$\displaystyle\langle 0,0|f(\phi_{1},\phi_{2})|4,0\rangle=\langle
0,0|f(\phi_{1},\phi_{2})|0,4\rangle=\langle
0,0|f(\phi_{1},\phi_{2})|2,2\rangle=0~{}.$ (82)

Then we only need to calculate,

$\displaystyle|\langle 0,0|f(\phi_{1},\phi_{2})|1,3\rangle|=|\langle
0,0|f(\phi_{1},\phi_{2})|3,1\rangle|=\frac{\sqrt{6}}{\omega^{2}}~{}.$ (83)

Now we have two non-vanishing $\tilde{A}$, which when summed together gives,

$\displaystyle\langle
n_{B}(0)n_{B}(t)\rangle=\frac{3\kappa^{2}}{m^{2}\omega^{4}}e^{-\frac{4m^{2}}{3H}t}~{}.$
(84)

The spatial correlation of $n_{B}$ is

$\displaystyle\langle n_{B}(0)n_{B}({\bf
x})\rangle\approx\frac{3\kappa^{2}}{m^{2}\omega^{4}}(aH{\bf|x|})^{-\frac{8m^{2}}{3H^{2}}}~{},$
(85)

which is consistent with the work Hook:2015foa . Now we find that the
correlation length of $n_{B}$ is

$\displaystyle R_{n_{B}}\approx
H^{-1}\exp{\left(\frac{H^{2}}{4m^{2}}\right)}~{}.$ (86)

which is much smaller than $R_{c}$. Note that even if $\langle
n_{B}\rangle=0$, once $R_{n_{B}}$ is much larger than our current universe, we
could live in a baryon-rich universe, which is the case for $m\ll 0.06H$ for
$N_{*}=60$.

To end, we give one example of a baryogenesis model which is successful
despite the angular mode having a $R_{c}$ that is much smaller than our
current universe. One such example is baryogenesis from sneutrino decay
Hamaguchi:2001gw , the total baryon number is

$\displaystyle{n_{B}}=\epsilon n_{\phi}\approx{\epsilon}m|\phi|^{2}~{}.$ (87)

Since $n_{B}$ is independent of the $\theta$, in the whole universe we could
have the same sign of $n_{B}$.

##  7 Discussion and conclusion

In this work, we calculated the correlation length of the angular mode for
models with an approximate $U(1)$ symmetry. We find that for a massive nearly
non-interacting fields, the correlation length of the angular mode is
determined by the mass parameter, which is approximately
$R_{c}=H^{-1}\exp(H^{2}/m^{2})$. Applying this result to baryogenesis through
the Affleck-Dine mechanism, if $m\sim H$, the correlation length of the
angular mode would be much smaller than our current horizon size and the
average of the baryon number in our universe would be nearly zero.

Figure 2: Illustration of the quantum jump for $\theta$.

The reason for this is simple. One can consider the quantum jump in each
e-folding $\delta\phi\approx\frac{H}{2\pi}$ within a horizon size $H^{-1}$,
then $\delta\theta\sim\frac{H}{2\pi\phi_{0}}$ where
$\phi_{0}\equiv\sqrt{\langle\varphi^{2}\rangle}=\sqrt{\frac{3}{2}}\frac{H^{2}}{2\pi
m}$. The breaking of the correlation length for $\theta_{0}\sim 1$ happens at
an e-fold $N$ satisfying $\sqrt{N}=\frac{2\pi\phi_{0}}{H}\sim H/m$, therefore
a natural correlation length for the $\theta_{0}$ is around
$H^{-1}\exp\left(N\right)=H^{-1}\exp\left(\frac{H^{2}}{m^{2}}\right)$ which is
consistent with our calculation. For $m\lesssim H$, we have $\phi_{0}\sim
H/2\pi$, then $\theta$ would soon become randomly distributed with a
correlation length $H^{-1}$. Now we conclude that only when
$m\ll\mathcal{O}(0.1)H$ (assuming $N_{*}=60$) can the correlation length of
the baryon number density be much larger than our current horizon size, such
that we live in the baryon-rich region333The baryonic isocurvature from CMB
sets a stronger limit on $m$.. For the case $m\sim H$ which is considered in
the work Wu:2020pej ; Wu:2021gtd ; Wu:2021mwy , at early times our universe
would consist of numerous patches of baryon-rich and anti-baryon-rich regions
with the average baryon number being nearly zero.

To end, we give a comment on the correlation length of $\theta$ in the
original Affleck-Dine baryogenesis scenario, where the Affleck-Dine field gets
a negative mass from the inflationary field Dine:1995kz . In this case, $\phi$
is trapped at a much larger value $\phi_{0}\gg H$. Thus, the quantum jump is
just $\delta\theta\sim\frac{H}{2\pi\phi_{0}}$ which is much smaller than the
$\theta_{0}\sim\mathcal{O}(1)$. In this case, the correlation length of
$\theta$ indeed could be much larger than our current horizon size.

###### Acknowledgements.

C.H. acknowledges Neil D. Barrie and Shi Pi for their helpful discussions. C.
H. is supported by the Guangzhou Basic and Applied Basic Research Foundation
under Grant No. 202102020885, and the Sun Yat-Sen University Science
Foundation.

## References

  * [1] R. Brout, F. Englert, and E. Gunzig.  The Creation of the Universe as a Quantum Phenomenon.  Annals Phys., 115:78, 1978.
  * [2] K. Sato.  First Order Phase Transition of a Vacuum and Expansion of the Universe.  Mon. Not. Roy. Astron. Soc., 195:467–479, 1981.
  * [3] Alan H. Guth.  The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems.  Phys. Rev. D, 23:347–356, 1981.
  * [4] Andrei D. Linde.  A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems.  Phys. Lett. B, 108:389–393, 1982.
  * [5] Andreas Albrecht, Paul J. Steinhardt, Michael S. Turner, and Frank Wilczek.  Reheating an Inflationary Universe.  Phys. Rev. Lett., 48:1437, 1982.
  * [6] T. S. Bunch and P. C. W. Davies.  Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting.  Proc. Roy. Soc. Lond. A, 360:117–134, 1978.
  * [7] Alexei A. Starobinsky and Junichi Yokoyama.  Equilibrium state of a selfinteracting scalar field in the De Sitter background.  Phys. Rev. D, 50:6357–6368, 1994.
  * [8] Ian Affleck and Michael Dine.  A New Mechanism for Baryogenesis.  Nucl. Phys. B, 249:361–380, 1985.
  * [9] Michael Dine, Lisa Randall, and Scott D. Thomas.  Supersymmetry breaking in the early universe.  Phys. Rev. Lett., 75:398–401, 1995.
  * [10] Michael Dine, Lisa Randall, and Scott D. Thomas.  Baryogenesis from flat directions of the supersymmetric standard model.  Nucl. Phys. B, 458:291–326, 1996.
  * [11] Andrei D. Linde.  The New Mechanism of Baryogenesis and the Inflationary Universe.  Phys. Lett. B, 160:243–248, 1985.
  * [12] Yi-Peng Wu and Kalliopi Petraki.  Stochastic Baryogenesis.  JCAP, 01:022, 2021.
  * [13] Yi-Peng Wu, Elena Pinetti, Kalliopi Petraki, and Joseph Silk.  Baryogenesis from ultra-slow-roll inflation.  JHEP, 01:015, 2022.
  * [14] Yi-Peng Wu, Elena Pinetti, and Joseph Silk.  Cosmic Coincidences of Primordial-Black-Hole Dark Matter.  Phys. Rev. Lett., 128(3):031102, 2022.
  * [15] Anson Hook.  Baryogenesis in a CP invariant theory.  JHEP, 11:143, 2015.
  * [16] Koichi Hamaguchi, Hitoshi Murayama, and T. Yanagida.  Leptogenesis from N dominated early universe.  Phys. Rev. D, 65:043512, 2002.

##  Appendix A Numerical calculation of $R_{n,l}$

Although Mathematica 12 provides a function “NDEigensystem” to solve the
eigensystem, we can directly apply it to the $R_{n,l}$ function. The reason is
that the normalization of the $R_{n,l}$ function is

$\displaystyle\int R_{n,l}(r)R_{n,l}(r)rdr=1~{}.$ (88)

To preform the numerical calculation, we can define
$R_{n,l}=\chi_{n,r}r^{-1/2}$, then the normalization of $\chi$ becomes

$\displaystyle\int\chi_{n,l}(r)\chi_{n,l}(r)dr=1~{},$ (89)

then

$\displaystyle R^{\prime}$ $\displaystyle=$
$\displaystyle\chi^{\prime}r^{-1/2}-\frac{1}{2}\chi r^{-3/2}~{},$ (90)
$\displaystyle R^{\prime\prime}$ $\displaystyle=$
$\displaystyle\chi^{\prime\prime}r^{-1/2}-\chi^{\prime}r^{-3/2}+\frac{3}{4}\chi
r^{-5/2}~{}.$ (91)

Notice we have

$\displaystyle-\left[R^{\prime\prime}_{n,l}+\frac{R^{\prime}_{n,l}}{r}-\frac{l^{2}}{r^{2}}R_{n,l}\right]+\left[(v^{\prime})^{2}-v^{\prime\prime}-\frac{v^{\prime}}{r}\right]R_{n,l}=\frac{8\pi^{2}\Lambda_{n,l}}{H^{3}}R_{n,l}~{},$
(92)

and

$\displaystyle-\left[\chi^{\prime\prime}_{n,l}+\frac{1}{4r^{2}}\chi_{n,l}-\frac{l^{2}}{r^{2}}\chi_{n,l}\right]+\left[(v^{\prime})^{2}-v^{\prime\prime}-\frac{v^{\prime}}{r}\right]\chi_{n,l}=\frac{8\pi^{2}\Lambda_{n,l}}{H^{3}}\chi_{n,l}~{}.$
(93)

Then we can use $\chi_{n,l}$ to perform all the numerical calculations instead
of $R_{n,l}$.