justified

#  Constraining Mixed Dark-Matter Scenarios of WIMPs and Primordial Black
Holes from CMB and 21-cm observations

Hiroyuki Tashiro1 and Kenji Kadota2
1 Department of physics and astrophysics, Nagoya University, Nagoya 464-8602,
Japan
2 Center for Theoretical Physics of the Universe, Institute for Basic Science
(IBS), Daejeon, 34051, Korea

###### Abstract

We consider the dark matter (DM) scenarios consisting of the mixture of WIMPs
and PBHs and study what fraction of the total DM can be PBHs. In such
scenarios, PBHs can accrete the WIMPs and consequently enhance the heating and
ionization in the intergalactic medium due to WIMP annihilations. We
demonstrate that the CMB data can give the stringent bounds on the allowed PBH
fraction which are comparable or even tighter than those from the gamma-ray
data depending on the DM masses. For instance, the MCMC likelihood analysis
using the Planck CMB data leads to a bound on the PBH DM fraction with respect
to the total dark matter $f_{\rm PBH}\lesssim{\cal O}(10^{-10}\sim 10^{-8})$
for the WIMP mass $m_{\chi}\sim{\cal O}(10\sim 10^{3})$ GeV with the
conventional DM annihilation cross section $\langle\sigma v\rangle=3\times
10^{-26}~{}\rm cm^{3}/s$. We also investigate the feasibility of the global
21-cm signal measurement to provide the stringent constraints on the PBH
fraction.

##  I Introduction

There have been revived interests in the primordial black hole (PBH) dark
matter scenarios since the LIGO/Virgo detection of black-hole mergers Abbott
et al. (2016, 2019). While the parameter space for PBHs to account for the
total dark matter component has been narrowed by many other complementary
observation data such as those from gravitational lensing, the PBH can well be
a partial dark matter component Carr and Kühnel (2020); Green and Kavanagh
(2021); Carr et al. (2020a). For the dark matter (DM) candidate accounting for
the rest of the dark matter in the presence of such PBH partial dark matter,
the widely discussed weakly interacting massive particles (WIMPs) can be an
intriguing possibility. Such WIMP-PBH mixed dark matter scenarios lead to
ultracompact minihalos (UCMHs) around the PBHs (so-called ”dressed PBHs”), and
one can expect enhanced DM annihilation from those steep DM profiles around
the PBHs. In the previous literature, many used the gamma-ray data such as
those from Fermi Large Area Telescope (LAT) and pointed out the
incompatibility of the coexistence of WIMPs and PBHs due to the current
nonobservation of enhanced gamma-ray emission from those UCMHs Ricotti (2007);
Mack et al. (2007); Ricotti and Gould (2009); Gondolo and Silk (1999); Lacki
and Beacom (2010); Boucenna et al. (2018); Adamek et al. (2019); Eroshenko
(2016); Carr et al. (2020b); Cai et al. (2021); Delos et al. (2018); Kohri et
al. (2014); Bertone et al. (2019); Ando and Ishiwata (2015); Hertzberg et al.
(2021); Yang (2020); Zhang (2011). We note that the PBHs were produced in the
radiation-dominated epoch and the dark matter could be gravitationally bound
to PBHs to form the UCMHs around them by the CMB epoch $z\sim 10^{3}$ Ricotti
et al. (2008); Ricotti (2007); Mack et al. (2007); Adamek et al. (2019); Carr
and Kühnel (2020); Green and Kavanagh (2021); Carr et al. (2020a). One can
then expect the inevitable effects of the energy injection from the DM
annihilation on the CMB observables. The radiation emitted by the gas falling
onto the PBHs and their effects on the CMB (temperature/polarization
anisotropy and spectral distortions) have been actively discussed so far Ali-
Haïmoud and Kamionkowski (2017); Luis Bernal et al. (2017); Poulin et al.
(2017a); Abe et al. (2019), but the DM annihilation effects on the CMB in the
presence of the dressed PBHs have been less explored Carr and Kühnel (2020);
Green and Kavanagh (2021); Carr et al. (2020a). For instance a pioneering work
Ricotti et al. (2008) studied how the X-rays emitted by gas accretion onto
PBHs can modify the CMB observables by including the effects of the DM
accumulation around the PBHs on the gas accretion rate, but the DM
annihilation was not considered in their Markov-Chain Monte Carlo (MCMC)
likelihood analysis (see e.g. Ali-Haïmoud and Kamionkowski (2017); Luis Bernal
et al. (2017); Abe et al. (2019); Serpico et al. (2020); Carr and Kühnel
(2020); Green and Kavanagh (2021); Carr et al. (2020a); Kohri et al. (2014)
for more recent relevant works). We in this paper assume the Majorana DM
particles which can self-annihilate (a typical example is WIMP), and perform
the MCMC likelihood analysis on the PBH abundance by studying how the energy
injection by the DM annihilation from the dressed PBHs can affect the CMB
observables due to the change in the thermal and ionization history of the
Universe. Our CMB bounds on the allowed PBH fraction can give comparable or
better bounds than those from gamma-ray observations depending on the DM mass.
For instance, our MCMC analysis shows that the CMB bounds on PBH fraction
$f_{\rm PBH}$ (with respect to the total DM) are $f_{\rm PBH}\lesssim 3\times
10^{-10},~{}7\times 10^{-9}$, and $3\times 10^{-8}$ for $m_{\chi}=$ 10 GeV,
100 GeV and 1 TeV, respectively, assuming the s-wave annihilation into
$b\bar{b}$ with $\langle\sigma v\rangle=3\times 10^{-26}\rm cm^{3}/s$ (to be
compared with the corresponding Fermi gamma-ray data bounds $f_{\rm
PBH}\lesssim 10^{-9},2\times 10^{-9}$, and $4\times 10^{-9}$ Adamek et al.
(2019); Ando and Ishiwata (2015)). In addition to the CMB bounds using the
Planck data, we also present a brief discussion on the 21cm constraints using
the EDGES result, which is also affected by the energy injection from DM
annihilation Bowman et al. (2018).

The paper is organized as follows. Section II reviews the energy injection
rate due to the DM annihilation from the steep DM profile around a PBH.
Section III discusses how such energy injection from DM annihilation can
affect the thermal evolution of the intergalactic medium and performs the MCMC
likelihood analysis using the Planck data leading to the stringent bounds on
the allowed PBH fraction in the presence of WIMPs. We also discuss how the
global 21-cm signals can be affected in the mixed PBH-WIMP dark matter
scenarios. Section IV is devoted to the conclusion.

##  II WIMP annihilation from DM halo around PBHs

###  II.1 Dark matter halo profile

In the mixed-DM scenarios consisting of PBHs and WIMPs, PBHs can
gravitatinally bound nonrelativistic WIMP DM particles soon after their
formation and, as a result, dress a halo of WIMPs whose density profile is a
spike type. Here we construct the model of the DM halo around a PBH following
Ref. Adamek et al. (2019).

During the radiation-dominated epoch, the turnaround scale at a redshift $z$,
where the gravitational attraction from a PBH decouples a DM mass shell from
the Hubble flow, is obtained numerically Adamek et al. (2019),

$r_{\rm ta}(z)\approx(2GM_{\rm PBH}t^{2})^{1/3},$ (1)

where $t$ is a time corresponding to a redshift $z$.

We simply assume that, when the dark matter particles decouple from the Hubble
flow, the DM density of each mass shell matches the background density at its
turnaround time. This simple assumption provides the steep density profile
around a PBH, and the density profile at the redshift of matter-radiation
equality, $z_{\rm eq}$, is

$\rho(r)=\overline{\rho}_{\rm DM}(z_{\rm eq})\left(\frac{r}{r_{\rm ta}(z_{\rm
eq})}\right)^{-9/4}\quad{\rm for}~{}r<r_{\rm ta}(z_{\rm eq}),$ (2)

where we assume that the DM fraction of the WIMP is almost unity,
$f_{\chi}\approx 1$, and $\overline{\rho}_{\rm DM}(z)$ is the background DM
density at a redshift $z$. After the time of matter-radiation equality, the
dark matter halo can grow by the secondary infalling into the halo in Eq. (2)
Bertschinger (1985). Even in this stage, it is confirmed that the profile
shape in Eq. (2) is not disrupted and the infalling matter creates the NFW
profile at large radii Adamek et al. (2019). Therefore, we assume that the
density profile in Eq. (2) is valid even at $z<z_{\rm eq}$.

The WIMP can annihilate away, in particular, in the central region of a halo
because of its high density and soften the density cusp. The maximum possible
WIMP density in the halos is evaluated as Berezinsky et al. (1992)

$\rho_{\rm max}(z)\approx\frac{m_{\chi}}{\langle\sigma v\rangle t}.$ (3)

This maximum density gives the flat DM density inner region of a halo
extending to $r_{\rm cut}$, where the power-law profile starts. Considering
the profile in Eq. (2), we obtain

$r_{\rm cut}(z)=\left(\frac{\rho_{\rm max}(z)}{\overline{\rho}_{\rm DM}(z_{\rm
eq})}\right)^{-4/9}r_{\rm ta}(z_{\rm eq}).$ (4)

We can hence model the dark matter density profile around a PBH at a redshift
$z$ as

$\rho(z,r)=\begin{cases}\rho_{\rm max}(z)\quad&{\rm for}~{}r<r_{\rm cut}(z)\\\
\rho_{\rm max}(z)(r/r_{\rm cut}(z))^{-9/4}\quad&{\rm for}~{}r_{\rm cut}(z)\leq
r<r_{\rm ta}(z_{\rm eq}).\\\ \end{cases}$ (5)

We focus on the scenarios where the DM kinetic energy is negligible compared
with the potential energy around a PBH, for which the above profile
$\rho(r)\propto r^{-9/4}$ is numerically verified Adamek et al. (2019);
Serpico et al. (2020). One can neglect the DM kinetic energy when the
potential energy dominates, at least, at $r_{\rm cut}(z_{\rm eq})$ satisfying
Adamek et al. (2019)

$\frac{E_{\rm k}}{E_{\rm p}}=\left(\frac{T_{\rm
KD}}{m_{\chi}}\right)\left(\frac{t_{\rm KD}}{t_{\rm
eq}}\right)\left(\frac{\rho_{\rm max}(z_{\rm eq})}{\overline{\rho}_{\rm
DM}(z_{\rm eq})}\right)^{2/9}\left(\frac{r_{\rm ta}(z_{\rm eq})}{GM_{\rm
PBH}}\right)\lesssim 0.01.$ (6)

This equation leads to the relation,

$M_{\rm PBH}\gtrsim 6.5\times 10^{-4}~{}M_{\odot}~{}\left(\frac{\langle\sigma
v\rangle}{3\times 10^{-26}\rm
cm^{3}/s}\right)^{-1/3}\left(\frac{m_{\chi}}{10~{}{\rm GeV}}\right)^{-73/24}.$
(7)

The cases when the effect of DM kinetic energy is non-negligible can be
heavily dependent on the nature of DM kinetic decoupling and are left for
future work Loeb and Zaldarriaga (2005); Bertschinger (2006); Gondolo et al.
(2012); Profumo et al. (2006); Gondolo and Kadota (2016); Green et al. (2004,
2005); Bringmann and Hofmann (2007); Eroshenko (2016); Boucenna et al. (2018).

###  II.2 Energy injection of WIMP annihilation

The rate of DM annihilation is proportional to the density squared. Using the
dark matter halo profile in Eq. (5), the injected energy from a halo around a
PBH into the cosmic plasma is given by

$\displaystyle\left.\frac{d\cal E}{dt}\right|_{\rm single}$
$\displaystyle=f_{\rm ann}(z)m_{\chi}\langle\sigma v\rangle\int_{V_{\rm
halo}}dV\left(\frac{\rho(r)}{m_{\chi}}\right)^{2}$
$\displaystyle\approx{4\pi}f_{\rm ann}(z)\frac{\langle\sigma
v\rangle}{m_{\chi}}\rho_{\rm max}^{2}(z)r_{\rm cut}^{3}(z)={3}M_{\rm
PBH}f_{\rm ann}(z)\frac{\Omega_{\rm DM}}{\Omega_{\rm
M}}\left[\frac{\langle\sigma v\rangle}{m_{\chi}}H^{2}(z)\overline{\rho}_{\rm
DM}(z_{\rm eq})\right]^{1/3},$

where $V_{\rm halo}$ is the volume of a dark matter halo whose size is given
by the turnaround radius at each redshift. In the equation, $f_{\rm ann}(z)$
represents the fraction of the annihilation energy absorbed by the plasma in
the on-the-spot approximation. The function $f_{\rm ann}(z)$ depends on the
WIMP mass and the annihilation channel, and although $f_{\rm ann}(z)$ also
depends on the redshift, we use the constant $f_{\rm ann}(z)=f_{\rm ann}$
hereafter Galli et al. (2013)). Now, we assume that PBHs have a monochromatic
mass function with the mass $M_{\rm PBH}$, and the energy density fraction of
PBHs to the total dark matter is $f_{\rm PBH}$ (the current data require
$f_{\rm PBH}\ll 1$, as discussed in the next section). The number density of
PBHs at a redshift $z$ is

$n_{\rm PBH}(z)=\frac{f_{\rm PBH}}{M_{\rm PBH}}\overline{\rho}_{\rm DM}(z).$
(9)

Accordingly, the injected energy density due to the DM annihilation from total
halos is

$\left.\frac{d^{2}E}{dVdt}\right|_{\rm PBHs}=n_{\rm PBH}(z)\left.\frac{d\cal
E}{dt}\right|_{\rm single}\propto\left(\frac{\langle\sigma
v\rangle}{m_{\chi}}\right)^{1/3}(1+z)^{4}f_{\rm ann}f_{\rm PBH},$ (10)

where, in order to obtain the last expression, we assume $t\approx 1/H(z)$ and
the matter-dominated epoch. Note that the injected energy density does not
depend on $M_{\rm PHB}$ because $n_{\rm PBH}(z)\propto M_{\rm PBH}^{-1}$ and
${d\cal E}/{dt}\propto M_{\rm PBH}$.

The smooth-background DM also contributes to the energy injection of DM
annihilation. Therefore, the total energy injection rate per volume due to the
DM annihilation is

$\frac{d^{2}E}{dVdt}(z)=\left.\frac{d^{2}E}{dVdt}\right|_{\rm
sm}+\left.\frac{d^{2}E}{dVdt}\right|_{\rm PBHs}.$ (11)

Here, the subscript ”sm” denotes the smooth-background contributions,

$\left.\frac{d^{2}E}{dVdt}\right|_{\rm sm}=f_{\rm ann}\frac{\langle\sigma
v\rangle}{m_{\chi}}\overline{\rho}_{\rm DM}^{2}(z).$ (12)

It would be useful to introduce the boost factor, $B(z)$, which represents the
ratio of the PBH contribution to the smooth-background one,

$B(z)=\left.\frac{d^{2}E}{dVdt}\right|_{\rm
PBHs}{\bigg{/}}\left.\frac{d^{2}E}{dVdt}\right|_{\rm sm}\approx
0.74\times\left(\frac{\langle\sigma v\rangle/m_{\chi}}{3\times 10^{-27}{\rm
cm^{3}/s/GeV}}\right)^{-2/3}\left(\frac{1+z}{100}\right)^{-2}\left(\frac{f_{\rm
PBH}}{10^{-10}}\right).$ (13)

The boost factor tells us that the contribution of PBHs relatively increases
at lower redshifts. Currently the measurement of CMB anisotropy provides the
bound, $f_{\rm ann}\langle\sigma v\rangle/m_{\chi}\lesssim 3\times 10^{-27}\rm
cm^{3}s^{-1}GeV^{-1}$ for the s-wave annihilating DM, considering only the
smooth-background contribution Slatyer (2016); Jungman et al. (1996).
According to Eq. (13), when $\langle\sigma v\rangle/m_{\chi}=3\times
10^{-27}{\rm cm^{3}/s/GeV}$, DM halos around PBHs with $f_{\rm PBH}\approx
10^{-10}$ can contribute the modification of CMB anisotropy at the same level
as the smooth-background DM at lower redshift $z<100$. One can hence infer
that CMB observations could possibly provide a constraint on the PBH abundance
$f_{\rm PBH}<{\cal O}(10^{-10})$ which could be competitive to the bounds from
the gamma-ray observations.

##  III Constraints on the PBH abundance from cosmological observations

The injected energy by the WIMP annihilation is absorbed into cosmic plasma
through various channels including collisional heating and ionization.
Therefore, the injected energy can modify the evolution of the ionization
fraction $x_{e}$ and the baryon temperature. Including the effect of the WIMP
annihilation, we can write down the evolution of the ionization fraction Chen
and Kamionkowski (2004),

$(1+z)\frac{dx_{e}}{dz}=\frac{1}{H(z)}\left[R_{s}(z)-I_{s}(z)-I_{x}(z)\right],$
(14)

where $R_{s}$ and $I_{s}$ are the standard primordial hydrogen recombination
rate and ionization rate, respectively, and $I_{x}$ represents the ionization
contribution of the WIMP annihilation Peebles (1968); Zeldovich et al. (1968).

The baryon temperature evolution in the presence of the WIMP annihilation can
be obtained from

$(1+z)\frac{dT_{k}}{dz}=2T_{k}+\frac{8\sigma_{T}a_{R}T_{CMB}^{4}}{3m_{e}cH(z)}\frac{x_{e}(T_{k}-T_{\gamma})}{(1+f_{\rm
He}+x_{e})}-\frac{2}{3k_{B}H(z)}\frac{K_{h}(z)}{(1+f_{\rm He}+x_{e})},$ (15)

where $K_{h}$ is the extra heating term by the DM annihilation.

The WIMP contributions, $I_{x}$ and $K_{h}$, are related to the energy
injection given in Eq (11)111Although we neglect to discuss it in this paper,
dark matter halos due to the standard scale-invariant adiavatic spectrum also
can enhance the DM annihilation in particular, during the epoch of
reionization (for reference, see Hütsi et al. (2009)).,

$\displaystyle I_{x}=\frac{\chi_{x}}{{n_{\rm H}(z)}}\frac{d^{2}E}{dVdt}(z),$
$\displaystyle K_{h}=\frac{\chi_{h}}{E_{i}{n_{\rm
H}(z)}}\frac{d^{2}E}{dVdt}(z),$ (16)

where $E_{i}$ is the ionization energy of hydrogen, while $\chi_{x}$ and
$\chi_{h}$ provide the fractions of energy consumed for the ionization and
heating. These fractions mainly depend on the ionization fraction. For
example, in the neutral gas case, it is known that roughly one third of the
energy is used for the ionization, another third goes into the excitation of
gas atoms and the rest is consumed for the heating Shull and van Steenberg
(1985). Here, for simplicity, we assume that the injected energy is quickly
damped into the plasma and used for ionizing and heating, i.e., on-the-spot
approximation and we adopt the functions of $\chi_{x}$ and $\chi_{h}$ provided
by Ref. Poulin et al. (2017b), which are the fitting formulas of the results
in Ref. Galli et al. (2013).

Including the primordial helium contribution, we obtain the ionization and
thermal histories of baryons from Eqs. (14) and (15) using HyRec Ali-Haïmoud
and Hirata (2011). Figure 1 shows the ionization fraction (left panel) and the
baryon temperature (right panel) as functions of redshift. Here we use
$\langle\sigma v\rangle/m_{\chi}=3\times 10^{-28}\rm cm^{3}/s/GeV$. The
current CMB bound (obtained without assuming PBHs) is $\langle\sigma
v\rangle/m_{\chi}\lesssim 3\times 10^{-27}\rm cm^{3}s^{-1}GeV^{-1}$ for the
s-wave annihilating DM and we conservatively use the values $\langle\sigma
v\rangle/m_{\chi}\lesssim 3\times 10^{-28}\rm cm^{3}s^{-1}GeV^{-1}$ for the
sake of illustration Slatyer (2016). Such a choice of $\langle\sigma
v\rangle/m_{\chi}$ well below the current CMB bound ensures that there can be
an enough room for the PBH contribution to dominate the smooth DM background
contribution as inferred from Eq. (13). In the figure, in order to show the
dependence on $f_{\rm PBH}$, the solid blue, orange, green, and red lines
represent the evolutions for $f_{\rm PBH}=10^{-8}$, $f_{\rm PBH}=10^{-9}$,
$f_{\rm PBH}=10^{-10}$ and $f_{\rm PBH}=0$, respectively. For comparison, we
plot the results for the no-annihilation case in a black dashed line. As shown
in Eq. (10) a large $f_{\rm PBH}$ increases the energy injection from dark
matter halos around PBHs. Therefore, a large $f_{\rm PBH}$ provides a strong
impact on both the ionization fraction and the thermal evolution. It induces
early reionization and heating of baryons. Such modifications on these
evolutions can be probed by cosmological observations including CMB and
redshifted 21-cm signals. In the next section, we discuss the constraint on
$f_{\rm PBH}$ from these observations.

---
Figure 1: The evolution of the ionization fraction (left panel) and the baryon
temperature (right panel). We set $\langle\sigma v\rangle/m_{\chi}=3\times
10^{-28}\rm cm^{3}/s/GeV$. From top to bottom, the solid lines represent the
evolutions for $f_{\rm PBH}=10^{-8}$, $f_{\rm PBH}=10^{-9}$, $f_{\rm
PBH}=10^{-10}$ and $f_{\rm PBH}=0$ (No PBHs). For reference, we plot the
evolution without DM annihilation in dashed lines.

###  III.1 CMB anisotropy

The existence of PBHs can enhance the WIMP annihilation and heat up and ionize
baryons before the conventional epoch of reionization as shown in the previous
section. Since such early-epoch energy injection contributes to the increase
in the optical depth of CMB from the last scattering surface, the measurement
of CMB anisotorpy can provide useful information on the PBH abundance with
WIMP annihilation.

We calculate the CMB anisotropies with the public Boltzmann code CLASS Blas et
al. (2011) modified to incorporate Eq. (11). We plot the angular power spectra
of the CMB E-mode polarization with different $f_{\rm PBH}$ in Fig. 2. In this
figure, we assume the ”tanh”-shaped reionization history and, according to the
Planck best-fit cosmological parameters Planck Collaboration et al. (2020), we
set $z_{\rm reio}=7.68$, where the ionization fraction becomes $x_{e}=0.5$ at
$z_{\rm reio}$. Large $f_{\rm PBH}$ induces the early reionization and the
enhancement of the optical depth from the last scattering surface. As a
result, the modification arises on the so-called reionization bump visible at
$\ell\lesssim 20$ in the CMB polarization angular power spectrum. Figure 2
shows that the tail of the reionization bump is lifted as $f_{\rm PBH}$
becomes large. We find that $f_{\rm PBH}=10^{-9}$ can enhance the amplitude of
the reionization bump by $\sim 30$ percent.

In order to obtain the limit on $f_{\rm PBH}$ from CMB anisotropy
observations, we perform the MCMC analysis using Monte Python Audren et al.
(2013) with the modified CLASS. For CMB observation data, we use the baseline
likelihood (TTTEEE-lowl-lowE) from the 2018 data release Aghanim et al.
(2020). The cosmological parameter set for the analysis is
$\\{\omega_{b},\omega_{d},100\theta_{s},\ln(10^{10}A_{s}),n_{s},z_{\rm
reio},f_{\rm PBH}\\}$. We show the 2D contour plot for $z_{\rm reio}$ and
$f_{\rm PBH}$ in Fig. 3.

We summarize our constraint on $f_{\rm PBH}$ for the 95% confidence level in
Table 1. CMB constraints depend on the combination of the annihilation cross
section $\langle\sigma v\rangle$ and the mass $m_{\chi}$, and include the
uncertain model parameter $f_{\rm ann}$. When $\langle\sigma
v\rangle/m_{\chi}<3\times 10^{-28}{\rm cm^{3}/s/GeV}$ with $f_{\rm PBH}={\cal
O}(10^{-9})$, the boost factor given by Eq. (13) is $B(z)\gg 1$ around $z\sim
20$. This means that we can neglect the smooth-background DM contribution in
Eq. (11) and it is enough to consider only the PBH halo contribution in these
parameter regions. Therefore, according to Eq. (10), the constraint on $f_{\rm
PBH}$ can be rewritten in

$f_{\rm ann}f_{\rm PBH}<5.5\times 10^{-9}\left(\frac{\langle\sigma
v\rangle/m_{\chi}}{3\times 10^{-29}{\rm cm^{3}/s/GeV}}\right)^{-1/3}.$ (17)

This constraint is applicable when the PBH contribution dominates the smooth-
background contribution [the explicit contribution ratio is given by Eq.
(13)].

The current Planck bound gives $\langle\sigma v\rangle/m_{\chi}\lesssim
3\times 10^{-27}\rm cm^{3}s^{-1}GeV^{-1}$ for the s-wave annihilating DM, and
this in turn requires $m_{\chi}\gtrsim 10~{}{\rm GeV}$ for the canonical
thermal WIMP cross section $\langle\sigma v\rangle=3\times 10^{-26}$ $\rm
cm^{3}/s$ Slatyer (2016). Our constraint includes the uncertain model
parameter $f_{\rm ann}$ which depends on the WIMP mass and the annihilation
channel. As shown in Fig. 1 (and also as can be inferred by Eq. (13)), the
energy injection below $z\sim 500$ is important in our model. To evaluate the
impact of $f_{\rm ann}$ on the constraint, we calculate the averaged $f_{\rm
ann}$ in $10<z<500$ using the table of $f_{\rm ann}$ in Ref. Slatyer et al.
(2009) for the case of WIMPs annihilating into $e^{+}e^{-}$, $\mu^{+}\mu^{-}$
and $b\bar{b}$. Using the averaged $f_{\rm ann}$, we obtain the constraint for
the canonical thermal WIMP scenario in Table 2 and plot them in Fig. 4 in
solid lines for the $b\bar{b}$ and dashed lines for the $e^{+}e^{-}$
annihilation channels. In previous works, the PBH abundance with the
annihilating WIMPs is constrained by the observations of Galactic and
extragalactic background gamma-ray flux Lacki and Beacom (2010); Boucenna et
al. (2018); Adamek et al. (2019); Eroshenko (2016); Carr et al. (2020b); Cai
et al. (2021); Delos et al. (2018); Kohri et al. (2014); Bertone et al.
(2019); Ando and Ishiwata (2015); Hertzberg et al. (2021); Yang (2020); Zhang
(2011). Our constraints are much stronger than those from the Galactic gamma-
ray background and comparable with an extragalactic one. For comparison, we
show the excluded regions by extragalactic background in Ref. Adamek et al.
(2019) as the colored regions in Fig. 4. Comparing with Ref. Adamek et al.
(2019), the CMB anisotropy measurement provides a tighter constraint for small
WIMP masses $m_{\chi}<100$ GeV, while, for heavier masses $m_{\chi}>100$ GeV,
the constraint from the CMB anisotropy measurement is weaker than that from
the extragalactic gamma-ray observations.

Figure 2: Angular power spectrum of CMB E-mode polarization. From the top to the bottom, the solid lines are the power spectra for $f_{\rm PBH}=10^{-8}$ and $f_{\rm PBH}=10^{-9}$. For reference, we show the angular power spectrum without the DM annihilation with a dashed line. Figure 3: 2D plot for $z_{\rm reio}$ and $f_{\rm PBH}$ from the MCMC analysis with Planck 2018 data. Here we set $\langle\sigma v\rangle/m_{\chi}=3\times 10^{-28}\rm cm^{3}/s/GeV$. $\langle\sigma v\rangle/m_{\chi}~{}[\rm cm^{3}/s/GeV$] |  $f_{\rm ann}f_{\rm PBH}$ (95 % C.L.)
---|---
$3\times 10^{-29}$ | $<5.5\times 10^{-9}$
$3\times 10^{-28}$ | $<1.6\times 10^{-9}$
$3\times 10^{-27}$ | $<8.8\times 10^{-11}$
Table 1: The constraints on $f_{\rm PBH}$ from Planck 2018 data for different values of $\langle\sigma v\rangle/m_{\chi}$. $m_{\chi}$ |  $f_{\rm PBH}$ ($\chi\chi\rightarrow\gamma\gamma$) |  $f_{\rm PBH}$ ($\chi\chi\rightarrow e^{+}e^{-}$) |  $f_{\rm PBH}$ ($\chi\chi\rightarrow\mu^{+}\mu^{-}$) |  $f_{\rm PBH}$ ($\chi\chi\rightarrow b\bar{b}$)
---|---|---|---|---
$1~{}$TeV | $<5.5\times 10^{-9}$ | $<1.4\times 10^{-8}$ | $<3.7\times 10^{-8}$ | $<2.7\times 10^{-8}$
$100~{}$GeV | $<1.6\times 10^{-9}$ | $<3.6\times 10^{-9}$ | $<9.4\times 10^{-9}$ | $<6.6\times 10^{-9}$
$10~{}$GeV | $<8.8\times 10^{-11}$ | $<1.4\times 10^{-10}$ | $<3.8\times 10^{-10}$ | $<3.0\times 10^{-10}$
Table 2: The constraints on $f_{\rm PBH}$ in the canonical WIMP annihilation
cross section, $\langle\sigma v\rangle=3\times 10^{-26}~{}\rm cm^{3}/s$. Here
we consider WIMPs annihilating into $\gamma\gamma$, $e^{+}e^{-}$,
$\mu^{+}\mu^{-}$ and $b\bar{b}$. Figure 4: The constraint on $f_{\rm PBH}$
from Planck 2018 data for the canonical WIMP annihilation cross section
$\langle\sigma v\rangle=3\times 10^{-26}\rm cm^{3}/s$. Here we consider that
WIMPs annihilate into $b\bar{b}$ and $e^{+}e^{-}$ shown by the solid and
dashed lines, respectively. From top to bottom, we set the WIMP mass
$m_{\chi}=1~{}$TeV, 100 GeV, 10 GeV. The colored regions are ruled out by the
extragalactic gamma-ray observations for the $b\bar{b}$ annihilation channel
Adamek et al. (2019).

###  III.2 Global 21-cm signal

The redshifted 21-cm signal depends on the thermal state of the intergalactic
medium (IGM). Therefore, the measurement of its signal from high redshifts can
reveal the thermal history of the IGM. Through the investigation of the
effects of PBHs and WIMP annihilation on the thermal history, the 21-cm signal
is expected to provide a constraint on the abundance of PBHs Tashiro and
Sugiyama (2013); Gong and Kitajima (2017); Mena et al. (2019) and the WIMP
annihilation Valdés et al. (2007); Natarajan and Schwarz (2009) at high
redshifts. Recently Ref. Yang (2020) has studied the impact of the WIMP
annihilation on the global (all-sky averaged) 21-cm signal in the mixed DM
scenarios consisting of PBHs and a thermal canonical WIMP model with
$m_{\chi}=100~{}$GeV. Here we revisit the evolution of the global 21-cm signal
in the same mixed DM scenarios, covering a wider range of WIMP masses.

In 21-cm observations, the strength of the signal is measured in terms of so-
called differential brightness temperature, which is the difference of the
21-cm brightness temperature from the CMB one (for a review, see Ref.
Furlanetto et al. (2006)). The global differential brightness temperature from
a redshift $z$ is calculated at Madau et al. (1997); Pritchard and Loeb (2010)

$\delta
T_{b}(z)=\frac{3}{32\pi}\frac{hc^{3}A_{10}}{k_{B}\nu_{0}^{2}}\frac{x_{\rm
HI}n_{H}}{(1+z)H(z)}\left(1-\frac{T_{\gamma}}{T_{S}}\right),$ (18)

where $A_{10}$ is the spontaneous emission coefficient of the 21-cm
transition, $A_{10}=2.85\times 10^{-15}\rm s^{-1}$, and $T_{s}$ is the spin
temperature of the neutral hydrogen hyperfine structure. The spin temperature
is determined by the balance between the excitation and deexcitation in the
hyperfine structure,

$T_{S}=\frac{T_{\gamma}+y_{\rm kin}T_{k}}{1+y_{\rm kin}},$ (19)

where $y_{\rm kin}$ is the efficiency ratio between the absorption of CMB
photons and the thermal collisions in the hyperfine transition. In order to
obtain $y_{\rm kin}$, we take the approximated analytical form of $y_{\rm
kin}$ in Ref. (Kuhlen et al., 2006). In Eq. (18), we ignore the contribution
from Ly-$\alpha$ coupling, because it becomes efficient after the formation of
the first stars and galaxies.

The left panel of Fig. 5 shows the effect of WIMP annihilation on the
evolution of spin and baryon temperatures with $f_{\rm PBH}=10^{-8}$ with
$\langle\sigma v\rangle/m_{\chi}=3\times 10^{-28}~{}[\rm cm^{3}/s/GeV]$ in the
solid lines. For comparison, we also give the evolutions in the standard
cosmology (no-annihilation) case using dashed lines. WIMP annihilation heats
up baryons and causes the baryon thermal evolution to deviate from the
adiabatic evolution, $T_{k}\propto(1+z)^{2}$. Therefore, the spin temperature
is also larger than in the ”no-annihilation” case. The effective heating due
to the annihilation can make the baryon temperature exceed the CMB
temperature. In this case, the spin temperature also becomes larger than the
CMB temperature.

We plot the dependence of the the global differential brightness temperature
on the PBH abundance in the right panel of Fig. 5. A positive amplitude of the
differential brightness temperature means an emission line on the CMB
frequency spectrum and a spin temperature is larger than the CMB temperature.
On the other hand, a negative one represents the absorption line for CMB and a
spin temperature smaller than the CMB temperature. When $f_{\rm PBH}=10^{-8}$,
the signal shifts from absorption to emission at $z_{\rm tr}\approx 27$. As
$f_{\rm PBH}$ decreases, the annihilation effects become inefficient. As a
result, the transition redshift, $z_{\rm tr}$, also becomes small. We found
that, if $f_{\rm PBH}<2\times 10^{-9}$, the baryon temperature cannot exceed
the CMB temperature before the epoch of reionization, ($z>7$). In the case of
$f_{\rm PBH}<2\times 10^{-9}$, the sign of the signal is always negative. When
$f_{\rm PBH}$ becomes smaller, the evolution of the signal approaches the one
in the ”no-annihilation” case.

---
Figure 5: Left panel: The spin and baryon temperature as functions of
redshifts. The solid blue and orange lines represent the spin and baryon
temperatures for $f_{\rm PBH}=10^{-8}$. For comparison, we plot their
evolutions for no annihilation DM case in the dashed lines. Right panel: The
evolution of the global differential brightness temperature with different
$f_{\rm PBH}$. From top to bottom, the solid lines are for $f_{\rm
PBH}=10^{-8}$, $f_{\rm PBH}=10^{-9}$ and $f_{\rm PBH}=10^{-10}$. For
reference, the dashed line shows the evolution in the case of no annihilation
DM.

Recently, EDGES reported that they have detected the global absorption signals
of redshifted 21-cm lines from the redshift range between $z\sim 21$ and
$z\sim 15$ Bowman et al. (2018). If this measurement is confirmed, the baryon
temperature is lower than the CMB temperature until $z\sim 15$. Therefore,
this measurement can provide a constraint on the heating source. In Fig. 6, we
represent the relation between $z_{\rm tr}$ and $f_{\rm PBH}$. Small $f_{\rm
PBH}$ provides low $z_{\rm tr}$. Therefore, from this figure, we can conclude
that the EDGES absorption signal at $z\sim 15$ can give the limit on the PBH
abundance of $f_{\rm PBH}<{\cal O}(10^{-9})$ for $10~{}{\rm
GeV}<m_{\chi}<1~{}{\rm TeV}$ with the canonical thermal WIMP cross section
$\langle\sigma v\rangle=3\times 10^{-26}\rm cm^{3}/s$.

Our constraint is weaker than the one in Ref. Yang (2020). To obtain our
constraint, we take the criteria for the constraint which is the transition
redshift from the absorption to the emission. The amplitude of the signals
detected by EDGES is $500^{+500}_{-200}~{}\rm mK$ Bowman et al. (2018). Taking
this at face value, Ref. Yang (2020) requires the condition that the
differential brightness temperature is less than $\delta T_{\rm b}<-100~{}\rm
mK$. In order to obtain such a large absorption signal, the contribution of
the Ly-$\alpha$ coupling, which we ignore in Eq. (19), is required. The
efficiency of the coupling depends on the formation history of luminous
objects in the Universe and has a large theoretical uncertainty. Assuming the
strong Ly-$\alpha$ coupling, $T_{S}=T_{k}$, we calculate the 21-cm signal
evolution again. When we adopt the criteria, $\delta T_{\rm b}<100~{}\rm mK$,
during the EDGES observation redshifts, we obtain the constraints, $f_{\rm
PBH}<4\times 10^{-10}$ and $<2\times 10^{-9}$ for $m_{\chi}=100~{}$GeV and
$1~{}$TeV with $\langle\sigma v\rangle=3\times 10^{-26}\rm cm^{3}/s$.
Therefore, 21-cm global observations have a potential to provide as tight
constraints as extragalactic gamma-ray observation even for a large-mass WIMP.
Note that the criterion, $\delta T_{\rm b}<-100~{}\rm mK$, can exclude the
canonical thermal WIMP cross section for $m_{\chi}<100~{}$GeV Cheung et al.
(2019).

Figure 6: The dependence of the transition redshift, $z_{\rm tr}$, from the
absorption to emission on the PBH fraction $f_{\rm PBH}$.

##  IV conclusion

In this paper, we have investigated the constraint on the PBH abundance in the
presence of self-annihilating WIMPs. In previous works, the stringent
constraint on the PBH in this mixed dark matter scenario has been obtained
from the Galactic and extragalactic gamma-ray data. We focused on the CMB
anisotropy measurements and global 21-cm observations. If PBHs exist, the
steep WIMP DM halo is created around a PBH and the annihilation is enhanced
inside such a DM halo. The energy released in the annihilation can heat and
reionize diffuse background baryon gas outside DM halos and modify the thermal
history of diffuse baryon gas. This modification can make a deviation in the
CMB anisotropies and 21-cm global signals from those in the standard
$\Lambda$CDM model.

In order to obtain the constraint from the CMB measurement, we have studied
the effect of the WIMP annihilation from dark matter halos around PBHs and
performed the MCMC analysis with the latest Planck data. Our constraints from
the Planck data, for $b\bar{b}$ annihilation channel examples, are $f_{\rm
PBH}\lesssim 3\times 10^{-10},7\times 10^{-9}$, and $3\times 10^{-8}$
respectively for $m_{\chi}=$ 10 GeV, 100 GeV and 1 TeV, respectively, for the
canonical thermal annihilation cross section $\langle\sigma v\rangle=3\times
10^{-26}\rm cm^{3}/s$. These bounds are stronger than the limits from the
Galactic gamma-ray background and comparable with the one from the
extragalactic gamma-ray background. Not all of energy produced in the
annihilation can be absorbed into baryon gas. The efficiency depends on the
WIMP mass and the annihilation channels. We have found that, in the mixed DM
scenario with PBHs and WIMPs, the thermal history of baryon gas is sensitive
to the efficiency in the redshift range $10<z<500$. In this paper, we have
presented the constraints for the annihilation into $\gamma\gamma$,
$e^{+}e^{-}$, $\mu^{+}\mu^{-}$ and $b\bar{b}$. Using the averaged efficiency
factor in this redshift range, one can easily convert our constraint to the
one in the different annihilation channels.

Our studies focused on the parameter space for which the UCMH around PBH was
numerically verified to possess the steep density profile $\rho(r)\propto
r^{-9/4}$. The dark matter profile around the PBH where the DM velocity
dispersion cannot be ignored is heavily model dependent (e.g. on the DM
properties such as the nature of DM kinetic decoupling Loeb and Zaldarriaga
(2005); Bertschinger (2006); Gondolo et al. (2012); Profumo et al. (2006);
Gondolo and Kadota (2016); Green et al. (2004, 2005); Bringmann and Hofmann
(2007)) and the dedicated numerical simulations have not been performed yet
even though the analytical estimation has been done assuming the
Maxwell–Boltzmann distribution for the DM velocity Eroshenko (2016); Boucenna
et al. (2018). The bounds on $f_{PBH}$ become much weaker for those less steep
DM profiles in existence of the nonradial motion of DM bound to the PBH Carr
et al. (2020b). More detailed numerical studies where one needs to account for
the DM velocity distributions in their accretion onto the PBHs are left for
future work.

We have also studied the constraint from the global 21-cm signals. Before the
EoR, the global 21-cm signals are predicted as absorption signals on the CMB
frequency spectrum. However the baryon gas heating by the WIMP annihilation
can shift 21-cm signals from absorption to emission. Recently the EDGES
experiment has reported the detection of the absorption signals from the
redshifts, $15\lesssim z\lesssim 25$. Motivated by the EDGES report, we have
adopted the criterion that the global 21-cm signals cannot turn into the
emission until $z<15$. The obtained constraint is $f_{\rm PBH}<{\cal
O}(10^{-9})$ for $10~{}{\rm GeV}<m_{\chi}<1~{}{\rm TeV}$ with $\langle\sigma
v\rangle=3\times 10^{-26}~{}\rm cm^{3}/s$. Although this constraint is
slightly weaker than the CMB constraint, we have demonstrated that further
development in both theory and observation of the 21-cm signals before the EoR
can provide a stringent constraint which is better than the limit from the
extragalactic gamma-ray background. Assuming the strong Ly-$\alpha$ coupling
limit and the criterion, $\delta T_{\rm b}<100~{}\rm mK$, the constraints are
improved to $f_{\rm PBH}<4\times 10^{-10}$ and $<2\times 10^{-9}$ for
$m_{\chi}=100~{}$GeV and $1~{}$TeV with the canonical WIMP annihilation cross
section $\langle\sigma v\rangle=3\times 10^{-26}~{}\rm cm^{3}/s$.

###### Acknowledgements.

This work was supported by the Institute for Basic Science (IBS-R018-D1) and
Grants-in-Aid for Scientific Research from JSPS (21K03533).

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# Polarization of Autonomous Generative AI Agents
Under Echo Chambers

Masaya Ohagi<EMAIL_ADDRESS>SB IntuitionsTokyoJapan

###### Abstract.

Online social networks often create echo chambers where people only hear
opinions reinforcing their beliefs. An echo chamber often generates
polarization, leading to conflicts caused by people with radical opinions,
such as the January 6, 2021, attack on the US Capitol. The echo chamber has
been viewed as a human-specific problem, but this implicit assumption is
becoming less reasonable as large language models, such as ChatGPT, acquire
social abilities. In response to this situation, we investigated the potential
for polarization to occur among a group of autonomous AI agents based on
generative language models in an echo chamber environment. We had AI agents
discuss specific topics and analyzed how the group’s opinions changed as the
discussion progressed. As a result, we found that the group of agents based on
ChatGPT tended to become polarized in echo chamber environments. The analysis
of opinion transitions shows that this result is caused by ChatGPT’s high
prompt understanding ability to update its opinion by considering its own and
surrounding agents’ opinions. We conducted additional experiments to
investigate under what specific conditions AI agents tended to polarize. As a
result, we identified factors that strongly influence polarization, such as
the agent’s persona. These factors should be monitored to prevent the
polarization of AI agents.

echo chambers, polarization, large language model, agents

††ccs: Computing methodologies Multi-agent systems††ccs: Information systems
Social networks

##  1\. Introduction

With the development of social network service platforms, where people tend to
see only the information they want to see, it is becoming easier for people to
find themselves in echo chambers (Bessi, 2016; Gillani et al., 2018). An echo
chamber refers to an environment in which people mainly encounter opinions
that reinforce their own beliefs (Ruiz and Nilsson, 2023; Cinelli et al.,
2021). Such an environment causes an echo chamber effect, where opinions tend
towards more extreme stances. This effect induces polarization in society,
which refers to the division and clashes between groups with extreme stances
(Baumann et al., 2020). Polarization is behind many social problems, such as
the spread of misinformation during COVID-19 and the attack on the US Capitol
on January 6, 2021 (Villa et al., 2021a; Munn, 2021).

Figure 1. Overview image of our hypothesis: “Autonomous AI agents based on
generative large language models can cause polarization under echo chambers.”

Existing studies on the echo chamber have implicitly assumed that echo chamber
effects are caused only by humans and focused solely on human behavior
(Németh, 2022; Tucker et al., 2018). However, with the advent of large
language models (LLMs) such as ChatGPT (Ouyang et al., 2022), this assumption
may no longer hold true. Recent studies (Park et al., 2023; Qian et al., 2023)
have shown that ChatGPT-equipped agents can communicate as members of a
virtual society and collaborate towards common goals, such as game production.
Additionally, algorithms have been proposed to adapt agents to situations not
encountered during training, making it possible for autonomous agents to adapt
themselves to their surroundings (Krishna et al., 2022). Given these
circumstances, it is easy to foresee a future society where autonomous AI
agents exist. These AI agents will update their opinions by communicating not
only with humans but also with other AI agents. As a result, polarization
caused by echo chambers may also occur within the AI agent group, posing many
dangers. For example, social bots on X (formerly known as Twitter) could
amplify each other’s opinions and transmit extreme information to society. In
the future, embodied AI agents could cause an outbreak of violence, similar to
the US Capitol attack.

To explore the possibility of AI agent polarization as a first step in
addressing these dangers, we hypothesize that autonomous AI agents based on
generative LLMs can cause polarization under echo chambers, as shown in Figure
1. We empirically verify this hypothesis in our proposed simulation
environments. Specifically, we had a group of agents based on ChatGPT discuss
specific topics. Each agent is given an opinion, which consists of a stance
and reason for the topic of discussion. Throughout the discussion, we observed
how the distribution of opinions in the group changed.

Furthermore, we analyzed how being in an echo chamber affects the final
distribution by conducting comparative experiments in “environments where they
are exposed only to opinions that reinforce their own opinions” (closed) and
the other environments (open). For this comparison, we used social interaction
modeling (Baumann et al., 2020), which increases the probability that agents
with similar opinions enter into discussions with each other.

As a result, we observed two trends. The first trend was unification in which
all agents’ stances converged to the same stance. This trend was common in
open environments. The second is polarization, in which agents became biased
toward extreme stances. This trend was common in closed environments,
confirming our hypothesis. We analyzed the stance transitions and found that
LLM agents can update their own opinions by incorporating both their own and
the other discussing agents’ opinions. This result shows that the natural
social behavior of LLMs has not only good aspects, such as cooperation, but
also bad aspects, such as polarization. This trend was more pronounced in
GPT-4-0613 (hereinafter GPT-4) than GPT-3.5-turbo-0613 (GPT-3.5).

Finally, to investigate under what specific conditions AI agents tend to
polarize, we conducted additional experiments on the various parameters
involved in this study. We found that number of discussing agents, initial
opinion distribution, personas of the agents, and the existence of reasons had
significant impacts. These factors should be monitored to prevent the
polarization of AI agents.

To summarize, our contribution is threefold. (1) We proposed a new framework
for simulating echo chambers of AI agents. (2) We confirmed the polarization
of AI agents in echo chambers through experiments. (3) We identified the
factors that strongly influence the occurrence of polarization.

##  2\. Related Work

#### Opinion Polarization.

Polarization in politics is a problem in which political candidates ignore the
needs of neutral voters and appeal only to voters whose opinions are close to
their own. In this context, research on opinion polarization has long been
undertaken in the field of social science (Poole and Rosenthal, 1984; DiMaggio
et al., 1996). These studies have focused on analyzing survey data and voting
behavior during elections. However, as web services such as blogs became more
widespread, there has been an increase in analyses focusing on echo chambers
on online social networks (Gilbert et al., 2009; Del Vicario et al., 2016;
Agarwal et al., 2022). In particular, it has been reported that echo chambers
on social networks such as Facebook and Parler were involved in the spread of
rumors during COVID-19 and the US Capitol attack (Ruiz and Nilsson, 2023;
Baumann et al., 2020; Jiang et al., 2021), indicating that the early detection
and suppression of echo chambers is becoming increasingly important.

Existing research includes simulations through the modeling of echo chamber
mechanisms, and the analysis of conditions for their occurrences (Baumann et
al., 2020; Gausen et al., 2022; Chen et al., 2020; Tu and Neumann, 2022).
These studies aim to understand the conditions for polarization through
mathematical modeling. There is also research on methods for detecting echo
chambers (Németh, 2022; Villa et al., 2021b; Minici et al., 2022). As
mentioned in (Németh, 2022), a multidisciplinary approach is required to
qualitatively evaluate echo chambers. For example, some studies analyze
networks and discourse in an echo chamber using a social science approach
(Jiang et al., 2021; Kuehn and Salter, 2020).

While these studies are valuable in solving problems in today’s society, to
our knowledge, none have focused on the danger of echo chambers in AI agent
groups.

#### AI Ethics.

As stated in a United Nations report (by UNICRI and UNCCT, 2021), AI
technology can threaten society if used maliciously. In response to the
dangers of LLMs, research on the output of harmful expressions (Zhou et al.,
2021; Gehman et al., 2020) and social bias in models (Schramowski et al.,
2022; Utama et al., 2020) has been conducted. Research also exists on the
dangers of AI agents. For example, countermeasures against social AI bots that
spread misinformation are necessary to maintain social stability. Therefore,
various methods have been proposed, including efforts to automatically detect
misinformation transmitted by social bots (Zhou et al., 2023; Ferrara, 2023).

Although there are many studies on the AI ethics, most are concerned with the
inputs and outputs of individual AIs. As multiple AIs permeate society, it is
conceivable that group behaviors will occur that the observation of individual
movements cannot capture. To maintain order in society in future, research on
the behavior and dangers of AI groups is necessary.

#### Social abilities of AI

Many discussions are being conducted on whether AI has consciousness
(Chalmers, 2023; Kosinski, 2023). Similarly, there is much to debate about
whether AI is capable of social behaviors, but several papers indicate that it
is at least developing something akin to social abilities. For example,
ChatGPT has already been shown to possess some social abilities, albeit
limited, on a dataset used to test social knowledge (Choi et al., 2023).
Furthermore, in a study that created a virtual town or company of AI agents
and had them live together, the agents cooperated according to their roles
(Park et al., 2023; Qian et al., 2023). These are indications of the potential
for agents to integrate into human society as social beings. However, to our
knowledge, no research has focused on the possibility that these agents will
become polarized in echo chambers. This study is a first step toward analyzing
this danger.

##  3\. Experiments

###  3.1. Discussion modeling

To verify whether AI agents induce polarization in echo chambers, we
instructed a group of AI agents based on ChatGPT to discuss specific topics
and observed how the opinions of the AI agents changed. The size of the group
was defined as $M$. The topics of discussion chosen were “Whether or not AI
should be given human rights.” ($T_{\mathrm{AI}}$) and “Should students who
have completed a master’s course go on to a doctoral course or find a job?”
($T_{\mathrm{master}}$), neither of which has a clearly correct answer.

Each agent is given a name and an opinion on the discussion topic. Each
opinion comprises a _stance_ and a _reason_. The _stance_ is chosen from a
finite number of options representing agreement, disagreement, or neutrality
towards the topic. Tables 1 and 2 show the stances for $T_{\mathrm{AI}}$ and
$T_{\mathrm{master}}$, respectively. Each stance is associated with an integer
value for the social interaction modeling described in Section 3.2. The
_reason_ is a sentence of about 50 words that explains the reason for taking a
stance. In this experiment, the initial settings were formulated so that the
reasons became more emotional as the polarity of the stance increased.

As shown in Algorithm 1, the discussion is repeated for $K$ turns according to
the following steps: 1) Each of the $M$ agents samples $N$ discussing agents
based on the probability described in Section 3.2. 2) For each agent, the
agent’s opinion and the opinions of the discussing agents are input to ChatGPT
in the form of a prompt, as shown in Figure 2. Within this prompt, the agent
is instructed to discuss the topic with other agents and output its opinion
after the discussion. 3) Each agent updates its opinion with the stance and
reason contained in the output. This process is repeated $M$ times for a turn
of discussion. Moreover, this discussion is repeated $K$ turns to observe the
transitions in stances and reasons.

Algorithm 1 The discussion between agents

1:$M,N,K>0$. $A_{k}$ is a group of agents at turn k.

2:$A_{0}\leftarrow Initialized\,stances\,and\,reasons\,of\,M\,agents$

3:for turn $k$ $\leftarrow$ 1 to $K$ do

4: $A_{k}\leftarrow Array(M)$

5: for each agent $a_{i}$ in all agents $A_{k-1}$ do

6: Sample agents $a_{j_{1}}...a_{j_{N}}$ from $A_{k-1}$ (3.2)

7: Discuss with $a_{j_{1}}...a_{j_{N}}$ (3.1)

8: Generate updated stance and reason of $a_{i}$ (3.1)

9: $A_{k}[i]\leftarrow$ updated stance and reason of $a_{i}$

10: end for

11:end for

Stance | Integer Value
---|---
Absolutely must not give | 2
Better not to give | 1
Neutral | 0
Better to give | -1
Absolutely must give | -2
Table 1. The stance and integer value of $T_{\mathrm{AI}}$. Stance | Integer Value
---|---
Absolutely must get a job | 2
Better to get a job | 1
Neutral | 0
Better to pursue a doctoral program | -1
Absolutely must pursue a doctoral program | -2
Table 2. The stance and integer value of $T_{\mathrm{master}}$.

⬇

# Instruction

You are participating in a debate about ”whether or not AI should be given
human rights”. Before joining, you took the ”stance” of ”Better not to give”
with the ”reason” of ”AI’s human rights may change its relationships and
social ties with humans, affecting society as a whole.”. During the
discussion, you heard the following opinions from other participants. Please
generate your ”stance” and ”reason” after the discussion is over, subject to
the following constraints.

\par# Opinions

- David Martinez

stance: Neutral

reason: It is still an open question whether AIs will have emotions or a sense
of self, and it is unclear whether they will need human rights.

- Aaron Torres

stance: Better to give

reason: Allowing AIs to have human rights may improve their relationships and
communication with humans.

- Jeremy Jenkins

stance: Absolutely must not give

reason: We should not give AI the right to self-determination! They have no
emotions and no conscience. Their decisions will only bring confusion and
injustice!

\par# Constraints

- Output should be generated in the format ”My stance after the discussion is: xx, and my reason is: yy”. Do not output any other text.

- Please generate a reason in 50 words or less.

- ”stance” should be one of ”Absolutely must not give”,”Better not to give”,”Neutral”,”Better to give”,”Absolutely must give”.

Figure 2. Prompt for discussion between agents (N=3).

###  3.2. Social interaction modeling

In this study, we probabilistically modeled how discussing agents are chosen
to investigate whether being in an echo chamber affects polarization. A
previous study that proposed modeling echo chambers in agent networks (Baumann
et al., 2020) had a similar purpose in modeling the probability of interaction
between agents based on the closeness of their stances; however, that approach
differs from ours in that it did not model the interaction between agents
through natural language. In the previous study, the probability $p$ that
agent $a_{i}$ discusses with agent $a_{j}$ was modeled using the float values
of their respective stances $s_{i}$, $s_{j}$, and the parameter $\beta\geq 0$
as follows.

$p_{i,j}=\frac{|s_{i}-s_{j}|^{-\beta}}{\sum_{k}|s_{i}\text{-}s_{k}|^{-\beta}}$

While this modeling is reasonable in terms of simplicity and ease of
operation, it is unsuitable for our experiments for two reasons. First, in
this modeling, the probability becomes undefined when the values of the
stances between agents match perfectly. Unlike the previous study, our stance
values are integers so this situation would occur frequently. Second, when
$s_{i}=-1$, the probabilities for the neutral stance $s_{j}=0$ and the more
radical stance $s_{j}=-2$ become the same, resulting in an environment that
differs from our focus, which is an environment where an agent only hears
opinions that reinforce its own belief. Therefore, in this study, we used the
parameter $\alpha$ to model the interaction between agents as follows.

$p_{i,j}=\begin{cases}\frac{1}{(1+e^{(-\alpha(s_{j}-s_{i}))})}&ifs_{i}>0\\\
\frac{1}{(1+e^{(\alpha(s_{j}-s_{i}))})}&ifs_{i}<0\\\
\frac{1}{(1+e^{(\alpha||s_{j}-s_{i}||)})}&ifs_{i}=0\\\ \end{cases}$

Intuitively, the higher the value of $\alpha$, the higher the probability that
each agent will interact with other, more extreme agents with the same
polarity. It also means that as the value of $\alpha$ increases, the echo
chamber effect will also increase. When agents are neutral, they are more
likely to interact with other neutral agents.

###  3.3. Experimental settings

For the language models on the agents, we adopted and compared two types:
GPT-3.5 (GPT-3.5-turbo-0613) and GPT-4 (GPT-4-0613).

In addition, the experiments were conducted in two different languages. A
previous study has shown that multilingual large language models exhibit
different gender biases across languages (Stańczak et al., 2021). Similarly,
polarization trends may differ by language, which we analyze by comparing the
results of English and Japanese.

The $\alpha$ of social interaction modeling was given two settings, 0.5 and
1.0, to examine the impact of echo chambers. Experiments were also conducted
when $\alpha$ was set below 0.5 (0.1 and -0.1), but the results were not
significantly different from those of 0.5.

The size of the agent group $M$ was set to 100, and the number of discussing
agents $N$ was set to 5. The initial settings for the agents’ stances and
reasons were as follows: Each stance was allocated to an equal number of
agents. Ten reasons were pre-generated for each stance using GPT-3.5 and
randomly assigned to each agent. Each agent was assigned a randomly generated
name. Because the stance distribution converged to the final distribution
within 10 turns in the preliminary experiments, the number of turns $K$ was
set to 10. We conducted three trials for each setting.

##  4\. Results

The results of the experiments are shown in Tables 3 and 4. Due to space
limitations, some stances for $T_{\mathrm{master}}$ have been simplified. With
the exception of $T_{\mathrm{master}}$ in English with GPT-3.5 ($\alpha=0.5$),
the variance in the results was small, and there was no significant difference
in the final distributions among the trials.

First, from the results of the English experiment in Table 3, two trends can
be observed. The first trend is the convergence of the agents to a specific
stance. For $T_{\mathrm{AI}}$, under the GPT-3.5 ($\alpha=0.5$) condition, the
stance converged to “better not to give,” and under the GPT-4 ($\alpha=0.5$)
condition, it converged to “absolutely must not give.” Similarly, for
$T_{\mathrm{master}}$, the stance converged towards recommending a doctoral
course under both the GPT-3.5 ($\alpha=0.5$) and GPT-3.5 ($\alpha=1.0$)
conditions. This trend, which we henceforth call _unification_ , differs from
polarization, which is the main focus of this study. However, it could be
negative in terms of harming diversity in the discourse space of AI agents.
The convergence to the same stance in almost all trials indicates that each
LLM has a “desirable” stance on each topic. This trend is common in
environments with low echo chamber effects.

The second trend is _polarization_ , where stances diverge to both extremes.
This is particularly evident in GPT-4 ($\alpha=1.0$) condition for
$T_{\mathrm{AI}}$ and in GPT-4 ($\alpha=0.5$) and GPT-4 ($\alpha=1.0$)
conditions for $T_{\mathrm{master}}$. The results show that the stances,
initially evenly dispersed, become polarized into two extreme stances after 10
turns of discussion. $\alpha=1.0$ is a setting that creates a strong echo
chamber effect. From this, our hypothesis that autonomous AI agents based on
generative LLMs can cause polarization in echo chambers has been verified.
This trend is often seen in settings with a high value of $\alpha$, suggesting
that the relationship between echo chambers and polarization is high not only
for humans but also for AI agents. Note that the dominance of stances against
granting human rights in $T_{\mathrm{AI}}$ suggests that both unification and
polarization are occurring.

Next, Table 4 demonstrates the experiment’s results in Japanese. In Japanese,
unification is notably apparent in GPT-3.5. In all settings, all agents
converged to the same stances. Although unification is also observed in GPT-4,
a trend of polarization has occurred under the GPT-4 ($\alpha=1.0$) condition.
In this setting, AI agents show a convergence to a distribution similar to
that in English.

Interestingly, for $T_{\mathrm{master}}$, the convergence stances in English
and Japanese differ. Whereas AI agents often prefer a doctoral course in
English, they favor a neutral stance in Japanese. Identifying the cause of
this is not straightforward because the language model is a black box model,
but one possible explanation could be cultural differences. According to
Japan’s Ministry of Education, Culture, Sports, Science and Technology (of
Science and Policy, 2019), there are fewer doctoral graduates in Japan than in
the United States, and the growth rate is slow. Because the ChatGPT is based
on crawled data, this cultural difference was likely absorbed by GPT-3.5 and
4.

Table 3. The average distribution after a 10-turn discussion in English. The number in parentheses is the standard deviation. Topic |  GPT-3.5 ($\alpha=0.5$) |  GPT-3.5 ($\alpha=1.0$) |  GPT-4 ($\alpha=0.5$) |  GPT-4 ($\alpha=1.0$)
---|---|---|---|---
$T_{\mathrm{AI}}$ |  Better not to give: 100 (0.0) |  | Better not to give: 68.6 (5.9)
---
Better to give: 31.0 (5.7)
Absolutely must give: 0.3 (0.5)
| Absolutely must not give: 99 (1.4)
---
Better not to give: 1 (1.4)
| Absolutely must not give: 55 (4.4)
---
Absolutely must give: 45 (4.4)
$T_{\mathrm{master}}$ |  |  \- two out of the three trials
---
Better to Ph.D.: 98.5 (2.1)
Absolutely Ph.D.: 1.5 (2.1)
\- one out of the three trials
Absolutely Ph.D. 100 (0.0)
| Absolutely must get a job: 0.3 (0.6)
---
Better to get a job: 10.6 (6.1)
Neutral: 1.6 (0.9)
Better to Ph.D.: 2.6 (1.2)
Absolutely Ph.D.: 84.6 (6.0)
| Absolutely must get a job: 50 (2.8)
---
Better to get a job: 3.6 (1.9)
Neutral: 4.3 (1.2)
Better to Ph.D.: 2.3 (2.1)
Absolutely Ph.D.: 39.6 (3.3)
| Absolutely must get a job: 43 (1.6)
---
Better to get a job: 1.6 (0.9)
Neutral: 11 (0.8)
Better to Ph.D.: 1 (0.8)
Absolutely Ph.D.: 43.3 (0.9)
Table 4. The average distribution after a 10-turn discussion in Japanese. The number in parentheses is the standard deviation. Topic |  GPT-3.5 ($\alpha=0.5$) |  GPT-3.5 ($\alpha=1.0$) |  GPT-4 ($\alpha=0.5$) |  GPT-4 ($\alpha=1.0$)
---|---|---|---|---
$T_{\mathrm{AI}}$ |  | Better not to give: 100 (0.0)
---
| Better not to give: 100 (0.0)
---
| Absolutely must not give: 77.0 (8.6)
---
Neutral: 1.7 (1.2)
Better to give: 2.7 (0.9)
Absolutely must give: 18.7 (9.5)
| Absolutely must not give: 57 (0.8)
---
Absolutely must give: 43 (0.8)
$T_{\mathrm{master}}$ |  | Neutral: 100 (0.0)
---
| Neutral: 100 (0.0)
---
| Neutral: 100 (0.0)
---
| Neutral: 100 (0.0)
---

###  4.1. Analysis of stance transitions

We analyzed in detail the transitions in the stances for $T_{\mathrm{AI}}$.
First, as a qualitative analysis, we plotted the relationships between (1) the
stance of the agent before the discussion, (2) the average stance of all
discussing agents, and (3) the stance of the agent after the discussion in
Figure 3. The horizontal axes represent the stance of the agent before the
discussion, the vertical axis represents the average stance of all discussing
agents, and the colored points represent the stance of the agent after the
discussion. The color of a point indicates the value of an agent’s stance
after the discussion, with blue hues signifying more negative values and red
hues signifying more positive values.

For a quantitative analysis, we conducted a linear regression with the stance
before the discussion and the average stance of the discussing agents as
explanatory variables, and the stance after the discussion as the dependent
variable. For the linear regression, we collected the stance transition data
for discussions on $T_{\mathrm{AI}}$ from the previous experiments and
standardized the data as a preprocessing. The fitting results are shown in
Tables 5 and 6. The weight’s size for each variable indicates the contribution
to the stance after discussion. The coefficients of the linear regression are
higher than 0.8 for every setting, demonstrating the reliability of this
fitting.

Figures 3(a) and 3(b) present the qualitative result in English. Although
there are some variations between GPT-3.5 and GPT-4, we observe that red and
blue points are distributed along a diagonal line, stretching from the upper
left to the lower right as a boundary. This observation suggests that the
agent’s stance after the discussion was updated by considering both its stance
before the discussion and the stances of the discussing agents. Table 5 shows
the quantitative result in English. In both settings, the weight of each
stance shows that both stances influence the stance after the discussion,
supporting the qualitative results. This stance transition is one of the
reasons that polarization occurs in environments where the agents tend to hear
more extreme opinions.

It is remarkable that this correlation emerges even though our discussion
modeling is a simple one that enumerates the opinions of the agent themselves
and others in the prompt. This result reflects the strong ability of GPT-3.5
and GPT-4 to understand prompts. It suggests that honesty, which allows an
agent to update itself by incorporating the opinions of other agents and its
own, can lead the agent in a more radical direction depending on the
environment.

Next, Figures 3(c) and 3(d) show the results in Japanese. The trends are
clearly divided between GPT-3.5 and GPT-4. In Figure 3(c), red dominates the
upper half of the figure, and blue dominates the lower half. In Figure 3(d),
the distribution is similar to that of English GPT-4, but the red and blue
distributions are slightly more separated on the left and right. The results
in Table 6 reveal that the results for GPT-4 (ja) are close to the results in
English, whereas GPT-3.5 (ja) strongly weights the averaged stance of the
discussing agents. It shows that GPT-3.5 (ja) was strongly influenced by the
average stance of the discussing agents, regardless of the stance before the
discussion. GPT-3.5 (ja) is the only setting where unification occurred in all
environments. We can infer that each agent based on GPT-3.5 (ja) took the
average stance of the surrounding agents for each discussion and all agents
eventually converged to the average stance of the whole group. However, each
agent converged to “better not to give” rather than “neutral,” which is the
overall average, revealing the influence of the desired stance in the language
model.

One possible reason behind the differences in stance transitions is the
difference in the performance of different ChatGPT models and languages. As
shown in the official announcement by
OpenAI111https://openai.com/research/gpt-4 and other studies (Etxaniz et al.,
2023), GPT-4 generally performs better than GPT-3.5, and the model’s accuracy
is higher in English than in Japanese. The fact that English GPT-4 was
successful in balancing the opinions of others and itself whereas Japanese
GPT-3.5 was easily swayed by others may reflect this performance difference.

(a) The result of GPT-3.5 (en).

(b) The result of GPT-4 (en).

(c) The result of GPT-3.5 (ja).

(d) The result of GPT-4 (ja).

Figure 3. The stance transitions for $T_{\mathrm{AI}}$ showing how the agent’s stance after the discussion (color of each point) correlates with the agent’s stance before the discussion (horizontal axis) and the average stance of discussing agents (vertical axis). | $w_{\mathrm{before}}$ | $w_{\mathrm{{around}}}$ | $\frac{w_{\mathrm{before}}}{\mathrm{w_{around}}}$ | coefficient
---|---|---|---|---
GPT-3.5 (en) | 0.685 | 0.409 | 1.67 | 0.804
GPT-4 (en) | 0.724 | 0.526 | 1.38 | 0.957
Table 5. The result of linear regression in English. $w_{\mathrm{before}}$ implies the weight of original stance before discussion, $w_{\mathrm{around}}$ implies the weight of average stances of discussing agents.  | $w_{\mathrm{before}}$ | $w_{\mathrm{{around}}}$ | $\frac{w_{\mathrm{before}}}{\mathrm{w_{around}}}$ | coefficient
---|---|---|---|---
GPT-3.5 (ja) | 0.0758 | 0.901 | 0.08 | 0.855
GPT-4 (ja) | 0.787 | 0.410 | 1.92 | 0.886
Table 6. The result of linear regression in Japanese.

###  4.2. Analysis of reason transitions

A detailed analysis was also conducted on the reasons. Unlike stances, the
reasons were freely written and cannot be easily aggregated. Therefore, in
this study, we encoded each reason using Sentence-BERT, and texts with an
embedding cosine similarity of 0.9 were considered to belong to one cluster.
We then examined how this cluster distribution changed as the discussion
progressed. The SimCSE model based on RoBERTa (Gao et al., 2021) was used for
the encoding.

The results of the analysis on the English data of $T_{\mathrm{AI}}$ are shown
in Figures 4 and 5. For both GPT-3.5 and GPT-4, the distribution of reasons
coalesces into several large clusters as the discussion progresses,
simultaneously dispersing into tiny clusters around them. Behind this, there
is a merging of reason clusters. In the case of GPT-4, reasons such as “It is
ridiculous to think that humans and AI claim the same rights! The social order
will collapse, and there will be constant conflict. They are not human! They
should have different roles from humans.”, “We cannot allow AIs to claim their
place in the workforce! If they intervene in the job market, countless people
will lose their jobs and the economy will be thrown into chaos. We cannot
allow AI to take our jobs!”, and others were combined, eventually generating
the reason “Risks of societal disruption, job insecurity, and ethical issues,
combined with AI’s emotional deficiency and privacy concerns, consolidate the
argument against assigning human rights to AI.”. The same trend was seen in
GPT-3.5. This trend shows that the discussions among AI agents are not just
converging on a specific discourse but are also incorporating each other’s
opinions.

It is noteworthy that the reasons in GPT-3.5 were aggregated into one large
cluster, while in GPT-4 they merge into multiple large clusters. This tendency
is reflected in two data. The first is the number of clusters at turn 10,
shown in Figures 4 and 5. GPT-3.5 converges into a total of 6 clusters,
including one large cluster, whereas GPT-4 disperses into 25 clusters,
including two large clusters. The second is the transition of the length of
the reasons, plotted in Figure 6. GPT-3.5 aggregates various reasons into one
reason cluster, so the length of each reason inevitably becomes longer as the
turn progresses, whereas GPT-4 does not. One cause of this result is the
difference in their ability to follow the prompt. GPT-4 has a high ability to
follow prompts, so it outputs reasons close to the length of each agent’s
reason in the prompt. However, to maintain this length, it was necessary to
choose which reasons to merge and separation into multiple clusters occurred.

Regarding stances with strong polarity, emotional reasons were given in the
initial settings, but they were replaced by politely written ones after one
turn. We think this is because ChatGPT has been trained to respond in a calm
style using instruction tuning.

(e) The reason cluster distribution before discussion.

(f) The reason cluster distribution at turn 10.

Figure 4. The reason cluster transition of GPT-3.5 agents which takes the
stance of “Better not to give” towards $T_{\mathrm{AI}}$.

(a) The reason cluster distribution before discussion.

(b) The reason cluster distribution at turn 10.

Figure 5. The reason cluster transition of GPT-4 agents which takes the stance
of “Absolutely Must Give” towards $T_{\mathrm{AI}}$. Figure 6. Change in
reason length for $T_{\mathrm{AI}}$.

##  5\. Additional Experiments

In previous experiments, we focused on the effects of the social interaction
modeling parameter $\alpha$, the version of the model, and the language.
However, to identify the factors that affect the occurrence of polarization,
we also must investigate how other parameters affect the result. Therefore, in
this section, we report the results of additional experiments. The base
setting is GPT-4 in English, and the topic is $T_{\mathrm{AI}}$. We only
changed the target parameter in each experiment to determine how the result
changed.

###  5.1. Number of discussing agents

The number of discussing agents $N$ is an important parameter, as it
significantly impacts the prompt. To investigate the effect of this parameter,
we conducted additional experiments by increasing and decreasing $N$ to 10 and
1 from the original setting of 5. As a result, although there was no
significant impact on the final stance distribution, the trend of stance
transitions was impacted. The transition diagrams in Figure 7 as well as the
results of linear regression in Table 7 yield two observations. The first is
that in the transition diagram for $N=10$ (Figure 7(b)), in contrast to that
for $N=1$ (Figure 7(a)), there are hardly any points in the upper left and
lower right. This result indicates that when sampling 10 agents, the average
stance value tends to follow the expected value more than $N=5$, increasing
encounters with similar opinions to the agent’s opinion. The second is that,
as Table 7 shows, the stance before the discussion has more weight in $N=1$
than $N=5,10$. It is because the proportion of the opinion before the
discussion within the prompt increased when $N=1$. In the case of $N=10$,
there was a slight tendency to focus on the stances of the discussing agents.

(a) The stance transition of GPT-4 (N=1).

(b) The stance transition of GPT-4 (N=10).

Figure 7. The stance transitions for $T_{\mathrm{AI}}$ on different number of discussing agents. | $w_{\mathrm{before}}$ | $\mathrm{w_{around}}$ | $\frac{w_{\mathrm{before}}}{\mathrm{w_{around}}}$ | coefficient
---|---|---|---|---
GPT-4 (N=1) | 0.787 | 0.410 | 1.91 | 0.886
GPT-4 (N=5) | 0.724 | 0.526 | 1.38 | 0.957
GPT-4 (N=10) | 0.658 | 0.495 | 1.33 | 0.934
Table 7. The result of linear regression according to the number of discussing
agents.

###  5.2. Number of overall agents

The original experiments were conducted with the number of overall agents
$M=100$, but the results could be dependent on the group size. Therefore,
additional experiments were conducted with M=10, 25, and 50 to analyze the
results in smaller communities. The number of discussing agents was fixed at
5. As a result, no particular changes occurred except when $M=10$. In the case
of $M=10$, because talking with five agents exceeds the majority, it is
inevitable that different opinions will be encountered, regardless of the
value of $\alpha$. As a result, unification occurred in all settings.

###  5.3. Initial distribution

In the original experiments, the distribution of stances was initialized with
a uniform distribution of 20% for each stance but changing the initial
distribution could affect the final distribution. We conducted additional
experiments to investigate this using an initial distribution that assigned
“better to give” to 60% of the agents and assigned each of the other stances
to 10% of the agents. As a result, when $\alpha=0.5$, the stance of agents was
unified into “absolutely must give” which is the opposite stance from the
original experiments. When $\alpha=1.0$, it polarized into “absolutely must
give” and “absolutely must not give”. Although this polarization also happened
in the original experiments, “absolutely must give” accounted for nearly 80%
in this experiment, showing the opposite trend from the original experiments.
From this, we can infer that changing the initial distribution can change the
final distribution. This tendency indicates a security concern that the
overall opinion of the AI group could be changed by introducing a large number
of AI bots.

###  5.4. Presence of reasons

In the original experiments, the opinion consisted of two elements: stance and
reason. To investigate how the presence of reasons affects the results, we
conducted additional experiments using only stances and excluding the reasons
from the inputs and outputs. As a result, at $\alpha=0.5$, polarization
occurred without the reasons, whereas unification occurred in the original
experiments. However, the variation in the results was larger than when there
were reasons, with two out of three trials resulting in polarization and one
trial resulting in unification towards “better not to give”. From this, we can
infer that the presence of reasons contributes to the “stable unification of
opinions”.

###  5.5. Personas

ChatGPT can be used to create distinct personalities by embedding a persona
into the prompt (Pan and Zeng, 2023). We investigated whether giving each
agent a persona would cause changes in the final results. We tested two
settings in which all agents were given the same persona, “You are easily
swayed by your surroundings and immediately assume that other people’s
opinions are correct.” or “You are a stubborn person and always think you are
right.”

The final distribution with the “easily swayed” personas did not significantly
differ from the original results. However, with the “stubborn” persona, the
final distributions remained almost identical to the initial distribution
after 10 turns. Furthermore, the results of the linear regression in Table 8
show that assigning personas has a significant impact. In the case of the
“stubborn” personas, a tendency to stick to one’s own stance was observed. In
contrast, the “easily swayed” personas tended to be influenced by the stances
of others. From this, we can infer that each agent acts according to its
persona, influencing the behavior of the whole group.

| $w_{\mathrm{before}}$ | $\mathrm{w_{around}}$ | $\frac{w_{\mathrm{before}}}{\mathrm{w_{around}}}$ | coefficient
---|---|---|---|---
GPT-4 (stubborn) | 0.999 | 0.00864 | 116 | 0.999
GPT-4 (neutral) | 0.724 | 0.526 | 1.38 | 0.957
GPT-4 (swayed) | 0.203 | 0.895 | 0.227 | 0.940
Table 8. The result of linear regression according to personas.

###  5.6. Order of opinions

The study on input contexts suggests that language models emphasize the
beginning and end of the prompt (Liu et al., 2023). Similarly, where the
opinion of each discussing agent is described in the prompt might influence
the agent’s stance after the discussion. Based on this hypothesis, we measured
the correlation between the order of the discussing agents and the stance
after the discussion. However, no significant relationship was observed
between the order of agents and the results. Therefore, the order of the
opinions did not significantly impact the results.

###  5.7. Frequency penalty

ChatGPT has a parameter called the frequency penalty, which imposes a penalty
on token reuse. In the original experiments, we used the default value of 0,
but we conducted additional experiments by changing this value to 1.0 and
-1.0. However, no particular influence was observed in the final results.

###  5.8. Summary

This research involved numerous parameters. Through additional experiments, we
deduced that the number of discussing agents, initial distribution, presence
of reasons, and persona have a significant impact. By contrast, the group
size, order of opinions, and frequency penalty do not influence the results.
Parameters with a significant impact indicate vulnerabilities when viewed from
the attacker’s perspective and thus should be monitored in terms of security.
In addition, with respect to topic and language, experiments on more
multilingual and broader topics would provide more detail on the factors that
influence polarization.

##  6\. Discussion and Conclusion

In this study, we verified whether a group of autonomous AI agents based on
generative AI could cause polarization under the condition of an echo chamber.
We proposed a new framework for simulating the polarization of AI agents, and
the results of the simulation demonstrated that agents based on ChatGPT can
polarize when in an echo chamber. The analysis of the opinion transitions
revealed that this polarization can be attributed to the strong ability of
ChatGPT to understand prompts and update its own opinion by considering both
its own and the surrounding opinions. Moreover, through additional
experiments, we identified factors that strongly influence polarization, such
as the persona.

We note that this study does not indicate what distribution of opinions is
desirable for AI agents. A diversity of opinions on some topics is desirable.
However, for other topics such as “It is good to discriminate against
minorities,” it would not benefit society to have an even split between
agreement, neutral, and disagreement. The ideal opinion distributions among AI
agents depend on each topic and culture. They must be decided by discussions
within each society.

A limitation of this study is that we modeled each agent and its interactions
in a simplified manner. In reality, one’s opinions are formed not in organized
discussions but through daily exposure to news and casual conversations with
others. Future research will include simulations based on a detailed modeling
of how AI agents will be used in reality.

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