# Leptogenesis via Inflaton Mass Terms in Non-Minimally Coupled Inflation

Kit Lloyd-Stubbs and John McDonald<EMAIL_ADDRESS><EMAIL_ADDRESS>Dept. of Physics, Lancaster University, Lancaster
LA1 4YB, UK

###### Abstract

We consider a model of baryogenesis based on adding lepton number-violating
quadratic mass terms to the inflaton potential of a non-minimally coupled
inflation model. The $L$-violating mass terms generate a lepton asymmetry in a
complex inflaton field via the mass term Affleck-Dine mechanism, which is
transferred to the Standard Model (SM) sector when the inflaton decays to
right-handed (RH) neutrinos. The model is minimal in that it requires only the
SM sector, RH neutrinos, and a non-minimally coupled inflaton sector. We find
that baryon isocurvature fluctuations can be observable in metric inflation
but are negligible in Palatini inflation. The model is compatible with
reheating temperatures that may be detectable in the observable primordial
gravitational waves predicted by metric inflation.

###### pacs:

##  I Introduction

Baryogenesis is an essential feature of a complete model of particle
cosmology. In the absence of a fundamental theory of particle physics, a
minimal model-building approach seems most likely to lead to the correct
solution, by virtue of its fewer assumptions. A minimal model of particle
physics and cosmology consists of the Standard Model sector, neutrino masses
via right-handed (RH) neutrinos, a dark matter candidate, and an inflaton
sector. In this case there is a limited range of possibilities for generating
the baryon asymmetry. The most commonly considered are electroweak
baryogenesis ew  and leptogenesis via out-of-equilibrium decays of RH
neutrinos lept ; lept2 ; lept3 .

Another possibility is the generation of the asymmetry via the decay of the
inflaton. This could occur via out-of-equilibrium decay of the inflaton if it
has the required CP- and B-violating decay modes infb ; infb2 . Alternatively,
the asymmetry could first be generated in the inflaton field itself via the
Affleck-Dine (AD) ad  mechanism adc0 ; adc1 ; adc1a ; adc2 ; adc3 ; cline ;
clineL ; kls1 ; kawa . This is arguably the simplest mechanism for generating
the baryon asymmetry, requiring only $(B-L)$-violating terms in the potential
of a complex scalar inflaton and a decay mode to transfer the asymmetry to the
SM sector.

In a previous paper kls1  we discussed a new implementation of AD inflation,
in which the asymmetry is generated at late times by $(B-L)$-violating
quadratic potential terms (mass terms) during the rapid coherent oscillations
of a complex inflaton111The AD mechanism based on mass terms was previously
considered in a different context in gorb . This analysis does not include the
case of final asymmetry generation from averaging over a rapidly oscillating
AD asymmetry.. This results in an oscillating asymmetry in the inflaton
condensate, with mean value equal to zero. Nevertheless, when the asymmetry is
transferred to the SM sector, the net asymmetry is non-zero.

Inflation models are constrained by present bounds on the range of scalar
spectral index values and the upper bound on the tensor-to-scalar ratio. One
class of models that provides a good fit to observations are the non-minimally
coupled inflation models bbs . There are two favoured implementations of this
model, based on either the metric formalism of General Relativity (GR), as in
conventional Higgs Inflation ks , or the Palatini formalism palatiniHI . Such
models can be driven by a renormalisable $\lambda\phi^{4}$ inflaton potential
with a value for $\lambda$ that is not very small compared to 1. Non-minimally
coupled inflation therefore provides a good candidate for the inflaton sector
of a minimal model of particle cosmology.

In our previous paper kls1  we discussed the general idea of baryogenesis via
inflaton mass terms. However, we did not consider our results in the context
of a specific inflation model. In this paper we will present a minimal model
based on non-minimally coupled inflation. Inflaton mass term AD baryogenesis
and leptogenesis has also recently been applied in the context of models which
relate inflation and baryon asymmetry generation to neutrino-antineutrino
oscillations mo1 ; to neutrino masses combined with pseudo-Goldstone dark
matter mo2 , WIMP dark matter mo3  and axion dark matter mo4 ; and to
cogenesis of the baryon asymmetry and dark matter density okada .

The minimal model we consider consists of the SM sector extended by RH
neutrinos plus a complex inflaton sector, with a global $U(1)_{L}$ symmetry.
To this we add $U(1)_{L}$-breaking mass terms for the inflaton and the RH
neutrinos. The baryon asymmetry is generated by creating a lepton asymmetry in
the inflaton field via the mass term AD mechanism, which is subsequently
transferred to the SM sector via inflaton decays to RH neutrinos.

The paper is organised as follows. In Section 2 we introduce our leptogenesis
model. In Section 3 we review the essential results of non-minimally coupled
inflation in the metric and Palatini formalisms. In Section 4 we review and
elaborate upon the general results of kls1  for the mass term AD mechanism and
apply the results to our leptogenesis model. In Section 5 we discuss
condensate decay, reheating and the conditions to ensure no asymmetry washout.
We also discuss the consistency of the assumptions made in the derivation of
the analytical results. In Section 6 we discuss the possibility of generating
observable baryon isocurvature perturbations. In Section 7 we discuss the
conditions under which the quadratic $U(1)_{L}$-breaking inflaton potential
terms dominate the quartic $U(1)_{L}$-breaking terms. In Section 8 we present
our conclusions.

##  II A Minimal Model for Affleck-Dine Leptogenesis via Inflaton Mass Terms

###  II.1 The Observable Sector

A complex inflaton $\Phi$ has a general renormalisable potential given by222We
do not include a linear term as this can always be eliminated by a redefiniton
of $\Phi$ and the $A$, $B$ and $C$ terms.

$\mbox{$$}V(\Phi)=m_{\phi}^{2}|\Phi|^{2}+\lambda_{\Phi}|\Phi|^{4}-(A\Phi^{2}+{\rm
h.\,c.})-(B\Phi^{3}+{\rm h.\,c.})-(C\Phi^{4}+{\rm h.\,c.})~{}.\vspace{0.1cm}$
(1)

In the limit $A=B=C=0$ this has a global $U(1)$ symmetry, $\Phi\rightarrow
e^{i\alpha}\Phi$, which corresponds to lepton number in the model we are
considering. The $A$, $B$ and $C$ terms are $L$-violating terms that can be
used to generate a lepton asymmetry in the inflaton condensate via the AD
mechanism. For simplicity, we will assume that there is an unbroken discrete
symmetry of the potential, $\Phi\leftrightarrow-\Phi$, so that $B=0$.
Generation of the baryon asymmetry in non-minimally coupled inflation via the
quartic $C$-term has previously been considered in cline  and clineL . Here we
will focus on generation via the $A$-term kls1 , which has a quite different
dynamics.

In order to transfer the asymmetry to the SM sector, we couple the inflaton to
the RH neutrinos via

$\mbox{$$}{\cal
L}_{int}=-y_{\Phi}\Phi\overline{N_{R}^{c}}N_{R}-h_{\nu}\overline{N_{R}^{c}}HL-
m_{N}\overline{N_{R}^{c}}N_{R}+{\rm h.\,c.}~{},\vspace{0.1cm}$ (2)

where $\Phi$ has lepton number $-2$. We have also included the Yukawa coupling
$h_{\nu}$ of the RH neutrino to the Higgs $H$ and the lepton $L$ doublets, and
an $L$-violating mass for the RH neutrinos. These result in a Type-I neutrino
mass matrix once the Higgs expectation value $v$ is introduced. For the case
of a single generation, which we will use as a representative example, this
results in a heavy eigenvalue with mass approximately equal to $m_{N}$ and a
light eigenvalue $m_{\nu}\approx m_{D}^{2}/m_{N}$ corresponding to the mass of
the observed neutrinos, where $m_{D}=h_{\nu}v/\sqrt{2}$. $h_{\nu}$ is then
related to the observed neutrino masses by

$\mbox{$$}h_{\nu}=\frac{\sqrt{2}}{v}\left(m_{\nu}m_{N}\right)^{1/2}~{},\vspace{0.1cm}$
(3)

where $v=246{\rm\ GeV}$ is the Higgs vacuum expectation value.

In addition to Eq. (2), there can be a natural L-conserving portal coupling to
the Higgs doublet of the form $\lambda_{\Phi H}|\Phi|^{2}|H|^{2}$. Therefore
the process of reheating could be due to either perturbative decay of the
$\Phi$ condensate scalars to RH neutrinos or parametric resonance to Higgs
bosons pairs. Other possibilities are decay or annihilation of the inflaton
via the non-minimal coupling or graviton exchange. Here we will focus on
reheating via perturbative decay to RH neutrinos. We will comment on the
conditions for perturbative decay to RH neutrinos to dominate reheating and on
how the results could change if reheating was dominated by the portal coupling
or other processes.

###  II.2 The Inflation Sector

The renormalisable potential Eq. (1) is naturally compatible with non-
minimally coupled inflation. The action of the inflaton sector is

$\mbox{$$}S=\int
d^{4}x\sqrt{-g}\left[\left(1+\frac{2\xi|\Phi|^{2}}{M_{Pl}^{2}}\right)\frac{M_{Pl}^{2}R}{2}-\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi-V(\Phi)\right]~{},\vspace{0.1cm}$
(4)

with signature $(-,+,+,+)$. In general, inflation will occur along a random
value of $\theta$, where $\Phi=\phi e^{i\theta}/\sqrt{2}$. During inflation we
will consider the $A$-terms and $C$-terms to be negligible, in which case the
action can be written as

$\mbox{$$}S=\int
d^{4}x\sqrt{-g}\left[\left(1+\frac{\xi\phi^{2}}{M_{Pl}^{2}}\right)\frac{M_{Pl}^{2}R}{2}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right]~{},\vspace{0.1cm}$
(5)

where

$\mbox{$$}V(\phi)=\frac{m_{\Phi}^{2}\phi^{2}}{2}+\frac{\lambda_{\Phi}\phi^{4}}{4}~{}.\vspace{0.1cm}$
(6)

Inflation is analysed in the Einstein frame, with action

$\mbox{$$}S_{E}=\int
d^{4}x\sqrt{-\tilde{g}}\left[\frac{M_{Pl}^{2}\tilde{R}}{2}-\frac{1}{2}\left(\Omega^{2}+\frac{6s\xi^{2}\phi^{2}}{M_{Pl}^{2}}\right)\frac{\partial_{\mu}\phi\partial^{\mu}\phi}{\Omega^{4}}-V_{E}(\phi)\right]~{},\vspace{0.1cm}$
(7)

where $s=1$ for the metric formalism and $s=0$ for the Palatini formalism, and
the Einstein frame potential is

$\mbox{$$}V_{E}(\phi)=\frac{V(\phi)}{\Omega^{4}}~{}.\vspace{0.1cm}$ (8)

The contraction of indices and $\tilde{R}$ are now defined in terms of
$\tilde{g}_{\mu\;\nu}$, where $\tilde{g}_{\mu\;\nu}=\Omega^{2}g_{\mu\;\nu}$
and

$\mbox{$$}\Omega^{2}=1+\frac{\xi\phi^{2}}{M_{Pl}^{2}}~{}.\vspace{0.1cm}$ (9)

##  III Non-Minimally Coupled Inflation in the metric and Palatini formalisms

In this section we summarise the key results of non-minimally coupled
inflation that are relevant to the leptogenesis model.

###  III.1 Inflation Observables

The amplitude of the power spectrum is

$\mbox{$$}A_{s}=\frac{\lambda_{\Phi}N^{2}}{72\pi^{2}\xi^{2}}\;\;\;\;{\rm(Metric)}\;\;\;\;;\;\;\;\;A_{s}=\frac{\lambda_{\Phi}N^{2}}{12\pi^{2}\xi}\;\;\;\;{\rm(Palatini)}~{},\vspace{0.1cm}$
(10)

where $N$ is the number of e-foldings corresponding to the pivot scale. The
scalar spectral index is the same for both metric and Palatini

$\mbox{$$}n_{s}\approx 1-\frac{2}{N}~{}.\vspace{0.1cm}$ (11)

The tensor-to-scalar ratio is

$\mbox{$$}r=\frac{12}{N^{2}}\;\;\;\;{\rm(Metric)}\;\;\;\;;\;\;\;\;r=\frac{2}{\xi
N^{2}}\;\;\;\;{\rm(Palatini)}~{}.\vspace{0.1cm}$ (12)

###  III.2 The non-minimal coupling

From the observed amplitude of the power spectrum, $A_{s}=2.1\times 10^{-9}$,
it follows from Eq. (10) that:

$\mbox{$$}\xi=820N\sqrt{\lambda_{\Phi}}\;\;\;\;{\rm(Metric)}~{}\vspace{0.1cm}$
(13)

and

$\mbox{$$}\xi=4.0\times
10^{6}N^{2}\lambda_{\Phi}\;\;\;\;{\rm(Palatini)}~{}.\vspace{0.1cm}$ (14)

The corresponding values for $N=55$ are $\xi=4.5\times
10^{4}\sqrt{\lambda_{\Phi}}$ for the metric case and $\xi=1.2\times
10^{10}\lambda_{\Phi}$ for the Palatini case.

Using these, Eq. (12) with $N=55$ gives for the tensor-to-scalar ratio
$r\approx 0.004$ in the metric model and $r\approx 5.5\times
10^{-14}/\lambda_{\Phi}$ for the Palatini model.

###  III.3 $\phi$ for which non-minimally coupled dynamics are negligible

The values of $\phi$ at which the Einstein and Jordan frame actions become
completely equivalent are

$\mbox{$$}\phi<\phi_{c}=\frac{M_{Pl}}{\sqrt{6}\xi}\;\;\;\;{\rm(Metric)}~{}\vspace{0.1cm}$
(15)

and

$\mbox{$$}\phi<\phi_{c}=\frac{M_{Pl}}{\sqrt{\xi}}\;\;\;\;{\rm(Palatini)}~{}.\vspace{0.1cm}$
(16)

##  IV Affleck-Dine via inflaton mass terms

In kls1  we introduced AD baryogenesis via quadratic inflaton potential terms.
In this section we will review and expand upon the analytical results of kls1
. In particular, we will compare the analytic predictions for the
baryon/lepton number-to-entropy ratio with the results of a complete numerical
calculation based on solving the field equations, where we will show almost
perfect agreement with the analytical results. We will then apply our
analytical results to the leptogenesis model.

###  IV.1 Analytical Lepton Asymmetry

In terms of $\Phi=(\phi_{1}+i\phi_{2})/\sqrt{2}$, the inflaton potential is

$\mbox{$$}V(\Phi)=\frac{1}{2}(m_{\Phi}^{2}-2A)\phi_{1}^{2}+\frac{1}{2}(m_{\Phi}^{2}+2A)\phi_{2}^{2}+\frac{\lambda_{\Phi}}{4}(\phi_{1}^{2}+\phi_{2}^{2})^{2}~{},\vspace{0.1cm}$
(17)

where without loss of generality we can define $A$ to be real and positive by
a $U(1)_{L}$ rotation of $\Phi$. The field equations, including the decay
terms, are

$\mbox{$$}\ddot{\phi}_{1}+3H\dot{\phi}_{1}+\Gamma_{\Phi}\dot{\phi_{1}}=-m_{1}^{2}\phi_{1}-\lambda_{\Phi}(\phi_{1}^{2}+\phi_{2}^{2})\phi_{1}~{}\vspace{0.1cm}$
(18)

and

$\mbox{$$}\ddot{\phi}_{2}+3H\dot{\phi}_{2}+\Gamma_{\Phi}\dot{\phi}_{2}=-m_{2}^{2}\phi_{2}-\lambda_{\Phi}(\phi_{1}^{2}+\phi_{2}^{2})\phi_{2}~{},\vspace{0.1cm}$
(19)

where

$\mbox{$$}m_{1}^{2}=m_{\Phi}^{2}-2A\;\;\;,\;\;\;\;\;m_{2}^{2}=m_{\Phi}^{2}+2A~{},\vspace{0.1cm}$
(20)

and $\Gamma_{\Phi}$ is the inflaton decay rate. In the limit
$\lambda_{\Phi}\rightarrow 0$ the equations for $\phi_{1}$ and $\phi_{2}$ are
decoupled from each other, with coherently oscillating solutions for
$\phi_{1}$ and $\phi_{2}$ with different angular frequencies $m_{1}$ and
$m_{2}$.

As long as the $A$-terms do not become dynamically significant until after the
coherent oscillations are dominated by the $|\Phi|^{2}$ term in the potential,
and neglecting the decay terms for now, the solution for the coherently
oscillating field is accurately described by considering only the mass terms
in the potential,

$\mbox{$$}V(\Phi)=m_{\Phi}^{2}|\Phi|^{2}-(A\Phi^{2}+{\rm
h.c.})\;\;\;;\;\;\;\;\;\phi<\phi_{*}~{},\vspace{0.1cm}$ (21)

where $\phi_{*}=m_{\Phi}/\sqrt{\lambda_{\Phi}}$ is the value of $\phi$ (with
$\Phi=\phi e^{i\theta}/\sqrt{2}$) below which the field equations become
dominated by the $|\Phi|^{2}$ term. The field at late times, $t>t_{*}$, where
$\phi=\phi_{*}$ at $t=t_{*}$, is then

$\mbox{$$}\phi_{1}=\phi_{1,\;*}\left(\frac{a_{*}}{a}\right)^{3/2}\cos(m_{1}(t-t_{*}))\;\;;\;\;\phi_{2}=\phi_{2,\;*}\left(\frac{a_{*}}{a}\right)^{3/2}\cos(m_{2}(t-t_{*}))~{},\vspace{0.1cm}$
(22)

assuming that the asymmetry at $t\leq t_{*}$ is zero. The lepton asymmetry in
the condensate (with $L(\Phi)=-2$) is given by

$\mbox{$$}n_{L}(t)=iL(\Phi)\left(\Phi^{\dagger}\dot{\Phi}-\dot{\Phi}^{\dagger}\Phi\right)=-2\left(\dot{\phi}_{1}\phi_{2}-\dot{\phi}_{2}\phi_{1}\right)~{}.\vspace{0.1cm}$
(23)

Assuming that $2A\ll m_{\Phi}^{2}$, and neglecting inflaton decay for the
moment, the lepton asymmetry to leading order in $A/m_{\Phi}^{2}$ is

$\mbox{$$}n_{L}(t)=-2\phi_{1,\;*}\phi_{2,\;*}\left(\frac{a_{*}}{a}\right)^{3}\left[m_{\Phi}\sin\left(\frac{2A(t-t_{*})}{m_{\Phi}}\right)+\frac{A}{m_{\Phi}}\sin\left(2m_{\Phi}(t-t_{*})\right)\right]~{}.\vspace{0.1cm}$
(24)

On averaging over the rapid coherent oscillations of $\phi$, the second term
in Eq. (24) averages to zero and the condensate lepton asymmetry at $t>t_{*}$
is

$\mbox{$$}n_{L}^{0}(t)=-2\phi_{1,\;*}\phi_{2,\;*}\left(\frac{a_{*}}{a}\right)^{3}m_{\Phi}\sin\left(\frac{2A(t-t_{*})}{m_{\Phi}}\right)~{},\vspace{0.1cm}$
(25)

where the superscript $0$ denotes the absence of decays. Therefore the
asymmetry in the condensate will oscillate about zero with period $T_{asy}$
given by

$\mbox{$$}T_{asy}=\frac{\pi m_{\Phi}}{A}~{}.\vspace{0.1cm}$ (26)

Defining the comoving lepton asymmetry in the inflaton field by
$n_{L\;c}(t)\equiv(a(t)/a_{*})^{3}n_{L}(t)$, where $a_{*}$ is the scale factor
at $t_{*}$ ($n_{L\;c}(t)$ is constant when there is no production or decay of
the asymmetry), we obtain in the absence of decays kls1

$\mbox{$$}n_{L\;c}^{0}(t)=-\frac{\phi_{*}^{2}m_{\Phi}\sin(2\theta)}{2}\sin\left(\frac{2A(t-t_{*})}{m_{\Phi}}\right)~{}.\vspace{0.1cm}$
(27)

In the limit where $\Gamma_{\Phi}^{2}\ll m^{2}$ and $H^{2}\ll m^{2}$, the
inclusion of the decay terms multiplies the solutions for $\phi_{1}$ and
$\phi_{2}$ by a factor $e^{-\Gamma_{\Phi}(t-t_{*})/2}$ and $n_{L\;c}^{0}$ by a
factor $e^{-\Gamma_{\Phi}(t-t_{*})}$, therefore

$\mbox{$$}n_{L\;c}(t)=-\frac{\phi_{*}^{2}\sin(2\theta)m_{\Phi}e^{-\Gamma_{\Phi}(t-t_{*})}}{2}\sin\left(\frac{2A(t-t_{*})}{m_{\Phi}}\right)~{}.\vspace{0.1cm}$
(28)

The lepton asymmetry transferred to the SM sector by inflaton decays is then
given by

$\mbox{$$}\hat{n}_{L\;c}(t)=\int_{t_{*}}^{t}\Gamma_{\Phi}n_{L\;c}(t)dt=-\frac{\Gamma_{\Phi}\phi_{*}^{2}\sin(2\theta)m_{\Phi}}{2}\int_{t_{*}}^{t}e^{-\Gamma_{\Phi}(t-t_{*})}\sin\left(\frac{2A(t-t_{*})}{m_{\Phi}}\right)dt~{},\vspace{0.1cm}$
(29)

where ‘hat’ denotes the transferred asymmetry. The total comoving asymmetry
transferred to the SM sector in the limit $t\rightarrow\infty$ is then

$\mbox{$$}\hat{n}_{L\;c,\;tot}=\frac{-\Gamma_{\Phi}\phi_{*}^{2}\sin(2\theta)m_{\Phi}^{2}}{2A}\left(1+\left(\frac{\Gamma_{\Phi}m_{\Phi}}{2A}\right)^{2}\right)^{-1}~{}.\vspace{0.1cm}$
(30)

Figure 1: Numerical comoving condensate lepton asymmetry for the case
$\tau_{\Phi}=T_{asy}$, with $m_{\Phi}=10^{16}{\rm\ GeV}$,
$A^{1/2}=10^{13}{\rm\ GeV}$, $\lambda_{\Phi}=0.1$ and $\sin(2\theta)=-1$,
illustrating the oscillations of the condensate asymmetry about zero. Figure
2: Comoving transferred lepton asymmetry for the case $\tau_{\Phi}=T_{asy}$,
showing the non-zero net transferred asymmetry. Figure 3: The lepton-number to
entropy ratio for the case $\tau_{\Phi}=T_{asy}$. The late-time final value
almost coincides with the analytical prediction from Eq. (34).  Figure 4:
Numerical comoving condensate lepton asymmetry for the case
$\tau_{\Phi}=5T_{asy}$, with $m_{\Phi}=10^{16}{\rm\ GeV}$,
$A^{1/2}=10^{13}{\rm\ GeV}$, $\lambda_{\Phi}=0.1$ and $\sin(2\theta)=-1$ and
$T_{R}=2.1\times 10^{13}{\rm\ GeV}$, illustrating more rapid oscillations of
the condensate asymmetry. Figure 5: Comoving transferred lepton asymmetry for
the case $\tau_{\Phi}=5T_{asy}$, showing the non-zero net transferred
asymmetry. Figure 6: The lepton-number to entropy ratio for the case
$\tau_{\Phi}=5T_{asy}$. The late-time final value almost coincides with the
analytical prediction from Eq. (34). (We have shown the values of $n_{L}/s$
starting from $At/m_{\Phi}=10$ in order to emphasize the late-time value.)

There are two regimes for $A$, corresponding to the cases where the inflaton
lifetime $\tau_{\Phi}$ is short ($\Gamma_{\Phi}\gg 2A/m_{\Phi}$) and where it
long ($\Gamma_{\Phi}\ll 2A/m_{\Phi}$) compared to $2A/m_{\Phi}\equiv
T_{asy}/2\pi$. The resulting transferred asymmetry is

$\mbox{$$}\hat{n}_{c,\;tot}=\frac{-2A\phi_{*}^{2}\sin(2\theta)}{\Gamma_{\Phi}}\;\;\;\;\left(\tau_{\Phi}\ll\frac{T_{asy}}{2\pi}\right)~{}\vspace{0.1cm}$
(31)

and

$\mbox{$$}\hat{n}_{c,\;tot}=\frac{-\Gamma_{\Phi}\phi_{*}^{2}m_{\Phi}^{2}\sin(2\theta)}{2A}\;\;\;\;\left(\tau_{\Phi}\gg\frac{T_{asy}}{2\pi}\right)~{}.\vspace{0.1cm}$
(32)

The maximum possible lepton asymmetry is at $\Gamma_{\Phi}m_{\Phi}/2A=1$,
corresponding to $\tau_{\Phi}=T_{asy}/2\pi$,

$\mbox{$$}\hat{n}_{c,\;max}=-\frac{m_{\Phi}\phi_{*}^{2}\sin(2\theta)}{2}~{}.\vspace{0.1cm}$
(33)

In order to have an analytical relation for the baryon number to entropy of
the condensate, we will use the approximation that the Universe is matter
dominated until $\Gamma_{\phi}=H$, at which time the energy in the condensate
and the total final lepton number asymmetry is instantly transferred to
radiation and the SM sector lepton number asymmetry. We will check the
accuracy of this approximation later in this section. In this case, the decay
rate of the field is related to the reheating temperature $T_{R}$ by
$\Gamma_{\Phi}=k_{T_{R}}T_{R}^{2}/M_{Pl}$, where
$k_{T_{R}}=(\pi^{2}g(T_{R})/90)^{1/2}$ and $g(T)$ is the number of
relativistic degrees of freedom. To convert total the comoving lepton
asymmetry to the physical asymmetry at $T\leq T_{R}$ we multiply
$\hat{n}_{L\;tot}$ by
$(a_{*}/a(T_{R}))^{3}=H(T_{R})^{2}/H_{*}^{2}=6k_{T_{R}}^{2}T_{R}^{4}/m_{\Phi}^{2}\phi_{*}^{2}$.
The final lepton number to entropy after inflaton decay at $T_{R}$, where the
entropy density is given by $s=4k_{T}^{2}T^{3}$, is then

$\mbox{$$}\frac{n_{L}}{s}=-\frac{3}{4}\frac{k_{T_{R}}T_{R}^{3}\sin\left(2\theta\right)}{AM_{Pl}}\;\;\;\;\left(\tau_{\Phi}\gg\frac{T_{asy}}{2\pi}\right)~{}\vspace{0.1cm}$
(34)

and

$\mbox{$$}\frac{n_{L}}{s}=-\frac{3AM_{Pl}\sin(2\theta)}{k_{T_{R}}T_{R}m_{\phi}^{2}}\;\;\;\;\left(\tau_{\Phi}\ll\frac{T_{asy}}{2\pi}\right)~{}.\vspace{0.1cm}$
(35)

The maximum possible asymmetry is

$\mbox{$$}\frac{n_{L,\;max}}{s}=-\frac{3T_{R}\sin\left(2\theta\right)}{4m_{\Phi}}~{}.\vspace{0.1cm}$
(36)

The transferred lepton asymmetry is partially converted to a baryon asymmetry
via $(B+L)$-violating sphaleron fluctuations, with the final baryon asymmetry
to entropy ratio $n_{B}/s$ given by sph ; sph2

$\mbox{$$}\frac{n_{B}}{s}=-\frac{28}{79}\frac{\hat{n}_{L}}{s}~{}.\vspace{0.1cm}$
(37)

The final baryon asymmetry is then

$\mbox{$$}\frac{n_{B}}{s}=3.7\times
10^{-21}\,\frac{m_{\Phi}^{2}}{A}\left(\frac{T_{R}}{10^{8}{\rm\
GeV}}\right)^{3}\left(\frac{10^{13}{\rm\
GeV}}{m_{\Phi}}\right)^{2}\sin\left(2\theta\right)\;\;\;\;\left(\tau_{\Phi}\gg\frac{T_{asy}}{2\pi}\right)~{}\vspace{0.1cm}$
(38)

and

$\mbox{$$}\frac{n_{B}}{s}=7.7\times
10^{9}\,\frac{A}{m_{\Phi}^{2}}\left(\frac{10^{8}{\rm\
GeV}}{T_{R}}\right)\sin\left(2\theta\right)\;\;\;\;\left(\tau_{\Phi}\ll\frac{T_{asy}}{2\pi}\right)~{},\vspace{0.1cm}$
(39)

where we have normalised the expression to some representative values of
$T_{R}$ and $m_{\Phi}$.The maximum possible asymmetry is

$\mbox{$$}\frac{n_{B,\;max}}{s}=2.7\times
10^{-6}\,\left(\frac{T_{R}}{10^{8}{\rm\ GeV}}\right)\left(\frac{10^{13}{\rm\
GeV}}{m_{\Phi}}\right)\sin\left(2\theta\right)~{}.\vspace{0.1cm}$ (40)

The observed baryon-to-entropy ratio is $(n_{B}/s)_{obs}=0.861\pm 0.005\times
10^{-10}$. In order to account for the observed asymmetry, we therefore
require that

$\mbox{$$}\frac{A^{1/2}}{m_{\Phi}}=6.7\times
10^{-6}\sin^{1/2}\left(2\theta\right)\left(\frac{10^{13}{\rm\
GeV}}{m_{\Phi}}\right)\left(\frac{T_{R}}{10^{8}{\rm\
GeV}}\right)^{3/2}\;\;\;\;\left(\tau_{\Phi}\gg\frac{T_{asy}}{2\pi}\right)~{}\vspace{0.1cm}$
(41)

and

$\mbox{$$}\frac{A^{1/2}}{m_{\Phi}}=1.1\times
10^{-10}\,\left(\frac{T_{R}}{10^{8}{\rm\
GeV}}\right)^{1/2}\left(\frac{1}{\sin\left(2\theta\right)}\right)^{1/2}\;\;\;\;\left(\tau_{\Phi}\ll\frac{T_{asy}}{2\pi}\right)~{}.\vspace{0.1cm}$
(42)

###  IV.2 Numerical Lepton Asymmetry

In order to check our analytical expressions, we have solved the complete
field equations and included the continuous transfer of energy from the
decaying inflaton condensate to radiation via

$\mbox{$$}\frac{d\rho_{rad}}{dt}+4H\rho_{rad}=\Gamma_{\Phi}\rho_{\Phi}~{}.\vspace{0.1cm}$
(43)

In the figures we show the exact numerical solution for two cases,
$\tau_{\Phi}=T_{asy}$ and $\tau_{\Phi}=5T_{asy}$. In Figures 1-3 we show the
evolution of the comoving condensate asymmetry, the comoving transferred
asymmetry, and the lepton number-to-entropy for the case
$\tau_{\Phi}=T_{asy}$, with $m_{\Phi}=10^{16}{\rm\ GeV}$,
$A^{1/2}=10^{13}{\rm\ GeV}$ and $\lambda_{\Phi}=0.1$. (The comoving
asymmetries are given in Planck units.) For this case $T_{R}=4.8\times
10^{13}{\rm\ GeV}$ using the instantaneous decay expression. In Figures 4-6 we
show the corresponding results for the case with $\tau_{\Phi}=5T_{asy}$, in
which case $T_{R}=2.1\times 10^{13}{\rm\ GeV}$. In order to test the
analytical predictions we have chosen values of $m_{\Phi}$ and $\Gamma_{\Phi}$
which allow for numerical solution of the field equations without extreme
differences between the field oscillation time and the $\Phi$ lifetime, even
though these values do not produce realistic lepton asymmetries.

The analytical prediction for the lepton number-to-entropy ratio from Eq. (34)
for the case $\tau_{\Phi}=T_{asy}$ is

$\mbox{$$}\left(\frac{n_{L}}{s}\right)_{analytical}=1.15\times
10^{-3}~{},\vspace{0.1cm}$ (44)

where we have used $g(T_{R})=100$ and set $\sin(2\theta)=-1$ to have a
positive lepton asymmetry. The corresponding numerical result from Figure 3 is

$\mbox{$$}\left(\frac{n_{L}}{s}\right)_{numerical}=1.106\times
10^{-3}~{}.\vspace{0.1cm}$ (45)

The analytical prediction for the lepton number-to-entropy ratio for the case
$\tau_{\Phi}=5T_{asy}$ is

$\mbox{$$}\left(\frac{n_{L}}{s}\right)_{analytical}=1.03\times
10^{-4}~{}.\vspace{0.1cm}$ (46)

The corresponding numerical result from Figure 6 is

$\mbox{$$}\left(\frac{n_{L}}{s}\right)_{numerical}=9.80\times
10^{-5}~{}.\vspace{0.1cm}$ (47)

Therefore the analytical solution gives a very accurate estimate of the final
lepton-to-entropy ratio.

###  IV.3 Consistency of the assumption that AD leptogenesis occurs during
$|\Phi|^{2}$ domination

The above analysis assumes that the lepton asymmetry is entirely generated
when the potential is dominated by the $|\Phi|^{2}$ term. The condition for
this assumption to be valid is that the $A$-terms do not significantly
influence the motion of the $\phi_{1}$ and $\phi_{2}$ fields until
$\phi<\phi_{*}$, when the potential is dominated by the $|\Phi|^{2}$ term.
This is true if the mass of the angular field perturbations about the minimum
of the potential as a function of $\theta$ for a given $\phi$ is less than $H$
when $\phi\geq\phi_{*}$. This can be understood by considering the field
equations in terms of the radial and angular variables $\Phi=\phi
e^{i\theta}/\sqrt{2}$.

$\mbox{$$}\ddot{\phi}+3H\dot{\phi}=-(m_{\Phi}^{2}+2\lambda_{\Phi}|\Phi|^{2})\phi-2A\phi\cos(2\theta)~{}\vspace{0.1cm}$
(48)

and

$\mbox{$$}\phi\ddot{\theta}+2\dot{\theta}\dot{\phi}+3H\phi\dot{\theta}=2A\phi\sin(2\theta)~{}.\vspace{0.1cm}$
(49)

For $A$ real and positive, the minimum of the potential is along the direction
$\theta=\pi/2$. To estimate the condition for the angle $\theta$ to be
unaffected during the $\phi$ oscillations, we can consider a small
perturbation of $\theta$ from the minimum direction. For convenience, we
change variable to $\hat{\theta}=\pi/2-\theta$ so that the minimum direction
becomes $\hat{\theta}=0$. The equations then become

$\mbox{$$}\ddot{\phi}+3H\dot{\phi}=-(m_{\Phi}^{2}+2\lambda_{\Phi}|\Phi|^{2})\phi+2A\phi\cos(2\hat{\theta})~{}\vspace{0.1cm}$
(50)

and

$\mbox{$$}\phi\ddot{\hat{\theta}}+2\dot{\hat{\theta}}\dot{\phi}+3H\phi\dot{\hat{\theta}}=-2A\phi\sin(2\hat{\theta})~{}.\vspace{0.1cm}$
(51)

Then, for $\hat{\theta}\ll 1$, we have

$\mbox{$$}\ddot{\phi}+3H\dot{\phi}=-(m_{\Phi}^{2}-2\lambda_{\Phi}|\Phi|^{2}-2A)\phi~{}\vspace{0.1cm}$
(52)

and

$\mbox{$$}\phi\ddot{\hat{\theta}}+\left(3H+2\frac{\dot{\phi}}{\phi}\right)\phi\dot{\hat{\theta}}=4A\phi\hat{\theta}~{}.\vspace{0.1cm}$
(53)

From the RHS of Eq. (53), if we average over the $\phi$ oscillations and
consider $\phi$ to be a constant, and define a field
$\sigma=\phi\hat{\theta}$, then the mass of the $\sigma$ field is
$m_{\sigma}=\sqrt{4A}$.

So we can expect the dynamics of the angular field to be unaffected by the
$A$-term if $m_{\sigma}^{2}<H^{2}$. If true, we can consider effectively $A=0$
in Eq. (53) and so $\theta$ equal to a constant will be a solution. Therefore,
as long as $4A<H^{2}$, the AD dynamics due to the $A$-term will be negligible.
We therefore require that this is satisfied at
$\phi\;^{>}_{\sim}\;\phi_{*}=m_{\Phi}/\sqrt{\lambda_{\Phi}}$. Let $\phi_{AD}$
be the value of the field at which $4A=H^{2}$ and the phase field evolution
(and so Affleck-Dine dynamics) becomes important. We then require that
$\phi_{AD}<\phi_{*}$, which requires that

$\mbox{$$}4A<H_{*}^{2}~{}.\vspace{0.1cm}$ (54)

At $\phi=\phi_{*}$,
$V(\phi_{*})=m_{\Phi}^{2}\phi_{*}^{2}/2+\lambda_{\Phi}\phi_{*}^{4}/4=3m_{\Phi}^{4}/4\lambda_{\Phi}$
and
$H_{*}=(V(\phi_{*})/3M_{Pl}^{2})^{1/2}=m_{\Phi}^{2}/2\sqrt{\lambda_{\Phi}}M_{Pl}$.
The condition Eq. (54) then becomes

$\mbox{$$}\frac{A^{1/2}}{m_{\Phi}}<\frac{A_{th}^{1/2}}{m_{\Phi}}=\frac{m}{4\sqrt{\lambda_{\Phi}}M_{Pl}}=1.04\times
10^{-6}\lambda_{\Phi}^{-1/2}\left(\frac{m_{\Phi}}{10^{13}{\rm\
GeV}}\right)~{},\vspace{0.1cm}$ (55)

where $A_{th}$ is the value of $A$ below which the threshold approximation is
valid. For $A<A_{th}$, the asymmetry will be entirely generated during
$\phi^{2}$ oscillations and the analytical approximation is valid. In kls1  it
was confirmed numerically that if Eq. (55) is satisfied then the analytical
prediction for the asymmetry based on $|\Phi|^{2}$ domination is valid. If
this condition is not satisfied and the $L$-violating mass terms become
dynamical during the $\phi^{4}$ oscillation-dominated era, then there is an
additional suppression of the asymmetry due to the evolution of the phase
during $\phi^{4}$-dominated oscillations kls1 .

###  IV.4 Consistency with Non-Minimally Coupled Inflation

In our calculation of the asymmetry we have assumed that the effect of the
non-minimal coupling is negligible. This requires that the
$\phi^{2}$-dominated evolution begins after the modification of the dynamics
due to the non-minimal coupling becomes negligible. We therefore require that
$\phi_{*}<\phi_{c}$, where $\phi_{c}$ is the value of the inflaton field at
which the non-minimal coupling strongly modifies the dynamics. For the metric
case $\phi_{c}$ is given by Eq. (15) and for the Palatini case by Eq. (16).

The condition $\phi_{*}=m_{\Phi}/\sqrt{\lambda_{\Phi}}<\phi_{c}$ then gives
for the metric case

$\mbox{$$}m_{\Phi}<\frac{\sqrt{\lambda_{\Phi}}M_{Pl}}{\sqrt{6}\xi}~{}\vspace{0.1cm}$
(56)

and for the Palatini case

$\mbox{$$}m_{\Phi}<\frac{\sqrt{\lambda_{\Phi}}M_{Pl}}{\sqrt{\xi}}~{}.\vspace{0.1cm}$
(57)

Using the expressions Eq. (13) and Eq. (14) for $\xi$ at $N=55$, which give
$\xi=4.5\times 10^{4}\sqrt{\lambda_{\Phi}}$ for the metric case and
$\xi=1.2\times 10^{10}\,\lambda_{\Phi}$ for the Palatini case, we obtain the
same condition for both the metric and Palatini cases

$\mbox{$$}m_{\Phi}<2.2\times 10^{13}{\rm\ GeV}~{}.\vspace{0.1cm}$ (58)

##  V Lepton Asymmetry Transfer, Reheating and Washout

###  V.1 Lepton Asymmetry Transfer and Reheating

We assume that reheating occurs through the perturbative decay of the
condensate scalars to RH neutrinos. In this case, reheating and lepton number
transfer to the SM sector occur simultaneously. The inflaton decay rate for
$\Phi\rightarrow N_{R}N_{R}$ is

$\mbox{$$}\Gamma_{\Phi}=\frac{y_{\Phi}^{2}m_{\Phi}}{8\pi}~{}.\vspace{0.1cm}$
(59)

As usual, we define $T_{R}$ to be the effective temperature at which the
Universe becomes dominated by relativistic particles after $\Phi$ decay, in
this case right-handed neutrinos. The RH neutrinos will subsequently decay to
light SM particles via $N_{R}\rightarrow HL$, at which time the Universe
rapidly thermalises to a temperature $T_{th}\leq T_{R}$. If the $N_{R}$ decay
rate is fast enough that they decay immediately after being produced, then
$T_{th}=T_{R}$.

As a specific example to show that reheating and lepton asymmetry transfer can
easily be achieved without washout occurring due to L-violating scattering
processes, we consider the case with $m_{\Phi}=10^{13}{\rm\ GeV}$ and
$T_{R}=10^{8}{\rm\ GeV}$, with $m_{N}$ assumed to be small compared to the
inflaton mass but large compared to the reheating temperature,
$m_{N}=10^{10}{\rm\ GeV}$. In this case the number density of RH neutrinos in
thermal equilibrium is strongly Boltzmann suppressed. The reheating
temperature is determined by
$\Gamma_{\Phi}=H(T_{R})=k_{T_{R}}T_{R}^{2}/M_{Pl}$. The coupling $y_{\Phi}$
required for a given effective decay temperature is then

$\mbox{$$}y_{\Phi}=1.9\times 10^{-7}\left(\frac{10^{13}{\rm\
GeV}}{m_{\Phi}}\right)^{1/2}\left(\frac{T_{R}}{10^{8}{\rm\
GeV}}\right)~{}.\vspace{0.1cm}$ (60)

The rest frame $N_{R}\rightarrow HL$ decay rate is

$\mbox{$$}\Gamma_{N_{R}}=\frac{h_{\nu}^{2}m_{N}}{8\pi}=\frac{m_{\nu}m_{N}^{2}}{4\pi
v^{2}}~{},\vspace{0.1cm}$ (61)

where we have used Eq. (3). The energy of $N_{R}$ immediately after pair
production will be $E_{N}=m_{\Phi}/2$. Assuming that $m_{\Phi}\gg m_{N}$, the
produced $N_{R}$ are relativistic, $E_{N}\gg m_{N}$, and so the decay rate of
the $N_{R}$ will be reduced by the time-dilation factor $m_{N}/E_{N}$. The
condition that $N_{R}$ can decay immediately after production is then

$\mbox{$$}\frac{m_{N}}{E_{N}}\Gamma_{N_{R}}>H(T_{R})\Rightarrow\frac{m_{\nu}m_{N}^{3}}{2\pi
v^{2}m_{\Phi}}>\frac{k_{T}T_{R}^{2}}{M_{Pl}}~{}.\vspace{0.1cm}$ (62)

This is satisfied if

$\mbox{$$}m_{\nu}>5.2\times 10^{-5}{\rm\
eV}\times\left(\frac{m_{\Phi}}{10^{13}{\rm\
GeV}}\right)\left(\frac{10^{10}{\rm\
GeV}}{m_{N}}\right)^{3}\left(\frac{T_{R}}{10^{8}{\rm\
GeV}}\right)^{2}~{}.\vspace{0.1cm}$ (63)

Thus if we consider $m_{\nu}$ to be of the order of the observed neutrino mass
splittings, $m_{\nu}\sim 0.01-0.1{\rm\ eV}$, then in this example both
reheating and the thermalisation of the SM thermal background will occur
immediately when the inflaton decays, since the RH neutrinos immediately decay
to SM particles.

###  V.2 Absence of Washout

We next consider the condition for the SM lepton asymmetry to avoid washout
after being transferred from the inflaton condensate. Processes involving
thermal background RH neutrinos will be very strongly Boltzmann suppressed,
since $m_{N}/T_{R}=100$ for this example. $\Delta L=2$ scattering processes
can occur via heavy $N$ exchange between $LH$. The resulting Weinberg operator
is

$\mbox{$$}\frac{h_{\nu}^{2}}{M_{N}}LHLH~{}\vspace{0.1cm}$ (64)

and the thermal scattering rate is wein

$\mbox{$$}\Gamma_{W}=\frac{h_{\nu}^{4}}{4\pi}\frac{T^{3}}{m_{N}^{2}}=\frac{m_{\nu}^{2}T^{3}}{\pi
v^{4}}~{},\vspace{0.1cm}$ (65)

where have used the relation Eq. (3). Requiring that this is less than $H$ at
reheating gives the bound

$\mbox{$$}T_{R}<\frac{k_{T}\pi v^{4}}{m_{\nu}^{2}M_{Pl}}=1.6\times
10^{12}\left(\frac{0.1{\rm\ eV}}{m_{\nu}}\right)^{2}{\rm\
GeV}~{}.\vspace{0.1cm}$ (66)

Therefore, for $m_{N}=10^{10}{\rm\ GeV}$, $T_{R}=10^{8}{\rm\ GeV}$ and
$m_{\nu}\;^{<}{}_{\sim}\;0.1{\rm\ eV}$, there is no danger of washout of the
lepton asymmetry after it transfers to the SM sector.

Thus with $m_{\Phi}=10^{13}$ GeV, $m_{N}=10^{10}{\rm\ GeV}$ and $T_{R}=10^{8}$
GeV, the inflaton condensate will decay to relativistic RH neutrinos pairs
which immediately decay to $LH$ pairs and thermalise the SM thermal
background. Thermal L-violating scattering processes are ineffective,
therefore the lepton asymmetry successfully transfers to the SM sector without
danger of washout. The final baryon asymmetry is produced by $(B+L)$-violating
sphaleron fluctuations, which partially convert the lepton asymmetry to a
baryon asymmetry.

We note that reheating temperature in this example, $T_{R}=10^{8}{\rm\ GeV}$,
is within the range $10^{6}-10^{8}{\rm\ GeV}$ that may be observable in the
spectrum of primordial gravitational waves with $r=0.001-0.1$ gravrh . This is
in contrast to the case of thermal leptogenesis, which requires that
$T_{R}\;^{>}{}_{\sim}\;10^{9}{\rm\ GeV}$ lept ; lept2 ; lept3 . For the metric
non-minimally coupled inflation model, the tensor-to-scalar index is
$r=0.004$, which is large enough to be observed by next generation telescopes
such as LiteBird ltb . Therefore the model could explain both the observed
baryon asymmetry and a future observation of primordial gravitational waves
with a reheating signature in their spectrum.

###  V.3 Condensate Decay via the Higgs portal, via the Non-Minimal Coupling
and via Gravitational Decay

####  V.3.1 Decay via the Higgs Portal

In the above analysis, we have considered reheating to occur through
perturbative decays of the inflaton to right-handed neutrinos. In general, we
expect a Higgs portal coupling of the inflaton to the Higgs doublet bosons to
exist, as it cannot be suppressed by any symmetry. This can allow condensate
decay via annihilation of the condensate scalars. Here we will estimate an
upper limit on the coupling of the inflaton to the Higgs boson doublet for
condensate decay to right-handed neutrinos to dominate reheating.

The portal coupling is

$\mbox{$$}\lambda_{\Phi H}|\Phi|^{2}|H|^{2}=\frac{\lambda_{\Phi
H}}{4}\phi^{2}\sum_{i=1}^{4}h_{i}^{2}~{},\vspace{0.1cm}$ (67)

where $h_{i}$ are the real scalars of the Higgs doublet. To estimate the decay
of the condensate to the Higgs doublet scalars, we will use the results given
in Appendix A of cond . For a single real Higgs scalar in Eq. (67), the
condensate decay rate has the form

$\mbox{$$}\Gamma_{portal}=\frac{C\,\lambda_{\Phi H}^{2}\rho_{\phi}}{256\pi
m_{eff}^{3}}~{},\vspace{0.1cm}$ (68)

where the effective mass of the condensate scalars is
$m_{eff}^{2}=V^{{}^{\prime\prime}}(\phi)$ and $\rho_{\phi}\equiv V(\phi)$,
where $\phi$ is the amplitude of oscillation. The constant $C$ depends on the
potential. For the $\phi^{2}$ potential, $m_{eff}=m_{\phi}$ and $C=1$. For the
$\phi^{4}$ potential, $m_{eff}=\sqrt{3\lambda_{\Phi}}\phi$ and $C\approx 18$
cond .

We wish to ensure that the condensate decay rate via the Higgs portal is less
than $H$ for all $\phi$. We first derive the condition on $\lambda_{\Phi H}$
for this to be true at $\phi<\phi_{*}$. We then show that if decay does not
occur at $\phi<\phi_{*}$ then it will not occur at $\phi>\phi_{*}$.

At $\phi<\phi_{*}$ we have a $\phi^{2}$ potential. Therefore
$\rho_{\phi}=m_{\Phi}^{2}\phi^{2}/2$ and

$\mbox{$$}\Gamma_{portal}\approx\frac{\lambda_{\Phi H}^{2}\phi^{2}}{128\pi
m_{\Phi}}~{},\vspace{0.1cm}$ (69)

where we have summed over all four real scalars of the Higgs doublet. The
condition that $\Gamma_{portal}\;^{<}{}_{\sim}\;H=m\phi/\sqrt{6}M_{Pl}$ at
$\phi<\phi_{*}$ is strongest at $\phi=\phi_{*}$, where it requires that

$\mbox{$$}\lambda_{\Phi
H}\;^{<}{}_{\sim}\;\left(\frac{128\pi}{\sqrt{6}}\right)^{1/2}\left(\frac{m_{\Phi}}{M_{Pl}}\right)^{1/2}\lambda_{\Phi}^{1/4}~{}.\vspace{0.1cm}$
(70)

Therefore

$\mbox{$$}\lambda_{\Phi H}\;^{<}{}_{\sim}\;2.6\times
10^{-2}\,\lambda_{\Phi}^{1/4}\left(\frac{m_{\Phi}}{10^{13}{\rm\
GeV}}\right)^{1/2}~{}.\vspace{0.1cm}$ (71)

Thus only a moderate suppression of the Higgs portal coupling is required in
the example we are considering.

At $\phi>\phi_{*}$ we have $m_{eff}\propto\phi$ and $V\propto\phi^{4}$.
Therefore $\Gamma_{portal}\propto V(\phi)/m_{eff}^{3}\propto\phi$. Since
$H\propto V^{1/2}\propto\phi^{2}$, it follows that if $\Gamma_{portal}<H$ at
$\phi=\phi_{*}$, then this will also be true at $\phi>\phi_{*}$. Thus Eq. (71)
is sufficient to ensure that decay via the Higgs portal is negligible.

If the portal coupling were to dominate reheating, then the symmetric
component of the inflaton condensate could decay earlier, but the asymmetric
component would remain in the form of a maximally asymmetric $\Phi$
condensate, with a circular orbit in the field space, until decay to RH
neutrinos occurred. As a result, there could be two separate reheating events,
with initial reheating leaving a maximally asymmetric inflaton condensate,
which could later come to dominate the radiation density and eventually decay
to RH neutrinos, reheating the Universe a second time.

####  V.3.2 Decay via the Non-Minimal Coupling

The inflaton has a large non-minimal quadratic coupling to $R$, suggesting
that annihilations to SM particles via the non-minimal coupling could be
important. To estimate the decay rate of the condensate to SM particles, we
will extend the non-minimally coupled action to include the SM Higgs doublet.

In the Einstein frame, even in the absence of perturbative interactions
between the inflaton and SM sectors, the non-minimal coupling will create non-
renormalisable interactions. The important terms in the Einstein frame
Lagrangian are

$\mbox{$$}-\frac{1}{\Omega^{2}}\partial_{\mu}H^{\dagger}\partial^{\mu}H-\frac{\lambda_{h}|H|^{4}}{\Omega^{4}}~{}.\vspace{0.1cm}$
(72)

During $\phi$ oscillations after inflation we have $\xi\phi^{2}/M_{Pl}^{2}\ll
1$ and we can therefore expand the conformal factors. We then obtain for the
leading order non-renormalisable interactions between $\phi$ and each real
Higgs scalar $h_{i}$,

$\mbox{$$}\frac{\xi}{2M_{Pl}^{2}}\phi^{2}\sum_{i=1}^{4}\partial_{\mu}h_{i}\partial^{\mu}h_{i}+\frac{\lambda_{h}\xi}{2M_{Pl}^{2}}\phi^{2}\left(\sum_{i=1}^{4}h_{i}^{2}\right)^{2}~{}.\vspace{0.1cm}$
(73)

These interactions allow the scalars in the condensate to annihilate to Higgs
scalars.

To estimate the annihilation rate, we will consider the first interaction in
Eq. (73); the second should give a similar annihilation rate, having a
suppression due to $\lambda_{h}$ but an enhancement due to the larger phase
space for decay to 4$h_{i}$. Since we have annihilation of zero momentum
scalars, the final state Higgs will have energy $E_{h}=m_{\Phi}/2$ and momenta
$k_{1}$ and $k_{2}$ such that $k_{1}.k_{2}=2E_{h}^{2}=m_{\Phi}^{2}/2$. We can
therefore replace the first term in Eq. (73) by an effective portal coupling

$\mbox{$$}\sum_{i=1}^{4}\frac{\xi
m_{\Phi}^{2}}{4M_{Pl}^{2}}\phi^{2}h_{i}^{2}~{}.\vspace{0.1cm}$ (74)

We can then apply the previous analysis of the portal coupling with
$\lambda_{\Phi H}\rightarrow\xi m_{\Phi}^{2}/M_{Pl}^{2}$. From Eq. (70) the
condition for annihilations to Higgs bosons via the non-minimal coupling to be
negligible is

$\mbox{$$}m_{\Phi}\;^{<}{}_{\sim}\;\frac{\lambda_{\Phi}^{1/6}}{\xi^{2/3}}\left(\frac{128\pi}{\sqrt{6}}\right)^{1/3}M_{Pl}~{}\vspace{0.1cm}$
(75)

For metric inflation with $N=55$ we have $\xi=4.5\times
10^{4}\sqrt{\lambda_{\Phi}}$. Therefore Eq. (75) becomes

$\mbox{$$}m_{\Phi}\;^{<}{}_{\sim}\;\frac{1.0\times 10^{16}{\rm\
GeV}}{\lambda_{\Phi}^{1/6}}~{}.\vspace{0.1cm}$ (76)

For Palatini inflation with $N=55$ we have $\xi=1.2\times
10^{10}\lambda_{\Phi}$ and Eq. (75) becomes

$\mbox{$$}m_{\Phi}\;^{<}{}_{\sim}\;\frac{2.5\times 10^{12}{\rm\
GeV}}{\lambda_{\Phi}^{1/2}}~{}.\vspace{0.1cm}$ (77)

The bound on $m_{\Phi}$ is easily satisfied for metric inflation when
$m_{\Phi}$ is small enough for non-minimal dynamics to be negligible,
$m_{\Phi}<2.2\times 10^{13}{\rm\ GeV}$. For Palatini inflation, a mild
suppression of $\lambda_{\Phi}$, with $\lambda_{\Phi}$ less than about 0.01,
would be necessary to satisfy Eq. (77) at the dynamical upper limit of
$m_{\Phi}$.

####  V.3.3 Gravitational Decay of the Condensate

Assuming that the $\phi\leftrightarrow-\phi$ symmetry of the potential is
unbroken by gravitational interactions, any purely gravitational decay of the
condensate will involve graviton exchange processes with $\phi^{2}$ and
therefore annihilation of the condensate scalars to SM particles. In this case
we expect that annihilation via the non-minimal coupling, which is
characterised by a mass scale $M_{Pl}/\sqrt{\xi}$, will easily dominate any
purely gravitational annihilation mode scaled by $M_{Pl}$. Therefore we expect
that direct gravitational decay of the condensate will generally be negligible
if decay via the non-minimal coupling is negligible.

It is known that gravitational instantons can break global symmetries, which
could lead to $U(1)_{L}$\- and $Z_{2}$-breaking couplings of the inflaton to
SM particles. However, being a non-perturbative effect, the strength of the
couplings is strongly dependent upon the UV completion of gravity, with an
exponential suppression $e^{-S}$ due to the instanton tunnelling action $S$ in
addition to suppression by the Planck scale. Therefore there is no reason to
assume that this effect will significantly modify reheating.

##  VI Baryon Isocurvature Perturbations

We next consider the possibility of observable baryon isocurvature
perturbations in this model. For $A^{1/2}<m_{\Phi}\ll H$ during inflation, the
phase $\theta$ of $\Phi$ is effectively a massless field. Therefore quantum
fluctuations of $\theta$ will give rise to baryon isocurvature perturbations.
The magnitude of the isocurvature perturbations will depend upon the specific
non-minimally coupled inflation model being considered: metric or Palatini. In
non-minimally coupled inflation models in the Einstein frame, the kinetic term
for $\phi_{2}$ (where $\Phi=(\phi_{1}+i\phi_{2})/\sqrt{2}$) is

$\mbox{$$}\frac{1}{2\Omega^{2}}\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}~{},\vspace{0.1cm}$
(78)

where

$\mbox{$$}\Omega^{2}=1+\frac{\xi(\phi_{1}^{2}+\phi_{2}^{2})}{M_{Pl}^{2}}~{}.\vspace{0.1cm}$
(79)

During inflation, the inflaton, which we choose to be $\phi_{1}$, has a slow-
rolling background value $\overline{\phi}_{1}$, whilst by defining $\theta$ to
be zero along $\phi_{1}$ we can consider $\phi_{2}$ to be purely due to
quantum fluctuations. Treating $\overline{\phi}_{1}$ as effectively constant
on the time scales over which the $\phi_{2}$ quantum fluctuations are
produced, and with $\xi\overline{\phi}_{1}^{2}/M_{Pl}^{2}\gg 1$ during
inflation, the $\phi_{2}$ kinetic term is

$\mbox{$$}\frac{M_{Pl}^{2}}{2\xi\overline{\phi}_{1}^{2}}\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}~{}.\vspace{0.1cm}$
(80)

The fluctuations of the phase $\theta$ are related to the fluctuations of
$\phi_{2}$ by

$\mbox{$$}\Phi=\frac{\phi}{\sqrt{2}}e^{i\theta}=\frac{1}{\sqrt{2}}(\phi_{1}+i\phi_{2})\Rightarrow\frac{\overline{\phi}}{\sqrt{2}}(1+i\delta\theta)=\frac{1}{\sqrt{2}}(\overline{\phi}_{1}+i\delta\phi_{2})\Rightarrow\delta\theta=\frac{\delta\phi_{2}}{\overline{\phi}_{1}}\;\;,\;\;\overline{\phi}_{1}=\overline{\phi}~{}.\vspace{0.1cm}$
(81)

To obtain the quantum fluctuation of $\phi_{2}$, which will give the
fluctuation of the phase, we transform to a canonically normalised field
$\chi_{2}$, where

$\mbox{$$}\chi_{2}=\frac{M_{Pl}}{\sqrt{\xi}\overline{\phi}_{1}}\phi_{2}~{}.\vspace{0.1cm}$
(82)

Therefore

$\mbox{$$}\delta\theta=\frac{\sqrt{\xi}}{M_{Pl}}\delta\chi_{2}~{}.\vspace{0.1cm}$
(83)

Since $\chi_{2}$ is a massless field, it will develop a quantum fluctuation
with the standard power spectrum

$\mbox{$$}{\cal P}_{\delta\chi_{2}}=\frac{H^{2}}{4\pi^{2}}~{}.\vspace{0.1cm}$
(84)

The corresponding power spectrum of the phase fluctuations is then

$\mbox{$$}{\cal P}_{\delta\theta}=\frac{\xi}{M_{Pl}^{2}}{\cal
P}_{\delta\chi_{2}}=\frac{\xi H^{2}}{4\pi^{2}M_{Pl}^{2}}~{}.\vspace{0.1cm}$
(85)

The global $U(1)_{L}$ symmetry ensures that this fluctuation applies at all
values of $\theta$. From Eq. (34) and Eq. (35), the lepton number and hence
baryon number is proportional to $\sin(2\theta)$, therefore

$\mbox{$$}\frac{\delta
n_{B}}{n_{B}}=\frac{2\delta\theta}{\tan(2\theta)}~{}.\vspace{0.1cm}$ (86)

Fluctuations of $\theta$ will therefore produce an isocurvature perturbation

$\mbox{$$}{\cal I}=\frac{\Omega_{b}}{\Omega_{dm}}\frac{\delta
n_{B}}{n_{B}}=\frac{\Omega_{b}}{\Omega_{dm}}\frac{2\delta\theta}{\tan(2\theta)}~{},\vspace{0.1cm}$
(87)

where we have written this as an equivalent CDM isocurvature perturbation
${\cal I}$ for comparison with the results of Planck planckiso . Planck gives
constraints in terms of the parameter $\beta_{iso}$ for CDM isocurvature
perturbations, where

$\mbox{$$}\beta_{iso}=\frac{{\cal P}_{I}}{{\cal P}_{R}+{\cal
P}_{I}}~{}\vspace{0.1cm}$ (88)

and ${\cal P}_{R}\equiv A_{s}$ is the curvature perturbation power. From Eq.
(87),

$\mbox{$$}{\cal
P_{I}}=\left(\frac{\Omega_{b}}{\Omega_{dm}}\right)^{2}\frac{4}{\tan^{2}(2\theta)}{\cal
P}_{\delta\theta}=\left(\frac{\Omega_{b}}{\Omega_{dm}}\right)^{2}\frac{1}{\tan^{2}(2\theta)}\frac{\xi
H^{2}}{\pi^{2}M_{Pl}^{2}}~{}.\vspace{0.1cm}$ (89)

The ratio ${\cal P}_{I}/{\cal P}_{R}$ is related to $\beta_{iso}$ by

$\mbox{$$}\frac{{\cal P}_{I}}{{\cal
P}_{R}}=\frac{\beta_{iso}}{1-\beta_{iso}}~{}.\vspace{0.1cm}$ (90)

Assuming that $\beta_{iso}\ll 1$, the prediction for $\beta_{iso}$ is

$\mbox{$$}\beta_{iso}=\frac{\xi
H^{2}}{\pi^{2}M_{Pl}^{2}A_{s}\tan^{2}(2\theta)}\left(\frac{\Omega_{b}}{\Omega_{dm}}\right)^{2}~{}.\vspace{0.1cm}$
(91)

In both the metric and the Palatini models, the value of the Einstein frame
potential on the plateau during inflation is given by

$\mbox{$$}V_{E}=\frac{\lambda_{\Phi}M_{Pl}^{4}}{4\xi^{2}}~{}.\vspace{0.1cm}$
(92)

Thus

$\mbox{$$}H=\left(\frac{\lambda_{\Phi}}{12}\right)^{1/2}\frac{M_{Pl}}{\xi}~{}\vspace{0.1cm}$
(93)

Therefore the model predicts that

$\mbox{$$}\beta_{iso}=\frac{\lambda_{\Phi}}{12\pi^{2}\xi
A_{s}\tan^{2}(2\theta)}\left(\frac{\Omega_{b}}{\Omega_{dm}}\right)^{2}~{}.\vspace{0.1cm}$
(94)

The value of $\xi$ at $N=55$ in the metric model is $\xi=4.5\times
10^{4}\,\sqrt{\lambda_{\Phi}}$. Therefore the metric model predicts that

$\mbox{$$}\beta_{iso,\,metric}=\frac{3.1\sqrt{\lambda_{\Phi}}}{\tan^{2}(2\theta)}~{}.\vspace{0.1cm}$
(95)

Planck (2018) obtains the 2-$\sigma$ upper bound $\beta_{iso}<0.038$ planckiso
. Therefore the metric model satisfies the isocurvature bound if

$\mbox{$$}\lambda_{\Phi}<1.5\times
10^{-4}\,\tan^{4}(2\theta)~{}.\vspace{0.1cm}$ (96)

The value of $\xi$ at $N=55$ in the Palatini model is $\xi=1.2\times
10^{10}\,\lambda_{\Phi}$. Therefore the Palatini model predicts that

$\mbox{$$}\beta_{iso,\;Palatini}=\frac{1.2\times
10^{-5}}{\tan^{2}(2\theta)}~{}.\vspace{0.1cm}$ (97)

Thus the Palatini model easily satisfies the present observational bound
independently of $\lambda_{\Phi}$, assuming that $\tan(2\theta)$ is not
unusually small.

Therefore there is an upper bound on $\lambda_{\Phi}$ for the metric model to
be consistent with the present bound on isocurvature perturbations.
Significantly, baryon isocurvature perturbations close to the present
observational limit are possible in the metric model if $\lambda_{\Phi}\sim
10^{-4}\tan^{4}(2\theta)$. Therefore the metric model would be able to explain
a future observation of baryon isocurvature perturbations close to the present
limit. In contrast, the Palatini model prediction is much smaller than the
present isocurvature bound independently of $\lambda_{\Phi}$, assuming that
$\tan(2\theta)$ is not unusually small.

##  VII Quadratic versus Quartic $U(1)_{L}$-Breaking Terms

In our analysis we have assumed that leptogenesis is due to the quadratic
A-term. In general, we would also expect a quartic C-term to exist. AD
leptogenesis via a C-term in non-minimally coupled inflation has been studied
in cline . The dynamics in this case is quite different, with the asymmetry
being generated during inflation rather than at late times. Here we consider
the limits on $C$ for which the A-term will dominate leptogenesis. The
$U(1)_{L}$-breaking potential terms are

$\mbox{$$}A(\Phi^{2}+\Phi^{\dagger\,2})+C(\Phi^{4}+\Phi^{\dagger\,4})~{}.\vspace{0.1cm}$
(98)

Thus the C-term, for a given $\phi$, will be less important to the field
dynamics than the A-term if $C\phi^{2}\;{}^{<}_{\sim}\;A$. We will first
consider this condition at $\phi\leq\phi_{*}$, when the potential is
$\phi^{2}$ dominated. $C\phi^{2}\;{}^{<}_{\sim}\;A$ at $\phi=\phi_{*}$
requires that

$\mbox{$$}C\phi_{*}^{2}=\frac{Cm_{\Phi}^{2}}{\lambda_{\Phi}}\;^{<}_{\sim}\;A\Rightarrow\frac{C}{\lambda_{\Phi}}\;^{<}_{\sim}\;\frac{A}{m_{\Phi}^{2}}~{}.\vspace{0.1cm}$
(99)

In other words, if the relative contribution of the C-term to the quartic term
is no greater than the contribution of the A-term to the quadratic term, the
A-term will be dominant at $\phi\leq\phi_{*}$. It is still possible that at
$\phi>\phi_{*}$ the quartic term could become important for large enough
$\phi$. However, to influence the evolution of the phase and so the baryon
asymmetry, $C\phi^{2}$ would also have to become larger than $H^{2}$, where we
are considering $C\phi^{2}$ to be approximately the mass squared of the phase
field. At $\phi=\phi_{*}$, since we assume that $\phi_{AD}<\phi_{*}$, the
phase is not dynamical and so $A\;^{<}_{\sim}\;H^{2}$ at $\phi=\phi_{*}$.
Therefore $C\phi_{*}^{2}$ is also less than $H^{2}$ at $\phi=\phi_{*}$. Then
since $H^{2}\propto\phi^{4}$ at $\phi>\phi_{*}$, it follows that $C\phi^{2}$
can never exceed $H^{2}$ at $\phi>\phi_{*}$ if it is less than $H^{2}$ at
$\phi=\phi_{*}$. Therefore the condition Eq. (99) is sufficient for the A-term
to dominate the dynamics of the phase field throughout and so to dominate
asymmetry generation.

##  VIII Conclusions

We have discussed a minimal leptogenesis model based on the inflaton mass term
Affleck-Dine mechanism introduced in kls1 . The inflaton sector is a non-
minimally coupled complex inflaton, which couples to the Standard Model via RH
neutrinos. We have reviewed the analytical predictions for the baryon
asymmetry and compared these to some examples of complete numerical solutions,
confirming their accuracy. We have also derived conditions for the consistency
of the analytical results. Using the analytical expressions, we have shown
that the model can easily generate the observed baryon asymmetry. In the case
of the model with a metric non-minimally coupled inflaton sector, the model
can produce baryon isocurvature perturbations that are close to the present
observational bound. The model is also consistent with the range of reheating
temperatures, $T_{R}=10^{6}-10^{8}{\rm\ GeV}$, that could be detected in the
spectrum of observable primordial gravitational waves predicted by the metric
model. This is in contrast to the case of conventional thermal leptogenesis,
which requires that $T_{R}\;^{>}{}_{\sim}\;10^{9}{\rm\ GeV}$. The Palatini
model predicts that both the baryon isocurvature perturbations and primordial
gravitational waves are much smaller than the present and the expected future
observational limits.

## Acknowledgements

KLS was funded by STFC while this project was undertaken.

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# Iterated Brownian motion ad libitum is not the pseudo-arc

Jérôme Casse and Nicolas Curien Université Paris-Dauphine. Email:
<EMAIL_ADDRESS>Paris-Saclay and Institut Universitaire
de France. E-mail<EMAIL_ADDRESS>
###### Abstract

We show that the construction of a random continuum $\mathcal{C}$ from
independent two-sided Brownian motions as considered in [11] almost surely
yields a non-degenerate indecomposable but not-hereditary indecomposable
continuum. In particular $\mathcal{C}$ is (unfortunately) not the pseudo-arc.

##  1 Introduction

#### Iterated Brownian motions ad libitum.

Let $(\mathfrak{B}_{i})_{i\geq 1}$ be a sequence of i.i.d. two-sided Brownian
motions (BM), i.e. $(\mathfrak{B}_{i}(t))_{t\geq 0}$ and
$(\mathfrak{B}_{i}(-t))_{t\geq 0})$ are independent standard linear Brownian
motions started from $0$. The $n$th iterated BM is

$I^{(n)}=\mathfrak{B}_{1}\circ\dots\circ\mathfrak{B}_{n}.$ (1)

The doubly iterated Brownian motion $I^{(2)}$ has been deeply studied in the
90’s. It permits to construct solutions to partial differential equations [9]
and lots of results about its probabilistic and analytic properties can be
found in [1, 4, 5, 8, 10, 16, 17] and references therein. Of course $I^{(n)}$
is wilder and wilder as $n$ increases (see Figure 1) but in [7], second author
and Konstantopoulos proved that _the occupation measure_ of $I^{(n)}$ over
$[0,1]$ converges as $n\to\infty$ towards a random probability measure $\Xi$
which can be though of as iterated Brownian motions ad libitum. This object
has then been studied in [6] by the first author and Marckert, and they gave a
description of $\Xi$ using invariant measure of an iterated functions system
(IFS). However, many distributional properties of $\Xi$ remain open.

Figure 1: Simulations of $I^{{(1)}},I^{{(2)}}$ and $I^{{(3)}}$, the first
three iteration of independent two-sided Brownian motions. The article studies
random continuum build out the sequence of $(I^{{(n)}}:n\geq 1)$.

#### Continuum and pseudo-arc.

In a recent work, Kiss and Solecki used iterated Brownian motions to define a
random _continuum_. Recall that a _continuum_ is a nonempty, compact,
connected metric space. They were interested by the so-called _pseudo-arc_.
The _pseudo-arc_ is a homogeneous continuum which is similar to an arc, so
similar, that its existence was unclear in the beginning of the last century.
A continuum $C$ is

  * •

_chainable_ (also called _arc-like_ , see [15, Theorem 12.11]), if for each
$\varepsilon>0$, there exists a continuous function $f:C\to[0,1]$ such that
the pre-images of points under $f$ have diameter less than $\varepsilon$.

  * •

_decomposable_ , if there exist $A$ and $B$ two subcontinua of $C$ such that
$A,B\neq C$ and $C=A\cup B$. A non decomposable continuum is called
_indecomposable_.

  * •

_hereditarily indecomposable_ if any of its subcontinuum (non reduced to a
singleton) is indecomposable.

By [3], the _pseudo-arc_ is the unique (up to homeomorphisms) chainable and
hereditarily indecomposable continuum non reduced to a singleton. In
particular, any subcontinuum (non reduced to a singleton) of a pseudo-arc is a
pseudo-arc. Its name “pseudo-arc” comes from this property because arcs have
the same property, in the sense that any subcontinuum (non reduced to a
singleton) of an arc is an arc. For more information on pseudo-arc, we refer
the interested reader to the second paragraph of [15, Chapter XII] and to [2,
3, 12, 13]. Sadly, it is very complicated to get a “drawing” of the pseudo-arc
due to its complicated crocked structure, see [15, Exercise 1.23]. Following
the works of Bing, one can wonder whether the pseudo-arc is typical among arc-
like continua and ask whether there is a natural probabilistic construction of
the pseudo-arc.

Let us recall the construction of continua from inverse limits used in [11],
see [15, Section II.2] for details. Suppose we are given a sequence

$\cdots\xrightarrow[]{f_{3}}X_{3}\xrightarrow[]{f_{2}}X_{2}\xrightarrow[]{f_{1}}X_{1}$

where for any $i\geq 1$, the metric space $(X_{i},d_{i})$ is compact and
$f_{i}:X_{i+1}\to X_{i}$ is a continuous surjective function. Then the
_inverse limit_ of $(\\{X_{i},f_{i}\\})_{i\geq 1}$ is the subspace of
$\prod_{i\geq 1}X_{i}$ defined by

$\varprojlim(f_{i},X_{i}:i\geq 1)=\left\\{(x_{i})_{i\geq 1}\in\prod_{i\geq
1}X_{i}:f_{i}(x_{i+1})=x_{i}\right\\}.$ (2)

In the application below $X_{i}$ are compact intervals of $\mathbb{R}$ and in
this case, by [15, Theorems 2.4 and 12.19], the inverse limit is a chainable
continuum. In [11], Kiss and Solecki constructed a system as above using two-
sided independent Brownian motions $(\mathfrak{B}_{i}:i\geq 1)$. More
precisely, they proved that for any interval $J$ of $\mathbb{R}$ with $0\in J$
and $J\neq\\{0\\}$, the following limit exists almost surely

$\mathcal{I}_{i}=\lim_{m\to\infty}\mathfrak{B}_{i}\left(\mathfrak{B}_{i+1}\left(\dots\left(\mathfrak{B}_{i+m}\left(J\right)\right)\dots\right)\right),$
(3)

and does not depend on $J$, so that we can consider the random chainable
continuum $\mathcal{C}$ obtained as the inverse limit of the system

$\cdots\xrightarrow[]{\mathfrak{B}_{3}}\mathcal{I}_{3}\xrightarrow[]{\mathfrak{B}_{2}}\mathcal{I}_{2}\xrightarrow[]{\mathfrak{B}_{1}}\mathcal{I}_{1}.$

Kiss and Solecki proved [11, Theorem 2.1] that the random chainable continuum
$\mathcal{C}$ is almost surely non-degenerate and indecomposable. This note
answers negatively the obvious question the preceding result triggers:

######  Theorem 1.

Almost surely, the random continuum $\mathcal{C}$ is _not_ hereditary
indecomposable (hence is not the pseudo-arc).

The proof below could be adapted to prove that a random continuum constructed
similarly from a sequence of i.i.d.  _reflected_ Brownian motions is neither a
pseudo-arc, answering a question in [11, Section 3.1.1]. Although almost
surely not homeomorphic to the pseudo-arc, the random continuum $\mathcal{C}$
is interesting in itself and one could ask about its topological property,
e.g. we wonder whether the topology of $\mathcal{C}$ is almost surely constant
and if it is easy to characterise.

Acknowledgements: We acknowledge support from the ERC 740943 “GeoBrown” and
ANR 16-CE93-0003 “MALIN”.

##  2 Finding good intervals

In the rest of the article the Brownian motions $\mathfrak{B}_{i}$ are fixed
and we recall the definition of $\mathcal{I}_{i}$ in (3) and of the continuum
$\mathcal{C}$. We will show that Theorem 1 follows from the proposition below
stated in terms of images of intervals under the flow of independent Brownian
motions whose proof occupy the remaining of the article:

######  Proposition 2.

For any $\varepsilon>0$ small enough, with probability at least

$p_{\varepsilon}=\prod_{i=1}^{\infty}1-2\left(\varepsilon^{{(5/4)}^{i-1}}\right)^{1/8}>0,$

there exists two sequences $(U_{i})_{i\geq 1}$ and $(V_{i})_{i\geq 1}$ of
subintervals of $\mathbb{R}$ such that, for any $i\geq 1$, the five following
conditions are satisfied

  1. 1.

$U_{i},V_{i}\subset\mathcal{I}_{i}$ where $\mathcal{I}_{i}$ is defined in (3),

  2. 2.

$U_{i}\nsubseteq V_{i}$ and $V_{i}\nsubseteq U_{i}$,

  3. 3.

$U_{i}\cap V_{i}\neq\emptyset$,

  4. 4.

$U_{i}=\mathfrak{B}_{i}(U_{i+1})$ and $V_{i}=\mathfrak{B}_{i}(V_{i+1})$,

  5. 5.

$|U_{i}|,|V_{i}|\leq\varepsilon^{(5/4)^{i-1}}$.

###### Proof of Theorem 1 given Proposition 2..

In the proof, since we are always working with the functions
$\mathfrak{B}_{i}$ we write $\varprojlim(W_{i}:i\geq 1)$ for the inverse limit
previously denoted by $\varprojlim(\mathfrak{B}_{i},W_{i}:i\geq 1)$ for any
sequence of intervals $W_{1},W_{2},...$ such that
$W_{i+1}\xrightarrow{\mathfrak{B}_{i}}W_{i}$. On the event described in the
above proposition we have with probability at least $p_{\varepsilon}>0$:

  * •

For any $i\geq 1$, $\mathfrak{B}_{i}(U_{i+1}\cup V_{i+1})=U_{i}\cup V_{i}$
(point 4) and $U_{i}\cup V_{i}\subset\mathcal{I}_{i}$ (point 1) and $U_{i}\cup
V_{i}$ is an interval (point 3), so by Lemma 2.6 of [15],
$\varprojlim(U_{i}\cup V_{i}:i\geq 1)$ is a subcontinuum of $\mathcal{C}$.

  * •

By Lemma 2.6 of [15], both $\varprojlim(U_{i}:i\geq 1)$ and
$\varprojlim(V_{i}:i\geq 1)$ are also subcontinua of $\varprojlim(U_{i}\cup
V_{i}:i\geq 1)$.

  * •

Let $x=(x_{i})_{i\geq 1}\in\varprojlim(U_{i}\cup V_{i}:i\geq 1)$, then

    * –

either, for any $i$, we have $x_{i}\in U_{i}\cap V_{i}$, and so
$x\in\varprojlim(U_{i}:i\geq 1)$ and $x\in\varprojlim(V_{i}:i\geq 1)$,

    * –

or there exists $j\geq 1$ such that $x_{j}\in U_{j}$ and $x_{j}\notin V_{j}$,
but then by point $4$ we have $x_{i}\in U_{i}$ for all $i\geq j$ and so
$x\in\varprojlim(U_{i}:i\geq 1)$,

    * –

or there exists $j\geq 1$ such that $x_{j}\notin U_{j}$ and $x_{j}\in V_{j}$
and similarly we deduce that $x\in\varprojlim(V_{i}:i\geq 1)$.

Hence, $\varprojlim(U_{i}\cup V_{i}:i\geq 1)\subset\varprojlim(U_{i}:i\geq
1)\cup\varprojlim(V_{i}:i\geq 1)$ and the reverse inclusion is obvious.

  * •

$\varprojlim(U_{i}\cup V_{i}:i\geq 1)\neq\varprojlim(U_{i}:i\geq 1)$ nor
$\varprojlim(U_{i}\cup V_{i}:i\geq 1)\neq\varprojlim(V_{i}:i\geq 1)$ by
combining point $2$ and point 4.

All of these points imply that $\varprojlim(U_{i}\cup V_{i}:i\geq 1)$ is a
decomposable subcontinuum of $\mathcal{C}=\varprojlim(\mathcal{I}_{i}:i\geq
1)$. That implies that $\mathcal{C}$ is not a pseudo-arc with probability at
least $p_{\varepsilon}$ for any $\varepsilon>0$. As $p_{\varepsilon}\to 1$
when $\varepsilon\to 0$, it is not a pseudo-arc with probability one. ∎

###  2.1 Construction of a decomposable subcontinuum using good shape
excursions

Let us now explain the idea behind the construction of the intervals of
Proposition 2. This relies on the concept of excursions with a good shape.
Imagine that we have a sequence of non trivial intervals
$[u_{i},v_{i}]\subset[0,1]$ such that
$\mathfrak{B}_{i}([u_{i+1},v_{i+1}])=[u_{i},v_{i}]$ and furthermore that
$\mathfrak{B}_{i}(u_{i+1})=u_{i}$ and $\mathfrak{B}_{i}(v_{i+1})=v_{i}$ and
$\mathfrak{B}_{i}(t)\in(u_{i},v_{i})$ for $t\in(u_{i+1},v_{i+1})$. In words,
over the time interval $[u_{i+1},v_{i+1}]$, the Brownian motion
$\mathfrak{B}_{i}$ makes an excursion from $u_{i}$ to $v_{i}$. We say that
this excursion has a _good shape_ if it stays in the pentomino of Figure 2.

Figure 2: An excursion from $u_{i}$ to $v_{i}$ over the time interval
$[u_{i+1},v_{i+1}]$ has a good shape if it stays in the light grey region.

If we have such a sequence of intervals and excursions, then one can define a
sequence of intervals $U_{i},V_{i}$ by setting for any $i\geq 1$,

$\displaystyle
U_{i}=\lim_{n\to\infty}\underbrace{\left(\mathfrak{B}_{i}\circ\mathfrak{B}_{i+1}\circ\dots\circ\mathfrak{B}_{i+n-1}\right)\left(\left[u_{i+n},\frac{u_{i+n}+2v_{i+n}}{3}\right]\right)}_{U_{i,n}}\text{
and }$ $\displaystyle
V_{i}=\lim_{n\to\infty}\left(\mathfrak{B}_{i}\circ\mathfrak{B}_{i+1}\circ\dots\circ\mathfrak{B}_{i+n-1}\right)\left(\left[\frac{2u_{i+n}+v_{i+n}}{3},v_{i+n}\right]\right).$

First, these two limits exist a.s. and are closed intervals a.s. because they
are limits of a sequence of decreasing closed intervals. Indeed, because
$\mathfrak{B}_{i+n}$ performs a good shape excursion from $u_{i+n}$ to
$v_{i+n}$ over $[u_{i+n+1},v_{i+n+1}]$ we have

$\displaystyle\mathfrak{B}_{i+n}\left(\left[u_{i+n+1},\frac{u_{i+n+1}+2v_{i+n+1}}{3}\right]\right)$
$\displaystyle\subset\left[u_{i+n},\frac{u_{i+n}+2v_{i+n}}{3}\right],\text{
and so}$ $\displaystyle U_{i,n+1}$ $\displaystyle\subset U_{i,n},$

and $U_{i,n}$ are intervals because the BM is continuous a.s. It is then an
easy matter to check that the interval constructed above satisfies points 2-4
of Proposition 2. Our task is thus to construct the sequence $u_{i},v_{i}$ so
that $\mathfrak{B}_{i}$ performs a good shape excursion from $u_{i}$ to
$v_{i}$ over $[u_{i+1},v_{i+1}]$ and to ensure points $1$ and $5$ of
Proposition 2. The key idea is to look for these intervals in the vicinity of
$0$ because any given small interval close to $0$ has MANY pre-images close to
$0$ by a Brownian motion. These many pre-images enable us to select one with a
good shape.

###  2.2 Pre-images of a small interval by a Brownian motion

In the following lemma the dependence in $i$ is superfluous but we keep it to
make the connection with the preceding discussion easier to understand.

######  Lemma 3.

Let $a_{i}$ be any real positive number small enough. Fix
$[u_{i},v_{i}]\subset[0,a_{i}]$. Then with probability at least

$1-2a_{i}^{1/8}$

we can find $[u_{i+1},v_{i+1}]\subset[0,a_{i}^{5/4}]$ so that
$\mathfrak{B}_{i}$ performs an excursion with a good shape from $u_{i}$ to
$v_{i}$ over the time interval $[u_{i+1},v_{i+1}]$.

###### Proof.

Fix $0<u_{i}<v_{i}$ and consider the successive excursions
$\mathcal{E}_{1},\mathcal{E}_{2},...$ that the Brownian motion
$\mathfrak{B}_{i}$ performs from $u_{i}$ to $v_{i}$ over the respective time
intervals
$[u_{i+1}^{(1)},v_{i+1}^{(1)}],[u_{i+1}^{(2)},v_{i+1}^{(2)}],\cdots$. By the
Markov property of Brownian motion and standard argument in excursion theory,
these excursions are i.i.d. We claim that

$r=\mathbb{P}(\mathcal{E}\mbox{ has a good shape})>0.$

Indeed, since the law of Brownian motion has full support in the space of
continuous functions (with the topology of uniform convergence over all
compacts of $\mathbb{R}_{+}$), the first excursion from $u_{i}$ to $v_{i}$
might be close to any prescribed continuous function and in particular, the
probability to have a good shape is strictly positive. See Figure 3.

Figure 3: For any given continuous function $f$ starting from $0$ and any
$\varepsilon>0$, the Brownian motion may stay within distance $\varepsilon>0$
of $f$ up to time $1$ with a positive probability. Choosing $f$ carefully, we
deduce that the first excursion from $u_{i}$ to $v_{i}$ has a good shape with
positive probability.

Hence, the probability that at least one of the $k$ first excursions has a
good shape is at least

$1-(1-r)^{k}.$

Figure 4: In red, blue and orange, the excursion from $u_{i}$ to $v_{i}$ we
consider.

To control the number of excursions from $u_{i}$ to $v_{i}$ performed up to
time $a_{i}$ by $\mathfrak{B}_{i}$, we introduce the auxiliary stopping times
defined by $w_{i+1}^{(1)}=\inf\\{t\geq 0:\mathfrak{B}_{i}(t)=u_{i}\\}$ and for
$k\geq 2$

$w_{i+1}^{(k)}=\inf\\{t\geq v_{i+1}^{(k-1)}:\mathfrak{B}_{i}(t)=u_{i}\\}.$

Hence $w_{i+1}^{(1)}<v_{i+1}^{(1)}<w_{i+1}^{(2)}<v_{i+1}^{(2)}<\cdots$ are the
successive hitting times of $u_{i},v_{i},u_{i},v_{i}$ by $\mathfrak{B}_{i}$,
see Figure 4. For $a\geq 0$, we let $\mathcal{T}_{a}=\inf\\{t\geq
0:\mathfrak{B}_{i}(t)=a\\}$ the hitting time of $a$ by a standard linear
Brownian motion. It is classic (see e.g. [14, Theorem 2.35]) that for $a>0$ we
have $\mathcal{T}_{a}=a^{2}\cdot\mathcal{T}_{1}$ in law where
$\mathcal{T}_{1}$ is distributed according to the Lévy law

$\mathcal{T}_{1}\underset{(d)}{=}\frac{\mathrm{d}t}{\sqrt{2\pi
t^{3}}}\mathrm{exp}\left(-\frac{1}{2t}\right)\mathbf{1}_{t>0}.$

In our case, applying the strong Markov property at time
$w_{i+1}^{(1)}<v_{i+1}^{(1)}<w_{i+1}^{(2)}<v_{i+1}^{(2)}<\cdots$ and using
invariance by symmetry we deduce that we have the equalities in distribution

$w_{i+1}^{(1)}\stackrel{{\scriptstyle(d)}}{{=}}\mathcal{T}_{u_{i}},\quad
v_{i+1}^{(1)}\stackrel{{\scriptstyle(d)}}{{=}}\mathcal{T}_{u_{i}+|v_{i}-u_{i}|},\quad
w^{(2)}_{i+1}\stackrel{{\scriptstyle(d)}}{{=}}\mathcal{T}_{u_{i}+2|v_{i}-u_{i}|},\cdots,\quad
v^{(k)}_{i+1}\stackrel{{\scriptstyle(d)}}{{=}}\mathcal{T}_{u_{i}+(2k-1)|v_{i}-u_{i}|},$

for $k\geq 2$. Since
$\mathcal{T}_{u_{i}+(2k-1)|v_{i}-u_{i}|}\leq\mathcal{T}_{2ka_{i}}$, the
probability that the first $k$ excursions of $\mathfrak{B}_{i}$ occurs before
$a_{i}^{5/4}$ is at least

$\mathbb{P}\left(\mathcal{T}_{2ka_{i}}<a_{i}^{5/4}\right)=\mathbb{P}\left(\mathcal{T}_{1}<\left(\frac{1}{2k\,a_{i}^{3/8}}\right)\right)\geq
1-\sqrt{\frac{2}{\pi}}2ka_{i}^{3/8}\text{ (for $ka_{i}^{3/8}$ small enough)}.$

Gathering-up the above remarks and taking $k=\lfloor a_{i}^{-1/4}\rfloor$, we
deduce that the probability to do not find an excursion from $u_{i}$ to
$v_{i}$ with a good shape in $[0,a_{i}^{5/4}]$ is bounded above by

$(1-r)^{\lfloor a_{i}^{-1/4}\rfloor}+2\sqrt{\frac{2}{\pi}}\lfloor
a_{i}^{-1/4}\rfloor a_{i}^{3/8}\leq 2a_{i}^{-1/8}\text{ (for $a_{i}$ small
enough)}.\qed$

##  3 Proof of Proposition 2

Let $(\mathfrak{B}_{i})_{i\geq 1}$ be a sequence of i.i.d. two-sided Brownian
motions, and $\varepsilon$ be any real positive number small enough. For any
$i\geq 1$, take $a_{i}=\varepsilon^{(5/4)^{i-1}}$.

Firstly, we put $[u_{1},v_{1}]=[0,\varepsilon]=[0,a_{1}]$, by Lemma 3, with
probability at least $1-2a_{1}^{1/8}$, there exists an interval
$[u_{2},v_{2}]\subset[0,a_{1}^{5/4}]=[0,a_{2}]$ such that $\mathfrak{B}_{1}$
performs a good shape excursion from $u_{1}$ to $v_{1}$ over the time interval
$[u_{2},v_{2}]$. Now, we apply Lemma 3 to $[u_{2},v_{2}]\subset[0,a_{2}]$,
etc. At the end, with probability at least

$\prod_{i=1}^{\infty}1-2\left(\varepsilon^{{(5/4)}^{i-1}}\right)^{1/8},$

we obtain a sequence of non trivial intervals $([u_{i},v_{i}])_{i\geq 1}$ such
that for any $i$, $\mathfrak{B}_{i}$ makes a good shape excursion from $u_{i}$
to $v_{i}$ over $[u_{i+1},v_{i+1}]$. By Section 2.1, we can then construct two
sequences of intervals $U_{i}$, $V_{i}$ that satisfy points 2-4 of Proposition
2. Moreover, by construction,
$U_{i},V_{i}\subset[u_{i},v_{i}]\subset[0,a_{i}]$, hence point 5 is also
satisfied.

Finally, to obtain point 1, just remark that, for any $i,n\geq 1$,
$[u_{i+n},v_{i+n}]\subset[0,a_{i+n}]\subset[0,1]$, so

$\displaystyle U_{i}$
$\displaystyle=\lim_{n\to\infty}\left(\mathfrak{B}_{i}\circ\mathfrak{B}_{i+1}\circ\dots\circ\mathfrak{B}_{i+n}\right)\left(\left[u_{i+n+1},\frac{u_{i+n+1}+2v_{i+n+1}}{3}\right]\right)$
$\displaystyle\subset\lim_{n\to\infty}\left(\mathfrak{B}_{i}\circ\mathfrak{B}_{i+1}\circ\dots\circ\mathfrak{B}_{i+n}\right)\left(\left[0,1\right]\right)=\mathcal{I}_{i}\text{
(by\leavevmode\nobreak\ \eqref{eq:interval})}.$

Similarly, $V_{i}\subset\mathcal{I}_{i}$. ∎

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