# Asymmetric reheating from a symmetric inflationary potential

James M. Cline<EMAIL_ADDRESS>Jean-Samuel Roux
<EMAIL_ADDRESS>McGill University, Department of Physics, 3600
University St., Montréal, QC H3A2T8 Canada

###### Abstract

We explore a model of two-field inflation with nonminimal kinetic terms in
which two identical matter sectors decoupled from each other may reheat to
different temperatures while preserving the symmetry of the Lagrangian. This
scenario is motivated by mirror dark matter models in which the temperature of
the mirror sector is constrained to be $T^{\prime}\lesssim 0.5T$ by big bang
nucleosynthesis and the cosmic microwave background. For a given class of
nonminimal kinematic terms, we find that the symmetric field trajectory $X=Y$
is a repeller solution, such that any randomly-occurring asymmetry in the
initial conditions is amplified by many orders of magnitude during inflation,
far beyond what canonical power-law models can achieve. Isocurvature
fluctuations are strongly suppressed in this model, but a
$\mathcal{O}(0.03$–$0.07$) tensor-to-scalar ratio could be observed in the
near future. The range of potential parameters compatible with Planck
constraints is shown to be much larger than in corresponding single-field
models. This occurs through a mechanism for lowering the spectral index that
we dub CTHC: curved trajectory at horizon crossing.

##  I Introduction

Mirror models were perhaps the original paradigm for a hidden sector in
particle physics Kobzarev:1966qya ; Okun:2006eb ; Foot:2014mia . A copy of the
standard model (SM) field content and gauge group is hypothesized, which in
its simplest form is exact, so that the two sectors are related by a discrete
$Z_{2}$ mirror symmetry. This form is subject to significant cosmological
constraints on the additional relativistic degrees of freedom—mirror photons
$\gamma^{\prime}$ and neutrinos $\nu^{\prime}$—that would increase the Hubble
expansion rate at early times, in contradiction to the successful predictions
of big bang nucleosynthesis (BBN) and the cosmic microwave background (CMB).
To circumvent the bound on the effective number of neutrino species $N_{\rm
eff}$, one must either break the mirror symmetry by making $\gamma^{\prime}$
and $\nu^{\prime}$ massive, or in the case of exact $Z_{2}$ symmetry, arrange
for the initial conditions of the Universe to create a lower temperature
$T^{\prime}$ in the mirror sector than in the visible one, with $x\equiv
T^{\prime}/T\lesssim 0.5$ Berezhiani:2000gw ; Roux:2020wkp .

Similar frameworks like Twin Higgs models (see Curtin:2021alk  for a recent
example) or parity solutions to the strong CP problem Craig:2020bnv  break the
mirror symmetry at late times. However, if at least one mirror species like
$\gamma^{\prime}$ remains massless, those scenarios also require $x\lesssim
0.6$ to satisfy cosmological bounds on $N_{\rm eff}$. The breaking of the
symmetry might facilitate entropy transfers that cool the mirror sector, but
it is also possible that mirror and visible species decouple when the symmetry
still holds, in which case the temperature hierarchy must originate from early
universe dynamics, when the Lagrangian was still symmetric.

One may wonder how likely it is to realize perfect mirror symmetry in a
complete model including inflation, such that the relative temperatures in the
two sectors differ as required by the constraints. In Refs. Berezhiani:2000gw
; Berezhiani:1995am ; Berezinsky:1999az , it was noted that asymmetric
reheating would generically occur in models with two inflatons, one for each
sector, due to differences in the initial conditions. Here we revisit this
idea, in the light of current CMB constraints from Planck Akrami:2018odb
.111An early proposal for getting asymmetric reheating was given in ref.
1985Natur.314..415K , which proposed a ‘double-bubble inflation’ model where
the ordinary and mirror inflatons finish inflation by bubble nucleation at
different (random) times. In this case the first sector to undergo reheating
gets exponentially redshifted until the second field nucleates a bubble of
true vacuum. However this is in the context of “old inflation” driven by false
vacua, which is untenable because the phase transitions never complete.

We consider two-field chaotic inflation with decoupled potentials of the form

$V_{\rm
tot}=V(X)+V(Y)=\lambda\frac{|X|^{p}}{m_{P}^{p-4}}+\lambda\frac{|Y|^{p}}{m_{P}^{p-4}}\,,$
(1)

(where $m_{P}$ is the reduced Planck mass) plus respective couplings of each
field to its own sector’s matter particles, to accomplish reheating. In the
case of a purely quadratic potential ($p=2$) the solutions are such that the
ratio of the two inflatons $Y/X$ remains constant during inflation
Berezinsky:1999az . Then the ratio of the reheating temperatures goes as
$T^{\prime}/T=(Y^{2}/X^{2})^{1/4}$, and is thereby analytically determined in
terms of the initial conditions. This example is now ruled out by Planck data
Akrami:2018odb , which strongly disfavors chaotic inflation models that have
convex potentials.

In the following, we study a model with noncanonical inflaton kinetic terms,
proposed in Ref. Roux:2020wkp , that generates a temperature hierarchy by the
spontaneous breaking of mirror symmetry by the initial values of the
inflatons. At large field values, it is equivalent to Eq. (1) with fractional
values of $p$, and we therefore consider both kinds of models. Such fractional
power-law potentials have been proposed in the context of string theory
Silverstein:2008sg ; McAllister:2008hb ; Dong:2010in ; Gur-Ari:2013sba ;
McAllister:2014mpa ; Marchesano:2014mla ; Bielleman:2016grv ; Landete:2017amp
or supergravity Gao:2014fha . We will show, somewhat surprisingly, that the
initial values $X_{i},Y_{i}$ cannot be too different from each other, while
remaining consistent with Planck constraints on the CMB observables;
nevertheless, a small initial asymmetry is typically amplified during
inflation into a very large asymmetry in the final temperature ratio. We will
also show that the two-field inflationary scenarios lead to much better
agreement with the CMB than their single-field counterparts, due to the effect
of curvature in the inflaton trajectory in the $X$-$Y$ field space, at the
time of horizon crossing. We abbreviate this effect by “CTHC.” It was
previously observed in the context of multifield inflation with fractional
power law potentials in Ref. Wenren:2014cga , with emphasis on many inflatons
having random potential parameters.

In Section II we introduce the noncanonical two-field models, and the
corresponding canonical models that are equivalent at large field values. The
numerical techniques used are described there, along with analytical
approximations that explain the qualitative behavior of the exact solutions.
We illustrate our results for three benchmark parameter choices in Section
III. This is followed in Section IV by a description of a Monte Carlo search
of the full parameter space for models that are consistent with all
constraints. We give conclusions in Section V.

##  II Noncanonical two-field inflation

Planck data favor concave inflaton potentials, whereas convex ones occur more
generically. A popular solution to this problem is through nonminimal coupling
to gravity (see e.g., Ref. Linde:2011nh ), which in our framework would take
the form

${\cal L}\ni{\textstyle{1\over 2}}m_{P}^{2}\,\zeta\,R\,(X^{2}+Y^{2})\,,$ (2)

where $R$ is the Ricci curvature. Transforming to the Einstein frame, the
inflaton potential gets rescaled by $V(X,Y)\to V(X,Y)/\Omega^{4}$ where
$\Omega^{2}=1+\zeta(X^{2}+Y^{2})$, so that $V$ becomes concave at large field
values.

However this simple device, while reconciling chaotic inflation potentials
with CMB data, introduces a potentially strong coupling between the two
fields, through the factor $\Omega^{-4}$. By numerical investigation we found
that whenever $\zeta$ is large enough to resolve the tensions with CMB
observations, it also causes the trajectories to align in the inflationary
attractor solutions, $X\cong Y$, such that the effect of random initial
differences $X_{i}\neq Y_{i}$ gets erased rather than enhanced during
inflation, leading to nearly equal reheat temperatures in the two sectors.
This is not compatible with the goal of the present work.

###  II.1 Noncanonical models

Another way of reconciling chaotic inflation potentials with CMB constraints
is to use nonminimal kinetic terms, for instance Lee:2014spa ,

$\displaystyle{\cal L}$
$\displaystyle=\frac{1}{2}\left(1+f{\frac{X^{n}}{m_{P}^{n}}}\right)(\partial
X)^{2}+\frac{1}{2}\left(1+f{\frac{Y^{n}}{m_{P}^{n}}}\right)(\partial Y)^{2}$
$\displaystyle+\frac{1}{2}\,m^{2}\left(X^{2}+Y^{2}\right)\,.$ (3)

For $X,Y\gg m_{P}$, the canonically normalized fields are $U\sim X^{1+n/2}$,
$W\sim Y^{1+n/2}$, so that the potential becomes proportional to
$(U^{4/(n+2)}+W^{4/(n+2)})$. Here we will take the noncanonical fields to have
a quadratic potential, as the simplest and most generic example.

The model of Eq. (3) essentially coincides with the power-law potential of Eq.
(1) at large field values, with $p=4/(2+n)$. The two scenarios only differ
significantly near the end of inflation when the fields are close to the
minimum of the potential, and most results from Ref. Berezinsky:1999az  also
apply to our noncanonical scenario.

In the slow roll approximation, the equations of motion $3H\dot{X}=-\partial
V/\partial X$ and $3H\dot{Y}=-\partial V/\partial Y$ for the potential of Eq.
(1) can be integrated to show that Berezinsky:1999az

$\displaystyle X/Y$ $\displaystyle={\rm const},\quad p=2;$ $\displaystyle
X^{2-p}-Y^{2-p}$ $\displaystyle={\rm const},\quad p\neq 2.$ (4)

Unlike the quadratic scenario, for $p\neq 2$ the inflaton trajectory is not a
straight line in the $(X,Y)$ field space, unless the initial values $X_{i}$
and $Y_{i}$ are taken to be equal (or if one of them remains zero throughout
inflation). If $p>2$, the two fields become synchronized before the end of
inflation, since $X$ and $Y$ are both decreasing, and $X\approx Y$ when
reheating takes place. This generally leads to both sectors reaching thermal
equilibrium by the time of BBN, a possibility that is ruled out by the current
bounds on $N_{\rm eff}$.

Figure 1: Illustrative trajectory for inflatons starting from nearly symmetric
initial conditions in the model of Eq. (3), but achieving asymmetric reheating
due to inflation ending earlier for the mirror inflaton $Y$. The inset
highlights the end of inflation.

The case $p<2$ is the most appealing for our purposes, since the fields are
driven away from the symmetric trajectory $X=Y$. Then any initial randomly-
occurring asymmetry is amplified by inflation, possibly leading to a
difference of many orders of magnitude between the respective reheat
temperatures. Moreover, potentials with $p\sim 2/3$ McAllister:2008hb ;
Silverstein:2008sg  are generally consistent with Planck constraints on the
tensor-to-scalar ratio $r$ and spectral index $n_{s}$ Akrami:2018odb .
However, in single-field inflation this agreement is marginal at best and is
limited to a small range of powers $p$, as flatter potentials lead to to too-
large predictions for $n_{s}$, and steeper ones give too-large $r$
Planck:2013jfk . We will show that in the two-field models the consistency is
improved, for both the canonical and noncanonical implementations, and for a
larger range of $p$ relative to single-field models.

In Fig. 1 we show an example of field trajectories for the noncanonical
Lagrangian of Eq. (3) with an effective $p=2/3$ power-law potential ($n=4$)
and parameter values222Throughout this work we fix the inflaton mass to
$m=10^{-6}$ in the noncanonical model (no explicit mass term is present in the
canonical version). Its value does not impact the inflationary dynamics during
slow roll, but in a complete model it would set the scale of reheating, the
details of which are not considered here.

$f=1,\ X_{i}=3,\ Y_{i}=2.9\,,$ (5)

where here and henceforth we work in Planck units with $m_{P}=1$, and assume
vanishing initial velocities. Even though the initial values in this example
differ only by 3%, slow roll ends somewhat earlier in the $Y$ field, and its
amplitude gets damped by a factor of $\sim 25$ relative to that of $X$,
leading to a temperature ratio of $x\sim 0.2$, in agreement with the
cosmological bound $x\lesssim 0.5$. Although we do not discuss the detailed
mechanism of reheating in this work, the assumptions of unbroken mirror
symmetry and complete conversion of the inflaton energies into radiation in
the two respective sectors allows us to estimate the temperature ratio as

$x\equiv{T^{\prime}\over T}=\left(\rho^{\prime}\over\rho\right)^{1/4}\,,$ (6)

by evaluating the energy densities at the end of inflation.

###  II.2 Numerical solution

To solve for the inflationary dynamics, we use the first order equations of
motion for numerical integration, in terms of the canonical momenta
$\pi_{x}=\partial{\cal L}/\partial\dot{X}=F(X)\dot{X}$, $\pi_{y}=\partial{\cal
L}/\partial\dot{Y}=F(Y)\dot{Y}$, with $F(X)\equiv 1+fX^{n}$ for noncanonical
models. It is convenient to take the number of $e$-foldings $N$ as the
independent variable, defined through $dN=Hdt$ in terms of the Hubble
parameter $H$. Denoting $d/dN$ by primes, the system of equations is

$\displaystyle X^{\prime}$ $\displaystyle=$ $\displaystyle{\pi_{x}\over
F(X)H}\,,$ $\displaystyle\pi_{x}^{\prime}$ $\displaystyle=$
$\displaystyle-3\pi_{x}+{1\over H}\left({F^{\prime}(X)\over
2}\left(\pi_{x}\over F(X)\right)^{2}-m^{2}X\right)\,,$ $\displaystyle
Y^{\prime}$ $\displaystyle=$ $\displaystyle{\pi_{y}\over F(Y)H}\,,$
$\displaystyle\pi_{y}^{\prime}$ $\displaystyle=$
$\displaystyle-3\pi_{y}+{1\over H}\left({F^{\prime}(Y)\over
2}\left(\pi_{y}\over F(Y)\right)^{2}-m^{2}Y\right)\,,$ $\displaystyle H$
$\displaystyle=$ $\displaystyle{1\over\sqrt{6}}\left({\pi_{x}^{2}\over
F(X)}+{\pi_{y}^{2}\over F(Y)}+m^{2}(X^{2}+Y^{2})\right)^{1/2}\,.$ (7)

Since we assume the two fields are decoupled from each other, each inflaton
decays into its own matter sector during reheating. The resulting temperature
ratio can therefore be estimated as in Eq. (6),

$x=\left(\frac{\frac{1}{2}F(Y)(\partial Y)^{2}+V(Y)}{\frac{1}{2}F(X)(\partial
X)^{2}+V(X)}\right)^{1/4}\,,$ (8)

evaluated at the end of inflation. Without loss of generality, we will assume
$Y<X$ initially, so that $x<1$ and $X$ denotes the visible sector inflaton.

###  II.3 Slow-roll parameters

To compute the slow-roll parameters and inflationary observables it is
convenient to work in the canonical basis, with fields denoted by $(U,W)$.
Since the two kinetic terms in Eq. (3) are decoupled from each other, the
Jacobian matrix $Z$ is diagonal,

$\begin{pmatrix}\dot{X}\\\
\dot{Y}\end{pmatrix}=\begin{pmatrix}F(X)^{-1/2}&0\\\
0&F(Y)^{-1/2}\end{pmatrix}\begin{pmatrix}\dot{U}\\\ \dot{W}\end{pmatrix}\equiv
Z\begin{pmatrix}\dot{U}\\\ \dot{W}\end{pmatrix}.$ (9)

The slow-roll parameters computed in the canonical basis (indices $m,n$) are
related to derivatives with respect to fields in the the original basis
(indices $i,j$) by

$\displaystyle\epsilon_{m}$
$\displaystyle=\frac{(Z_{im}\partial_{i}V)^{2}}{2V^{2}}\,,$
$\displaystyle\eta_{mn}$
$\displaystyle=Z_{im}Z_{jn}\frac{\partial_{ij}V}{V}+Z_{im}\partial_{i}Z_{jn}\frac{\partial_{j}V}{V}.$
(10)

For numerical purposes, we modify these definitions by replacing
$V\to\rho=3H^{2}$ in the denominators. During inflation, this makes a
negligible difference, whereas at the end of inflation while the inflaton is
oscillating around its minimum, it avoids the artificial singularities that
result from $V$ passing through zero.

The formalism for computing the spectral index $n_{s}$ of primordial adiabatic
perturbations and the tensor-to-scalar ratio $r$ in two-field models was
developed in Refs. Gordon:2000hv ; Byrnes:2006fr . One first introduces the
adiabatic/entropy basis $(\sigma,s)$, defined by

$\displaystyle d\sigma$ $\displaystyle=$ $\displaystyle
c_{\alpha}dU+s_{\alpha}dW\,,$ $\displaystyle ds$ $\displaystyle=$
$\displaystyle-s_{\alpha}dU+c_{\alpha}dW\,,$ (11)

where $\alpha=\tan^{-1}(\dot{W}/\dot{U})$ is the instantaneous slope of the
field trajectory in the $U$-$W$ plane. The slow-roll parameters in the
$(\sigma,s)$ basis are computed using Gordon:2000hv

$\displaystyle\epsilon_{\sigma}$ $\displaystyle=$
$\displaystyle(c_{\alpha}\partial_{U}V+s_{\alpha}\partial_{W}V)^{2}/(2V^{2})$
$\displaystyle\epsilon_{s}$ $\displaystyle\simeq$ $\displaystyle 0$
$\displaystyle\eta_{\sigma\sigma}$ $\displaystyle=$ $\displaystyle
c_{\alpha}^{2}\eta_{UU}+s_{\alpha}^{2}\eta_{WW}$ $\displaystyle\eta_{ss}$
$\displaystyle=$ $\displaystyle
s_{\alpha}^{2}\eta_{UU}+c_{\alpha}^{2}\eta_{WW}$ $\displaystyle\eta_{\sigma
s}$ $\displaystyle=$ $\displaystyle
c_{\alpha}s_{\alpha}(\eta_{WW}-\eta_{UU}).$ (12)

and the fact that $\eta_{UW}=0$ because $Z$ is diagonal [_cf._ Eq. (9)]. The
derivatives of the potential in the $(U,W)$ basis are computed using the
Jacobian matrix, $\partial_{m}V=Z_{im}\partial_{i}V$. Then to leading order in
the slow-roll expansion, the scalar spectral index and tensor-to-scalar ratio
are Byrnes:2006fr

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!n_{s}-1$ $\displaystyle=$
$\displaystyle-(6-4c_{\Delta}^{2})\epsilon_{\sigma}\\!+\\!2s_{\Delta}^{2}\eta_{\sigma\sigma}\\!+\\!4s_{\Delta}c_{\Delta}\eta_{\sigma
s}\\!+\\!2c_{\Delta}^{2}\eta_{ss}\,,$ $\displaystyle r$ $\displaystyle=$
$\displaystyle 16\epsilon_{\sigma}\,,$ (13)

where $c_{\Delta}=-2{\cal C}\,\eta_{\sigma s}$,
$s_{\Delta}=+\sqrt{1-c_{\Delta}^{2}}$, ${\cal C}=2-\ln 2-\gamma\simeq 0.73$
($\gamma$ is the Euler constant).

For the interpretation of the following results, it is useful to derive
approximate expressions for $n_{s}$ and $r$ in terms of the background field
amplitude and the canonical field ratio, defined respectively as

$\sigma\equiv(U^{2}+W^{2})^{1/2},\quad\xi\equiv{W\over U}\,.$ (14)

This will help to elucidate how the two-field model improves over single-field
inflation. In the large field limit, $\xi\cong\dot{W}/\dot{U}$, which is
equivalent to approximating the instantaneous field velocity as being radial
in the $U$-$W$ plane. This implies that $\cos\alpha\approx U/\sigma$ and
$\sin\alpha\approx W/\sigma$ in Eqs. (11-12). One can further approximate
$\eta_{\sigma s}\ll 1$ so that $\cos^{2}\Delta\sim 0$ in Eq. (13). Hence we
obtain

$\displaystyle n_{s}-1$
$\displaystyle\approx-\frac{p(p+2)}{\sigma_{*}^{2}}-\frac{8\mathcal{C}p^{2}(p-1)^{2}}{\sigma_{*}^{4}}\left(\frac{\xi_{*}^{p}-\xi_{*}^{2}}{\xi_{*}(1+\xi_{*}^{p})}\right)^{2},$
$\displaystyle r$ $\displaystyle\approx\frac{8p^{2}}{\sigma_{*}^{2}},$ (15)

where $\sigma_{*}$ and $\xi_{*}$ are evaluated at horizon crossing,
$N_{*}\approx\sigma_{*}^{2}/2p$. The extra term in $n_{s}-1$, which is
negative and arises from the term $4s_{\Delta}c_{\Delta}\eta_{\sigma s}$ in
Eq. (15)), explains why the spectral index comes into better agreement with
Planck data than in single-field chaotic inflation. This term is explicitly
related to the curvature of the potential and ensuing slow-roll trajectory,
inspiring our CTHC (curved trajectory at horizon crossing) acronym. The
expression for $r$ is identical to that for single-field inflation
Planck:2013jfk .333In the numerical results, $r$ also depends weakly on
$\xi_{*}$, but this dependence does not appear in Eq. (15) at the level of
approximations we have used; to leading order $r$ is determined by $N_{*}$.

These are rough estimates, since Eq. (4) indicates $\dot{W}/\dot{U}=\xi^{p-1}$
during slow roll, in contrast to the approximation we have used. For small
values of $\xi_{*}\lesssim 0.1$, the mirror inflaton is no longer slowly
rolling and then higher order contributions to Eq. (13) become important. One
can clearly see this breakdown in the approximations in the limit $\xi_{*}\to
0$, where the predictions for single-field inflation would be recovered in an
exact expression. Nevertheless, Eq. (15) accurately describes the correlations
between $\sigma_{*}$, $\xi_{*}$ and the cosmological observables when the two
scalar fields are both slowly rolling during horizon exit, which is satisfied
for $\xi_{*}$ not too much less than unity.

|  | Noncanonical | Canonical
---|---|---|---
$p$ | $x_{i}$ | $F(X)$ | $X_{i}$ | $Y_{i}$ | $f$ | $N_{\rm tot}$ | $x_{f}$ | $\beta$ | $X_{i}$ | $Y_{i}$ | $N_{\rm tot}$ | $x_{f}$
$2/3$ | $0.91$ | $1+fX^{4}$ | $3.0$ | $2.48$ | $0.78$ | $68.1$ | $4.2\times 10^{-3}$ | $5.8\times 10^{-15}$ | $8.4$ | $4.77$ | $71.6$ | $2.8\times 10^{-4}$
$1/2$ | $0.96$ | $1+fX^{6}$ | $2.0$ | $1.84$ | $4.5$ | $112.6$ | $5.2\times 10^{-7}$ | $1.6\times 10^{-20}$ | $8.6$ | $6.16$ | $113.8$ | $1.2\times 10^{-3}$
$1/3$ | $0.987$ | $1+fX^{10}$ | $1.6$ | $1.56$ | $12$ | $247.9$ | $2.1\times 10^{-8}$ | $1.8\times 10^{-32}$ | $9.7$ | $8.33$ | $247.8$ | $4.5\times 10^{-3}$
Table 1: Parameters and initial values for three noncanonical benchmark models
and their corresponding canonical power-law scenarios, $V\sim|X|^{p}+|Y|^{p}$,
where $p=4/(n+2)$. Parameters include the total number of $e$-foldings of
inflation $N_{\rm tot}$, the initial and final values of the temperature ratio
$x$, and the amplitude of the isocurvature power spectrum on large scales
$\beta$.

###  II.4 Isocurvature fluctuations

Two-field inflation models have the potential of generating isocurvature
contributions to the power spectrum $\mathcal{P}_{S}(k)$, which are
constrained by Planck. Thus, we compute the evolution of entropy perturbations
to estimate their amplitude after inflation.

One can relate the adiabatic and entropy fluctuations at the end of inflation
to their values at horizon crossing using a matrix of transfer functions
Byrnes:2006fr ; Amendola:2001ni ; Wands:2007bd ,

$\begin{pmatrix}\mathcal{R}\\\
\mathcal{S}\end{pmatrix}=\begin{pmatrix}T_{RR}&T_{RS}\\\
T_{SR}&T_{SS}\end{pmatrix}\begin{pmatrix}\mathcal{R}_{*}\\\
\mathcal{S}_{*}\end{pmatrix},$ (16)

where the dimensionless adiabatic and isocurvature fluctuations are given by
$\mathcal{R}=H\delta\sigma/\dot{\sigma}$ and $\mathcal{S}=H\delta
s/\dot{\sigma}$ in the spatially flat gauge, and the star subscript indicates
the time of horizon crossing. The matrix elements in the upper and lower rows
are obtained by solving the perturbed equations of motion with initial
conditions $(\mathcal{R}_{*},\mathcal{S}_{*})=(1,0)$ and $(0,1)$ respectively.

To compute the transfer functions of Eq. (16), it is more convenient to
instead consider the evolution of the canonical fields’ perturbations $\delta
U,\delta W$ by making use of Eq. (11). Their equations of motion are
Cline:2019fxx

$\displaystyle\delta U^{\prime\prime}$ $\displaystyle=-C_{1}\,\delta
U^{\prime}-3\eta_{UU}\,\delta U+U^{\prime}\,\delta C$
$\displaystyle+(U^{\prime 2})^{\prime}\,\delta
U+(U^{\prime}W^{\prime})^{\prime}\,\delta W$ $\displaystyle\delta
W^{\prime\prime}$ $\displaystyle=-C_{1}\,\delta W^{\prime}-3\eta_{WW}\,\delta
W+W^{\prime}\,\delta C$ (17) $\displaystyle+(W^{\prime 2})^{\prime}\,\delta
W+(U^{\prime}W^{\prime})^{\prime}\,\delta U.$

Primes denote $d/dN$, $C_{1}=3+H^{\prime}/H$ and $\delta
C=C_{1}(U^{\prime}\,\delta U+W^{\prime}\,\delta W)$.

The amplitude of the isocurvature power spectrum is characterized by the
scale-dependent primordial isocurvature fraction,
$\beta(k)=\mathcal{P}_{S}(k)/[\mathcal{P}_{S}(k)+\mathcal{P}_{R}(k)]$, where
$\mathcal{P}_{R}(k)$ denotes the adiabatic power spectrum. On large scales,
the leading contribution is Byrnes:2006fr

$\beta\cong\frac{T_{SS}^{2}}{1+T_{SS}^{2}+T_{RS}^{2}}\lesssim\mathcal{O}(10^{-3}\text{--}10^{-2})\,.$
(18)

The experimental upper limit from Planck Akrami:2018odb  depends on the
assumptions of the fit and which datasets are used.

##  III Benchmark models

We start by considering three sets of benchmark parameters that illustrate the
possibilities for phenomenologically successful inflation with a large
reduction of temperature in the mirror sector. The results are presented for
both the noncanonical models and their corresponding canonical versions in
Table 1 and in Fig. 2. Initial conditions were chosen such that the initial
temperature ratios $x_{i}=(\rho^{\prime}_{i}/\rho_{i})^{1/4}\sim 0.91-0.99$
were the same in both types of models. It can be seen in the figure that
corresponding models lead to similar predictions for $n_{s}$ and $r$ (solid
versus dashed lines).

The canonically normalized models with fractional power potentials are
numerically challenging to evolve at late times while the fields are
oscillating. During this regime, the fields undergo damped anharmonic motion
and their energy density redshifts as Turner:1983he

$\rho\sim a^{-6p/(2+p)},$ (19)

where $a$ is the scale factor. As the amplitudes decrease, their frequency
increases rapidly, impeding accurate numerical evolution. Hence for the
canonical models we stop following the evolution of the mirror inflaton $Y$
once it reaches its minimum and instead extrapolate its energy density from
the onset of oscillations to later times using Eq. (19).

The amplitude of the isocurvature power spectrum $\beta$ in noncanonical
models was estimated by assuming a number of $e$-foldings at horizon crossing
$N_{*}=55$. In every case, the predicted amplitude is much smaller than
current bounds set by Planck.444A similar conclusion applies for the model of
Ref. Cline:2019fxx , which was not recognized in that work. This can be
understood from the field trajectory of Fig. 1: since $Y$ reaches its minimum
before the end of inflation, its perturbations are strongly diluted by the
continuing exponential expansion of the universe. Although we could not
determine $\beta$ very precisely in canonical power-law models, one expects
that since the general dynamics are similar to noncanonical models, they
should likewise lead to insignificant isocurvature perturbations.

Figure 2: Spectral index and tensor-to-scalar ratio for the models of Table 1
for $N_{*}\in[50,60]$. The shaded regions show the 68 and 95% C.L. limits set
by Planck. For noncanonical models, the effective power $p$ corresponds to
$n=4/p-2$ in Eq. (3). The thick lines to the right show the predictions in
corresponding single-field chaotic inflation with the same fractional power-
law potential.

All of the benchmark noncanonical models and their canonical counterparts lead
to temperature ratios that are safely below the cosmological bound
$x_{f}<0.5$, despite the initial ratio being close to $1$ at the beginning of
inflation. However, for $p<2/3$, noncanonical models lead to values of $x_{f}$
drastically lower than their canonical counterparts, $x\ll 10^{-3}$, a trend
we will confirm below in a Monte Carlo analysis. In this limit the mirror
sector would be almost unpopulated after reheating, an assumption that was
made in e.g. Refs. Foot:2014mia ; Foot:2011ve . To our knowledge, the scenario
presented here is the first one that can consistently predict such a cold
mirror sector, consistent with CMB constraints, without requiring significant
fine tuning in the initial conditions.

Figure 3: Left: correlation of initial conditions in the MCMC chains. Dark
(light) colors correspond to canonical (noncanonical) models. The inset shows
the noncanonical models only. Right: correlation of $f$ versus $X_{i}$ for
noncanonical models.

###  III.1 Comparison with single-field power law inflation

For a fixed initial temperature ratio $x_{i}$, the two types of models predict
similar values for $n_{s}$ and $r$ if initial conditions are chosen such that
the total duration of inflation $N_{\rm tot}$ is approximately the same, as
can be seen in Table 1 and in Fig. 2. This follows from the fact that they
coincide in the large field limit, and differ only in their behavior near the
end of inflation. As predicted by Eq. (15), lower values of $p$ lead to lower
predictions for the tensor-to-scalar ratio $r$, which tends to zero in the
limit $p\to 0$ ($n\to\infty$). We can extrapolate our results to predict that
a value of $r$ below $0.01$ would require a power-law potential with
$p\lesssim 0.15$, that is, $n\gtrsim 25$.

It is striking that the two-field inflationary scenarios are generally in much
better agreement with Planck data than their single-field counterparts,
represented by thick lines in Fig. 2. In single-field models, a flatter
potential (lower $p$) decreases $r$ but also makes $n_{s}$ closer to 1, such
that $p=2/3$ and $p=1/2$ models are marginally consistent with Planck while
$p=1/3$ is disfavored Planck:2013jfk .

With two scalar fields rolling together, the Hubble parameter is increased by
a factor of $\sqrt{2}$, effectively making each field roll more slowly at a
given amplitude.555‘Assisted inflation’ and ‘N-flation’ models are based on
this idea Liddle:1998jc ; Dimopoulos:2005ac . With $N$ independent scalar
fields, $H$ is increased by $\sqrt{N}$, and the dynamics mimics that of a
field at a higher scale. However this does not explain the better agreement of
our models, since this effect preserves the dependences of $n_{s}$ and $r$ on
$N_{*}$. In fact, if we set initial conditions $X_{i}=Y_{i}$, the failure of
the single-field models persists, as can be seen by letting $\xi_{*}=1$ in Eq.
(15).

Instead, the values of $n_{s}$ and $r$ in the two-field models depend on the
difference in times for the end of inflation in the two sectors. As the mirror
inflaton approaches its potential minimum, the field trajectory deviates from
a straight line, and the extra curvature of the potential contributes to
making the spectral index lower than what single-field inflation can achieve.
This is the CTHC mechanism described by Eq. (15): since the field amplitude at
horizon exit $\sigma_{*}$ is set by $N_{*}$, the main quantity that impacts
$n_{s}$ is the canonical field ratio $\xi_{*}=W_{*}/U_{*}$. $\xi$ is strictly
decreasing during inflation, and Eq. (15) is a monotonically increasing
function of $\xi$ in the interval $[0,1]$. This implies that $\xi_{*}$ must
fall within a precise window of values at $N_{*}$ to give optimal agreement
with CMB observations [see Eq. (20)]. Moreover, if the field ratio at the
beginning of inflation $\xi_{i}$ is too small, that is, if the initial
asymmetry between the fields is too big, the value of $n_{s}$ will be too
small. In other words, the agreement between cosmological observations and the
predictions of our model imposes a lower bound on the field ratio at the
beginning of inflation (assuming that $Y_{i}<X_{i}$). We will quantify this
bound and the optimal range of $\xi_{*}$ at horizon crossing in the next
section.

Figure 4: Scatter plots from the Markov Chain Monte Carlo (MCMC) analysis.
Left: correlation between $n_{s}$ and $r$. Right: correlation of the initial
temperature asymmetry $x_{i}=T^{\prime}_{i}/T_{i}$ with the total number of
$e$-foldings during inflation $N_{tot}$. Dark (light) colors correspond to
canonical (noncanonical) models, but there is no significant distinction
between the two types of scenarios within these plots. All models are within
the 95% C.L. limits of Planck.

However, if $\xi_{*}\ll 1$ at horizon crossing, the approximate results of Eq.
(15) are invalid and we instead recover the single-field limit (cf the thick
lines in Fig. 2), since the mirror field makes a negligible contribution to
inflation. Here the final temperature ratio can be arbitrarily small and
isocurvature perturbations are essentially nonexistent, but the spectral index
and the tensor-to-scalar ratio are at best marginally consistent with CMB
constraints. In the remainder of this work we will focus on the two-field
inflationary models, where the mirror inflaton is still slowly rolling at
horizon crossing.

Because the field ratio $\xi$ is closely related to the temperature ratio
$x=T^{\prime}/T\approx(W/U)^{p/4}$, Eq. (15) further implies a correlation
between $n_{s}$ and $x_{f}$. This will allow us to identify ranges of $x_{f}$
that minimize the $\chi^{2}$ of observed versus predicted $n_{s}$ and $r$
values in the following section, that is, ranges of $x_{f}$ that are most
consistent with CMB observations.

Figure 5: Correlation between the final temperature ratio $x_{f}$ and
$\chi^{2}$ for noncanonical (left) and canonical (right) models from the MCMC
analysis, that respect the 95% C.L. limits of Planck. The grey shading at the
top indicates the excluded region $x_{f}>0.5$. For comparison, the vertical
lines show the minimal value of $\chi^{2}$ for single-field $p=1/2$ (green)
and $p=2/3$ (orange) inflation models. Data points above the dashed line in
the left panel correspond to models where the oscillations of $Y$ are
overdamped, as explained in the text.

##  IV Markov Chain Monte Carlo

To more completely explore the parameter space, we performed a Markov Chain
Monte Carlo (MCMC) analysis for both the canonical and noncanonical models.
The parameters that were allowed to vary were the initial field amplitudes and
the kinetic parameter $f$ in Eq. (3), and we ran separate MCMCs for the values
$p=1/3,\,1/2,\,2/3$, selecting models that fell within the Planck 95%
confidence limits. Scatter plots showing correlations of the input parameter
values from the resulting MCMC chains are presented in Fig. 3. They
demonstrate the requirement of a small initial asymmetry $Y_{i}\cong X_{i}$,
and an inverse correlation between $f$ and $X_{i}$ that is necessary for
achieving a long enough period of inflation. Fig. 3 shows that smaller initial
field values are required in the noncanonical models, but in terms of the
canonically normalized fields, e.g., $U=\int F(X)^{1/2}dX$, the initial values
are of the same order ($\gtrsim 10$) for both kinds of models.

###  IV.1 Inflationary observables

As explained above, canonical and noncanonical models lead to very similar
predictions for $n_{s}$ and $r$, differing primarily in their predicted final
temperature ratios. Fig. 4 shows these predictions, as well as the correlation
between the total duration of inflation $N_{\rm tot}$ and the initial
temperature asymmetry $x_{i}$, for both kinds of models. There is no apparent
distinction between the two kinds of models in these plots. For the value of
$N_{*}$ at which $n_{s}$ and $r$ were evaluated in each model, we chose a
random number in the interval $[50,60]$, to reflect the uncertainty in the
overall scale of inflation and reheat temperatures.

We observe a significant variation for the value of the spectral index (and,
to a lesser extent, for the tensor-to-scalar ratio) within each class of
models depending on the parameters and initial conditions. This is the result
of the dependence of $n_{s}$ on both $N_{*}$ and the field ratio $\xi_{*}$ at
horizon crossing described by Eq. (15), and it demonstrates the versatility of
the two-field models in contrast to single-field inflation.

The right panel of Fig. 4 can be understood in terms of the CTHC effect
described previously: if the fields are close to each other at the beginning
of inflation ($x_{i}=\xi_{i}^{p/4}\approx 1$), then inflation must last longer
in order for the trajectory to bend enough to reach the ratio
$\xi_{*}=(Y_{*}/X_{*})^{2/p}$ at horizon crossing that corresponds to optimal
agreement with CMB observations. One can achieve a longer period of slow-roll
either by increasing the initial values of the fields or the nonmiminal
kinetic parameter $f$. Numerically, we find that

$0.4\lesssim\xi_{*}\lesssim 0.7$ (20)

is required at horizon exit for $n_{s}$ to fall within Planck’s 68% confidence
limit, $n_{s}=0.9649\pm 0.0042$ Akrami:2018odb , although the precise interval
depends mildly on $p$.

It is noteworthy that for a given value of $p$, there exists a minimum value
of $x_{i}$, below which no models were retained in our MCMC chains. This can
be understood in terms of the optimal range (20) for $\xi_{*}\cong
x_{*}^{p/4}$. If $x_{i}$ falls below the corresponding lower bound of Eq.
(20), $\xi_{i}\lesssim 0.4^{p/4}$, then $x(t)$ continues to decrease during
inflation, resulting in a value of $n_{s}$ that is too small no matter how
long inflation lasts, in conflict with Planck data. These minimum values are
rather close to 1: $x_{i}\gtrsim 0.86$, $0.92$ and $0.94$ for $p=2/3$, $1/2$
and $1/3$, respectively. Therefore, the flatter the potential is, the closer
the energy densities must be initially. Whether this could be considered as a
mild fine tuning of initial conditions cannot be quantified in the absence of
a complete theory for the probability distribution of the field amplitudes at
the beginning of inflation. It is surprising and somewhat ironic that our
search for a large temperature asymmetry reveals the requirement of a moderate
level of symmetry in the initial conditions, to achieve agreement with current
cosmological observations.

###  IV.2 Temperature ratio

It is striking that our mechanism can produce final temperature asymmetries as
great as $x_{f}\sim 10^{-10}$ (for the noncanonical models), and it rarely
gives values $x_{f}\gtrsim 0.5$ that are in the cosmologically excluded
region. This is illustrated in Fig. 5, which shows the correlation between
$x_{f}$ and the value of $\chi^{2}$ with respect to $n_{s}$ and $r$. The most
important feature is the difference of scale between the vertical axes of the
two panels: the noncanonical models can lead to dramatically lower $x_{f}$
than the canonical ones.

The pertinent difference between the two kinds of models arises from the
equation of state of the fields near the minimum of the potential. Since we
assume the canonical potential retains its fractional power-law shape in the
oscillatory regime, the energy density of the mirror inflaton $Y$ redshifts as
in Eq. (19) once it starts oscillating. One can see that higher values of $p$
make $Y$ decay faster, which is why $p=2/3$ canonical models may lead to a
temperature ratio as low as $x_{f}\sim 10^{-5}$ while $p=1/3$ models are
limited to $x_{f}\sim 10^{-3}$ at best. These conclusions could change if the
shape of the potential was different near its minimum.666For instance, axion
monodromy models McAllister:2008hb  typically have a potential of the form
$V(X)\sim(X^{2}+\epsilon^{2})^{p/2}$, which looks like a power-law when
$X\gg\epsilon$ while being approximately quadratic near the minimum, when
$X\ll\epsilon$. This scenario would be very similar to our noncanonical model
since they would both yield the same equation of state at small field
amplitude.  In contrast, the noncanonical models have a quadratic potential at
small field values, leading to same equation of state as cold dark matter
after the end of inflation: $\rho\sim a^{-3}$. Hence in these models the
mirror inflaton generally decays much faster than in canonical models,
explaining why the final temperature ratio can be as low as $\sim 10^{-10}$.

Exceptionally, the left panel of Fig. 5 reveals some noncanonical models that
lead to small values of $\chi^{2}$, yet with much larger temperature ratios.
These points are separated from the rest by a dashed line to highlight the
distinct correlations. They correspond to models where the nonminimal kinetic
parameter $f$ is small ($\lesssim 1$, cf. the lower part in the second panel
of Fig. 3). Such values of $f$ generally require larger initial amplitudes
$X_{i}$ and $Y_{i}$ to maintain the requirement (20) on $\xi_{*}$. This in
turn makes the Hubble parameter larger, increasing the impact of damping on
the oscillations of $Y$ in its oscillating phase.

Elaborating on this point, by neglecting the nonminimal kinetic term after $Y$
leaves the slow rolling regime, the equation of motion of the mirror inflaton
is

$Y^{\prime\prime}+3Y^{\prime}+\frac{m^{2}}{H^{2}}Y\approx 0,$ (21)

which describes a damped harmonic oscillator. When $H/m>2/3$, the oscillations
of $Y$ are overdamped, making its amplitude decay very slowly. By contrast, if
$H/m<2/3$ when $Y$ enters its oscillating phase it is underdamped and it
decays exponentially. Models above the dashed line in the right panel of Fig.
5 all correspond to the overdamped case, which is why the final temperature
ratio is many orders of magnitude above the low-amplitude cases. This behavior
occurs only in the noncanonical models because there both $f$ and the initial
conditions $X_{i},Y_{i}$ impact the duration of inflation; in canonical
scenarios only the latter play a role.

###  IV.3 Predictions for $x_{f}$ and isocurvature

Although the final temperature ratios $x_{f}$ span many orders of magnitude in
the MCMC results, if future CMB measurements converge on values of $n_{s}$ and
$r$ near the Planck best-fit values, this range can become narrower. This is
due to the correlation between the spectral index and the field ratio given by
Eq. (15), which implies a correlation between $x_{f}$ and $n_{s}$. Hence only
certain values of $x_{f}$ are predicted by models that yield $n_{s}=0.9649\pm
0.0042$. The right panel of Fig. 5, which refers to the canonical models,
predicts e.g. $\sim 10^{-3}$–$10^{-2}$ for $p=1/3$. Wider intervals are
allowed for the underdamped noncanonical models (below the dashed line of the
left panel), e.g., the data points cluster near $\chi^{2}\approx 0.6$ for
$x_{f}\sim 10^{-10}$–$10^{-6}$ for $p=1/3$, while the possibility of
overdamped models extends the most likely range of $x_{f}$ to up to $\sim
10^{-2}$.

Fig. 5 shows, by the vertical lines, the minimal values of $\chi^{2}$
obtainable in single-field $p=2/3$ and $p=1/2$ inflationary model. Neither
model gives a good fit to the data, and the $p=1/3$ case is disfavored at
$>95\%$ C.L. (see Fig. 2), so its corresponding horizontal line falls outside
the plotted region. The single-field models also describe two-field inflation
in the regime where $Y\ll X$ at horizon crossing, so that $x_{f}$ can be
arbitrarily small in those scenarios. However, most points from our MCMC
analysis give better fits than the single-field models.

Figure 6: Correlation between the amplitude of the isocurvature spectrum
$\beta$ on large scales and the final temperature ratio $x_{f}$ in the
noncanonical models. The grey bands are the excluded region $x_{f}>0.5$ and
the limit on the Planck sensitivity $\beta>10^{-3}$.

Similarly to the benchmark models of section III, we computed the amplitude of
the isocurvature power spectrum $\beta$ for noncanonical models in the MCMC
chains. Fig. 6 shows the resulting correlation between $\beta$ and $x_{f}$. As
anticipated, the $\beta$ is too small to be observable in most cases, with
only a handful of $p=2/3$ models approaching the sensitivity of Planck. We
empirically observe a scaling relation $\beta\sim x_{f}^{4}$. Even though
models with observable isocurvature are rare, these examples are interesting
because they suggest that a detection of isocurvature could be correlated with
an observable deviation in $N_{\rm eff}$.

##  V Conclusion

In this work we have revisited a mechanism for generating a cooler mirror
sector from inflation, due to random initial conditions, while maintaning
exact mirror symmetry at the Lagrangian level. It was shown that two-field
inflation models with a fractional power-law potential can efficiently amplify
a small asymmetry between the visible sector and its mirror counterpart,
leading to a temperature ratio as low as $x_{f}\sim 10^{-10}$ at the end of
inflation. Models with a nonminimal kinetic term generally lead to values of
$x_{f}$ that are much lower than the corresponding canonically normalized
scenarios, due to the difference in the equation of state of the fields when
they are oscillating around the minimum of the potential.

While both kinds of models can be in much better agreement with Planck data
than single-field inflation, somewhat surprisingly this requires the initial
temperature asymmetry between the two sectors to be relatively small,
$x_{i}\gtrsim 0.86$. The value of the tensor-to-scalar ratio $r$ depends on
the effective power $p$ of the potential, lying in the range $\sim
0.03$–$0.07$ for $p$ between $1/3$ and $2/3$, which could be observed in
upcoming CMB experiments Hazumi:2019lys .

A key assumption is that the two sectors are decoupled or very weakly
interacting with each other. For example, a
$\lambda_{HH^{\prime}}|H|^{2}|H^{\prime}|^{2}$ coupling mixing the Higgs
bosons of the two sectors must have $\lambda_{HH^{\prime}}\lesssim 10^{-8}$ to
avoid equilibration of the temperatures after reheating.777By demanding the
scattering rate $\Gamma\sim\lambda_{HH^{\prime}}^{2}T<H\sim T^{2}/M_{p}$ down
to the weak scale $T\sim m_{H}$.  This is technically natural since
$\lambda_{HH^{\prime}}$ is only multiplicatively renormalized.

On the other hand we have also assumed the possible interaction
$\lambda_{XY}X^{2}Y^{2}$ to be absent. One might expect that its presence
could synchronize the two fields during inflation and make $X\approx Y$ at the
onset of reheating. In a preliminary investigation we find the opposite
behavior: nonvanishing $\lambda_{XY}$ instead tends to enhance the final
temperature asymmetry, naively estimated as we have done throughout this work.
However whether this would be a good estimate in the present case is
questionable, because of the possibility of $Y$ particle productions via
parametric resonance or $XX\to YY$ scattering during reheating. This question
is beyond the scope of the present work, but could be interesting for future
study.

Acknowledgment. We thank Eva Silverstein for helpful correspondence. This work
was supported by NSERC (Natural Sciences and Engineering Research Council,
Canada).

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