# Adiabatic Cooper pair splitter

Fredrik Brange  Department of Applied Physics, Aalto University, 00076 Aalto,
Finland  Riya Baruah  Department of Applied Physics, Aalto University, 00076
Aalto, Finland  Christian Flindt  Department of Applied Physics, Aalto
University, 00076 Aalto, Finland

###### Abstract

Recent experiments have observed Cooper pair splitting in quantum dots coupled
to superconductors, and efficient schemes for controlling and timing the
splitting process are now called for. Here, we propose and analyze an
adiabatic Cooper pair splitter that can produce a regular flow of spin-
entangled electrons in response to a time-dependent and periodic gate voltage.
The splitting process is controlled by moving adiabatically back and forth
along an avoided crossing between the empty state and the singlet state of two
quantum dots that are coupled to a superconductor, followed by the emission of
the split Cooper pairs into two normal-state drains. The scheme does not rely
on fine-tuned resonance conditions and is therefore robust against
experimental imperfections in the driving signal. We identify a range of
driving frequencies, where the output currents are quantized and proportional
to the driving frequency combined with suppressed low-frequency noise. We also
discuss the main sources of cycle-missing events and evaluate the statistics
of electrons emitted within a period of the drive as well as the distribution
of waiting times between them. Realistic parameter estimates indicate that the
Cooper pair splitter can be operated in the gigahertz regime.

_Introduction._ — Cooper pair splitters are experiencing a surge of interest
as several promising experiments have brought the field closer to the ultimate
goal of detecting and exploiting the non-local entanglement of split Cooper
pairs [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 3, 5, 7, 4, 11,
12, 15, 8, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Since entanglement is a
critical resource for most quantum applications, Cooper pair splitters have
attracted attention across a wide range of fields in quantum science and
technology. Recently, Cooper pair splitting has been observed with charge
detectors [16, 20] and dispersive read-out [23], correlations between spin
currents have been measured [21], and Cooper pair splitters with triplet
pairing have been realized [22, 24]. The thermoelectric properties of Cooper
pairs splitter have also been explored in theory [26, 27, 28, 29] and
experiment [15]. Cooper pair splitters have been implemented in a variety of
architectures based on nanowires [3, 5, 8, 9, 11, 14, 18, 19, 21, 22, 23],
carbon nanotubes [4, 6, 7, 13], graphene [10, 12, 15, 17], semiconductor
quantum dots [23], and metallic islands [16, 20]. Very recently, setups with
several quantum dots and superconductors have also been experimentally
realized [25].

These experimental advances have reduced the gap between experiment and
theory, and several theoretical ideas for future experiments may soon be
within reach. As an example, the distribution of waiting times was already
measured following a recent suggestion [30, 16]. There are also proposals for
observing the entanglement of the split Cooper pairs by either violating a
Bell inequality [31, 32, 33, 34] or by using an entanglement witness
formulated in terms of cross-correlation measurements of the outgoing spin
currents [35, 36, 37]. In addition, several Cooper pair splitters may be
combined to create a Kitaev chain with Majorana bound states forming at the
ends [38, 39, 40, 22, 25]. Moreover, while experiments have focused on static
devices, there are also proposals to control the splitting of Cooper pairs
using time-dependent drives [41, 42]. In this context, it is an open question
how one should design the driving scheme in the best way.

Figure 1: Adiabatic Cooper pair splitter. (a) The device consists of a
nanowire (green) with gate-defined quantum dots coupled to a superconductor
($S$, blue). The amplitude for Cooper pair splitting is denoted by $\gamma$,
while $\Gamma$ is the rate at which electrons are emitted into the normal-
state electrodes ($N$, red). A time-dependent gate voltage, $V_{g}(t)$, is
used to control the left quantum dot level. (b) The superconducting gap is
denoted by $\Delta$, and $\varepsilon_{L/R}$ are the tunable level positions.
(c) Current as a function of the level positions for a static device with
$\hbar\Gamma/\gamma=0.01$. In our scheme, we move the left level back and
forth across the peak in the current. (d) The amplitude for Cooper pair
splitting leads to an avoided crossing between the singlet state $|S\rangle$
and the empty state $|0\rangle$. We move back and forth along the avoided
crossing, and a split Cooper pair is emitted into the drains after each
crossing. Figure 2: Adiabatic driving scheme, average current, and low-
frequency noise. (a) The period of the drive is divided into two splitting
phases and two emissions phases, each of duration $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$, respectively, such that
$\mathcal{T}=2(\mathcal{T}_{1}+\mathcal{T}_{2})$. During the splitting phases,
the left level is moved across the resonance,
$\varepsilon_{L}=-\varepsilon_{R}$, and a Cooper pair is split. During the
emission phases, the levels are kept far off resonance, and the split Cooper
pair tunnels into the drains. (b) Average current as a function of the driving
frequency. The parameters are $\kappa=\gamma$, $\varepsilon_{1}=50\gamma$,
$\varepsilon_{2}=100\gamma$, $\varepsilon_{R}=-100\gamma$, and
$\hbar\Gamma=0.001\gamma$ (red), $0.002\gamma$ (green), and $0.003\gamma$
(blue), and we have defined $f_{0}=\alpha\gamma/2\pi\hbar$ with
$\alpha=\mathcal{T}_{1}/\mathcal{T}=0.01$. The adiabatic regime, where the
current should taken on the value $I_{\ell}=2f$, is indicated for the red
curve by the shaded area according to Eq. (3). (c) The Fano factor,
$F_{\ell}=S_{\ell}/I_{\ell}$, as a function of the driving frequency. The
three circles in panel (b) indicate the frequencies used in Figs. 3 and 4.

In this Letter, we propose and analyze an adiabatic Cooper pair splitter that
operates by driving two quantum dots coupled to a superconductor back and
forth along an avoided crossing between the empty state and the singlet state
of the quantum dots, see Fig. 1. Each time the dots are filled by a split
Cooper pair from the superconductor, the system is taken back to the empty
state as the electrons are emitted into the drain electrodes, and the process
can repeat. The avoided crossing occurs because of the amplitude of Cooper
pair splitting, which couples the singlet state of the quantum dots with the
empty state, and the driving can be implemented with an external gate. When
operated adiabatically, the Cooper pair splitter delivers a regular and low-
noise flow of split Cooper pairs as shown in Fig. 2. The scheme does not rely
on fine-tuned resonance conditions or accurate timing and may be realized
based on recent experiments.

_Adiabatic Cooper pair splitter._ — Figure 1(a) shows the Cooper pair splitter
consisting of two single-level quantum dots coupled to a superconductor. We
here consider a setup based on quantum dots along a nanowire, similar to
recent experiments [3, 19, 21, 22, 23], but our proposal would also work for
other architectures. With a large superconducting gap, the dynamics of the
quantum dots can be described by the effective Hamiltonian [43, 44, 41, 45,
42]

$\hat{H}=\sum_{\ell\sigma}\varepsilon_{\ell}\hat{d}_{\ell\sigma}^{\dagger}\hat{d}_{\ell\sigma}^{\phantom{\dagger}}-\gamma(\hat{d}_{S}^{\dagger}+\hat{d}_{S})-\kappa\sum_{\sigma}(\hat{d}_{L\sigma}^{\dagger}\hat{d}_{R\sigma}^{\phantom{\dagger}}+\mathrm{h.c.}),$
(1)

where $\varepsilon_{\ell}$ are the energy levels of the quantum dots,
$\ell=L,R$, which can be tuned by external gates to control the splitting of
Cooper pairs. The amplitudes for Cooper pair splitting and elastic cotunneling
are denoted by $\gamma$ and $\kappa$, respectively. The operator
$\hat{d}_{\ell\sigma}^{\dagger}$ creates electrons with spin
$\sigma=\,\uparrow,\downarrow$ in either of the dots, while
$\hat{d}_{S}^{\dagger}\equiv(\hat{d}_{L\downarrow}^{\dagger}\hat{d}_{R\uparrow}^{\dagger}-\hat{d}_{L\uparrow}^{\dagger}\hat{d}_{R\downarrow}^{\dagger})/\sqrt{2}$
describes a singlet state that is delocalized between them. Strong Coulomb
interactions on the quantum dots prevent each of them from being doubly
occupied, which ensures that the electrons from a split Cooper pair tunnel
into different quantum dots. We work with a large detuning of the dot levels,
$|\varepsilon_{L}-\varepsilon_{R}|\gg\kappa$, to suppress elastic cotunneling
between them. The empty state of the quantum dots, $|0\rangle$, with zero
energy is coherently coupled to the singlet state,
$|S\rangle=\hat{d}_{S}^{\dagger}|0\rangle$, with energy
$\varepsilon_{L}(t)+\varepsilon_{R}$ by the amplitude for Cooper pair
splitting, $\gamma$, and the energy of the singlet state is controlled by a
time-dependent gate voltage on the left quantum dot.

Figure 3: Time-dependent currents. We show the time-dependent currents
corresponding to the three points marked with circles in Fig. 2. (a) At low
frequencies, more than one electron is emitted per half-period, and emissions
occur already in the splitting phase, see inset. (b) In the adiabatic regime,
one electron is emitted at every half-period, and the leakage current in the
splitting phase is suppressed. (c) At high frequencies, the regularity is
gradually lost, and there is always a finite current running.

As shown in Fig. 1(b), large voltages are applied to the normal-state
electrodes, so that they function as drains for the dots. Without a time-
dependent drive, the (particle) currents running into the drains are

$I_{L/R}=\frac{2\Gamma\gamma^{2}}{(\varepsilon_{L}+\varepsilon_{R})^{2}+(\hbar\Gamma)^{2}+4\gamma^{2}},$
(2)

where $\Gamma$ is the tunneling rate into the drains, which we assume to be
the same for the two drains to keep the discussion simple [46, 44, 45]. In
Fig. 1(c), we show the current as a function of the level positions, and we
see a peak along the diagonal $\varepsilon_{R}=-\varepsilon_{L}$, where the
singlet state is on resonance with the empty state. A similar dependence was
observed in the recent experiments of Refs. [22, 24].

_Driving scheme._ — To describe the adiabatic driving scheme, we show in Fig.
1(d) the energy of the empty state and the singlet state as a function of
$\varepsilon_{L}$, and we observe an avoided crossing between them at
$\varepsilon_{L}=-\varepsilon_{R}$ because of the coupling $\gamma$. Thus, if
we start with a large value of $\varepsilon_{L}$, the empty state will have
the lowest energy, and as we move through the avoided crossing by decreasing
$\varepsilon_{L}$, the quantum dots will eventually become occupied by a split
Cooper pair. At the same time, the probability increases for the electrons to
leave the dots via the drains. The system thereby returns to the empty state,
which now has a higher energy than the singlet state. After that, we increase
the energy of the left level, and we again move from the empty state to the
singlet state, but this time following the excited state of the system.
Eventually, the quantum dots are again occupied by a split Cooper pair, and
once again the system is taken back to the empty state as the electrons tunnel
into the drains. By doing so periodically, two split Cooper pairs should be
produced per period of the drive.

The driving scheme in Fig. 2(a) is now designed with the following
requirements in mind. To formulate them, we divide the period of the drive
into four phases, two splitting phases, each of duration $\mathcal{T}_{1}$,
and two emission phases, each of duration $\mathcal{T}_{2}$. The period of the
drive is then $\mathcal{T}=2(\mathcal{T}_{1}+\mathcal{T}_{2})=1/f$, where $f$
is the driving frequency. Our requirements for the drive are now:

  1. 1.

_Adiabatic splitting:_ The splitting phase should start off resonance, so that
$\gamma/\varepsilon_{1}\ll 1$, and the drive should be slow, so that $\gamma
T_{1}/\hbar\times\gamma/\varepsilon_{1}\gg 1$, where
$\varepsilon_{1}=\varepsilon_{L}+\varepsilon_{R}$ is the singlet energy at the
onset [47, 48].

  2. 2.

_No leakage:_ To make sure that no electrons are emitted during the splitting
phase, we need $\Gamma\mathcal{T}_{1}\ll 1$.

  3. 3.

_Emission:_ To ensure that the electrons are emitted during the emission
phase, we need $\Gamma\mathcal{T}_{2}\gg 1$. Also, during the emission phase,
Cooper pair splitting should be off resonance, so that
$|\varepsilon_{L}+\varepsilon_{R}|\gg\gamma$.

These requirements can be combined into the inequality

$\alpha\Gamma\ll
f\ll\mathrm{min}\\{\alpha\gamma^{2}/\hbar\varepsilon_{1},\Gamma/2\\},$ (3)

which specifies the range of possible driving frequencies for the adiabatic
Cooper pair splitter given a fixed ratio of the splitting time over the period
of the drive, $\alpha=\mathcal{T}_{1}/\mathcal{T}$. To provide realistic
estimates, we note that the amplitude for Cooper pair splitting can be on the
order of $\gamma=40$ $\mu$eV together with tunneling rates of $\hbar\Gamma=4$
$\mu$eV (or 1 GHz) [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 3, 5, 7,
4, 11, 12, 15, 8, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Taking the duration
of the splitting phase so that $\alpha=0.1$, combined with a singlet energy of
$\varepsilon_{1}=100$ $\mu$eV at the onset, the inequality (3) predicts
adiabatic frequencies in the range 50 MHz $\ll f\ll$ 500 MHz. For example,
with a driving frequency of $f=100$ MHz, we would expect currents of about 20
pA, since two electrons are emitted into each drain per period of the drive.
We may also take $\varepsilon_{2}=100$ $\mu$eV in Fig. 2(a) to suppress Cooper
pair splitting during the emission phase.

_Average current._ — To illustrate the operation of the Cooper pair splitter,
we calculate the drain currents. To this end, we consider the density matrix
of the dots, $\hat{\rho}(t)$, whose dynamics obeys the Lindblad equation [44,
45, 49]

$\frac{d}{dt}\hat{\rho}(t)=\mathcal{L}(t)\hat{\rho}(t)=\frac{1}{i\hbar}[\hat{H}(t),\hat{\rho}(t)]+\mathcal{D}\hat{\rho}(t).$
(4)

Here, tunneling to the drains is described by the term

$\mathcal{D}\hat{\rho}(t)=\Gamma\sum_{\ell\sigma}\big{(}\hat{d}_{\ell\sigma}^{\phantom{\dagger}}\hat{\rho}(t)\hat{d}_{\ell\sigma}^{\dagger}-\frac{1}{2}\\{\hat{\rho}(t),\hat{d}_{\ell\sigma}^{\dagger}\hat{d}_{\ell\sigma}^{\phantom{\dagger}}\\}\big{)},$
(5)

and the Hamiltonian $\hat{H}(t)$ is given by Eq. (1) with time-dependent
levels. Because of the large voltages, the chemical potential of the drains is
placed well below the quantum dot levels, so that the thermal smearing due to
a finite temperature becomes unimportant. Single-electron excitations above
the gap are exponentially suppressed in the ratio of the superconducting gap
over the temperature as $\exp(-\Delta/k_{B}T)$, allowing us to ignore such
excitations. Realistically, the gap can be up to $\Delta\simeq 1$ meV
(corresponding to a temperature of about 10 K), which indeed is much higher
than typical experimental temperatures of around $T=100$ mK.

Figure 4: Distribution of waiting times. We show distributions corresponding
to the three points marked with circles in Fig. 2. (a) At low frequencies, a
peak develops at short waiting times. (b) In the adiabatic regime, a single
peak at half the period shows that Cooper pairs are being split periodically.
(c) At high frequencies, cycle-missing events give rise to several peaks.

Figure 2(b) shows the current as a function of the driving frequency [46].
When operated in the adiabatic regime, the device should deliver two split
Cooper pairs per period of the drive, and the drain currents should take on
the quantized value $I_{\ell}=2f$. This expectation is confirmed by our
calculations, which show a quantized current in the adiabatic regime defined
by Eq. (3). At higher frequencies, the number of emitted electrons drops off,
since the driving becomes too fast, and a Cooper pair is not split in each
crossing of the resonance. The current does not vanish at low frequencies,
since the system is biased, and a current will run even without the drive. For
this reason, the ratio of the current over the frequency can become large at
very low frequencies. Experimentally, the plateau in Fig. 2(b) would
demonstrate the adiabatic splitting of Cooper pairs.

_Noise and Fano factor._ — To further analyze the splitting of Cooper pairs,
we show in Fig. 2(c) the low-frequency noise $S_{\ell}$ of the drain currents,
quantified by the Fano factor, $F_{\ell}=S_{\ell}/I_{\ell}$ [46, 50, 51, 52,
53, 54, 55]. In the adiabatic regime, we expect a strong suppression of the
noise, which indeed is confirmed by our calculations. By contrast, at lower
frequencies, the Fano factor increases and comes closer to the values for a
static device [45]. At high frequencies, the splitting of Cooper pairs becomes
rare and uncorrelated, and the Fano factor approaches one. We also observe
oscillations in the current and the Fano factor, which can be attributed to an
interplay between the amplitude of Cooper pair splitting and the driving
frequency. However, for our purposes, we focus on the noise in the adiabatic
regime, which provides another experimental signature of the regular splitting
of Cooper pairs. Unlike the current, which should be measured over a range of
frequencies to observe the plateau in Fig. 2(b), the low noise can be measured
at just a single frequency.

_Cycle-missing events._ — As the driving frequency is increased beyond the
adiabatic regime, we expect cycle-missing events to occur because of non-
adiabatic excitations [47, 48]. In particular, the system may make transitions
between the instantaneous eigenstates, if we move too fast along the avoided
crossing. Also, if the unloading phase is too short compared with the escape
time to the drains, a split Cooper pair may not be emitted into the drains,
and it might be transferred back into the superconductor. If we denote the
small probability of a cycle-missing event by $p\ll 1$, the current will be
reduced to $I_{\ell}=2f(1-p)$, while the noise increases from zero to
$S_{\ell}=2fp$ [56]. The Fano factor can then be approximated as
$F_{\ell}\simeq p$, showing that it directly measures the probability of
cycle-missing events. In Fig. 2(c), the Fano factor becomes as small as one
percent, noting that we are aiming for a periodic emitter of entangled
electrons rather than metrological applications, which often require error
rates below parts per million [57].

_Time-dependent current._ — It is instructive also to consider the time-
dependent currents $I_{\ell}(t)$, which provide information about the
statistics of electrons emitted within a period of the drive. In Fig. 3, we
show the time-dependent currents for the three points marked with circles in
Fig. 2. At low frequencies, we enter the quasi-static regime, where the
current approaches the static result in Eq. (2) with the time-dependent level
position inserted. By contrast, in the adiabatic regime, the time-dependent
current shows how the quantum dots are periodically filled by a split Cooper
pair, followed by the emission of the electrons into the drains. At higher
frequencies, the driving becomes non-adiabatic, such that the quantum dots are
not filled or emptied in every half-period, and there is always a finite
current running.

_Distribution of waiting times._ — Finally, we turn to the distribution of
electron waiting times [58, 59], which were recently measured for a static
Cooper pair splitter [30, 16]. Here, we consider the distribution of the time
that passes between electrons tunneling into one of the drains [58, 30, 42,
46]. In Fig. 4, we show distributions for the three points marked with circles
in Fig. 2. At low frequencies, a peak develops at short times, corresponding
to several emissions occurring as the current resonance is crossed. On the
other hand, in the adiabatic regime, a single peak at half the period shows
that Cooper pairs are split periodically, with the width of the peak given by
the tunneling rate to the drains. Finally, in the non-adiabatic regime, peaks
appear at multiples of the half-period, since cycle-missing events start to
occur.

_Conclusions._ — We have proposed and analyzed an adiabatic Cooper pair
splitter that operates by moving a quantum dot level back and forth along an
avoided crossing. Each time the resonance is crossed, a Cooper pair is split
and emitted from the quantum dots. When operated in the adiabatic regime, the
device generates a regular flow of spin-entangled electrons with currents that
are proportional to the driving frequency combined with vanishing low-
frequency noise. Our proposal appears feasible in the light of recent
experiments, and it can be extended in many directions. For example, it may be
possible to increase the driving frequency with a shortcut to adiabaticity
[60]. Moreover, in materials like InAs, one may use the spin-orbit coupling
combined with time-dependent gates to rotate the spins in the dots [61, 62].
One may also envision Cooper pair splitters that are coupled to ballistic
conductors so that the entangled electrons can be transferred to other parts
of a solid-state circuit for further operations, manipulation, and read-out.

###### Acknowledgements.

_Acknowledgements._ — We acknowledge support from the Nokia Industrial
Doctoral School in Quantum Technology and the Research Council of Finland
through the Finnish Centre of Excellence in Quantum Technology (grant number
352925) and grant number 331737.

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Challenges for Inflaton Dark Matter

Oleg Lebedev and Jong-Hyun Yoon

Department of Physics and Helsinki Institute of Physics,
Gustaf Hällströmin katu 2a, FI-00014 Helsinki, Finland

Abstract

We examine an intriguing possibility that a single field is responsible for
both inflation and dark matter, focussing on the minimal set–up where
inflation is driven by a scalar coupling to curvature. We study in detail the
reheating process in this framework, which amounts mainly to particle
production in a quartic potential, and distinguish thermal and non–thermal
dark matter options. In the non–thermal case, the reheating is impeded by
backreaction and rescattering, making this possibility unrealistic. On the
other hand, thermalized dark matter is viable, yet the unitarity bound forces
the inflaton mass into a narrow window close to half the Higgs mass.

##  1 Introduction

The absence of spectacular signals of new physics in particle experiments
motivates one to explore scenarios based on minimalism. One economical
possibility is to account for both inflation and dark matter with just a
single new degree of freedom. The corresponding Lagrangian would also be
minimalistic: it is allowed to contain only renormalizable interactions
augmented with a scalar coupling to curvature [1], which is in any case
induced by quantum effects. This is a rigid framework, yet it may account for
some of the most puzzling aspects of modern cosmology.

The discussion of possible inflaton and dark matter unification in a more
general (often non–minimal) setting started with papers by Liddle et al.
[2],[3]. The simplest concrete model based on a non–minimal scalar coupling to
curvature was presented in [4], where the thermal DM option was studied.
Non–thermal inflaton dark matter in a similar setting was recently considered
in [5],[6],[7]. Other possibilities for inflaton and dark matter unification
were explored in [8]–[20].

If the inflaton is stable, the Standard Model particles are produced during
the inflaton oscillation epoch. The decay of the inflaton background and the
dynamics of the system, in general, depend crucially on collective effects.
These manifest themselves in resonant particle production [21],[22],[23] as
well as significant backreaction and rescattering [24],[25],[26]. Perturbative
estimates are often inadequate and can misrepresent the system behaviour by
orders of magnitude. This concerns, in particular, the decay of the inflaton
zero mode, the reheating temperature and other related quantities. Although
certain aspects of non–perturbative phenomena can be treated analytically with
the help of semi–classical methods [21], to account for backreaction and
rescattering properly requires lattice simulations.

In this work, we study in detail the reheating processes in the minimal
inflaton dark matter model, taking into account the relevant collective
phenomena with the help of lattice simulations. We find that these make a
crucial impact on the viability of the model.

##  2 Minimal model

The minimal inflaton dark matter model contains a real scalar $\phi$ in
addition to the Standard Model fields. The interactions of $\phi$ are subject
to the parity symmetry

$\phi\rightarrow-\phi\;,$ (1)

which makes $\phi$ stable. All renormalizable interactions consistent with
this symmetry are to be included. To account for inflation, one also includes
the non–minimal scalar coupling to curvature $\phi^{2}\hat{R}$. This
interaction is expected on general grounds and is induced by loop effects. The
resulting inflationary potential at large field values fits the PLANCK
observations very well [27], while the challenge is to understand whether
$\phi$ can also fit the observed dark matter abundance. This is the main
subject of the present work.

###  2.1 Set–up

The Higgs–inflaton system with non–minimal scalar–gravity couplings is
described by the Lagrangian [4]

${\cal L}_{J}=\sqrt{-\hat{g}}\left(-{1\over 2}\Omega\hat{R}\,+{1\over
2}\,\partial_{\mu}\phi\partial^{\mu}\phi+\,(D_{\mu}H)^{\dagger}D^{\mu}H-{V(\phi,H)}\right)\;,$
(2)

where $\hat{g}^{\mu\nu}$ is the Jordan frame metric and $\hat{R}$ is the
corresponding scalar curvature. In the unitary gauge,

$H(x)={1\over\sqrt{2}}\left(\begin{matrix}0\\\ h(x)\end{matrix}\right)\;,$ (3)

the $Z_{2}$–symmetric potential has the form

$V(\phi,h)={1\over 4}\lambda_{h}h^{4}+{1\over 4}\lambda_{\phi
h}h^{2}\phi^{2}+{1\over 4}\lambda_{\phi}\phi^{4}+{1\over
2}m_{h}^{2}h^{2}+{1\over 2}m_{\phi}^{2}\phi^{2}\;.$ (4)

We assume the mass parameters to be far below the Planck scale. The function
$\Omega$ includes the lowest order non–minimal scalar–gravity couplings. In
Planck units ($M_{\rm Pl}=1$), it reads

$\Omega=1+\xi_{h}h^{2}+\xi_{\phi}\phi^{2}\;.$ (5)

In what follows, we take $\xi_{\phi},\xi_{h}\geq 0$ to avoid a singularity at
large field values. Cosmological implications of the Higgs portal models of
this type have been reviewed in [28].

###  2.2 Singlet–driven inflation

The scalar couplings to gravity can be eliminated by a conformal metric
rescaling

$g_{\mu\nu}=\Omega\,\hat{g}_{\mu\nu}\;,$ (6)

which brings us to the Einstein frame. In this frame, the scalar curvature
term is canonical, while the kinetic terms and the potential are rescaled
according to [29]

$\displaystyle K^{ij}={3\over
2}\;{\partial\log\Omega\over\partial\phi_{i}}\,{\partial\log\Omega\over\partial\phi_{j}}+{\delta^{ij}\over\Omega}\;,$
$\displaystyle V_{E}={V\over\Omega^{2}}\;,$ (7)

where $i,j$ label scalar fields. In the large field limit,

$\xi_{h}h^{2}+\xi_{\phi}\phi^{2}\gg 1\;,$ (8)

the frame function can be approximated by
$\Omega\simeq\xi_{h}h^{2}+\xi_{\phi}\phi^{2}$ and the Einstein frame
Lagrangian takes the form

${\cal L}={3\over
4}\Bigl{(}\partial_{\mu}\ln(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})\Bigr{)}^{2}+{1\over
2}{1\over\xi_{h}h^{2}+\xi_{\phi}\phi^{2}}\Bigl{(}(\partial_{\mu}h)^{2}+(\partial_{\mu}\phi)^{2}\Bigr{)}-{V\over(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})^{2}}\;.$
(9)

Introducing the variables [30]

$\displaystyle\chi=\sqrt{3\over 2}\ln(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})\;,$
$\displaystyle\tau={h\over\phi}\;,$ (10)

one may rewrite the kinetic terms as

$\displaystyle{\cal L}_{\rm kin}$ $\displaystyle=$ $\displaystyle{1\over
2}\left(1+{1\over
6}{\tau^{2}+1\over\xi_{h}\tau^{2}+\xi_{\phi}}\right)\,(\partial_{\mu}\chi)^{2}+{1\over\sqrt{6}}{(\xi_{\phi}-\xi_{h})\tau\over(\xi_{h}\tau^{2}+\xi_{\phi})^{2}}\,(\partial_{\mu}\chi)(\partial^{\mu}\tau)$
(11) $\displaystyle+$ $\displaystyle{1\over
2}{\xi_{h}^{2}\tau^{2}+\xi_{\phi}^{2}\over(\xi_{h}\tau^{2}+\xi_{\phi})^{3}}\,(\partial_{\mu}\tau)^{2}\;.$

In general, $\tau$ and $\chi$ mix, while if the (local) minimum of the
potential is at

$\tau_{\rm min}=0\;,$ (12)

the mixing vanishes. Inspecting the Einstein frame potential at large $\chi$,

$V_{E}={\lambda_{h}\tau^{4}+\lambda_{\phi h}\tau^{2}+\lambda_{\phi}\over
4(\xi_{h}\tau^{2}+\xi_{\phi})^{2}}\;,$ (13)

one finds that $\tau=0$ is a local minimum if

$\lambda_{\phi h}\xi_{\phi}-2\lambda_{\phi}\xi_{h}>0\;.$ (14)

The canonically normalized fields at this point are

$\chi^{\prime}={\chi\;\sqrt{1+{1\over
6\xi_{\phi}}}}~{}~{},~{}~{}\tau^{\prime}={\tau\over\sqrt{\xi_{\phi}}}\;.$ (15)

At leading order in $1/(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})$, the potential is
flat with respect to $\chi$: $V_{E}=\lambda_{\phi}/(4\xi_{\phi}^{2})$. The
$\tau^{\prime}$ field behaves as a heavy spectator if
$m_{\tau^{\prime}}\gtrsim H$, where $H$ is the inflationary Hubble rate.
Computing $m_{\tau^{\prime}}$ from the above potential, one finds that
$\tau^{\prime}$ can be integrated out when

$\lambda_{\phi h}\gtrsim\lambda_{\phi}\;{12\xi_{h}+1\over 6\xi_{\phi}}\;.$
(16)

If this inequality is violated, the dynamics of the system do not reduce to
that of a single field $\chi^{\prime}$. We note that the choice $\xi_{h}=0$ is
allowed by this constraint, if $\lambda_{h\phi}$ is sufficiently large.

Suppose that the constraint (16) is satisfied. Since the potential is almost
flat in the $\chi^{\prime}$ direction, it is a natural inflaton candidate.
Retaining the next to leading term in the
$1/(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})$ expansion of (7), one finds

$V_{E}={\lambda_{\phi}\over
4\xi_{\phi}^{2}}\left(1+\exp\left(-{2\gamma\chi^{\prime}\over\sqrt{6}}\right)\right)^{-2}\;,$
(17)

where

$\gamma=\sqrt{6\xi_{\phi}\over 6\xi_{\phi}+1}\;.$ (18)

There is no contribution to the potential from the $\tau-\chi$ mixing at this
order since $\partial V_{E}/\partial\tau=0$ at the minimum. The above
potential is the same as that for Higgs inflation [27] which requires
$\gamma\simeq 1$, while in our case $\gamma$ can be below 1.

The inflationary predictions are read off from the slow roll parameters,

$\displaystyle\epsilon={1\over 2}\left({\partial
V_{E}/\partial\chi^{\prime}\over V_{E}}\right)^{2}\;,$
$\displaystyle\eta={\partial^{2}V_{E}/\partial\chi^{\prime\,2}\over V_{E}}\;.$
(19)

Inflation ends when $\epsilon\simeq 1$, which determines $\chi^{\prime}_{\rm
end}$. The number of $e$–folds is given by

$N=\int_{\rm in}^{\rm end}H\,dt=-\int^{\rm end}_{\rm in}{V_{E}\over\partial
V_{E}/\partial\chi}\,d\chi\;.$ (20)

For a given $N$, this equation defines the initial $\chi^{\prime}_{\rm in}$.
The COBE constraint on inflationary perturbations requires
$V_{E}/\epsilon\simeq 0.027^{4}$ at $\chi^{\prime}_{\rm in}$, which implies

${\lambda_{\phi}\over 4\xi_{\phi}^{2}}=4\times
10^{-7}\,{1\over\gamma^{2}N^{2}}\;.$ (21)

The spectral index $n$ and the tensor to scalar ratio $r$ are computed via

$\displaystyle n=1-6\epsilon+2\eta\simeq 1-{2\over N}-{9\over
2\gamma^{2}N^{2}}\;,$ $\displaystyle
r=16\epsilon\simeq{12\over\gamma^{2}N^{2}}\;.$ (22)

These predictions fit very well the PLANCK data for $N=50$ to 60 [31].

It is important to note that $\gamma\ll 1$ is inconsistent with our
approximations, i.e. the expansion in
$1/(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})=\exp\left(-{2\gamma\chi^{\prime}/\sqrt{6}}\right)$.
For $\gamma\lesssim 1/\sqrt{N}$, inflation takes place in the regime
$\exp\left(-{2\gamma\chi^{\prime}/\sqrt{6}}\right)\gtrsim 1$, where higher
order terms are important.

Further constraints on the model are imposed by unitarity considerations. A
non–minimal coupling to gravity corresponds to a non–renormalizable dim–5
operator, which implies that the model is meaningful up to the unitarity
cutoff, namely, $\Lambda\sim 1/\xi_{\phi}$ in Planck units [32],[33]. In
particular, the inflationary scale $(\lambda_{\phi}/4\xi_{\phi}^{2})^{1/4}$
must be below the cutoff. It should be noted that $\Lambda$ depends on the
inflaton background value [34], however at the end of inflation and beginning
of reheating, this background becomes insignificant, such that the cutoff is
around $1/\xi_{\phi}$ (see also [35]). Combined with (21), this requires at
the inflationary scale [36]

$\lambda_{\phi}(H)<4\times 10^{-5}$ (23)

and $\xi_{\phi}(H)<300$. Here we have set $\gamma\sim 1$ since the bound is
only relevant for large $\xi_{\phi}$.

###  2.3 Non–thermal inflaton dark matter and reheating

The critical question in this model is how the inflaton energy gets converted
into SM radiation. Since the direct inflaton decay is forbidden, this energy
transfer can only happen during the inflaton oscillation epoch. After that,
the total number of the inflaton quanta remains constant due to its feeble
interactions.

Since the presence of a non–trivial $\xi_{h}$ is inessential for our
discussion, let us now focus on the case $\xi_{h}=0$ and $\lambda_{\phi
h}\gtrsim\lambda_{\phi}/(6\xi_{\phi})$, which implies that the Higgs is a
heavy spectator stabilized at $h\simeq 0$ during inflation. At the end of
inflation, the inflaton field value is around the Planck scale. Therefore, the
Planck mass cannot be neglected in $\Omega$ leading to a complicated
dependence of the canonically normalized inflaton $\chi$ on $\phi$,

${d\chi\over
d\phi}=\sqrt{1+\xi_{\phi}(1+6\xi_{\phi})\phi^{2}\over(1+\xi_{\phi}\phi^{2})^{2}}\;.$
(24)

This equation is solved by [37]

$\,\chi(\phi)=\sqrt{1+6\xi_{\phi}\over\xi_{\phi}}\,\sinh^{-1}\left(\sqrt{(1+6\xi_{\phi})\xi_{\phi}}\,\phi\right)-\sqrt{6}\,\sinh^{-1}\left({\sqrt{6}\xi_{\phi}\phi\over\sqrt{1+\xi_{\phi}\phi^{2}}}\right)\;.$
(25)

Shortly after inflation, the inflaton amplitude decreases such that
$\xi_{\phi}\phi^{2},\xi^{2}_{\phi}\phi^{2}<1$ and

$\chi\simeq\phi\;.$ (26)

Therefore, the inflaton oscillates in the quartic potential,111We assume the
bare inflaton mass to be far below the Planck scale so that it can be
neglected during (p)reheating. At large $\xi_{\phi}$, however, there is a
brief period in which the inflaton potential is quadratic:
$V_{E}\simeq\lambda_{\phi}\chi^{2}/(6\xi_{\phi}^{2})$ for the field range
$1/(2\xi_{\phi})\ll|\chi|\ll 1$. We find that this feature is unimportant for
our purposes (see also [36]).

$V_{E}(\phi)\simeq{1\over 4}\lambda_{\phi}\,\phi^{4}\;.$ (27)

At this stage, the Higgs quanta start getting produced by a time dependent
inflaton background. Since the fields are effectively massless, particle
production takes place in a quasi–conformal regime, which means that all the
relevant quantities redshift in the same fashion leading to strong
Bose–Einstein enhancement of the process [23].

Let us consider the main features of particle production in a $\phi^{4}$
potential [23]. The classical inflaton background obeys the equation of motion

$\ddot{\phi}+3H\dot{\phi}+\lambda_{\phi}\,\phi^{3}=0\;.$ (28)

After a few oscillations, the solution takes the form of a Jacobi cosine,

$\phi(t)={\Phi_{0}\over a(t)}\;{\rm cn}\left(x,{1\over\sqrt{2}}\right)\;,$
(29)

where $x=(48\lambda_{\phi})^{1/4}\sqrt{t}$ is the conformal time. The scale
factor is fixed by $a(0)=1$ and $a(t)\propto\sqrt{t}$ shortly thereafter.

The quantum Higgs field $\hat{h}$ can be expressed in terms of the creation
and annihilation operators $\hat{a}_{k}^{\dagger},\hat{a}_{k}$ as

$\hat{h}={1\over(2\pi)^{3/2}}\int d^{3}k\left(\hat{a}_{k}h_{k}(t)e^{-i{\bf
k\cdot x}}+\hat{a}_{k}^{\dagger}h_{k}(t)^{*}e^{i{\bf k\cdot x}}\right)\;,$
(30)

where $h_{k}$ are the Fourier modes with comoving 3–momentum ${\bf k}={\bf
p}/a$. The equations of motion for the mode functions simplify in terms of the
rescaled variables $X_{k}(t)\equiv a(t)h_{k}(t)$,

$X_{k}^{\prime\prime}+\left[\kappa^{2}+{\lambda_{\phi h}\over
2\lambda_{\phi}}\;{\rm
cn}^{2}\left(x,{1\over\sqrt{2}}\right)\right]\,X_{k}=0\;,$ (31)

where the prime denotes differentiation with respect to $x$ and

$\kappa^{2}={k^{2}\over\lambda_{\phi}\Phi_{0}^{2}}\;.$ (32)

This is known as the Lamé equation. Since the Jacobi cosine is a periodic
function, the Floquet theory applies and the solutions either grow
exponentially or oscillate in $x$, depending on $\kappa^{2}$ and
$q=\lambda_{\phi h}/(2\lambda_{\phi})$. The corresponding stability chart is
shown in Fig. 1. In the “unstable” regions, the amplitude grows exponentially
representing parametric resonance and leading to Higgs production. The Higgs
$k$–mode occupation numbers are then found via

$n_{k}={\omega_{k}\over
2}\,\left({|\dot{X}_{k}|^{2}\over\omega_{k}^{2}}+|X_{k}|^{2}\right)-{1\over
2}\;,$ (33)

where $\omega_{k}^{2}=\kappa^{2}+{\lambda_{\phi h}\over 2\lambda_{\phi}}\;{\rm
cn}^{2}\left(x,{1\over\sqrt{2}}\right)$. The corresponding $comoving$ variance
and energy density (in the Hartree approximation) are given by

$\langle
X^{2}\rangle=\int{d^{3}k\over(2\pi)^{3}}\;|X_{k}|^{2}\simeq\int{d^{3}k\over(2\pi)^{3}}\;{n_{k}\over\omega_{k}}~{}~{}~{},~{}~{}~{}\rho_{X}=\int{d^{3}k\over(2\pi)^{3}}\;\omega_{k}n_{k}\;.$
(34)

Figure 1:  Stability chart of the Lamé equation [23] with $q=\lambda_{\phi
h}/(2\lambda_{\phi})$. The solution grows exponentially in the areas enclosed
by the contours. The Floquet exponent is constant along the contours and
decreases outwards.

Note that the behaviour of the solution is determined by the $ratio$ of the
couplings, $\lambda_{\phi h}/(2\lambda_{\phi})$. Therefore, even for small
couplings the resonance can be strong, especially if $\lambda_{\phi
h}/(2\lambda_{\phi})\gg 1$. Since the system is conformal, the resonance does
not stop due to the Universe expansion. It only terminates due to backreaction
of the produced particles and rescattering. For $\lambda_{\phi
h}/(2\lambda_{\phi})\ll 1$, the resonance becomes narrow, with a suppressed
Floquet exponent, and the exponential growth of the amplitude is only seen on
a large timescale.

Due to self–interaction, an oscillating inflaton background also generates
inflaton quanta. Indeed, expanding $\phi=\langle\phi\rangle+\delta\phi$ and
quantizing $\delta\phi$, one can write down an equation of motion for the
inflaton analog of $X_{k}$, which we denote by $\varphi_{k}$. Since at
quadratic order $\phi^{4}\rightarrow 6\langle\phi\rangle^{2}\,\delta\phi^{2}$,
we find

$\varphi_{k}^{\prime\prime}+\left[\kappa^{2}+3\;{\rm
cn}^{2}\left(x,{1\over\sqrt{2}}\right)\right]\,\varphi_{k}=0\;.$ (35)

Therefore, the inflaton quanta are generated with the effective $q$–parameter
$\lambda_{\phi h}/(2\lambda_{\phi})\rightarrow 3$.

For $\lambda_{\phi h}/(2\lambda_{\phi})\ll 3$, the inflaton quanta production
is more efficient than Higgs production. In this case, most of the energy
remains in the dark sector and no successful reheating occurs. Therefore, in
what follows we exclude this possibility and focus on moderate and large
$\lambda_{\phi h}/(2\lambda_{\phi})$.

In reality, particle production is complicated by backreaction and
rescattering of the produced particles [24],[26], which is not captured by the
Lamé equation. Therefore, lattice simulations are often necessary for
understanding the dynamics of the system.

###  2.4 Lattice simulations

In the coupling range of interest, $\lambda_{\phi h}\gtrsim\lambda_{\phi}$,
the parametric resonance is sufficiently strong such that the perturbative
estimates are inadequate. In this regime, the occupation numbers are large
which enables the use of classical lattice simulations. These are essential
for capturing the backreaction and rescattering effects, especially when the
dynamics become highly non–linear. Related lattice studies and tools have
recently appeared in [38],[39].

Our main goal is to understand how much energy can be transferred from the
inflaton to the Higgs field, which determines the composition of the Universe.
In a realistic scenario, almost all of the inflaton energy must be converted
in the SM radiation. Since the efficiency of particle production depends on
the ratio of the couplings $\lambda_{\phi h}/\lambda_{\phi}$, in our
simulations we set $\lambda_{\phi}=10^{-13}$ and vary $\lambda_{\phi h}$. The
allowed values of $\lambda_{\phi h}$ are bounded from above for non–thermal
dark matter. First, DM should not thermalize and, second, the Higgs thermal
bath should not produce too much DM via freeze–in [40], which is operative for
reheating temperatures above 20 GeV or so. The second constraint is stronger:
it requires $\lambda_{\phi h}<2\times 10^{-11}$ for $m_{\phi}\gtrsim m_{h}$
and $\lambda_{\phi h}<10^{-11}\;\sqrt{{\rm GeV}/m_{\phi}}$ for $m_{\phi}\ll
m_{h}$ [41]. This implies

$\lambda_{\phi h}\lesssim 10^{-9}$ (36)

as long as dark matter is warm or cold, $m_{\phi}\gtrsim 10$ keV, as required
by structure formation. If the reheating temperature is very low, the bound
gets weaker, although this does not affect our analysis of particle
production.

The realistic system is very complicated: the Higgs production is accompanied
by its decay into other SM states and their thermalization. To account for all
of the effects is a (currently) unsurmountable task, hence we have to resort
to simplifications. We will consider the limiting cases, where either Higgs
decay or Higgs production dominates. The result also depends on the Higgs
self–coupling as it can induce significant backreaction. The value of the
coupling at high energies is uncertain due to the uncertainty in the top quark
mass, hence we take two representative values $\lambda_{h}=0$ and
$\lambda_{h}=0.01$.

The simulations are performed in conformal time $z$ defined by
$dz=\sqrt{\lambda_{\phi}}\,\Phi_{0}\,dt/a(t)$. It is equivalent to the
variable $x$ at late times, which in practice means after a few oscillations.
At early times, this relation is integrated numerically.

####  2.4.1 Fast Higgs decay: no resonance

If the Higgs quanta decay faster than they are generated, no resonant
production takes place. Since $\langle h\rangle\ll\sqrt{\lambda_{\phi
h}}\phi$, the quarks are effectively massless compared to the Higgs and the
main decay channel is $h\rightarrow\bar{t}t$. The corresponding decay width is

$\Gamma_{h}={3y_{t}^{2}\over 16\pi}m_{h}^{\rm eff}\;,$ (37)

where $m_{h}^{\rm eff}={\sqrt{\lambda_{\phi h}/2}}\,\phi$. Accounting for the
time–dependence of $\Gamma_{h}$, this results in the particle number decrease
as $\exp(-2\Gamma_{h}\,t)$. On the other hand, the $k$–mode occupation number
during the resonance grows as $\exp(2\mu_{k}z)$, where $\mu_{k}$ is the
Floquet exponent. Using $z=(48\lambda_{\phi})^{1/4}\sqrt{t}$ and
$\phi\sim(3/\lambda_{\phi})^{1/4}\,t^{-1/2}$ at late times, we find that the
decay dominates for

$\lambda_{\phi h}\gtrsim 10^{3}\lambda_{\phi}\;,$ (38)

with a typical value $\mu_{k}\sim 10^{-1}$. For smaller $\mu_{k}$, the bound
decreases as $\mu_{k}^{2}$. Although this argument neglects possible
backreaction of the produced quarks and effects of thermalization, it is clear
that, at very large $\lambda_{\phi h}$, the Higgs decay is important making
resonant Higgs production inefficient.

Figure 2:  Decay of the classical inflaton background $\langle\varphi\rangle$
in the comoving frame ($\varphi=a\,\phi$) due to emission of the inflaton
quanta and rescattering. The conformal time $z$ is defined by
$dz=\sqrt{\lambda_{\phi}}\,\Phi_{0}\,dt/a(t)$; $\lambda_{\phi}=10^{-13}$ and
$\Phi_{0}=1.7$ in Planck units. The amplitude is normalized to 1 at the
initial point. The simulation is performed with LATTICEEASY [42].

On the other hand, the resonant production of the inflaton quanta proceeds
unimpeded according to Eq. 35. Simulations show that the resonance is
terminated by backreaction at [24]

$z_{*}=76-14.3\ln\lambda_{\phi}\;.$ (39)

After that, the inflaton zero mode decays due to rescattering. Specifically,
defining the comoving inflaton background as $\varphi=a\phi$, one finds
$\varphi\propto z^{-1/3}$. Since the energy density of the zero mode scales as
$\varphi^{4}$, within conformal time $2z_{*}$ most of the energy is contained
in the $fluctuations$. An example of the zero mode decay for
$\lambda_{\phi}=10^{-13}$ obtained with LATTICEEASY [42] simulations is shown
in Fig. 2. This simulation includes the inflaton field only. We find that the
resonance ends around $z_{*}\simeq 500$ and most of the energy gets converted
into inflaton fluctuations by $z\sim 800$.

The perturbative Higgs production is highly suppressed in this regime. The
Higgs pairs are generated by an oscillating classical background [43],[44],
and the corresponding inflaton decay rate for 4 $massless$ Higgs d.o.f. is
given by [28]

$\Gamma_{\phi}^{\rm pert}=C\,{\lambda_{\phi h}^{2}\over 16\pi}\,{\Phi_{0}\over
a(t)\sqrt{\lambda_{\phi}}}\;,$ (40)

where $C\simeq 0.4$. Requiring $\Gamma_{\phi}^{\rm pert}\,t\sim 1$ translates
into the conformal decay time of order $z\sim
10^{2}\lambda_{\phi}/\lambda_{\phi h}^{2}$. This is already much longer than
the characteristic decay time into inflaton fluctuations, $z\sim 10^{3}$. In
reality, the Higgs production is much more suppressed since the Higgs is
heavy: the effective Higgs mass is much greater than the principal oscillation
frequency of the inflaton, $m_{h}^{\rm eff}={\sqrt{\lambda_{\phi
h}/2}}\,\phi\gg\omega\sim\sqrt{\lambda_{\phi}}\,\phi$. This leads to
exponential suppression of the decay rate by $e^{-2\pi\,m_{h}^{\rm
eff}/\omega}$ [28]. Thus, perturbative decay can be neglected.

We conclude that the inflaton background decays primarily into inflaton
fluctuations. Feebleness of their interactions ensures that the reaction rates
are slower than the Hubble expansion222Relevant thermalization constraints
have been considered carefully in [45],[46]. so that the inflaton quanta get
never converted into SM radiation and

$\rho_{\phi}\gg\rho_{\rm SM}\;.$ (41)

These quantities scale as radiation until the inflaton becomes
non–relativistic, which makes the inequality even stronger. The resulting
Universe is therefore dark and unrealistic.

####  2.4.2 Slow Higgs decay, $\lambda_{h}=0$

Consider now another idealized situation: suppose the Higgs decay can be
neglected on the timescale of the resonance and rescattering, i.e. when
$\lambda_{\phi h}$ is not too large or Higgs decay is impeded by backreaction.
For $\lambda_{\phi h}\gtrsim\lambda_{\phi}$, the resonance is strong and
results in explosive Higgs production. As shown in Fig. 3, a large fraction of
the initial inflaton energy density can be converted into the Higgs quanta.
Here we include all 4 Higgs degrees of freedom $h_{i}$ available at high
energies. We observe that the inflaton zero mode decays quickly and converts
into fluctuations.

Figure 3:  Upper row: fraction of the energy density carried by 4 Higgs d.o.f.
and the inflaton as a function of the conformal time $z$,
$dz=\sqrt{\lambda_{\phi}}\,\Phi_{0}\,dt/a(t)$ and $\Phi_{0}=1.7$ in Planck
units, obtained with LATTICEEASY. Lower row: decay of the classical inflaton
background $\langle\varphi\rangle^{2}$ in the comoving frame
($\varphi=a\,\phi$). The amplitude is normalized to 1 at the initial point.

At large couplings, $\lambda_{\phi h}\sim 10^{-10}$, the system tends to
quasi–equilibrium on a relatively short time scale, $z\sim 10^{3}$. Although
the distribution is not yet Bose–Einstein, the energy is distributed almost
equally among the available degrees of freedom. For $\lambda_{\phi h}\gtrsim
10^{-10}$, roughly 80% of the energy is carried by the Higgs quanta and 20% by
the inflaton, while for smaller couplings the inflaton fraction is higher
(Fig. 3). We have verified that with a single Higgs d.o.f., the maximal Higgs
fraction is about 50%.

It should be noted that, according to the previous subsection, $\lambda_{\phi
h}$ values above $10^{-10}$ are expected to lead to fast Higgs decay.
Nevertheless, we still consider such large couplings in this subsection since
the Higgs decay may be blocked by backreaction, e.g. thermal masses of the
decay products. Such effects are hard to evaluate precisely at this stage
given the multitude of possible processes and timescales, hence one should not
exclude the range $\lambda_{\phi h}>10^{-10}$ from consideration.

In a realistic situation, we expect the Higgs field to channel at least some
of its energy into other SM states. If this process is fast enough, the system
may reach quasi–equilibrium where the energy would be shared almost
democratically by the relativistic degrees of freedom,

${\rho_{\phi}\over\rho_{\rm tot}}\sim{1\over\\#\,{\rm d.o.f.}}$ (42)

When rescattering stops, the total number of the inflaton quanta remains
approximately constant. This allows us to obtain the $lower$ bound on the dark
matter abundance $Y$, which also remains invariant after this stage. It is
defined by $Y=n_{\phi}/s_{\rm SM}$, where $n_{\phi}$ is the inflaton number
density and $s_{\rm SM}$ is the entropy density of the SM fields. For a
thermalised SM sector in the relativistic regime, $s_{\rm SM}$ is close the SM
quanta number density, up to a factor of a few: $s_{\rm SM}\sim 4n_{\rm SM}$.
Since $n_{\phi}/n_{\rm SM}\sim\rho_{\phi}/\rho_{\rm SM}$, for $107$ SM degrees
of freedom at high temperature, we find

$Y\gtrsim 10^{-3}\;.$ (43)

This number is far above the observed value $Y_{\rm obs}=4\times
10^{-10}\;{\rm GeV}/m_{\phi}$, given that $m_{\phi}\gtrsim 10$ keV to be
consistent with the structure formation constraints. Thus, the emerging
Universe is again unacceptably dark.

####  2.4.3 Slow Higgs decay, $\lambda_{h}=0.01$

The Higgs self–interaction has a profound effect on the dynamics of the system
[26]. A significant self–coupling induces a large effective mass term shutting
down the resonance. This occurs when the Higgs fluctuations, characterized by
the variance $\langle h^{2}\rangle$, grow large such that the effective
inflaton and Higgs masses become comparable,

$\lambda_{\phi}\phi^{2}\sim\lambda_{h}\langle h^{2}\rangle\;.$ (44)

After that, the explosive growth of the Higgs amplitude stops. Subsequently,
$\langle h^{2}\rangle$ evolves non–linearly: it decreases and increases again
before stabilizing eventually. As shown in Fig. 4, only a tiny fraction of the
inflaton energy can be converted into the Higgses for a realistic
self–coupling.333This ceases to be true at very large $\lambda_{\phi
h}\gtrsim\lambda_{h}$, which however leads to thermal dark matter. Since
$\phi^{2}$ and $\langle h^{2}\rangle$ redshift the same way, this remains true
at later times as well.

Although the Higgs production is hindered by backreaction, the background
decay into the inflaton quanta continues. We thus end up with a situation
similar to that discussed in Section 2.4.1,

$\rho_{\phi}\gg\rho_{\rm SM}\;.$ (45)

If one were to account for Higgs decay which reduces $\langle h^{2}\rangle$,
the system would interpolate between those of Section 2.4.1 and Section 2.4.2.
Either way, the resulting Universe is unrealistic.

We conclude that backreaction and rescattering make a crucial impact on the
reheating process in the framework of inflaton dark matter, rendering the
non–thermal dark matter option unrealistic. We find similar results for a
$\phi^{2}$ inflaton potential: for a small Higgs portal coupling, the energy
transfer to the SM radiation is suppressed, while for a large coupling, the
system reaches quasi–equilibrium and the inflaton retains too large a fraction
of the total energy.

Figure 4:  Fraction of the energy density carried by 4 Higgs d.o.f. with
self–interaction and the inflaton as a function of the conformal time $z$,
obtained with LATTICEEASY. $\Phi_{0}=1.7$ in Planck units.

##  3 Thermal inflaton dark matter

If $\lambda_{\phi h}$ is sufficiently large, the system reaches thermal
equilibrium via processes like $\phi h\rightarrow\phi h,\phi\phi\rightarrow
hh$, etc. Its precursors are already seen in Fig. 3, right panels. The dark
matter abundance is then determined by the temperature and the usual
freeze–out approach applies [47],[48],[49]. The correct relic abundance can be
obtained for parameter values allowing for efficient dark matter annihilation,
subject to direct DM detection and perturbativity constraints.

Away from the narrow resonance region $m_{\phi}\simeq 62$ GeV, efficient DM
annihilation $\phi\phi\rightarrow\;$SM combined with the XENON1T bound
requires [50]

$\lambda_{\phi h}(1\,{\rm TeV})\gtrsim 0.25\;.$ (46)

This bound applies at the TeV scale, while the couplings at the inflation
scale are obtained by the solving the renormalization group (RG) equations.
The list of the RG equations can be found in [28], while the most important
one for us reads

$16\pi^{2}{d\lambda_{\phi}\over dt}=2\lambda_{\phi
h}^{2}+18\lambda_{\phi}^{2}\;,$ (47)

where $t=\ln\,\mu$ with $\mu$ being the RG energy scale. This implies, in
particular, that the inflaton self–coupling at least of the size
$\lambda_{\phi h}^{2}/(8\pi^{2})$ gets generated (ignoring a large $log$),

$\lambda_{\phi}(H)\gtrsim 10^{-3}\;.$ (48)

In other words, the Higgs–inflaton coupling induces the Coleman–Weinberg
contribution to the (Jordan frame) inflaton potential,

$\Delta V_{\rm 1-loop}\simeq{\lambda_{\phi h}^{2}\over
64\pi^{2}}\,\phi^{4}\ln{\phi^{2}\over\phi_{*}^{2}}\;,$ (49)

where $\phi_{*}$ is a reference field value and 4 Higgs degrees of freedom
have been included. By choosing $\phi_{*}$ appropriately, one can suppress the
correction at one point, but not over the entire field range where the last 60
e–folds of inflation take place. Clearly, the generated coupling
$\lambda_{\phi}$ is far above the unitarity bound (23), which signals
inconsistency of the model.

The above conclusion is evaded at the Higgs resonance,

$m_{\phi}\simeq m_{h_{0}}/2\;,$ (50)

where $m_{h_{0}}=125$ GeV is the physical Higgs mass. In this case, resonant
DM annihilation $\phi\phi\rightarrow h\rightarrow{\rm SM}$ is efficient even
for small couplings $\lambda_{\phi h}\gtrsim 10^{-4}$ [50], although possible
complications related to early kinetic decoupling should be kept in mind [51].
The resulting correction to the inflaton self–coupling is negligible and all
of the constraints are satisfied. We note however that $|m_{\phi}-m_{h}/2|$
must be below a few GeV, which appears rather unnatural yet not impossible.

##  4 Extensions

Our results depend critically on the minimality assumption, which serves as
the main motivation for inflaton dark matter. In particular, we require the
minimal number of degrees of freedom consistent with observations.
Nevertheless, it is worthwhile to explore further options.

The unitarity problem can be evaded in the Palatini formalism, i.e. at the
price of adding extra (gravitational) degrees of freedom [52]. In this case,
the connection $\Gamma_{\alpha\beta}^{\lambda}$ is promoted to a dynamical
variable along with the metric. One finds that the unitarity cutoff of this
theory is of order (in Planck units) [53]

$\Lambda_{\rm Pal}\sim{1\over\sqrt{\xi_{\phi}}}\;,$ (51)

which is above the inflationary scale. As a result, a large inflaton
self–coupling can be consistent with unitarity, thus opening the possibility
of a TeV scale thermal inflaton DM. A somewhat uncomfortable aspect of this
model is that the requisite $\xi_{\phi}$ has to be very large, of order
$10^{7}$ or above.

In our reheating analysis, we have set the non–minimal Higgs coupling to
gravity $\xi_{h}$ to zero. Nevertheless, its non–zero value subject to (16) is
not expected to affect the results in any significant way, even if it makes
the Higgs production more efficient [54],[55]. Indeed, the main issue we find
is that efficient particle production leads to quasi–equilibrium, where the
energy is distributed almost equally among the constituents. Thus, the
inflaton carries at least 1% of the energy of the system. At weak coupling and
for realistic Higgs self–interaction, the resonance is inefficient and this
fraction is much larger. In both cases, dark matter is overabundant unless it
is capable of annihilating efficiently.

One may also relax the assumption that $\phi$ alone drives inflation. In
general, the inflaton may be a combination of the Higgs and the DM field
$\phi$. At large field values, there exists a stable flat direction [30]

${h\over\phi}=\sqrt{2\lambda_{\phi}\xi_{h}-\lambda_{\phi h}\xi_{\phi}\over
2\lambda_{h}\xi_{\phi}-\lambda_{\phi h}\xi_{h}}$ (52)

if both the numerator and denominator under the square root are positive, and
$\xi_{\phi}+\xi_{h}\gg 1$. Slow roll along this direction would lead to
inflation with predictions close to those of Eq. 22. The efficiency of
reheating can then be increased if the Higgs proportion is large, $h/\phi\gg
1$. In that case, however, inflation is driven mostly by the Higgs such that
one faces the usual Higgs inflation unitarity problem.444A mixed Higg–singlet
inflaton scenario was explored in [13]. The above conclusion does not apply to
this model since the singlet is unstable by construction.

##  5 Conclusion

The concept of inflaton dark matter is interesting in that it is economical: a
single field is responsible for both inflation and dark matter. We have
considered the minimal framework which fits the inflationary data very well
and where inflation is driven by a non–minimal scalar coupling to gravity. The
parity symmetry guarantees that the inflaton remains as a stable relic and can
potentially play the role of dark matter.

The focus of this work is on understanding the reheating processes in such a
framework. We have examined both the thermal and non–thermal dark matter
options. One of the important features of the system is that the inflaton
background decays non–perturbatively, thus necessitating lattice simulations
to describe. We find that, for non–thermal DM, the reheating is hindered by
backreaction and rescattering effects resulting in overabundance of dark
matter. For thermal dark matter, radiative corrections to the inflaton
potential play an important role. In this case, the correct relic abundance
can be produced, yet the unitarity constraint forces the inflaton mass into a
very narrow window close to half the Higgs mass.

These conclusions can be evaded in less minimalistic set–ups which invoke
additional degrees of freedom.

Acknowledgements. We are thankful to Yohei Ema for helpful discussions and
sharing some of his results.

##  Appendix A Simulation details

In this work, we have performed lattice simulations with CLUSTEREASY, the
parallel computing version of LATTICEEASY. For most purposes, the dimension of
the lattice was set to $D=3$ and the number of the grid points per edge was
fixed at $N=128$ ($128^{3}$ in total). The simulations mainly target the late
time behaviour of the system, for which the UV momentum spectrum is essential.
To capture the relevant features, we have made the upper bound of the momentum
space $k_{max}$ (in LATTICEEASY convention) dynamical, i.e. $\lambda_{\phi
h}$–dependent. The size of the box $L$ in rescaled distance units was set to

$L=\frac{\pi\sqrt{D}N}{40}\left(\frac{\lambda_{\phi
h}}{\lambda_{\phi}(4\pi^{2})}\right)^{-0.25}\;,$ (53)

such that

$k_{max}=k_{min}\times\frac{\sqrt{D}}{2}N=\frac{2\pi}{L}\frac{\sqrt{D}}{2}N=40\times\left(\frac{\lambda_{\phi
h}}{\lambda_{\phi}(4\pi^{2})}\right)^{0.25}\;,$ (54)

where $k_{min}$ represents the lower bound of the momentum space in rescaled
units and the prefactor 40 has been determined empirically. To verify
reliability of our results, we have run extended 2D simulations with $N=1024$
which also capture the relevant IR physics. We find that the late time
distributions are indeed consistent.

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