# Robustness of inflation to kinetic inhomogeneities

Matthew Elley<EMAIL_ADDRESS>Department of Physics, University of
Basque Country, UPV/EHU, 48080, Bilbao, Spain  Theoretical Particle Physics
and Cosmology Group, Physics Department, King’s College London, Strand, London
WC2R 2LS, United Kingdom  Josu C. Aurrekoetxea
<EMAIL_ADDRESS>Astrophysics, University of Oxford, DWB,
Keble Road, Oxford OX1 3RH, United Kingdom  Katy Clough<EMAIL_ADDRESS>Geometry, Analysis and Gravitation, School of Mathematical Sciences, Queen
Mary University of London, Mile End Road, London E1 4NS, United Kingdom
Raphael Flauger<EMAIL_ADDRESS>UC San Diego, Department of Physics, 9500
Gilman Rd, La Jolla, CA, 92093, USA  Panagiotis Giannadakis
<EMAIL_ADDRESS>Theoretical Particle Physics and Cosmology
Group, Physics Department, King’s College London, Strand, London WC2R 2LS,
United Kingdom  Eugene A. Lim<EMAIL_ADDRESS>Theoretical Particle
Physics and Cosmology Group, Physics Department, King’s College London,
Strand, London WC2R 2LS, United Kingdom

###### Abstract

We investigate the effects of large inhomogeneities in both the inflaton field
and its momentum. We find that in general, large kinetic perturbations reduce
the number of e-folds of inflation. In particular, we observe that
inflationary models with sub-Planckian characteristic scales are not robust
even to kinetic energy densities that are sub-dominant to the potential energy
density. This strengthens the results of our previous work. In inflationary
models with super-Planckian characteristic scales, despite a reduction in the
number of e-folds, inflation is robust even when the potential energy density
is initially sub-dominant. For the cases we study, the robustness of inflation
strongly depends on whether the inflaton field is driven into the reheating
phase by the inhomogeneous scalar dynamics.

##  I Introduction

The theory of cosmic inflation [1, 2, 3, 4, 5] is most widely studied paradigm
of the very early universe. It posits a period of nearly exponential
expansion, driving the universe to a state of spatial flatness and
homogeneity. Furthermore, it provides a natural mechanism to generate a
approximately scale-invariant spectrum of primordial perturbations which seed
structure formation, consistent with observations [6]. The simplest mechanism
to drive inflation is a single scalar field, the inflaton, slowly rolling down
its potential. The expectation of the inflationary paradigm is that
sufficiently “generic” initial data will lead to trajectories in which the
scalar field potential dominates and the universe undergoes a period of nearly
exponential expansion. Since generic initial data is inhomogeneous, studying
to what extent the homogeneous inflationary trajectory is an attractor
requires solving the full general relativistic equations of motion.

The problem of initial conditions for inflation has been extensively studied
using analytic and semi-analytic methods [7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59]. In particular, early numerical relativity work on this
effort was led by Goldwirth and Piran [60, 61, 62], which focused on the
spherical collapse of inhomogeneities in an inflationary theory. These
simulations suggested that inflation would begin as long as there is initially
a homogeneous region of radius $\sim H_{0}^{-1}$, where $H_{0}$ is the
inflationary scale.

There are several different ways single-field models of inflation are
classified. For example, one may classify them as high-scale or low-scale
models, depending on the energy scale associated with inflation, or one may
classify them as large-field or small-field models depending on their field
range, and some care is needed in interpreting the different terms [63]. For
high-scale inflation, it has been argued that inflation can naturally occur
under generic initial conditions [5, 64, 65, 66], and a number of simulations
have supported this claim. The first 3+1D simulations by [67, 68] showed that
inflation could occur even if kinetic and gradient energies are initially
comparable to the potential energy density. However, these simulations were
limited to initial conditions that did not lead to black hole formation. More
general initial conditions, including initial conditions that do lead to black
hole formation, were considered in [69, 70, 71, 72] and also showed that high-
scale inflation can occur in the presence of large inhomogeneities. Some
recent work has opposed this conclusion on the basis that the initial
conditions and boundary choices made may bias the results towards successful
inflation in some “non-generic” regions of low initial curvature [73, 74].
Addressing these concerns properly goes beyond the scope of this work, but we
provide some brief comments in the Summary and Outlook section to set our work
in context with respect to these challenges.

The robustness of low-scale models to inhomogeneous initial conditions is an
even more important issue. As the energy scale is smaller, the size of the
causally disconnected patches is larger, and the potential energy may be
orders of magnitudes smaller than other components. Consequently, it is
possible that large inhomogeneities could spoil inflation before it begins.
This has motivated a wealth of studies on the effects of initial conditions on
low-scale inflation, many of which involve simulating the inflating spacetime
with numerical relativity in full 3+1D [75, 69, 70, 71, 76, 72].

In contrast to the above classifications, we follow our previous work [71] and
classify models as models with sub-Planckian and super-Planckian
characteristic scales,111The Planckian scale in the nomenclature refers to the
reduced Planck scale. depending on the scale in field space over which the
potential varies appreciably. This quantity was introduced by one of the
authors in [77], and is a more accurate indicator both of the robustness to
large inhomogeneities [71] and of the amount of tensor fluctuations a model
produces and hence its detectability [77]. For a brief review see [71]. The
simulations in [71] showed that for models with a sub-Planckian characteristic
scale even small excursions of the field can cause inflation to fail by
dragging the rest of the field to the minimum. In contrast, models with super-
Planckian characteristic scales were shown to be robust to large initial
gradient energies, at least for the class of initial conditions studied. Even
if the initial configuration of the field reaches the minimum, the gradient
pressure of the field can pull it back into the part of the potential that
supports inflation.

Here we extend the work of [69, 70, 71] by investigating the effect of large
and non-uniform initial inflaton velocity in addition to an inhomogeneous
initial field profile. This was first studied by [72] for the large field case
in both single and two-field inflationary models, and it was found that these
models are generally robust. In our work, we consider two plateau models that
are members of the $\alpha$-attractor family [78, 79]: the Starobinsky model
[10] with a super-Planckian characteristic scale, and an $\alpha$-attractor
model with a sub-Planckian characteristic scale. We use grchombo, a
multipurpose numerical relativity code [80]. Our conclusions are as follows

  * •

Models with a super-Planckian characteristic scale are significantly more
robust to momentum fluctuations than models with a sub-Planckian
characteristic scale. For models with a sub-Planckian characteristic scale,
even kinetic energy densities that are subdominant to the potential energy
density can end inflation, similar to field fluctuations.

  * •

The shorter the wavelength of the momentum fluctuation, the more energy is
required for it to alter the field dynamics and end inflation. Again this is
similar to what we observed for the field fluctuations [69]. So generally
momentum modes with long wavelengths are more dangerous to inflation.

  * •

The relative phase between the peaks of the field and momentum fluctuations
significantly affects the energy required to end inflation - the closer they
are aligned the less energy is needed.

The paper is organised as follows. In Section II, we introduce the the models
of inflation we consider and discuss the space of initial conditions we
explore. In Section III, we discuss the numerical results. We conclude in
Section IV. We set $c=\hbar=1$ throughout, and define the (non-reduced) Planck
mass $G=M_{\mathrm{Pl}}^{-2}$.

##  II Theory and Methodology

###  II.1 Space of models

We consider a model of single-field inflation with action given by

$S=\int\mathrm{d}^{4}x\sqrt{-g}\left(\frac{M_{\mathrm{Pl}}^{2}}{16\pi}R-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi-V(\phi)\right)\,.$
(1)

We consider the $\alpha$-attractor potential [78, 79]

$V(\phi)=\Lambda^{4}\left(1-e^{\phi/\mu}\right)^{2}\,,$ (2)

but we expect our results to be valid for other potentials [81, 82]. For
negative field values, the potential exponentially approaches a plateau, and
the approach to the plateau is governed by the characteristic scale $\mu$. The
parameter $\Lambda$ determines the energy-scale of inflation. For a given
value $\mu$, $\Lambda$ is determined by the observed amplitude of the power
spectrum of primordial density perturbations

$\Delta_{R}^{2}=\frac{H^{2}}{\pi M_{\mathrm{Pl}}^{2}\epsilon}\approx 2\times
10^{-9}\,.$ (3)

Here $H$ is the Hubble parameter during inflation and $\epsilon$ is the first
slow-roll parameter given by

$\epsilon=\frac{M_{\mathrm{Pl}}^{2}}{16\pi}\left(\frac{V^{\prime}}{V}\right)^{2}~{}.$
(4)

###  II.2 Space of initial conditions

To investigate the effects of inhomogeneities in the scalar-field initial data
on the dynamics we consider sinusoidal perturbations of the scalar field and
its momentum in three spatial directions

$\displaystyle\phi_{\mathrm{init}}(\textbf{x})$
$\displaystyle=\phi_{0}+\frac{\Delta\phi}{3}\sum_{i=1}^{3}\cos{\left(\frac{2\pi
N_{\phi}x_{i}}{L}\right)}\,,$ (5)
$\displaystyle\Pi_{\mathrm{init}}(\textbf{x})$
$\displaystyle=\Pi_{0}+\frac{\Delta\Pi}{3}\sum_{i=1}^{3}\cos{\left(\frac{2\pi
N_{\Pi}x_{i}}{L}+\theta\right)}\,.$ (6)

Here $\phi_{0}$ and $\Pi_{0}$ are the initial homogeneous components for the
scalar field and conjugate momentum, and we impose periodic boundary
conditions. $\phi_{0}$ was chosen such that in the absence of inhomogeneities
and $\Pi_{0}=0$, inflation will last around 100 e-folds. The values
$\Delta\phi$ and $\Delta\Pi$ are the amplitudes of inhomogeneities of the
scalar field and conjugate momentum. Additionally, we include a relative phase
$\theta$ between the initial configurations of scalar field $\phi(x)$ and
momentum $\Pi(x)$. The dimensionless wavenumbers $N_{\phi}$ and $N_{\Pi}$
determine the wavelengths of the $\phi$ and $\Pi$ fields together with the box
size $L$, which we choose to be the initial Hubble length $H^{-1}_{0}$ in the
absence of inhomogeneities, so that

$L=\frac{3M_{\mathrm{Pl}}}{\sqrt{24\pi V\left(\phi_{0}\right)}}\,.$ (7)

We foliate the spacetime into spatial hypersurfaces such that the metric can
be written as

$\mathrm{d}s^{2}=-\alpha^{2}\mathrm{d}t^{2}+\gamma_{ij}(\mathrm{d}x^{i}+\beta^{i}\mathrm{d}t)(\mathrm{d}x^{j}+\beta^{j}\mathrm{d}t)\,.$
(8)

Here $\gamma_{ij}$ is the 3-dimensional spatial metric of the hypersurfaces,
which we evolve along with the extrinsic curvature
$K_{ij}=\partial_{t}\gamma_{ij}+2D_{(i}\beta_{j)}$ as our physical degrees of
freedom. The lapse function $\alpha$ and shift vector $\beta^{i}$ are evolved
as gauge choices representing the inherent coordinate freedom of GR. Following
the BSSN formulation [83, 84], we decompose the extrinsic curvature into a
trace $K=\gamma^{ij}K_{ij}$ and traceless part $A_{ij}$. We then extract the
conformal factor $\chi$ from the spatial metric to yield the conformally
related metric $\tilde{\gamma}_{ij}=\chi\gamma_{ij}$.

Given an initial scalar field configuration, we solve the Hamiltonian and
momentum constraints of general relativity using the CTTK method [85]. We
assume an initially conformally flat metric $\tilde{\gamma}_{ij}=\delta_{ij}$
and choose $\chi=1$. Subsequently, one obtains an algebraic equation for the
Hamiltonian constraint and a Poisson-like equation for the momentum constraint
which can be solved to obtain the mean expansion $K$ and trace-free extrinsic
curvature $A_{ij}$ respectively, see [85] for more details. We use periodic
boundary conditions, which enforces integrability conditions on the constraint
equations for the initial data [85]. That is, one requires the integral of the
constraints over the domain to vanish. With an initially constant conformal
factor, this leads to the condition $\int S_{i}\mathrm{d}V=0$, where
$S_{i}=\Pi\partial_{i}\phi$ over our numerical domain. Given the sinusoidal
configurations that we choose, this condition restricts the value of the phase
$\theta=\\{0,\pi\\}$ for equal wavemodes $N_{\Pi}=N_{\phi}$.

In the FLRW limit, the trace of the extrinsic curvature $K$ is related to the
Hubble parameter as $K=-3H$ and the scale factor is given by
$a(t)=\chi^{-1/2}$. Given our choice of initial data, in the inhomogeneous
case we define the average number of e-folds in terms of the conformal factor
as

$\langle\mathcal{N}\rangle=-\frac{1}{2}\langle\ln\chi\rangle\,,$ (9)

where the angle brackets $\langle\rangle$ denote proper volume average over
the hypersurfaces defined by the $t$ coordinate

$\langle X\rangle\equiv\int_{V}\mathrm{d}^{3}x\sqrt{\gamma}X~{}.$ (10)

###  II.3 Evolution and diagnostics

We use the BSSN formulation [83, 86, 84] to evolve the spacetime quantities.
We employ a modified form of moving puncture gauge [87, 88]. We evolve the
lapse according to

$\partial_{t}\alpha=-\alpha e^{-\alpha}K+\beta^{i}\partial_{i}\alpha\,,$ (11)

and invoke the standard Gamma-driver condition for the shift.

The exponential damping prevents the lapse from growing too large for
expanding regions (i.e. where $K<0$), but approaches the standard 1+log form
for collapsing regions.

In [71] we described two outcomes when evolving a scalar field configuration:
(i) pull-back when the field restores itself to a more homogeneous
configuration and (ii) drag-down when some part of the field rolls down the
potential and drags the rest of the field configuration with it, preventing
inflation from starting. In the absence of initial momentum, we found that for
inflationary models with sub-Planckian characteristic scales there exists a
critical amplitude beyond which the field is dragged down
$\Delta\phi>\Delta\phi_{\mathrm{crit}}$. In this paper we fix the
inhomogeneous scalar field configuration $\phi(x)$ and study the impact of
varying its initial conjugate momentum $\Pi(x)$. In Fig. 1 we show two example
simulations, one in which the field is pulled back (top panel) and one in
which the field is dragged down to the minimum of the potential as $\Pi(x)$ is
increased (bottom panel). Most of the dynamics occurs within the first few
e-folds, after which the inhomogeneities exit the horizon (if inflation gets
started), and subsequently slowly rolls down the potential (black line in Fig.
2).

Figure 1: Evolution of the scalar field across the largest diagonal of our
numerical domain $(0,0,0)\rightarrow(L,L,L)$. Both panels start from the same
scalar field configuration $N_{\phi}=N_{\Pi}=1$ and phase $\theta=0$, but
different $\Delta\Pi$. The top panel shows the pull-back, when the scalar
field homogenizes. The bottom panel shows the effect of larger initial
momentum, which can push the field sufficiently down the potential to remain
stuck at the minimum (dashed) and drag down the rest with it.

A typical simulation leads to disjoint regions of space, with some regions
where the scalar field has reached the reheating minimum (ending inflation)
and some regions in which the scalar field is approximately homogeneous and on
the inflating plateau. In the regions in which the field reaches the minimum,
the field oscillates. Because of the much shorter timescale of this
oscillation, we rapidly lose accuracy and hence we assumed that such regions
are no longer inflating. For regions in which the field remains in the
plateau, inflation persists, although in each such region the field may take
different (approximately homogeneous) field values.

To estimate the expected number of e-folds in these latter regions, we extend
the evolution of $\phi_{\mathrm{max}}$ and $\phi_{\mathrm{min}}$ by solving
the _homogeneous_ evolution equations

$\displaystyle\ddot{\phi}$ $\displaystyle+3H\dot{\phi}+V^{\prime}(\phi)=0\,,$
(12) $\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}$
$\displaystyle=\frac{8\pi}{3M_{\mathrm{Pl}}^{2}}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right)\,,$
(13)

where we take as initial conditions the values of $\phi$ and $\dot{\phi}$
extracted from numerical relativity simulations. We can use the homogeneous
equations as these regions have spatial dimensions equal to or exceed the
Hubble radius associated with the inflationary vacuum $\geq H_{0}^{-1}$. We
have checked in [71] that this gives excellent agreement with numerical
results.

The result is shown in Fig. 2, where we plot the evolution of the maximum and
minimum values of the scalar field (dashed blue lines), extending the results
from numerical relativity simulations (solid black lines). This allows us to
define a range of expected number of e-folds as
$\Delta\mathcal{N}\equiv\mathcal{N}_{\mathrm{max}}-\mathcal{N}_{\mathrm{min}}$.

Figure 2: Left: illustration of the $\alpha$-attractor potential and and
initial minimum $\phi_{\mathrm{min}}$ and maximum $\phi_{\mathrm{max}}$ values
of the scalar field. Right: evolution of $\phi_{\mathrm{min}}$ and
$\phi_{\mathrm{max}}$ with the local number of e-folds $\mathcal{N}$ for the
entirety of inflation. The values in black lines are extracted from numerical
relativity simulations. As the field homogenizes significantly after a few
e-folds, we evolve the field extrema to the end of slow-roll $\epsilon\approx
1$ via Eqn. (12-13) (dashed blue lines). We use this to predict the range of
expected number of e-folds
$\Delta\mathcal{N}\equiv\mathcal{N}_{\mathrm{min}}-\mathcal{N}_{\mathrm{max}}$.

Writing the Hamiltonian constraint as

$K^{2}=\frac{3}{2}\Big{(}16\pi\rho+A_{ij}A^{ij}-R\Big{)}\,.$ (14)

and noting that for a spatially homogeneous and isotropic universe,
$A_{ij}A^{ij}$ vanishes, the Friedmann equation is recovered via the
definitions $K=-3H$ and $R=6k/a^{2}$ where $k=0$ in our case. Splitting the
scalar-field energy density $\rho$ into its respective components and taking
the volume average yields

$1=\Omega_{\mathrm{pot}}+\Omega_{\mathrm{grad}}+\Omega_{\mathrm{kin}}+\Omega_{A_{ij}}+\Omega_{R}\,,$
(15)

where we have defined the density parameters

$\Omega_{i}=c_{i}\frac{\langle\rho_{i}\rangle}{\langle K^{2}\rangle}\,,$ (16)

for the respective densities $\rho_{i}$, including the potential
$V\left(\phi\right)$ and curvature terms $A_{ij}A^{ij}$ and $R$. Here $c_{i}$
are the associated factors i.e. $c_{i}=24\pi$ for the matter density terms and
$c_{i}=3/2$ for the curvature terms.

As long as the potential energy dominates ($\Omega_{\mathrm{pot}}\approx 1$)
over a large enough volume and sufficient time, the exponential expansion
irons out any inhomogeneities and the region approaches the quasi-de Sitter
solution. We vary the initial gradient and kinetic energies by choosing
different profiles of the field $\phi(x)$ and conjugate momentum $\Pi(x)$.
Solving the Hamiltonian and momentum constraints sources additional shear
contributions via $A_{ij}A^{ij}$, which _a priori_ are not known. For this
reason, we parameterize the amplitude of inhomogeneities in terms of the
departure from potential energy domination $(1-\Omega^{0}_{\mathrm{pot}})$,
the total amount of energy densities working against inflation. We also track
the evolution of diagnostic quantities constructed from the Weyl tensor, the
Weyl curvature and Chern-Pontryagin invariants, see [89] for more details.

Figure 3: The effect of varying the field momentum for different wavelengths
($N_{\Pi}$) and relative phases ($\theta$) with the scalar field configuration
on the expected number of e-folds $\Delta\mathcal{N}$ for the case with sub-
Planckian characteristic scale. Left: We fix the relative phase $\theta=0$ and
study the effect of different wavelengths $N_{\Pi}$. The lowest mode
$N_{\Pi}=0$ (uniform kick) is the most dangerous case, quickly reducing
$\Delta\mathcal{N}$. The higher the wavelength, the more robust the
configuration. Right: We fix the wavelength $N_{\Pi}=2$ and study the effect
of relative phases $\theta$ between the field and momentum. The configuration
for which the maxima are aligned ($\theta=0$ for $(N_{\phi},N_{\Pi})=(1,2)$)
is the most dangerous case.

##  III Numerical results

###  III.1 Sub-Planckian characteristic scale

For the model with sub-Planckian characteristic scale, we fix the model
parameters so that $\mu=5\times 10^{-4}M_{\mathrm{Pl}}$ and
$\Lambda^{4}=1.18\times 10^{-18}M_{\mathrm{Pl}}^{4}$. We also fix the initial
scalar field configuration choosing $\phi_{0}=-6.33\times
10^{-2}M_{\mathrm{Pl}}$, $\Delta\phi=4.5\times 10^{-2}M_{\mathrm{Pl}}$ and
$N_{\phi}=1$. We vary parameters of the initial conjugate momentum: the
amplitude $\Delta\Pi$, mode $N_{\Pi}$ and relative phase $\theta$, which
generate configurations with different initial
$(1-\Omega^{0}_{\mathrm{pot}})$. The precise values are _a priori_ not known
as they depend on the solution of the Hamiltonian and momentum constraints for
given $\phi^{0}(x)$ and $\Pi^{0}(x)$ configurations. However, for models with
sub-Planckian characteristic scale, even small gradient and kinetic energy
densities prevent inflation from starting. They do not source a significant
shear contribution via $A_{ij}A^{ij}$, so that
$1-\Omega^{0}_{\mathrm{pot}}\approx\Omega^{0}_{\mathrm{grad}}+\Omega^{0}_{\mathrm{kin}}\ll
1$.

In agreement with our previous studies that did not include initial momentum
[69, 71], we find that models with sub-Planckian characteristic scale are not
robust and fail to provide a sufficient number of e-folds even when the
potential dominates initially $\Omega^{0}_{\mathrm{pot}}\approx 1$. Adding
momentum does not change the overall conclusion that sub-Planckian models are
susceptible to inhomogeneities. In Fig. 3 we plot how the expected number of
e-folds $\Delta\mathcal{N}$ varies as we decrease
($1-\Omega^{0}_{\mathrm{pot}}$).

_In-phase perturbations_. In the left panel we fix the relative phase
$\theta=0$ and study the impact of increasing $(\Pi_{0},\Delta\Pi)$ for three
dimensionless wavenumbers $N_{\Pi}=\\{0,1,2\\}$ of the scalar momentum. The
$N_{\Pi}=0$ case exhibits the most effective reduction in e-folds, as the
field is uniformly boosted towards the potential minimum. For $N_{\Pi}=1$ we
observe an initial increase in the maximum number of e-folds as the kinetic
energy is increased. This is because the momentum fluctuation is in phase with
the field fluctuation and at late times one finds an even larger spread of the
field. However, increasing the kinetic energy eventually prevents the pull-
back of the field up the plateau by gradients. Subsequently, the field rolls
down to the potential minimum ending inflation prematurely. Lastly, we find
that for $N_{\Pi}=2$ there is no significant change in $\Delta\mathcal{N}$ for
the range of energies plotted in the left panel.

_Out-of-phase perturbations_. In the right panel we focus on the most robust
wavelength we have simulated ($N_{\Pi}=2$) and study the impact of varying the
relative phase $\theta$ between the field $\phi$ and momentum $\Pi$
configurations. This results in different momentum density profiles and
amplitudes $S_{i}\sim\Pi\partial_{i}\phi$. The relative phase of the initial
$\phi$ and $\Pi$ profiles significantly impact the likeliness of failure. We
find that if the perturbations of $\Pi$ and $\phi$ are out of phase, then the
effects can even cancel each other, resulting in more robust inflation with an
number of e-folds that can be larger than for the homogeneous initial
conditions.

We conclude that modes with larger wavenumbers result in a larger expected
number of e-folds and thus, similarly to perturbations of the scalar amplitude
[69, 71], longer wavelength kinetic inhomogeneities are the most dangerous.
This agrees with what one might expect as smaller wavelength momentum modes
source shorter wavelength inhomogeneities in the scalar field, which
experience larger gradient forces that subsequently pull back and homogenize
the field. While the larger gradients source larger curvatures, at least
locally, the primary dynamic that determine inflationary failure is the motion
of the scalar field (as opposed to dynamics of the curvature), which is
consistent with our previous results. In light of this, our results
unsurprisingly show that the most dangerous phases are those for which the
field and momentum maxima are in phase.

###  III.2 Super-Planckian characteristic scale

Previous studies [69, 71, 75] showed that large field inflation is robust to
large initial scalar field inhomogeneities. Furthermore, in [72], it was also
shown that this resilience carries through to large initial scalar field
momenta222See also [54] for an early investigation.. In this section, we
confirm the latter results.

As a concrete example, we study the robustness of Starobinsky inflation, for
which $\mu=\sqrt{3/16\pi}M_{\mathrm{Pl}}$ and $\Lambda^{4}=2.64\times
10^{-15}M_{\mathrm{Pl}}^{4}$. Given the results from simulations with sub-
Planckian characteristic scale, we only focus on the most dangerous scenario:
uniform kicks ($N_{\Pi}=0$) towards the potential minimum. We compare this
effect for two scalar field configurations with
$\phi_{0}/M_{\mathrm{Pl}}=-1.2$ and
$\Delta\phi/M_{\mathrm{Pl}}=\\{0.19,0.38\\}$, resulting in gradient energies
densities $\rho^{0}_{\mathrm{grad}}\approx V(\phi)$ and
$\rho^{0}_{\mathrm{grad}}\approx 3V(\phi)$ respectively. Note that the initial
density parameter
$1-\Omega^{0}_{\mathrm{pot}}\approx\Omega^{0}_{\mathrm{grad}}+\Omega^{0}_{\mathrm{kin}}+\Omega^{0}_{A_{ij}}$
is a priori unknown before solving the constraints, because for models with a
super-Planckian characteristic scale the shear contribution given by
$A_{ij}A^{ij}$ is significant. This shear contribution limits the range of
initial conditions that we can explore, as it results in collapsing regions in
which the evolution breaks down. We believe that the breakdown at these points
is not physical and is instead related to numerical problems in evolving
collapsing planar gravitational wave structures. Such problems have been seen
elsewhere in critical collapse simulations of gravitational waves, and while
progress has been made, such scenarios remain challenging to evolve in the
standard gauges that we currently use [90]. We hope to employ the recent
advances in critical collapse simulations [91, 92, 93] in future work, as
discussed further in the Outlook section below.

In Fig. 4, we plot the reduction in the expected number of e-folds
$\Delta\mathcal{N}$ as the momentum is increased (and
$\Omega^{0}_{\mathrm{pot}}$ is reduced). In contrast to the model with sub-
Planckian characteristic scale, we observe significant robustness even to
strong uniform boosts for which the potential vacuum energy is highly
subdominant $\Omega^{0}_{\mathrm{pot}}\approx 0.05$. So models of inflation
with a super-Planckian characteristic scale are robust even to large initial
momenta, consistent with the results of [72]. Moreover, we find that for fixed
$1-\Omega^{0}_{\mathrm{pot}}$, _somewhat counter-intuitively larger gradient
energy densities make the model more robust_ – again this was first noted by
[72]. We agree with their argument that this is a consequence of higher
gradient restoring forces, which are more efficient in homogenizing the scalar
field.

We conclude that large fluctuations in the conjugate momentum are more
dangerous than large perturbations in the scalar field itself. This may not be
surprising as the restoring forces from scalar gradients act to homogenize the
configuration to the mean value, whereas the conjugate momentum actively
pushes the field to a more inhomogeneous distribution, and possibly towards
the potential minimum to end inflation.

Figure 4: The effect of increasing the momentum (decreasing
$\Omega^{0}_{\mathrm{pot}}$) on the range of the expected number of e-folds
$\Delta\mathcal{N}$ for the Starobinsky model (super-Planckian characteristic
scale). We show two sets of runs with different initial gradient energy
densities $\rho_{\mathrm{grad}}$ beginning with $\Pi=0$. We conclude that
Starobinsky inflation is robust, as they can support significant
inhomogeneities even when the initial potential energy is subdominant
$\Omega^{0}_{\mathrm{pot}}<0.5$.

##  IV Summary and Outlook

Figure 5: Top: Evolution of the potential density parameters for models with
sub-Planckian (solid) and super-Planckian (dashed) characteristic scales.
Bottom: The evolution of the maximum values of the Weyl (red) and Chern-
Pontryagin (blue) curvature scalars. Both panels show the failure of inflation
for the model with sub-Planckian characteristic scale even when the potential
initially dominates and the initial curvature scalars are small. On the other
hand, the model with super-Planckian characteristic scale (Starobinsky
inflation) is successful in starting inflation and homogenizing the field for
subdominant potential energy densities and in the presence of $\mathcal{O}(1)$
curvature invariants.

We have studied the robustness of inflation to large inhomogeneities in the
scalar field and its conjugate momentum. In general, introducing momentum
fluctuations in the scalar field makes inflation more likely to fail,
especially for low frequency perturbations. Furthermore, the relative phase
between the scalar field and conjugate momentum plays an important role. The
case in which the phases of the inhomogeneities in field and in the momentum
are aligned is the most likely to result in failure. Nevertheless, the
introduction of momentum fluctuations does not significantly alter the
evolution from that of a purely inhomogeneous field configuration. The results
are consistent with our previous work which suggests it is scalar field
dynamics, i.e. how the field is being pushed to the reheating zone via the
initial inhomogeneities, that determines the robustness of inflation. On the
other hand, we find that a combination of both kinetic and scalar field
perturbations can interact in interesting ways. For example, we find that for
fixed $\Omega^{0}_{\mathrm{pot}}$, increasing the gradient energy density
makes inflation more robust. That is, departures from
$\Omega^{0}_{\mathrm{pot}}=1$ are more robust if they are contained in
$\Omega^{0}_{\mathrm{grad}}$ than in $\Omega^{0}_{\mathrm{kin}}$.

Previous results [75, 69, 71, 76, 72] have shown that models with super-
Planckian characteristic scale are robust to initial perturbations, and we
find that this broad conclusion remains true in the presence of large kinetic
fluctuations. Models of inflation with sub-Planckian characteristic scale
remain susceptible to a variety of initial inhomogeneities. Even though we
have focused on two particular cases of $\alpha$-attractor inflation, we
expect these results to apply more generally. In Fig. 5 we illustrate this,
plotting the evolution of the potential density parameter in the top panel and
the maximum values of the (rescaled) Weyl curvature scalar

$\mathcal{\tilde{C}}_{\mathrm{W}}=\frac{\mathcal{C}^{\mu\nu\rho\omega}\mathcal{C}_{\mu\nu\rho\omega}}{\langle
K^{4}\rangle}\,,$ (17)

and Chern-Pontryagin invariant

$\mathcal{\tilde{C}}_{\mathrm{P}}=\frac{{}^{*}\mathcal{C}^{\mu\nu\rho\omega}\mathcal{C}_{\mu\nu\rho\omega}}{\langle
K^{4}\rangle}\,,$ (18)

in the bottom panel. Here $\mathcal{C}_{\mu\nu\rho\omega}$ is the Weyl
curvature tensor and ${}^{*}\mathcal{C}^{\mu\nu\rho\omega}$ its dual.
Employing such curvature invariants was suggested as a diagnostic to determine
the success of inflation in [89].

Models with sub- and super-Planckian characteristic scales are denoted by the
solid and dashed lines, respectively. We find that while models with sub-
Planckian characteristic scale can fail even when the potential density
parameter dominates $\Omega^{0}_{\mathrm{pot}}\approx 1$ and the curvature
invariants are small, models with super-Planckian characteristic scale show
greater robustness, homogenizing the spacetime even starting from large
departures with $\Omega^{0}_{\mathrm{pot}}\approx\mathcal{O}(0.01)$. We see
that for a super-Planckian characteristic scale, we can drive the Weyl
curvature invariants to subdominant values. The case we study appears roughly
comparable to the “best” inflation case studied in [74], with maximum
curvature scalars that take O(1) values during the simulation. As discussed in
[73], this finding does not address the question of whether the inflation that
we observe is being dominated by “non generic” initial regions with lower
values of the curvature - to do this would require a more accurate tracking of
observers through the evolution to map initial to final regions. As in [74],
we are limited in our choice of initial data to zero average spatial
curvature, and as a result even the maximum initial values of the Chern-
Pontryagin invariant are small.

The strong dependence of our results on the characteristic scale are
particularly noteworthy because upcoming CMB experiments are designed to
detect or exclude the imprint of primordial gravitational waves on the CMB for
models of inflation with a super-Planckian characteristic scale [77]. As a
consequence the absence of a detection may exclude the models that are robust
to inhomogeneities. In the context of inflation, this would imply the need for
some mechanism that sets up initial conditions that are suitable for the
models with sub-Planckian characteristic scale. Such a mechanism could be as
simple as an earlier period of inflation. Nevertheless, this suggests that
observations could exclude single-field inflation models. Here we have studied
a model with a super-Planckian characteristic scale, and one with a
characteristic scale that is well below the Planck scale. We will sharpen
these statements by studying the transition in more detail in future work.

Our work so far has focused on conformally flat initial conditions and it
would be natural to relax this assumption in future work. In addition, one
could try to avoid the use of periodic boundary conditions altogether, and the
resulting constraints on initial data, by using spherically adapted
coordinates that asymptote to a specific cosmology. These changes would
potentially permit the study of larger initial curvature invariants, which
would provide more stringent tests of models with a super-Planckian
characteristic scale. More work could also consider how superhorizon modes, as
part of a power spectrum of initial perturbations, can affect the overall
robustness.

The effect of gauge also merits further study. In our simulations, the
foliation of space-time is decided by a choice of the lapse $\alpha$ and shift
$\beta_{i}$ parameters, which are dynamically evolved using an adapted form of
the moving puncture gauge [87, 88]. We have detailed above the challenges this
introduces when we follow the evolution through multiple e-folds in regions
with very different expansion rates, and the instabilities that appear to
arise with the collapse of strong tensor perturbations in the larger field
case. In [73, 89, 94], a different choice of gauge corresponding to constant
mean curvature slices is employed, which enable the authors to keep track of
the evolution with large numbers of e-folds. Other ideas for improved gauge
conditions have also been proposed in [95, 91]. It would be valuable to
explore whether such strategies can also be exploited or adapted to the BSSN
formalism.

###### Acknowledgements.

We acknowledge useful conversations with Maxence Corman, William East, Alan
Guth, David Kaiser and Sonia Paban. We would also like to thank the GRChombo
team (http://www.grchombo.org/). This work is supported by a Research Project
Grant RPG-2021-423 from Leverhulme Trust. JCA acknowledges funding from the
Beecroft Trust and The Queen’s College via an extraordinary Junior Research
Fellowship (eJRF). RF was supported in part by the Department of Energy under
Grant No. DE-SC0009919. This work was performed using the DiRAC@Durham
facility managed by the Institute for Computational Cosmology on behalf of the
STFC DiRAC HPC Facility (www.dirac.ac.uk) under DiRAC RAC13 Grant ACTP238 and
DiRAC RAC15 Grant ACTP316. The equipment was funded by BEIS capital funding
via STFC capital grants ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham
University and STFC operations grant ST/R000832/1. This work also used the
DiRAC Data Intensive service at Leicester, operated by the University of
Leicester IT Services, which forms part of the STFC DiRAC HPC Facility
(www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC
capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant
ST/R001014/1. DiRAC is part of the National e-Infrastructure.

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## Summary of simulations and code validation

We set the length of our computational domain $L=32M$ and impose periodic
boundary conditions. Our standard runs have $N^{3}=128^{3}$ number of coarse
grid points of the model with sub-Planckian characteristic scale and
$N^{3}=256^{3}$ for the super-Planckian case. In Table 1 we list the summary
of simulations. We fix $\phi_{0}=-6.33\times 10^{-2}M_{\mathrm{Pl}}$ and
$\Delta\phi=4.5\times 10^{-2}M_{\mathrm{Pl}}$ for sub-Planckian runs, and
$\phi_{0}=-1.2M_{\mathrm{Pl}}$ and
$\Delta\phi=\\{0.19M_{\mathrm{Pl}},0.38M_{\mathrm{Pl}}\\}$ for super-
Planckian.

Sub-Planckian runs:  Run $N_{\Pi}$ $\theta$ $\Pi_{0}$ [$M_{\mathrm{Pl}}^{2}$]
$\Delta\Pi$ [$M_{\mathrm{Pl}}^{2}$] $1-\Omega^{0}_{\mathrm{pot}}$ SF$01$ $0$
N/A $0$ $0$ $5.30\times 10^{-2}$ SF$02$ $0$ N/A $7.68\times 10^{-11}$ $0$
$5.52\times 10^{-2}$ SF$03$ $0$ N/A $1.09\times 10^{-10}$ $0$ $5.75\times
10^{-2}$ SF$04$ $0$ N/A $1.33\times 10^{-10}$ $0$ $5.97\times 10^{-2}$ SF$05$
$0$ N/A $1.54\times 10^{-10}$ $0$ $6.20\times 10^{-2}$ SF$11$ $1$ $0$ $0$
$1.88\times 10^{-10}$ $5.52\times 10^{-2}$ SF$12$ $1$ $0$ $0$ $2.77\times
10^{-10}$ $5.74\times 10^{-2}$ SF$13$ $1$ $0$ $0$ $3.26\times 10^{-10}$
$5.97\times 10^{-2}$ SF$14$ $1$ $0$ $0$ $3.76\times 10^{-10}$ $6.19\times
10^{-2}$ SF$21$ $2$ $0$ $0$ $5.95\times 10^{-10}$ $7.51\times 10^{-2}$ SF$22$
$2$ $0$ $0$ $8.41\times 10^{-10}$ $9.66\times 10^{-2}$ SF$23$ $2$ $0$ $0$
$1.03\times 10^{-9}$ $1.17\times 10^{-1}$ SF$24$ $2$ $0$ $0$ $1.19\times
10^{-9}$ $1.37\times 10^{-1}$ SF$21\pi$ $2$ $\pi$ $0$ $5.95\times 10^{-10}$
$7.51\times 10^{-2}$ SF$22\pi$ $2$ $\pi$ $0$ $8.41\times 10^{-10}$ $9.66\times
10^{-2}$ SF$23\pi$ $2$ $\pi$ $0$ $1.03\times 10^{-9}$ $1.17\times 10^{-1}$
SF$24\pi$ $2$ $\pi$ $0$ $1.19\times 10^{-9}$ $1.37\times 10^{-1}$ SF$21\pi 2$
$2$ $\pi/2$ $0$ $5.95\times 10^{-10}$ $7.51\times 10^{-2}$ SF$22\pi 2$ $2$
$\pi/2$ $0$ $8.41\times 10^{-10}$ $9.66\times 10^{-2}$ SF$23\pi 2$ $2$ $\pi/2$
$0$ $1.03\times 10^{-9}$ $1.17\times 10^{-1}$ SF$24\pi 2$ $2$ $\pi/2$ $0$
$1.19\times 10^{-9}$ $1.37\times 10^{-1}$

Super-Planckian runs:
Run $\Delta\phi$ [$M_{\mathrm{Pl}}$] $\Pi_{0}$ [$M_{\mathrm{Pl}}^{2}$]
$1-\Omega^{0}_{\mathrm{pot}}$ LF$11$ $1.90\times 10^{-1}$ 0 $5.04\times
10^{-1}$ LF$12$ $1.90\times 10^{-1}$ $1.67\times 10^{-7}$ $6.45\times 10^{-1}$
LF$13$ $1.90\times 10^{-1}$ $3.34\times 10^{-7}$ $8.10\times 10^{-1}$ LF$14$
$1.90\times 10^{-1}$ $4.68\times 10^{-7}$ $8.81\times 10^{-1}$ LF$15$
$1.90\times 10^{-1}$ $7.35\times 10^{-7}$ $9.44\times 10^{-1}$ LF$16$
$1.90\times 10^{-1}$ $1.27\times 10^{-6}$ $9.80\times 10^{-1}$ LF$21$
$3.80\times 10^{-1}$ $0$ $8.05\times 10^{-1}$ LF$22$ $3.80\times 10^{-1}$
$2.67\times 10^{-7}$ $8.78\times 10^{-1}$ LF$23$ $3.80\times 10^{-1}$
$5.34\times 10^{-7}$ $9.45\times 10^{-1}$ LF$24$ $3.80\times 10^{-1}$
$1.07\times 10^{-6}$ $9.83\times 10^{-1}$

Table 1: Summary of our simulations with sub-Planckian and super-Planckian
characteristic scales.

To convergence test our results, in Fig. 6 we compare the evolution of the
maximum of the scalar field for simulations with three resolution
$N^{3}=\\{192^{3},224^{3},256^{3}\\}$. We find that the error decreases
consistent with 2nd order convergence.

We also monitor the evolution of the Hamiltonian and momentum constraints,
shown in Fig. 7. At $\langle N\rangle\approx 1$ we observe a sharp peak in
both constraints due to black hole formation, which is inflated away shortly
after.

Figure 6: Convergence test of the error in the evolution of the maximum of the
scalar field $\mathrm{err}(\phi_{\mathrm{max}})$ between simulations with
different resolutions, showing 2nd order convergence. The labels refer to the
differences between low-resolution (LR), medium-resolution (MR) and high-
resolution (HR). Figure 7: Evolution of the averaged Hamiltonian and momentum
constraints for the simulation with super-Planckian characteristic scale, and
of largest gradient and kinetic energy.