11institutetext: School of Physics and Astronomy, University of Nottingham,
Nottingham NG7 2RD, UK

# CMB constraints on monodromy inflation at strong coupling

Edmund J. Copeland 1 Francesc Cunillera 1 Adam Moss 1 Antonio Padilla
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We carry out a thorough numerical examination of field theory monodromy
inflation at strong coupling. We perform an MCMC analysis using a Gaussian
likelihood, fitting multiparameter models using CMB constraints on the
spectral index and the tensor to scalar ratio. We show that models with
uniquely positive Wilson coefficients are ruled out. If there are coefficients
that can take on both signs, there can be a cancellation of terms that
flattens the potentials and allows one to satisfy current data, and forecasts
with strong constraints on the tensor to scalar ratio. Models of field theory
monodromy are naturally enhanced to include a mechanism for canceling off
radiative corrections to vacuum energy, via vacuum energy sequestering (VES).
Although they include a much larger parameter space, we find that a similar
numerical examination yields no significant change in the Bayesian evidence
for VES enhanced models, with naturalness considerations making them more
attractive from a theoretical perspective.

##  1 Introduction

Inflation provides the most successful explanation for the isotropy and
homogeneity observed in the cosmic microwave background (CMB) to date. Indeed,
modern observations of the CMB 1807.06211  are in agreement with an early
epoch of accelerated expansion driven by a slowly rolling scalar field. On top
of this, the quantum fluctuations during this inflationary epoch lead to
large-scale structure formation which is consistent with the structure we
observe in the CMB and in galaxy distributions (for nice reviews see
Baumann:2009ds  and Weinberg:2008zzc ).

Single field inflationary models are quite remarkable in their simplicity. For
a canonical scalar the observational consequences of these models are
determined by the first and second derivative of the potential. Of particular
interest are large field inflation models which provide some of the simplest
and most calculable models of the early universe Linde:1983gd ; Freese:1990rb
. Due to the super-Planckian excursions, these models are very sensitive to
CMB measurements. On the other hand, protecting the effective description from
quantum corrections is non-trivial due to the large values of the inflaton
field. A possible approach is to use symmetries that protect the scalar field
from UV effects. One such approach within string theory is monodromy inflation
0803.3085 ; 0808.0706 . In these scenarios the inflaton is an axion, a
pseudoscalar with a discrete shift symmetry allowing one to explore large
field values without losing control of the effective description.

A field theory description of axion monodromy was first introduced in hep-
th/0410286 ; hep-th/0507215  and its implications for inflation later explored
in arXiv:0810.5346 ; arXiv:0811.1989 ; arXiv:1101.0026 . The effects of higher
order corrections and strong coupling were investigated in detail in
1607.06105  through the Kaloper-Lawrence (KL) Lagrangian (cf. Eq. (5))
allowing constraints based on effective monomial potentials to be derived in
1709.07014 . It should be noted that although it is a consistent EFT, it
remains unclear whether or not there exists a consistent string embedding of
the KL model 1611.00394 .

In this paper, we begin by generalising the analysis of 1709.07014  to derive
the observational implications of the KL models. In particular, we do not
limit our analysis to effective monomials but rather consider the full
Lagrangian in Eq. (5) and show that one can consistently truncate its sums to
polynomials of some degree. The exact degree of the polynomials is fixed by
demanding that the error associated to the truncation of the infinite sums,
i.e. the higher order operators that we neglect, is smaller than the
observational error margins of the data we use to fit the observables, in our
case the Planck 2018 data of 1807.06211 . This leads to an effective
description of KL in terms of degree 20 polynomials and the ability to discuss
the predictions of a large class of these models in full generality. We will
limit ourselves, however, to two-derivative theories, meaning that the
relevant cosmological observables that limit the parameter space of the theory
will be: the spectral index $n_{s}$, the tensor-to-scalar ratio $r$ and the
amplitude of scalar perturbations $\mathcal{A}_{s}$. It would be interesting
to consider the introduction of higher order derivatives operators since it
would add further bounds to the parameter space coming from, say, non-
gaussianities and the speed of propagation of scalar perturbations. This is
left for future work.

We also extend our study in another important direction, allowing us to probe
a proposed mechanism Kaloper:2013zca ; Kaloper:2014dqa ; Kaloper:2014fca ;
Kaloper:2015jra ; Kaloper:2016yfa ; Kaloper:2016jsd ; DAmico:2017ngr ;
Niedermann:2017cel ; Coltman:2019mql  for solving the cosmological constant
problem Padilla:2015aaa . This follows from the realisation that monodromy
inflation models could be deformed in a natural way to give rise to vacuum
energy sequestering (VES) 1806.04740 . Within the VES framework, radiative
corrections to vacuum energy are reabsorbed through new rigid degrees of
freedom. The theory remains locally indistinguishable from General Relativity,
while global dynamics are modified. The mechanism itself, initially inspired
by Tseyltin’s model of duality symmetric strings Tseytlin:1990hn , is
reminiscent of so-called decapitation, at least in a field theory context hep-
th/0209226 ; 1706.04778 .

In 1806.04740 , it was shown that the KL model could be made compatible with
VES by introducing a second monodromy into the theory. The EFT would then
consist of a heavy monodromy, whose scalar would play the role of a constant
dilaton enforcing the global VES constraints on the system, and a light
sector, which would contain the inflaton. Then, the cancellation of the vacuum
energy loops can be accomplished via the sequestering mechanism at low
energies, consistent with a particular structure for the inflationary sector.
The recipe for this EFT description requires multiple axions, together with
their respective discrete shift symmetries, and a hierarchy between a heavy
and a light sector. These ingredients appear naturally in string theory, which
prompts the interesting question of whether a similar mechanism would be
available to string axion monodromy models of inflation. One of the main goals
of this paper is to investigate the impact of VES deformations on KL models of
monodromy inflation, via their imprint on the CMB.

Overall, after the numerical analysis, we find that KL models both with and
without VES deformations, are able to fit current Planck data and forecasted
CMB Stage 3 and Stage 4 pseudo-data. We remark that this is in contrast with
the effective monomial results of 1709.07014  which would be disfavoured at a
$2\sigma$ level by current observations. The key difference is in the more
general form of the Lagrangian allowed by our analysis. Critically, fitting
the Planck data requires both positive and negative coefficients in Eq. (5).
The monomial potentials of 1709.07014  can be mapped to our analysis with a
single positive coefficient in the potential, all other terms being turned
off. As can be seen in Fig. 1, the monomial line lies outside of the data and
crosses the region with only positive coefficients. Allowing for negative
coefficients provides a much more efficient way of lowering $r$ while keeping
$n_{s}$ on the observational range. We are also able to show that the
introduction of VES is not in tension with observations. Therefore, VES
enchanced KL models have the added benefit of including a mechanism to solve
the cosmological constant problem in a natural way without any observational
drawbacks.

Figure 1: Planck data (red) along with KL models: with only positive
coefficients (green) and with positive-negative coefficients (blue).
Superimposed is the monomial line, i.e. the $n_{s}$-$r$ predictions for
potentials of the form $V\sim\phi^{p}$, with $p=2$ and $N_{\star}=60$ given by
the filled circle.

The structure of the paper is as follows: in Sec. 2 we introduce the KL
models, then in Sec. 3 we follow 1806.04740  in introducing a second monodromy
and present the form of the KL Lagrangian with VES enhancement. In Sec. 4 we
present the relevant equations for discussing slow roll monodromy inflation at
strong coupling, with and without VES enhancements. In Sec. 5 and Sec. 6 we
describe our numerical approach in some detail and present the results for the
predictions of the KL models, with and without a VES sector, for current
Planck data and forecasted CMB Stage 3 and Stage 4 pseudo-data. Finally, we
conclude in Sec. 7.

##  2 The Kaloper-Lawrence model

In 1607.06105 ; 1101.0026 ; 1709.07014 , the authors build upon earlier work
hep-th/0410286 ; hep-th/0507215  to construct an inflation model inspired by
axion flux monodromy 0803.3085 ; 0808.0706 . The starting point for the KL
model is a massive $U(1)$ gauge theory given by

$\mathcal{L}=-\frac{1}{48}F^{2}-\frac{m^{2}}{12}\left(A_{\mu\nu\alpha}-h_{\mu\nu\alpha}\right)^{2}+\frac{m}{6}\phi\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}\partial_{\mu}h_{\nu\alpha\beta}\
,$ (1)

where $F_{\mu\nu\alpha\beta}=4\partial_{[\mu}A_{\nu\alpha\beta]}$ is the four-
form field strength, $\epsilon^{\mu\nu\alpha\beta}$ is the totally
antisymmetric Levi-Civita symbol defined such that $\epsilon^{0123}=1$, and
$\phi$ is an axion with periodicity $\phi\to\phi+2\pi f$, with $f$ left to be
determined. Indices are raised and lowered with respect to the metric
$g_{\mu\nu}$. The equation of motion for $\phi$ yields $h=db$, with
$b_{\mu\nu}$ the Stueckelberg field invariant under $b\rightarrow b+d\lambda$,
and $h$ is it’s three form field strength. Integrating out the Stueckelberg
field through its equation of motion yields

$\mathcal{L}=-\frac{1}{48}F^{2}-\frac{1}{2}\left(\partial\phi\right)^{2}+\frac{m}{24}\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}\phi
F_{\mu\nu\alpha\beta}\ .$ (2)

We now introduce a Lagrange multiplier $Q$, fixing
$F_{\mu\nu\alpha\beta}=4\partial_{[\mu}A_{\nu\alpha\beta]}$ on shell via a
term $\int
d^{4}x\frac{1}{24}Q\epsilon^{\mu\nu\alpha\beta}(F_{\mu\nu\alpha\beta}-4\partial_{[\mu}A_{\nu\alpha\beta]})$.
As the four-form now enters the Lagrangian algebraically, it can be integrated
out, with $Q$ now identified with its magnetic dual. This allows us to rewrite
the theory in terms of a pair of pseudo-scalars,

$\mathcal{L}=-\frac{1}{2}(\partial\phi)^{2}-\frac{m^{2}}{2}\left(\phi+\frac{Q}{m}\right)^{2}-\frac{Q}{6}\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}\partial_{\mu}A_{\nu\alpha\beta}\
,$ (3)

where $Q$ is constrained to be constant on-shell. In the presence of
membranes, it is quantised in units of the membrane charge $q$, $\langle
Q\rangle=2\pi Nq$ for integers $N$. The Lagrangian is invariant under a
discrete shift symmetry

$\phi\ \rightarrow\ \phi+2\pi f\ ,\qquad Q\ \rightarrow\ Q-2\pi q\ ,$ (4)

where $f=q/m$.

The theory Eq. (3) describes quadratic inflation, which would be in tension
with current observational bounds 1807.06211 . One can ease the tension with
observations by exploiting corrections due to higher order operators.
Completing the theory, by writing all the possible higher order terms allowed
by symmetries, one finds 1607.06105

$\displaystyle\mathcal{L}_{KL}$
$\displaystyle=-\frac{1}{2}(\partial\phi)^{2}-\frac{m^{2}}{2}\left(\phi+\frac{Q}{m}\right)^{2}-\sum_{k>1}\frac{a_{k}}{2^{k}\,k!}\left(\frac{4\pi}{M^{2}}\right)^{2k-2}\left(\partial\phi\right)^{2k}$
$\displaystyle-\sum_{\begin{subarray}{c}n>2\\\
\end{subarray}}\frac{b_{n}}{n!}\left(\frac{4\pi}{M^{2}}\right)^{n-2}\left(m\phi+Q\right)^{n}-\sum_{\begin{subarray}{c}k,n>1\\\
\end{subarray}}\frac{c_{k,n}}{2^{k}\,k!n!}\left(\frac{4\pi}{M^{2}}\right)^{2k+n-2}\left(m\phi+Q\right)^{n}\left(\partial\phi\right)^{2k}\
$
$\displaystyle-\frac{Q}{6}\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}\partial_{\mu}A_{\nu\alpha\beta}\
,$ (5)

where all the coefficients are taken to be $\mathcal{O}(1)$ by naturalness and
$M$ is the EFT cut-off. By using naive dimensional analysis (NDA)
Manohar:1983md ; 1601.07551  one can find the appropriate factors of $4\pi$
that guarantee that the higher order corrections remain under control as long
as we probe energies below the cut-off.

Crucially, the theory permits a small window $M/\sqrt{4\pi}<m\phi+Q<M^{2}$
where it is strongly coupled (lower bound) but the effective description
remains under control (upper bound). In this strongly coupled regime, the
corrections become important and can flatten the overall potential. This
argument was used in 1709.07014  to motivate a description for the potential
as a shallower-than-quadratic monomial, i.e. $V\sim\varphi^{p}$ with $p<2$
where $\varphi$ is the canonical inflaton. A priori, monomial inflationary
potentials with $p<2$ tend to push the models to higher values of the spectral
index $n_{s}$ putting them in tension with data. As we will show, the results
of 1709.07014  are disfavoured by Planck 2018\. Nonetheless, one might hope
that the more general character of Eq. (5) might capture regimes that are not
trivially given by the monomial potentials, significantly altering their
location in the $n_{s}-r$ plane, as we will show in the later sections.

##  3 The Kaloper-Lawrence model with VES enhancement

The monodromy inflation models of the previous section can be deformed in a
very natural way, giving rise to an emergent mechanism for solving the
cosmological constant problem 1806.04740 . The mechanism for cancelling off
radiative corrections to the vacuum energy is achieved via vacuum energy
sequestering Kaloper:2013zca ; Kaloper:2014dqa ; Kaloper:2014fca ;
Kaloper:2015jra ; Kaloper:2016yfa ; Kaloper:2016jsd ; DAmico:2017ngr ;
Niedermann:2017cel ; Coltman:2019mql  . In this section, we shall review how
field theory monodromy can be deformed to give rise to VES and how this yields
the desired cancellation of radiative corrections to the cosmological
constant. VES enhanced monodromies can also be thought of as an extension of
VES to high energies, perhaps pointing the way to an embedding of VES in
fundamental theory.

We start with the leading order KL theory Eq. (2) and introduce a small
symmetry breaking deformation parametrised by a spurion term

$\mathcal{L}=-\frac{1}{48}F^{2}-\frac{1}{2}\left(\partial\phi\right)^{2}+\frac{m}{24}\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}\phi
F_{\mu\nu\alpha\beta}-\frac{1}{2}\bar{m}^{2}\phi^{2}\ ,$ (6)

where $\bar{m}$ is the mass for the spurion and
$\left(\frac{\bar{m}}{m}\right)\ll 1$. Such a term may arise from integrating
out loops of heavy matter if there exists a suitable coupling between this and
the inflaton. In any event, one can now dualise the deformed Lagrangian and
apply the NDA scheme to find

$\displaystyle\mathcal{L}$ $\displaystyle=-\sum_{k\geq
1}\frac{a_{k}}{2^{k}\,k!}\left(\frac{4\pi}{M^{2}}\right)^{2k-2}\left(\partial\phi\right)^{2k}-\sum_{\begin{subarray}{c}n\geq
2\\\
\end{subarray}}\frac{b_{n}}{n!}\left(\frac{4\pi}{M^{2}}\right)^{n-2}\left(m\phi+Q\right)^{n}$
$\displaystyle-\sum_{\begin{subarray}{c}k,n\geq 1\\\
\end{subarray}}\frac{c_{k,n}}{2^{k}\,k!n!}\left(\frac{4\pi}{M^{2}}\right)^{2k+n-2}\left(m\phi+Q\right)^{n}\left(\partial\phi\right)^{2k}-\sum_{l\geq
2}\frac{d_{l}}{l!}\left(\frac{4\pi}{M^{2}}\right)^{l-2}\left(\bar{m}\phi\right)^{l}$
$\displaystyle-\sum_{k,l\geq
1}\frac{\mathfrak{a}_{k,l}}{2^{k}\,k!l!}\left(\frac{4\pi}{M^{2}}\right)^{2k+l-2}\left(\bar{m}\phi\right)^{l}\left(\partial\phi\right)^{2k}-\sum_{\begin{subarray}{c}l\geq
1,\\\ n\geq
2\end{subarray}}\frac{\mathfrak{b}_{n,l}}{l!n!}\left(\frac{4\pi}{M^{2}}\right)^{l+n-2}\left(\bar{m}\phi\right)^{l}\left(m\phi+Q\right)^{n}$
$\displaystyle-\sum_{\begin{subarray}{c}k,n,l\geq 1\\\
\end{subarray}}\frac{\mathfrak{c}_{k,n,l}}{2^{n}\,k!l!n!}\left(\frac{4\pi}{M^{2}}\right)^{2k+l+n-2}\left(\bar{m}\phi\right)^{l}\left(m\phi+Q\right)^{n}\left(\partial\phi\right)^{2k}-\frac{Q}{6}\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}\partial_{\mu}A_{\nu\alpha\beta}\
\ ,$ (7)

and the labels for the coefficients comply with

$\displaystyle 2k+l+n\geq 3\ .$

The Lagrangian contains a pseudo symmetry: it will remain invariant under
shifts of the axion field and the magnetic dual as in Eq. (4) as long as we
transform the spurion

$\bar{m}\rightarrow\bar{m}-\frac{2\pi q}{\phi}\frac{\bar{m}}{m}\ .$ (8)

To simplify the notation henceforth, it will be convenient to define
$\mu\equiv{M\over\sqrt{4\pi}}$ and the following dimensionless quantities

$\displaystyle\kappa\equiv\frac{\bar{m}}{m}\
,\qquad\xi\equiv\frac{m\varphi}{\mu^{2}}\
,\qquad\xi_{\phi}\equiv\frac{m\phi}{\mu^{2}}\ ,$ (9)

where $\varphi\equiv\phi+Q/m$ will be our inflaton field. If we now perform a
derivative expansion and truncate the Lagrangian at second order, the EFT
becomes

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{2}\mathcal{Z}_{\text{eff}}\left(\xi,\kappa\,\xi_{\phi}\right)(\partial\varphi)^{2}-\mu^{4}\mathcal{V}_{\text{eff}}\left(\xi,\kappa\,\xi_{\phi}\right)-\frac{Q}{6}\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}\partial_{\mu}A_{\nu\alpha\beta}\
$ (10)

where the explicit forms of the dimensionless wavefunction renormalisation and
the potential are respectively given by

$\displaystyle\mathcal{Z}_{\text{eff}}\equiv 1+\sum_{n\geq
1}\frac{c_{1,n}}{n!}\xi^{n}+\frac{\mathfrak{a}_{1,n}}{n!}\left(\kappa\,\xi_{\phi}\right)^{n}+\sum_{l,n\geq
1}\frac{\mathfrak{c}_{1,n,l}}{l!n!}\xi^{n}\left(\kappa\,\xi_{\phi}\right)^{l}\
,$ (11) $\displaystyle\mathcal{V}_{\text{eff}}\equiv\sum_{n\geq
2}\frac{b_{n}}{n!}\xi^{n}+\frac{d_{n}}{n!}\left(\kappa\,\xi_{\phi}\right)^{n}+\sum_{\begin{subarray}{c}n\geq
2\\\ l\geq
1\end{subarray}}\frac{\mathfrak{b}_{n,l}}{l!n!}\xi^{n}\left(\kappa\,\xi_{\phi}\right)^{l}\
,$ (12)

with $b_{2}=1$.

To make contact with VES, we introduce a second monodromy where the field
$\hat{\phi}$ is given a very heavy mass $\hat{m}$

$\hat{\mathcal{L}}=\frac{M_{g}^{2}}{2}R-\frac{1}{48}\hat{F}_{\mu\nu\alpha\beta}^{2}-\frac{1}{2}(\partial\hat{\phi})^{2}+\frac{\hat{m}}{4!}\hat{\phi}\epsilon^{\mu\nu\alpha\beta}\hat{F}_{\mu\nu\alpha\beta}+\frac{1}{2}\hat{g}^{2}R\hat{\phi}^{2}\
,\qquad$ (13)

where $\hat{g}$ is a dimensionless coupling constant of the axion with
gravity, which we expect to be $\hat{g}\leq 1$ for energies below the UV
cutoff. We will also demand that $\hat{m}\gg M$, thus for the range of
validity of the EFT the field $\hat{\phi}$ is forced to lie at the minimum of
its potential. This implies that once we find the dual theory to this
Lagrangian in much the same way as we did for Eq. (6), the gravitational
contribution goes as

$\hat{\mathcal{L}}=\frac{M_{g}^{2}}{2}\left[1+\left(\frac{\hat{Q}}{\hat{m}\hat{M}}\right)^{2}\right]R-\frac{\hat{Q}}{6}\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}\partial_{\mu}\hat{A}_{\nu\alpha\beta}\
,$ (14)

where we have defined the energy scale $\hat{M}=\frac{M_{g}}{\hat{g}}$. As
promised, from the viewpoint of the EFT, adding the second monodromy is
equivalent to minimally coupling the inflaton to gravity. The full action for
the theory is now given by 1806.04740

$S=S_{\text{eff}}+S_{g}+S_{F}+S_{m}\ ,$ (15)

where

$\displaystyle S_{\text{eff}}=\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}\mathcal{Z}_{\text{eff}}\left(\xi,\kappa\,\xi_{\phi}\right)\left(\partial\varphi\right)^{2}-\mu^{4}\mathcal{V}_{\text{eff}}\left(\xi,\kappa\,\xi_{\phi}\right)\right]\
,$ (16) $\displaystyle S_{g}=\int d^{4}x\sqrt{-g}\
\frac{M_{g}^{2}}{2}\left[1+\left(\frac{\hat{Q}}{\hat{m}\hat{M}}\right)^{2}\right]R\
,$ (17) $\displaystyle S_{F}=-\int
d^{4}x\frac{\epsilon^{\mu\nu\alpha\beta}}{6}\left[Q\partial_{\mu}A_{\nu\alpha\beta}+\hat{Q}\partial_{\mu}\hat{A}_{\nu\alpha\beta}\right]\
.$ (18)

and $S_{m}$ is the action for all additional matter fields, including the
Standard Model, minimally coupled to the metric $g_{\mu\nu}$. To understand
how vacuum energy is cancelled, consider the path integral, and in particular
the integration over the threes forms and the Lagrange multipliers,

$Z=\int\ldots[\mathcal{D}Q][\mathcal{D}\hat{Q}][\mathcal{D}A][\mathcal{D}\hat{A}]e^{i(S_{\text{eff}}+S_{g}+S_{F}+S_{m})}.$

Integrating out the three forms suppresses all local variations of the
Lagrange multipliers, $\partial_{\mu}Q=\partial_{\mu}\hat{Q}=0$, such that the
path integral is reduced to

$Z=\int\ldots[dQ][d\hat{Q}]e^{i(S_{\text{eff}}+S_{g}+S_{m}-Qc-\hat{Q}\hat{c})}.$

where the Lagrange multipliers are fixed to be spacetime constants, although
we still allow for their global variation in the path integral, and $c=\int
F$, $\hat{c}=\int\hat{F}$ correspond to the global values of the flux (see
Appendix A of 1903.07612  for more details).The equations of motion include
two local equations coming from the variation of the metric, $g_{\mu\nu}$, and
the inflaton, $\varphi$,

$\displaystyle
M_{g}^{2}\left[1+\left(\frac{\hat{Q}}{\hat{m}\hat{M}}\right)\right]^{2}G_{\mu\nu}=T_{\mu\nu}+T^{\varphi}_{\mu\nu}\
,$ (19)
$\displaystyle\nabla_{\mu}\left[\mathcal{Z}_{\text{eff}}\nabla^{\mu}\varphi\right]-\frac{m}{2\mu^{2}}(\mathcal{Z}_{\text{eff},1}+\kappa\mathcal{Z}_{\text{eff},2})(\partial\varphi)^{2}-m\mu^{2}(\mathcal{V}_{\text{eff},1}+\kappa\mathcal{V}_{\text{eff},2})=0$
(20)

where $\mathcal{Z}_{\text{eff},i}$ denotes the partial derivative of
$\mathcal{Z}_{\text{eff}}$ with respect to its i-th argument (similarly for
$\mathcal{V}_{\text{eff}}$). The energy momentum tensors for the matter fields
and the inflaton are given respectively by $T_{\mu\nu}$ and

$T^{\varphi}_{\mu\nu}=\mathcal{Z}_{\text{eff}}\partial_{\mu}\varphi\partial_{\nu}\varphi+p_{\varphi}g_{\mu\nu}$

where
$p_{\varphi}=-\frac{1}{2}\mathcal{Z}_{\text{eff}}(\partial\varphi)^{2}-\mu^{4}\mathcal{V}_{\text{eff}}$
is the inflaton pressure. In addition to these local equations of motion,
there are two global constraints coming from the global variation of the
Lagrange multipliers, giving

$\displaystyle c$ $\displaystyle=\frac{\partial S_{\text{eff}}}{\partial
Q}=-\frac{\kappa}{\mu^{2}}\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}\mathcal{Z}_{\text{eff},2}\left(\partial\varphi\right)^{2}-\mu^{4}\mathcal{V}_{\text{eff},2}\right],$
(21) $\displaystyle\hat{c}$ $\displaystyle=\frac{\partial
S_{\text{g}}}{\partial\hat{Q}}=\int
d^{4}x\sqrt{-g}R\left(\frac{M_{g}}{\hat{M}}\right)^{2}\frac{\hat{Q}}{\hat{m}^{2}}\
,$ (22)

Defining the normalised spacetime average $\langle.\rangle$ as

$\langle Y\rangle\equiv\frac{\int d^{4}x\sqrt{-g}\ Y}{\int d^{4}x\sqrt{-g}}\
,$ (23)

and taking ratios of Eq. (21) and Eq. (22) we arrive at the following
constraint on the long wavelength mode of the Ricci scalar

$\langle R\rangle=\frac{\kappa}{\hat{Q}}\left(\frac{\hat{m}\hat{M}}{\mu
M_{g}}\right)^{2}\left\langle\frac{1}{2}\mathcal{Z}_{\text{eff},2}\left(\partial\varphi\right)^{2}+\mu^{4}\mathcal{V}_{\text{eff},2}\right\rangle\frac{\hat{c}}{c}\equiv
R_{\infty}(c,\hat{c})\ .$ (24)

This constraint ensures that the large scale scalar curvature is controlled by
the flux terms and is not affected by large radiative corrections to the
vacuum energy. To see the implications of this, we take traces and spacetime
expectation values of the field equations, to arrive at the VES version of the
Einstein equations

$\kappa_{g}^{2}G_{\mu\nu}=T_{\mu\nu}-\frac{1}{4}\langle T\rangle
g_{\mu\nu}+\mathcal{Z}_{\text{eff}}\partial_{\mu}\varphi\partial_{\nu}\varphi-
g_{\mu\nu}\left(\delta\lambda+\Delta\Lambda_{\text{eff}}\right)\ .$ (25)

Here $\kappa_{g}^{2}\equiv
M_{g}^{2}\left(1+\left(\frac{\hat{Q}}{\hat{m}\hat{M}}\right)^{2}\right)$ is
the effective gravitational coupling, $\delta\lambda=\langle
p_{\varphi}\rangle-p_{\varphi}$ measures local fluctuations of the inflation
pressure and

$\Delta\Lambda=\frac{1}{4}\left[\kappa_{g}^{2}R_{\infty}(c,\hat{c})+\langle\mathcal{Z}_{\text{eff}}(\partial\varphi)^{2}\rangle\right]$
(26)

is a global cosmological constant term that depends on the global flux, but is
independent of the vacuum energy. To see how this equation is independent of
the Standard Model vacuum energy, we decompose the energy-momentum tensor of
matter into a vacuum energy piece and local excitations (likes stars and
planets) i.e. $T_{\mu\nu}=-V_{\text{vac}}g_{\mu\nu}+\tau_{\mu\nu}$. One can
easily verify that $V_{\text{vac}}$ drops out of the VES Einstein equations
Eq. (25), giving

$\kappa_{g}^{2}G_{\mu\nu}=\tau_{\mu\nu}-\Lambda_{\text{eff}}g_{\mu\nu}+\mathcal{Z}_{\text{eff}}\partial_{\mu}\varphi\partial_{\nu}\varphi-\delta\lambda
g_{\mu\nu}\ .$ (27)

where the residual cosmological constant
$\Lambda_{\text{eff}}=\Delta\Lambda+\frac{1}{4}\langle\tau\rangle$. Large
radiative corrections to the vacuum energy arising from matter loops do not
induce large corrections to $\Lambda_{\text{eff}}$. Note that the VES
mechanism does not explain why the residual cosmological constant is small but
it does explain why it is radiatively stable. The scenario is now reminiscent
of chiral symmetry and fermion masses in effective field theories - this
explains why radiative corrections to the mass are under control but does not
explain or predict the value of the mass itself.

##  4 Equations for slow-roll inflation at strong coupling

From the previous sections, we see that the dynamics of strongly coupled
monodromy inflation, with and without VES enhancement, is captured by an
effective Einstein equation of the form

$\kappa_{g}^{2}G_{\mu\nu}=-\Lambda_{\star}g_{\mu\nu}+\mathcal{Z}_{\text{eff}}\partial_{\mu}\varphi\partial_{\nu}\varphi-
g_{\mu\nu}\left[\frac{1}{2}\mathcal{Z}_{\text{eff}}(\partial\varphi)^{2}+\mu^{4}\mathcal{V}_{\text{eff}}\right]\
.$ (28)

and an inflaton equation

$\nabla_{\mu}\left[\mathcal{Z}_{\text{eff}}\nabla^{\mu}\varphi\right]-\frac{1}{2}\mathcal{Z}_{\text{eff},\varphi}(\partial\varphi)^{2}-\mu^{4}\mathcal{V}_{\text{eff},\varphi}=0$
(29)

The original Kaloper-Lawrence model, coupled to General Relativity, is
obtained by taking the $\kappa\to 0$ limit for $\mathcal{Z}_{\text{eff}}$ and
$\mathcal{V}_{\text{eff}}$, and tuning $\Lambda_{\star}$ to vanish. Radiative
corrections to the vacuum energy would destablise the latter, which is, of
course, the statement of the cosmolgical constant problem Padilla:2015aaa . In
terms of our numerical analysis, the relevant Wilson coefficients in the two
potentials are $\\{b_{n},c_{n}\\}$. In contrast, when we include VES
enhancement, the cosmological constant is given by

$\Lambda_{\star}=\frac{1}{4}\kappa_{g}^{2}R_{\infty}(c,\hat{c})-\frac{1}{4}\langle\mathcal{Z}_{\text{eff}}(\partial\phi)^{2}+4\mu^{4}\mathcal{V}_{\text{eff}}\rangle$
(30)

Once again we fix $\Lambda_{\star}$ to vanish, although this result now
remains stable under radiative corrections to the vacuum energy thanks to the
VES mechanism. The potentials also admit corrections from the VES
deformations, parametrised by the combination

$\kappa\,\xi_{\phi}\equiv\frac{m\kappa}{\mu^{2}}\left(\varphi-\frac{Q}{m}\right)=\frac{\bar{m}}{\mu^{2}}\phi\
,$

which comes from the fact that the spurion term is related to the axion field
$\phi$ rather than the inflaton field $\varphi$. Since the inflaton is
dominated by the flux contribution, the VES deformation terms are heavily
suppressed with respect to the KL terms, i.e. $\bar{m}\phi\ll m\varphi$. One
can then approximate $\mathcal{Z}_{\rm eff}\simeq\mathcal{Z}_{\rm eff}(\xi,0)$
and $\mathcal{V}_{\rm eff}\simeq\mathcal{V}_{\rm eff}(\xi,0)$, neglecting the
VES contributions altogether. In contrast, derivatives of these quantities
will see a non-trivial contribution from the VES corrections. For example,
under the previous considerations the effective potential becomes

$\mathcal{V}_{\text{eff}}=\sum_{n\geq
2}\left(b_{n}+\mathfrak{b}_{n,1}\kappa\,\xi_{\phi}\right)\frac{\xi^{n}}{n!}+\mathcal{O}(\kappa^{2}\xi_{\phi}^{2})\simeq\mathcal{V}_{\text{eff}}(\xi,0)\
,$ (31)

since $\kappa\,\xi_{\phi}\ll 1$. However, its first derivative receives a non-
trivial contribution, at leading order in $\kappa$, from the
$\mathfrak{b}_{n,1}$ term

${\partial\over\partial\varphi}\mathcal{V}_{\text{eff}}={m\over\mu^{2}}\sum_{n\geq
2}\left(n\,b_{n}+\mathfrak{b}_{n,1}\kappa\,\xi\right)\frac{\xi^{n-1}}{n!}+\mathcal{O}(\kappa\xi_{\phi})\
,$ (32)

which is only suppressed by $\kappa$. In general, the p-th derivative of the
potential will obtain a contribution of the form (for $p\geq 2$)

${\partial^{p}\over\partial\varphi^{p}}\mathcal{V}_{\text{eff}}=\left({m\over\mu^{2}}\right)^{p}\sum_{n\geq
p}\left[(n-p+1)\,b_{n}+p\,\mathfrak{b}_{n,1}\kappa\,\xi\right]\frac{\xi^{n-p}}{(n-p+1)!}+\mathcal{O}(\kappa\xi_{\phi})\
.$ (33)

A similar discussion applies to derivatives of $\mathcal{Z}_{\rm eff}$.
Overall, we find that the relevant Wilson coefficients for our numerical
analysis are

$\\{b_{n},\mathfrak{b}_{n},c_{n},\mathfrak{c}_{n}\\}\ $ (34)

where we have set $\mathfrak{b}_{n}\equiv\mathfrak{b}_{n,1}$, $c_{1,n}\equiv
c_{n}$ and $\mathfrak{c}_{1,n,1}\equiv\mathfrak{c}_{n}$ to avoid cluttering
the notation in the remainder of this paper. The VES enhancement is contained
in the $\mathfrak{b}_{n}$ and $\mathfrak{c}_{n}$.

For general values of the coefficients in the effective Lagrangian of Eq. (7),
there is no closed form for the integral defining the canonical inflaton field

$\psi\equiv\int\sqrt{\mathcal{Z}_{\text{eff}}}\;d\varphi=\frac{\mu^{2}}{m}\int\sqrt{1+\sum_{n\geq
1}\frac{c_{n}}{n!}\xi^{n}}\;d\xi\ .$ (35)

For this reason it is convenient to study the dynamics of slow-roll inflation
in a non-canonical frame. Let us now briefly review the relevant formulae,
beginning with the geometrical slow-roll parameter

$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{\dot{\varphi}}{2H}\left[\frac{\frac{m}{2\mu^{2}}\mathcal{Z}^{\prime}_{\text{eff}}\dot{\varphi}^{2}+\mu^{2}m\
\mathcal{V}^{\prime}_{\text{eff}}+\mathcal{Z}_{\text{eff}}\ddot{\varphi}}{3H^{2}}\right]\simeq\frac{1}{2}\frac{1}{\mathcal{Z}_{\text{eff}}}\left(\frac{\mathcal{V}^{\prime}_{\text{eff}}}{\mathcal{V}_{\text{eff}}}\right)^{2}\
,$ (36)

where prime denotes $\partial_{\phi}$. In the last step we have used the slow
roll approximation for the non-canonical field $\varphi$

$\displaystyle\hskip
25.00003pt\frac{1}{2}\mathcal{Z}_{\text{eff}}\dot{\varphi}^{2}\ll\mu^{4}\left(\mathcal{V}_{\text{eff}}\right)\
,$
$\displaystyle\frac{m}{2\mu^{2}}\mathcal{Z}^{\prime}_{\text{eff}}\left(\dot{\varphi}\right)^{2}+\mathcal{Z}_{\text{eff}}\ddot{\varphi}\ll\mu^{2}m\mathcal{V}^{\prime}_{\text{eff}}\
,$ (37)

Similarly, one can derive the form of the slow-roll parameter $\eta$ from its
definition as

$\eta\equiv\frac{\dot{\epsilon}}{H\epsilon}\simeq\frac{M_{pl}^{2}\,m^{2}}{\mu^{4}}\frac{1}{\mathcal{Z}_{\text{eff}}}\left(\frac{\mathcal{V}_{\text{eff}}^{\prime\prime}}{\mathcal{V}_{\text{eff}}}-\frac{1}{2}\frac{\mathcal{V}_{\text{eff}}^{\prime}\mathcal{Z}^{\prime}_{\text{eff}}}{\mathcal{V}_{\text{eff}}\mathcal{Z}_{\text{eff}}}\right)\
.$ (38)

It is now straightforward to express the spectral index $n_{s}$, the scalar-
to-tensor ratio $r$ and the amplitude of scalar perturbations
$\mathcal{A}_{s}$ using the usual formulae

$\displaystyle n_{s}=1-6\epsilon+2\eta\ ,$ (39) $\displaystyle r=16\epsilon\
,$ (40)
$\displaystyle\mathcal{A}_{s}=\frac{\mu^{4}}{24\pi^{2}M_{pl}^{4}}\frac{\mathcal{V}_{\text{eff}}}{\epsilon}\left(1+\frac{2}{3}\epsilon+\mathcal{O}(\epsilon^{2})\right)\
.$ (41)

where the form of the slow-roll parameters is taken as in Eq. (36) and Eq.
(38) .

Finally, the number of e-foldings is given by

$dN\equiv\frac{1}{\epsilon
H}dH\underset{\text{SR}}{\simeq}d\psi\frac{V_{\text{eff}}}{V_{\text{eff},\psi}}\
,$ (42)

where the final expression comes from assuming slow roll. This yields an
expression

$N_{\star}=-\frac{\mu^{4}}{M_{pl}^{2}m^{2}}\int_{\psi(\xi=\xi_{\star})}^{\psi(\xi=\xi_{e})}d\xi\
\mathcal{Z}_{\text{eff}}\frac{\mathcal{V}_{\text{eff}}}{\mathcal{V}_{\text{eff},\xi}}\
.$ (43)

computed in the non-canonical frame. Here $\xi_{e}$ is the value of the field
at the end of inflation and $\xi_{\star}$ its value when the pivot scale,
$k_{\star}=0.05\text{ Mpc}^{-1}$ crosses the horizon.

##  5 Numerical strategy and caveats

The most immediate obstacle to developing a sound numerical strategy in the
strongly coupled regime is the infinite tower of Wilson coefficients that
appear in the relevant potentials. Indeed, for the KL model, both the wave
renormalisation factor, $\mathcal{Z}_{\text{eff}}$, and the effective
potential, $\mathcal{V}_{\text{eff}}$, can be schematically reduced to an
infinite series of the form

$\mathcal{C}_{\text{eff}}(\xi)=\sum_{n}\frac{c_{n}}{n!}\xi^{n}\ .$ (44)

VES enhanced models also contain an additional series of the same schematic
form, given in terms of $\xi_{\phi}$ as opposed to $\xi$. Of course, series of
the form Eq. (44) converge for $c_{n}\sim\mathcal{O}(1)$. However, for
arbitrary coefficients, they cannot be expressed in closed form in terms of
elementary functions. To get around this, we truncate the potentials at some
finite order $n_{\text{obs}}$, replacing $\mathcal{C}_{\text{eff}}(\xi)$ with

$\bar{\mathcal{C}}_{{\rm eff}}(\xi)=\sum_{n\leq n_{\rm
obs}}\frac{c_{n}}{n!}\xi^{n}$ (45)

and compute the corresponding observables $\bar{n}_{s}$ ands $\bar{r}$. Such a
truncation will inevitably induce an error in the potentials, given
schematically as

$\Delta\mathcal{C}_{{\rm eff}}=\sum_{n>n_{\rm obs}}\frac{c_{n}}{n!}\xi^{n}\ ,$
(46)

which is passed on to the observational predictions. In particular, the
spectral index and tensor-to-scalar ratio are expected to be off by an amount
$\Delta n_{s}$ and $\Delta r$ respectively, where

$\frac{|\Delta n_{s}|}{1-n_{s}}\sim\frac{|\Delta
r|}{r}\sim\frac{\Delta\mathcal{C}_{{\rm eff}}}{\mathcal{C}_{{\rm eff}}}$ (47)

The key point is that the data itself contains errors in the values of these
parameters. Of relevance to us, the combined measurement of Planck
TT,TE,EE+lowE+lensing+BK15+BAO 1807.06211  yields

$n_{s}=0.9668\pm 0.0037,\qquad r<0.063\ ,$ (48)

where the confidence interval is to $1\sigma$ and the upper bound on $r$ to
$2\sigma$. The errors introduced through our truncation will be less than the
observational errors in the data provided $\Delta\mathcal{C}_{{\rm
eff}}/\mathcal{C}_{{\rm eff}}\lesssim 0.1$ for the generic potentials (and
their derivatives). In this work we truncate the potentials at 20th order, for
which $n_{\rm obs}=20$. To get a flavour of the size of
$\Delta\mathcal{C}_{{\rm eff}}/\mathcal{C}_{{\rm eff}}$, consider the case
where $\mathcal{C}_{{\rm eff}}=e^{\xi}$. In that case, the generic error can
be expressed in terms of incomplete Gamma functions,

$\frac{\Delta\mathcal{C}_{{\rm eff}}}{\mathcal{C}_{{\rm
eff}}}=1-\frac{\Gamma(n_{\rm obs}+1,\xi)}{n_{\rm obs}!}$ (49)

For $n_{\rm obs}=20$ and $\xi$ bounded by the strong coupling limit, $\xi\leq
4\pi$, we find that $\Delta\mathcal{C}_{{\rm eff}}/\mathcal{C}_{{\rm eff}}\leq
1-\frac{\Gamma(21,4\pi)}{20!}\approx 0.018$. This places the error well below
the observational limits.

For the truncated polynomials, the slow-roll equations were solved numerically
using Python. For a given model, the procedure we adopt is as follows:

  1. 1.

Initially, all roots $\mathcal{V}_{\text{eff}}(\xi)=0$ in the range
$\xi\in[0.01,4\pi]$ are identified, and any intervals with
$\mathcal{V}_{\text{eff}}<0$ are discarded. The lower bound in $\xi$ arises
from noting that for $\xi<1$ the potential becomes nearly quadratic which
would be in tension with data. As a sanity check, we allow a region
$0.01\leq\xi\leq 1$ where we expect quadratic inflation to occur and show that
observationally viable inflation requires $\xi>1$.

  2. 2.

Within each interval, $\xi\in[\xi_{\rm lower},\xi_{\rm upper}]$, we solve for
all possible field values that will lead to the end of inflation, specified by
$\epsilon(\xi_{e})=1$.

  3. 3.

For each $\xi_{e}$, we evaluate whether $N_{\star}$ can be satisfied using
(43). This is achieved by choosing a trial value for
$\xi_{\star}=\xi_{e}+\Delta\xi$, and increasing $\Delta\xi$ until sufficient
e-folds are achieved, or $\xi_{\star}>\xi_{\rm upper}$. If successful, a
further root finding step is performed to find $\epsilon_{\star}$.

  4. 4.

Finally, we perform an additional check that both slow-roll conditions are
satisfied,
$|\eta|<1,\epsilon<1\,\forall\xi\in\left[\xi_{e},\xi_{\star}\right]$.

Passing these conditions ensures there is a sufficient period of slow-roll
inflation. Each model may give rise to either no, single or multiple
inflationary regions. Before discussing the generic predictions for these
families of models, let us point out a few important caveats as well as
fleshing out the numerical strategy that we followed. In the following, we
will only consider models that comply with the following:

  1. i.

Positivity of $\mathcal{Z}_{\text{eff}}$: In order to avoid ghosts in the
theory, we require that

$\mathcal{Z}_{\text{eff}}(\xi)>0\
\forall\xi\in\left[\xi_{e},\xi_{\star}\right]\ .$

  2. ii.

After the end of inflation, the potential monotonically decreases to the
Minkowski vacuum defined at $\xi=0$. The original Kaloper-Sorbo theory is
defined as an expansion around the Minkowski vacuum at $\xi=0$, the weakly
coupled regime. We limit ourselves to those potentials that allow access to
this vacuum classically.

  3. iii.

Related to the previous point, we only consider the classical evolution of the
field. Even though one might be tempted to argue in favour of keeping models
with false de Sitter vacua, since in some finite time we would tunnel to the
true Minkowski vacuum, the technicalities of the calculation involving
tunnelling probabilities disallow the statistical treatment of the next
section.

  4. iv.

We only consider a single phase of inflation. Although models with multiple
phases of inflation could satisfy the observational constraints on the length
of inflation, these models require further study of the dynamics and the
spectrum of perturbations. This implies that, in the case of multiple regions
that satisfy the above constraints, we choose the region that is closest to
the Minkowski vacuum, as this would be the first N e-folds of inflation that
we would observe.

Figure 2: Random potentials, $\mathcal{V}_{\text{eff}}$ v $\xi$, for the
Kaloper-Lawrence model. The banded regions show
$\left[\xi_{e},\xi_{\star}\right]$. The red regions are ruled out by the
positivity of $\mathcal{Z}_{\text{eff}}$ and the orange regions by the
requirement that the final phase of evolution must decrease monotonically to
the Minkowski vacuum.

In order to gain some intuition on the range of possible solutions, we
generate 10k Kaloper-Lawrence models drawn randomly from a mass prior of
$m\sim\mathcal{U}(0.1,1)$, and $b_{n>2}$,
$c_{n>0}\sim\mathcal{U}\left[-3,-0.1\right]\cup\mathcal{U}\left[0.1,3\right]$.
The remaining parameters are set by their values in Eq. (5), i.e.
$b_{0},b_{1}=0$, $b_{2}=1$ and $c_{0}=1$. These priors are chosen to give
natural $\mathcal{O}(1)$ values. We also did a run with coefficients in the
range $\mathcal{U}\left[-3,3\right]$ with no major change in results. During
sampling, we choose $N_{\star}\sim\mathcal{U}(50,60)$ to give a representative
range consistent with standard reheating. As discussed above, we truncate the
series at $n_{\rm obs}=20$.

In total, we find 39% of models satisfy slow-roll and give a sufficient number
of e-folds, but only 12% are viable, due to conditions (i) and (ii) above111We
do not consider observations when determining if a model is viable.. We plot a
selection of models in Fig. 2. If one repeats the analysis considering only
positive coefficients, we find viable inflation occurs for all 10k random
models. Many of these models, of course, will not give consistent
observations, so in Fig. 3 we show the fraction of viable models that survive
a given observational bound on $r$. Of particular note is that for KL models
both with and without VES enhancement, if we demand only positive
coefficients, none of them in the random sample survive $r<0.1$, whereas, when
we allow for both positive and negative coefficients, $\sim 30$% of the
previously viable models survive the very stringent CMB bound of $r<0.001$.

To evaluate the impact of VES, we use priors of
$\mathfrak{b}_{n>1},\mathfrak{c}_{n>1}\sim\mathcal{U}\left[-3,-0.1\right]\cup\mathcal{U}\left[0.1,3\right]$
and $\log_{10}\kappa\sim\mathcal{U}(-5,-1)$. The ranges on $\kappa$ are fixed
so that the condition $\kappa:=\bar{m}/m\ll 1$ is satisfied, with the upper
bound marginally doing so. One could consider even lower values of $\kappa$,
however that would make the contributions from the VES sector even weaker. For
the range $10^{-5}<\kappa<10^{-1}$, it is already the case that including VES
does not significantly change these results and one finds a similar number of
viable models. Therefore, we would expect that pushing the bound on $\kappa$
to lower values would not yield any interesting new results.

The small impact arising from the VES corrections are not entirely unexpected.
For example, from Eq. (32) and using that the sum peaks around $\xi\simeq n+1$
we see that the first derivative of the potential is given by

$\mathcal{V}^{\prime}_{\text{eff}}={m\over\mu^{2}}\sum_{n\geq
2}\left(n\,b_{n}+\mathfrak{b}_{n,1}\kappa\,\xi\right)\frac{\xi^{n-1}}{n!}\sim{m\over\mu^{2}}\sum_{n\geq
2}n(1+\kappa)\frac{\xi^{n-1}}{n!}\sim\mathcal{V}^{\prime}_{\text{eff}}(\xi,0)(1+\kappa)\
,$ (50)

with the corrections to the cosmological observables being, roughly,
$n_{s}\sim n_{s}^{\rm ns}(1+\kappa)$ and $r\sim r^{\rm ns}(1+\kappa)$, with
$n_{s}^{\rm ns},r^{\rm ns}$ the original non-VES KL predictions. Nonetheless,
it is very interesting to point out that despite more than doubling the number
of parameters in the theory, the VES models are not disfavoured by a Bayesian
analysis, as shown in Sec. 6. From a theoretical point of view, the VES models
might even be preferred since they include a mechanism to cancel vacuum energy
loop contributions.

Figure 3: Fraction of viable inflationary models that survive an observational
bound on $r$. On the left we show positive coefficient models only, and on the
right both positive and negative coefficients. The same model but with VES
included is given by the dashed red curve.

In the following section we assess more concretely whether these solutions
give rise to slow-roll parameters consistent with observations.

##  6 Testing with observations and forecasting

We now ask whether the models under investigation give rise to slow-roll
parameters consistent with observations. We first consider the case of
strictly positive Wilson coefficients using Planck 2018 Planck:2018jri  and
Bicep-Keck (BK15) Ade:2018gkx  data. As suggested by Fig. 3, we expect these
class of models to be under tension due to the paucity of consistent solutions
satisfying $r<0.1$. However, these were generated by random sampling from the
prior, and it may be possible that regions of parameter space are more
consistent with observations.

For this analysis, we perform a Markov Chain Monte Carlo (MCMC) analysis using
a Gaussian likelihood. We fit the ($n_{\mathrm{s}}$, $r$) covariance matrix
from the Planck TT, TE, EE + lowE + BAO + BK15 chains, which gives a good
approximation to the true likelihood. In total we fit for 39 parameters in the
Kaloper-Lawrence model, and 79 including VES enhancements, using the same
prior ranges as in the previous section and fixing $N_{\star}=60$. We use the
ensemble sampler emcee 2013PASP..125..306F , using 200 walkers in the ensemble
and a combination of the affine invariant stretch 2010CAMCS…5…65G  and
differential evolution moves. The reason for this is when generalising to
negative Wilson coefficients we find a bi-modal posterior, with a mode of
lower likelihood running along the monodromy line, and this combination of
MCMC moves gives improved mixing. Running the ensemble for a chain length of
20 autocorrelation times, the resulting posterior is shown in Fig. 4. As
expected, models with strictly positive Wilson coefficients are unable to fit
observational data and can be ruled out at the $\sim 2.7\sigma$ level.

Figure 4: Planck data (red) along with predictions (blue) for Kaloper-Lawrence
inflation with strictly positive Wilson coefficients, both without (left) and
with (right) VES contributions. We also show the $V\sim\phi^{p}$ monodromy
line, with $p=2$ and $N_{\star}=60$ given by the filled circle.

The marginalised parameters and best-fits are similar both with and without
VES, as expected from the small $\sim(1+\kappa)$ corrections to $r$ and
$n_{\mathrm{s}}$. This is supported by inspecting the posterior distribution
for $\kappa$, which only becomes non-uniform for $\log_{10}\kappa\gtrsim-2$.
Given the large number of additional parameters in the fit, it is natural to
ask if VES is disfavoured by model comparison. In order to quantity this, we
compute the Bayesian Evidence by performing nested sampling. For ease of
integration with our code we use the dynesty sampler Speagle:2019ivv  with
1000 live points. The Bayesian Evidence ($B$) values are given in Tab. 1. We
find the difference between VES and no VES to be $\log B=0.6$, which is
inconclusive on the Jeffreys scale. Whilst this may seem counter-intuitive,
VES is compatible with data over almost all of its prior range, so is not
penalised by Bayesian Evidence as very little additional parameter space is
wasted.

Dataset | Model | $\log B$
---|---|---
Planck | Positive Wilson coefficients | -4.8
Planck | Positive Wilson coefficients with VES | -4.2
Planck | Positive and negative Wilson coefficients | -4.4
Planck | Positive and negative Wilson coefficients with VES | -3.8
CMB S3 | Positive Wilson coefficients | -40.5
CMB S3 | Positive Wilson coefficients with VES | -41.0
CMB S3 | Positive and negative Wilson coefficients | -26.9
CMB S3 | Positive and negative Wilson coefficients with VES | -24.3
CMB S4 | Positive Wilson coefficients | -231.7
CMB S4 | Positive Wilson coefficients with VES | -229.5
CMB S4 | Positive and negative Wilson coefficients | -27.2
CMB S4 | Positive and negative Wilson coefficients with VES | -28.0

Table 1: Log of the Bayesian Evidence $B$ for various data and model
combinations.

The result of Fig. 3 indicates that negative Wilson coefficients are needed.
This is related to the fact that observationally successful models of
inflation require the presence of a turning point in the potential, i.e. some
degree of cancellation between neighbouring coefficients. Running the MCMC, we
find this class of model has no issue in fitting current data, so we also
check that it is compatible with forecasted Stage 3 (S3) and Stage 4 (S4)
constraints. To do this, we assume an uncorrelated Gaussian likelihood for
($n_{\mathrm{s}}$, $r$), using $1\sigma$ upper bounds on $r$ of 0.01 and 0.003
respectively. Since these experiments will primarily target the large-angle
polarization signal, we use existing Planck constraints on $n_{\mathrm{s}}$.
In Figs. 5 and 6 we show the resulting posterior distributions. As expected
from the previous section, these models can naturally give rise to small $r$
values and cannot be ruled out, even by a future S4 experiment.

To see why, it is worth examining the reconstruction of the potential. In Fig.
7 we show $\mathcal{V}_{\text{eff}}(\xi)$, $\mathcal{Z}_{\text{eff}}(\xi)$ and
$\mathcal{V}_{\text{eff}}(\varphi)$ for samples from the MCMC chains for the
Kaloper-Lawrence model with positive Wilson coefficients and Planck data. In
Fig. 8 we show the same but for positive and negative coefficients and a S4
experiment. The crucial role played by the negative Wilson coefficients
becomes clear – they have the effect of flattening the potential from its
otherwise rapdily monotonic decline, in particular leading to a point of
inflection from where inflation can begin, staying flat enough to give rise to
at least sixty efolds of acceleration expansion. This flatness slows the
inflaton down, decreasing $\epsilon$, and leading to a much smaller value for
$r$.

Plotting a histogram of the number of negative coefficients, we find this is
strongly peaked at $\sim n/2$. The requirement that a degree of cancellation
is required between coefficients is not ideal from a model building
perspective. This can be quantified by comparing the Bayesian Evidence between
models with positive and negative coefficients, compared to strictly positive
coefficients. From Tab. 1 one can see that for Planck, even though there is an
improvement in the fit, there is only weak evidence for including negative
coefficients. For a stage 3 experiment, however, if there is no detection of
tensor modes, there would be strong evidence for negative coefficients in the
context of monodromy models.

Figure 5:  CMB S3 experimental data (red) along with predictions (blue) for
Kaloper-Lawrence inflation with positive and negative Wilson coefficients,
both without (left) and with (right) VES contributions. Figure 6: CMB S4
experimental data (red) along with predictions (blue) for Kaloper-Lawrence
inflation with positive and negative Wilson coefficients, both without (left)
and with (right) VES contributions. Figure 7:  (Left)
$\mathcal{V}_{\text{eff}}(\xi)$ from MCMC samples for the Kaloper-Lawrence
model with positive Wilson coefficients and Planck data. (Middle)
$\mathcal{Z}_{\text{eff}}(\xi)$. (Right) Potential for the canonical field,
$\mathcal{V}_{\text{eff}}(\psi)$. The banded regions show the final 60 e-folds
of inflation.  Figure 8: Same as Fig. 7 but for positive and negative Wilson
coefficients with S4 constraints. The combination of positive and negative
coefficients flattens the potential during the crucial final 60 efoldings of
inflation region, leading naturally to small values of $r$ consistent with the
data.

##  7 Conclusions

Field theory models of monodromy inflation allow one to consistently probe
large values of the inflaton field whilst retaining control of the effective
description. Of course, such models are guaranteed to be ruled out by
observation unless there is some additional flattening of the potential,
something which can be achieved by pushing the dynamics into a strongly
coupled regime, without losing control of the theory. Although an heuristic
analysis of the observational tests of these models was performed in
1709.07014 , a thorough numerical examination was still lacking. In this
paper, we have carried out that examination and revealed some new features
required by the family of models in order for them to satisfy the
observational constraints. In particular, the tower of Wilson coefficients
that define the higher order operators of the theory cannot all have the same
sign. Indeed, in order for the potential to be sufficiently flat to yield
enough inflation and be compatible with bounds on the spectral index and
tensor to scalar ratio, there needs to be cancellations between terms. This
requires a combination of positive and negative coefficients. Once this is the
case, these coefficients can be order one in absolute value, consistent with
naturalness, and remain compatible with CMB data, both current and forecasted
Stage 3 and Stage 4 with strong constraints on the tensor to scalar ratio.
Although the best models seem to require a roughly equal number of positive
and negative coefficients, we were not able to identify any further structure
in the array of coefficients.

These monodromy models can be enhanced, from a theoretical perspective by
including the corrections suggested by vacuum energy sequestering
Kaloper:2013zca ; Kaloper:2014dqa ; Kaloper:2014fca ; Kaloper:2015jra ;
Kaloper:2016yfa ; Kaloper:2016jsd ; DAmico:2017ngr ; Niedermann:2017cel ;
Coltman:2019mql  . These are desirable since they guarantee the cancellation
of vacuum energy loops, allowing us to set the vev of the potential to vanish
without having to fine tune the result against large radiative corrections.
From a Bayesian perspective, we might have expected the addition of the VES
corrections to have been disfavoured because we introduce many more parameters
into the theory. As it happens, this doesn’t matter: VES enhanced models are
not significantly favoured or disfavoured relative to the original models of
monodromy inflation, at least as far as the data is concerned. This is
encouraging as it suggests an emergent mechanism for solving the cosmological
constant problem can be incorporated into inflationary dynamics without any
cost to the likelihood.

We might ask whether there is any prospect for picking out VES enhanced models
versus say the KL models? These initial results suggest not at the level of
the inflaton itself, as the two models agree so closely. However there may be
features arising during the reheating phase, arising from the extra couplings
associated with VES models that would lead to observational features on small
scales. We might also ask, what are the weaknesses and strengths of the
scenarios we have investigated. A clear strength of the VES enhanced models is
that they contain a mechanism to address the cosmological constant problem. A
potential strength of both sets of models, with and without VES enhancement,
arises from the fact that when allowing for positive and negative
coefficients, these can conspire to flatten the potential as seen in Fig. 7.
It opens up the intriguing possibility that they may cause the potential to
flatten to such a degree that the inflaton field could enter a period of ultra
slow roll inflation, leading to the associated production of primordial black
holes, and therefore providing, both, a new observational signature for the
model and a new set of constraints on the allowed values of the coefficients.
It would also arguably be the first particle inspired model that leads
naturally to a period of ultra slow roll inflation occurring during the final
e-foldings of inflation. Perhaps a weakness in probing both sets of models is
that we don’t know precise values of the Wilson coefficients and so the best
we can do is to sample from a broad class of possible values. However, we are
not alone with this problem, it is true of any effective theory unless there
is some additional symmetry controlling the relative structure of the higher
order operators. Not knowing these coefficients makes it hard to say much
about the end of inflation if you want to think about the formation of
oscillons, axion stars, and possible links to dark matter. But it remains the
case that we may be able to make more general statements about the likely
formation mechanisms and the associated production of primordial gravitational
waves - an exciting prospect.

###### Acknowledgements.

AP, EJC and AM were supported by an STFC consolidated grant number
ST/T000732/1 and FC by a University of Nottingham studentship. EJC is also
supported by a Leverhulme Research Fellowship reference: RF-2021-312. AM is
also supported by a Royal Society University Research Fellowship.
For the purpose of open access, the authors have applied a CC BY public
copyright licence to any Author Accepted Manuscript version arising.
Data Availability Statement: The code and data produced in this paper are
available upon reasonable request to AM.

## References

  * (1) Planck collaboration, _Planck 2018 results. X. Constraints on inflation_ , _Astron. Astrophys._ 641 (2020) A10 [1807.06211].
  * (2) D. Baumann, _Inflation_ , in _Theoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small_ , pp. 523–686, 2011, DOI [0907.5424].
  * (3) S. Weinberg, _Cosmology_ (2008).
  * (4) A.D. Linde, _Chaotic Inflation_ , _Phys. Lett. B_ 129 (1983) 177.
  * (5) K. Freese, J.A. Frieman and A.V. Olinto, _Natural inflation with pseudo - Nambu-Goldstone bosons_ , _Phys. Rev. Lett._ 65 (1990) 3233.
  * (6) E. Silverstein and A. Westphal, _Monodromy in the CMB: Gravity Waves and String Inflation_ , _Phys. Rev. D_ 78 (2008) 106003 [0803.3085].
  * (7) L. McAllister, E. Silverstein and A. Westphal, _Gravity Waves and Linear Inflation from Axion Monodromy_ , _Phys. Rev. D_ 82 (2010) 046003 [0808.0706].
  * (8) G. Dvali, _Large hierarchies from attractor vacua_ , _Phys. Rev. D_ 74 (2006) 025018 [hep-th/0410286].
  * (9) G. Dvali, _Three-form gauging of axion symmetries and gravity_ , hep-th/0507215.
  * (10) N. Kaloper and L. Sorbo, _Where in the String Landscape is Quintessence_ , _Phys. Rev. D_ 79 (2009) 043528 [0810.5346].
  * (11) N. Kaloper and L. Sorbo, _A Natural Framework for Chaotic Inflation_ , _Phys. Rev. Lett._ 102 (2009) 121301 [0811.1989].
  * (12) N. Kaloper, A. Lawrence and L. Sorbo, _An Ignoble Approach to Large Field Inflation_ , _JCAP_ 03 (2011) 023 [1101.0026].
  * (13) N. Kaloper and A. Lawrence, _London equation for monodromy inflation_ , _Phys. Rev. D_ 95 (2017) 063526 [1607.06105].
  * (14) G. D’Amico, N. Kaloper and A. Lawrence, _Monodromy Inflation in the Strong Coupling Regime of the Effective Field Theory_ , _Phys. Rev. Lett._ 121 (2018) 091301 [1709.07014].
  * (15) I. Valenzuela, _Backreaction Issues in Axion Monodromy and Minkowski 4-forms_ , _JHEP_ 06 (2017) 098 [1611.00394].
  * (16) N. Kaloper and A. Padilla, _Sequestering the Standard Model Vacuum Energy_ , _Phys. Rev. Lett._ 112 (2014) 091304 [1309.6562].
  * (17) N. Kaloper and A. Padilla, _Vacuum Energy Sequestering: The Framework and Its Cosmological Consequences_ , _Phys. Rev. D_ 90 (2014) 084023 [1406.0711].
  * (18) N. Kaloper and A. Padilla, _Sequestration of Vacuum Energy and the End of the Universe_ , _Phys. Rev. Lett._ 114 (2015) 101302 [1409.7073].
  * (19) N. Kaloper, A. Padilla, D. Stefanyszyn and G. Zahariade, _Manifestly Local Theory of Vacuum Energy Sequestering_ , _Phys. Rev. Lett._ 116 (2016) 051302 [1505.01492].
  * (20) N. Kaloper, A. Padilla and D. Stefanyszyn, _Sequestering effects on and of vacuum decay_ , _Phys. Rev. D_ 94 (2016) 025022 [1604.04000].
  * (21) N. Kaloper and A. Padilla, _Vacuum Energy Sequestering and Graviton Loops_ , _Phys. Rev. Lett._ 118 (2017) 061303 [1606.04958].
  * (22) G. D’Amico, N. Kaloper, A. Padilla, D. Stefanyszyn, A. Westphal and G. Zahariade, _An étude on global vacuum energy sequester_ , _JHEP_ 09 (2017) 074 [1705.08950].
  * (23) F. Niedermann and A. Padilla, _Gravitational Mechanisms to Self-Tune the Cosmological Constant: Obstructions and Ways Forward_ , _Phys. Rev. Lett._ 119 (2017) 251306 [1706.04778].
  * (24) B. Coltman, Y. Li and A. Padilla, _Cosmological consequences of Omnia Sequestra_ , _JCAP_ 06 (2019) 017 [1903.02829].
  * (25) A. Padilla, _Lectures on the Cosmological Constant Problem_ , 1502.05296.
  * (26) A. Padilla, _Monodromy inflation and an emergent mechanism for stabilising the cosmological constant_ , _JHEP_ 01 (2019) 175 [1806.04740].
  * (27) A.A. Tseytlin, _Duality symmetric string theory and the cosmological constant problem_ , _Phys. Rev. Lett._ 66 (1991) 545.
  * (28) A. Adams, J. McGreevy and E. Silverstein, _Decapitating tadpoles_ , hep-th/0209226.
  * (29) F. Niedermann and A. Padilla, _Gravitational Mechanisms to Self-Tune the Cosmological Constant: Obstructions and Ways Forward_ , _Phys. Rev. Lett._ 119 (2017) 251306 [1706.04778].
  * (30) N. Kaloper, A. Lawrence and L. Sorbo, _An Ignoble Approach to Large Field Inflation_ , _JCAP_ 03 (2011) 023 [1101.0026].
  * (31) A. Manohar and H. Georgi, _Chiral Quarks and the Nonrelativistic Quark Model_ , _Nucl. Phys. B_ 234 (1984) 189.
  * (32) B.M. Gavela, E.E. Jenkins, A.V. Manohar and L. Merlo, _Analysis of General Power Counting Rules in Effective Field Theory_ , _Eur. Phys. J. C_ 76 (2016) 485 [1601.07551].
  * (33) B.K. El-Menoufi, S. Nagy, F. Niedermann and A. Padilla, _Quantum corrections to vacuum energy sequestering (with monodromy)_ , _Class. Quant. Grav._ 36 (2019) 215014 [1903.07612].
  * (34) Planck collaboration, _Planck 2018 results. X. Constraints on inflation_ , _Astron. Astrophys._ 641 (2020) A10 [1807.06211].
  * (35) BICEP2, Keck Array collaboration, _BICEP2 / Keck Array x: Constraints on Primordial Gravitational Waves using Planck, WMAP, and New BICEP2/Keck Observations through the 2015 Season_ , _Phys. Rev. Lett._ 121 (2018) 221301 [1810.05216].
  * (36) D. Foreman-Mackey, D.W. Hogg, D. Lang and J. Goodman, _emcee: The MCMC Hammer_ , _Publ. Astron. Soc. Pac._ 125 (2013) 306 [1202.3665].
  * (37) J. Goodman and J. Weare, _Ensemble samplers with affine invariance_ , _Communications in Applied Mathematics and Computational Science_ 5 (2010) 65.
  * (38) J.S. Speagle, _dynesty: a dynamic nested sampling package for estimating Bayesian posteriors and evidences_ , _Mon. Not. Roy. Astron. Soc._ 493 (2020) 3132 [1904.02180].