# Dilatonic Couplings and the Relic Abundance of Ultralight Dark Matter

Ahmad Alachkar  Theoretical Particle Physics and Cosmology, King’s College
London, Strand, London, WC2R 2LS, United Kingdom  Malcolm Fairbairn
Theoretical Particle Physics and Cosmology, King’s College London, Strand,
London, WC2R 2LS, United Kingdom  David J. E. Marsh  Theoretical Particle
Physics and Cosmology, King’s College London, Strand, London, WC2R 2LS, United
Kingdom

###### Abstract

Models of scalar field dark matter where the scalar is a dilaton have a
special behaviour, since non-trivial couplings, $d$, to matter result in a
contribution to the potential for the field which is proportional to the trace
of the stress-energy tensor. We look in more detail at the dilaton mass,
$m_{\phi}$, and initial conditions required to yield the correct relic
abundance for couplings that are not already excluded by terrestrial
experiments. In minimal models with only couplings accessible to terrestrial
searches, we find that dilaton dark matter with $m_{\phi}\gtrsim 10^{-10}$ eV
requires couplings suppressed compared to constraints from equivalence
principle (EP) tests and fifth force searches in order to not produce too much
dark matter, improving on the strongest current experimental constraints by up
to $\sim{\cal O}(10)$, with consequences for the proposed mechanical resonator
dilaton DM searches. In non-minimal or universally coupled models, the
unconstrained couplings of the dilaton to e.g. the top quark can strongly
influence the relic abundance at all masses. In particular, this implies that
atom interferometry searches at masses $m_{\phi}\approx 10^{-19}\text{ eV}$
are unable to constrain the early Universe behaviour or UV physics of the
dilaton. We also find that dilatonic couplings allow for compatibility of
$m_{\phi}\gtrsim 10^{-7}\text{ eV}$ with an observably large tensor-to-scalar
ratio in the cosmic microwave background, which is not possible for a
decoupled scalar of the same mass.

Einstein’s theory of general relativity is extremely successful
will2014confrontation ; reynaud2008tests ; GRtest1 ; GRtest2 , as is the
Standard Model (SM) of particle physics odom2006new ; aad2012observation . The
combination of the two theories leads to the prediction of a hot big bang with
primordial nucleosynthesis Steigman:2007xt  and Cosmic Microwave Background
(CMB) radiation spergel2003first ; aghanim2020planck ; akrami2020planck  that
are observed to have occurred in our Universe. Despite these huge successes,
multiple independent observations have also indicated the presence of some so
far undetected dark matter (DM) Marsh:2024ury  which is not part of the
Standard Model of particle physics. Furthermore, the ultraviolet (UV)
completion of both gravity and quantum field theory is still a subject of much
debate. One leading candidate is string/M-theory. Since string theory contains
no dimensionful constants, and requires dimensional reduction to describe our
3+1 dimensional world, the low energy effective field theory (EFT) generically
contains a plethora of scalar (and pseudoscalar) degrees of freedom. If these
degrees of freedom are light, then they provide candidates for DM, and
possible IR modifications of gravity. The possible existence of such particles
is the subject of intense experimental study at present 2168507 ;
arvanitaki2015searching ; Adams:2022pbo .

General Relativity is obtained by varying the Einstein Hilbert action plus a
matter action with canonical kinetic terms. This is referred to as the case of
minimal coupling between matter and gravity. Theories of gravity with non-
minimal couplings between matter and curvature, and non-minimal matter kinetic
terms, have been explored extensively over the past decades as possible
explanations of, or being related to, dark matter arvanitaki2015searching ;
banerjee2023phenomenology ; brzeminski2021time , dark energy brax2010dilaton ;
khoury2004chameleon  and the UV completion of QFT and gravity narain1989new ;
taylor1988dilaton .

One example of such investigations is the idea of dilatonic dark matter (DDM),
where the non-minimal coupling between gravity and the SM is given by a
dynamical scalar field, the oscillation of which can provide stress energy
compatible with the equation of state of DM. The DDM relic abundance can be
provided by a process similar to vacuum realignment of axion DM
preskill1983cosmology ; abbott1983cosmological ; dine1983not , which is
specified by the potential of the dilaton, and the initial conditions of the
field. (See hubisz2024note  for a critique of the generation of cosmologically
relevant dilatonic dark matter via such mechanisms.)

As with any successful and phenomenologically interesting theory of DM, a
connection between the relic abundance and the couplings constrained in the
laboratory is desirable if we are to use observations to learn about the
underlying fundamental theory of DM, and its connection to early Universe and
UV physics. In the following we will show that, on the one hand at very low
dilaton masses $m_{\phi}\approx 10^{-19}\text{ eV}$ there are significant
unknowns associated with making such a connection which prevent one from
linking direct detection to early Universe physics and the initial conditions
of the dilaton in the UV. On the other hand, however, at larger dilaton masses
$m_{\phi}\approx 10^{-7}\text{ eV}$, the unique physics of the dilaton can
allow compatibility between DDM and certain models of inflation, which is not
possible for generic scalar DM of the same mass.

The low energy EFT of DDM can be modelled by a generalised scalar-tensor
theory defined by the following action

$\displaystyle S(g_{\mu\nu},\varphi,\Psi_{i})$
$\displaystyle=\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}\Big{[}\varphi
R-\frac{\omega}{\varphi}(\nabla\varphi)^{2}$ (1)
$\displaystyle-U(\varphi)\Big{]}+S_{m}(g_{\mu\nu},\varphi,\Psi_{i}),$

where $R$ is the Ricci scalar constructed from the metric $g_{\mu\nu}$ with
determinant $g$, $\omega=-1$ is a dimensionless coupling constant called the
dilatonic Brans-Dicke parameter, $\kappa\equiv\sqrt{8\pi G}$ with $G$ the bare
gravitational constant (i.e. the gravitational constant in the absence of the
scalar interaction), $U(\varphi)$ is the scalar field potential defined in
this conformal frame, referred to hereinafter as the matter frame, $S_{m}$ is
the matter action and $\Psi_{i}$ denotes the matter fields in the SM. An
equivalent formulation of the theory (modulo boundary terms) is given in the
Einstein frame (here and throughout this work, Einstein frame quantities are
labelled with a tilde), where the kinetic term for the metric is of the
Einstein-Hilbert form, via a Weyl rescaling
$g_{\mu\nu}\longrightarrow\tilde{g}_{\mu\nu}=\Omega^{2}(x){g}_{\mu\nu}$, with
$\Omega^{2}\equiv F(\varphi)=\varphi\equiv\kappa\phi$, and a redefinition
$\tilde{\phi}=\sqrt{\frac{|2\omega+3|}{2\kappa^{2}}}\ln\left(\frac{\phi}{\phi_{0}}\right)$
such that the field is canonically normalised. $\phi_{0}$ is chosen to be
$\kappa^{-1}$ so that the scalar field gives the correct value of $G$ today.

The effective low-energy, matter-gravity-dilaton action in the Einstein frame
is then

$\displaystyle S(\tilde{g}_{\mu\nu},\tilde{\phi},\Psi_{i})$
$\displaystyle=\int
d^{4}x\sqrt{-\tilde{g}}\Big{[}\frac{1}{2\kappa^{2}}\tilde{R}-\frac{1}{2}\tilde{g}^{\mu\nu}\nabla_{\mu}\tilde{\phi}\nabla_{\nu}\tilde{\phi}$
(2)
$\displaystyle-V(\tilde{\phi})\Big{]}+S_{m}(\Omega^{-2}\tilde{g}_{\mu\nu},\tilde{\phi},\Psi_{i}),$

where $V(\tilde{\phi})=U(\varphi)/(2\kappa^{2}\varphi^{2})$.

Figure 1: Bottom: the effective potential as a function of expectation value
for the field for DDM with mass $m_{\phi}=10^{-19}$ eV, where the couplings
sensitive to current EP and fifth force constraints are set at their maximum
permissible values, for different values of temperature. The dotted line is
the purely harmonic potential. Top left: the phase portrait for different
initial field values that give rise to an attractor solution, corresponding to
the linear regime in the effective potential. Top right: The cosmological
evolution of the dilaton, for the same model parameters, as function of
temperature (right), for a range of initial field values (see Fig. 4), showing
the kicks due to the interaction with SM species.
$\tilde{\phi}^{\mathrm{DM}}_{\mathrm{in}}$ is the initial field value that
gives the measured DM density today.

_Dilaton evolution and relic abundance:_ We solve the dilaton evolution in the
Einstein frame effective potential using the Friedmann-Robertson-Walker (FRW)
background during the radiation dominated era, with scale factor $\tilde{a}$
and Hubble parameter $\tilde{H}=\dot{\tilde{a}}/\tilde{a}$. When the dilaton
mass is smaller than the Hubble scale during inflation, this leads to a
uniform initial field value, $\tilde{\phi}_{\mathrm{in}}$, and
$d\tilde{\phi}/d\tilde{t}|_{\tilde{t}=\tilde{t}_{\mathrm{in}}}\approx 0$. The
effective potential and evolution of the dilaton is illustrated in Fig. 1.

The dynamics of the dilaton are governed by the Klein-Gordon equation,
$\tilde{\Box}\tilde{\phi}=-\partial
V/\partial\tilde{\phi}-\beta(\tilde{\phi})\tilde{T}^{\mu}_{~{}\mu}-\tilde{\sigma}$,
where we define the Einstein frame scalar charge density of the sources
$\tilde{\sigma}$ by the variation with respect to $\tilde{\phi}$ as
$\tilde{\sigma}\equiv\frac{1}{\sqrt{-\tilde{g}}}\frac{\delta
S_{m}}{\delta\tilde{\phi}}$ hees2018violation ; campbell2021searching , and
$\tilde{\Box}\equiv\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}$
is the d’Alembert operator for the covariant derivative
$\tilde{\nabla}_{\mu}$, with Levi-Civita connection associated with the
rescaled metric. The parameter $\beta(\tilde{\phi})\equiv\frac{1}{2}d\ln
F(\phi)/d\tilde{\phi}=\sqrt{\kappa^{2}/(2(2\omega+3))}$ is a universal
coupling that characterizes the departure of this generalised scalar-tensor
theory from GR, in the absence of $\tilde{\sigma}$. The Einstein frame stress-
energy tensor is given by
$\tilde{T}_{\mu\nu}\equiv(-2/\sqrt{-\tilde{g}})\delta
S_{m}/\delta\tilde{g}^{\mu\nu}$, which sources the Friedmann equation:

$3\tilde{H}^{2}=\kappa^{2}\left[\tilde{\rho}+\frac{1}{2}\Big{(}\frac{d\tilde{\phi}}{d\tilde{t}}\Big{)}^{2}+V(\tilde{\phi})\right],$
(3)

and the associated (redundant) acceleration equation for
$d\tilde{H}/d\tilde{t}$. We drop the dilaton contribution to the right hand
side when computing the relic abundance (DM should be sub-dominant during the
radiation era), but note that the dilaton affects $\tilde{\rho}$ implicitly
via the matter frame temperature $T$.

In flat FRW the dilaton field equation is:

$\frac{d^{2}\tilde{\phi}}{d\tilde{t}^{2}}+3\tilde{H}\frac{d\tilde{\phi}}{d\tilde{t}}+\frac{dV}{d\tilde{\phi}}=\beta(\tilde{\phi})\sum_{i}\Big{(}-\tilde{\rho}_{i}+3\tilde{p}_{i}\Big{)}-\tilde{\sigma},$
(4)

where the index $i$ spans the various Standard Model components of the
Universe (hadrons, radiation, etc.) with corresponding densities $\rho_{i}$
and pressures $p_{i}$. The parameter $\tilde{\sigma}$ is given by

$\begin{split}\tilde{\sigma}&=\left(\kappa^{2}\phi_{0}^{2}\exp\left(2\sqrt{\frac{2\kappa^{2}}{2\omega+3}}\tilde{\phi}\right)\right)^{-1}\Bigg{[}\kappa
d_{m_{e}}(\rho_{e}-3p_{e})\\\ &+\kappa d_{m_{u}}(\rho_{u}-3p_{u})+\kappa
d_{m_{d}}(\rho_{d}-3p_{d})+\kappa
d_{e}\frac{\pi^{2}}{3}\frac{\alpha}{4\pi}T^{4}\Bigg{]}.\end{split}$ (5)

We include the contribution to the induced dilaton mass from thermal field
theory bouley2023constraints , where the leading diagram appears at two loops.

We consider a simple harmonic potential to stabilise the dilaton given by
$V(\tilde{\phi})=m_{\phi}^{2}\tilde{\phi}^{2}/2$, and determine the dynamics
by numerically solving the coupled system of equations comprising Eq. (3) and
(4). This system of equations is closed when we specify how the matter-frame
temperature $T$ depends on $\tilde{a}$ and $\tilde{\phi}$, following the
prescription of erickcek2014chameleons . We evaluate the number of degrees of
freedom, $g_{\ast S}(T)$ for the SM particle spectrum, obtained from
saikawa2018primordial , treating the QCD phase transition as instantaneous at
a temperature of $T_{\mathrm{QCD}}=150$ MeV.

At early times deep in the radiation era, $\tilde{H}^{2}\gg
m^{2}_{\mathrm{eff}}$, where $m_{\text{eff}}$ is the effective dilaton mass
given by the total potential, damping any initial velocity possessed by the
field. We compute the evolution of $\tilde{\phi}$ for a range of masses
$m_{\phi}$, initial conditions $\tilde{\phi}_{\mathrm{in}}$ and coupling
strengths $d_{i}$ and determine the relic abundance using a WKB approximation
to redshift to the present once oscillations are within the harmonic regime.
The evolution of the dilaton can generally be split into three regimes
depending on which of the following three terms dominates Eq. (4): the Hubble
friction term $\tilde{H}$, the bare mass $m_{\phi}$ or the induced mass due to
$\beta(\tilde{\phi})\tilde{T}^{\mu}_{~{}\mu}+\tilde{\sigma}$.

The effective potential, $V_{\mathrm{eff}}$, governing the dynamics, plotted
in Fig 1 for $m_{\phi}=10^{-19}$ eV with SM couplings set to existing EP and
fifth force constraints (see below for definition), consists of the zero
temperature quadratic term (bare mass) as well as a temperature-dependent
linear term due to the trace of stress-energy tensor (induced potential),
exponentially suppressed by
$2\sqrt{\frac{2\kappa^{2}}{2\omega+3}}\tilde{\phi}$ due to the conformal
mapping of physical quantities in the matter frame. The field value of
$\tilde{\phi}^{0}_{\mathrm{in}}=1/(2\sqrt{\frac{2\kappa^{2}}{2\omega+3}}\tilde{\phi})\sim
9\times 10^{17}$ GeV separates two regimes for fixed $T\gtrsim 5\times 10^{4}$
eV. At $\tilde{\phi}_{\mathrm{in}}>\tilde{\phi}^{0}_{\mathrm{in}}$, the field
rolls down $V_{\mathrm{eff}}$ towards larger values initially before the bare
potential dominates, where $\Delta\tilde{\phi}<0$, and $\tilde{\phi}$
henceforth exhibits the standard behaviour of a damped (by $\tilde{H}$)
harmonic oscillator. For
$\tilde{\phi}_{\mathrm{in}}\leq\tilde{\phi}^{0}_{\mathrm{in}}$, we have
$\Delta\tilde{\phi}<0$ throughout the evolution, along the constant-slope
induced potential $\propto\ln F(\phi(\tilde{\phi}))$. Now fixing
$\tilde{\phi}_{\mathrm{in}}$, for $T<5\times 10^{4}$ eV , the field excursion
is always negative, towards $\tilde{\phi}_{\mathrm{min}}$ which coincides with
the bare potential minimum at low enough $T$ (late times).

For a radiation fluid, the trace of the stress- energy tensor is zero during
most of the early Universe. However, as the radiation bath cools, whenever
$T\sim m_{i}$ for a SM species with rest mass $m_{i}$ there is a non-zero
contribution for about one e-fold of expansion as the particle $i$ becomes
non-relativistic and its pressure decreases at a faster rate than its energy
density. This generates a ‘kick’ in the forcing term imparting a significant
velocity to the dilaton- an effect first studied in damour1993tensor ;
damour1994string , in the context of scalar-tensor models. For
$\tilde{\phi}_{\mathrm{in}}\leq\tilde{\phi}^{0}_{\mathrm{in}}$ these kicks,
along with the linear part of the potential, give rise to an attractor
solution. This is demonstrated in the phase portrait shown in top left Fig. 1.
The kicks are visible in the field evolution shown in the top right of Fig. 1.

_Minimal Fifth-Force Consistent Couplings._ We have seen above how the
couplings between the dilaton and SM fields modify the effective potential of
the dilaton, leading to attractor behaviour in the evolution of the field.
Direct detection of DDM relies on these same couplings, and so it is natural
to ask to what level the dilaton evolution is affected by observably large
values for the couplings. We consider the following couplings to the light
fields of the SM (at leading order), which are those probed, e.g., by atom
interferometers:

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{\mathrm{SM}}+\mathcal{L}_{\mathrm{int}\phi}$ (6)
$\displaystyle=\mathcal{L}_{\mathrm{SM}}+\kappa\tilde{\phi}\Big{[}+\frac{d_{e}}{4e^{2}}F_{\mu\nu}F^{\mu\nu}-\frac{d_{g}\beta_{3}}{2g_{3}}F_{\mu\nu}^{A}F^{A\mu\nu}$
$\displaystyle-\sum_{i=e,u,d}\left(d_{m_{i}}+\gamma_{m_{i}}d_{g}\right)m_{i}\bar{\psi}_{i}\psi_{i}\Big{]},$

introducing linear interaction terms to the effective Lagrangian describing
the physics of ground state nuclei, $\mathcal{L}_{\mathrm{SM}}$. $F_{\mu\nu}$
is the electromagnetic Faraday tensor, $F_{\mu\nu}^{A}$ is the gluon field
strength tensor, $\beta_{3}$ is the $\beta$-function for the running of
$g_{3}$, $m_{i}$ is the mass of the fermions, $\gamma_{m_{i}}$ is the
anomalous dimension giving the energy running of the masses of the quarks, and
$\psi_{i}$ denotes the Weyl fermion spinors. A $\tilde{\phi}$-dependent
coupling to the kinetic term of the fermion,
$f(\tilde{\phi})\bar{\psi_{i}}i\not{D}\psi_{i}$, is omitted as it can be
absorbed in a suitable $\tilde{\phi}$–dependent rescaling of $\psi_{i}$.

At low energies of $\lesssim 1$ GeV, one has integrated out the effect of weak
interactions, and the heavy quarks $c$, $b$ and $t$ (see damour2010equivalence
for a discussion on the issue of $\tilde{\phi}$ sensitivity of effects linked
to the strange quark). This leaves us with the electron $e$, the $u$ quark,
and the $d$ quark, with interactions mediated by the electromagnetic
($A_{\mu}$) and gluonic ($A^{A}_{\mu}$) fields.

Up to linear order, each of the five terms in $\mathcal{L}_{\mathrm{SM}}$ can
couple to $\kappa\tilde{\phi}$ via a dimensionless coefficient, $d_{i}$. Our
parameterisation of the couplings $d_{i}$ is relative to gravity
damour2010equivalence , so $d_{i}=1$ would be the universal couplings of a
scalar graviton. This normalisation implies that the physical meaning of the
five dilaton-coupling coefficients ${d_{i}}$ is to introduce a
$\tilde{\phi}$-dependence (and hence temporal variation) in the parameters
entering the low-energy physics of the form
$X(\tilde{\phi})=\left(1+d_{i}\kappa\tilde{\phi}\right)X$, where
$X\in(\Lambda_{3},\alpha,m_{j})$, corresponding to $i\in(g,e,j)$ in that
order, for $j=e,u,d$, $\alpha$ denotes the fine structure constant and
$\Lambda_{3}$ is the QCD mass scale.

Figure 2: Relic abundance vs initial value of scalar for masses at the two
opposite ends of the range investigated. Solid lines show the case of the
dilaton with minimal EP and fifth-force consistent couplings, while dotted
lines show the case of a purely harmonic potential. The grey region would be
excluded for hypothetical detection of tensor-to-scalar ratio at the current
upper limit $r=0.032$. Figure 3: Maximum coupling strength $d_{e}$
corresponding to the correct DM relic abundance vs. dilaton mass $m_{\phi}$,
with the other three couplings of the minimal model set at the maximum
permissible values $d_{5th}$, given the existing EP and fifth force
constraints. The dotted line is an extrapolation to higher masses, which are
not accessible by our simulations. Overlaid are current experimental
constraints on $d_{e}$ given by the E$\ddot{o}$t-Wash equivalence principle
test. The solid circles are projections for the minimum detectable coupling
for three compact mechanical resonators proposed as candidate detectors for
scalar dark matter manley2020searching : a superfluid helium bar resonator
(light blue) de2017ultra , a cylindrical HEM® sapphire test mass
rowan2000investigation  (green) and a sapphire micropillar (dark blue)
resonator Neuhaus . The orange region is natural for coupling to the fine
structure constant with a 10 TeV cutoff manley2020searching ;
arvanitaki2016sound .

For a minimal scenario where only the above couplings are present, not just in
the IR but at all scales, we fix the couplings to have the largest values
allowed by EP and fifth force constraints wagner2012torsion ;
schlamminger2008test ; smith1999short ; berge2018microscope ;
banerjee2023phenomenology ; bouley2023constraints , and switch off couplings
to all other SM species. We then calculate the dilaton relic abundance as a
function of initial field value $\tilde{\phi}_{\mathrm{in}}$, for a range of
masses $m_{\phi}\in\left[10^{-22},10^{-7}\right]$ eV, which we illustrate with
two examples in Fig. 2.

We find that lower masses (in the range $10^{-22}-10^{-12}$ eV) give solutions
for the relic abundance for $\tilde{\phi}_{\mathrm{in}}$ that are very close
to the case of a quadratic potential (zero dilatonic couplings), whereas
higher masses give solutions that deviate strongly from this. This is because
for the relatively low values of $\tilde{\phi}_{\mathrm{in}}$ required at high
mass, the relic density stays constant and the system’s behaviour is
relatively invariant over a range of initial conditions due to the scalar
field attractor, leading to the relic abundance fixed only by $m_{\phi}$
(holding couplings fixed). At the large initial field values required at low
dilaton mass, the induced potential is sub-dominant to the bare potential and
$\Omega_{\phi}h^{2}\propto\tilde{\phi}_{\mathrm{in}}^{2}$, giving solutions
that align with that of a purely quadratic potential. Thus the early Universe
behaviour of minimal DDM at low mass is insensitive to the exact values of the
couplings in the minimal model, once they are below EP and fifth force
constraints.

On the other hand, DDM at larger masses $m_{\phi}\gtrsim 10^{-10}\text{ eV}$
requires couplings suppressed with respect to their EP and fifth force values
in order to give a consistent relic abundance, due to small field values and
the dominance of the attractor. This is illustrated in more detail in Fig. 3,
where we compute the maximum allowed value of $d_{e}$ (the coupling to the
fine structure constant, which has the dominant effect on the cosmological
evolution) required in order to not produce too much DM. At the largest mass
we have considered, $m_{\phi}=10^{-7}\text{ eV}$, $d_{e}$ should be suppressed
by a factor of 10 compared to the EP constraints. Extrapolating our numerical
results to higher dilaton masses $m_{\phi}\gtrsim 10^{-6}\text{ eV}$ the
constraint from DM overproduction reaches the ‘natural’ values of couplings
expected with 10 TeV UV cut-off manley2020searching ; arvanitaki2016sound ,
which is a regime defined by the criterion that loop corrections to the
dilaton mass are less than the physical mass $m_{\phi}$
dimopoulos1996macroscopic .

_The Case of Unconstrained Couplings:_ We now consider the effect of the
coupling of the dilaton to other SM species that are not constrained
experimentally, for example to the top quark. This is shown for
$m_{\phi}=10^{-19}$ eV in Fig. 4, where in addition to the minimal model
couplings at EP and fifth force constraints level, we add a coupling
$d_{\mathrm{top}}$ at 0.1 and 1, or a universal coupling to all SM species. If
such an unconstrained $d$ is switched on, we find different solutions for the
relic abundance $\Omega_{\phi}h^{2}$ as a function of the initial conditions
$\tilde{\phi}_{\mathrm{in}}$ and mass $m_{\phi}$. We thus conclude that,
without a strong reason to exclude dilaton couplings to the heavy SM fields,
it is generally impossible to make inferences about the early Universe
behaviour and UV physics of DDM based on results informed only by existing
experiments.

Figure 4: Four different scenarios all with DDM with mass $m_{\phi}=10^{-19}$
eV where all the couplings sensitive to current EP and fifth force constraints
are set at their maximum permissible values, denoted by $d_{5th}$. The dotted
line again denotes the vanilla model with completely minimal coupling to
gravity (no dilatonic couplings to SM), corresponding to a purely harmonic
potential. Any slight variation in the unconstrained coupling to top quarks or
indeed a universal coupling of unity to all SM particles not probed by EP or
fifth force experiments greatly change the value of
$\tilde{\phi}_{\mathrm{in}}$ to obtain good relic abundance
($i=u,q,e,\gamma$).

_Relation to models of inflation:_ Given the precise measurement of the scalar
amplitude of the CMB aghanim2020planck , the value of the tensor-to-scalar
ratio, $r$, determines the Hubble parameter during inflation,
$H_{\mathrm{\ast}}$, as $H_{\ast}=8\times 10^{13}\sqrt{r/0.1}$ GeV. The
existing upper limit on $r$ is $r<0.032$ at $95\%$ confidence
tristram2022improved ; ade2021improved ; collaboration2020planck .

As a spectator field, the expectation value of the DDM will be excited during
inflation, carrying out a random walk with rms amplitude
$\Theta\simeq\sqrt{N}H_{\ast}/(2\pi)$ enqvist2012spectator , where $N$ is the
number of efolds, which we fix to 60. A higher energy scale during inflation
will result in a greater expectation value for $\tilde{\phi}_{\mathrm{in}}$,
ruling out scenarios where relic abundance requires lower
$\tilde{\phi}_{\mathrm{in}}$. The grey region on Fig. 2 represents the region
of parameter space which would be ruled out if $r=0.032$, i.e. measured near
the current upper limit. We observe that scalar DM with no dilatonic couplings
(dashed line) and $m_{\phi}>10^{-7}\text{ eV}$ would produce too much relic
abundance and thus cannot compose the DM in this case (note that a vanilla
scalar field dark matter model with completely minimal coupling to gravity
would be also in tension with a value of $r=0.032$, c.f. Ref. Marsh:2014qoa ).
The introduction of dilatonic couplings can tune the location and size of the
minimum in the relic abundance at larger $\tilde{\phi}_{\mathrm{in}}$,
potentially leading to a scenario where scalar DM with $m_{\phi}>10^{-7}\text{
eV}$ is consistent with high scale inflation.

_Conclusion:_ Our study of the DDM relic abundance has two main implications
for DM direct detection at relatively high and relatively low dilaton mass.
Firstly, at low masses, probed for example by AION badurina2020aion , the
existence of unconstrained couplings to unstable higher mass particles which
do not contribute to EP limits prevents a completely final conclusion being
drawn on the order of magnitude of $\tilde{\phi}_{\mathrm{in}}$ based on a
detection of DDM with fixed mass. This means, in contrast to the case of the
axion for example, it is slightly more complicated to connect DDM direct
detection to initial conditions in the Early Universe. At high masses, the DDM
relic abundance is affected by the dilatonic couplings and the energy scale of
inflation. Simultaneous measurement of scalar DM and a large tensor-to-scalar
ratio in the CMB, corresponding to high scale inflation, is inconsistent if
the scalar DM has no interactions beyond $m_{\phi}^{2}\phi^{2}$, but can be
made consistent by introducing dilatonic couplings. In the happy event of this
dual detection, it is possible to make inferences about the presently
unconstrained dilatonic couplings necessary for consistency. Lastly, we showed
how the coupling to the fine structrue constant can be constrained by the
requirement that DM is not overabundant, improving on existing limits by up to
an order of magnitude, probing natural values of the coupling, and with
consequences for mechanical resonator DDM searches manley2020searching .

## Acknowledgements

The work of AA was supported by a UK STFC studentship, that of MF was
supported in part by STFC Grants ST/P000258/1 and ST/T000759/1. The work of
DJEM is supported by STFC grant ST/T004037/1. We are grateful to Edmund
Copeland for discussions and correspondence, and to Angelo Maggi for
collaboration at an early stage of this work.

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