# Two-field Screening and its Cosmological Dynamics

Philippe Brax  Institut de Physique Théorique, Université Paris-Saclay,CEA,
CNRS, F-91191 Gif-sur-Yvette Cedex, France.  CERN, Theoretical Physics
Department, Geneva, Switzerland.  Ayoub Ouazzani  Institut de Physique
Théorique, Université Paris-Saclay,CEA, CNRS, F-91191 Gif-sur-Yvette Cedex,
France.  Département de Physique de l’Ecole Normale Supérieure, Paris.

###### Abstract

We consider the screening of the axio-dilaton fields when both the dilaton and
the axion couple to matter with Yukawa couplings. We analyse the screening of
the dilaton in the vicinity of a compact object and find that this can only
take place when special boundary conditions at infinity are imposed. We study
the cosmological dynamics of the axio-dilaton system when coupled to matter
linearly and find that the special boundary conditions at infinity, which
guarantee the screening of compact objects, do not generically emerge from
cosmology. We analyse the background cosmology and the cosmological
perturbations at late time in these models and show that the Baryon Acoustic
Oscillations constrain the coupling of the dilaton to matter to be smaller
than in its natural supergravity realisation. Moreover we find that the Hubble
rate in the present Universe could deviate from the normalised Planck value,
although by an amount too small to account for the $H_{0}$ tension, and that
the growth of structure is generically reduced compared to $\Lambda$CDM.

##  I Introduction

General Relativity (GR) passes all the current gravitational tests in the
solar system with flying coloursWill (2014). Yet there are still tensions on
larger scales which may eventually necessitate to challenge Einstein’s theory
of gravity. These are related to two classes of observations. First, there are
long standing astrophysical results that imply the existence of dark matter
Feng (2010). Second, there is now robust evidence in favour of the
acceleration of the expansion of the Universe Aghanim  _et al._ (2020);
Copeland  _et al._ (2006). The latter is best accounted for in GR by adding a
cosmological constant to the Lagrangian of the theory. However, understanding
the origin of this constant energy density, e.g. from quantum field theoretic
considerations, has proven to be fraught with difficulties Weinberg (1989).
Modifying GR is not an easy task either de Rham  _et al._ (2011). We know from
Lovelock’s theorem Lovelock (1972) and, in the effective field theory context,
from Weinberg’s theorem Weinberg (1965) that GR is unique in four dimensions
provided Lorentz invariance and the masslessness of the graviton hold Clifton
_et al._ (2012).

One way to modify GR is to add additional fields. This is the route taken by
scalar-tensor theories Damour and Esposito-Farese (1992) in which one or more
additional scalar fields are added to the setting and couple to matter. Such
scalar fields appear in many proposed dynamical models attempting to account
for the late cosmic acceleration Brax (2017). They also find theoretical
motivation in the fact that they appear naturally in UV-complete theories such
as string theory Damour  _et al._ (2002); Burgess and Quevedo (2022). In
particular, the swampland conjectures favour an explanation of the late-time
acceleration of the Universe resulting from the non-trivial dynamics of string
moduli Vafa (2005). We will present the axio-dilaton which is one such scalar-
tensor theories whose origin can be traced to a compactification of ten to
four dimensions Burgess and Quevedo (2022).

Unfortunately, most scalar-tensor theories are typically ruled out by solar
system tests of gravity. In their most naive form, consistency of scalar-
tensor theories either requires them to have a small coupling to matter or the
scalar must be stabilised with large masses leading to short ranged
interactions Brax  _et al._ (2022a).

This can be illustrated with the Brans-Dicke theory (BD) and the action Will
(2014); Avilez and Skordis (2014)

$\displaystyle\mathcal{S}_{\mathrm{BD}}=\int
d^{4}x\sqrt{-g}M_{p}^{2}\left(\frac{\mathcal{R}}{2}-\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi-V(\varphi)\right)$
$\displaystyle+\mathcal{S}_{m}\left({g}^{J}_{\mu\nu},\,\psi_{m}\right)$ (1)

where $\psi_{m}$ represents the ordinary matter fields and the Jordan frame
metric is given by

${g}_{\mu\nu}^{J}=A^{2}(\varphi){g}_{\mu\nu}\quad\text{with}\quad
A(\varphi)=e^{\mathfrak{g}\varphi}$ (2)

where $\mathfrak{g}$ is a coupling constant. The scalar couples to matter only
via the metric ${g}_{\mu_{\nu}}^{J}$. The metric ${g}_{\mu\nu}$ is the
Einstein metric for which the Einstein equations take their usual form, i.e.
matter “feels” the geometry of the Jordan frame.

Constraints from solar system tests like those using measurements by the
Cassini probe Bertotti  _et al._ (2003) can be understood using the
Parameterized Post-Newtonian (PPN) formalism. It is enough here to define two
important PPN parameters. We introduce the gravitational mass of the source
$M$, and the PPN parameters $\gamma_{PPN}$, $\beta_{PPN}$ via the following
parameterisation of the Jordan metric element in isotropic coordinates around
a given compact object

$\displaystyle
ds_{J}^{2}=-\left(1-\frac{2GM}{r}+2\beta_{PPN}\left(\frac{GM}{r}\right)^{2}+{\cal
O}\left(\frac{1}{r^{3}}\right)\right)dt^{2}$
$\displaystyle+\left(1+2\gamma_{PPN}\frac{GM}{r}+{\cal
O}\left(\frac{1}{r^{2}}\right)\right)\left(dr^{2}+r^{2}d\Omega^{2}\right)~{}.$

In GR $\gamma_{PPN}=\beta_{PPN}=1$ whilst for the Brans-Dicke theory the
deviation from GR is captured by $\gamma_{PPN}$, i.e. $\beta_{PPN}=1$ and
Burgess and Quevedo (2022)

$\gamma_{PPN}=\frac{1-2\mathfrak{g}^{2}}{1+2\mathfrak{g}^{2}}~{}.$ (4)

The Cassini probe on the other hand gives Bertotti  _et al._ (2003)

$|{\gamma_{PPN}-1}|<2.3\times 10^{-5}~{}.$ (5)

This constrains the coupling $\mathfrak{g}^{2}$ to be less than $10^{-5}$.

Larger deviations from GR can be reached in certain regimes when the theories
are subject to a screening mechanism, i.e. a modification which allows the
theory to evade solar system tests while having an order unity coupling to
matter whose role is important on large scales. There are now at least three
types of screening mechanisms for single field models, i.e. the chameleon,
K-mouflage and Vainshtein mechanisms Khoury and Weltman (2004); Babichev  _et
al._ (2009); Vainshtein (1972). They all rely on higher derivatives and/or
non-linearities of the kinetic terms and interacting potentials of the models
Brax  _et al._ (2022a). In Burgess and Quevedo (2022), a new screening
mechanism was introduced, which relies on a second field that has a non-zero
but small coupling to matter. The mechanism depends crucially on the interplay
between the scalar profiles inside and outside matter Brax  _et al._ (2022b);
Lacombe and Mukohyama (2023). Here, we consider the situations where the two
fields, the dilaton and the axion, have a linear coupling to matter that we
denote by $\kappa$ and $\epsilon$. We focus on the small $\epsilon$ regime and
vary $\kappa$ from small values up to unity which corresponds to its value in
supergravity where the dilaton plays the role of the volume modulus of string
compactifications Cicoli  _et al._ (2023). We find that for $\kappa=1$,
screening of the dilaton around compact objects only takes place when the
fields take particular values at infinity. These values should emerge from the
cosmological dynamics. We then focus on the cosmology of these values where
both $\kappa$ and the values of the coupling $\epsilon$ to both baryons and
cold dark matter are varied. Generically, the field values in the present
Universe do not satisfy the screening conditions. In fact, one must resort to
yet unknown screening mechanisms for the axio-dilaton system in order to
accommodate both local solar system tests and cosmological constraints such as
the Baryon Acoustic Oscillations Alam  _et al._ (2021).

The coupling of the axion to dark matter $\epsilon_{C}$ plays a crucial role
cosmologically and can be of order unity. On the other hand, we find that the
coupling of the dilation $\kappa$ must be reduced locally to small values in
order to pass solar system tests. In this paper, we will consider two likely
scenarios. The first one is that the coupling $\kappa$ is determined locally
to be extremely small, of the order of $10^{-3}$, and that most of the
cosmological dynamics are due to the coupling of the axion $\epsilon_{C}$ in a
manner reminiscent of coupled quintessence Amendola (2000) although in a
multi-field setting here Amendola  _et al._ (2014). This could be achieved if
another field $\chi$ drove the coupling $\kappa$ to such values dynamically in
the whole Universe. Another possibility could be that the coupling $\kappa$ is
small locally but allowed to take larger values cosmologically. This could
happen if the field $\chi$ only made $\kappa$ small locally. In this second
scenario, we find that the BAO constraints on late time cosmology are so
stringent that $\kappa$ cannot be taken of order unity as in the original
supergravity model. In this setting, we consider the allowed deviations of
$H_{0}$ from their values as calibrated by the Planck satellite experiment and
find that only a few percent of discrepancy are allowed. This is much less
than the current $H_{0}$ tension Verde  _et al._ (2019); Valentino  _et al._
(2021). Finally, we notice that the linear growth in these models is reduced
compared to the $\Lambda$CDM case, despite the existence of attractive fifth
forces due to the dilaton and the axion. This could provide a solution to the
$\sigma_{8}$ tension Barros  _et al._ (2019) where the observed amount of
clustering is reduced compared to the expected one from early times Di
Valentino  _et al._ (2021); Nunes and Vagnozzi (2021). The precise study of
this possibility is left to future work.

The scenario that we present here enlarges the usual single-field class of
models for late time cosmology. Such multi-field generalisations could prove
useful in view of future measurement and present cosmological tensions
Amendola  _et al._ (2018).

The paper is arranged as follows. In section II, we present the axio-dilaton
model. In section III, we consider the screening of a compact objects. We then
discuss the cosmology of these models in section IV.

##  II The axio-dilaton theory

###  II.1 The Lagrangian

The axio-dilaton theory contains two scalar fields, the dilaton $\tau>0$ and
the axion $a$. The dilaton $\tau$ couples to matter only through the Jordan
frame metric while $a$ is directly coupled to matter. The difference will be
made clear below where we will construct an effective metric which will
mediate the coupling of both scalars to matter. The action of the theory is

$\mathcal{S}=\int
d^{4}x\sqrt{-g}M_{p}^{2}\left[\frac{\mathcal{R}}{2}-\frac{3}{4}\left(\frac{\partial^{\mu}\tau\partial_{\mu}\tau+\partial^{\mu}a\partial_{\mu}a}{\tau^{2}}\right)\right]+\mathcal{S}_{m}~{},$
(6)

with

$\displaystyle\mathcal{S}_{m}=\mathcal{S}_{m}({g}_{\mu\nu}^{J},\,a,\,\psi_{m}),\qquad{g}_{\mu_{\nu}}^{J}=A(\tau)^{2}{g}_{\mu\nu}$
$\displaystyle A(\tau)=\tau^{-\kappa/2}.$

$\kappa$ is a coupling constant. The theory studied in Burgess and Quevedo
(2022) corresponds to $\kappa=1$ and is associated to a supergravity model of
string theory origin with a Kähler potential
$K=-3\ln(\mathcal{T}+\overline{\mathcal{T}})$ and a coupling to matter
determined by $A=e^{K/6}$. In this setting, the dilaton can be seen as the
volume modulus of a 6d compatification of string theory from 10d to 4d.
Introducing $\kappa$ enables one to tune the matter-dilaton coupling and make
the solar system tests of gravity easier to satisfy. We will see that
screening is compulsory for the model to pass the solar system tests of
gravitation.

The two fields can be viewed as the real and imaginary parts of a complex
field $\mathcal{T}=\frac{1}{2}(\tau+ia)$ whose Lagrangian is

$\mathcal{L}=\sqrt{-g}M_{p}^{2}\left(\frac{\mathcal{R}}{2}-\frac{3\partial^{\mu}\mathcal{T}\partial_{\mu}\overline{\mathcal{T}}}{(\mathcal{T}+\overline{\mathcal{T}})^{2}}\right)+\mathcal{L}_{m}~{}.$
(8)

In this model, we define the usual stress-energy tensor and the coupling of
$a$ to matter

$T^{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta S_{m}}{\delta g_{\mu\nu}},\
\mathcal{A}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta S_{m}}{\delta a}~{}.$ (9)

We will use for the trace the notation $T\equiv g_{\mu\nu}T^{\mu\nu}$. Notice
that the matter action depends on the axion field in a non-trivial manner and
not via the Jordan metric. This will have drastic consequences that we will
unravel below.

There are two Klein-Gordon equations for the two scalar fields, i.e.

$\Box\tau-\frac{1}{\tau}(\partial_{\mu}\tau\partial^{\mu}\tau-\partial_{\mu}a\partial^{\mu}a)-\kappa\frac{\tau}{3M_{p}^{2}}T=0~{},$
(10)

and

$\Box
a-\frac{2}{\tau}\partial_{\mu}\tau\partial^{\mu}a+\frac{\tau^{2}}{3M_{p}^{2}}\mathcal{A}=0~{}.$
(11)

It is convenient to introduce the dilaton field $\varphi$ such that
$\tau=e^{\varphi}$ leading to

$\Box\varphi+(\partial_{\mu}a\partial^{\mu}a)e^{-2\varphi}-\frac{\kappa}{3M_{p}^{2}}T=0~{}.$
(12)

The Einstein equation is simply

$\mathcal{R}_{\mu\nu}-\frac{3}{2\tau^{2}}(\partial_{\mu}\tau\partial_{\nu}\tau+\partial_{\mu}a\partial_{\nu}a)-\frac{1}{M_{p}^{2}}(T_{\mu\nu}-\frac{1}{2}T{g}_{\mu\nu})=0~{}.$
(13)

where we have separated the matter energy momentum tensor from the scalar one.

###  II.2 The symmetries

In the absence of matter, the theory is invariant under a $SL(2,\mathbb{R})$
group whose origin can be traced back to supergravity. Indeed, the kinetic
term of the fields is invariant under

$\mathcal{T}\longrightarrow\frac{a\mathcal{T}-ib}{ic\mathcal{T}+d}\qquad\text{provided}\qquad
ad-bd=1$ (14)

corresponding to a Kähler transformation of the theory. There are thus three
conserved currents, corresponding to the dimension of the symmetry group in
the absence of matter. We can choose as a basis for these currents

  * •

The axion shift symmetry $\mathcal{T}\rightarrow\mathcal{T}-ib$ ($a=c=0$,
$d=1$):

$J^{\mu}_{A}=\frac{\partial^{\mu}a}{\tau^{2}}~{}.$ (15)

  * •

The rescaling symmetry $\mathcal{T}\rightarrow a\mathcal{T}$ ($b=c=0$, $d=1$):

$J^{\mu}_{S}=\frac{\partial^{\mu}\tau}{\tau}+\frac{a\partial^{\mu}a}{\tau^{2}}~{}.$
(16)

  * •

The non-linear symmetry $\mathcal{T}\rightarrow\mathcal{T}-ic\mathcal{T}^{2}$
($a=d=1$, $b=0$, $c\ll 1$):

$J^{\mu}_{N}=\frac{\tau^{2}-a^{2}}{\tau^{2}}\partial^{\mu}a-2a\frac{\partial^{\mu}\tau}{\tau}~{}.$
(17)

From the Klein-Gordon equations we can directly obtain the (non-)conservation
laws

$\displaystyle\nabla_{\mu}J^{\mu}_{A}=-\frac{\mathcal{A}}{3M_{p}^{2}}~{},\qquad\nabla_{\mu}J^{\mu}_{S}=\frac{\kappa
T-a{\cal A}}{3M_{p}^{2}}~{},$
$\displaystyle\nabla_{\mu}J^{\mu}_{N}=\frac{(a^{2}-\tau^{2})\mathcal{A}-2a\kappa
T}{3M_{p}^{2}}~{}.$

As can be seen, matter breaks the whole symmetry group as none of the three
currents is conserved anymore.

When $\mathcal{S}_{m}$ does not depend on $a$, i.e. $\mathcal{A}=0$, the axio-
dilation theory is equivalent to a Brans-Dicke theory. Indeed when
$\mathcal{A}=0$ we have the axion solution $a=\text{cste}$, and so Eq.(12)
becomes

$\Box\varphi-\frac{\kappa}{3M_{p}^{2}}T=0~{}.$ (19)

Similarly, the BD Lagrangian (50) gives the Klein-Gordon equation

$\Box\varphi_{\mathrm{BD}}+\frac{\mathfrak{g}}{M_{p}^{2}}T=0~{}.$ (20)

Matching the two Weyl factors
$e^{-\kappa\varphi/2}=e^{\mathfrak{g}\varphi_{\mathrm{BD}}}$ we get
$\varphi_{\mathrm{BD}}=(-\kappa/2\mathfrak{g})\varphi$. Combining the two
Klein-Gordon equations

$\left(\frac{-\kappa}{2\mathfrak{g}}+\frac{3\mathfrak{g}}{\kappa}\right)\Box\varphi=0\implies\mathfrak{g}^{2}=\kappa^{2}/6~{}.$
(21)

When $\kappa=1$, we find that the coupling reduces to $1/\sqrt{6}$, i.e. the
same as for $f(R)$ and massive gravity.

##  III Non-relativistic source and screening

Screening requires to study the gravitational physics around objects like the
Sun. We model the Sun and other compact objects such as the Earth or the Moon
as non-relativistic sources, i.e the only non-zero component of their stress-
energy tensor is $T_{00}\equiv\rho$. We also assume that they are static and
spherically symmetric. We look for space-time solutions with the same
symmetries that we chart with isotropic coordinates

$g_{\mu\nu}dx^{\mu}dx^{\nu}=-e^{2u(r)}dt^{2}+e^{2w(r)}\left(dr^{2}+r^{2}d\Omega^{2}\right)~{}.$
(22)

The Jordan metric is obtained by multiplying this line element by the coupling
function $A^{2}$.

###  III.1 Exterior solution

We will simplify the setting by considering non-relativistic objects with a
small Newtonian potential. As a result, we will approximate the Klein-Gordon
equations using a flat background $\boldsymbol{g}=\boldsymbol{\eta}$ where
$\eta_{\mu\nu}$ is the Minkowski metric tensor. The validity of the
approximation is evaluated a posteriori.

In the following, primes will refer to derivatives with respect to $r$. We
assume spherical symmetry. The resulting Klein-Gordon equations are

$\tau^{\prime\prime}+\frac{2\tau^{\prime}}{r}-\frac{\tau^{\prime
2}-{a^{\prime}}^{2}}{\tau}+\kappa\frac{\tau\rho}{3M_{p}^{2}}=0~{},$ (23)

and

$a^{\prime\prime}+\frac{2a^{\prime}}{r}-\frac{2\tau^{\prime}a^{\prime}}{\tau}+\frac{\tau^{2}\mathcal{A}}{3M_{p}^{2}}=0~{}.$
(24)

Here $\rho$ is the matter density inside or outside the object. Instead of
using these equations directly, we can use the equations for the currents
(II.2). In the absence of matter, i.e. outside the objects, the currents are
conserved

$\displaystyle r^{2}\frac{a^{\prime}}{\tau^{2}}=C_{A}~{},$ (25) $\displaystyle
r^{2}\left(\frac{\tau^{\prime}}{\tau}+\frac{aa^{\prime}}{\tau^{2}}\right)=C_{S}~{},$
(26) $\displaystyle
r^{2}\left(\frac{(\tau^{2}-a^{2})a^{\prime}}{\tau^{2}}-\frac{2a\tau^{\prime}}{\tau}\right)=C_{N}~{}.$
(27)

These constants are fixed by the conditions inside the source.

We are interested in the case where $a\neq\text{cste}$. So $C_{A}\neq 0$.
Defining $\gamma\equiv C_{A}$, $\alpha\equiv C_{S}/C_{A}$,
$\beta\equiv(C_{S}/C_{A})^{2}+C_{N}/C_{A}$, we obtain

$\tau^{2}+(a-\alpha)^{2}=\beta^{2}~{}.$ (28)

where $\tau$ and $a$ thus evolve on a circle in the $\tau-a$ plane. We can
thus eliminate $\tau$ in Eq.(25) and get

$a^{\prime}=\frac{\gamma}{r^{2}}(\beta^{2}-(a-\alpha)^{2})~{}.$ (29)

We integrate this to obtain the axion profile

$a=\alpha-\beta\tanh{X(r)}\quad\text{with}\quad
X(r)=\frac{\gamma\beta}{r}+\delta~{},$ (30)

where $\delta$ is a new integration constant. Using again Eq.(25) we finally
get

$\tau=\frac{\beta}{\cosh{X(r)}}~{}.$ (31)

Some of the integration constants are fixed by the boundary conditions inside
the source. Indeed, from the currents (II.2), we see that

$\gamma=C_{A}=-\frac{1}{3M_{p}^{2}}\int_{0}^{R}drr^{2}\mathcal{A}(r)~{},$ (32)
$\gamma\alpha=C_{S}=-\frac{1}{3M_{p}^{2}}\int_{0}^{R}drr^{2}(\kappa{}\rho(r)+a(r)\mathcal{A}(r))~{}.$
(33)

On the other hand, $\beta$ and $\delta$ can only be fixed by the values of the
fields at infinity

$a_{\infty}=\alpha-\beta\tanh{\delta},\
\beta=\tau_{\infty}\cosh{\delta}=\sqrt{\tau_{\infty}^{2}+(\alpha-
a_{\infty})^{2}}~{}.$ (34)

As shown in appendix A, these solutions in the flat background approximation
are valid as long as

$r\gg GM\qquad\text{and}\qquad r\gg|\gamma\beta|~{}.$ (35)

The first condition corresponds to being far from the Schwarzschild radius.

###  III.2 Screening

We first investigate screening in the Jordan frame where we will extract the
PPN parameters from the Jordan metric

$\displaystyle{g}_{\mu\nu}^{J}dx^{\mu}dx^{\nu}=A^{2}{g}_{\mu\nu}dx^{\mu}dx^{\nu}$
(36)
$\displaystyle=-\left(1-\frac{2GM_{g}}{r}+2\beta_{PPN}\left(\frac{GM_{g}}{r}\right)^{2}+{\cal{O}}\left(\frac{1}{r^{3}}\right)\right)dt^{2}$
$\displaystyle+\left(1+2\gamma_{PPN}\frac{GM_{g}}{r}+{\cal{O}}\left(\frac{1}{r^{2}}\right)\right)(dr^{2}+r^{2}d\Omega^{2})~{}.$

where $A^{2}=1/\tau^{\kappa}$ and $g_{\mu\nu}$ is given by (22). $M_{g}$ is
the gravitational mass as defined in the PPN formalism. We have that
$M_{g}\neq M=\int_{\text{source}}\rho$, i.e. the mass of the object in the
Jordan frame is renormalised by the presence of the scalar fields.

In the absence of fields, the Einstein frame metric $g_{\mu\nu}$ is the
Schwarzschild metric. In the presence of the fields the Einstein equations are
modified and we expand

$\displaystyle e^{2u}$
$\displaystyle=1-\frac{2l}{r}+\frac{2l^{2}}{r^{2}}+{\cal
O}\left(\frac{1}{r^{3}}\right)~{},$ (37) $\displaystyle e^{2w}$
$\displaystyle=1+\frac{2l}{r}+{\cal O}\left(\frac{1}{r^{3}}\right)~{}.$ (38)

where $l=GM$. Expanding the conformal factor $A^{2}$ in inverse powers of the
distance, we have

$A^{2}=A_{\infty}^{2}\left(1-\alpha_{1}/{r}+{\alpha_{2}}/{r^{2}}+{\cal
O}\left(\frac{1}{r^{3}}\right)\right)~{},$ (39)

from which we deduce that

$\displaystyle\gamma_{PPN}$
$\displaystyle=\frac{1-\frac{\alpha_{1}}{2l}}{1+\frac{\alpha_{1}}{2l}}~{},$
(40) $\displaystyle\beta_{PPN}$
$\displaystyle=\frac{l^{2}+l\alpha_{1}+\frac{1}{2}\alpha_{2}}{(l+\frac{1}{2}\alpha_{1})^{2}}~{}.$
(41)

For the axio-dilaton theory the conformal factor is given by

$A^{2}=\tau^{-\kappa}=\left(\frac{\cosh(\frac{\beta\gamma}{r}+\delta)}{\beta}\right)^{\kappa}$
(42)

where we have used the explicit solution for $\tau(r)$. Expanding in $1/r$ we
find the coefficients

$\displaystyle\alpha_{1}=-\kappa\beta\gamma\tanh{\delta}\qquad,$
$\displaystyle\alpha_{2}=\frac{\kappa\beta^{2}\gamma^{2}}{2}((\kappa-1)\tanh^{2}{\delta}+1)~{}.$
(43)

In the following, we will always choose the following ansatz following Burgess
and Quevedo (2022)

$\mathcal{A}=-\epsilon T~{},$ (44)

where $\epsilon$ is a small constant and $T$ the trace of the energy momentum
tensor. In the case of static sources this reduces to
$\mathcal{A}=\epsilon\rho$. As we shall see, this choice is not innocent as
this will bring back the two-field model within the realm of the scalar-tensor
theories with an effective coupling to both the dilaton and the axion. We will
make this explicit in the following. From Eq.(32) we obtain

$\gamma=-\frac{2\epsilon GM}{3}=-\frac{2\epsilon l}{3}~{}.$ (45)

and in the Jordan frame the PPN parameters become

$\gamma_{PPN}=\frac{3-\epsilon\kappa\beta\tanh{\delta}}{3+\epsilon\kappa\beta\tanh{\delta}}~{},$
(46)

and

$\beta_{PPN}=1+\frac{\kappa\epsilon^{2}\beta^{2}(1-\tanh^{2}{\delta})}{(3+\kappa\epsilon\beta\tanh{\delta})^{2}}~{}.\\\
$

Hence by taking $\epsilon$ small GR is recovered as long as $\beta$ does not
increase accordingly. This is the essence of this new type of screening which
corresponds to a non-uniform limit $\epsilon\to 0$. Indeed, when $\epsilon=0$
the model is equivalent to a Brans-Dicke model with a coupling
$\kappa/\sqrt{6}$ which needs to be small enough to satisfy the solar system
tests of gravity. But it turns out that the non-uniform limit $\epsilon\to 0$
requires a particular choice of the boundary conditions at infinity which are
not generic. We will come back to this point in the next section.

###  III.3 The interior solution

We now solve Eq.(23) and Eq.(24) inside the source of radius $R$, with the
boundary conditions $\tau^{\prime}(0)=a^{\prime}(0)=0$ at the origin. We
assume a uniform density inside the body as a simplifying assumption and a
coupling $\mathcal{A}$ with the same profile as $\rho$

$\left\\{\begin{aligned} \rho&\equiv\rho_{0}=\frac{M}{\frac{4}{3}\pi R^{3}}\\\
\mathcal{A}&=\epsilon\rho_{0}\end{aligned}\right.~{}.$ (47)

The dynamical equations cannot be solved exactly. We will get perturbative
solutions in $\epsilon$ as we have seen that a small coupling $\epsilon$ is
required to screen in the Jordan frame.

####  III.3.1 The dilaton dynamics

The equation for the current $J_{A}^{\mu}$ from Eq.(II.2) gives

$\left(r^{2}\frac{a^{\prime}}{\tau^{2}}\right)^{\prime}=-\frac{r^{2}\mathcal{A}}{3M_{p}^{2}}~{}.$
(48)

Assuming regularity for the fields at the origin, we obtain

$\frac{a^{\prime}(r)}{\tau^{2}(r)}=-\frac{\epsilon\rho_{0}}{3M_{p}^{2}r^{2}}\frac{r^{3}}{3}~{}.$
(49)

We define $m^{2}=\rho_{0}/3M_{p}^{2}$ leading to

$a^{\prime}=-\frac{\epsilon m^{2}}{3}r\tau^{2}~{}.$ (50)

We can substitute this expression in Eq.(23) to get

$\tau^{\prime\prime}+\frac{2\tau^{\prime}}{r}-\frac{(\tau^{\prime})^{2}}{\tau}+\frac{\epsilon^{2}m^{4}}{9}r^{2}\tau^{3}+\kappa
m^{2}\tau=0~{}.$ (51)

In terms of the dilaton $\varphi=\ln{\tau}$, the Klein-Gordon equation Eq.(12)
then becomes

$\varphi^{\prime\prime}+\frac{2}{r}\varphi^{\prime}+\frac{\epsilon^{2}m^{4}}{9}r^{2}e^{2\varphi}+\kappa
m^{2}=0$ (52)

where we impose the boundary equation $\varphi^{\prime}(0)=0$.

####  III.3.2 Perturbative expansion in $\epsilon$

Up to now, everything is exact. We now expand in powers of $\epsilon$:

$\displaystyle\varphi$
$\displaystyle=\varphi^{(0)}+\epsilon\varphi^{(1)}+\dots~{},$ (53)
$\displaystyle a$ $\displaystyle=a^{(0)}+\epsilon a^{(1)}+\dots~{},$ (54)

and impose the boundary conditions at each order.

The advantage of this perturbative method is that the problematic term
$e^{2\varphi}$ in Eq.(52) appears only at second order in $\epsilon$. Indeed
we get at orders 0 and 1

$\displaystyle{\varphi^{(0)}}^{\prime\prime}+\frac{2}{r}{\varphi^{(0)}}^{\prime}+\kappa
m^{2}$ $\displaystyle=0~{},$ (55)
$\displaystyle{\varphi^{(1)}}^{\prime\prime}+\frac{2}{r}{\varphi^{(1)}}^{\prime}$
$\displaystyle=0~{}.$ (56)

The solution for $\varphi^{(1)}$ is then

$\varphi^{(1)}=c_{0}+\frac{c_{1}}{r}~{}.$ (57)

For $\varphi^{(0)}$ we have

$\varphi^{(0)}=d_{0}+\frac{d_{1}}{r}-\frac{\kappa m^{2}}{6}r^{2}~{}.$ (58)

The boundary conditions ${\varphi^{(n)}}^{\prime}(0)=0$, impose that the
fields are regular at 0, and so we have no $1/r$ term, i.e. $c_{1}=d_{1}=0$,
and finally

$\tau=(\tau_{0}+\epsilon\tau_{1})e^{-\frac{\kappa m^{2}}{6}r^{2}}+\dots$ (59)

where we have redefined the constants of integration. Inside the source and
for small $\epsilon$, $\tau$ decreases exponentially fast.

We can now obtain $a$. Using Eq.(50) we have

${a^{(0)}}^{\prime}+\epsilon{a^{(1)}}^{\prime}=\epsilon\frac{\tau_{0}^{2}}{2}\left(e^{-\frac{\kappa
m^{2}}{3}r^{2}}\right)^{\prime}~{},$ (60)

and therefore

$\displaystyle a^{(0)}$ $\displaystyle=\text{cste}~{},$ (61) $\displaystyle
a^{(1)}$ $\displaystyle=\frac{\tau_{0}^{2}}{2}e^{-\frac{\kappa
m^{2}}{3}r^{2}}+\text{cste}~{}.$ (62)

The axion field is then given by

$a=a_{0}+\epsilon\left(a_{1}+\frac{\tau_{0}^{2}}{2}e^{-\frac{\kappa
m^{2}}{3}r^{2}}\right)+\dots~{}.$ (63)

We see that the axion and dilaton fields only evolve when $\epsilon\neq 0$.

####  III.3.3 Matching to the exterior solution

The continuity of $\tau$ and $a$ at $r=R$ reads

$\displaystyle(\tau_{0}+\epsilon\tau_{1})e^{-\frac{\kappa m^{2}}{6}R^{2}}$
$\displaystyle=\frac{\beta}{\cosh(\frac{\beta\gamma}{R}+\delta)}~{},$ (64)
$\displaystyle
a_{0}+\epsilon\left(a_{1}+\frac{\tau_{0}^{2}}{2}e^{-\frac{\kappa
m^{2}}{3}R^{2}}\right)$
$\displaystyle=\alpha-\beta\tanh(\frac{\beta\gamma}{R}+\delta)~{}.$ (65)

We use the continuity equations for $\varphi^{\prime}=\tau^{\prime}/\tau$ and
$a^{\prime}/\tau^{2}$

$\displaystyle-\frac{\kappa m^{2}}{3}R$
$\displaystyle=\tanh(\frac{\beta\gamma}{R}+\delta)\frac{\beta\gamma}{R^{2}}~{},$
(66)
$\displaystyle-\frac{\epsilon\kappa\tau_{0}^{2}m^{2}}{3(\tau_{0}+\epsilon\tau_{1})^{2}}R$
$\displaystyle=\frac{\gamma}{R^{2}}~{}.$ (67)

We have thus eight integration constants, i.e. $\alpha$, $\beta$, $\gamma$,
$\delta$ for the exterior solution and $\tau_{0}$, $\tau_{1}$, $a_{0}$,
$a_{1}$ for the interior solution. Recall that

$\displaystyle\gamma$ $\displaystyle=-\frac{\epsilon m^{2}}{3}R^{3}~{},$ (68)
$\displaystyle\gamma\alpha$
$\displaystyle=-m^{2}\int_{0}^{R}drr^{2}(\kappa+\epsilon a(r))~{}.$ (69)

So $\gamma$ is fixed independently of the others and $\alpha$ depends on
$a_{0}$ and $a_{1}$. We are left with 6 parameters. With a system of 4
continuity constraints as given above, we end up with 2 degrees of freedom.
These can be parameterised by the values of the fields at infinity
$\tau_{\infty}$ and $a_{\infty}$ which determine the full solution.

###  III.4 Screening revisited

We can now revisit the conditions under which the gravitational deviation from
GR in the Jordan frame is small. Using

$\gamma\alpha=-\frac{1}{3M_{p}^{2}}\int_{0}^{R}drr^{2}(\kappa\rho(r)+a(r)\mathcal{A}(r))~{},$
(70)

and $\gamma=-{\epsilon m^{2}R^{3}}/{3}$ we obtain the identity

$\displaystyle\frac{-\epsilon
m^{2}R^{3}}{3}\alpha=-m^{2}\left(\kappa\frac{R^{3}}{3}+\epsilon\int_{0}^{R}drr^{2}a^{(0)}(r)\right.$
$\displaystyle+\left.\epsilon^{2}\int_{0}^{R}drr^{2}a^{(1)}(r)\right)~{}.$
(71)

As a result, the expansion of $\alpha$ is singular in the limit $\epsilon\ll
1$ and becomes

$\alpha=\frac{\kappa}{\epsilon}+a_{0}+\epsilon a_{1}+\dots$ (72)

implying that the outside solution is very sensitive to small values of
$\epsilon$. In particular, we see that the limit of small $\epsilon\ll 1$
leads to a large value for the exterior axion field. Using Eq.(34) we obtain

$\beta=\sqrt{\tau_{\infty}^{2}+\left(\frac{\kappa}{\epsilon}+a_{0}-a^{(0)}_{\infty}\right)^{2}}$
(73)

where we have neglected the terms of order $\epsilon$. Now unless $a_{\infty}$
turns out to be of order $1/\epsilon$ and cancels exactly the term in
$\kappa/\epsilon$, we find that for generic boundary values at infinity

$\beta=\frac{\kappa}{\epsilon}+a_{0}-a_{\infty}^{(0)}+\dots$ (74)

when $\kappa={\cal O}(1)$ and $\epsilon\ll 1$. The matching conditions at
$r=R$ simplify as we notice that $m^{2}R^{2}/2=G_{N}M/R$ where $M$ is the mass
of the object. This is nothing but the Newtonian potential of the compact
object at its surface which is always small in our Newtonian approximation,
its value being close to $10^{-6}$ for the Sun. We then deduce using (65) that
$a_{0}=a_{\infty}^{(0)}$ to leading order and similarly

$\tanh{\delta}=1-\frac{\epsilon^{2}}{\kappa}\frac{\tau_{0}^{2}}{2}+\dots$ (75)

implying that $\delta$ is always large. As a result we have
$\epsilon\beta\tanh{\delta}=\kappa+\dots~{}.$ and the PPN parameters are
simply the ones of a scalar tensor theory with a coupling $\kappa/\sqrt{6}$

$\gamma_{PPN}=\frac{3-\kappa^{2}}{3+\kappa^{2}}+{\cal O}(\epsilon)~{},$ (76)

and

$\beta_{PPN}=1+{\cal O}(\epsilon^{2})~{}.$ (77)

In this limit $\beta_{PPN}$ can be arbitrarily close to $1$ for small
$\epsilon$, but not $\gamma_{PPN}$. Small deviations of $\gamma_{PPN}$ from
unity are only achieved for very small $\kappa$, a result which does not
differ from Brans-Dicke’s and signals that screening does not take place.
Screening can only take place when

$a_{\infty}^{(0)}-a_{0}=\frac{\kappa}{\epsilon}.$ (78)

which corresponds to a specific choice for the axion field at infinity. As the
theory has no scalar potential, the value of the axion field at infinity is
not obtained by minimising an effective potential like in the chameleon
mechanism. Hence, the boundary value of the axion field must emerge from the
cosmological dynamics. We will study this below.

Our analysis has assumed that $\mathcal{A}=\epsilon\rho$. As soon as the
dependence of the matter action on the axion is weak, i.e.
$\mathcal{A}/\rho\sim\epsilon\ll 1$, the same qualitative results follow as
$\gamma$ will be of order $\epsilon$ and $\gamma\alpha$ of order $\kappa$.
This reasoning is independent of the details of the model inside the source.

###  III.5 The effective metric

As we have seen, the generic absence of screening leads to the coupling of a
compact of object to be equivalent to the one of a point particle with
coupling $\kappa/\sqrt{6}$. This is the coupling of matter to the dilaton. The
fact that the axion couples to the matter action too implies that compact
objects do not follow the geodesics of the Jordan metric but the ones of an
effective metric whose presence can be inferred from the small field expansion

$\delta S_{m}=-\int d^{4}x\sqrt{-g}\left(\frac{\partial\ln
A}{\partial\varphi}\delta\varphi-\frac{\epsilon}{2}\delta a\right)T$ (79)

where the variation of the fields is taken around the background values for
the dilaton and the axion. Notice that the axion and the dilaton fields both
couple to the trace of the energy momentum tensor. Let us define

$B(\varphi,a)=A(\varphi)e^{-\epsilon a/2}$ (80)

then the coupling to matter can be written as

$\delta S_{m}=-\int d^{4}x\sqrt{-g}\left(\frac{\partial\ln
B}{\partial\varphi}\delta\varphi+\frac{\partial\ln B}{\partial a}\delta
a\right)T$ (81)

corresponding to the coupling of a two-field scalar-tensor theory where the
effective metric is

$g_{\mu\nu}^{\rm eff}=B^{2}(\varphi,a)g_{\mu\nu}.$ (82)

As a result, compact objects evolve along the geodesics of the effective
metric and not the Jordan metric.

This can be confirmed by analysing the geodesic equations for pressureless
matter for the axio-dilaton theories. Indeed the Klein-Gordon equations and
the Bianchi identity $\nabla^{\mu}(R_{\mu\nu}-\frac{R}{2}g_{\mu\nu})=0$ imply
the non-conservation equation

$\nabla_{\mu}{T}^{\mu\nu}=\frac{1}{2}(\kappa
T\partial^{\nu}\varphi-\mathcal{A}\partial^{\nu}a)~{}.$ (83)

For non-relativistic matter, the energy momentum tensor is simply

${T}_{\mu\nu}=\rho u_{\mu}u_{\nu}\quad\text{with}\quad u_{\mu}u^{\mu}=-1~{}.$
(84)

In this section, dots will be denoting the time derivative along the particle
lines defined by $u^{\mu}$, i.e.
$\dot{X}\equiv\nabla_{u}X=u^{\mu}\nabla_{\mu}X~{}.$ For scalar quantities $X$,
this also corresponds to the derivative with respect to the proper time of a
particle moving with velocity $u^{\mu}$. We define the local Hubble rate as
$3h\equiv\nabla_{\mu}u^{\mu}~{}.$ The non-conservation equation then becomes

$\dot{\rho}u^{\mu}+3h\rho
u^{\mu}=\frac{1}{2}(\kappa\rho\partial^{\mu}\varphi+\mathcal{A}\partial^{\mu}a)~{}.$
(85)

Contracting with $u_{\mu}$ and using $u^{\mu}u_{\mu}=-1$ we get the
generalised continuity equation

$\dot{\rho}+3h\rho=-\frac{1}{2}(\kappa\rho\dot{\varphi}+\mathcal{A}\dot{a})~{}.$
(86)

We recognise the coupling function $B$

$B\equiv e^{-\frac{1}{2}(\kappa{\varphi}+\epsilon{a})}$ (87)

when $\epsilon=\text{cste}$. We can define a conserved density
$\rho_{\text{con}}$ in the Einstein frame such that

$\rho=B\rho_{\text{con}}$ (88)

as

$\dot{\rho}_{\text{con}}+3h\rho_{\text{con}}=0~{}.$ (89)

This is the conserved matter density in the axion-dilaton setting. Combining
Eq.(85) and Eq.(86), we obtain the modified Newton’s Law

$\dot{u}^{\mu}-\frac{1}{2}(\kappa\dot{\varphi}+\epsilon\dot{a})u^{\mu}=\frac{1}{2}(\kappa\partial^{\mu}{\varphi}+\epsilon\partial^{\mu}{a})~{}.$
(90)

This reads

$\dot{u}^{\mu}+\frac{d\ln B}{d\eta}u^{\mu}=-\partial^{\mu}\ln B.$ (91)

where $\eta$ is the proper time. Defining by $m_{0}$ the mass of the
particles, we find that the effective mass of these particles in the Einstein
frame is dressed by the scalar field and becomes

$m=B(\varphi,a)m_{0}.$ (92)

This follows from the identification
$\rho=m\delta^{(4)}(x^{\mu}-x^{\mu}(\tau))$ along the particle’s trajectory
and $\rho_{\rm con}=m_{0}\delta^{(4)}(x^{\mu}-x^{\mu}(\tau))$. The momentum of
each particle becomes

$p^{\mu}=mu^{\mu}.$ (93)

Newton’s law then becomes

$\dot{p}^{\mu}=-m\partial^{\mu}\ln B.$ (94)

As a result, a force deriving from the potential $\ln B$ is exerted on each
particle whose mass is also field dependent. For instance in the non-
relativistic limit and in the presence of gravity, Newton’s law becomes

$\frac{dp^{i}}{dt}=-m\partial^{i}\Phi$ (95)

where $\Phi_{N}$ is Newton’s potential and

$\Phi=\Phi_{N}+\ln B$ (96)

combines the effects of gravity and the scalar field. This modification of
Newton’s law is nothing but the one which can be derived from the coupling of
matter to the effective metric $g_{\mu\nu}^{\rm eff}$. As a result we have
confirmed that compact objects do not follow the geodesics of the Jordan
metric but the one of the effective metric. We will analyse the cosmological
consequences of this result below.

###  III.6 The effective charge

Let us come back to the effective scalar charge carried by a compact object.
We have seen that in the $\epsilon\ll 1$ limit and unless the fields at
infinity take special values, which should be adjusted cosmologically, the
objects are not screened. Far away from a given object we expect the
acceleration of another object due to the scalar field to fall off as

$a^{i}\simeq-\frac{2Q^{2}G_{N}M}{r^{3}}r^{i}$ (97)

where $Q$ is the scalar charge of both objects. Here $M$ is the mass of the
object responsible for the acceleration of the second body. The charges of
both objects are equal as no screening takes place. Using

$\ln B=-\frac{1}{2}(\kappa\varphi+\epsilon a)\supset\frac{1}{2}(\kappa\ln\cosh
X(r)+\epsilon\beta\tanh X(r))$ (98)

and identifying this to $-2Q^{2}G_{N}M/r$ at large distance we find that far
away from the object

$2Q^{2}G_{N}M=-\frac{\gamma\beta}{2}(\kappa\tanh\delta+\epsilon\beta(1-\tanh^{2}\delta))$
(99)

and for $\epsilon\ll 1$ we retrieve that

$Q=\frac{\kappa}{\sqrt{6}}$ (100)

up to corrections of order $\epsilon^{2}$. The resulting interaction including
gravity is equivalent to rescaling Newton’s constant as

$G_{\rm eff}=(1+2Q^{2})G_{N}$ (101)

with $\Phi=-G_{\rm eff}M/r$. As expected in this limit corresponding to
$\epsilon\ll\kappa$, the coupling of the axion field to matter becomes
negligible for far-away objects and the coupling is the same as in the Jordan
frame.

In conclusion, we find that $\kappa\lesssim 10^{-3}$ for the Cassini test to
be evaded. This is a very small value which could only be avoided if the
cosmological values of the axion and dilaton fields were tuned cosmologically.

###  III.7 Numerical integration

In this section, we will find numerical solutions of the equations around a
massive sphere.

####  III.7.1 Setting the numerical problem

The Klein-Gordon equations with the constant source inside a ball of radius
$R$ have the form

$\tau^{\prime\prime}+\frac{2\tau^{\prime}}{r}-\frac{(\tau^{\prime})^{2}}{\tau}+\frac{(a^{\prime})^{2}}{\tau}+\frac{\kappa\rho_{0}\tau}{3M_{p}^{2}}\theta(r)=0~{},$
(102)
$a^{\prime\prime}+\frac{2a^{\prime}}{r}-\frac{2a^{\prime}\tau^{\prime}}{\tau}+\frac{\epsilon\rho_{0}\tau^{2}}{3M_{p}^{2}}\theta(r)=0~{},$
(103)

where $\theta(r)$ is the step function that goes from 1 to 0 at the radius of
the source $R$. We introduce the characteristic length
$L=\sqrt{3M_{p}^{2}/\rho_{0}}=m^{-1}$, and write $r=\hat{r}L$. We obtain the
dimensionless equations:

${\tau}^{\prime\prime}+\frac{2{\tau}^{\prime}}{\hat{r}}-\frac{({\tau}^{\prime})^{2}}{{\tau}}+\frac{({a}^{\prime})^{2}}{{\tau}}+\kappa{\theta}(\hat{r}){\tau}=0~{},$
(104)
${a}^{\prime\prime}+\frac{2{a}^{\prime}}{\hat{r}}-\frac{2{a}^{\prime}{\tau}^{\prime}}{{\tau}}+\epsilon{\theta}(\hat{r}){\tau}^{2}=0~{},$
(105)

which we solve with the initial conditions
${\tau}^{\prime}(0)={a}^{\prime}(0)=0$. We also regularise the step function,
e.g.:
$\hat{\theta}(\hat{r})=\frac{1}{2}(\tanh(N\frac{\hat{R}-\hat{r}}{\hat{R}})+1)$
to have a transition of width $\sim\hat{R}/N$. Notice that
$\hat{R}=R/L=\sqrt{2GM/R}$, i.e the square root of the ratio of the
Schwarzschild radius to the radius of the source. For the Sun
$\hat{R}_{\odot}=2.05\times 10^{-3}$.

####  III.7.2 Results

We see in the example in Fig.1 that we obtain a perfect match with the
exterior solution, but also with the first order interior solution. The
variations of the fields are relatively small, i.e. of order $10^{-6}$
relative to the value at the centre. For other sets of parameters the
variation increases with the initial value at the centre and of course the
value of $\kappa$ and $\epsilon$.

Figure 1: The field profiles for $\hat{R}=\hat{R}_{\odot}$, $\kappa=1$ and
$\epsilon=0.5$. The initial values are taken such that $\tau_{0}=a_{0}=1$,
$\tau^{\prime}(0)=a^{\prime}(0)=0$.

The equations that we have used in the flat metric approximation are valid
only for

$r\gg GM\quad,\quad r\gg|\gamma\beta|~{}.$ (106)

With $R=\hat{R}L$, we have

$r\gg GM\iff\hat{r}\gg\frac{\hat{R}^{3}}{2}~{}.$ (107)

For the Sun, $\hat{R}_{\odot}^{3}\sim 10^{-8}$. One can trust our description
of the static solution apart from a small region around the origin. Similarly,
we have $\gamma=-\epsilon\hat{R}^{3}L/3$, so we get

$r\gg|\gamma\beta|\iff\hat{r}\gg\hat{R}^{3}\frac{\epsilon\beta}{3}~{},$ (108)

which excludes a very small region around the origin.

##  IV Axio-dilaton cosmology

###  IV.1 Two fluid model

In the following, we will concentrate on models where two fluids are present,
i.e. the baryons and Cold Dark Matter (CDM) with couplings determined by
$\kappa$ to the dilaton and $\epsilon_{B,C}$ to the axion. We have seen that
$\kappa$ and $\epsilon_{B}$, i.e. the coupling of the baryons to the axion,
must be small to comply with the solar system tests unless the fields take
special values at infinity. This translates into a choice of boundary
conditions for the fields on cosmological scales now. We will see that this
situation is not generic and that starting from initial conditions in the
radiation era which do not perturb the background cosmology, the cosmological
dynamics do not drive the fields to special values now.

The energy-momentum tensor of matter is taken to be

${T}_{\mu\nu}^{B,C}=\rho_{B,C}u_{\mu}^{B,C}u_{\nu}^{B,C}$ (109)

corresponding to the baryons $B$ and CDM $C$. We will also assume here for
definiteness

$\mathcal{A}=-\epsilon_{C}T_{C}-\epsilon_{B}T_{B}$ (110)

where $T_{B,C}=-\rho_{B,C}$ are the baryonic and CDM densities respectively.
The matter energy momentum tensors are not conserved but satisfy the non-
conservation equations

$\nabla_{\mu}{T}^{\mu\nu}_{i}=\frac{1}{2}(\kappa
T_{i}\partial^{\nu}\varphi+\epsilon_{i}T_{i}\partial^{\nu}a)$ (111)

where $i=B,C$. This implies that the total energy momentum tensor

$T^{\mu\nu}=T_{B}^{\mu\nu}+T^{\mu\nu}_{C}$ (112)

satisfies the non-conservation equation

$\nabla_{\mu}{T}^{\mu\nu}=\frac{1}{2}(\kappa
T\partial^{\nu}\varphi-\mathcal{A}\partial^{\nu}a)$ (113)

which is a consequence of the Bianchi identity.

In this section, we will denote the time derivative along the particle lines,
defined by $u^{\mu}_{i}$, by $d/d\tau_{i}=u^{\mu}_{i}\nabla_{\mu}$ We define
the local Hubble rate as $3h_{i}\equiv\nabla_{\mu}u^{\mu}_{i}~{}.$ Notice that
the covariant derivatives are calculated in the Einstein frame, hence this is
the local Hubble rate along the particle lines as measured using the geometry
of the Einstein frame. The non-conservation equations for each species then
become

$\frac{d\rho_{i}}{d\tau_{i}}+3h_{i}\rho_{i}=-\frac{1}{2}(\kappa\dot{\varphi}+\epsilon_{i}\dot{a})\rho_{i}~{}.$
(114)

We define the coupling function $B_{i}$

$B_{i}\equiv e^{-\frac{1}{2}(\kappa{\varphi}+\epsilon_{i}{a})}$ (115)

when $\epsilon_{i}=\text{cste}$. We can now introduce a conserved density
$\rho_{\text{con,i}}$ in the Einstein frame such that

$\rho_{i}=B_{i}\rho_{\text{con,i}}$ (116)

and

$\frac{d\rho_{{\rm con},i}}{d\tau_{i}}+3h_{i}\rho_{\rm{con},i}=0~{}.$ (117)

This is the conserved matter density in the axio-dilaton setting. Similarly we
obtain the modified Newton’s Law

$\frac{d{u}^{\mu}_{i}}{d\tau_{i}}+\frac{d\chi_{i}}{d\tau_{i}}u^{\mu}_{i}=-\partial^{\mu}\chi_{i}$
(118)

where

$\chi_{i}\equiv\ln B_{i}.$ (119)

For each species, we can define an effective metric

$g^{i}_{\mu\nu}=B_{i}^{2}g_{\mu\nu}$ (120)

which corresponds to the Jordan frame for the given species. As $B_{B}\neq
B_{C}$, we see that the Jordan frames for CDM and the baryons do not coincide.

We will apply this formalism first to the background cosmological case and
then to the cosmological perturbations.

###  IV.2 Spatially flat cosmology

We are interested in the cosmology of a homogeneous and isotropic Universe in
the presence of the axio-dilaton fields. The FLRW (Friedmann-Lemaître-
Robertson-Walker) metric reads

$\displaystyle{g}_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=-dt^{2}+R^{2}(t)\gamma_{ij}dx^{i}dx^{j}$ (121)
$\displaystyle=-dt^{2}+R^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2})$

where $R$ is the scale factor. We define as usual the Hubble rate as
$H=\dot{R}/R$. In the following we will focus on the spatially flat case
$k=0$.

We assume that the fields are irrelevant in the early Universe up until some
redshift $z_{i}$ which will be typically the matter-radiation equality.
Indeed, in the radiation era the matter density is negligible and therefore
the fields are hardly influenced by their matter couplings. As a result they
remain constant if their initial velocities vanish. This also guarantees that
the influence of the axio-dilaton system on Big Bang Nucleosynthesis (BBN) is
minimal. We will choose the initial conditions for the fields $\varphi$ , $a$
and their derivatives such that they vanish at $z_{i}$. This does not
correspond to the values of the fields for which screening takes place. On the
other hand, this entails a vanishing energy density for the axion and dilatons
initially.

Each species has its own Jordan frame with the background metric

$g_{\mu\nu}^{i}dx^{\mu}dx^{\nu}=-dt_{i}^{2}+R^{2}_{i}dx^{i}dx_{i}$ (122)

where the cosmic time and the scale factor are defined by

$dt_{i}=B_{i}dt,\ \ R_{i}=B_{i}R.$ (123)

As CDM is not subject to the precision tests in the solar system, we will
allow for large values of $\epsilon_{C}$. Moreover we will consider the
effects on the scale factor in the Jordan frame of the baryons $R_{B}=B_{B}R$
corresponding to the conservation of baryonic matter. We will use the
convention that $R_{B}=1$ today and identify the redshift as detected from
transition lines of atoms by

$1+z=R_{B}^{-1}.$ (124)

Although we will focus on the dynamics in the baryon frame, we will first
study the equations of motion in the Einstein frame.

###  IV.3 The Klein-Gordon equations

We look for time-dependent solutions for the scalar fields $\tau(t)$ and
$a(t)$ of the Klein-Gordon equations

$\ddot{\tau}+3H\dot{\tau}-\frac{\dot{\tau}^{2}-\dot{a}^{2}}{\tau}-\frac{\kappa\tau\rho}{3M_{p}^{2}}=0~{},$
(125)

and

$\ddot{a}+3H\dot{a}-\frac{2\dot{\tau}\dot{a}}{\tau}-\frac{\tau^{2}\mathcal{A}}{3M_{p}^{2}}=0~{}.$
(126)

Here the total matter density is $\rho=\rho_{B}+\rho_{C}$ and ${\cal
A}=\epsilon_{B}\rho_{B}+\epsilon_{C}\rho_{C}$. Notice that these equations are
valid both in the radiation and matter eras. In the radiation era, the source
terms depend on the matter density only as the trace of the radiation energy
momentum tensor vanishes. As a first approximation, we will neglect the source
terms in the radiation era. This implies that $\dot{a}\approx 0$ and
$\dot{\phi}\approx 0$ and the field hardly move during the radiation era. In
our numerical analysis, we will be interested in the physics in the matter era
and will fix the initial conditions at matter-radiation equality. A detailed
analysis of the model from the end of inflation through the radiation to the
matter era is left for future work. We will consider that the fields start
evolving significantly when the matter era begins.

###  IV.4 The continuity equations

The energy density and pressure carried by the two scalar fields are given by

$\rho_{f}=\frac{3M_{p}^{2}}{4}(\frac{\dot{\tau}^{2}+\dot{a}^{2}}{\tau^{2}})~{},$
(127)

and

$p_{f}=\frac{3M_{p}^{2}}{4}(\frac{\dot{\tau}^{2}+\dot{a}^{2}}{\tau^{2}})~{},$
(128)

corresponding to a perfect fluid with equation of state $\omega_{f}=1$. Using
the Bianchi identity and the Einstein equation in the Einstein frame, we
obtain that the total energy is conserved, i.e.

$\dot{\rho}+\dot{\rho}_{f}+3H(\rho+\rho_{f}+P+P_{f})=0~{},$ (129)

where the pressure terms have been included. This leads to

$\dot{\rho}+\dot{\rho}_{f}+3H(\rho+2\rho_{f})=0~{}.$ (130)

Using the identity

$\dot{\rho}_{f}=\frac{3M_{p}^{2}}{4}\frac{2}{\tau^{3}}(\tau\dot{\tau}\ddot{\tau}+\tau\dot{a}\ddot{a}-\dot{\tau}^{3}-\dot{\tau}\dot{a}^{2})$
(131)

and the expression of $\ddot{\tau}$ and $\ddot{a}$ in the Klein-Gordon
equations Eq.(125) and Eq.(126), we get

$\dot{\rho}_{f}+6H\rho_{f}=\frac{1}{2}\left(\kappa\rho\frac{\dot{\tau}}{\tau}+\mathcal{A}\dot{a}\right).$
(132)

Finally using the continuity equation we obtain

$\dot{\rho}+3H\rho+\frac{1}{2}(\kappa\rho\dot{\varphi}+\mathcal{A}\dot{a})=0$
(133)

as we have argued previously. This is associated to the non-conservation
equations per species

$\dot{\rho_{i}}+3H\rho_{i}+\frac{1}{2}\rho_{i}(\kappa\dot{\varphi}+\epsilon_{i}\dot{a})=0.$
(134)

This can be integrated exactly and leads to

$\rho=\rho_{B}+\rho_{C}=B_{B}\frac{\rho_{0B}}{R^{3}}+B_{C}\frac{\rho_{0C}}{R^{3}}.$
(135)

Notice that if we define

$\rho=B\frac{\rho_{0}}{R^{3}}$ (136)

then $B=\lambda_{B}B_{B}+\lambda_{C}B_{C}$ where the fraction of baryons and
CDM are $\lambda_{i}=\frac{\rho_{0i}}{\rho_{0}}$ where
$\rho_{0}=\rho_{0B}+\rho_{0C}$. In practice we have
$\lambda_{B}=\frac{\Omega_{0B}}{\Omega_{0B}+\Omega_{0C}}$ where
$\Omega_{0B}\simeq 0.022$ and $\Omega_{0C}\simeq 0.12$ from the Planck
experiment Aghanim  _et al._ (2020). Here and in the following we normalise
the density $\rho_{0}$ to the Planck value as deduced from early time physics
compared to the late time effects on the matter density that we will study
below.

###  IV.5 The Friedmann equations

The Friedmann equation is obtained from the $(00)$ component of the Einstein
equation and becomes

$H^{2}=\frac{\rho}{3M_{p}^{2}}+\frac{\rho_{f}}{3M_{p}^{2}}+\frac{\rho_{\Lambda}}{3M_{p}^{2}}~{},$
(137)

where we have introduced a cosmological constant associated to the energy
density $\rho_{\Lambda}$. The Raychaudhuri equation from the $(ii)$ Einstein
equation reads

$\frac{\ddot{R}}{R}=-\frac{1}{6M_{p}^{2}}(\rho+\rho_{f}+\rho_{\Lambda}+3(P+P_{f}+P_{\Lambda}))~{},$
(138)

leading to

$\frac{\ddot{R}}{R}=-\frac{1}{6M_{p}^{2}}(\rho+4\rho_{f}-2\rho_{\Lambda})~{}.$
(139)

These equations are defined in the Einstein frame and will be transformed into
the effective frame for baryons below.

We thus have the following system of differential equations for $R$, $\tau$
and $a$ only:

$\displaystyle\ddot{\tau}+3H\dot{\tau}-\frac{\dot{\tau}^{2}-\dot{a}^{2}}{\tau}-{\kappa}\frac{\rho(\tau,a)}{3M_{p}^{2}}\tau$
$\displaystyle=0~{},$ (140)
$\displaystyle\ddot{a}+3H\dot{a}-\frac{2\dot{\tau}\dot{a}}{\tau}-\epsilon\frac{\rho(\tau,a)}{3M_{p}^{2}}\tau^{2}$
$\displaystyle=0~{},$ $\displaystyle\dot{R}=H(\tau,a)R~{},$

with the Friedmann equation

$\displaystyle
H^{2}=\frac{\rho}{3M_{p}^{2}}+\frac{\rho_{f}}{3M_{p}^{2}}+\frac{\rho_{\Lambda}}{3M_{p}^{2}}~{},$
$\displaystyle\rho=B_{B}\frac{\rho_{0B}}{{R^{3}}}+B_{C}\frac{\rho_{0C}}{{R^{3}}},\
\rho_{f}=\frac{3M_{p}^{2}}{4}\left(\frac{\dot{\tau}^{2}+\dot{a}^{2}}{\tau^{2}}\right)~{},$
$\displaystyle\frac{d\ln
B_{i}}{dt}=-\frac{1}{2}(\kappa\dot{\varphi}+\epsilon_{i}\dot{a})~{}.$

We will solve these equations numerically for different values of $\epsilon$
and $\kappa$.

###  IV.6 Dynamics in the effective baryon frame

In the baryon frame, the Hubble rate is

$H_{B}\equiv\frac{d\ln
R_{B}}{dt_{B}}=\frac{H}{B_{B}}+\frac{d\chi_{B}}{dt_{B}}~{}.$ (142)

The conserved baryon density in the baryon frame is simply

$\tilde{\rho}_{B}\equiv\rho_{{\rm con}B}=\frac{\rho_{0B}}{R_{B}^{3}}.$ (143)

In this frame, CDM is not conserved but an observer fitting the evolution of
the Universe with a prior that CDM is also conserved in the same frame as the
baryons would identify the conserved CDM density as

$\tilde{\rho}_{C}=\frac{\rho_{0C}}{R_{B}^{3}},$ (144)

and would write an effective Friedmann equation in the baryonic frame

$H_{B}^{2}\equiv\frac{8\pi G_{B}}{3}\tilde{\rho}_{B}+\frac{8\pi
G_{C}}{3}\tilde{\rho}_{C}+\frac{8\pi G_{N}}{3}\tilde{\rho}_{\Lambda}~{},$
(145)

where we have used $8\pi G_{N}=1/M_{\rm p}^{2}$. This allows one to identify
the effective Newton constants $G_{B,C}$ and the dark energy component
$\bar{\rho}_{\Lambda}$. None of these constants are constant in the baryon
frame. In practice, the effective Newton constants are determined by

$G_{B}=B_{B}^{2}{G_{N}},\ G_{C}=B_{B}B_{C}G_{N}$ (146)

i.e. the two Newtonian constants evolve differently. The dark energy component
is simply defined as the complement to the baryon and CDM contributions in the
baryonic Friedmann equation.

As long as the fields do not evolve rapidly, i.e. at the beginning of the
matter era we have $H_{B}\approx H/B_{B}$. The dark energy component becomes

$\tilde{\rho}_{\Lambda}=\frac{\rho_{f}+\rho_{\Lambda}}{B_{B}^{2}}.$ (147)

In the late Universe, this identification is not valid anymore and a numerical
integration of the equations of motion is necessary.

The same Friedmann equation in the baryon frame can be written as

$\Omega_{B}^{B}+\Omega_{C}^{B}+\Omega_{\Lambda}^{B}=1$ (148)

where the energy fractions $\Omega^{B}_{i}$, $i=B,C,\Lambda$ are identified in
the baryon frame and are such that

$\Omega_{B,C}^{B}=\frac{8\pi G_{B,C}\tilde{\rho}_{B,C}}{3H_{B}^{2}},\
\Omega_{\Lambda}^{B}=\frac{8\pi G_{N}\tilde{\rho}_{\Lambda}}{3H_{B}^{2}}.$
(149)

The deviations of the energy fractions from $\Lambda$CDM are represented in
Fig. 5.

###  IV.7 Deviations from $\Lambda\mathrm{CDM}$ and observational constraints

We will first define the effective gravitational constant that an observation
would measure. For example, Big Bang Nucleosynthesis (BBN) puts constraints on
the variation of such a $G_{\rm eff}$ Accetta  _et al._ (1990); Uzan (2003):

$\absolutevalue{\Delta
G/G}\equiv\absolutevalue{\frac{G_{\text{eff}}^{\text{today}}-G_{\text{eff}}^{\text{BBN}}}{G_{\text{eff}}^{\text{BBN}}}}<0.4~{}.$
(150)

This assumes a Hubble evolution similar to that of $\Lambda\mathrm{CDM}$ in
the matter-dominated era. In our model, since observations are made in the
baryon frame, the corresponding $G_{\text{eff}}$ satisfies

$H_{B}^{2}=\frac{8\pi}{3}G_{\text{eff}}\bqty{\frac{c_{\rho}}{R_{B}^{3}}+\rho_{\Lambda,B}}~{},$
(151)

for some $c_{\rho}$. In order to have $G_{\text{eff}}(z_{i})=G_{N}$ initially,
we set $c_{\rho}=\rho_{0}$, as defined by Eq.(136), i.e we normalise Newton’s
constant by using the Planck normalisation in the early Universe. Since the
physics between BBN and $z=z_{i}$ is the same as in the standard model,
$G_{\text{eff}}=G_{N}$ till $z=z_{i}$. Later the relative variation of the
effective coupling to baryons can be computed between $z=z_{i}$ and today:

$\evaluated{\frac{\Delta G_{B}}{G_{B}}}_{{\rm BBN}\rightarrow{\rm
today}}=\evaluated{\frac{\Delta G_{B}}{G_{B}}}_{z_{i}\rightarrow{\rm
today}}~{}.$ (152)

We can further constrain the possible deviations of the Hubble rate from the
standard model by imposing that this should be less than the discrepancy
appearing in the $H_{0}$ tension. Indeed, there are two diverging
determinations of the present time Hubble rate $H_{0}$ with a relative
difference of order $10\%$ Valentino  _et al._ (2021). In the axio-dilaton
theory, the fact that Newton’s constant varies implies that the Hubble now
differs from the corresponding Hubble rate in the standard model . We have
normalised the Hubble rates to coincide at the beginning of the matter era.
This motivates looking for parameters that satisfy

$\absolutevalue{\frac{\Delta H_{B}}{H_{B}}}_{\rm
tension}\equiv\absolutevalue{\frac{H_{B}(z=0)-H_{\rm SM}(z=0)}{H_{\rm
SM}(z=0)}}<0.1~{},$ (153)

where $H_{\rm SM}$ is the Hubble rate in the standard model. Another stringent
constraint comes from BAO (Baryon Acoustic Oscillations) Alam  _et al._ (2021)
which specify that the deviations of $H_{B}(z)$ for $0.2\lesssim z\lesssim
2.5$ should less than around 3 percent Alam  _et al._ (2021)

$\displaystyle\absolutevalue{\frac{\Delta H_{B}}{H_{B}}}_{\rm BAO}\equiv$
$\displaystyle\absolutevalue{\frac{H_{B}(z\in[0.2,2.5])-H_{B}^{\rm
SM}(z\in[0.2,2.5])}{H_{B}^{\text{SM}}(z\in[0.2,2.5])}}<0.03~{}.$

This implies that the differences between $\Lambda$-CDM and the axio-dilaton
models must appear late in the evolution of the Universe. We will see that the
BAO constraint is the most stringent one amongst the ones we have selected. Of
course a much more precise numerical study is required to constrain the
parameter space. This is left to future work.

We also consider the effective equation of state of dark energy. In GR with
matter in addition to a fluid $X$ with equation of state $w$, the deceleration
parameter

$q_{0}:=-\absolutevalue{\frac{\ddot{R}}{RH^{2}}}_{\text{today}}~{}.$ (155)

is given by

$q_{0}=\frac{1}{2}(\Omega_{m,0}+(1+3\omega)\Omega_{X,0})$ (156)

where $\Omega_{i,0}=\rho_{i}(z=0)/3M_{p}^{2}H_{0}^{2}$. Observations give thus
an estimate for $w$ depending on $\Omega_{m,0}$. For $\Omega_{m,0}\sim 0.3$,
which can be obtained independently, this leads to Seljak  _et al._ (2005);
Peter and Uzan (2005)

$w\sim-1\pm 0.1~{}.$ (157)

Recent constraints and future prospects can be found in Linder (2023). We now
define the effective equation of state

$w_{\mathrm{eff}}\equiv\frac{1}{3}{\frac{2q_{B,0}-\Omega_{m,0}}{\Omega_{\Lambda,0}}-1}~{},$
(158)

where:

$q_{B,0}=-\evaluated{\frac{\partial_{t_{B}}^{2}{R_{B}}}{R_{B}H_{B}^{2}}}_{\text{today}}~{}.$
(159)

Taking $\Omega_{m,0}\simeq 0.3$, we must impose the constraint

$|{\Delta w}|\equiv|{w_{\mathrm{eff}}+1}|\lesssim 0.1~{}.$ (160)

We will use this bound in what follows as a guiding principle. We are not
trying to give a precise fit to the data but an indication on the parameter
space compatible with cosmology.

###  IV.8 Numerical integration

####  IV.8.1 Dimensionless equations

In the following, we will use $L=\sqrt{3M_{P}^{2}/\rho_{0}}$ as the unit of
time and length. Here $\rho_{0}$ is defined via Eq.(135). We will obtain
quantities as functions of redshift starting at matter-radiation equality. We
define the system of dynamical equations with the number of efolds $N$ and
$\varphi$ such that $R=e^{N}$ and $\tau=e^{\varphi}$. We get

$\displaystyle\ddot{\varphi}=-3\hat{H}\dot{\varphi}-\dot{a}^{2}e^{-2\varphi}+{\kappa}\hat{\rho}$
$\displaystyle\ddot{a}=-3\hat{H}\dot{a}+2\dot{\varphi}\dot{a}+\epsilon\hat{\rho}e^{2\varphi}$
$\displaystyle\dot{N}=\hat{H}\ $

with

$\displaystyle\hat{H}^{2}=\hat{\rho}+\hat{\rho}_{f}+\hat{\rho}_{\Lambda},\quad\hat{\rho}=Be^{-3N},$
$\displaystyle\hat{\rho}_{f}=\frac{1}{4}(\dot{\varphi}^{2}+\dot{a}^{2}e^{-2\varphi})~{}.$

Here $\hat{\rho}_{\Lambda}$ corresponds to $\rho_{\Lambda}/\rho_{0}$ where
$\rho_{\Lambda}$ is the density associated to the cosmological constant. In
$\Lambda\mathrm{CDM}$, $\rho_{0}$ is also the value of the matter density
today, so that $\hat{\rho}_{\Lambda}=\Omega_{\Lambda,0}/\Omega_{m,0}\simeq
7/3$ in $\Lambda\mathrm{CDM}$. It is convenient to work in conformal time
$\eta$, such that $Rd\eta=dt$, implying that

$\displaystyle{\varphi}^{\prime\prime}=-2\hat{\mathcal{H}}{\varphi}^{\prime}-{a^{\prime}}^{2}e^{-2\varphi}+\kappa\tilde{\rho}$
$\displaystyle{a}^{\prime\prime}=-2\hat{\mathcal{H}}{a}^{\prime}+2{\varphi^{\prime}}{a^{\prime}}+\epsilon\tilde{\rho}e^{2\varphi}$
$\displaystyle{N}^{\prime}=\hat{\mathcal{H}}$

with

$\hat{\mathcal{H}}^{2}=\check{\rho}+\check{\rho}_{f}+\check{\rho}_{\Lambda}~{},$
(163)

where the derivatives are now with respect to ${\eta}$. We have rescaled all
the densities as $\check{\rho}=e^{2N}\hat{\rho}$ and similarly for the scalar
and dark energy parts.

####  IV.8.2 Results

Figure 2: We show the values of the parameters which satisfy the cosmological
tests for $\kappa=10^{-3}$ and $\epsilon_{B}=10^{-3}$. $602\times 602$ points
are plotted. The coloured regions satisfy the four constraints: $|{\Delta
H_{0}/H_{0}}|<0.1$, $|{\Delta w}|<0.1$, $|{\Delta G/G}|<0.4$, $|\Delta
H/H|_{\text{BAO}}<0.03$. Figure 3: Positions of a selection of parameters in
the region of parameter space where the cosmological tests are satisfied:
$\epsilon_{C}=0.05,\hat{\rho}_{\Lambda}=2.2$;
$\epsilon_{C}=0.05,\hat{\rho}_{\Lambda}=2.5$;
$\epsilon_{C}=0.15,\hat{\rho}_{\Lambda}=2.3$;
$\epsilon_{C}=0.17,\hat{\rho}_{\Lambda}=2.55$;
$\epsilon_{C}=0.22,\hat{\rho}_{\Lambda}=2.65$.

Figure 4: The Field evolutions for $\kappa=10^{-3}$ and $\epsilon_{B}=10^{-3}$
as a function of the redshift $z$ defined in the baryon frame. The values of
$\epsilon_{C}$ and $\hat{\rho}_{\Lambda}$ for each colour are given in Fig.3.

Figure 5: Evolution of energy contents of the Universe in the Einstein frame
(left) and baryon frame (top three figures on the right) as a function of the
redshift of the baryon frame. The $\Lambda\mathrm{CDM}$ case is represented in
the bottom right figure. Deviations from $\Lambda$CDM are represented in the
other figures for $\kappa=10^{-3}$, $\epsilon_{B}=10^{-3}$ and values of
$\epsilon_{C}$ and $\hat{\rho}_{\Lambda}$ given in Fig.3.

Figure 6: Evolution of the cosmological quantities $H_{B}$, $G_{\rm eff}$ and
$w_{\rm eff}$ as a function of the redshift in the baryon frame for
$\kappa=10^{-3}$ and $\epsilon_{B}=10^{-3}$. We zoom on the late evolution on
the right. The values of $\epsilon_{C}$ and $\hat{\rho}_{\Lambda}$ for each
colour are given in Fig.3.

Figure 7: Deviations of the Hubble rate from $\Lambda$CDM for different values
of $\kappa$, $\epsilon_{C}=\epsilon_{B}=10^{-3}$ and
$\hat{\rho}_{\Lambda}=7/3$, as a function of the redshift in the baryon frame.

There are four main parameters: $\kappa$, $\epsilon_{B}$, $\epsilon_{C}$ and
$\hat{\rho}_{\Lambda}$. Following our discussion about screening in §III.4 and
our identification of the effective metric in §IV.1, we will first take
$\kappa$ and $\epsilon_{B}$ small. Taking $\kappa=\epsilon_{B}=10^{-3}$ turns
out to be enough to satisfy solar system constraints. Taking one of them to be
one order of magnitude higher leads to a violation of the cosmological
constraints when $\epsilon_{C}$ is large enough. Numerically we study the
cosmological evolution for a large range of values of $\epsilon_{C}$ and
$\hat{\rho}_{\Lambda}$. We always start the cosmological evolution at
$z_{i}=3400\approx z_{\mathrm{eq}}$.

A first picture of the deviations from $\Lambda\mathrm{CDM}$ can be obtained
by focusing on the cosmological tests (Newton’s constant, the equation of
state, the Hubble parameter today and in $z\in[0.2,2.5]$ relative to that of
the standard model) we defined above and their consequences as viewed in the
$\epsilon-\hat{\rho}_{\Lambda}$ plane. This is shown in Fig.2. We show their
values only for the region in which the four of them satisfy the observational
constraints. It turns out that the constraint from BAO is always the
strongest. We illustrate the cosmological evolution using five sets of
parameters in Figs.(4,6,5). Standard model quantities are computed
analytically from the $\Lambda\mathrm{CDM}$ solutions in the matter dominated
era with $\Omega_{\Lambda,0}/\Omega_{m,0}=7/3$.

We recover that the smallest deviations to $\Lambda\mathrm{CDM}$ are for
$\epsilon_{C}=0$, $\hat{\rho}_{\Lambda}\simeq 7/3$. We observe that deviations
of $H_{0}$ depend mostly on $\hat{\rho}_{\Lambda}$ while deviations of $G$ are
essentially given by $\epsilon_{C}$. Additionally, we see deviations which are
both positive and negative for $H_{0}$, the sign depending on the sign of the
deviation of $\hat{\rho}_{\Lambda}$ compared to $\approx 7/3$. On the other
hand, there are only negative deviations of $G$. As we observe in Fig.4,
$\epsilon a$ increases and so $G_{B}$ decreases.

For both $H_{0}$ and $w_{\mathrm{eff}}$, the deviations due to $\epsilon_{C}$
tend to be compensated by deviations of $\hat{\rho}_{\Lambda}$. This can be
understood from the fact that the fields act as a fluid of equation of state
$w_{f}=1$ opposite to that of the cosmological constant $w_{\Lambda}=-1$. They
have opposite effects on the cosmic acceleration.

Moving on to the dynamical curves, we see in Fig.4 the evolution of the
fields. The axion $a$ increases with $z$, which can be expected as the source
term in its Klein-Gordon equation (126) has a factor $\epsilon>0$ and
initially dominates. On the other hand, $\varphi$ slightly increases at first
but eventually decreases much more. This can be expected, as the source term
in its Klein-Gordon equation has a factor $\kappa>0$ which initially
dominates, and later the source term is dominated by the axion term
proportional to $-\dot{a}^{2}$. The deviation from the constant and vanishing
fields increases with $\epsilon$ as we can expect.

Fig.5 shows that the evolution of the energy content of the Universe in both
the Einstein and baryon frames. As the couplings $\kappa$ and $\epsilon_{B}$
are small, the difference between the Einstein and the baryon frame quantities
is negligible. In the matter-dominated era we have a slight increase in the
field density and a corresponding decrease of the matter density. In the very
late Universe close to a vanishing redshift the proportion of the cosmological
constant increases and we get the usual values 0.3 and 0.7 for matter and
$\Lambda$ with some deviations of the order of 0.01.

As a result we can see that generically the matter contents of the Universe
are modified. First of all, the axion and dilaton energy densities evolve from
being negligible initially to a long phase in the matter era where they remain
nearly constant before dropping to lower values in the last few efoldings of
the Universe. This has important consequences as the dynamics of the axio-
dilaton system, and in particular the variation of the conformal factors
$B_{B,C}$ imply that the matter fractions of the Universe deviate from their
$\Lambda$CDM values. We also notice that the deviation can be negative by a
few percent. This is important as this will hamper the growth of structure and
entail a compensation of the extra growth due to the attractive scalar forces
by the lower amount of matter in the Universe. This will result in a reduced
growth of structure in these models.

Fig.6 shows the evolution for the cosmological quantities of interest. In
$\Lambda\mathrm{CDM}$, both $\Delta H_{0}/H_{0}$ and $\Delta G/G$ vanish. We
observe deviations that gets stronger as the parameters move away from the
light regions and into the darker ones of Fig.2. Both also present a maximal
negative deviation around $z\sim 2$. At smaller redshifts the deviation
shrinks. This can be attributed to the effect of the cosmological constant
which becomes non-negligible at late times. In the case of the red, orange and
purple curves, the deviation crosses $0$ and becomes positive. They have in
common that their values for the cosmological constant are stronger than the
standard model value: $\hat{\rho}_{\Lambda}\gtrsim 7/3$. Finally, notice that
the evolution of $G_{\rm eff}$ shows a negative deviation from Newton’s
constant in GR. This has two origins which can be traced to its definition
(151). First of all, as the effective Newtonian constant is normalised with
the Planck normalisation for $\rho_{0}$, the fact that in Fig. 5 we find that
there is less matter in the recent Universe than in $\Lambda$CDM implies that
$G_{\rm eff}/G_{N}$ should be less than unity. Another important effect is
that the conformal factors $B_{B,C}$ are less than unity too, implying a
reduction of $G_{\rm eff}$ compared to $G_{N}$. These effects could have been
compensated by the extra pull arising from the scalar forces. This is not the
case.

Fig.6 also shows $w_{\mathrm{eff}}$. It is defined by generalizing (158)

$w_{\mathrm{eff}}(z):=\frac{1}{3}\pqty{\frac{2q_{J}(z)-\Omega_{m,0}}{\Omega_{\Lambda,0}}-1}~{}.$
(164)

This can simply be seen as a rescaling of the deceleration parameter $q(z)$.
In $\Lambda\mathrm{CDM}$ it is easy to see that $w_{\mathrm{eff}}$ goes from
$0$ to $-1$ (and equivalently, $q$ from $0.5$ to $-0.55$). Fig.6 shows that
for parameters consistent with observation the deviations are not drastic.

####  IV.8.3 Cosmological constraint on $\kappa$

We now concentrate on another scenario where the coupling of the dilaton is
relaxed from its solar system bound. This is of interest if the dilaton is
screened in the solar system and partially on cosmological scales. The largest
dilaton coupling allowed by cosmological data is smaller than the supergravity
motivated value $\kappa=1$.

We fix $\epsilon_{B}=10^{-3}$, and we look for the highest value of $\kappa$
such that there are some values of $\epsilon_{C}$ and $\hat{\rho}_{\Lambda}$
such that the observables are within the observational bounds. We use a
precision of the order $10^{-3}$ relative to the order of magnitude of the
parameters. We find that the maximal order of magnitude of $\kappa$ is $0.1$.
More precisely, if we fix also $\epsilon_{C}$ and try to find some valid
values of $\hat{\rho}_{\Lambda}$, we find that for $\epsilon_{C}=0.1$,
$\kappa_{\rm lim}=0.110$; and for $\epsilon_{C}=0.001$, $\kappa_{\rm
lim}=0.124$.

This is illustrated in Fig.7. Since BAO is the strongest constraint, we look
only at the deviations of the Hubble rate in the BAO interval. We can see that
for favourable values of the other parameters
($\epsilon_{C}=\epsilon_{B}=10^{-3}$, $\hat{\rho}_{\Lambda}=7/3$), $\kappa$ is
allowed at least up to $0.12$. But for $\kappa=0.13$ the BAO constraint is no
longer satisfied. This is of course much less than unity and signals that the
dilaton must be cosmologically screeened.

##  V Cosmological perturbations

In this section, we study the growth of perturbations for the axio-dilaton
models in the matter era in the sub-horizon and quasi-static approximation
Pogosian and Silvestri (2016). We will obtain the equation governing the
evolution of the density contrast $\delta\rho/\rho$ Zhao  _et al._ (2009).

###  V.1 Cosmological perturbations

We focus on small perturbations to the background solutions. We are interested
in structure formation and will evaluate the time evolution of the baryon and
CDM density contrasts simultaneously Huterer  _et al._ (2015). We only
consider the scalar modes of the perturbation of the metric. Using Newton’s
gauge in the Einstein frame we have

${g}_{\mu\nu}dx^{\mu}dx^{\nu}=R^{2}(\eta)(-({1+2\Phi_{N}})d\eta^{2}+(1-2\Phi_{N})\gamma_{ij}dx^{i}dx^{j})$
(165)

where $\gamma_{ij}=\delta_{ij}$. The cosmological perturbations are defined by

$\rho_{i}=\overline{\rho}_{i}+\delta\rho_{i}=\overline{\rho}_{i}(1+\delta_{i}),\
u^{\mu}_{i}=\frac{1}{R}(1+\delta v^{0}\,,\,\vec{v}_{i})$ (166)

for each field. We also perturb the fields and the axion-matter coupling:

$\varphi=\overline{\varphi}+\delta\varphi~{},$ (167) $a=\overline{a}+\delta
a~{},$ (168) $\mathcal{A}=\overline{\mathcal{A}}+\delta\mathcal{A}~{}.$ (169)

All the symbols with a bar denote background values. The velocity
$u^{\mu}_{i}$ of the two fluids are given by

$u^{\mu}_{i}=\frac{1}{R}(1-\Phi_{N}\,,\,\vec{v}_{i})~{}.$ (170)

at linear order. The perturbed Newton’s law becomes for each species

$\partial_{\eta}\vec{v}_{i}+(\mathcal{H}+\partial_{\eta}\overline{\chi}_{i})\vec{v}_{i}=-\frac{\vec{\nabla}}{R^{2}}(\Phi_{N}+\delta\chi_{i}).$
(171)

We recognise the scalar force in the Euler’s equation due to the interaction
with dark matter and the friction term whose origin is the modified Hubble
rate ${\cal H}_{i}=\mathcal{H}+\partial_{\eta}\overline{\chi}_{i}$ in
conformal time and in the Einstein frame.

Figure 8: Growth rate for a few allowed parameters and in
$\Lambda\mathrm{CDM}$ shown for redshifts in the baryon frame $z\leq 3$.
Figure 9: Growth rate at $z\leq 3$ in the baryon frame for different values of
$\kappa$ and $\epsilon_{C}=\epsilon_{B}=10^{-3}$, $\hat{\rho}_{\Lambda}=7/3$.

###  V.2 Density contrast of sub-horizon modes in quasi-static regime

We focus on sub-horizon modes such that $k\gg\mathcal{H}$ and in the quasi-
static regime $\partial_{\eta}\sim\mathcal{H}$. Using the Einstein equation

$\mathcal{R}_{00}-\frac{3}{2\tau^{2}}(\partial_{\eta}\tau\partial_{\eta}\tau+\partial_{\eta}a\partial_{\eta}a)-\frac{1}{M_{p}^{2}}(T_{00}-\frac{1}{2}Tg_{00})=0~{},$
(172)

and the curvature perturbation

$\delta\mathcal{R}_{00}=3\mathcal{H}\Phi^{\prime}_{N}+\Delta\Phi_{N}\approx\Delta\Phi_{N}~{},$
(173)

which reduces to the Laplacian of Newton’s potential as
$3\mathcal{H}\Phi^{\prime}_{N}\sim\mathcal{H}^{2}\Phi_{N}\ll
k^{2}\Phi_{N}\sim\Delta\Phi_{N}$, the Poisson equation then becomes

$\Delta\Phi_{N}=\frac{1}{2M_{p}^{2}}R^{2}(\bar{\rho}_{B}\delta_{B}+\bar{\rho}_{C}\delta_{C})~{}.$
(174)

The perturbed Klein-Gordon equation for $\varphi$ is

$\Delta\delta\varphi=-\frac{\kappa}{3M_{p}^{2}}R^{2}(\bar{\rho}_{B}\delta_{B}+\bar{\rho}_{C}\delta_{C})~{}.$
(175)

Its structure is similar to the Poisson equation. Similarly we obtain for the
axion field

$\Delta\delta
a=-\frac{e^{2\overline{\varphi}}}{3M_{p}^{2}}R^{2}(\bar{\rho}_{B}\epsilon_{B}\delta_{B}+\bar{\rho}_{C}\epsilon_{C}\delta_{C})$
(176)

where we have systematically used the sub-horizon and quasi-static
approximations. The conservation equation for each species implies that
$\rho_{i}=B_{i}\rho_{i\rm cons}$ where the conserved density satisfies

$\frac{d\rho_{i\rm con}}{d\tau_{i}}+3h_{i}\rho_{i\rm con}=0.$ (177)

In the subhorizon limit, we can identify $\delta_{i}\simeq\delta_{i\rm cons}$.
This is also the density constrast in the baryon frame as the contributions of
both $\delta a$ and $\delta\phi$ to the change of frame are negligible in the
subhorizon limit. This implies that the perturbed conservation equation
becomes

$\delta^{\prime}_{i}=-\vec{\nabla}.\vec{v}_{i}~{}.$ (178)

Now we apply $\vec{\nabla}$ to Eq.(171) and obtain

$-\delta_{i}^{\prime\prime}-(\mathcal{H}+\overline{\chi}_{i}^{\prime})\delta_{i}^{\prime}=-\frac{\Delta}{R^{2}}(\Phi_{N}+\delta\chi_{i})~{}.$
(179)

Using the Laplacians from Eq.(174), Eq.(175), and Eq.(176) we finally deduce
the growth equation for each species in the subhorizon limit

$\displaystyle\delta_{i}^{\prime\prime}+\left(\mathcal{H}+\overline{\chi}_{i}^{\prime}\right)\delta_{i}^{\prime}-\frac{3}{2}\Omega_{B}\mathcal{H}^{2}\left(1+\frac{\kappa^{2}+e^{2\overline{\varphi}}\epsilon_{B}^{2}}{3}\right)\delta_{B}$
$\displaystyle-\frac{3}{2}\Omega_{C}\mathcal{H}^{2}\left(1+\frac{\kappa^{2}+e^{2\overline{\varphi}}\epsilon_{C}^{2}}{3}\right)\delta_{C}=0$

in terms of the matter fraction $\Omega_{i}$ in the Einstein frame. We find
that the deviations from the standard model have two origins. First there is
the friction term depending on ${\cal H}_{i}={\cal
H}+\overline{\chi}_{i}^{\prime}$ which is specific to each species as the two
fluids couple differently to the axion. There is also a modification of
Newton’s constant for each species on the two effective couplings

$G_{N}^{i}=(1+2Q_{i}^{2})G_{N}$ (181)

such that the perturbation equations become

$\displaystyle\delta_{i}^{\prime\prime}+(\mathcal{H}+\overline{\chi}_{i}^{\prime})\delta_{i}^{\prime}$
$\displaystyle-\frac{3}{2}\Omega_{B}\mathcal{H}^{2}(1+2Q_{B}^{2})\delta_{B}-\frac{3}{2}\Omega_{C}\mathcal{H}^{2}(1+2Q_{C}^{2})\delta_{C}=0~{}.$

We have defined the effective couplings

$Q_{i}^{2}=\frac{\kappa^{2}+e^{2\overline{\varphi}}\epsilon^{2}_{i}}{6}$ (183)

which parameterise the deviations from $\Lambda$CDM. We retrieve that in the
absence of the $\epsilon_{i}$’s, gravity is modified by a factor
$\kappa^{2}/6$ like in the static regime around a compact object. In the
following, we will solve these equations numerically as a way of investigating
the growth of structure for the baryons and CDM.

###  V.3 Growth rate

We now focus on the growth rate for small redshifts DESI Collaboration (2016)
as defined by

$f_{i}=\frac{\mathrm{d}\,\ln{\delta_{i}}}{\mathrm{d}\,\ln{R_{B}}}$ (184)

where the redshift is deduced from the baryonic scale factor. We represent the
growth rates in Fig.8. We choose as initial conditions $\delta_{i}(z_{ini})\ll
1$ and
$\delta_{i}^{\prime}(z_{ini})/\delta_{i}(z_{ini})\sim\mathcal{H}_{ini}$. We
notice that the growth can be either enhanced or disfavoured at small
redshifts depending on $\epsilon_{C}$ and $\rho_{\Lambda}$. On the other hand,
the maximal deviation from $\Lambda$CDM is at most five percent. We have also
represented the growth factors when $\kappa$ is varied and both
$\epsilon_{B,C}$ are small. Notice that when $\kappa$ is varied, the growth
rate at small redshift becomes smaller and smaller. This is also the case when
$\kappa$ is fixed and $\epsilon_{C}$ is increased. As the effective Newton
constants should increase the growth thanks to the presence of fifth forces
between particles, we conclude that the background evolution and the effective
friction have a drastic effect on the growth of structure. This can be
observed in Fig.5 where the matter density in the late Universe decreases
compared to $\Lambda$CDM as in Barros  _et al._ (2019) where a similar effect
was obtained and used to alleviate the $\sigma_{8}$ tension. It would
certainly be interesting to see if this trend can be also present in the non-
linear regime and could have some consequences for the $S_{8}$ tension where
less matter clustering is observed at late time than inferred in the
$\Lambda$CDM scenario Poulin  _et al._ (2022) from the Planck data Aghanim
_et al._ (2020). The analysis of the $S_{8}$ tension in this scenario is left
for future work.

Finally let us remark that we have not taken into account an important effect
which would result from both the fact that the effective equation of state
$w_{\rm eff}$ (see Fig. 6) is not strictly equal to zero deep into the matter
era for $z\gtrsim 2$ and that the perturbations do not behave like in the
Einstein-de Sitter Universe with $f\neq 1$ (see Fig. 8). This would imply that
the Newtonian potential $\Phi_{N}$ in (174) is not strictly constant in the
matter era. This could lead to a large Integrated Sachs Wolfe effect (ISW)
which is tightly constrained and could appear in the galaxy counts versus the
CMB (Cosmic Microwave Background) cross-correlations Crittenden and Turok
(1996); Khosravi  _et al._ (2016); Benevento  _et al._ (2018). This will
certainly restrict the available parameter space and constrain the possible
deviations of the growth factor from $\Lambda$CDM. A detailed study of this
effect is left for future work.

##  VI Conclusion

The axio-dilaton model has a clear origin in string theory. We have focused on
the screening mechanism introduced in Burgess and Quevedo (2022) for a
constant coupling of the axion to matter and explicitly shown how it can only
be effective when the field values at infinity are tuned to specific values.
These values should be determined cosmologically. We have studied the
background cosmology of these models and shown that the cosmological values do
not correspond to the tuned values generically.

In the absence of explicit screening for the axio-dilaton system which may
require to introduce non-linear couplings to matter and/or new fields Burgess
_et al._ (2022) whose dynamics would drive the couplings of the dilaton and
the axion to small values in the solar system and larger values
cosmologically, we have employed a simple alternative and considered two
scenarios. In the first one, the coupling of the dilaton and the axion to
baryons is taken to satisfy the solar system constraints and remain identical
to these values on large scales. Only the coupling to cold dark matter is
allowed to take much larger values. In this case, we find that the coupling to
cold dark matter must be bounded. It turns out that the constraints from
Baryonic Acoustic Oscillations (BAO) at small redshift are the tightest and
the present day Hubble rate does not deviate from the Planck normalised one by
more than three percent. This is not enough to account for the $H_{0}$
tension, which lies at the ten percent level. Similarly, the growth of
structure is affected at the five percent level. Interestingly, in these
models growth is not always enhanced and effectively a decrease in the growth
rate is observed for a large part of the parameter space of the model. This
follows from the fact that the growth increase due to the scalar forces is
compensated by the decrease of the matter density. This could have some
relevance to the $\sigma_{8}$ tension. We hope to come back to this suggestion
in the near future. We also consider the case where the axion does not couple
significantly to matter and the dilaton couples with a strength $\kappa$
reduced from the string theory motivated example. We find that $\kappa$ cannot
be allowed values of order unity and must be bounded around $0.1$. This
entails that the dilaton must not only be screened locally in the solar
system, but also cosmologically.

Of course, our examples can be modified and the resulting physics very
different. For instance, the couplings to matter of both the axion and the
dilaton could become non-linear and therefore lead to screening mechanisms
akin to the ones of the symmetron model for instance. Another possibility
would be that other fields could relax to values whereby the couplings of the
dilaton and the axion would become very small in the solar system and small
cosmologically. The construction of these models is left for future work.

Phenomenologically, the scenarios we have introduced fall within the category
of late time dark energy models where the evolution of the fields at small
redshift would modify both the background cosmology and the growth of
structure. As expected, we find that the BAO bound entails a tight constraint
on both the possible deviations of the present Hubble rate from its Planck
value and the growth factor from its $\Lambda$CDM counterpart. We notice that
the allowed deviations of the growth factor could reach a few percent and
therefore may become detectable by future large scale surveys. The detailed
study of the phenomenology of these models is left for future work.

Acknowledgments: We would like to thank E. Lindner and L. Pogosian for
fruitful comments. We would also like to thank A. T. Ortiz for running some of
the long numerical computations. A.O. would like to thank IPhT for funding
during his internship, as well as the physics department of the Ecole Normale
Supérieure for its scholarship.

##  Appendix A Validity of the flat background approximation

We consider here the conditions for the flat background approximation used in
§III to hold. We assume the exterior solution of §III.1 and find in which
regime the terms neglected by the approximation are indeed negligible. The
metric of §III can be written:

$ds^{2}=-({1+2\Phi})dt^{2}+(1-2\Psi)(dr^{2}+r^{2}d\Omega^{2})~{}.$ (185)

The Christoffel symbols, the Ricci and Einstein tensors for this metric can be
found in e.g. Carroll (2004) §7. The $(tt)$ and $(rr)$ equations from Eq.(13)
are

$\Delta\Phi=\frac{\rho}{2M_{p}^{2}}~{},$ (186)
$(\Delta-\partial_{r}^{2})({\Phi-\Psi})=\frac{3}{4}\frac{{(\tau^{\prime})^{2}+(a^{\prime})^{2}}}{\tau^{2}}~{},$
(187)

where $\Delta$ is the Laplacian. The exterior solution gives

$\frac{{(\tau^{\prime})^{2}+(a^{\prime})^{2}}}{\tau^{2}}=\frac{\gamma^{2}\beta^{2}}{r^{4}}~{}.$
(188)

Integration gives us then

$\Phi=-\frac{GM}{r}\qquad,\qquad\Psi-\Phi=\frac{3}{16}\frac{\gamma^{2}\beta^{2}}{r^{2}}~{}.$
(189)

The deviation from the flat metric are therefore negligible for

$\absolutevalue{\Psi}\ll 1\qquad,\qquad\Phi\simeq\Psi~{}.$ (190)

This is verified in the regime

$r\gg GM\qquad,\qquad r\gg\absolutevalue{\gamma\beta}~{}.$ (191)

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