Constraints on Late Time Violations of the Equivalence Principle in the Dark Sector Cameron C. Thomas and Carsten van de Bruck Consortium for Fundamental Physics School of Mathematics and Statistics

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Constraints on Late Time Violations of the Equivalence Principle in the Dark Sector

Cameron C. Thomas and Carsten van de Bruck

Consortium for Fundamental Physics, School of Mathematics and Statistics,

University of Sheﬃeld, Hounsﬁeld Road, Sheﬃeld S3 7RH, United Kingdom

(Dated: November 17, 2022)

If dark energy is dynamical due to the evolution of a scalar ﬁeld, then in general it is expected

that the scalar is coupled to matter. While couplings to the standard model particles are highly

constrained by local experiments, bounds on couplings to dark matter (DM) are only obtained

from cosmological observations and they are consequently weaker. It has recently been pointed

out that the coupling itself can become non-zero only at the time of dark energy domination, due

to the evolution of dark energy itself, leading to a violation of the equivalence principle (EP) in

the dark sector at late times. In this paper we study a speciﬁc model and show that such late-

time violations of the EP in the DM sector are not strongly constrained by the evolution of the

cosmological background and by observables in the linear regime (e.g. from the cosmic microwave

background radiation). A study of perturbations in non-linear regime is necessary to constrain

late–time violations of the equivalence principle much more strongly.

I. INTRODUCTION

There are several reasons to study alternative theories to

the cosmological constant as a model for dark energy (DE).

Firstly, if DE is due to a non-vanishing cosmological con-

stant, its value has to be very small to ﬁt the data. The

expected theoretical value, however, is much larger. This

problem has not been solved, but there are attempts to ad-

dress this problem (see [1–3] and references therein). Sec-

ondly, there are several tensions between data sets, provid-

ing tantalising hints that the standard model of cosmology,

the Λ-Cold-Dark-Matter(ΛCDM) model, may be in need

of corrections (we refer to [4] for a recent overview over

the tensions and [5] for an overview of attempts to solve

the tensions). Among the extensions of the ΛCDM model

which remain the best motivated ones are scalar-ﬁeld mod-

els of DE, in which the accelerated expansion is driven by a

scalar ﬁeld [6–8]. It is expected that, in general, the scalar

is coupled to at least one species of matter, unless there is

a symmetry which forbids such couplings [9]. Such a cou-

pling results in an additional force mediated between the

coupled species. Since the interaction between DE and or-

dinary matter is strongly constrained, in some models only

the coupling to cold dark matter (CDM) is of cosmological

signiﬁcance (see e.g.[10–14] and references therein). It is

this type of theory which we consider in this paper.

It has recently been suggested, that the potential energy

of scalar ﬁelds appearing in string theory cannot be arbi-

trarily ﬂat [15,16], see [17] for an overview of the swamp-

land programme. If true, the accelerated expansion cannot

be driven by a cosmological constant (de Sitter space is

not realised in string theory) and the equation of state of

dark energy is not constant and deviates potentially sig-

niﬁcantly from the value expected in the ΛCDM model.

Additionally, a coupling of the scalar to some sectors in

the theory are expected. Based on these observations, sev-

eral phenomenological models have been proposed recently

[18–23]. In this paper we study speciﬁc a model in which

the coupling function between the dark energy ﬁeld and

dark matter has a minimum [18]. As a result of the mini-

mum, the coupling switches on only at late times, at the

beginning of the dark energy dominated epoch. One of

our main results of this paper is that the regime in which

linear perturbation theory is valid does not constrain the

parameter of the model greatly. In other words, late time

violations of the equivalence principle in the dark sector

are not strongly constrained by studying the background

evolution or CMB anisotropies. Instead, to obtain stronger

constraints a study of the non-linear regime in considerable

detail is needed.

The paper is organised as follows. In Section II we

present the model and its parameter. In Section III we

describe our methodology, describe the data sets used and

present the constraints on the model. We conclude in Sec-

tion IV.

II. INTERACTING DARK ENERGY

The model we consider consists of the gravitational sec-

tor described by the Einstein–Hilbert action without cos-

mological constant, a part which describes the standard

model (SM) particles and a part for DE described by a

scalar ﬁeld φwith potential energy V(φ). Finally, the in-

teraction between DE and DM is described by a conformal

coupling. The full action reads

S=Zd4x√−gM2

2R − 1

2gµν ∂µφ∂νφ−V(φ)

+Zd4x√−gLSM(g, Ψi) + Zd4xp−˜gLDM(˜g, χ) (1)

where MPl is the reduced Planck mass, Ris the Ricci–

scalar, the SM ﬁelds are denoted by Ψiand χis the DM

ﬁeld (assuming here for simplicity that dark matter con-

sists of only one species). The metrics gand ˜gare related

by a conformal transformation ˜gµν =C(φ)gµν . Such the-

ories have been discussed in considerable length in the lit-

erature, but the new ingredient in this paper is that the

arXiv:2210.09732v2 [hep-th] 16 Nov 2022

function C(φ) has a minimum. That the coupling func-

tions in string theory could possess a minimum due to non-

perturbative eﬀects was suggested in the works by Damour

and Polyakov [24] and has been used in [25] to construct a

dilaton-model of DE. Combining these theoretical develop-

ments motivated us in [18] to consider a speciﬁc model in

which C(φ) has a minimum at some value scalar ﬁeld value

φ∗. To be concrete, in this paper we consider the following

function for C(φ):

C(φ) = cosh √α(φ−φ∗)/MPl,(2)

where φ∗denotes the minimum of the function C(φ) and α

is a constant. In this paper we choose φ∗= 1 MPl without

loss of generality. In [18] it was pointed out that even if

the ﬁeld starts away from the minimum in the very early

universe, there are attractor mechanisms at work in the

early universe which drive the ﬁeld towards the minimum

quickly. Nevertheless, in our analysis below we allow the

ﬁeld to start away from the minimum value at φ∗to ﬁnd

constraints on the initial conditions of the DE ﬁeld. In our

analysis we choose an exponential potential with

V(φ) = V0e−λφ/MPl ,(3)

where λdenotes the slope of the potential, which in string

theory according to [15] cannot be arbitrarily small and

should be O(1). Finally, we assume in the following that

the universe is spatially ﬂat.

Because of the coupling, there is an exchange of energy

between DM and the DE ﬁeld. As a result, the evolution of

the DM density and the modiﬁed Klein–Gordon equation

are given by

˙ρDM + 3HρDM =βM−1

Pl ˙

φρDM

and

φ+ 3H˙

φ+Vφ=−βM−1

Pl ρDM.

In these equations we have deﬁned

β=MPl

d ln C

dφ.

The eﬀective gravitational constant between two DM par-

ticles is given by [18,26]

Geﬀ =GN1+2β2.(4)

The evolution of Geﬀ is shown in Figure 1 for various choices

of parameters αand λ. In general, the additional force be-

tween DM particles due to the scalar ﬁeld only becomes

signiﬁcant at redshifts z < 1, when the DE ﬁeld starts to

evolve due to the inﬂuence of the potential, thereby dis-

placing it from the minimum of the coupling function. We

emphasize that the eﬀective gravitational coupling (Equa-

tion (4)) between DM particles depends on αas well as on

the scalar ﬁeld.

To summarize, the parameters of the model we seek to

constrain are the slope of the potential λ, the parameter

α, which is related to strength of the coupling between

DM and DE, and the initial ﬁeld value φini deep inside the

radiation dominated epoch.

FIG. 1. The evolution of the eﬀective gravitational constant,

deﬁned in Equation (4), for models with a diﬀerent value of α

but same value of λ= 0.1 (top) and for models with a diﬀer-

ent value of λbut same value of 10−4α= 1 (bottom). Where

applicable, the values of our cosmological parameters are taken

from the best ﬁt values of a ΛCDM cosmology based on P lanck

TTTEEE+lowE, such as in Table 2 of [27].

III. METHODOLOGY, DATA AND RESULTS

TABLE I. Flat priors for the cosmological parameters sampled

in our analysis.

Parameter Prior

Ωbh2[0.005,0.1]

Ωcdmh2[0.001,0.99]

100θs[0.5,10.0]

ln 1010As[2.7,4.0]

ns[0.9,1.1]

τreio [0.01,0.8]

λ[0,2]

10−4α[0,50]

φini/MPl [0,2]

In our analysis, the IDE model is described by a set

of nine parameters whose priors are speciﬁed in Table I

and where his the reduced Hubble constant deﬁned by

H0= 100hkms−1Mpc−1. These parameters are the re-

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ConstraintsonLateTimeViolationsoftheEquivalencePrincipleintheDarkSectorCameronC.ThomasandCarstenvandeBruckConsortiumforFundamentalPhysics,SchoolofMathematicsandStatistics,UniversityofSheeld,HounseldRoad,SheeldS37RH,UnitedKingdom(Dated:November17,2022)Ifdarkenergyisdynamicalduetotheevolutionofasca...

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Constraints on Late Time Violations of the Equivalence Principle in the Dark Sector Cameron C. Thomas and Carsten van de Bruck Consortium for Fundamental Physics School of Mathematics and Statistics

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