Searching for dilaton elds in the Ly forest

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Searching for dilaton ﬁelds in the Lyαforest

Louis Hamaide∗

King’s College London, Strand, London, WC2R 2LS, UK

Hendrik M¨uller†

Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, D-53121 Bonn (Endenich), Germany

David J. E. Marsh

King’s College London, Strand, London, WC2R 2LS, UK

(Dated: October 10, 2022)

Dilatons (and moduli) couple to the masses and coupling constants of ordinary matter, and these

quantities are ﬁxed by the local value of the dilaton ﬁeld. If, in addition, the dilaton with mass mφ

contributes to the cosmic dark matter density, then such quantities oscillate in time at the dilaton

Compton frequency. We show how these oscillations lead to broadening and shifting of the Voigt

proﬁle of the Lyαforest, in a manner that is correlated with the local dark matter density. We

further show how tomographic methods allow the eﬀect to be reconstructed by observing the Lyα

forest spectrum of distant quasars. We then simulate a large number of quasar lines of sight using

the lognormal density ﬁeld, and forecast the ability of future astronomical surveys to measure this

eﬀect. We ﬁnd that in the ultra low mass range 10−32 eV ≤mφ≤10−28 eV upcoming observations

can improve over existing limits to the dilaton electron mass and ﬁne structure constant couplings

set by ﬁfth force searches by up to ﬁve orders of magnitude. Our projected limits apply assuming

that the ultralight dilaton makes up a few percent of the dark matter density, consistent with upper

limits set by the cosmic microwave background anisotropies.

I. INTRODUCTION

Dilatons and moduli (including the volume modulus,

and radions) are scalar degrees of freedom of string theory

and other extra-dimensional theories, which arise in the

low energy eﬀective theory after compactiﬁcation [1,2].

These ﬁelds appear in the scalar potential, which can in

some cases lead to their having extremely small masses.

Couplings to the Standard Model arise in a variety of

ways. Moduli, for example, appear in the gauge ki-

netic function, with the scalar moduli giving the value

of the ﬁne structure constant (pseudoscalar axions ﬁx

the Chern-Simons term). The dilaton itself couples to all

ﬁelds via the Einstein frame metric. For brevity in what

follows we refer to such ﬁelds collectively as dilatons.

More generally, since string theory contains no dimen-

sionful constants, all the properties of low energy physics

must be determined by the values of (scalar) ﬁelds. The

observed low energy “constants” therefore only appear

so, with the values ﬁxed only due to the ﬁeld taking a

particular location in some local minimum of the scalar

potential. The ﬁelds would generically be displaced from

this minimum at early times (due to e.g. thermal or

quantum ﬂuctuations). Motion from the initial loca-

tion to the local minimum results in damped oscillations

about the minimum. If such a ﬁeld, φ, is furthermore sta-

ble on cosmological time scales, then today the relic oscil-

lations behave as a contribution to the dark matter den-

sity of the Universe [3]. If initial displacements of φfrom

∗louis.hamaide@kcl.ac.uk

†Both ﬁrst authors have contributed equally to this work.

the vacuum are of order of the GUT scale, the correct

relic abundance is achieved for masses mφ≈10−20 eV. 1

In such a scenario, the constants of nature oscillate with

a frequency given by the dark matter mass, and an ampli-

tude related to the local dark matter density. A number

of surveys have already searched for such eﬀects for dark

matter fully composed of dilatons, unsuccessfully (for a

review see e.g. [5], or novel ideas in [6]). However, we are

encouraged by Webb et al.’s [7,8] searches for dipole vari-

ations in αon cosmological scales, as this could caused

by a mφ≈ O(10−32) eV dilaton. In the following, we

show how to search for higher mode oscillations of αus-

ing the Lyαforest and consequently probe higher dilaton

masses.

The Lyαforest is a prominent absorption feature in

the spectra of distant quasars bluewards of the Lyαab-

sorption line. It consists of densely packed, narrow ab-

sorption lines caused by the absorption of quasar light by

intervening neutral hydrogen in the intergalactic medium

(IGM) along the line of sight [9–11]. The optical depth

of the Lyαabsorption in a single absorption proﬁle is

proportional to the column density of neutral hydrogen.

Hence, the Lyαforest is an excellent tomographic tracer

for cosmic large scale structures [12,13]. Many studies

targeted the line proﬁle of absorption lines in the Lyα

forest to study IGM physics, e.g. by using its curvature

[14–16], by a wavelet analysis [16–19], with the 1D-ﬂux

power spectrum [16,20] and the distribution of Doppler

1For heavy, unstable particles, the corresponding phenomena re-

sult in the “cosmological moduli problem” [4], restricting unsta-

ble moduli to be heavier than around 100 TeV.

arXiv:2210.03705v1 [astro-ph.CO] 7 Oct 2022

parameters [16,21,22]. Recently we developed novel

methods [23] and software [24] to compute the direct de-

convolution of the neutral hydrogen ﬂuctuations from the

Voigt-proﬁle for highest resolution (R&50000) spectra

and applied this procedure to UVES SQUAD data [25]

to obtain measurements of the IGM temperature both

consistent with and more accurate then existing meth-

ods [26].

As we will show, the oscillation of the ﬁne-structure

constant induced by dilaton DM aﬀects the Lyαforest

by shifting the wavelength of the Lyαtransition. For

larger dilaton masses (mφ&10−28 eV) the dilaton under-

goes several oscillations while a photon travels through

an overdensity in the IGM. In this case oscillation of αor

meappears as an additional broadening of the absorp-

tion line, similar to thermal broadening. However, for

the smallest masses (mφ.10−28 eV), i.e. the longest

oscillation times, only partial oscillation occurs while a

photon passes through an overdensity, leading to a sys-

tematic shift of the absorption lines in redshift space.

Both eﬀects modify the absorption proﬁles along the line

of sight in the Lyαforest, and hence may be detectable

in high resolution spectra of quasars, using the tools we

developed in Refs. [23,26].

Following this we create synthetic Lyαforest spectra

for a wide range of dilaton masses (ranging from mφ=

10−20 eV to mφ= 10−32 eV) and compute mock con-

straints and forecasts for upcoming surveys. Our method

gives the strongest constraints for smallest masses, and

for the surveys UVES and SKA could signiﬁcantly out-

perform “ﬁfth force” constraints such as Ref. [27]. Our

main results concerning the dilaton coupling to αand me

are summarized in Figs. 1&2respectively.

The paper is structured as follows: in Sec. II we

present the theoretical basics and discuss the eﬀects of

dilatons on the Lyαforest. In Sec. IV we present our

software and synthetic data set. We present our results in

Sec. Vand discuss possible extensions, future directions

and drawbacks in VI.

For the rest of the manuscript we use Planck 18 cos-

mology [29].

II. THEORY

A. Dilaton Dark Matter

The dilaton couplings to the Standard Model arise

from the action (we adopt some of the notation of [2,30]

in the Einstein frame):

S=Zdx4p|g|1

2∂µφ∂µφ−V(φ) + LSM +Lφ,int

Lφ,int =√4π

MPl

φde

4e2Fµν Fµν −dmemee¯e(1)

where |g|is the determinant of the metric, LSM and Lφ,int

are the Standard Model and dilaton Lagrangians respec-

tively, Fµν is the electromagnetic ﬁeld tensor, ¯e=γ0e

where eis an electron spinor wavefunction, and φis the

canonically normalised dilaton ﬁeld from a local change

of variables. Except when introducing Lyαforest and

the Voigt proﬁle in Secs. II D and II E, we use units

where ~=c= 1. We take the dilaton potential to be

V(φ) = 1

2m2

φφ2, i.e. a simple mass term, valid for small

displacements from the vacuum. Implications of more

general potentials have been explored in [2,31]. We

have purposefully omitted other linear scalar couplings

such as kinetic gauge ﬁeld terms of the form φGA

µν GA

µν ,

quark mass terms φψi†ψi, or a Higgs portal term φH†H

[2,32,33] and quadratic (or higher order) terms. This

is because we are interested in looking for changes in the

energies of atomic states. We mention here that, as dis-

cussed in [2] and references therein, in the presence of

CP violating supersymmetric physics axions and axion-

like particles can couple to an electron mass term as well.

The above couplings (de, dme) can be absorbed into a

rescaled electromagnetic and electron mass terms such

that:

LEM =−1−deκφ

4e2Fµν Fµν ≈ − 1

4(1 + deκφ)e2Fµν Fµν

α→α+δα =α(1 + deκφ) (2)

and

me¯ee →me(1 + dmeκφ)¯ee (3)

where κ=√4π

MPl ,α=e2/4πand e2here (and in rest

of this paper) refers to the squared electric charge and

¯ee refers to the norm of the electron spinor. We thus

see that the local value of the dilaton ﬁeld determines

the local observed value of the ﬁne structure constant.

Exchange of virtual dilaton particles also mediates new

Yukawa forces between Standard Model particles, which

we discuss brieﬂy in Sec. V.

A local displacement from the vacuum expectation

value of the dilaton ﬁeld arises if all or some of the ob-

served cosmic DM abundance, ΩDMh2= ¯ρDM/(8.07 ×

10−11 eV4) [29] (where overbar denotes spatial av-

erage, and his the reduced Hubble rate, H0=

100hkm s−1Mpc−1), is composed of dilatons. The large

occupation number of DM particles throughout the Uni-

verse (we consider cases where the dilaton composes more

than 1% of the total DM) permits a description in terms

of a classical ﬁeld.

Dilaton DM can be produced in the early Universe

by the misalignment mechanism, similarly to the well

known case of axion and scalar ﬁeld DM [3,34–36]. The

background homogeneous ﬁeld evolves according to the

Klein-Gordon equation:

φ+ 3H˙

φ+m2

φφ= 0 ,(4)

where H= ˙a/a is the Hubble parameter and athe cosmic

scale factor, and dots denote derivatives with respect to

MICROSCOPE

Atom spectroscopy

Ly-αUVES

21cm SKA

10-32 10-30 10-28 10-26 10-24 10-22

10-10

10-8

10-6

10-4

10-2

Dilaton Mass, mϕ[eV]

Dilaton-Electron Charge Coupling, de

FIG. 1. Projected 90%C.L. constraints on the dilaton coupling deas a function of dilaton mass mφpossible with Lyαforest

spectra. The projected constraints are derived for mock UVES SQUAD Lyαdata (red) and adapted for a 21 cm HI survey with

SKA-like imaging capabilities (orange). Note we assume dilaton DM fractions given in Table I, consistent with measurements

of the CMB and matter power spectra. Existing laboratory constraints from the MICROSCOPE ﬁfth force search [27] and

atomic spectroscopy [28] are shown in dark grey. Dashed lines indicate extrapolation of [28] for mφ.10−24 eV, adjusted for

the maximum allowed dilaton abundance in each mass bin (see Table I).

MICROSCOPE

Ly-αUVES

21cm SKA

10-32 10-31 10-30 10-29 10-28 10-27 10-26

10-10

10-8

10-6

10-4

10-2

Dilaton Mass, mϕ[eV]

Dilaton-Electron Mass Coupling, dme

FIG. 2. Projected and existing constraints on the dilaton coupling deas a function of dilaton mass mφ, in the 90% conﬁdence

limit. The projected constraints are from analyzing mock Lyαdata (red) and adapted for an SKA-like telescope’s imaging

capabilities (orange). Competing constraints are from MICROSCOPE [27].

cosmic time, t. The Hubble parameter is determined by the Friedmann equation:

H2=8πGN

3ρ , (5)

with ρthe energy density. We assume a standard ΛCDM

cosmology to ﬁx ρ, containing radiation, baryons, DM,

and the cosmological constant.

With initial condition φ(ti) = φiand ˙

φ(ti)≈0, the

energy density in φtoday (Ωφ) is found by solution of

Eqs. (4,5). Such an initial displacement is expected to

be generated, for example, during inﬂation, and follows

in any theory where, in accordance with observation, the

initial state of the hot big bang phase is not the vac-

uum. The Hubble term in Eq. (4) acts as a friction,

preventing φfrom moving to the vacuum until such a

time as H.mφ, after which φundergoes damped os-

cillations. At late times, the solution is approximated

by φ∝a−3/2cos mφt. The energy density scales as

ρφ∝a−3when Hmφ, thus leading to a relic den-

sity of dilaton DM (for approximate analytic formulae,

see Ref. [37]).

B. Structure Formation

Structure formation with dilaton DM proceeds as for

standard ΛCDM via gravitational instability from initial,

approximately scale invariant, curvature perturbations in

the primordial plasma [38]. The curvature perturbations

seed initial ﬂuctuations in the modes, δφk, of the dilaton

ﬁeld on all scales, and in the “growing mode”, such that

δ˙

φk>0 (detailed solutions can be found in Ref. [39]).

When H < mφ, all δφkmodes begin to oscillate. The

evolution can be approximated as:

δφk=ψkeimφt+ψ∗

ke−imφt,(6)

where ψkis a slowly evolving function of t, i.e. ˙

ψmφψ.

In the non-relativistic limit, the density of DM in mode

kis given by:

ρk=1

2m2

φ|ψk|2,(7)

Taking mkψk=√ρkeiθk(the “Madelung form”), the

Klein-Gordon equation can be reduced to ﬂuid equations

for the dilaton overdensity, δφ,k = (ρk−¯ρφ)/¯ρφ, and

velocity ﬁeld vk=ikθk. In the non-relativistic limit, the

dilaton ﬂuid has an eﬀective sound speed (see e.g. [40]):

s=k2

4m2

φa2; (k2mφa).(8)

The initial perturbations in the dilaton ﬁeld begin to

grow signiﬁcantly in the matter dominated era, z.3400

(where zis the cosmic redshift). The ﬂuctuations are de-

scribed by the matter power spectrum, P(k). The speed

of sound, Eq. (8), leads to some modes with small kbe-

having as cold (collisionless, pressureless) DM, with stan-

dard linear growth of ﬂuctuations. Small scale (large k)

modes, on the other hand, oscillate rather than grow.

The scale of separation between growing and oscillating

modes is called the Jeans scale [41], and can be thought

of as the cosmic de Broglie wavelength [39].

The Jeans scale can be found analytically for a Uni-

verse dominated by dilaton DM, and is given by (e.g.

Refs. [37,42]):

kJ= 66.5a1/4Ωφh2

0.12 1/4mφ

10−22 eV1/2

Mpc−1.(9)

The presence of the Jeans scale causes P(k) to be sup-

pressed in models with a component of dilaton DM com-

pared to pure CDM. This is because dilaton modes with

k > kJexperience less growth than those with k < kJ.

Moreover, the power spectrum contains damped oscilla-

tions at large k[43–45].

The power spectrum P(k) is used to place constraints

on DM composed entirely of dilatons (for a compila-

tion of P(k) measurements, see Refs. [46–48]). The

strongest constraint on pure dilaton DM is derived from

the Lyαforest ﬂux power spectrum, which demands

mφ>2×10−20 eV at 95% credibility [49]. A weaker, but

independent limit can be found using the weak lensing

galaxy shear correlation function [50]. A standard “rule

of thumb” limit is mφ&10−22 eV, which is conﬁrmed by

a variety of measurements, including high redshift galaxy

formation [51–53].

For mφ<2×10−20 eV, dilaton DM is permitted to

compose only a fraction of the total observed DM abun-

dance, Ωφ/Ωd<1. Our constraints on this scenario

are derived from cosmological observations of the CMB

power spectrum [39]. The CMB lensing, galaxy power

spectrum, and Lyαforest ﬂux power spectrum can also

be used to exclude the existence of steps in P(k) [43,54–

56]. The constraints on Ωφ/Ωdfrom this data is sum-

marized in Table I, in dilaton mass bins ranging from

10−20 eV to 10−32 eV.

C. Modelling the Dilaton Field

The classical dilaton ﬁeld can be expanded as:

φ(x, t) = X

φ0,k cos(ωφt+k·x+ϕk) (10)

where ωφ=mφc2/~is the Compton frequency and the

(k, φ0,k, ϕk) are the momentum modes, associated am-

plitudes and phase of the ﬁeld, which are determined

by the dilaton DM power spectrum and local distribu-

tion described below. The phases ϕkare ﬁxed in a given

physical realisation of this power spectrum.

The dilaton power spectrum, Pφ(k), and total matter

power spectrum, P(k), can be computed in linear cos-

mological perturbation theory using axionCAMB [57],

a modiﬁed version of CAMB [58]. axionCAMB follows

the procedure outlined above to follow the evolution of

the ﬁeld φfrom adiabatic initial conditions, resulting in

a prediction for P(k, z)2The resulting P(k) can be used

2axionCAMB is described as a model of axion DM, however the

Mass Ωφ/ΩdImplementation

10−20 eV 1 Eq. (20)

10−22 eV 0.2 Eq. (20)

10−24 eV 0.2 Eq. (19)

10−26 eV 0.03 Eq. (19)

10−27 eV 0.03 Eq. (19)

10−28 eV 0.02 Eq. (18)

10−29 eV 0.02 Eq. (18)

10−30 eV 0.02 Eq. (18)

10−31 eV 0.02 Eq. (18)

10−32 eV 0.06 Eq. (18)

TABLE I. Summary of the synthetic data sets that we test

within this work. We present the mass, the dark matter frac-

tions (taken from upper 1σlimits presented in [55,56])

and which equations we use for computing the spectra.

to generate a realization of the dilaton and CDM over-

density ﬁelds. This realization does not include the time

evolution on the Compton scale, δt ≤m−1

φ.

The sum over angular modes going from kto kin P(k)

leads to a coherence time given by:

tc≈λdB

vφ

=2π

mφv2

(11)

where the velocity is, in the case of a DM halo, approxi-

mated by the virial velocity.

The ﬁeld φevolves over three distinct timescales. On

the longest time scales, the amplitude evolves over the

scale of linear growth of structure in the Universe, i.e.

over a Hubble time. This evolution is captured totally

by the non-relativistic cosmological structure given by

P(k, z).

Over the coherence time, Eq. (11), the amplitude also

oscillates. In linear perturbation theory, oscillations over

the coherence time are captured by the temporal oscilla-

tions in the linear growth factor and dilaton power spec-

trum (see e.g. Refs. [37,39]). Oscillations in the linear

power spectrum, while in principle captured by axion-

CAMB, are in practice ignored since we do not generate

realizations of the density ﬁeld using P(k, t) sampled on

such short time scales. As such, we take a stochastic

approach to these scales, taking φ0,k to be Rayleigh dis-

tributed, which can be derived analytically for virialized

DM halos [59–61].

Finally, over the Compton time, m−1

φ, the amplitude

oscillates. This oscillation is completely factored out in

the non-relativistic approximation to structure forma-

tion. Noting that H0≈10−33 eV, we see that the Comp-

ton time, the linear growth time, and the Hubble time

are all approximately equal for mφ= 10−33 eV, which

only assumption is that the scalar potential is V(φ) = m2

φφ2/2,

which applies equally to pseudoscalars such as the axion, and

scalars such as the dilaton, if self-interactions and other interac-

tions are too weak to aﬀect P(k).

occurs in quintessence models of dark energy, in which

the entire Universe consists of a single coherent ﬁeld. We

do not treat this limit, since gauge issues arise when con-

sidering density perturbations on ultra large scales.

D. Lyαforest

The Lyαforest is an absorption feature occurring in

the spectra of distant galaxies bluewards of the Lyαemis-

sion line as a sequence of densely packed, narrow ab-

sorption lines. These absorption lines are caused by the

absorption of the illuminating light of the background

quasar by the IGM. The observed ﬂux, Fobs, in the Lyα

forest is often expressed as a normalized ﬂux F:

F=Fobs

Ftrans

,(12)

where Ftrans is the maximal ﬂux that would have been

observed at full transmission. The optical depth τis

deﬁned by the logarithm of the normalized ﬂux:

τ=−ln(F).(13)

The optical depth is related to the neutral hydrogen

density nHI by convolution with the line emission proﬁle,

a thermal broadened Voigt proﬁle V[9,10,62]:

τ(z0) = σ0cZLOS

dx(z)nHI(x, z)

1 + z

× V (vH(z0)−vH(z)−vpec(x, z), bT(x, z), γ).

(14)

Here σ0is the eﬀective Lyαcross section, cthe speed

of light, zand z0are denoting redshifts, x(z) is the co-

moving distance at redshift z,vHthe diﬀerential Hubble

velocity, γ=λ0

2πτLyα (where λ0is the ﬁducial wavelength

of the transition and τLyα the average time of transition)

and bTthe thermal broadening of the line. The thermal

broadening parameter is proportional to the square root

of the temperature Tof the IGM. In fact, it is [13]:

bT(x, z) = s2kBT(x, z)

,(15)

where kBis the Boltzmann constant and mpthe mass of

the proton.

In fact, the local IGM temperature depends on the

overdensity again, see also our detailed discussion of IGM

physics in Appendix A. Hence, Eq. (14) cannot be un-

derstood as a true convolution as the emission proﬁle

depends on the neutral hydrogen density again due to

the thermal broadening of the line. We will use the term

convolution nevertheless for the remainder of the paper.

E. Dilaton Modiﬁed Voigt proﬁle

We present now in this subsection how the dilaton af-

fects the absorption features in the Lyαforest, i.e. how

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SearchingfordilatoneldsintheLyforestLouisHamaideKing'sCollegeLondon,Strand,London,WC2R2LS,UKHendrikMulleryMax-Planck-InstitutfurRadioastronomie,AufdemHugel69,D-53121Bonn(Endenich),GermanyDavidJ.E.MarshKing'sCollegeLondon,Strand,London,WC2R2LS,UK(Dated:October10,2022)Dilatons(andmoduli)coupleto...

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