#### Scale-invariant scalar field dark matter through the Higgs portal

Received: March
Scale-invariant scalar eld dark matter through the
Catarina Cosme 0 1 2
Jo~ao G. Rosa 0 1 2
O. Bertolami 0 1 2
0 of the Universe
1 Campus de Santiago , 3810-183 Aveiro , Portugal
2 Rua do Campo Alegre 687 , 4169-007, Porto , Portugal
We discuss the dynamics and phenomenology of an oscillating scalar eld coupled to the Higgs boson that accounts for the dark matter in the Universe. The model assumes an underlying scale invariance such that the scalar eld only acquires mass after the electroweak phase transition, behaving as dark radiation before the latter takes place. While for a positive coupling to the Higgs eld the dark scalar is stable, for a negative coupling it acquires a vacuum expectation value after the electroweak phase transition and may decay into photon pairs, albeit with a mean lifetime much larger than the age We explore possible astrophysical and laboratory signatures of such a dark matter candidate in both cases, including annihilation and decay into photons, Higgs decay, photon-dark scalar oscillations and induced oscillations of fundamental constants. We nd that dark matter within this scenario will be generically di cult to detect in the near future, except for the promising case of a 7 keV dark scalar decaying into photons, which naturally explains the observed galactic and extra-galactic 3.5 keV X-ray line.
Beyond Standard Model; Cosmology of Theories beyond the SM
1 Introduction 2
Dynamics before electroweak symmetry breaking
2.1 In ation 2.2 2.3
Radiation era
Condensate evaporation eld
Dynamics after the electroweak symmetry breaking
3.1
3.2
Negative Higgs-portal coupling
Positive Higgs-portal coupling
Phenomenology
4.1
Astrophysical signatures 4.2
Laboratory signatures
4.1.1
4.1.2
4.2.1
4.2.2
4.2.3
Dark matter annihilation
Dark matter decay
Invisible Higgs decays into dark scalars
Light shining through walls
Oscillation of fundamental constants
Cosmological implications of the spontaneous symmetry breaking
2.3.1
2.3.2
Higgs annihilation into higher-momentum particles
Perturbative production of particles by the oscillating background
HJEP05(218)9
Dark matter is one of the most important open puzzles of modern cosmology and
funda
Universe and later decoupled to yield a frozen-out abundance.
The so-called \WIMP
miracle", where the relic WIMP abundance matches the present dark matter abundance
for weak-scale cross sections, makes such scenarios quite appealing, with a plethora of
candidates within extensions of the Standard Model at the TeV scale. However, the lack
of experimental evidence for such WIMPs and, in particular, the absence of novel particles
at the LHC, strongly motivates looking for alternative scenarios.
An interesting candidate for dark matter is a dynamical homogeneous scalar eld that
is oscillating about the minimum of its (quadratic) potential, and which can be seen as a
condensate of low-momentum particles acting coherently as non-relativistic matter. Scalar
is the Higgs portal, where dark matter only interacts directly with the Higgs eld, H. A
coupling of the form g2j j2jHj2 should generically appear for any complex or real scalar
eld, since it is not forbidden by any symmetries, except for the QCD axion and analogous
pseudo-scalars where such an interaction is forbidden by a shift symmetry.
The Higgs portal for dark matter has been thoroughly explored in the context of scalar
WIMP-like candidates [1{14], but only a few proposals in the literature discuss the case
of an oscillating scalar condensate [15{19]. In this work, we aim to ll in this gap and
consider a generic model for scalar eld dark matter where, like all other known particles,
the dark scalar acquires mass exclusively through the Higgs mechanism, i.e. no bare scalar
mass term in the Lagrangian is introduced for dark matter. While the Standard Model
gauge symmetries forbid bare masses for chiral fermions and gauge bosons, this is not so for
scalars, since j j2 is always a gauge-invariant operator. Scalar mass terms are, however,
forbidden if the theory is scale-invariant (or exhibits a conformal invariance). This has
arisen some interest in the recent literature, with the possibility of dynamically generating
both the Planck scale and the electroweak scale through a spontaneous breaking of
scaleinvariance. In fact, with the inclusion of non-minimal couplings to gravity allowed by
scale-invariance, one can generate large hierarchies between mass scales from hierarchies
between dimensionless couplings and naturally obtain an in ationary period in the early
Universe, as shown in refs. [20{23].For other scenarios with scale-invariance and viable dark
matter candidates, see also refs. [24{29].
In this work we will pursue this possibility, considering a model of scalar- eld dark
matter with scale-invariant Higgs-portal interactions.
We assume that scale invariance
is spontaneously broken by some unspeci ed mechanism that generates both the Planck
scale, MP , and a negative squared mass for the Higgs eld at the electroweak scale, thus
working within an e ective eld theory where these mass scales are non-dynamical. The
important assumption is that scale-invariance is preserved in the dark matter sector, such
that the dark scalar only acquires mass after the electroweak phase transition (EWPT). In
addition, our scenario has also a U(
1
) symmetry (or Z2 symmetry for a real scalar, as we
{ 2 {
discuss later on) that, if unbroken, ensures the stability of the dark scalar. Its dynamics
are thus fully determined by its interaction with the Higgs boson and gravity, as well as
its self-interactions, all parametrized by dimensionless couplings. The relevant interaction
Lagrangian density is thus given by:
where the Higgs potential, V (H), has the usual \mexican hat" shape, g is the coupling
between the Higgs and the dark scalar and
is the dark scalar's self-coupling. The
last term in eq. (1.1) corresponds to a non-minimal coupling of the dark matter eld to
density is an extension of the model that we considered in ref. [18] where self-interactions
played no role in the dynamics, giving origin to a very light dark scalar, m
and therefore a very small coupling to the Higgs boson, g
10 16. Such feeble interactions
make such a dark matter candidate nearly impossible to detect in the near future, and in
this work we will show that the inclusion of self-interactions allows for heavier and hence
O 10 5 eV ,
more easily detectable dark scalars.
On the one hand, if the Higgs-dark scalar interaction has a positive sign, the U(
1
)
symmetry remains unbroken in the vacuum and the dark scalar is stable. On the other
hand, if the interaction has a negative sign, the U(
1
) symmetry may be spontaneously
broken, which can lead to interesting astrophysical signatures, as we rst observed in
ref. [19]. In this work we wish to provide a thourough discussion of the dynamics and
phenomenology in both cases, highlighting their di erences and similarities and exploring
the potential to probe such a dark matter both in the laboratory and through astrophysical
observations.
This work is organized as follows. In the next section we explore the scalar eld
dynamics from the in ationary period to the EWPT, where the Higgs-portal coupling plays
a negligible role. In section 3 we discuss the evolution of the scalar eld after the EWPT,
analyzing separately the cases where the Higgs-portal coupling is positive or negative,
and computing the present dark matter abundance in both scenarios.
We explore the
phenomenology of these scenarios in section 4, discussing possible astrophysical signatures
and experiments that could test these scenarios in the laboratory.
We summarize our conclusions and discuss possible avenues for future work in this subject in section 6.
2
Dynamics before electroweak symmetry breaking
Before the EWPT, the dynamics of the eld does not depend on the sign of the coupling
to the Higgs, since it plays a sub-leading role. This allows us to describe the behavior of
the eld without making any distinction between the two cases. We will rst explore the
dynamics during the early period of in ation, where the non-minimal coupling to gravity
plays the dominant role, and then the subsequent evolution in the radiation era, where the
eld dynamics is mainly driven by its quartic self-coupling.
{ 3 {
We consider a dark scalar eld non-minimally coupled to gravity, with the full action being
S =
Z p
1
2
g d4x
MP2 l f ( ) R
1
2
(r )
2
V ( ) ;
(2.1)
p
where
= = 2 and f ( ) = 1
eld component in a at FRW Universe is thus:
2=MP2 l.1 The equation of motion for the homogeneous
+ 3H _ + V 0 ( ) + R
, such that the dynamics during in ation is mainly driven by
the non-minimal coupling to gravity. Since during in ation R ' 12 Hi2nf , where the nearly
constant Hubble parameter can be written in terms of the tensor-to-scalar ratio r:
the e ective eld mass is:
with m
> Hinf for
> 1=12.
Hinf (r) ' 2:5
1013
r
0:01
1=2
GeV ;
m
'
p12 Hinf ;
The mass of the dark scalar has to exceed the Hubble parameter during in ation,
since otherwise it develops signi cant
uctuations on super-horizon scales that may give
rise to observable cold dark matter isocurvature modes in the CMB anisotropy spectrum,
which are severely constrained by data [30]. This implies that the classical eld is driven
towards the origin during in ation, but its average value can never vanish due to its
de
Sitter quantum
uctuations on super-horizon scales. Any massive scalar
eld exhibits
quantum
uctuations that get stretched and ampli ed by the expansion of the Universe.
For m =Hinf > 3=2 ( > 3=16), the amplitude of each super-horizon momentum mode is
suppressed by the mass of the dark scalar eld, yielding a spectrum [31]:
2
j kj '
Hinf
2
2
Hinf
m
Hinf
2
2
1
p12
;
where a(t) is the scale factor. Integrating over the super-horizon comoving momentum
0 < k < aHinf , at the end of in ation, the homogeneous eld variance reads:
1Since since
MPl and
. 1, f ( ) ' 1 and there are no signi cant di erences between the Jordan
and Einstein frames. In addition,
is not the in aton and, therefore, the gravitational dynamics during
in ation is not altered by , regardless its non-minimal coupling to curvature.
{ 4 {
(2.3)
(2.4)
(2.5)
(2.6)
setting the average amplitude of the eld at the onset of the post-in ationary era, inf :
inf = p
h 2i '
< 3=16, m =Hinf < 3=2, the spectrum is given by [31]:
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
+ g
1
2
4
r
V ( )
R 2 + 2
2R
;
(2.15)
leading to a cold dark matter isocurvature power spectrum of the form:
with the corresponding dimensionless power spectrum:
PI (k) ' k3
the end of in ation, we obtain for the eld variance:
. As above, integrating over all super-horizon modes at
HJEP05(218)9
Isocurvature perturbations are bounded by the Planck collaboration in terms of the ratio
iso (k) =
R
2 (k) +
I2 (k)
;
where
2R ' 2:2
lower bound:
109 is the amplitude of the adiabatic curvature perturbation spectrum
generated by the in aton eld. Since the
uctuations in
and in the in aton
eld are
uncorrelated, we may use iso (kmid) < 0:037, for kmid = 0:050 Mpc 1, corresponding to
the upper bound on uncorrelated isocurvature perturbations imposed by Planck [30]. For
55 e-folds of in ation, this yields
. 1:3, implying that m
& 0:75 Hinf and setting a
which thus gives for the average homogeneous eld amplitude at the end of in ation:
inf = p
h 2i '
Hinf
is the Ricci tensor. The energy density and pressure of the eld are thus,
The eld is in an overdamped regime until the e ective eld mass, m
the Hubble parameter in this era. At this point, the eld begins oscillating about the
= p3
origin with an amplitude that decays as a 1
/ T and
radiation.
a 4, thus behaving like dark
The temperature at the onset of the post-in ationary eld oscillations can be found
by equating the e ective eld mass with the Hubble parameter:
Trad =
1=4 p
inf MPl
270
2 g
1=4
;
below the reheating temperature, Trh
its amplitude decreases with T :
where g is the number of relativistic degrees of freedom. The temperature Trad is thus
pMPl Hinf . Since the eld behaves like radiation,
Using _
m
for an underdamped
eld, we can easily see that
so that the dark scalar will not a ect the in ationary dynamics even for large values of
its non-minimal coupling to gravity. Note that this assumes also that the Higgs eld does
not acquire a large expectation value during in ation, which is natural since de Sitter
uctuations also generate an average Higgs value . Hinf [9].
After in ation, the
eld will oscillate about the minimum of its potential and its
e ective mass m
H, and we show in appendix A that, for
. 1, all modi cations to the
dark eld's energy density and pressure due to its non-minimal coupling to gravity become
sub-dominant or average out to zero, thus recovering the conventional form for
= 0. In
addition, since R = 0 in a radiation-dominated era and R
non-minimal coupling's contribution to the eld's mass also becomes negligible. Thus, we
may safely neglect the e ects of the non-minimal coupling to gravity in its post-in ationary
evolution, hence its only role is to make the eld su ciently heavy during in ation so to
prevent the generation of signi cant isocurvature modes in the CMB anisotropy spectrum.
O(H2) in subsequent eras, the
2.2
Radiation era
After in ation and the reheating period, which we assume for simplicity to be
\instantaneous", i.e. su ciently fast, the Universe undergoes a radiation-dominated era for which
R = 0. Above the electroweak scale, the potential is then dominated by the quartic term,
V ( ) '
4
4
:
rad (T ) =
Trad
inf T =
2 g
270
1=4
1=2
inf
MPl
T
1=4 :
{ 6 {
a
a
. Hi4nf for
(2.16)
in the vicinity of the origin, with average thermal uctuations h
2
T 2, as shown e.g.
in ref. [32]. We can use this to show that the Higgs-dark scalar eld interactions play a
subdominant role before the EWPT. In particular, in the parametric regime that we are
interested in (see eqs. (3.13) and (3.22)), g
1;
since h
2
MTP53l according to ref. [32].
the Higgs
The dark scalar continues to behave like radiation until the EWPT, at which point
eld acquires its vacuum expectation value, H = h=p2 = v=p2, generating
a mass for the dark scalar. The EWPT transition will be completed once the leading
thermal contributions to the Higgs potential become Boltzmann-suppressed, which occurs
approximately at TEW
mW , where mW is the W boson mass. Comparing the quadratic
and quartic terms in the dark scalar potential at TEW, we nd:
where we used that g S ' 86:25 for the number of relativistic degrees of freedom
contributing to entropy at TEW, and de ned EW
rad(TEW). Since r . 0:1 [30], in the parametric
regime g & 10 4 1=4,
. 1 , we conclude that the quadratic term is dominant at TEW,
implying that the eld starts behaving as non-relativistic matter already at the EWPT.
We will verify explicitly below that this is, in fact, the parametric regime of interest for
the eld to account for all the present dark matter abundance.
For T < TEW the dynamics of the eld is di erent depending on whether the
Higgsportal coupling is positive or negative, so that we will study these cases separately. We
need, however, to ensure that the coherent behaviour of the homogeneous scalar eld is
preserved until the EWPT, as we explore in detail below.
2.3
Condensate evaporation
If its interactions are su ciently suppressed, the dark scalar eld behaves as a long-lived
oscillating condensate, i.e, a set of particles with zero (or actually sub-Hubble) momentum
that exhibit a collective behavior, and is never in thermal equilibrium with the SM
particles. Thereby, we must analyze the constraints on g and
to prevent its thermalization
and evaporation into a WIMP-like candidate, the phenomenology of which was studied in
ref. [9]. There are two main processes that may lead to condensate evaporation, as we
describe in detail below - Higgs annihilation into higher-momentum
particles and the
perturbative production of
particles by the oscillating background eld.
2.3.1
Higgs annihilation into higher-momentum particles
Higgs bosons in the cosmic plasma may annihilate into pairs, with a rate given by, for
T & TEW:
hh!
= nh h vi ;
{ 7 {
(2.22)
(2.20)
(2.21)
For T < TEW, the Higgs bosons decay into Standard Model degrees of freedom and the
production of
stops, so that we must require hh!
of the dark scalar condensate for T > TEW. Since
hh!
. H to prevent the thermalization
/ T and H / T 2 , the strongest
constraint is at TEW, which leads to an upper bound on g:
g . 8
corresponding to an upper bound on the dark scalar's mass m
Another possibility for the condensate's evaporation is the production of particles from the coherent oscillations of the background condensate. Particle production from an oscillating background eld in a quartic potential has been studied in detail in refs. [17, 33, 34].
For T > TEW,
particles are massless and interact with the background eld. The coupling
between the background eld and particle uctuations,
, is:
Lint =
3
2
2
For a quartic potential, the oscillating condensate evolves as [34]:
(t) =
p
5
4
3
4
rad
1
X
n=1
ei(2n 1)!t + e i(2n 1)!t
e 2 (2n 1)
1 + e (2n 1) ;
with ! = 12 p
6 ( )
(
3
)
ticle production rate is then given by [34]:
and m
= p3
rad before the EWPT. The corresponding
parwhere v
c
1 and nh is the number density of Higgs bosons in the termal bath,
Before the EWPT, the momentum of the Higgs particles is of the order of the thermal bath
temperature, jpj
T , and the cross-section for the process is
nh =
(
23
) T 3 :
g
4
64
1
1 +
m2
h
m2
:
The energy density,
, can be averaged over eld oscillations:
!
=
' 8:86
{ 8 {
We have computed this numerically, using eq. (2.27), and obtained:
implying a particle production rate:
1
most stringent constraint is at TEW, where rad =
EW, which places an upper bound on
the dark scalar's self-coupling:
If the constraints (2.25) and (2.32) are satis ed, the dark scalar is never in thermal
equilibrium with the cosmic plasma, behaving like an oscillating condensate of zero-momentum
particles throughout its cosmic history. We will see below that eq. (2.32) yields the most
stringent constraint.
3
Dynamics after the electroweak symmetry breaking
At the EWPT, the eld starts behaving di erently depending on the sign of the
Higgsportal coupling. If this coupling is negative, the eld may acquire a vacuum expectation
value and thus become unstable, although su ciently long-lived to account for dark matter.
On the other hand, if the coupling is positive, the U(
1
) symmetry remains unbroken after
the EWPT and the eld never decays. In this section, we explore the eld dynamics in
each case.
3.1
Negative Higgs-portal coupling
This case has been explored in ref. [19], and here we intend to give a more detailed
description of the eld dynamics. The relevant interaction potential is:
V ( ; h) =
2 h2 +
4 +
h
4
h
2
v~
As soon as the temperature drops below the electroweak scale, both the Higgs eld and
the dark scalar acquire non-vanishing vacuum expectation values for g4 < 4
h, which
we take to be the relevant parametric regime:
h0 =
1
g
2
4
g
4
4
h
0
4
v~
1=2
g
2
p
g3
2
{ 9 {
where v = 246 GeV. This induces a mass-mixing between the \ avour" basis
and h
elds, described by the squared mass matrix
v;
0 =
g v
p2
;
p
2
2 h
For small mixing, the mass eigenvalues are approximately given by:
The mass eigenstates can then be written in terms of the avour-basis elds as:
~ =
h ;
h~ = h +
This mixing is extremely relevant for the direct and indirect detection of the dark scalar as
we discuss in section 4.2. In this scenario, di erentiating the Lagrangian twice with respect
to , yields the mass of the eld:
m
= g v :
As we have seen in section 2.2, the eld starts behaving as cold dark matter at TEW
mW . The amplitude of the oscillations at this stage is given by eq. (2.19), which can be
rewritten as:
EW '
From eq. (3.9), in the parametric regime g & 10 4 1=4, EW . 0, for
& 1. This implies
that the potential minimum will move smoothly from the origin to 0, where the eld starts
to oscillate with an amplitude DM = xDM 0, with xDM . 1. In fact, a rigorous study of
the dynamics of the eld for TEW < T < TCO, where TCO corresponds to the electroweak
crossover temperature, would necessarily involve numerical simulations that are beyond
the scope of this paper. We may nevertheless estimate the uncertainty associated to the
eld's amplitude. Since TEW . TCO by an O (
1
) factor, and given that
T while
behaving as radiation and
T 3=2 while behaving as non-relativistic matter, the eld's
amplitude might decrease by at most an O (
1
) factor as well. Therefore, we expect the
eld's amplitude to be smaller than
0 at TEW by xDM . 1. Notice that xDM is thus not
an additional parameter of the model, but only a theoretical uncertainty in our analysis
that does not a ect the order of magnitude of the dark matter abundance and lifetime, as
we shall see below.
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
As soon as the eld reaches
0, it stops behaving like dark radiation and starts to
oscillate as cold dark matter. The eld's amplitude of oscillations about 0 evolves
according to
At this point, the number of particles in a comoving volume becomes constant:
number density and g S ' 86:25 at TEW. Using this, we can compute the present dark
matter abundance,
;0 ' 0:26, obtaining the following relation between the latter and m :
m
= (6
;0)1=2
g S
where g S0, T0 and H0 are the present values of the number of relativistic degrees of freedom,
CMB temperature and Hubble parameter, respectively. We thus obtain the following relation between the couplings g and :
This expression satis es the constraint g4 < 4
h and
EW .
0
. Given this relation
between couplings, eq. (2.32) yields, as anticipated, the strongest constraint, limiting the
viable dark matter mass to be . 1 MeV.
It should be emphasized that the properties of our model depend strongly whether
the U(
1
) symmetry is global or local. In particular, if the symmetry is global, it leads to
tight constraints on the dark scalar mass. We study the cosmological implications of the
spontaneous symmetry breaking (both local and global) in section 5.
3.2
Positive Higgs-portal coupling
As we have seen before, for T > TEW, the eld is oscillating as radiation. If the coupling
between the Higgs and the SDFM has a positive sign, as soon as the EWPT occurs, the only
eld that undergoes spontaneous symmetry breaking is the Higgs eld. The corresponding
interaction Lagrangian is:
Lint = +
g
2
4
2 v2 +
4
4 +
h
4
h
2
v
The interactions with the Higgs are responsible for providing the dark scalar mass, which
is given in this case by
m
= p g v ;
1
2
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
HJEP05(218)9
+ =
g
so that the number of particles per comoving volume reads:
n
s
=
45
1
4 2 g S
m
2
inf
1
Tr2ad TEW
:
Analogously to the negative coupling case, we may compute the eld's mass, taking into
account the present dark matter abundance,
m
= p6
;0
H0 MPl
inf
g ' 9
di ering from the mass of the negative coupling case, eq. (3.8), by a factor of p2 . Although
there is no spontaneous U(
1
) symmetry breaking, there is in practice an e ective
massmixing between the Higgs and dark scalar elds, given that the latter's value is oscillating
about the origin but does not vanish exactly, yielding an e ective mixing parameter:
where
denotes the time-dependent eld amplitude. Since the dark matter energy density
depends on the amplitude of the eld,
HJEP05(218)9
Eq. (3.16) can be rewritten as
At TEW, the eld starts to behave as cold dark matter, and its amplitude varies as:
leading to the following relation between g and
:
Once again, eq. (2.32) yields the strongest constraint and, as in the case of the negative
coupling, the dark scalar's mass must be smaller than 1 MeV to account for the present
dark matter abundance.
It must be highlighted that, in this scenario, once the dark matter abundance is xed,
the value of g depends on all other parameters of the model,
, r and , contrary to
the negative coupling case, where the Higgs-portal coupling is only related to
as given
in eq. (3.13). In other words, in the positive coupling case the
eld's mass depends on
the initial conditions set by in ationary dynamics, whilst in the case with spontaneous
symmetry breaking the nal dark matter abundance is e ectively independent of the initial
conditions, and all dynamics depends essentially on the vacuum expectation value, 0
which sets the amplitude of eld oscillations.
Having discussed the dynamics of the dark scalar throughout the cosmic history and
determined the parametric ranges for which it may account for the present dark matter
abundance, in this section we discuss di erent physical phenomena that may be used to
probe the model. This includes searching for direct signatures in the laboratory, as well as
for indirect signals in astrophysical observations.
Given the smallness of the Higgs portal coupling required for preventing the scalar
condensate's evaporation before the EWPT, it is likely that any signatures of such a dark matter
candidate require large uxes of
particles or the particles it interacts with, which may be
di cult to produce in the laboratory. Astrophysical signatures may, however, be enhanced
by the large dark matter density within our galaxy and other astrophysical systems, and
thus we shall explore in this section the possibility of indirectly detecting the dark scalar
through its annihilation or decay.
Dark matter annihilation
galactic center.
tion is given by:
Over the past decades, the emission of a 511 keV -ray line has been observed, by several
experiments, in a region around the galactic center (see e.g. ref. [37] for a review). The
511 keV line is characteristic of electron-positron annihilation, as has been shown by the
INTEGRAL/SPI observations [38{40] and has a ux of
GC = 9:9
This photon excess in the galactic bulge is intriguing because, although it is common to
nd positron sources in the galaxy (namely, from core collapse of supernovae and low-mass
X-ray binaries), most of these astrophysical objects are localized in the galactic disk rather
than in the galactic center [37].
An alternative possibility to explain this line is considering dark matter annihilation
into an e e+ pair. Although light WIMPs are practically excluded as a possible
explanation [42], other plausible dark matter candidates could predict this photon excess. In
this context, we study the annihilation of
particles into e e+ and into photons,
presenting the predictions for the photon
ux associated with our dark matter candidate in the
In general, the ux of photons from an angular region
from dark matter
annihilaann =
2m2 N h vi Z
1
4
l:o:s
d ;
Z
where
is the dark matter density for a given pro le, r is the radial distance from the
galactic center,
is the angle between the direction of observation in the sky and the
galactic center, the integral is evaluated along the line-of-sight (l.o.s) and N is the number
of photons produced by the annihilation [43]. We can split eq. (4.1) into two parts: one
that only depends on the particle physics,
2m2 N h4 vi
1
;
(4.1)
(4.2)
and another one depending only on the astrophysical properties:
Z
d ;
Z
and analyze each one separately.
jpf j = m
q
1
Let us rst focus on the particle physics component. Consider the case in which the
particles annihilate via virtual Higgs exchange, H , and the latter decays into Standard
Model particles. Since m
. 1 MeV (by the constraint imposed on
, eq. (2.32)), the
possible
nal decay products are only electron-positron pairs and photons for non-relativistic
dark scalars. Following eq. (4.2), it is necessary to compute the cross section of the process.
In the center-of-mass frame, using E
' m , the 4-momentum of H
is ph = (2 m ; 0),
implying that the invariant mass of the virtual Higgs is mH
= 2 m . By energy
conservation, if the nal state is a fermion-antifermion pair, the momentum of each particle reads
mf2 =m2 , where mf is the fermion's mass. Using this, the cross section for
dark matter annihilation into a virtual Higgs and its subsequent decay into fermions reads:
h vrelifermions ' 8
Nc D
v2
m4
h
4m2
h h
where Nc is the number of colors (Nc = 1 for leptons and Nc = 3 for quarks),
h =
4:07
10 3 GeV [35] is the total Higgs decay rate, and D = 1 for the case where the U(
1
)
symmetry is spontaneously broken and D = 4 otherwise.
In the case where we have a pair of photons in the nal state, the momentum of each
photon is jp j = m . The cross-section for this process is:
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
HJEP05(218)9
where GF = 1:17
constant and
F =
X
f
Nc Qf2 A1H=2 ( f ) + A1H ( w) ' 3
11
is a factor that takes into account the loop contributions of all charged fermions and the
W boson to H
!
, with i = 4mi2=m2
1 for all particle species involved, for
m
MeV [44]. We point out that for the decay of a virtual Higgs all charged fermions
give essentially the same loop contribution, whilst for an on-shell Higgs boson only the top
quark contributes signi cantly.
Concerning the astrophysical part, one needs to consider the pro le that best describes
the dark matter distribution in a particular region of the Universe to compute the so-called
J -factor in eq. (4.3). In the case of the galactic center, the most appropriate models are
the cuspy pro le ones. Following ref. [35], using a generalized Navarro-Frenk-White pro le
(NFW) to describe the dark matter distribution in the galactic center,
h vreli ' 32p2 3 v2
D
m4
GF
QED m2
2
4m2
+ m2 2
h h
F 2 ;
10 5 GeV 2 is Fermi's constant,
QED ' 1=137 is the ne structure
(r) = s
r
rs
1 +
r
rs
3+
;
where r is the spherical distance from the galactic center,
rs = 19:5 kpc; hence, the corresponding J -factor is:
= 1:2, s = 0:74 GeV=cm3 and
Using the last expressions, the predicted ux of photons originated by the annihilation of
particles into a virtual Higgs and the decay of the latter into an electron-positron pair,
for the galactic center, is given by:
h
4m2
h h
m
v
2
me 2
v
1
me2 !3=2
m2
:
& 0:511 MeV so that it can annihilate into an electron-positron
pair, we can simplify this to give:
!e e+ ' 1
which is clearly insu cient to explain the galactic center excess but shows how the
electronpositron
ux varies with the mass of the dark scalar and hence the Higgs-portal coupling.
The
ux coming from the annihilation of
into a virtual Higgs and its decay into
photons reads:
!
' 10 31 D
m
MeV
4
cm 2 s 1 :
The uxes we have found for the annihilation of particles into a virtual Higgs and its decay into an electron-positron pair or photons are thus extremely suppressed, which makes its indirect detection through annihilation a very di cult challenge.
4.1.2
Dark matter decay
Our dark matter candidate exhibits a small mixing
with the Higgs boson, as we have
seen in section 3 and may, therefore, decay into the same channels as the Higgs boson,
provided that they are kinematically accessible, and the decay width is suppressed by a
factor 2 relative to the corresponding Higgs partial width,
!pp = 2
H !XX , where
X corresponds to a generic particle. The partial decay width of a virtual Higgs boson into
photons is [44]:
where F is de ned in eq. (4.6). This yields for the dark scalar's lifetime:
H !
=
GF
QED m3
2
in the case with spontaneous symmetry breaking and
(+)
7 keV
m
5
sec ;
(4.8)
(4.9)
for the positive coupling. As expected, the case with no spontaneous symmetry breaking
yields an e ectively stable dark matter candidate, with the probability of decay into photons
being tremendously suppressed. In the negative coupling case, the dark scalar's lifetime
is larger than the age of the Universe, but there is a realistic possibility of producing an
observable monochromatic line in the galactic spectrum.
In fact, the XMM-Newton X-ray observatory has recently detected a line at 3:5 keV
in the galactic center, Andromeda and Perseus cluster [45{48]. The origin of the line
is not well-established, although dark matter decay and/or annihilation are valid
possibilities [48{53]. There are other astrophysical processes that might explain the 3:5 keV
line [54]. However, some independent studies suggest that those processes cannot provide
a satisfactory explanation [55{57]. In addition, some groups state that such a photon
excess is not present in dwarf galaxies, such as Draco [58], while others assert that the line
is only too faint to be detected with the technology that we have at our current disposal,
not ruling out a possible dark matter decay interpretation [59].
According to refs. [47, 59], the line present in the galactic center, Andromeda and
Perseus can be explained by the decay of a dark matter particle with a mass of
7 keV
with a lifetime within the interval
(6
9)
1027 sec, which also explains the absence
of such a line in the blank-sky data set. Concerning the positive coupling scenario, the
dark scalar's lifetime (eq. (4.14)) is not compatible with the required range. However, the
scenario where
undergoes spontaneous symmetry breaking yields a lifetime (eq. (4.13))
that matches the above-mentioned range, up to an uncertainty on the value of the eld
amplitude after the EWPT parametrized by xDM . 1, for m
= 7 keV. Furthermore, for
this mass, we have g ' 3
10 8 and, from eq. (3.13),
constraints in eqs. (2.25) and (2.32), as we had already presented in ref. [19].
We should emphasize the uniqueness of the result for the spontaneous symmetry
breaking scenario. In this case, our model predicts that the decay of
into photons produces
a 3.5 keV line compatible with the observational data, with e ectively only a single free
parameter, either g or
. Recall that the initial conditions of the eld depend on the
parameters r and , the rst one determining the initial oscillation amplitude in the
radiation era, whilst the latter is responsible for suppressing cold dark matter isocurvature
perturbations. However, after the EWPT, the eld oscillates around
0, which only
depends on g and
. Its amplitude is of the order of 0, meaning that r and
do not play
a signi cant role below TEW. Therefore, we have three observables that rely on just two
parameters (g and
) - the present dark matter abundance, the dark scalar's mass and
its lifetime. However, assuming that the dark scalar eld accounts for all the dark matter
in the Universe imposes a relation between g and
(eq. (3.13)), implying that m
and
depend exclusively on the Higgs-portal coupling. Hence, the prediction for the magnitude
of the 3.5 keV line in di erent astrophysical objects is quite remarkable and, as far as we
are aware, it has not been achieved by other scenarios, where the dark matter's mass and
lifetime can be tuned by di erent free parameters.
4.2
. MeV, the coupling between the dark scalar and the Higgs boson, g, is extremely
small, which hampers the detection of dark matter candidates of this kind. However, the
small h
mass mixing may nevertheless lead to interesting experimental signals that one
could hope to probe in the laboratory with improvements in current technology. In this
section, we explore examples of such signatures, namely invisible Higgs decays, photon-dark
scalar oscillations in an external electromagnetic eld and dark matter-induced oscillations
of fundamental constants, such as the electron mass and
Invisible Higgs decays into dark scalars
The simplest and cleanest way to test the Higgs-portal scalar eld dark matter scenario
in collider experiments is to look for invisible Higgs decays into dark scalar pairs, with a
HJEP05(218)9
decay width:
where mh is the Higgs mass. The current experimental limit on the branching ratio for
Higgs decay into invisible particles is [36]:
h!
=
1 g4v2 s
8 4 mh
1
4m2
m2 ;
h
Br ( h!inv) =
h!inv
h + h!inv
Assuming that
h!inv =
h!
following bound on the Higgs-portal coupling g:
and using
h = 4:07
which is much less restrictive than the cosmological bound in eq. (2.25). Conversely, for
m
< MeV to ensure the survival of the dark scalar condensate up to the present day, we
obtain the following upper bound on the branching ratio:
Br ( h!inv) < 10 19 :
This is, unfortunately, too small to be measured with current technology, but may serve
as motivation for extremely precise measurements of the Higgs boson's width in future
collider experiments, given any other experimental or observational hints for light
Higgsportal scalar eld dark matter, such as, for instance, the 3.5 keV line discussed above.
4.2.2
Light shining through walls
The dark scalar exhibits a small coupling to photons through its mass mixing with the
Higgs boson, which couples indirectly to the electromagnetic eld via W -boson and charged
fermion loops [44]. This will allow us to explore experiments that make use of the coupling
to the electromagnetic eld to probe dark matter candidates, as the case of \light shining
through a wall" experiments (LSTW). These experiments are primarily designed to
detect WISPs - Weakly Interacting Sub-eV Particles, such as axions and axion-like particles
(4.15)
(4.16)
(4.17)
(4.18)
(ALPs), taking advantage of their small coupling to photons. Therefore, we intend to apply
the same detection principles to our dark scalar eld, given its similarities with ALPs, in
terms of small masses and couplings.
In LSTW experiments, photons are shone into an opaque absorber, and some of them
may be converted into WISPs, which traverse the absorber wall. Behind the wall, some
WISPs reconvert back into photons that may be detected. This can be achieved using an
external magnetic (or electric) eld, which works as a mixing agent, making the
photonWISP oscillations possible and allowing for the conversion of photons into WISPs and
vice-versa. Using the paradigmatic case of a pseudo-scalar axion/ALP, the interaction
term between the latter and photons is of the form:
its dual and a the axion/ALP eld. For
where F
is the electromagnetic eld tensor, F~
the QCD axion, the coupling ga
reads
with Fa ' 6
and K
O (1
1011
ma
with the electromagnetic eld is given by [63, 65]:
La
=
ga
4
a F
~
F
=
ga
a E:B ;
ga
=
QED K
2 Fa
10 15K
ma
GeV 1
;
In the speci c case of the QCD axion, for L
mately given by:
4!=m2a, the LSTW probability is
approxiP !a!
= 6
and, using the parameters of the LSTW experiment ALPS (\Any Light Particle Search
experiment) at DESY [63] (! = 2:33 eV, Bext = 5 T and L = 4:21 m), the probability of
La
a F
F
= ga
a B
2
E
In both cases, the presence of an external magnetic eld, Bext, gives rise to a mass mixing
term between a and photons, thus inducing photon-ALPs oscillations. In the relativistic
limit, the WISP wavenumber is ka t !
Using this, considering the conversion of a photon into a WISP in a constant magnetic
eld of length L, in a symmetric LSTW setup, the conversion probability of a photon into
a WISP, P !a, is the same as for the inverse process, Pa! , being given by [63, 64]:
2!
m2a , where ! is the photon angular frequency [63].
Therefore, the total conversion probability along the path is simply the square of
g
4
Be4xt !4
m8a
sin4
m2a L
4 !
:
:
1
=
ga
4
GeV denoting the axion decay constant, ma the axion mass
10) is a model-dependent factor [60{62]. For scalar WISPs the interaction
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
with ALPS 2010 results [64, 65].
the Lagrangian [66]:
where the coupling gh
has the form
shining a photon through a wall via an intermediate axion with ma
P !a!
' 10 53 and the coupling to photons is ga
10 5 eV is about
10 15 GeV 1
. The upgrade
of LSTW experiments intends to achieve a sensitivity in the photon-WISP coupling of
ga
< 10 11 GeV 1, corresponding to an improvement of 3 orders of magnitude comparing
In our case, the Higgs coupling to two photons is expressed by the following term in
with F given by eq. (4.6). Using the mixing
h in eq. (3.7) we can write
and we may de ne the e ective coupling of the dark scalar eld to photons, g
, as:
Lh
=
gh
4
h F
F
;
gh
=
QED F
v
;
L
gh
4
~ F
F
;
= gh
;
=
g
(4.25)
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
where
=
former, we nd
for the negative coupling between
and h, and
= + otherwise. For the
g
( )
10 26
m
xDM
0:5
1
GeV 1
;
while for the case without spontaneous symmetry breaking we obtain:
g(+)
' 8
m
where we have used the cosmological value of the dark matter energy density. Hence, even
for the most promising case with spontaneous symmetry breaking the e ective coupling
to photons of our dark scalar is 11 orders of magnitude below the corresponding
axionphoton coupling for the same mass, with the probability for LSTW thus being suppressed
by
10 44 with respect to the axion. This makes our dark scalar much harder to detect
in LSTW experiments, and hence requiring a very substantial improvement in technology.
4.2.3
Oscillation of fundamental constants
A light dark scalar, 10 22 eV < m
< 0:1 eV, behaves as a coherently oscillating eld
on galactic scales, which may cause the variation of fundamental constants, namely the
electron's mass, me, and the ne structure constant,
The Standard Model Yukawa interactions can be generically written as:
Lhff = pgf f f h ;
2
where gf is the Yukawa coupling. Using the
mass after the EWPT is thus given by:
h mass mixing in eq. (3.7), the electron's
me ' p
gf v
2
1
p
dme
CDM
;
where
CDM corresponds to the present amplitude of the scalar eld and dme is a
dimensionless quantity that works as an e ective \dilatonic" coupling and is normalized by the
Planck mass, such that [68, 69]:
In the case where
undergoes spontaneous symmetry breaking, the dilatonic coupling
d(me) is
d(me) ' 6
v
:
=
m
10 15 eV
d(m+e) ' 2
m
m
m
xDM
0:5
1
;
CDM
xDM
0:5
;
1
;
:
:
(4.33)
(4.32)
(4.34)
(4.35)
(4.36)
(4.37)
(4.38)
Similarly, we may compute the variation of the ne structure constant due to the oscillating
dark scalar eld. We have seen that the interaction with the Higgs boson introduces a small
coupling to the electromagnetic eld (eq. (4.27)), which will induce a variation on the ne
structure constant of the form [74]:
where, after the EWPT,
and
The dilatonic couplings dme and d have thus comparable values, both in the case with and
without spontaneous symmetry breaking, which is a distinctive prediction of our model.
Despite the smallness of these dilatonic couplings, there are a few ongoing experiments
designed to detect oscillations of fundamental couplings, using, for instance, atomic clock
spectroscopy measurements and resonant-mass detectors [69{77]. For instance, atomic
spectroscopy using dysprosium can probe the dme coupling in the mass range 10 24 eV .
m
. 10 15 eV, for 10 7 . dme . 10 1. In turn, the current AURIGA experiment
may reach 10 5 . dme ; d
. 10 3 in the narrow interval 10 12 eV . m
whereas the planned DUAL detector intends to achieve, for 10 12 eV . m
. 10 11 eV,
. 10 11 eV,
sensitivity to detect 10 6 . dme ; d
. 10 2. From eqs. (4.34){(4.38), we conclude that
current technology is not enough to probe our model, which may nevertheless serve as a
motivation to signi cantly improve the sensitivity of such experiments.
We should also point out that ultra-light scalars with mass below 10 10 eV can lead to
superradiant instabilities in astrophysical black holes that may lead to distinctive
observational signatures, such as gaps in the mass-spin Regge plot as rst noted in [78]. However,
these instabilities should only be able to distinguish our Higgs-portal dark matter candidate
from other ultra-light scalars, such as axions and axion-like particles, if non-gravitational
interactions also play a signi cant dynamical role (see e.g. ref. [62]).
5
Cosmological implications of the spontaneous symmetry breaking
Throughout our analysis, we have assumed that the model exhibits a U(
1
) symmetry which,
in the case of a negative sign for the coupling between the Higgs and the dark scalar, is
spontaneously broken at the EWPT. The implications of such symmetry breaking depend
on whether it is a global or a local symmetry. The Lagrangian density is of the form:
The third term on the right-hand-side of eq. (5.3) allows for the decay of into two massless
particles, which could imply the complete evaporation of the dark scalar condensate. The
corresponding decay width is then:
where Lint is given by eq. (1.1).
First, let us suppose that the Lagrangian density is invariant under a global U(
1
)
symmetry, i.e. that the complex dark scalar is invariant under the following transformation:
where
is a constant parameter. Expanding the kinetic term for
=
introducing the rescaled eld
= 0 , we nd the following kinetic and interaction terms
between the dark scalar, , and the associated Goldstone boson, :
p
ei = 2 and
such that, using the relation between g and
in eq. (3.13),
corresponding to
=
1
64
m3
2 ;
0
!
0:5
xDM 1=2
0:5
1=2 5=4 sec 1
;
5=4 sec :
Lint ;
! e
i
;
2
2
0
!
1021
(5.1)
(5.2)
(5.4)
(5.5)
(5.6)
(5.7)
1
2
y =
(5.3)
0
The eld is su ciently long-lived to account for dark matter if
placing an upper bound on
:
!
> tuniv
4 1017 sec,
;
and thus restricting the viable range for the dark matter mass to:
Therefore, if the U(
1
) symmetry were global, there would be a stricter constraint on the
value of
, and our dark matter model could not, in particular, explain the 3.5 keV line.
However, we may consider a local U(
1
) gauge symmetry, where the Lagrangian is
invariant under
which can be achieved by introducing a gauge eld and covariant derivative such that:
This remains nite even in the limit e0 ! 0 and, in fact, it tends to the value in eq. (5.4),
since in this limit the symmetry is global and the decay into massless Goldstone bosons
is allowed.
On the other hand, for m 0 > m =2, the dark scalar's decay into dark photon pairs
becomes kinematically forbidden. This requires:
e0 > p2
;
where e0 denotes the associated gauge coupling. In this case, expanding the scalar kinetic
term in the unitary gauge, where the Goldstone boson is manifestly absorbed into the
longitudinal component of the \dark photon" gauge eld, we get:
D
D
y =
1
2
2
1 e02 A A
2 +
2
where the second and the third terms correspond to dark scalar-dark photon interactions
and the last term gives the mass of the dark photon, 0:
For m 0 < m =2, the dark scalar may decay into two dark photons with a decay width
m 0 = e0 0 :
! 0 0 ' 16
1
v
u
ut1
4 m20
m2
m40
m
2
0
3 +
1 m4
! A +
i e0 A
which does not pose a signi cant constraint, given the magnitude of the scalar self-couplings
considered in our analysis. Note that even if this condition is satis ed the scalar
selfcoupling is stable against gauge radiative corrections, since
=
self-coupling is typically very small in the scenarios under consideration.
e04=
&
and the
It is important to mention that, similarly to the case studied in section 2.3.2, before
the EWPT, the dark scalar's oscillations may also lead to the production of dark photons,
through the second term in eq. (5.12). This leads to an upper bound on the squared gauge
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
coupling comparable to that obtained for the scalar self-interactions in eq. (2.32), and thus
nevertheless compatible with the stability condition eq. (5.15).
Notice that the spontaneous breaking of a U(
1
) symmetry can lead to the generation of
cosmic strings at the EWPT. The ratio between the energy density of such cosmic strings,
s, and the background density, c, reads [79]
s
c
G
where G is Newton's gravitational constant and
is the string's energy per unit length.
Nevertheless, as we reported in ref. [19], 0
instance, 0 is always smaller than 1016 GeV for
1016 GeV even for very suppressed
(for
> 10 66), implying that the ratio
eq. (5.16) is negligible and that, therefore, this does not pose additional constraints on
the model.
Finally, we note that all the dynamics and predictions of our model could be achieved,
however, by considering only a real scalar eld with a Z2 symmetry. Spontaneous breaking
of such a symmetry then leads to the generation of domain walls at the EWPT, which
could have disastrous consequences for the cosmological dynamics. However, it has been
argued that a statistical bias in the initial con guration of the scalar eld could e ectively
yield a preferred minimum and thus make the domain wall network decay [80]. In
particular, according to ref. [80], in ation itself may produce such a bias through the quantum
uctuations of the scalar eld that become frozen on super-horizon scales. Since our dark
scalar never thermalizes with the ambient cosmic plasma, such a bias could survive until
the EWPT and therefore lead to the destruction of any domain wall network generated
during the phase transition. A detailed study of the evolution of domain wall networks in
the context of the proposed scenario is beyond the scope of this work, but this nevertheless
suggests that a real scalar eld, with no additional undesired degrees of freedom, may yield
a consistent cosmological scenario for dark matter with spontaneous symmetry breaking.
In summary, we conclude that the cosmological consistency of the scenarios with
spontaneous symmetry breaking requires either a complex scalar
eld transforming under a
gauged U(
1
) symmetry with su ciently large gauge coupling or possibly a real scalar eld
transforming under a Z2 symmetry.
6
Conclusions
In this work we have analyzed the dynamics and the phenomenology of a non-thermal
dark matter candidate corresponding to an oscillating scalar eld that, as all other known
elementary particles, acquires mass solely through the Higgs mechanism. The model
assumes an underlying scale invariance of the interactions that is broken by an unspeci ed
mechanism to yield the electroweak and Planck scales.
The dynamics of the scalar
eld may be summarized in the following way. During
in ation, the eld acquires a Hubble-scale mass through a non-minimal coupling to
gravity that drives the classical eld towards the origin, while de Sitter quantum
uctuations
generate a sub-Hubble
eld value on average. The Hubble scale mass also suppresses
potentially signi cant cold dark matter isocurvature modes in the CMB anisotropies
spectrum. After in ation, this non-minimal coupling plays no signi cant role in the dynamics,
which is driven essentially by the scalar potential. After in ation, the eld oscillates in an
quartic potential, behaving as dark radiation, until the electroweak phase transition. At
this point the
eld acquires mass through the Higgs mechanism, and starts behaving as
non-relativistic matter.
If the Higgs portal coupling is positive, the dark scalar oscillates about the origin
in a quadratic potential until the present day, while for a negative coupling it undergoes
spontaneous symmetry breaking and oscillates about a non-vanishing vacuum expectation
value. Whereas in the former case the present dark matter abundance depends on all model
parameters, including the non-minimal coupling to gravity and the scale of in ation, in
the latter scenario it is determined uniquely the the Higgs-portal coupling and the dark
scalar's self-interactions. The suppression of particle production processes that could lead
to the oscillating condensate's evaporation places strong constraints on the value of these
couplings, and we generically
nd that the dark scalar's mass must lie below the MeV
scale. It should be pointed out that the aim of this paper is not to explain the smallness
of the dark matter couplings. In fact, in the parametric regime that we are interested in
(g= 1=4
10 3
10 2), both g and
are technically natural, since their relation is not
signi cantly a ected by radiative corrections, which makes them as natural as the electron
Yukawa coupling. Nevertheless, small couplings can be naturally achieved in theories with
extra dimensions, as we have shown in ref. [18].
While there are several phenomenological consequences of the Higgs-portal interactions
of the dark scalar that could allow for its detection, the required suppression of the latter
makes this rather challenging in practice. Possible laboratory signatures include invisible
Higgs decays, dark scalar-photon oscillations and induced oscillations of the ne structure
constant and the electron mass. Indirect astrophysical signatures are also possible, namely
dark matter annihilation or decay into photons. All these processes could lead to a robust
identi cation of our proposed dark matter candidate, but unfortunately they lie generically
below the reach of current technology, even for the case with spontaneous breaking that is
generically easier to detect experimentally. An interesting exception that we have already
reported in ref. [19] is the decay of a 7 keV dark scalar in the case with spontaneous
symmetry breaking, whose decay into photons may naturally explain 3.5 keV emission line
observed in the galactic centre, the Andromeda galaxy and the Perseus cluster.
In summary, the proposed oscillating scalar eld is a viable dark matter candidate with
distinctive observational and experimental signatures, constituting a promising alternative
to WIMPs, which have so far evaded detection. While it is certainly amongst the \darkest"
dark matter candidates available in the literature, there may already be astrophysical
hints for its existence, and we hope that this work may motivate future technological
developments that may allow for testing its implications in the laboratory.
Acknowledgments
C.C. is supported by the Fundac~ao para a Ci^encia e Tecnologia (FCT) grant
PD/BD/114453/2016. J.G.R. is supported by the FCT Investigator Grant No. IF/01597/
2015 and partially by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904 and
by the CIDMA Project No. UID/MAT/04106/2013. We would like to thank Igor Ivanov
and his colleagues from the CFTP, IST-Lisbon for interesting discussions.
A
E ects of the non-minimal coupling to gravity on an oscillating scalar
eld
Consider a homogeneous scalar eld that is oscillating about the minimum of a potential V ( ) much faster than Hubble expansion, i.e. in the regime where the e ective eld mass
m
H. In this regime we then have that:
in a stationary con guration for quantities averaged over the oscillating period, where
we have used the equation of motion =
expansion. We thus obtain the virial theorem:
V 0( ) discarding the sub-leading e ects of
(A.1)
(A.2)
m2
(A.3)
(A.4)
(A.5)
(A.6)
h _2i = h V 0 ( )i
= nhV ( )i;
where the second line is valid for a potential of the form V ( )
n. Since R
H2
and _
m
, we may discard the terms that include R or H explicitly in the expressions
for the eld's energy density and pressure given in eq. (2.16), assuming
. 1. We may
then write these quantities approximately as:
It is easy check, using the virial relation (A.2), that the term proportional to
vanishes
on average, thus yielding the usual expressions for the energy density and pressure of a
homogeneous scalar eld in minimally-coupled general relativity. This leads to the following
equation of state parameter w:
Also, for m H, the eld's equation of motion reads
w
p
n
+ 3H _ + V 0 ( ) = 0 ;
and, multiplying both sides by _, we obtain the standard continuity equation:
d
dt 2
1 _2 + V ( ) + 3H _2 = _ + 3H(
+ p ) = 0 :
Hence, the non-minimal coupling to gravity does not a ect the eld's dynamics and properties in an underdamped regime. { 25 {
Open Access.
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