Induced Gravity and the Attractor Dynamics of Dark Energy/Dark Matter
ABSTRACT Attractor solutions that give dynamical reasons for dark energy to act like the cosmological constant, or behavior close to it, are interesting possibilities to explain cosmic acceleration. Coupling the scalar field to matter or to gravity enlarges the dynamical behavior; we consider both couplings together, which can ameliorate some problems for each individually. Such theories have also been proposed in a Higgs-like fashion to induce gravity and unify dark energy and dark matter origins. We explore restrictions on such theories due to their dynamical behavior compared to observations of the cosmic expansion. Quartic potentials in particular have viable stability properties and asymptotically approach general relativity. Comment: 11 pages, 10 figures
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ABSTRACT: We study dynamics of non-minimally coupled scalar field cosmological models with Higgs-like potentials and a negative cosmological constant. In these models, the inflationary stage of the Universe evolution changes into a quasi-cyclic stage of the Universe evolution with oscillation behaviour of the Hubble parameter from positive to negative values. Depending on the initial conditions, the Hubble parameter can perform either one or several cycles before becoming negative forever.Classical and Quantum Gravity 06/2012; 31(6):065007. · 3.56 Impact Factor - SourceAvailable from: Haidar Sheikhahmadi[Show abstract] [Hide abstract]
ABSTRACT: We study the holographic dark energy (HDE) model in generalized Brans-Dicke scenario with a non-minimal coupling between the scalar field and matter lagrangian namely Chameleon Brans Dicke (CBD) mechanism. In this study we consider the interacting and non-interacting cases for two different cutoffs. The physical quantities of the model such as, equation of state (EoS) parameter, deceleration parameter and the evolution equation of dimensionless parameter of dark energy are obtained. We shall show that this model can describe the dynamical evolution of fraction parameter of dark energy in all epochs. Also we find the EoS parameter can cross the phantom divide line by suitable choices of parameters without any mines kinetic energy term.Astrophysics and Space Science 01/2013; · 2.06 Impact Factor - SourceAvailable from: Sergey Yu. Vernov[Show abstract] [Hide abstract]
ABSTRACT: We employ the superpotential technique for the reconstruction of cosmological models with a non-minimally coupled scalar field evolving on a spatially flat Friedmann-Robertson-Walker background. The key point in this method is that the Hubble parameter is considered as a function of the scalar field and this allows one to reconstruct the scalar field potential and determine the dynamics of the field itself, without a priori fixing the Hubble parameter as a function of time or of the scale factor. The scalar field potentials that lead to de Sitter or asymptotic de Sitter solutions, and those that reproduce the cosmological evolution given by Einstein-Hilbert action plus a barotropic perfect fluid, have been obtained.Physical Review D 11/2012; 87:063503. · 4.69 Impact Factor
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arXiv:1010.2237v2 [astro-ph.CO] 8 Dec 2010
Induced Gravity and the Attractor Dynamics of Dark Energy/Dark Matter
Jorge L. Cervantes-Cota∗
Depto. de F´ ısica, Instituto Nacional de Investigaciones Nucleares, M´ exico and
Berkeley Center for Cosmological Physics, University of California, Berkeley, CA, USA
Roland de Putter†
Berkeley Center for Cosmological Physics, University of California, Berkeley, CA, USA
IFIC, Universidad de Valencia-CSIC, Valencia, Spain and
Institut de Ciencies del Cosmos, Barcelona, Spain
Eric V. Linder‡
Berkeley Center for Cosmological Physics, University of California, Berkeley, CA, USA and
Institute for the Early Universe, Ewha Womans University, Seoul, Korea
(Dated: December 9, 2010)
Attractor solutions that give dynamical reasons for dark energy to act like the cosmological con-
stant, or behavior close to it, are interesting possibilities to explain cosmic acceleration. Coupling
the scalar field to matter or to gravity enlarges the dynamical behavior; we consider both couplings
together, which can ameliorate some problems for each individually. Such theories have also been
proposed in a Higgs-like fashion to induce gravity and unify dark energy and dark matter origins.
We explore restrictions on such theories due to their dynamical behavior compared to observa-
tions of the cosmic expansion. Quartic potentials in particular have viable stability properties and
asymptotically approach general relativity.
I.INTRODUCTION
Over a decade ago two supernovae groups, the Su-
pernova Cosmology Project and the High-Z Supernovae
Search Team, provided evidence for an accelerated ex-
pansion of the Universe [1]. In recent years this discovery
has gained more evidence from a variety of observations:
further supernova data [2], measurements of the cosmic
microwave background radiation [3] and galaxy surveys
[4]. One possibility to explain this acceleration is to intro-
duce a new component within the dynamics of General
Relativity (GR), either as a uniform constant (cosmologi-
cal constant) or as a scalar field evolving along a potential
(as in the inflationary scenario). Since the current accel-
eration seems to be a unique phenomenon [5], at least
since the time of primordial nucleosynthesis, this poses
a fine tuning problem unless some dynamical attractor
exists.
A different possibility is to look for acceleration as aris-
ing from modifications of gravity. In many cases this can
be viewed as coupling scalar fields nonminimally to grav-
ity, within the framework of scalar-tensor theories [6],
an approach called extended quintessence [7]. Attractor
mechanisms can work here as well [8–11], and also in
the case where a scalar field is coupled to (dark) matter
[12–16].
In the present work we investigate the influence of both
couplings on the attractor dynamics. From a phenomeno-
∗Electronic address: jorge.cervantes@inin.gob.mx
†Electronic address: rdeputter@berkeley.edu
‡Electronic address: evlinder@lbl.gov
logical point of view this enriches the phase space and
also can help with problems that arise from one coupling
or the other. By comparison with observations of the
cosmic expansion behavior we can constrain the allowed
parameter space. From a theoretical point of view sev-
eral models can lend motivation to such a combination
of couplings.
Induced gravity [17] is similar to standard scalar-tensor
theories, but gravity is induced by a Higgs-like field. One
motivation stems from Einstein’s original ideas to incor-
porate Mach’s principle into GR, by which the mass of a
particle should originate from the interaction with all the
particles of the universe, and so the interaction should be
the gravitational one since it couples to all particles, i.e.
to their masses or energies. To realize a stronger rela-
tionship with the material contents, Brans and Dicke [6]
introduced their scalar-tensor theory of gravity, making
the gravitational coupling, that is Newton’s constant, a
scalar function determined by the distribution of the cos-
mic content.
On the other hand, in modern particle physics the
inertial mass is generated by the interaction with the
Higgs field; the successful Higgs mechanism also lies pre-
cisely in the direction of Einstein’s idea of producing
mass by a gravitational interaction. One can show [18]
that the Higgs field as source of the inertial mass of the
elementary particles mediates a scalar gravitational in-
teraction of Yukawa type between those particles that
become massive as a consequence of the spontaneous
symmetry breaking. Due to the equivalence principle, it
seems natural to identify both approaches. For this rea-
son, [19, 20] proposed a scalar-tensor theory of gravity
where the Higgs field of elementary particles also plays
simultaneously the role of a variable gravitational con-
Page 2
2
stant, instead of the scalar field introduced by Brans and
Dicke.
Put another way, if there is an interaction between a
scalar (Higgs) field and matter, and of course there exists
coupling between matter and gravity, why not close the
loop by incorporating a (non-minimal) coupling between
the scalar field and gravity? Or conversely, if one explores
scalar-tensor theories, and gravity couples with matter,
why not include an explicit interaction between the scalar
field and matter?
In this work we employ these ideas merely to motivate
the couplings. For example, one might try to identify the
dark energy (DE) with a Higgs-type (but not the Higgs)
field that is coupled to some dark fermion sector. Ac-
cordingly, in our model, gravity, i.e. the Ricci scalar R,
couples to a scalar field φ through the non-minimal cou-
pling φ2R. As the field evolves to its energy minimum the
Higgs coupling might give rise to the mass of some dark
fermion that would account for the dark matter (DM) of
the model. The scalar field evolves to a constant, to gen-
erate the mass, and simultaneously generates Newton’s
constant through the non-minimal coupling to gravity.
The resulting theory is an induced gravity in which GR
is dynamically obtained through a Higgs mechanism from
a scalar-tensor theory [19, 20].
The proposed Higgs mechanism is at present hypothet-
ical, but phenomenologically interesting since it can ac-
count for the mass of the DM and since its field can
act as a DE to accelerate the cosmological expansion.
While many DM-DE interaction models aiming to unify
the two quantities have difficulties in getting them si-
multaneously to exist and match observations, this can
sometimes be made easier by adding a third element, such
as inflation [21, 22] or, as in the present work, gravity. In
any case, apart from the motivation, exploration of the
dynamics of the matter- and gravity coupled system is of
interest.
We begin by describing the general field equations
in Sec. II, identifying the contributions of the gravity-
scalar and scalar-matter couplings. This is then exam-
ined for the homogeneous and isotropic FRW universe
background in Sec. III, including the evolution equations
for the scalar field and matter. Section IV discusses the
effective potential, illustrating it for a symmetry break-
ing form. In Sec. V we solve for the cosmic evolution of
the field and matter, for general classes of potential and
coupling, leading to constraints on the allowed parameter
space from the cosmic expansion behavior.
II.FIELD EQUATIONS
The scalar-tensor theory Lagrangian used here is simi-
lar to the one studied in the past for inflationary dynam-
ics [20]:
L =
α
16πφ2R +1
2φ;µφ;µ− V (φ) + LM,(1)
where φ is a real scalar field and α is a dimensionless pa-
rameter. We use the metric signature (+−−−). Rewrit-
ing the Lagrangian as
L =
1
16πGNR +
+1
2φ;µφ;µ− V (φ) + LM,
1
16πGN
?φ2
v2
GR
− 1
?
R
(2)
one sees that, formally, the Einstein-Hilbert action with
the standard Newton’s constant GNcorresponds to φ2→
v2
GR≡ 1/(αGN). Note however that even if on average
φ = vGR, the theory will still be distinct from GR because
of perturbations in the field.
Varying Eq. (1), one obtains the field equations
Rµν−1
2Rgµν = −8π
αφ2[Tµν+ V (φ)gµν]
−8π
αφ2
−1
φ2
?
(φ2);µ;ν− (φ2);λ
φ;µφ;ν−1
2φ;λφ;λgµν
?
?
?
;λgµν
, (3)
where a semicolon stands for a covariant derivative and
Tµν ≡
grangian we begin with a general “Yukawa” coupling,
LM=¯ψ (˙ ıγµ∂µ− f(φ))ψ, similar to in [13, 15], and later
explore some particular forms. The dark matter thus has
a φ dependent mass f(φ).
The scalar field fulfills the generalized Klein-Gordon
equation
2
√−g
∂√−gLM
∂gµν
.For the dark matter (DM) La-
φ;λ
;λ+∂V
∂φ−α
8πRφ =∂LM
∂φ
.(4)
Note the term involving αR arises from the non-minimal
coupling to gravity, and the right hand side comes from
the coupling to dark matter.
Taking the trace of Eq. (3) and substituting it into
Eq. (4) to remove R, one obtains
φ2;λ
;λ+
2
1 +3α
4π
?
φ∂V
∂φ− 4V (φ) − T − φ∂LM
∂φ
?
= 0.
(5)
The first term in the parentheses is the normal GR term
for an uncoupled scalar field (although with an altered
prefactor), the T+4V of the second and third terms stem
from the nonminimal coupling to gravity (specifically the
trace of the first line of Eq. (3), and the last term is due
to the DM-DE interaction.
This equation can be recast as:
1
2φφ2;λ
;λ+ V′
eff(φ) = 0,(6)
where a prime stands for partial derivative with respect
to φ. The effective potential is
Veff(φ) ≡
1
1 +3α
4π
?
V (φ) − LM(φ) −
?φ
dϕ4V (ϕ) + T(ϕ)
ϕ
?
,
(7)
Page 3
3
from which the effective mass of the Higgs particle will
be identified later on. We write the argument ϕ explic-
itly for LM to remind that only the partial derivative
with respect to the field is relevant. If V (φ) = V0φ4, i.e.
there is no intrinsic mass, then one sees that the poten-
tial terms cancel out and the effective mass is completely
determined by the trace T from the nonminimal gravity
coupling and the DM-DE interaction term LM.
It is convenient to define the energy momentum tensors
associated with the different contributions to the right
hand side of Eq. (3):
Tµν(φ)≡ V (φ)δµν+ φ;µφ;ν−1
Tµν(Rφ)≡
8π
2φ;λφ;λδµν
(8)
α
?
(φ2);ν;λgλµ− (φ2);λ
;λδµν
?
,(9)
where the first is the standard scalar field contribution
and the second stems from the nonminimal coupling to
gravity. Again, note that the GR limit in which Tµν(Rφ)
vanishes is not α = 0 but φ →const. Due to the cou-
plings, the components of the energy momentum tensor
are not individually conserved but the total energy mo-
mentum tensor is:
?
−8π
αφ2
?
Tµν+ Tµν(φ)+ Tµν(Rφ)??
;µ
= 0. (10)
The conservation equation for matter can be derived from
this (using the equation of motion (4) for φ and the Ein-
stein equations (3)), giving
Tµν;ν= −φ;µ∂LM
∂φ
. (11)
We will use this equation in the following to determine
the evolution of the matter component.
III. FRW COSMOLOGY EQUATIONS
To study the background evolution of the universe we
use the FRW metric
ds2= dt2− a2(t)
?
dr2
1 − kr2+ r2dΩ2
?
, (12)
and assume that the DM fluid is given by Tµν= ρuµuν
with the energy density ρ = nf(φ), where n is the num-
ber density, and uµ ≡ dxµ/ds = δ0
velocity. The energy momentum tensor behaves as a
pressureless dust perfect fluid because we assume the
dark matter to consist of non-relativistic fermions. The
equivalence between the description in terms of the spinor
field lagrangian as specified in the previous section on
the one hand, and the “dust” description on the other
hand is demonstrated in [23] for a standard linear cou-
pling (f ∝ φ). There, it is shown how a fermion field
interacting with a scalar field φ, with action
µthe comoving 4-
S =
?
d4√−g(i¯ψγµ∂µψ − y(φ − φ∗)¯ψψ),(13)
where y and φ∗are constant, is equivalent to a model of
a classical gas of pointlike particles, with action
S = −
?
i
?
y(φ − φ∗)dsi, (14)
in the limiting situation where the fermions’ de Broglie
wavelengths are much smaller than the characteristic
length scale of variation of the φ field. Following the
steps of that demonstration one can see that it is valid to
replace the factor y(φ − φ∗) by an arbitrary function of
φ, see also Ref. [24]. In our case we start with a fermion
field with a coupling f(φ)¯ψψ. Then, a valid effective ac-
tion is
S = −
?
i
?
f(φ)dsi= −
?
d4x√−gf(φ)n(x),(15)
where
n =
?
i
?
dsiδ4(x − x(si))
√−g
. (16)
This is exactly the action of a dust fluid with energy
density ρ = f(φ)n.
Thus, the matter Lagrangian is proportional to f(φ),
i.e. LM = −T = −ρ ∝ f(φ). From the gravity field
Eq. (3) one obtains the generalized Friedmann equations
˙ a2+ k
a2
=
8π
3αφ2
?
ρ + V (φ) +1
2
˙φ2−3α
4πHφ˙φ
?
,(17)
2¨ a
a+˙ a2+ k
a2
=
8π
αφ2
?
?
V (φ) −1
2
˙φ2
?
˙φ2
φ
−2
φ
¨φ + 2H˙φ +
?
,(18)
where H ≡ ˙ a/a is the Hubble parameter.
Rearranging the terms, the acceleration equation is
¨ a
a
= −
4π
3αφ2[ρ − 2V (φ)]
?
−1 +8π
3α
?˙φ2
φ2− H
˙φ
φ−
¨φ
φ.
(19)
The redundant Klein-Gordon equation (6) becomes
¨φ +
˙φ2
φ+ 3H˙φ + Veff′= 0, (20)
where
Veff(φ) =
1
1 +3α
4π
?
V (φ) + ρ(φ) −
?φ
dϕ4V (ϕ) + ρ(ϕ)
ϕ
?
.
(21)
Since the field is coupled to matter (either employing a
Higgs or other mechanism), the source term Veff′involves
Page 4
4
the density of the interacting DM fluid as well as the
potential of the DE field.
The conservation equation, Eq. (11), yields
˙ ρ + 3Hρ = −LM′˙φ =˙φρf
′
f,
(22)
so that the matter behaves the same way as in GR with
a DM-DE interaction [13]. This equation can be directly
integrated to give
ρ =n0
a3f(φ),(23)
where n0 is the DM number density at present. This
simply tells us that DM particle number is conserved and
the change in energy per comoving volume is purely due
to the varying mass f(φ).
IV.EFFECTIVE POTENTIAL
The nonminimal coupling to gravity, and the coupling
to matter, in addition to adding symmetry to the rela-
tions between the scalar field, matter, and gravity, also
can create a nonzero vacuum expectation value (vev) –
an effective cosmological constant – that can adiabati-
cally evolve. The nonminimal gravity coupling can also
reduce the driving term Veff′(this coupling gives rise to
the negative term in Eq. 21), slowing the field down. This
slow roll can often alleviate instabilities in coupled mat-
ter perturbations [15].
We can examine these influences in terms of the effec-
tive potential of the theory, which alters the bare scalar
field potential through the coupling to dark matter and
to gravity. Using the Yukawa coupling f ∝ φ in Eq. (21),
the effective potential becomes
Veff(φ) =
1
1 +3α
4π
?
V (φ) − 4
?φ
dϕV (ϕ)
ϕ
?
.(24)
Note that the density terms in Eq. (21) – the interact-
ing DM term L
the nonminimal coupling φ2R – cancel out since ρ is lin-
early proportional to φ. If furthermore V (φ) = V0φ4,
then the effective potential vanishes and the dark en-
ergy field, even coupled, is massless and acquires infinite
range. This is because in this case the theory has no
explicit mass scale in it and is thus scale invariant.
For the Yukawa coupling, the effective potential does
not run with the density, so it differs from – and is ac-
tually simpler than – what happens in the GR case with
matter coupling. The effective mass is only determined
by the potential terms. The field acts like quintessence
with a Veff(φ) given by the full Eq. (24).
For a Landau-Ginzburg symmetry breaking form for
the Higgs potential,
′
Mand the trace term T stemming from
V (φ) =λa
24
?
φ2−6µ2
λa
?2
,(25)
where λais a dimensionless constant, µ√2 is the mass of
the field at the potential minimum, and the Higgs ground
state v, such that V (v) = 0, is given by
v2=6µ2
λa
.(26)
In induced gravity, the Higgs potential V (φ) generates
a time varying gravitational coupling (as in scalar-tensor
theory)
G(φ) =
1
αφ2
(27)
as φ rolls from an initial state to its ground state and
thus for a given field value determined by the potential,
α needs to be chosen such that G(φ) ≈ GN. For example,
we might choose α such that G(φmin) = GN.
Relating this to the particle physics of the Higgs mech-
anism, one has
α = 2π
?
gMPl
Mb
?2
, (28)
where MPl ≡ 1/√GN ≈ 1.2 × 1019GeV is the Planck
mass, Mbthe boson mass, and g a coupling constant. If
we were to consider the standard model Higgs, one has
Mb = Mw = 80GeV for the W-boson and g = 0.18,
therefore α ≈ 1033.
not pass cosmological constraints on α discussed below.
Therefore, we consider another Higgs-like particle with a
much larger mass than that of the Higgs of the standard
model of particle physics. The parameter α will need to
be determined through cosmological observations.
The effective potential for the Landau-Ginzburg plus
Yukawa coupling case is
Such a value is huge and would
Veff=
1
1 +3α
4π
λav4
12
?(φ2/v2) − ln(φ2/v2)?,(29)
and has a minimum at φ = v at all times, at which
the mass of Higgs particle is m2
λav2/(3[1 + 3α/(4π)]).
We show the difference between the bare potential and
the effective potentials in induced gravity and in GR with
Yukawa coupling in Fig. 1. Since the matter density con-
tribution cancels out in induced gravity, the effective po-
tential is time independent, while the GR case is not. The
induced gravity case gives a broad, nearly flat minimum.
When the field is very slowly rolling, the modifications
to the right hand side of Eq. (3) are small.
In this Ansatz the Higgs field, rolling along the effective
potential, plays the role of DE and the particle masses
produced by the Higgs mechanism provide the DM of the
model. However, as we will see the dynamics does not
generally favor such a form for the potential.
H= Veff′′(φ = v) =
Page 5
5
FIG. 1: The Landau-Ginzburg potential in the uncoupled,
GR case (solid curve), in the matter coupled but minimally
gravity coupled (GR) case for two values of the redshift z
(dotted lines), and in the nonminimally gravity coupled in-
duced gravity (IG) case (dashed curve). The matter coupling
is taken to be of the Yukawa form.
V. DYNAMICAL EQUATIONS
We now examine the dynamics of the dark energy, the
effective equation of state of the dark matter, and the
overall expansion behavior.
A.Variables and Equations of Motion
The dynamics of the scalar field, non-minimally cou-
pled to gravity and to dark matter, are described by an
autonomous system of equations which can be solved in
a straightforward manner. One can define the following
set of variables [25]:
x ≡
κ˙φ
√6H
;y ≡κ√V
√3H
λ ≡ −V
′
κV
= −
?
β
6
dlnV
dlnφ
;N ≡ lna,(30)
where κ =
3α/(4π).
?6/β/φ is a function of the field, with β ≡
Assuming spatial flatness, i.e. k = 0, the autonomous
system becomes
dx
dN
=−3
+1 +√β x
1 + β
2xy2−3
2x − 3
??
?
3
2λy2− C
β x2+3
2x3
?
, (31)
dy
dN
=y
?
−
√β
1 + β
?3
2λx +3
2
?x2− y2+ 1?− 2
2λy2− C
?
β x
+
??3
??
.(32)
Here
C ≡
κ
√6H2
√β
2
?
ρf′
f
−4V + ρ
φ
?
(33)
=
?(F − 1) Ωm(N) − 4y2?, (34)
where F ≡ dlnf/dlnφ. The quantity C encodes the
information on the matter coupling (in the first term in-
volving f) and the non-minimal gravitational coupling.
The fractional matter density comes directly from
Eq. (17) as
Ωm(N) ≡
Ωφ = x2+ y2− 2
κ2ρ
3H2= 1 − Ωφ
?
β x. (35)
To evolve the field φ appearing in λ or in dlnf/dlnφ,
one uses the auxiliary equation
dlnφ
dN
=
?
β x.(36)
We define the effective equation of state for a compo-
nent in terms of κ2ρi(since the sum of these quantities
is conserved, see Eq. 10), by
wi= −1 −1
3
dlnκ2ρi
dN
.(37)
This takes into account effective pressure terms due to
the interactions. In particular, the matter equation of
state gains an effective nonzero term, due to the non-
minimal coupling to both the scalar field directly and to
gravity through the φ2R term. One has
√β
3
One has wm= 0 only when the field is frozen (x = 0),
which restores GR as discussed in Sec. II, or when f ∼ φ2,
which counteracts the κ2term in Eq. (37).
The equation of state for the scalar field is more in-
volved:
?
+x2(β[2(F − 2)β + 3F − 2] + 3) − β(F − 1)
−x
??
wm=(2 − F)x.(38)
wφ =y2[β(F − U) − 3] − (F − 2)x3?
β(β + 1)
?
3(β + 1)
β?y2(F − 2)(β + 1) + β(2 + F) + 4 − F??
?
x2− 2x
?
β + y2??
,(39)
Page 6
6
which depends on the nonminimal gravity coupling
through β, on the interacting DM-DE term (F), and on
the potential (U ≡ dlnV/dlnφ).
The total equation of state
wtot ≡ −1 −1
= wm(a)Ωm(a) + w(a)Ωφ(a)
3
dlnH2
dN
(40)
(41)
takes the same form with or without coupling to matter.
The deceleration parameter q = (1 + 3wtot)/2 as usual.
B. Critical points
Investigating the critical (or fixed) points of the dy-
namical system, we find the relation (for yc?= 0)
y2
c= 1 +
?
βU + 2
3
xc+
?
1 + βU + 2
3
?
x2
c.(42)
We restrict to the cases of U and F being constant in
this analysis; otherwise the equations are transcendental.
The dark energy density and equation of state are
Ωφ,c = 1 −
√β
3
(4 − U)xc+1
√β
3
3[6 + β(2 + U)]x2
c(43)
wφ,c = −1 +
(2 − U)xc. (44)
The fixed point xc is given by a quadratic equation.
The first solution is
xc1 =
?
β
4 − U
6 + β(2 + U)
(45)
Ωφ,c1 = 1(46)
1 + wφ,c1 =
β
3
(2 − U)(4 − U)
6 + β(2 + U)
.(47)
This is a stable fixed point over a wide range of parame-
ters (see next section) and it is independent of the mat-
ter coupling f(φ). Note that we obtain a future de Sitter
state (w = −1) for V ∼ φ2or V ∼ φ4asymptotic behav-
ior. The U = 4 case represents an attractor to GR, since
in this case xc = 0 =˙φ. An xc = 0 solution was also
found in GR with no matter couplings [8, 24, 26], and
with an exponential coupling [16], and also in induced
gravity with no matter couplings [11, 27]. For 2 < U < 4
the fixed point gives a phantom (wc< −1) attractor; a
phantom solution was obtained in Ref. [28] for F = 0 and
a particular choice of U. The general attractor solution
is illustrated in Fig. 2. Note that there is a wide range
of U and α (and all of F, which does not enter) where
the attractor gives acceleration, and in fact stays close to
w = −1. In all cases α → 0 makes wφ,c1→ −1.
The second solution (mostly a saddle point) does de-
FIG. 2: The dark energy equation of state for the first critical
point is plotted as a function of the power law index U of the
potential and the value α of the gravity coupling (the matter
coupling does not affect this critical point). The value w = −1
is indicated by the green mesh plane. Note the attractor gives
acceleration, and w ≈ −1, over a wide range of values.
pend on f(φ) and is given by
xc2 =
3
(F − U)√β
18 + β(6 + 7U − 4F + F2− FU)
β(F − U)2
2 − F
F − U.
(48)
Ωφ,c2 =
(49)
wφ,c2 =
(50)
Again, wc = −1 when V ∼ φ2(except when F = 2
also, since for F = U this solution does not exist). A
solution within GR for an exponential potential and an
exponential coupling function [12] also has a critical point
depending on the equivalent of F and U.
Note that x2, y2, and Ωmcan all be greater than unity.
This is possible because of the negative term −Hφ˙φ in
the modified Friedmann equation (17), which permits the
effective dark energy density to go negative. This causes
no physical problems within our Ansatz, as we demon-
strate in Sec. VD.
Three further mathematical solutions exist for yc =
0. These have xc4,c5=√β ±√1 + β, which are mostly
unstable and inaccessible, but of more interest is
√β (1 − F)
3 + β(2 + F)
β(F − 1)[5 + F + 2β(2 + F)]
[3 + β(2 + F)]2
β(F − 1)(F − 2)
3[3 + β(2 + F)].
xc3 =
(51)
Ωφ,c3 =
(52)
wφ,c3 =
(53)
This solution is important because y ≈ 0 corresponds to
κ2V/(3H2) ≈ 0, which holds at early times. We indeed
find a metastable attractor to this behavior at high red-
shift (basically, dx/dN ≈ 0 but y keeps growing). Note
that this high redshift attractor is generally wφ = wm,
i.e. a scaling solution. However, V (and y) are not ac-
tually zero, just small compared to the other densities,
Page 7
7
so the case F = 1, which would give xc= 0, breaks the
condition yc≪ xcand instead here the dynamics forces
wφ= −1 and wm= 0.
C.Stability
Carrying out a linear stability analysis, we find that the
stability of the critical points generally depends on the
values of α (i.e. β), U, and F, making analytic statements
difficult. However, a reasonable rule of thumb is that for
values of these variables not too positive or too negative,
the first critical point is a stable node (both eigenvalues of
the perturbation matrix negative) and the second critical
point tends to be a saddle point (one eigenvalue positive,
one negative). See [25] for a general discussion of stability
analysis and classification.
We show the 3-dimensional eigenvalue surfaces for the
first critical point in Fig. 3, for three different values for
α. The region of stability exists for a broad interval of
U and F, including “natural” values. As α decreases,
the area of stability grows, as seen in the sequence of the
three plots. For α < 0.1 the system is stable for any
U ∈ [−20,20], and F ∈ [−10,10].
By contrast, the second critical point is a saddle point,
as seen in Fig. 4. One eigenvalue is always positive and
the other negative, independent of the chosen U and F.
As α decreases, the eigenvalue sheets tend to pull further
away from the zero plane. Since the qualitative nature of
one positive eigenvalue, one negative eigenvalue does not
change, we show two values of α together in the figure.
Also note that the eigenvalues diverge for F = U, as
expected from Eq. (48).
The third critical point is also a saddle point for
U ∈ [−10,20] and F ∈ [−10,10], but it has a stabil-
ity window for α = 1 in the region U ∈ [−10,−20] and
F ∈ [−1,−5]. As α → 0 the sheets flatten, one below and
above the µ = 0 plane, making this a saddle point for any
U and F. Finally, the fourth and fifth fixed points are
mostly unstable, with small regions of stability (for big
|F| and |U|) for α ≥ 0.5, but these solutions are anyway
uninteresting since they are nonaccelerating and require
y = 0 = V while Ωφ = 1, hence φ = 0 and vanishing
gravity.
A key point is that U = 4 always has a stable attractor
to the first critical point, with w = −1 and˙φ = 0, for all
values of α and F (even F ?= constant, as we discuss in
the next section). So for a quartic potential there is an
attractor leading to ΛCDM and a restoration of GR.
D. Dynamical History
Turning now to the full dynamics, some numerical so-
lutions are illustrated in Fig. 5. At high redshift, where
H2is dominated by the matter density, y is small and the
field quickly forgets the initial conditions and goes to the
scaling solution of the third critical point (except when
FIG. 3: Surfaces for the two eigenvalues µ1 (upper red) and
µ2 (lower blue) of the first critical point. The stability region
for the critical point is where both eigenvalues are negative,
i.e. below the (green horizontal) µ = 0 plane, and is a function
of the power law index U of the potential, F of the matter
coupling, and the value α of the gravity coupling. The three
panels shows how the surfaces change with α: the smaller α,
the more stable the system.
Page 8
8
FIG. 4: Surfaces for the two eigenvalues µ1(upper red) and µ2
(lower blue) of the second critical point. Since one eigenvalue
is positive, this critical point is not stable. As α decreases,
the sheets pull away from the zero plane.
F = 1, then wφ= −1). As the contribution of the dark
energy potential energy becomes relatively more impor-
tant, it then approaches the asymptotic attractor of the
first critical point.
We define the present (a = 1) by when Ωφ= 0.72. For
clarity we show in Fig. 5 the results for a value of the
gravity coupling α = 1; for small α the deviation from
w(a) = −1 will scale roughly as α. The high redshift
metastable attractor and the asymptotic future stable
attractor behaviors can clearly be seen. Note that for
the Yukawa coupling (F = 1) the dynamics stays close
to w = −1. For F < 1, the −Hφ˙φ term can drive the
effective dark energy density through zero, making w go
to ±∞. This has no physical pathology, since the dark
energy density is merely an effective quantity and as we
will soon see the matter and total equations of state are
well behaved. In the future, the attractor solutions for
V ∼ φ2and V ∼ φ4, i.e. U = 2 and 4, are the de Sitter
state w = −1. The latter case represents the attractor
to GR.
Examining the future attractor solution more closely,
we see from Eq. (47) that within the range 2 < U < 4,
the attractor is to a phantom state w < −1, while out-
side this range it is to w > −1 (unless U gets extremely
negative). In general, reasonable inverse power law po-
tentials do not provide w ≈ −1 and would be disfavored
by observations. Figure 6 shows the full behaviors for
a variety of U values, fixing the matter coupling to the
Yukawa form f ∼ φ (F = 1).
Regarding the early time behavior, as stated the dark
energy equation of state becomes undefined when the
dark energy density passes through zero, as can occur for
F < 1. Since the dark energy density vanishes, however,
this has no physical effects. In particular, the matter
equation of state wmand total equation of state wtotex-
hibit no sign of this occurrence. In Fig. 7 we see that at
high redshift wmgoes to its metastable attractor solution
(third critical point), and because of scaling wtot= wm.
In the Yukawa coupling case (F = 1), wm= 0, as it is
FIG. 5: The dark energy equation of state is plotted for three
power law indices of the potential and the matter coupling.
All solutions asymptote to the stable fixed point wc1(U), in-
dependent of F. Note both quadratic and quartic potentials
(U = 2, 4) reach the de Sitter state w = −1. The deviations
of w(a) from −1 scale linearly with α for small α.
FIG. 6: The evolution of the dark energy equation of state is
shown for various power law potentials V ∼ φU, with Yukawa
coupling f ∼ φ and gravity coupling α = 1. The attractor
values 1 + wc scale with α for small α. Note that φ2and φ4
potentials give w = −1 attractors, while power law indices in
between give phantom attractors.
Page 9
9
for the F = 2 case always as well. Both wm and wtot
then smoothly evolve toward the future stable attractor
solution. Both wmand 1 + wtotscale linearly with α for
small α. For F = 1, all quantities are well behaved and
follow ΛCDM until recently.
FIG. 7: The matter equation of state wm and total equation
of state wtot are plotted for the same potential and matter
coupling cases as in Fig. 5.
of wm and 1 + wtot scale linearly with α for small α. For
Yukawa coupling (F = 1), until recently wm was zero and
wtot followed the standard ΛCDM history (solid black line,
also for U = 4, F = 1) as well.
The asymptotic future values
When α is small, the equations of state of matter, dark
energy, and the total energy are nearly the same as for
ΛCDM, regardless of F and U. This is demonstrated in
Fig. 8 for α = 0.1.
A small value of α corresponds to φ greater than the
Planck energy to induce GN(see Eq. 27). However, there
is no actual energy density that we are taking to be
greater than the Planck scale, so it is not clear there is a
problem in treating this as a low energy effective theory.
In fact, our theory does not have an explicit Planck scale
in it so it is not even clear what the cutoff energy is above
which our theory is expected to break down.
Another issue is how to apply standard scalar-tensor
theory limits from the Solar system, e.g. on the Jordan-
Brans-Dicke parameter ωJBD. Without the matter cou-
pling and with U = 0, one would say that ωJBD= 2π/α
and so these constraints [29] require α < 10−4. In Ref.
[27] it is shown a compatibility of the model U = 4,
F = 0 with current cosmological parameters respecting
Solar system constraints. However, the matter coupling
might alter that conclusion. Ref. [30] considers the differ-
ences between matter coupled and nonminimally gravity
FIG. 8: As the U = 4 case of Fig. 7, but for α = 0.1. Devi-
ations of wm from zero, and wtot from the ΛCDM behavior
are less than 10−2.
coupled models, but so far no analysis has been carried
out including both effects. The equations for the field
perturbations with all the couplings are quite compli-
cated; we leave that analysis for future work and here
only consider effects on the cosmic expansion.
We can consider going beyond constant F and U. In
the case of an exponential matter coupling, the logarith-
mic derivative F is not constant. Figure 9 shows w(a)
for such a case, where f ∼ ebφ, so F = bφ (here the field
values are all in units of the Planck mass). The dynam-
ics is now dependent on the initial value of the field φi,
and resembles that of the constant F case with F = bφi
(a better approximation is evaluating this at a φ part-
way between φi and a later time value). In particular
this applies to the high redshift attractor. However, note
that the late time attractor given by Eq. (47) remains
the same, independent of F and φi.
One can also consider potentials where the logarith-
mic derivative U is not constant. Figure 10 illustrates
w(a) for a potential of Landau-Ginzburg form (25). In
this case, with varying U, the dynamics is quite different
from the constant U case, and no attractor solution is
apparent.
VI.CONCLUSIONS
The dynamics of a scalar field model incorporating
both nonminimal coupling to gravity and coupling to
matter is rich. The relations to induced gravity through a
Higgs mechanism and the symmetrical ideas of coupling
Page 10
10
FIG. 9: Constant U potentials with exponential coupling f ∼
ebφ, shown for b = 2. For U = 4 (thick, black curves), the
cases for three initial field values φi are shown; for U = 2 and
U = −2 only the bφi = 2 case is plotted. The w(a) dynamics
depends on φi (although it is quite similar to a corresponding
constant F) but the final attractor depends on neither f nor
φi.
FIG. 10: Comparison of the Landau-Ginzburg (LG) form,
with either constant F or non power law f, to the φ4potential
shows significant differences. In particular, the non power law
potential does not exhibit attractor behavior.
around the circle of scalar field, matter, and gravity are
interesting. Furthermore, adding both couplings can re-
lieve some problems of either individual coupling.
A set of attractor solutions for the dynamics is found,
with a stable attractor to a dark energy dominated uni-
verse with w either equal to or near −1. High redshift
behavior is standard ΛCDM for a Yukawa matter cou-
pling (better behaved than if there were no coupling),
and a scaling solution otherwise. Throughout cosmic his-
tory the matter and total equations of state can also be
acceptably near ΛCDM behavior, for α not much larger
than unity, depending on the couplings.
A simple quartic potential V ∼ φ4appears viable
(again better behaved than if the potential were con-
stant), with a stable de Sitter attractor for all values
of gravity and matter couplings. Gravity is asymptoti-
cally restored to general relativity (at least as far as the
background behavior is concerned). A quadratic poten-
tial also has a de Sitter attractor.
We identified the roles of the various couplings in the
evolution equations and the effective potential.
slowing of field dynamics due to the gravity coupling
likely alleviates problems with coupled matter instabil-
ities, through the adiabatic mechanism discussed by [15],
and the coupling to matter may help issues with gravity
tests, but the system of perturbation equations becomes
quite complicated due to the two extra scalar field cou-
plings and we leave that to future work. Here we concen-
trated solely on the field and expansion dynamics, which
in itself can constrain the parameter space, while point-
ing the way to interesting attractor behaviors and the
possibility of dynamically establishing w ≈ −1.
The
Acknowledgments
We gratefully acknowledge a UC-MEXUS-CONACYT
Visiting Fellowship for JLCC to spend a sabbatical at
Berkeley. JLCC thanks the Berkeley Center for Cosmo-
logical Physics for hospitality and CONACYT for grant
No. 84133-F. This work has been supported in part
by the Director, Office of Science, Office of High En-
ergy Physics, of the U.S. Department of Energy under
Contract No. DE-AC02-05CH11231,and the World Class
University grant R32-2009-000-10130-0 through the Na-
tional Research Foundation, Ministry of Education, Sci-
ence and Technology of Korea.
Page 11
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