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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-

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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)

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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

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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) =

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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)

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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,

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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.

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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.

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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

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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.

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