The Einstein static universe in Scalar-Fluid theories

Abstract

A new Lagrangian framework has recently been proposed to describe
interactions between relativistic perfect fluids and scalar fields. In this
paper we investigate the Einstein static universe in this new class of
theories, which have been named Scalar-Fluid theories. The stability of the
static solutions to both homogeneous and inhomogeneous perturbations is
analysed deriving the relevant cosmological perturbation equations at the
linear order. We can find several configurations corresponding to an Einstein
static universes which are stable against inhomogeneous perturbations, but
unstable against homogeneous perturbations. This shows the possible
applications of Scalar-Fluid theories to the inflationary emergent universe
scenario.

Full text

The Einstein static universe in Scalar-Fluid theories
Christian G. B¨ohmer,1, ∗Nicola Tamanini,2, †and Matthew Wright1, ‡
1Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK
2Institut de Physique Th´eorique, CEA-Saclay, F-91191, Gif-sur-Yvette, France
(Dated: October 6, 2015)
A new Lagrangian framework has recently been proposed to describe interactions between rela-
tivistic perfect fluids and scalar fields. In this paper we investigate the Einstein static universe in
this new class of theories, which have been named Scalar-Fluid theories. The stability of the static
solutions to both homogeneous and inhomogeneous perturbations is analysed deriving the relevant
cosmological perturbation equations at the linear order. We can find several configurations corre-
sponding to an Einstein static universes which are stable against inhomogeneous perturbations, but
unstable against homogeneous perturbations. This shows the possible applications of Scalar-Fluid
theories to the inflationary emergent universe scenario.
I. INTRODUCTION
Scalar fields have a prominent role in present cosmology not only since they provide simple inflationary solutions for
the early universe, but also for their applications to late-time cosmology. In fact simple scalar field models have been
employed to characterize both the inflaton, a hypothetical field introduced to drive the primordial inflationary phase,
and dark energy, the entity made responsible for the late-time cosmological acceleration. In general it is believed that
further degrees of freedom, beyond the ones of general relativity and Standard Model particles, are needed in order
to account for the observations at both early and late times. A scalar field represents thus the simplest way to add
just one dynamical degree of freedom into the cosmological framework, and moreover it is usually enough to describe
the large-scale effects of high-energy or modified gravity theories, at least at an effective level.
Scalar fields beyond the Standard Model are however expected to possess non negligible interactions with the known
matter particles [1], and thus to provide a fifth force deviation from the geodesic motion of freely falling bodies. Because
of this fifth force, experiments within the Solar System set stringent constraints on any scalar field model, unless a
screening mechanism, such as the well-known chameleon mechanism [8], is introduced. The interaction between the
scalar field and the remaining matter sources is commonly characterized by the use of conformal (sometimes disformal)
transformations, the case of Scalar-Tensor theories being the most popular. In a recent series of contributions a new
framework for coupling a scalar field to matter, including as a sub-class the conformal coupling, has been developed
[3–5]. This new paradigm uses Brown’s Lagrangian formulation of relativistic fluids [7] to describe the matter sources
at a Lagrangian level, providing in such a way new possibilities for coupling the scalar field to the matter sector (see
Sec. II). For this reason, and in analogy with Scalar-Tensor theories, the resulting new class of theories has been
dubbed Scalar-Fluid theories [5].
The scope of the present paper is to find static cosmological solutions, known as Einstein universes, within the
framework of Scalar-Fluid theories and to analyse their perturbations. This constitutes a simple application of cosmo-
logical linear perturbation theory for this newly introduced class of theories, which so far has only been considered for
models of interacting dark energy [3–6], but in practice it can be applied to other situations, for example primordial
inflation. In fact the connection between the Einstein static solution and early universe inflationary theories has been
made explicit in the so-called emergent universe scenario [2], where the initial-time singularity, namely the Big Bang,
is replaced by a past asymptotic Einstein universe.
When considering homogeneous and isotropic solutions of the Einstein field equations, one finds that a generic
solution will either correspond to an expanding or a contracting universe, the Einstein static universe being the
limiting case where the universe does not evolve. However, for this solution to exist in General Relativity, one must
introduce the cosmological constant, without it static solutions cannot be found. Following this line of thought, we
would expect to find static solutions which are in general unstable with respect to small perturbations which favour
a dynamically evolving universe, in agreement with observations.
The emergent universe paradigm has then stimulated the studies of Einstein static solutions, and their stability under
inhomogeneous perturbations, not only in general relativity [9], but also in modified gravity theories representing viable
∗c.boehmer@ucl.ac.uk
†nicola.tamanini@cea.fr
‡matthew.wright.13@ucl.ac.uk

2
alternatives to single field inflation. In particular the Einstein static universes have been analysed in f(R) theories
of gravity [10–12], f(T) gravity [13], Brans-Dicke theory [14], modified Gauss-Bonnet f(G) theories of gravity [15],
hybrid metric Palatini gravity [16], Einstein-Cartan theory [17] and non-constant pressure models [18]. Additionally
they have also been investigated in loop quantum cosmology [19], Horava-Lifshitz gravity [20], IR modified Horava
gravity [21] and non-minimal kinetic coupled gravity [22]. It is thus interesting to investigate this particular type of
solutions within the framework of Scalar-Fluid theories which might constitute alternative models of dark energy and
inflation as well.
The paper is organized as follows. In Sec. II the action of Scalar-Fluid theories will be presented reviewing the main
details of the formulation and deriving the relativistic field equations. In Sec. III models with an algebraic coupling
between the scalar field and the matter sector will be considered. The cosmological equations will be computed at both
background and perturbation levels, while Einstein static solutions will be found for some specific models and their
stability will be investigated. In Sec. IV the same analysis will be applied to Scalar-Fluid models with a derivative
coupling between the scalar field and the matter sources. Finally in Sec. V the results obtained in the preceding
sections will be discussed and the conclusions will be drawn.
Throughout the paper the (−,+,+,+) convention for the signature of the spacetime metric will be used, the speed
of light will be set to one c= 1, and κ2= 8πG.
II. SCALAR-FLUID THEORIES: ACTION AND FIELD EQUATIONS
In this section we review the variational approach to interacting dark energy that was formulated in [3, 4]. The
total action of our interacting dark energy system is
S=Z(Lgrav +LM+Lφ+Lint)d4x , (1)
where the gravitational sector Lgrav is given by the standard Einstein-Hilbert Lagrangian density
Lgrav =√−g
2κ2R , (2)
with Rbeing the Ricci scalar with respect to the metric gµν , and gdenotes its determinant. The Lagrangian density
of the scalar field is taken to be of the canonical (quintessence) type
Lφ=−√−g1
2∂µφ ∂µφ+V(φ),(3)
where Vis the scalar field potential depending only on φ. The Lagrangian for the relativistic fluid is described using
Brown’s formulation for LMgiven by [3, 7]
LM=−√−g ρ(n, s) + Jµϕ,µ +sθ,µ +βAαA
,µ,(4)
where ρis the energy density of the fluid prescribed as a function of n, the particle number density, and s, the entropy
density per particle. The fields ϕ,θand βAare all Lagrange multipliers with Ataking the values 1,2,3 and αAare
the Lagrangian coordinates of the fluid. The vector-density particle number flux Jµis related to nas
Jµ=√−g n U µ,|J|=p−gµνJµJν, n =|J|
√−g,(5)
where Uµis the fluid 4-velocity obeying the relation UµUµ=−1.
This just leaves us to determine the interaction Lagrangian Lint. We will consider two distinct types of couplings.
In [3] an algebraic coupling between matter and the scalar field was considered. There the interaction Lagrangian
took the form
Lint =−√−g f(n, s, φ),(6)
where f(n, s, φ) is an arbitrary function which will specify the particular model. In this paper we will consider only
one specific type of coupling and take only couplings of the form
f(n, s, φ) = f(ρ, φ).(7)

3
This means we will not consider models where the interaction can depend on the entropy density per particle.
Moreover, we only consider an implicit dependence on the particle number nthrough the density ρ. Despite these
restrictions, this framework is substantial.
In [4] a different interaction Lagrangian was considered; a coupling between the matter sector and first derivatives
of the scalar field were considered. This time the interacting Lagrangian was given by
Lint =f(n, s, φ)Jµ∂µφ , (8)
where fis again an arbitrary function of the three physical fields. This is the most general coupling term where only
one spacetime derivative of the scalar field appears. As with the algebraic coupling, we restrict ourselves to entropy
independent interactions and implicit particle number dependence.
Variation of the total Lagrangian with respect to the metric gives the following Einstein Equations
Gµν =κ2Tµν +T(φ)
µν +T(int)
µν ,(9)
where the different energy momentum tensors are defined as
Tµν =p gµν + (ρ+p)UµUν,(10)
T(φ)
µν =∂µφ ∂νφ−gµν 1
2∂µφ ∂µφ+V(φ),(11)
T(int)
µν =pint gµν + (pint +ρint)UµUν.(12)
Here the fluid pressure is defined as
p=n∂ρ
∂n −ρ . (13)
In the case of an algebraic coupling, the interacting pressure and energy density are defined as
ρint =f(n, φ) and pint =n∂f(n, φ)
∂n −f(n, φ),(14)
whereas in the case of the derivative coupling they are defined as
ρint = 0 and pint =−n2∂f
∂n Uλ∂λφ . (15)
In what follows we will investigate the Einstein static universe in both of these scenarios.
III. ALGEBRAIC COUPLING
In this section we will consider the Einstein static universe where we assume that the coupling between matter and
the scalar field is purely algebraic. This means that the interacting pressure can now be written in terms of fas
pint = (ρ+p)∂f
∂ρ −f . (16)
The cosmological applications for a few particular choices of such an fhave been considered in [3]. These models can
exhibit a range of interesting cosmological phenomena. Dark energy dominated late time attractors with a dynamical
crossing of the phantom barrier have been found, along with scaling solutions, early time matter dominated epochs
and a possible inflationary origin.
A. Background Equations
To begin with we will consider the background cosmology equations of such models to show that Einstein static
universe solutions to the field equations do indeed exist. Let us consider the standard Friedmann-Robertson-Walker
(FRW) line element given by
ds2=−dt2+a(t)2dr2
1−kr2+r2(dθ2+ sin2θ dφ2),(17)

4
where a(t) is the cosmological scale factor and k=−1,0,1 according to the spatial openness, flatness or closeness
of the constant time hypersurfaces, respectively. Inputting this into the Einstein field equations (9) we derive the
following two Friedmann equations
3k
a2+ 3H2=κ2ρ+1
2˙
φ2+V+f,(18)
k
a2+ 2 ˙
H+ 3H2=−κ2p+1
2˙
φ2−V+pint.(19)
The Klein-Gordon or scalar field equation reduces to
¨
φ+ 3H˙
φ+∂V
∂φ +∂ρint
∂φ = 0 .(20)
We will now look for an Einstein static universe solution. We set the scale factor a(t) = a0= const, which implies
that H=˙
H= 0. We will also set our scalar field to be a constant: φ=φ0, and assume that the perfect fluid
obeys a simple linear equation of state p=wρ where wis a constant lying in the range −1< w < 1 which is called
the equation of state (EoS) parameter. Inputting these assumptions into the two Friedmann equations (18) and (19)
yields
3k
a2
0
=κ2(ρ0+V(φ0) + f) (21)
k
a2
0
=−κ2(p0−V(φ0) + pint),(22)
with the Klein-Gordon equation (20) reducing to
V0(φ0) + ∂f
∂φ φ=φ0
= 0.(23)
Combining (21) and (22) gives us the simple relation
ρ0(1 + 3w) + f+ 3pint = 2V(φ0) (24)
between the potential and the energy and pressure of both the fluid and the interacting fluid. The above equations
give three algebraic equations for the three unknowns ρ0,a0and φ0, and thus, as long as k6= 0, we can find an
Einstein static universe solution. If k= 0, we note that a0is undetermined. Note that Eq. (23) implies that the
static configurations of the scalar field lie in the minima of the effective potential V(φ) + f(ρ, φ). Finding an explicit
solution will depend on the particular functional form of f. We see that, unlike in general relativity, potentially there
is also the possibility of a static open universe with k=−1 if the function f, determining the interacting energy
density, is sufficiently negative.
B. Perturbation equations
We now wish to explore whether the Einstein static universe solutions found above are stable under small perturba-
tions. The perturbed equations for scalar-fluid theories of this type were first derived in [5], using a slightly different
notation. Here we will review these equations, using our particular functional form for the interacting function f.
We will work with the metric in the Newtonian gauge, also called the longitudinal gauge, which is given by the line
element
ds2=−(1 + 2Φ)dt2+(1 −2Ψ) a(t)2
1 + 1
4k(x2+y2+z2)2dx2+dy2+dz2,(25)
where both Ψ and Φ are functions of all the coordinates. Our matter sources can be considered as perfect fluids and
so we expect no anisotropic stresses to appear. Hence we may consider the off-diagonal ij-components of the Einstein
field equations, which indeed read
∂i∂j1 + 1
4kx2+y2+z2(Ψ −Φ)= 0 .(26)

5
From this equation we immediately find that
Φ=Ψ,(27)
as expected since no anisotropies are present in the system. In what follows thus we will simplify all of the equations
considering that Ψ equals Φ. This statement is independent of the coupling function fand hence valid for all models
in this class.
We also must determine how the matter variables are perturbed. We perturb the quantities ρ,φ,pand Uµaccording
to
φ+δφ , ρ +δρ , p +δp , Uµ+δUµ,(28)
where φ,ρ,pand Uµare the background quantities and the perturbation of the four velocity reads
δUµ= (−Ψ, ∂iv).(29)
Here vis the scalar perturbation of the matter fluid’s velocity.
We are now in a position to derive the perturbed Einstein equations. We will give the equations directly in Fourier
space, so that we write the Laplacian as ∇27→ −q2, where qis the wave number of the fluctuation. For the spatially
closed case k= 1 we have that this wave number must equal q=n(n+ 2) for positive integer n= 0,1,2, ..., whereas
for the spatially open case k=−1 we simply have that qis any real number such that q > 1.
Inserting the perturbed metric into the Einstein equation (9), the 00-component reads
6k
a2Ψ−q2
a2Ψ−4π1 + ∂f
∂ρ δρ −4π∂f
∂φ +V0δφ −8π(ρ+V+f) Ψ −3H˙
Ψ−4π˙
φ˙
δφ = 0 ,(30)
where dots denote differentiation with respect to t. After integrating over dxi, the 0i-components are
8π(p+ρ)1 + ∂f
∂ρ v−8π˙
φδφ + 2 ˙
Ψ+2HΨ=0,(31)
which as usual gives the velocity perturbation vin terms of the other perturbed variables. The ii-components become,
after a simplification using the background equations,
4π(ρ+p)∂2f
∂ρ2δρ + 4π1 + ∂f
∂ρ δp + 4π(ρ+p)∂2f
∂ρ∂φ −∂f
∂φ −V0δφ
+ 4π˙
φ˙
δφ +k
a2−4π˙
φ2−2˙
H−3H2Ψ−4H˙
Ψ−¨
Ψ=0.(32)
And finally the perturbation of the scalar field equation (20) is
¨
δφ + 3H˙
δφ +q2
a2+∂2f
∂φ2+V00 δφ +∂2f
∂ρ∂φ δρ −2¨
φ+ 3H˙
φΨ−4˙
φ˙
Ψ=0.(33)
From equation (30) one can solve for δρ and substitute it into the equations (32) and (33), which will then provide
two dynamical equations for the variables Ψ and δφ.
We will now insert our background Einstein static universe solution into the perturbed equations above. We will
assume adiabatic perturbations, so that the pressure perturbation obeys the same equation of state as the background
pressure, δp =wδρ. Inserting the static solution into (30), we find
3k
a2
0
Ψ−q2
a2
0
Ψ−4π1 + ∂f
∂ρ δρ = 0 .(34)
The other diagonal equation (32) simplifies to
4π(w+ 1)ρ0
∂2f
∂ρ2δρ + 4π1 + ∂f
∂ρ wδρ + 4πρ0(w+ 1) ∂2f
∂ρ∂φ δφ +k
a2
0
Ψ−¨
Ψ = 0 ,(35)
and the Klein-Gordon equation (33) becomes
¨
δφ +q2
a2
0
+∂2f
∂φ2+V00 δφ +∂2f
∂ρ∂φ δρ = 0 .(36)

6
The off-diagonal equation will not be needed to analyse the stability of the perturbations since the velocity perturbation
does not appear in the other equations.
We can solve (34) for δρ and insert this back into (35) and (36). We can then write the resulting equations as a
coupled two dimensional linear system of second order ordinary differential equations



¨
Ψ
¨
δφ

=M


Ψ
δφ

,(37)
where Mis a 2 ×2 matrix with constant coefficients involving the background quantities.
The matrix Mhas the following components
M=


M11 M12
M21 M22


(38)
where the components of this matrix are given by
M11 =

(w+ 1)ρ0∂2f
∂ρ2
1 + ∂f
∂ρ
+w
3k−q2
a2
0+k
a2
0
,(39)
M12 = 4πρ0(w+ 1) ∂2f
∂ρ∂φ ,(40)
M21 =−1
4π(1 + ∂f
∂ρ )
∂2f
∂ρ∂φ 3k−q2
a2
0,(41)
M22 =−q2
a2
0
+∂2f
∂φ2+V00 .(42)
Since the terms in the matrix are quite involved, there is little hope making generic statement about stability and
instability for general f.
C. Stability of perturbations
The linear system of equations described by (37) is a coupled second order system of differential equations. Therefore
it will have four linearly independent solutions as a result of the two eigenvalues of M. Let us denote the eigenvalues
of Mby λ1and λ2. The solution to the system will involve the frequencies ±√λ1and ±√λ2, and hence in order for
the perturbations to be stable we require the following conditions
<λi<0,=λi= 0, i = 1,2.(43)
Now the components of the matrix Mare too complicated to say anything general about the stability of the Einstein
static universe for a generic coupling function f. One could attempt using Sylvester’s criterion instead of working
with the eigenvalues directly, however, the resulting equations are still too involved. Thus we will examine a few
specific models corresponding to different functional forms of f.
D. Models
We will consider three separate models assuming different forms for our interaction function f. Two of the models
were first considered in [3] where the background cosmological dynamics were analysed. The standard chameleon
model [8] can be derived from this interacting Lagrangian approach with algebraic coupling by making the choice for
the coupling function f=−ρ+ρeβκφ , and thus we are also able to discuss the stability of the Einstein static universe
in this model within this framework. The models we will consider are outlined in the table below

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f pint
Model I γ ραexp(−βκφ) [α(w+ 1) −1]f
Model II γκφρ wf
Chameleon field −ρ+ρeβκφ wf
1. Model I
We first consider model I where we take the interacting function fto be a coupling of a power law of the energy
density and an exponential in the scalar field, and the potential Vto be a standard cosmological exponential potential
f(ρ, φ) = γ ραe−βκφ , V (φ) = V0e−λκφ .(44)
The background cosmology of this model was analysed in [3] for the particular cases of α= 1 and α= 3. These
models have a range of interesting phenomenology. Late time accelerating attractor solutions were shown to exist for
a wide choice of parameters, which can describe dark energy. Scaling solutions were found which may be useful for
solving the cosmic coincidence problem, along with solutions undergoing transient inflationary epochs at early times.
Now let us solve the background equations (21)-(23) for an Einstein static universe in this model. The Klein Gordon
equation gives us the condition
ρα
0=−λV0
βγ e(β−λ)κφ0.(45)
and thus for a positive energy density we require the condition that βγ < 0. The Friedmann equations then admit
the solution
a2
0=kβ(1 + 3w)
κ2(1 + w)(β+λ(α−1))V(φ0)(46)
where φ0is given implicitly by solving
φ0=1
βκ log κβγ
V0(φ0)−αlog κβ(1 + 3w)
2βκV (φ0)+2−3(1 + w)αV 0(φ0) (47)
Now assuming that both λ, β ≥0, requiring that the scale factor is real tells us we will only have a static solution in
a closed universe k= 1 when the EoS paramater lies in the range w > −1/3. For the case of an open universe with
k=−1, the opposite situation arises, with wnow lying in the range −1< w < −1/3.
Let us first consider a closed universe, so that we require w > −1/3. For the choice of exponents α= 1,2,3,4,5, it
has been checked numerically that the regions of stability of the homogeneous perturbations (corresponding to q= 0)
and the n= 2 perturbation (corresponding to q=√8) do not coincide for any win the range −1/3< w < 1. We can
therefore conclude that the Einstein static universe is not stable for these choices of exponent αin this model. We
show an example plot of the stability regions of the n= 0 and n= 2 perturbations in Fig. 1 when the matter EoS
w= 1/3. The grey region indicates the region where the homogeneous perturbations are stable, the grey represents
the n= 2 inhomogeneous perturbation, which overlap nowhere in parameter space. A similar result is found for all
values of win the allowed range.
Now considering the case of an open universe k=−1, we find that for qclose to 1 there are regions of stability.
However as we increase qthese regions shrink and disappear. This can be seen by noting that for large qit is the case
that one of the eigenvalues must be positive. Thus we cannot find a stable static solution in an open universe either.
2. Model II
Let us now consider the second model, where we choose the interaction function to be a simple linear coupling
between matter and the scalar field
f(ρ, φ) = γκφρ . (48)

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FIG. 1. Stability of the perturbations in β−λparameter space when w= 2/3. The left panel corresponds to the case α= 1, the
middle panel corresponds to α= 2. The black region represents the stability of n= 0 perturbation, the grey region represents
the n= 2 perturbation.
The background dynamics of this model were also considered in [3], assuming a standard exponential potential. The
dynamics of this model was very similar to that of the case of general relativity with a canonical scalar field, and thus
can in principle replicate the background dynamics of a ΛCDM universe given a flat enough potential.
Let us solve the background equations for a static solution. The Klein-Gordon equation (23) allows us to solve for
the energy density
ρ0=−1
κ2γV0(φ0).(49)
Inserting this into the constraint (24) yields the following implicit equation for φ0
κφ0=−2κV (φ0)
(1 + 3w)V0(φ0)−1
γ.(50)
We can now easily solve one of the remaining Friedmann equations for the scalar factor a0to give
a2
0=1
κ2V(φ0)
k(1 + 3w)
1 + w.(51)
And thus positivity of this expression means we only have an Einstein static universe for k= 1 if w > −1/3, and
k=−1 if w < −1/3. To ensure a positive energy density, we require that V0(φ0)<0 if γ > 0 and V0(φ0)>0 if γ < 0.
First we consider the case of a closed universe, k= 1. Looking at the homogeneous perturbations and the n= 2,
(so that q=√8), inhomogeneous perturbations, we find that the regions of stability of these two overlap for a small
range of parameter values. We should note here that one does not need to consider the n= 1 perturbations, this
is simply a gauge degree of freedom. Now as the wave number nincreases, the area of stability grows in parameter
space, and thus the small region of stability will remain stable to higher wave number perturbations. An example plot
showing this behaviour is shown in Fig. 2 where the n= 0,2,3,4 perturbations are considered and the EoS was chosen
to be w= 3/4. This region of stability only appears for sufficiently big w, for instance the n= 2 inhomogeneous
perturbations are always unstable for w < 1/5, a result that matches the one obtained with a single sourcing fluid [9].
According to Fig. 2 the region of stability of inhomogeneous perturbations (n≥2) is quite large in the (V, V 0) space.
Although only in a small part of such region also the homogeneous perturbations are stable, for the well-known
emergent universe scenario only the stability of inhomogeneous perturbations is required in order to have a viable
alternative model of inflation [2, 9]. Fig. 2 thus shows that Model II can be applied consistently to early universe
phenomenology within the emergent universe framework.

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FIG. 2. Parameter space plot of V00(φ0) against V0(φ0) showing the regions of stability of the n= 0,2,3,4 perturbations in
Model II with the EoS given by w= 3/4. There is a small region in which all perturbations are stable. Increasing nincreases
the region of stability.
FIG. 3. Parameter space plot of V00(φ0) against V0(φ0) showing the regions of stability of the q= 1.01,2,3,4 perturbations
in Model II with the EoS given by w=−1/2. The regions do not all overlap, and increasing qeventually makes the stability
regions vanish.
In the case of an open universe k=−1, we are unable to find solution which are stable to all perturbations. An
example plot is shown in Fig. 3, where regions of stability for different values of qare shown for the case w=−1/2. It
is found that increasing qstops the regions of stability overlapping, and for large enough qthe stable regions disappear
altogether.
3. Chameleon Model
Now let us consider the third of our models. If we choose the interaction function fand the potential Vto be
f=−ρ+ρeβκφ , V (φ) = M4+α
φα,(52)
then we recover the standard chameleon mechanism [8] within this scalar-fluid framework. Here αand βare positive
constants and Mis a mass scale. Such an interaction is of great theoretical interest, since it masks the appearance of
the fifth force deviation from general relativity at solar system length scales.

10
FIG. 4. Parameter space plot of αagainst βshowing the regions of stability of the n= 2,3,4,5,6 perturbations in the
chameleon model with the EoS given by w= 1/3. The stability regions grow for increasing n, and there is a region in which
all inhomogeneous perturbations are stable.
Now looking for static solutions within this model we find the following solution
ρ0=2α+1αM α+4 e−1
2α(1+3w)
βκ βκ
α(1 + 3w)α+1
,(53)
a2
0=βk
2αM4+αακ(1 + w)α(1 + 3w)
βκ α+1
, φ0=α(1 + 3w)
2βκ ,(54)
Of course, this solution will only exist if a2
0is positive. Thus depending on the particular parameter choices, there
will either exist a k= +1 or k=−1 static universe. Typically αand βare both positive for this model to allow
for cosmic acceleration and the screening mechanism, and thus for a closed k= 1 static universe we require that
w > −1/3. There are no open universe solutions unless w < −1 so this will not be considered here further.
If we analyse the stability matrix of this model, numerically we find that when α > 0 the homogeneous perturbations
are always unstable. This means one cannot achieve a stable Einstein universe in these chameleon theories in the
context of scalar fluid theories. However, analysing the stability of the inhomogeneous perturbations, it is found they
are always stable for βsufficiently small, see for example Fig. 4, where the regions of stability of the n= 2,3,4,5,6
perturbations are plotted for w= 1/3. With increasing nthe stability region grows, and for approximately β < 1 all
of the inhomogeneous perturbations are stable. This means that the chameleon model is potentially applicable to the
emergent universe framework.
IV. DERIVATIVE COUPLING
In this section we will analyse the Einstein static universe in the context of a derivative coupling between the matter
sector and the scalar field. Such a model was considered in [4] in the context of dark energy interacting with dark
matter.
A. Background equations
First we will derive the equations governing the background cosmological evolution. As before we will assume the
Friedmann-Robertson-Walker metric (17). This time the Friedmann equations read
3k
a2+ 3H2=κ2(ρ+1
2˙
φ2+V),(55)
k
a2+2˙
H+ 3H2=−κ2(p+1
2˙
φ2−V) + n2∂f
∂n ˙
φ , (56)

11
whereas the scalar field equation is modified to
¨
φ+ 3H˙
φ+V0−n2∂f
∂n 3H= 0 .(57)
Now we look for an Einstein static universe solution in this model, so we assume that our scale factor and all other
physical fields are independent of time. The Klein-Gordon equation (57) contains time derivatives in every quantity
except the potential term, so this equation simply reduces to the condition
V0(φ0)=0,(58)
so that the scalar field of the static universe solution must lie at an extremum of the potential. The static universe
is completely independent of the form of the coupling function fat the background level, because the only place at
which fenters the field equations it is multiplied by a factor of ˙
φ. And hence the static solution will be the same as
a static solution in standard quintessence. However despite this the equations at the level of the perturbations are
different, and so the stability of the static universe should be investigated. The Friedmann equations can easily be
seen to reduce to the system
k
a2
0
=κ2V(φ0)1 + w
1+3w,(59)
ρ0=2V(φ0)
1+3w.(60)
In order for the energy density to be positive we will require that the EoS satisfies w > −1/3, and this in turn means
that for the scale factor to be real we only have a static solution in the case of a closed universe k= +1.
B. Perturbations
We will now derive the general perturbation equations of this derivative coupling model. As before will work in
the Newtonian gauge (25) and perturb our matter variables according to (28). Following [4] we will also make an
additional assumption on the form of the coupling function f, so that the equations are independent of the particle
number density nexplicitly, with a dependence only implicitly through ρ. This leads us to consider the following form
of f
f(n, φ) = F(ρ, φ)
n.(61)
This means that the interaction energy momentum tensor is given by
T(int)
µν =F−(ρ+p)∂F
∂ρ Uλ∂λφ(gµν +Uµν ).(62)
Once again looking at the off-diagonal ij-components of the field equations immediately gives
Φ=Ψ,(63)
since once again no anisotropies are present. Thus as is the algebraic coupling case, in what follows we will simplify
the equations considering that Φ equals Ψ and we will give the equations directly in the Fourier space: ∇27→ −q2.
The 00-component of the Einstein field equations reads
6k
a2−q2
a2−κ2(ρ+V)Ψ−3H˙
Ψ−κ2
2(δρ +V0δφ +˙
φ˙
δφ) = 0 ,(64)
which is independent of the coupling function f, exactly as in the background case. The 0i-component are (after
integrating over dxi)
κ2F˙
φ+ (p+ρ)1−˙
φ∂F
∂ρ v−κ2˙
φδφ + 2 ˙
Ψ+2HΨ=0.(65)

12
The ii-components are, after a simplification using the background equations
¨
Ψ+4H˙
Ψ + 2˙
H+ 3H2−k
a2+κ2
2(˙
φ2+F˙
φ−(ρ+p)∂F
∂ρ ˙
φ)Ψ
+κ2
2(ρ+p)˙
φ∂2F
∂ρ2δρ +κ2
2˙
φ∂F
∂ρ −1δp
+κ2
2(ρ+p)˙
φ∂2F
∂ρ∂φ −˙
φ∂F
∂φ +V0δφ +κ2
2((ρ+p)∂F
∂ρ −F−˙
φ)˙
δφ = 0 .(66)
And finally the perturbation of the scalar field equation reads
3(F−(ρ+p)∂F
∂ρ )+4˙
φ˙
Ψ + 2¨
φ+ 6H˙
φ−3H((ρ+p)∂F
∂ρ −F)Ψ
+ 3H(ρ+p)∂2F
∂ρ2δρ + 3H∂F
∂ρ δp −((ρ+p)∂F
∂ρ −F)q2
a2v
+−q2
a2+ 3H((ρ+p)∂2F
∂ρ∂φ −∂F
∂φ )−V00 δφ −3H˙
δφ −¨
δφ = 0 .(67)
C. Stability of the static universe
Now let us insert our Einstein static universe solution into the perturbation equations. The 00 equation (64) now
becomes
3k
a2
0−q2
a2
0Ψ = κ2
2δρ . (68)
The 0i-component (65) takes the particularly simple form
κ2ρ0(1 + w)v=−2˙
Ψ,(69)
which allows one to find the velocity perturbation easily in terms of the metric perturbation. The ii components (66)
reduce to
κ2
2(−δp + ((ρ+p)∂F
∂ρ −F)˙
δφ)−k
a2
0
Ψ + ¨
Ψ=0,(70)
while the scalar field equation (67) becomes
¨
δφ +q
a2
0
+V00δφ + 3((ρ+p)∂F
∂ρ −F)˙
Ψ−(F−(ρ+p)∂F
∂ρ )q2
a2
0
v= 0 .(71)
Once again we will now assume an adiabatic perturbation, so that δp =wδρ. Substituting the density perturbation
from (68) into (70) and the velocity perturbation (69) into (71) the system of equations reduce to the following two
dimensional system
κ2
2ρ0(w+ 1)∂F
∂ρ −F˙
δφ +wq2
a2
0−(3w+ 1)k
a2
0Ψ + ¨
Ψ=0,(72)
¨
δφ +q
a2
0
+V00δφ +3 + 2q2
κ2a2
0(1 + w)ρ0ρ0(w+ 1) ∂F
∂ρ −F˙
Ψ=0.(73)
Now let us introduce the following vector
X=


Ψ
δφ

,(74)

13
which means we can write the above system of equations (72), (73) as the following two dimensional matrix equation
¨
X+A˙
X+BX= 0 .(75)
Here the matrices Aand Bhave been defined as
A=


0κ2
2P
(3 + 2q2
κ2a2
0(1+w)ρ0)P0

, B =


wq2
a2
0−(3w+1)k
a2
0
0
0q2
a2
0
+V00

,(76)
where we have introduced the quantity P
P=ρ0(w+ 1)∂F
∂ρ −F . (77)
Now to reduce the equation to a first order system, we introduce the vector
Y=˙
X,(78)
so that the equation (75) can be written as the following first order autonomous system



˙
Y
˙
X

=

−A−B
I20




Y
X

,(79)
where I2denotes the 2 ×2 identity matrix. For the system to be stable we simply require that the eigenvalues of the
above system are purely imaginary. Let us write
A=


0a1
a20

, B =


b10
0b2


.(80)
Then the four eigenvalues of the system (79) in terms of aiand biare simply
λi=±1
√2qa1a2−b1−b2±p(b1+b2−a1a2)2−4b1b2.(81)
Now we immediately see from the definitions of a1and a2that a1a2≥0. Thus requiring the eigenvalues (81) are
imaginary reduces to the following conditions
b1>0, b2>0,(82)
a1>0, a2>0, a1a2< b1+b2−2pb1b2,or a1<0, a2<0, a1a2< b1+b2+ 2pb1b2.(83)
For the homogeneous perturbations we set q= 0. Then requiring b1>0 tells us we must have w < −1/3, and
b2>0 tells us the potential must lie at a minimum V00(φ0)>0. However, we have already seen that the static
universe solution requires that w > −1/3, and hence the homogeneous perturbation cannot be stable. And thus no
static universe will be homogeneously stable in these derivative coupled models.
Now let us examine the stability against inhomogeneous perturbations. b1>0 will be satisfied as long as q2>4
and w > 1/(q2−3) (so w > 1/5 for the n= 2 perturbation [9]). The first of these condition is always satisfied since
the smallest inhomogeneous perturbation is the n= 2 mode, which corresponds to q2= 8. The condition b2>0 will
be satisfied as long a we are at a minimum of the potential: V00 (φ0)>0. We can simplify the quantity a2= (3+q2)P.
Therefore the condition for stability is:
κ2
2(3 + q2)P2< sq2
a2
0
+V00 +pwq2−(3w+ 1)
a0!2
if P > 0,(84)

14
and
κ2
2(3 + q2)P2< sq2
a2
0
+V00 −pwq2−(3w+ 1)
a0!2
if P < 0.(85)
These conditions can be satisfied for all inhomogeneous perturbations. To show this we will derives some sufficient
conditions for these inequalities to be true. For positive Pa sufficient condition for (84) to be true is
κ2
2(3 + q2)P2<q2
a2
0
,(86)
so we need
κ2
2a2
0P2<q2
3 + q2,for all q > √8,(87)
which is satisfied if
κ2
2a2
0P2<8
11 ,(88)
which alternatively we can write as
1+3w
1 + wP2
2V(φ0)<8
11 .(89)
For P < 0 a sufficient condition for (85) to be true is for
κ2
2(3 + q2)P2<(1 −√w)2q2
a2
0
,(90)
holding when 0 < w < 1. This can then be reduced to the following condition which will ensure the stability for all
inhomogeneous perturbations
1+3w
(1 + w)(1 −√w)2P2
2V(φ0)<8
11 .(91)
This is a stricter condition than for positive P. Nonetheless it can still easily be satisfied for sufficiently small P.
Let us examine the form of Pfor different choices of coupling functions. If Fof the form F= const or F=F(φ),
then we simply have P=−F. If Fis a constant then the stability of inhomogeneous perturbations will depend
not only on the value of F, but also on V(φ0) according to the conditions (89) and (91). An interesting case is
F(φ) = ξpV(φ), whose background cosmology curiously results to be equivalent to the one analysed in [23] in the
context of k-essence, as shown in [4]. According to the conditions (89) and (91), in this situation we find that
the stability of inhomogeneous perturbations will no longer depend on the scalar field potential, but only on the
constant ξ. For example considering w= 1/3, which is expected for early universe applications, we will find stability
approximately if −0.416 < ξ < 0.985. A similar reduction applies in the case F=γ√ρ, which has been studied in
[4]. In this case using the background equation (60) the conditions (89) and (91) become again independent of the
scalar field and the stability of perturbations will be determined by the constant γ. For early universe applications
(w= 1/3) one finds stability of the inhomogeneous perturbations if approximately −1.25 < γ < 9.80.
These last examples shows that Scalar-Fluid models with derivative couplings can easily be used in the context of the
emergent universe scenario where the stability of inhomogeneous perturbations and the instability of the homogeneous
perturbations are required for the viability of this alternative model of inflation.
V. DISCUSSION
In this work we have analysed Einstein static universe solutions in the newly proposed framework of Scalar-Fluid
models, where an interaction between an effective perfect fluid and a scalar field is introduced directly at the level
of the action. We have shown that generically static solutions exist and we have studied their stability against both
homogeneous and inhomogeneous perturbations, deriving the relevant cosmological perturbation equations at the
linear level.

15
In the case of purely algebraic couplings, we are unable to find simple analytic conditions to determine the stability
of our static solutions. The reason for this is the complicated structure of the matrix whose eigenvalues determine the
stability properties of the perturbed solutions. We were thus forced to consider individual models and numerically
explore the regions of stability. We have analysed three particular models. The first of these, assuming a non-linear
exponential coupling, was shown to be generically unstable, to both homogeneous and inhomogeneous perturbations.
However when a simple linear coupling is considered, namely ρint ∝φρ, it is found that there is a small region of
parameter space where the Einstein static universe is stable to both homogeneous and inhomogeneous perturbations,
while there is a large region in parameter space where it is stable only against inhomogeneous perturbations. This
last situation is exactly the one required by the emergent universe paradigm, implying that such model can be applied
to early universe phenomenology as an alternative inflationary scenario. Similar results can be found with the third
Scalar-Fluid coupling, which reproduces the well-known chameleon mechanism. This incidentally suggests possible
applications of screening models to the emergent universe scenario which could be taken into account for future
analyses.
On the other hand, when considering an arbitrary coupling between the fluid’s four velocity and the derivative of
the scalar field, we are able to make some generic statements applicable to all models of this type. It is found that
homogeneous perturbations are always unstable, while the stability of inhomogeneous perturbations is determined by
simple inequalities (see Eqs. (89) and (91)) depending on the scalar field potential and the specific form of the Scalar-
Fluid derivative coupling. Particular models of this kind, the ones admitting stable inhomogeneous perturbations, are
thus suitable for applications within the context of the emergent universe scenario.
In general thus the results obtained in this work show that Scalar-Fluid theories might well constitute new interesting
inflationary paradigms. Their broad applications to early universe phenomenology, also as possible mechanisms of
reheating, deserves to be studied in future works. It thus appears that these theories offer the possibility of studying
early time and late time phenomenology using a single model based on a well defined Lagrangian approach.
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