# Vector field models of modified gravity and the dark sector

J. Zuntz1, T.G Zlosnik2, F. Bourliot3, P.G. Ferreira1, G.D. Starkman4
1Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road,
Oxford OX1 3RH, UK
2Perimeter Institute for Theoretical Physics, 31 Caroline Street North,
Waterloo, Ontario N2L 2Y5, Canada
3 CPHT, Ecole Polytechnique, 91128 Palaiseau cedex, France
4Department of Physics/CERCA/ISO,Case Western Reserve University,
Cleveland,OH, 44106-7079

###### Abstract

We present a comprehensive investigation of cosmological constraints on the
class of vector field formulations of modified gravity called Generalized
Einstein-Aether models. Using linear perturbation theory we generate cosmic
microwave background and large-scale structure spectra for general parameters
of the theory, and then constrain them in various ways. We investigate two
parameter regimes: a dark-matter candidate where the vector field sources
structure formation, and a dark-energy candidate where it causes late-time
acceleration. We find that the dark matter candidate does not fit the data,
and identify five physical problems that can restrict this and other theories
of dark matter. The dark energy candidate does fit the data, and we constrain
its fundamental parameters; most notably we find that the theory’s kinetic
index parameter $n_{\mathrm{ae}}$ can differ significantly from its
$\Lambda$CDM value.

###### pacs:

PACS Numbers :

††preprint:

##  I Introduction

Over the past few years, a new suite of models for the dark sector has been
proposed. They invoke a vector field which is normally constrained to lie
along the time-like direction and may lead to modifications to the
gravitational sector Zlosnik et al. (2007); Jiménez and Maroto (2008);
Dimopoulos et al. (2009); Koivisto and Mota (2008). Sometimes called Einstein-
Aether models, they tend to entangle two of the main paradigms currently being
considered: on the one hand modified gravity and on the other dark matter and
energy.

Vector field models are attractive because they seem to be able to resolve the
problem of the dark sector (i.e. dark matter and energy) in a unified way.
Most of the emphasis has gone into constraining vector field models that lead
to accelerated expansion Jiménez and Maroto (2008); Dimopoulos et al. (2009);
Koivisto and Mota (2008) although there is a fair amount of work for which the
the vector field leads to a relativistic version of Modified Newtonian
Dynamics (MOND) Milgrom (1983) or can even play the role of dark matter
Zlosnik et al. (2007); Zhao (2007). In fact vector field models seem to
incorporate what seems to be a generic feature of relativistic modified
gravity models Ferreira and Starkman (2009): that it is impossible to
construct relativistic models that just modify the gravitational sector
without introducing new degrees of freedom, which can then behave like either
dark matter or dark energy (although for other approaches see, for example,
Milgrom (2009); Blanchet and Tiec (2009, 2008); Soussa and Woodard (2004);
Bruneton et al. (2009); Sagi (2009)).

There has been significant progress in trying to constrain these models. For
example, at a fundamental level it has been shown that a broad class will lead
to instabilities and the formation of caustics, signaling a break down of the
fundamental theory Contaldi et al. (2008). It has also been shown that for a
general choice of kinetic terms, these theories will be plagued by ghosts or
tachyons Carroll et al. (2009); Eling (2006); Lim (2005). These pathologies
are worrying but do not entirely rule out vector field models- it has been
shown that modifications to the kinetic term, for example, can cure them.

Substantial work has been done on understanding how these fields in these
theories behave on macroscopic scales, either through their interaction with
matter to form galaxies and clusters Dai et al. (2008), or on the largest
scales, affecting the growth of structure and its effect on the CMB Zlosnik et
al. (2008). Indeed for a particular, “vanilla” version of the vector-field
model, detailed and definitive constraints have been placed on the various
coupling constants Zuntz et al. (2008); Jacobson (2007).

In Zlosnik et al. (2008), it was found that one of the key effects that
vectors would have would be to modify the growth rate of structure. This is
not surprising- theories that modify gravity tend to have this effect. We also
found that it lead to a mismatch between the two gravitational potentials a
potentially observable effect Zhang et al. (2007). In this paper we wish to
pursue this analysis and quantify how strong these effects are. Although we
focus on a particular (albeit broad) class of theories, we are interested in
extracting general lessons from these models. We believe that much of what we
learn by looking at these models will shed light on other models of modified
gravity (such as, for example, $f(R)$ theories Bean et al. (2007) and bimetric
theories Bekenstein (2004)).

The structure of this paper is as follows. In Section II we lay out the
essential ingredients for a reasonably broad class of vector-like models and
its background evolution, and specialize to the form used in the remainder of
the paper. In section III.1 we lay out the equations of the perturbed theory,
and how we implement them with theoretical constraints. In section IV we
discuss the problems with modelling dark matter with the theory. In section V
we find and constrain the parameters which let the theory act as dark energy.
In section VI we conclude and draw more general lessons about the dark sector.

##  II The Theory

###  II.1 Theory Definition

A general action for a vector field $A^{a}$ coupled to gravity can be written
in the form:

$\displaystyle S=\int d^{4}x\sqrt{-g}\left[\frac{R}{16\pi G}+{\cal
L}(g^{ab},A^{b})\right]+S_{M}$ (1)

where $g_{ab}$ is the metric, $R$ the Ricci scalar of that metric, $S_{M}$ the
matter action and $\cal{L}$ is constructed to be generally covariant and
local. By construction $S_{M}$ only couples to the metric, $g_{ab}$ and not to
$A^{a}$.

We will restrict ourselves to consider a Lagrangian that only depends on
covariant derivatives of $A$ and we will consider a unit time-like $A^{a}$.
Such a theory can be written in the form:

$\displaystyle{\cal L}(g^{ab},A^{a})$ $\displaystyle=$
$\displaystyle\frac{M^{2}}{16\pi G}F(K)+\frac{1}{16\pi
G}\lambda(A^{a}A_{a}+1)$ (2) $\displaystyle K$ $\displaystyle=$ $\displaystyle
M^{-2}K^{ab}_{\phantom{ab}mn}\nabla_{a}A^{m}\nabla_{b}A^{n}$ (3)
$\displaystyle K^{ab}_{\phantom{ab}mn}$ $\displaystyle=$ $\displaystyle
c_{1}g^{ab}g_{mn}+c_{2}\delta^{a}_{\phantom{a}m}\delta^{b}_{\phantom{b}n}+c_{3}\delta^{a}_{\phantom{a}n}\delta^{b}_{\phantom{b}m}$
(4)

where $c_{i}$ are dimensionless constants and $M$ has the dimension of mass.
We have removed an additional $c_{4}$ ‘acceleration’ term in accordance with
the transformation described in Eling and Jacobson (2006).

As was the case with TeVeS, the constant $G$ may be a different number from
the locally measured value of Newton’s gravitational constant. $\lambda$ is a
non-dynamical Lagrange-multiplier field with dimensions of mass-squared.

The gravitational field equations for this theory, obtained by varying the
action with respect to $g^{ab}$ (see Zlosnik et al. (2007)) are

$G_{ab}=\tilde{T}_{ab}+8\pi GT^{\mathrm{matter}}_{ab}$ (5)

where the stress-energy tensor for the vector field is given by

$\displaystyle\tilde{T}_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\nabla_{c}(F_{K}(J_{(a}^{\phantom{a}c}A_{b)}-J^{c}_{\phantom{c}(a}A_{b)}-J_{(ab)}A^{c}))$
(6) $\displaystyle-F_{K}Y_{(ab)}+\frac{1}{2}g_{ab}M^{2}F+\lambda A_{a}A_{b}$
$\displaystyle F_{K}$ $\displaystyle\equiv$ $\displaystyle\frac{dF}{dK}$ (7)
$\displaystyle J^{a}_{\phantom{a}c}$ $\displaystyle=$
$\displaystyle(K^{ab}_{\phantom{ab}cd}+K^{ba}_{\phantom{ba}dc})\nabla_{b}A^{d}$
(8)

Brackets around indices denote symmetrization222we adopt the convention
$X_{(ab)}=\frac{1}{2}(X_{ab}+X_{ba})$, $X_{[ab)}=\frac{1}{2}(X_{ab}-X_{ba})$
and $Y_{ab}$ is the functional derivative

$\displaystyle
Y_{ab}=\nabla_{c}A^{e}\nabla_{d}A^{f}\frac{\delta(K^{cd}_{\phantom{cd}ef})}{\delta
g^{ab}}$ (9)

The equations of motion for the vector field, obtained by varying with respect
to $A^{b}$ are

$\displaystyle\nabla_{a}(F_{K}J^{a}_{\phantom{a}b})+F_{K}y_{b}$
$\displaystyle=$ $\displaystyle 2\lambda A_{b}$ (10)

where once again we define the functional derivative

$\displaystyle
y_{b}=\nabla_{c}A^{e}\nabla_{d}A^{f}\frac{\delta(K^{cd}_{\phantom{cd}ef})}{\delta
A^{b}}$ (11)

Finally, variations of the action with respect to $\lambda$ will fix
$A^{b}A_{b}=-1$. By inspection, contracting both sides of (10) with $A^{b}$
leads to a solution for $\lambda$ in terms of the the vector field and its
covariant derivatives.

These equations allow us to study a general theory of the form presented in
equation 1 with a unit time-like vector field. For our particular, restricted
choice of $K$ we have
$Y_{ab}=-c_{1}\left[(\nabla_{c}A_{a}(\nabla^{c}A_{b})-(\nabla_{a}A_{c})(\nabla_{b}A^{c})\right]$
and $y_{b}=0$.

###  II.2 Background Cosmology

In this paper we will restrict ourselves to background cosmologies where the
spacetime is of the spatially flat Friedmann-Robertson-Walker (henceforth FRW)
form:

$g_{ab}dx^{a}dx^{b}=-{dt}^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}.$ (12)

The energy momentum content of each matter fluid $(i)$ in the background is
taken to be of the usual form

$\displaystyle T^{i}_{ab}=(P^{i}+\rho^{i})u^{i}_{a}u^{i}_{b}+P^{i}g_{ab}.$
(13)

Where $\rho^{i}$ and $P^{i}$ are the co-moving density and pressure of the
$i$th fluid respectively. We assume that all fluids have identical four
velocities $u^{a}=(1,0,0,0)$. Furthermore we define
$\rho\equiv\sum_{i}\rho^{(i)}$.

In spacetimes with FRW symmetries, the vector field must align with the
direction $\partial_{t}$ and so the vector field is entirely fixed to have
components $(1,0,0,0)$ in the co-ordinate system (12). Explicitly, the
background value of the scalar $K$ is given by:

$K_{\mathrm{FRW}}=3\frac{\alpha H^{2}}{M^{2}}$ (14)

Where $H\equiv\partial_{t}\ln(a(t))$ and:

$\alpha\equiv c_{1}+3c_{2}+c_{3}.$ (15)

The Friedmann equation then takes the form Zlosnik et al. (2008):

$\displaystyle\left[1-\alpha
K^{1/2}\frac{d}{dK}\left(\frac{F}{K^{1/2}}\right)\right]H^{2}$
$\displaystyle=$ $\displaystyle\frac{8\pi G}{3}\rho$ (16)

where $\rho$ still includes only the matter components.

We note that the combination of (16) and the aether stress energy tensor
allows us to write down an effective energy density $\rho_{\mathrm{ae}}$ and
pressure $P_{\mathrm{ae}}$ of the aether. From this we may define the
fractional energy density $\Omega_{\mathrm{ae}}\equiv 8\pi
G\rho_{\mathrm{ae}}/3H_{0}^{2}$, and the aether equation of state parameter
$w_{\mathrm{ae}}\equiv P_{\mathrm{ae}}/\rho_{\mathrm{ae}}$:

$\displaystyle\Omega_{\mathrm{ae}}$ $\displaystyle=$
$\displaystyle\frac{M^{2}}{6}\left[\frac{\partial}{\partial
H}\left(\frac{F}{H}\right)\right]_{H=H_{0}}$ (17)

$w_{\mathrm{ae}}=-1-\frac{1}{3H^{2}}\frac{\frac{d^{2}}{dtdH}F}{\frac{d}{dH}\left(\frac{F}{H}\right)}$
(18)

The effect of the vector field on the background expansion may be see as an
expansion rate dependent modification to Newton’s constant i.e. schematically
(16) is an equation of the form $3H^{2}=8\pi G(H^{2})\rho$. It was found in
Zlosnik et al. (2008) that various forms of the function $F$ permitted a wide
variety of cosmological dynamics: the presence of the vector field variously
leading to accelerated expansion, slowed expansion, rescaling of $G$, and
recollapse as summarized in Figure 1

Figure 1: A schematic representation of the types of the late-time background
evolution permitted by the functional form $F=\gamma(-K)^{n}$ as a function of
$(n,\gamma)$ for $n<1$.

###  II.3 The functional form of $F(K)$ in cosmology

We must now specify the form of the function $F(K)$ in equation (6). There is
an obvious set of candidates here - we could attempt to be directly consistent
with MOND and use the same branch of $F$ as it uses on small scales. In
appendix A we show that doing so would make it impossible to consistently
generate late-time acceleration behavior in the background cosmology. Instead,
we will use a simple and reasonable ansatz that works for the regime $|K|\gg
1$ that we consider here.

Existing functional forms for F(K) in the MOND regime typically are dominated
for a single monomial term for $|K|\gg 1$ (see for instance Famaey and Binney
(2005)) and so it seems reasonable to restrict the function to take this form:

$\displaystyle F=\gamma(-K)^{n_{\mathrm{ae}}}\ \ $ $\displaystyle K$
$\displaystyle<0$ $\displaystyle F=\gamma(+K)^{n_{\mathrm{ae}}}\ \ $
$\displaystyle K$ $\displaystyle>0$ (19)

where ${n_{\mathrm{ae}}}\leq 1$. This form has sufficient power to express a
wide variety of behavior, and the parameters $\gamma$ and $n$ shall be central
to our further analysis.

##  III Linear perturbation Theory

###  III.1 Formalism & Theory

We have seen that the vector field can have a significant effect on the
quasistatic, weak field limit and the background cosmological geometry.
Significant evidence for non-baryonic mass persists on the largest
cosmological scales Dodelson and Liguori (2006) therefore it is vital that a
relativistic theory of MOND can account for this. As mentioned, it has been
argued Dai et al. (2008) that even in the quasistatic, weak field limit, a
spatial tilt to the vector field may produce significant deviations under some
circumstances from the local MOND force law. Similarly it was shown Zlosnik et
al. (2008) that in the context of linear cosmological perturbations the energy
density associated with the projection of the vector field onto surfaces of
constant conformal time could, to a degree, act as a cosmological ‘dark
matter’ . In this paper we shall comprehensively address the question of
whether the Lagrangian (2) represents a viable model of the dark sector in
light of precision cosmology.

In the main body of the paper we will consider scalar perturbations. In
Appendix B we derive the equations of motion for the vector field’s two
divergenceless vector modes as well as the gravitational wave tensor modes
(the speed of propagation of which is in general modified by the vector
field). Requiring the stability of these modes puts constraints on (2) (see
Section III.2).

We shall work in the synchronous gauge (see for instance Ma and Bertschinger
(1995)) and so the metric takes the following form:

$g_{\mu\nu}dx^{\mu}dx^{\nu}=-a^{2}d\tau^{2}+a^{2}[\gamma_{ij}+h_{ij}]dx^{i}dx^{j}$
(20)

where $\tau$ is conformal time, $\gamma_{ij}$ is a spatially flat spacelike
3-metric perturbed by $h_{ij}$ which is built from two scalar potentials
$\eta$ and $h$:

$h_{ij}(\textbf{x},\tau)=\int
d^{3}ke^{i\textbf{k}\cdot\textbf{x}}[\hat{\textbf{k}}_{i}\hat{\textbf{k}}_{j}h(\textbf{k},\tau)+(\hat{\textbf{k}}_{i}\hat{\textbf{k}}_{j}-\frac{1}{3}\delta_{ij})6\eta(\textbf{k},\tau)]$
(21)

Similarly we will expand the aether field as:

$A^{\mu}=\frac{1}{a}(1,\partial_{i}V)$ (22)

The zeroth component of the aether field is, by virtue of the gauge choice and
the constraint, fixed as equal to $1$ up to second order in perturbations.

A slight complication in the field equations arises because of the presence of
the function $F$ which depends nonlinearly on the scalar $K$. We assume that
for modes of interest one may consistently regard the perturbation to $K$ as
being much less than unity. Thus one can expand $K$ as $K=K^{0}+K^{\epsilon}$
and $F$ as $F=F^{0}+F_{K}^{0}K^{\epsilon}$ where $K^{\epsilon}\ll 1$ where the
superscript $0$ denotes the quantity corresponding to a function’s background
value. Explicitly we have that:

$\displaystyle K^{0}$ $\displaystyle=$ $\displaystyle 3\frac{\alpha{\cal
H}^{2}}{a^{2}M^{2}}$ (23) $\displaystyle{\cal H}K^{\epsilon}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}K^{0}(k^{2}V-\frac{h^{\prime}}{2})$ (24)

Where primes denote derivatives with respect to conformal time and we have
used the conformal Hubble parameter ${\cal H}\equiv H/a$. We also make use of
the following identity:

$\displaystyle(K^{0})^{\prime}{\cal H}=-4K^{0}({\cal
H}^{2}-\frac{1}{2}\frac{a^{\prime\prime}}{a})$ (25)

Henceforth we will drop the superscripts on $K^{0}$ and $F^{0}$ i.e. $K$ and
$F$ shall be assumed to represent _background_ values of the fields.

Towards simplifying the form of the equations, we will will rather use the
field $\xi$ instead of $V$, where $\xi$ is defined as:

$\xi\equiv V-\frac{1}{2k^{2}}(h+6\eta)^{\prime}$ (26)

For further compactness of notation we define the variables:

$\displaystyle\hat{\alpha}$ $\displaystyle\equiv$
$\displaystyle\left(1+2\frac{F_{KK}}{F_{K}}K\right)\alpha$ (27)
$\displaystyle\hat{c}_{1}$ $\displaystyle\equiv$
$\displaystyle\left(1+2\frac{F_{KK}}{F_{K}}K\right)c_{1}$ (28)

The vector field equation of motion (10) becomes :

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
c_{1}(1+c_{13}F_{K})\frac{(F_{K}\xi^{\prime})^{\prime}}{F_{K}}+2{\cal
H}c_{1}(1+c_{13}F_{K})\frac{(F_{K}\xi)^{\prime}}{F_{K}}$ (29)
$\displaystyle+[2c_{1}(1+c_{13}F_{K})(\frac{a^{\prime\prime}}{a}-{\cal
H}^{2})+2(\hat{c}_{1}+\hat{\alpha})({\cal
H}^{2}-\frac{1}{2}\frac{a^{\prime\prime}}{a})$
$\displaystyle+c_{1}c_{13}(F_{KK}K^{\prime})^{\prime}+\frac{1}{3}(\hat{\alpha}+2c_{13})k^{2}]\xi$
$\displaystyle+(c_{1}+\hat{\alpha})\eta^{\prime}+(\hat{c}_{1}+\hat{\alpha})\frac{1}{k^{2}}({\cal
H}^{2}-\frac{1}{2}\frac{a^{\prime\prime}}{a})(h^{\prime}+6\eta^{\prime})$
$\displaystyle-\frac{3}{2}\frac{c_{1}}{k^{2}}\frac{(F_{K}\Sigma_{f})^{\prime}}{F_{K}}$

The relevant Einstein equations are:

$\displaystyle(1-\frac{1}{2}\hat{\alpha}F_{K})k^{2}\eta^{\prime}$
$\displaystyle=$ $\displaystyle 4\pi Ga^{2}ik^{j}\delta T^{0}_{\phantom{0}j}$
(30) $\displaystyle+\frac{1}{6}k^{4}(\hat{\alpha}+2c_{13})F_{K}\xi$

and

$\displaystyle(1+\frac{1}{2}c_{1}F_{K})({\cal H}h^{\prime}-2k^{2}\eta)$
$\displaystyle=$ $\displaystyle-8\pi Ga^{2}\delta T^{0}_{\phantom{0}0}$ (31)
$\displaystyle-\frac{1}{2}F_{K}(c_{1}+\hat{\alpha})6{\cal H}\eta^{\prime}$
$\displaystyle-2\alpha F_{KK}K{\cal H}k^{2}\xi$
$\displaystyle+\frac{F_{K}c_{1}k^{2}}{a^{2}}\left(a^{2}(1+c_{13}F_{K})\xi\right)^{\prime}$
$\displaystyle-\frac{3}{2}c_{1}F_{K}\Sigma_{f}$

Where we have used the fact that:

$\displaystyle(h+6\eta)^{\prime\prime}+2{\cal H}(h+6\eta)^{\prime}-2k^{2}\eta$
$\displaystyle=$ $\displaystyle-3\Sigma_{f}$ (32)
$\displaystyle+2c_{13}k^{2}[F_{K}(2{\cal H}\xi+\xi^{\prime})$
$\displaystyle+F_{KK}K^{\prime}\xi]$

The functions $\delta T^{0}_{\phantom{0}j}$ and $\delta T^{0}_{\phantom{0}0}$
are the first order perturbations to the corresponding components of the
matter fields’ stress energy tensors. Summation over field species is assumed.
The field $\Sigma_{f}$ is the scalar component of the total fluid shear i.e.
$\Sigma_{f}=-8\pi
Ga^{2}(\hat{k}_{i}\hat{k}^{j}-\frac{1}{3}\delta^{j}_{\phantom{j}i})\Sigma^{i}_{\phantom{i}j}$
and $\Sigma^{i}_{\phantom{i}j}\equiv\delta
T^{i}_{\phantom{i}j}-\frac{1}{3}\delta^{i}_{\phantom{i}j}\delta
T^{k}_{\phantom{k}k}$. A gauge invariant formulation of the theory’s equations
may be found in Li et al. (2008a).

###  III.2 Parameter Constraints

We can immediately see a number of constraints on the $c_{i}$ and the form of
the function $F$. From (29) it can be shown that for in the limit of
timescales shorter than a Hubble time the quantity:

$C_{S}^{2}=\frac{2}{3}\frac{(\frac{\hat{\alpha}}{2}+c_{13})}{c_{1}(1+c_{13}F_{K})}$
(33)

can be interpreted as the squared sound speed of the field $\xi$. The
avoidance of exponentially growing subhorizon modes dictates that $C_{S}^{2}$
should be positive definite.

Similarly, one may consider the field equations of the two divergenceless
‘vector’ modes of the vector field and the two transverse traceless ‘tensor’
modes of of the metric. Each respectively has a squared sound speed function
(named $C_{V}^{2}$ and $C_{T}^{2}$ respectively) which, as in the scalar case,
should be positive definite. These functions are calculated in Appendix B and
are as follows:

$C_{T}^{2}=\frac{1}{1+c_{13}F_{K}}$ (34)

$C_{V}^{2}=\frac{F_{K}}{2c_{1}}\frac{2c_{1}+F_{K}(c_{1}^{2}-c_{3}^{2})}{1+c_{13}F_{K}}$
(35)

Note that the gravitational wave tensor modes now generically have a time
dependent speed of propagation.

Collectively the three positivity constraints imply the following constraints
on the $c_{i}$ and function $F$ parameter space:

$\displaystyle 1+F_{K}c_{13}$ $\displaystyle>$ $\displaystyle 0$ (36)
$\displaystyle(\hat{\alpha}+2c_{13})/2c_{1}$ $\displaystyle>$ $\displaystyle
0$ (37) $\displaystyle F_{K}(1+F_{K}\frac{c_{1}^{2}-c_{3}^{2}}{2c_{1}})$
$\displaystyle>$ $\displaystyle 0$ (38)

Now we turn to the Einstein equations (30) and (31). Terms in the perturbed
vector field stress energy tensor $\delta\tilde{T}_{ab}$ may contain terms
proportional (up to a time-dependent function of the background fields) to
$\delta G_{ab}$ and indeed this can be seen in (30) and (31) through the
appearance of terms proportional to the $c_{i}$ on the left hand sides of the
equations. If the effect $\xi$ is negligible then the effect of the aether
stress energy terms proportional to components of the Einstein tensor may
always be absorbed into a redefinition of Newton’s constant as a time-
dependent effective gravitational coupling. Thus, even if $\xi$ has
comparatively little effect, there may be a considerable modification to the
link between the matter fields and the gravitational field. The resulting
gravitational couplings should be greater than or equal to zero otherwise the
gravitational field will interpret normal matter as violating energy
conditions, and so risking the appearance of instabilities. This restriction
implies the following constraints:

$\displaystyle(1-\frac{1}{2}\hat{\alpha}F_{K})>0$ (39)

$\displaystyle(1+\frac{1}{2}c_{1}F_{K})>0$ (40)

Throughout our analysis, we will only consider regions of the model’s
parameter space which satisfy these constraints.

###  III.3 Computation

To study the effects of the vector field in detail we have modified the
structure formation Boltzmann code CMBEASY Doran (2005). We add a Newton-
Raphson solver for the Hubble parameter, with added aether components. The
perturbation evolution is also modified to include the aether components $\xi$
and $\xi^{\prime}$, and their contribution to the density, pressure and shear
perturbations. We also include the altered metric perturbations in the
calculation of the CMB source function. We use adiabatic initial
conditionsZlosnik (2009). Since Boltzmann codes are very highly optimized for
$\Lambda$CDM models, care must be taken to ensure that modifications are
performed in a consistent manner - for example an unmodified Friedmann
equation is often assumed for computational efficiency.

To explore parameter spaces, CMBEASY is coupled to a Monte-Carlo Markov Chain
(MCMC) engine Doran and Mueller (2004). We extended this engine to include our
new Aether parameters: $\Omega_{\mathrm{ae}}$, $c_{+}=(c_{1}+c_{3})$,
$c_{-}=(c_{1}-c_{3})$, $C_{S}^{2}$, $n_{\mathrm{ae}}$ and $M$. We include the
full ranges of these parameters by allowing the kinetic term to take two
branches, for positive and negative $K$ as in (19).

We constrain models using both CMB data from the WMAP experiment Komatsu et
al. (2009) and large-scale structure from the SDSS survey Adelman-McCarthy et
al. (2008), though not always at the same time. As we shall see, some Aether
models are extremely poor fits to the combined data sets; to illustrate such
problems we want to find models that fit only the large scale structure data.
In regimes where fits are extremely poor, MCMC does not work particularly
well. We ameliorate such situations by running larger numbers of shorter
Markov chains and re-starting from their best-fit positions, and sometimes by
abandoning MCMC altogether and simply performing random searches for good
parameter combinations.

##  IV The vector field as Dark Matter

###  IV.1 Background evolution & doppler peak positions

An unusual property of the model considered here is that at the level of
cosmological perturbations the field can mimic a perturbed pressureless fluid
in the formation of large scale structure whilst behaving entirely differently
in the cosmological background Zlosnik et al. (2008) . In this paper we would
like to consider not just large scale structure but also other cosmological
probes. The anisotropies in the CMB temperature are sensitive to the
background dynamics and perturbed dynamics of source of cosmic mass
discrepancies in a largely distinguishable manner.

If the aether plays the role of dark matter only in the perturbations, then
the background expansion is dark energy-dominated at an earlier time. For a
given $H_{0}$ this reduces the expansion rate of the universe between
recombination and now and so decreases the angle subtended on the sky by given
distance at last scattering. This moves the CMB doppler peaks to higher
$\ell$. Although the aether can give a suitable time dependence to the
effective gravitational coupling $G$ in the Friedmann equations so as to yield
the same expansion rate as $\Lambda$CDM, the necessary functional forms of $F$
are extremely contrived. For instance these forms essentially contain a new
constant scale roughly equal to the Hubble parameter at matter-radiation
equality. Even in the event of such a construction, it may not be possible for
the squared sound speed of vector field perturbations $C_{S}^{2}$ to remain
sufficiently small as to behave like cold dark matter in perturbations Zlosnik
et al. (2007).

The other input to the peak position is the sound horizon at last scattering.
The physics of this is sufficiently robust that changes compensatory to the
alteration in the distance to last scattering are not feasible without
exceptional fine tuning of the perturbational behavior of the aether. Indeed,
it has been argued that only additional components that behave like non-
relativistic matter in the background might fix this problem Ferreira et al.
(2008).

Could there indeed be such a non-relativistic matter in the background which
allows acoustic peak positions which are consistent with the data whilst
leaving the aether in its role as the seed of the formation of large scale
structure? A appropriate candidate would appear to be massive neutrinos. A
suitable mass of such particles so as to account for the right effective
contribution to the dust component of the background typically implies that
the neutrinos themselves are unable to clump on small enough scales so as to
be a good candidate for all of the dark matter. Such a solution as has been
proposed in Skordis et al. (2006) and Angus (2008).

###  IV.2 Perturbation evolution

We now argue that even with the inclusion of massive neutrinos, one
generically expects the aether to have an unacceptable influence on the large
scale CMB anisotropy if it is to also play the dominant role in structure
formation.

The first requirement for successful perturbation evolution is that structure
can form at all. One necessary condition for this is that the sound speed of
the structure seed not be too large, since this would wash out structure. We
require that the sound horizon in the model be less than the smallest scales
where structure can form linearly: $C_{S}k_{\mathrm{max}}\tau\lesssim 1$,
where $k_{\mathrm{max}}\sim 0.2h/M\mathrm{pc}$. For matter power observations
at $\tau\sim 3\times 10^{4}$, the present epoch, this yields $C_{S}\lesssim
10^{-4}$.

There are two underlying physical processes that further constrain the models.

The first is a change to the growth rate of perturbation amplitude. This can
cause discrepancies between the amplitudes we expect in the matter power
spectrum and the CMB, since the evolution between the two is different. It can
also lead to an integrated Sachs-Wolfe (ISW - see below) effect during the
matter era since $\Phi$ will accrue a time dependence.

The second is the increased presence of a $\Phi-\Psi$ metric shear. This also
leads (directly) to a matter era ISW.

### Observable 1: ISW

Under the assumption of adiabaticity we have that the anisotropy in the CMB,
$\Delta T({\hat{n}})/T$ on large scales in a given spatial direction
${\hat{n}}$ is given by

$\frac{\Delta
T({\hat{n}})}{T}\simeq-\frac{1}{3}\Psi(\tau_{*},d_{*}{\hat{n}})-\int_{\tau_{*}}^{\tau_{0}}d\tilde{\tau}({\Psi^{\prime}}+{\Phi^{\prime}})[\tilde{\tau},(\tau_{0}-\tilde{\tau}){\hat{n}}]$
(41)

where $\tau_{*}$ is the conformal time of last scattering, $\tau_{0}$ is the
conformal time today, $d_{*}$ is the comoving radius of the surface of last
scattering, and $\Phi$ and $\Psi$ are the conformal Newtonian gauge
gravitational potentials. The integral in (41) is the integrated Sachs-Wolfe
(ISW) effect. Writing the integrand of (41) as
$\Psi^{\prime}+\Phi^{\prime}=-(\Phi-\Psi)^{\prime}+2\Phi^{\prime}$, we can see
this as time derivatives of a shear part and a growth rate part.

In the standard cosmological model, the field $\Phi$ has negligible time
dependence during matter domination. It gains a time dependence only when the
background starts accelerating, and only then can the resulting growth rate
ISW contribution be considerable. In the model considered here, the situation
may be rather different. It was found in Zlosnik et al. (2008) that
substantial contributions to the ISW may occur even during the matter era. For
the aether field to seed structure formation the field $\xi$ must have a
suitable growing mode solution in the matter era. Typically the corresponding
spatial curvature perturbation $k^{2}\Phi$ will then have a time dependence
via the Poisson equation.

The shear part can also gain a time dependence in the aether model, which in
the $\Lambda$CDM is very small even during acceleration.

Each of these effects depend on the functional form of $F$, the time-
dependence of the $\xi$ growing mode and the choice of the parameters $c_{i}$.
It is extremely challenging to find combinations of the parameters which allow
for a realistic growth of structure whilst making the ISW acceptably low. This
is most easily illustrated by considering the theory TeVeS Bekenstein (2004)
which has many of the same properties as the model considered here. It may be
shown that TeVeS can be written as a single metric theory with a timelike
vector field of unfixed norm Zlosnik et al. (2006),Skordis (2009). As in the
model considered here, the longitudinal component of the vector field can
source the growth of structure Skordis et al. (2006), Dodelson and Liguori
(2006). We will call this field $V_{T}$. We consider a matter dominated era
where $V_{T}$ is responsible for the dominant source in the Poisson and shear
equations. These equations respectively then areSkordis (2006):

$\displaystyle k^{2}\Phi$ $\displaystyle\approx$ $\displaystyle-f_{s}{\cal
H}k^{2}V_{T}-\frac{K_{B}}{2}k^{2}V_{T}^{\prime}$ (42) $\displaystyle
k^{2}(\Psi-\Phi)$ $\displaystyle\approx$ $\displaystyle f_{s}k^{2}(2{\cal
H}V_{T}-V_{T}^{\prime})$ (43) $\displaystyle f_{s}(\tau)$
$\displaystyle\equiv$ $\displaystyle\frac{(1-\bar{A}^{4})}{\bar{A}^{4}}$ (44)

where $K_{B}$ is a positive constant of the action and $\bar{A}^{4}$ is the
norm of the vector squared again (equal to unity in the fixed norm case, but
in TeVeS the deviation of $|\bar{A}^{2}|$ from unity is essentially the
background variation of the ‘scalar field’ degree of freedom). Therefore:

$\displaystyle\Psi+\Phi=-(f_{s}(\tau)+K_{B})V^{\prime}_{T}$ (45)

In this era the vector field equation is:

$V_{T}^{\prime\prime}+b_{1}\frac{V_{T}^{\prime}}{\tau}+b_{2}\frac{V_{T}}{\tau^{2}}=S[\Phi,\Psi]$
(46)

where

$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle 2(3-\bar{A}^{4})$ (47)
$\displaystyle b_{2}$ $\displaystyle=$ $\displaystyle
2(2-\bar{A}^{4})+\frac{12\bar{A}^{4}}{K_{B}}(1-\bar{A}^{4})$ (48)

and $S[\Phi,\Psi]$ is a source term.

For this situation to arise, there must be a growing mode in $V_{T}$ Dodelson
and Liguori (2006). Therefore we require that $b_{2}<0$. The function
$\bar{A}^{4}$ will be rather close to unity Bourliot et al. (2007) so

$b_{2}\sim 2+12f_{s}(\tau)/K_{B}$ (49)

Therefore we require that $f_{s}(\tau)/K_{B}\lesssim-1/6$ (recalling that
$K_{B}$ is a positive number). By (45) we see that the contribution to the ISW
will be proportional to $f_{s}(\tau)+K_{B}$. Meanwhile we see from (42) that
the comparative contribution to Poisson’s equation is also independently
weighted by $f_{s}$ and $K_{B}$. If there is no time dependence in the Poisson
equation (42) due to the vector field then the vector field will make no
contribution to the ISW effect. If there is a time dependence in the Poisson
equation, there may yet be no contribution to the ISW if $f_{s}(\tau)\sim-
K_{B}$ between last scattering and now (though the overdensities of the baryon
field will then not generally grow as $a$, thus contributing to the ISW
effect). This would be consistent with the condition for a growing mode in
$V_{T}$ but it does not guarantee that the resulting growing mode would be
suitable.

Indeed, it was found Skordis et al. (2006) that in seeding the growth of large
scale structure in TeVeS there was a significant danger of incurring
unacceptably high temperature anisotropies in the CMB on large scales.
Although involving a larger number of terms, the same reasoning carries over
to the model considered here - i.e. parameters which allow realistic structure
formation will typically lead to an unacceptable ISW effect. This is vividly
illustrated in Figure 4 which shows best fit models as compared to SDSS large
scale structure data. In every case, the corresponding temperature anisotropy
displays a dramatically poor fit to the data at low $\ell$.

### Observable 2: Amplitudes

The ratio of the observed amplitudes of the CMB anisotropy and the matter
power spectrum is consistent with a growth rate proportional to $a$ (though
see Bean (2009)). Any uncompensated change to this growth rate in the aether
model over this period would lead to a different ratio. The bias parameter
between the galaxy distribution and the underlying density field can be used
to rectify this difference, but only if the change is relatively small and
unphysically large bias parameters (larger than $\sim 10$) are not required.

###  IV.3 Summary

We have seen that although the vector field may play a number of the roles
that dark matter plays, it seemingly cannot do all at once 111It is
interesting to compare this with similar results found in Li et al. (2008b)
for a framework for generalizations of dark matter. It is not clear whether
the model considered in this paper fits within this framework.

The position of the acoustic peaks in the CMB temperature anisotropy should be
taken as a strong indication of additional nonbaryonic nonrelativistic mattter
present in the universe during matter domination. Such an effect can be
achieved in this model by a rescaling of the value of Newton’s constant.
However, this rescaling cannot persist into the radiation era Carroll and Lim
(2004). Thus, the functional form would have to be approaching the rescaling
solution only after one would expect matter (including cold dark
matter)-radiation equality to happen.

This implies the presence of a new scale in the theory, roughly corresponding
to the Hubble parameter $H_{eq}$ at this time. It seems fair to say in general
that a model such as that considered here, is more likely to be a
cosmologically viable candidate for dark matter if the scale $M$ in the theory
is closer to $H_{eq}$ and not $H_{0}$. It is tempting to speculate whether a
theory where the scale $M$ itself is dynamical may find more success, but that
will not be explored in this work.

Furthermore we have seen that even if the background is consistent with
observations, the effect on the evolution of perturbations may be
unacceptable, notably either through the ISW effect or comparing the
respective amplitudes of the CMB anisotropy and matter power spectrum today.

###  IV.4 Example Problem Spectra

It has previously been shown that the Einstein Aether can produce acceptable
matter power spectra with certain parameter combinations Zlosnik et al.
(2008). Here we show that such combinations do not provide an acceptable fit
to CMB measurements. Despite extensive searches we have been unable to find
any parameter set within the model that does fit the WMAP data well; this is
entirely in line with the problems discussed above.

We use a parameter set which is consistent with BBN limits on
$\Omega_{B}h^{2}$ and the HST key project measurement of $H_{0}$. The standard
cosmological parameters are: $\Omega_{b}h^{2}=0.0193$, $n_{s}=0.83$,
$H_{0}=89.3\,\mathrm{km/s/Mpc}$, $\Omega_{c}h^{2}=0$. The new Aether variables
are $c_{+}=-4.72$, $c_{-}=-6.11$, $n_{\mathrm{ae}}=0.34$,
$\Omega_{\mathrm{ae}}=0.82$, $M_{\mathrm{ae}}=111.3\,\mathrm{km/s/Mpc}$ with
$c_{2}$ set by requiring zero sound speed. This parameter combination is in no
sense optimal, but it does provide an illustration of all the problems that
arise here.

Figure 4 shows power spectra from our modified Boltzmann code for this
parameter set. The matter power is a realistic fit to the SDSS data (this was
the criterion for our choice of parameters). The CMB spectra shows various
problems. In the low-$\ell$ regime a large ISW effect is clearly present,
destroying the fit at large scales, as described in section (IV). The
positions of the peaks are poorly fit by the model, as expected and discussed
in section (IV.1). Finally, in the plot we have rescaled the amplitude of the
matter power spectrum by a factor $0.02$, corresponding to a galaxy bias of
$0.14$ in order to reconcile the relative amplitudes of the two spectra with
the data; such a scaling is unphysically small. This corresponds to the
changed growth rate described in section (IV). All these effects cause severe
problems when attempting to simultaneously fit the CMB and large scale
structure.

Figures 2 and 3 illustrate the sources of the extreme ISW effects shown in
Figure 4; the time derivatives of the metric quantities plotted create an ISW
effect as shown in equation (41). The onset of background acceleration in each
case is marked by a turnover in the curves at late time. The GEA universe
exhibits a dramatically increased $|\Phi-\Psi|$ and time dependence of
$|\Phi|$ during the matter era as compared to the $\Lambda$CDM universe.
Although the $|\Phi-\Psi|$ has a smaller magnitude its time dependence can be
significant for the total ISW effect. Note that values of $\tau$ between the
two universes do not correspond to the same physical time or redshift since
the universes expand at different rates.

Figure 2: Exotic behavior of metric potentials for $k\sim$ $10^{-2}$
$Mpc^{-1}$. The panels show the fields $|\Phi|$ and $|\Phi-\Psi|$ for a
$\Lambda$CDM universe (green solid line) and GEA universe (dashed blue line)
as a function of $k(\tau-\tau_{r})$ where $\tau_{r}$ is the conformal time of
recombination.

Figure 3: Equivalent of Figure 2 for k $\sim$ $10^{-4}$ $Mpc^{-1}$

.

Figure 4: Matter power (top) and CMB (bottom) power spectra for the $\Lambda
CDM$ (dashed green) and typical GEA (solid red) models, with WMAP and SDSS
constraints.

##  V The vector field as Dark Energy

###  V.1 Dark Energy Regime

As shown in figure 1, our vector field can produce late-time acceleration and
so play the role of dark energy. Indeed, for the form of the vector field used
here, as the index $n_{\mathrm{ae}}\rightarrow 0$ the theory becomes the same
as a cosmological constant for both the background and the perturbations.
Since the model can fit the data well we can use our MCMC engine to find
constraints on the parameters of the vector field, telling us exactly how
close to the $n_{\mathrm{ae}}=0$ cosmological constant case the theory must be
to fit the CMB and LSS data.

If the model can fit the data only extremely close to $n_{\mathrm{ae}}=0$ then
it does not provide a compelling alternative to the cosmological constant. If,
on the other hand, there is significant flexibility in the model and no fine
tuning, or if it can provide a better fit than $\Lambda$CDM, then it is
somewhat more interesting.

In this section we will consider the resulting background evolution, CMB
temperature anisotropy, and matter power spectrum for a universe containing
the vector field, cold dark matter, the conventional matter fields, and no
cosmological constant. The acceleration will arise solely from the vector
field’s modification to the Friedmann equation.

###  V.2 Constraints on aether dark energy from data

Our MCMC generated the constraints on the vector field parameters shown in
figures 5 to 8; these curves are the smoothed histograms from our combined
Markov chains.

Figure 5: Constraints on the three coupling terms of the theory.  Figure 6:
Constraints on the kinetic term power law index parameter.  Figure 7:
Constraints on the vector field sound speed parameter.  Figure 8: Constraints
on the parameter $\alpha=c_{1}+3c_{2}+c_{3}$

The most important trend evident in these results is closeness of
$n_{\mathrm{ae}}$ to the $\Lambda$ value of zero. We find the best fit value
$n_{\mathrm{ae}}=4.2\cdot 10^{-3}$, with a 95% upper limit
$n_{\mathrm{ae}}<0.126$.

The best fit value in the MCMC run is very close to the $\Lambda$CDM
likelihood of the same data, at the cost of six extra parameters, meaning that
it is unlikely to be favored by any model comparison exercise. It does,
however, demonstrate the validity of modified gravity-related dark energy
candidates.

Having obtained these constraints we can determine their origin. There are two
ways in which the vector field must behave like $\Lambda$ to provide a good
fit. The first is that the late-time acceleration should be close to that
given by $\Lambda$. The second is that any perturbations in the field (which
are not present in $\Lambda$) should not affect the observable spectra.

###  V.3 Constraint origins - acceleration rates

The consistency of the acceleration of the universe with the cosmological
constant equation of state $w=-1$ is being measured with increasing precision
in supernova and baryon acoustic oscillation experiments (which are beyond the
scope of this paper). Here, they will be constrained by the late-time ISW
effect induced by dark energy, and by the perturbation growth rate.

We can assess how closely vector-induced acceleration mimics $\Lambda$-driven
expansion at late times with the equation of state $w_{\mathrm{ae}}$ of the
vector field in the background:

$w_{\mathrm{ae}}=-1-\frac{1}{3H^{2}}\frac{\frac{d^{2}}{dtdH}F}{\frac{d}{dH}\left(\frac{F}{H}\right)}.$
(50)

For the monomial form of $F(K)$ we have that:

$\displaystyle w_{\mathrm{ae}}=-1-\frac{2n}{3(2n-1)}\frac{\dot{H}}{H^{2}}$
(51)

Thus the equation of state will generically deviate from $-1$ whenever $n\neq
0$ and so the acceleration for these values will not be degenerate with a
cosmological constant. We see immediately from equation (51) that
$w_{\mathrm{ae}}(\tau)<-1$ for $0<n<1/2$ and $w_{\mathrm{ae}}(\tau)>-1$ for
$n>1/2$. This is clearly visible in Figure 9.

Figure 9: The vector field’s equation of state as a function of redshift $z$,
for various values of the kinetic term index $n_{\mathrm{ae}}.$

###  V.4 Constraint origins - perturbation evolution

Even if the background expansion is rather close to the the $\Lambda$CDM
model, the evolution of perturbations need not be. This is most easily
illustrated by considering the Poisson equation on large scales (see Zlosnik
et al. (2007) for a derivation). On these large scales there is a time-
dependent re-scaling of the the metric perturbation $\Phi$, which we can cast
as a modification of the effective gravitational constant $G$:

$\displaystyle k^{2}\Phi$ $\displaystyle=$ $\displaystyle-4\pi
G^{(1)}_{\mathrm{eff}}a^{2}\sum_{i}\bar{\rho}_{i}\delta_{i}$ (52)
$\displaystyle G^{(1)}_{\mathrm{eff}}$ $\displaystyle\equiv$
$\displaystyle\frac{G}{1+\frac{c_{1}}{2}F_{K}}$ (53)

where we have assumed that terms proportional to the velocity divergence are
ignorable and provisionally considered the effect of the field $\xi$ to be
subdominant.

The Friedmann equation may be used to cast the above equation in a more
familiar form by eliminating he background $\rho_{i}$ in favour of background
expansion rate of the universe and the time-dependent fractional energy
density $\Omega_{i}(\tau)\equiv 8\pi G\rho(\tau)/(3H(\tau)^{2})$. This yields:

$\displaystyle k^{2}\Phi$ $\displaystyle=$ $\displaystyle-\frac{3}{2}{\cal
H}^{2}\frac{G^{(1)}_{\mathrm{eff}}}{G^{(0)}_{\mathrm{eff}}}\sum_{i}\Omega_{i}(\tau)\delta_{i}$
(54)

where

$\displaystyle G^{(0)}_{\mathrm{eff}}$ $\displaystyle\equiv$
$\displaystyle\frac{G}{1-\alpha
K^{\frac{1}{2}}\frac{d}{dK}\left(\frac{F}{K^{\frac{1}{2}}}\right)}$ (55)

The $n=0$ $\Lambda$CDM Poisson equation may be cast in this form by taking
$G^{(1)}_{\mathrm{eff}}=G$ and
$G^{(0)}_{\mathrm{eff}}=G/(1-\Lambda/(3H^{2}))$. For the case where $n$
differs from $0$, the function $G^{(1)}_{\mathrm{eff}}$ will generically
possess a time dependence during the background evolution. Therefore the link
between the time evolution of the functions $G^{(0)}_{\mathrm{eff}}$,
$\delta_{i}$ and $\Phi$ will differ from the case where acceleration is caused
by a cosmological constant. We may thus expect the ISW effect to be of a non-
standard form. This is vividly illustrated in Figure 10 where it can be seen
that for a given set of $(c_{i},\gamma,M)$, variation of $n$ results in a
considerable variation in the large scale CMB temperature anisotropy. Also
evident is a variation in the matter power spectrum amplitude with $n$,
evident on all scales.

Variation of parameters other than $n$ could also have a significant impact on
the success of the models. Given the results of the previous section, it seems
unlikely that any influence of the field $\xi$ would tend to improve the
models. This indeed seems to be the case. Figure 11 depicts various models
where the function is varied $C_{S}^{2}$ for fixed values of the other
parameters. In particular, $n$ takes the value $0.1$. The function $C_{S}^{2}$
is ultimately a measure of the ability of the field $\xi$ to sustain any
homogeneous growing behavior for $(k\tau)>1$; higher values will tend to limit
the effect of the vector field to larger and larger scales. The sets of
parameters were chosen so such a growing solution indeed existed on
superhorizon scales. The figure indicates that a growing $\xi$ field will
indeed have deleterious effects on scales where it is not suppressed. There is
a significant ISW effect evident for the red curve CMB; the corresponding
model must be considered as being at the edge of acceptability. The
corresponding large scale matter power spectrum exhibits exotic oscillations,
entirely unrelated to the baryon-acoustic oscillations which occur on other
scales. Their presence is in this model is thus reflective of dynamics in the
dark energy sector.

Figure 10: Matter power (top) and CMB (bottom) power spectra for the various
GEA dark energy models, with WMAP and SDSS constraints. The power-law
function’s exponent $n$ is varied

.

Figure 11: Matter power (top) and CMB (bottom) power spectra for the various
GEA dark energy models, with WMAP and SDSS constraints. The squared speed of
sound of vector field perturbations is varied.

##  VI Conclusions

###  VI.1 Being Dark Matter is hard

Generalized Einstein-Aether can, with different parameter choices, resemble
dark matter in some important ways but never all of them at once.
Specifically, the new degrees of freedom introduced by the model may conspire
to identically replicate one or more but not all of the following properties
of cold dark matter.

####  VI.1.1 Background dynamics

To accomplish identical background dynamics to cold dark matter, one must
introduce considerable fine tuning into the function $F$ of the theory.
Specifically the function must change form either side of (dark) matter-
radiation equality. The parameter we tune to make this happen is in the action
itself, unlike the usual case where we simply alter the abundance
$\Omega_{c}$. Changes to fit cosmological observations can therefore have a
larger impact on the small scale behavior of gravity.

####  VI.1.2 The speed of sound

If the speed of sound is too high then structure cannot form on small enough
scales. In this model we may reduce the sound speed to be close to zero, at
the cost of one of our parameters. When designing new gravity theories this is
perhaps the easiest structure formation constraint to investigate, and it
should be examined to see if it conflicts with other constraints needed to
make the theory useful - for an example, see Seifert (2007).

####  VI.1.3 Growth rate of ‘overdensity’

Theories of modified gravity designed to replace dark matter must necessarily
have growing modes of fluctuation in at least one of the new degrees of
freedom they introduce, in order to sufficiently source gravitational collapse
and structure formation on scales within their own sound horizon. There
remains some flexibility in the perturbation growth rate, since the bias on
the galaxy power spectrum measurements is a free factor. As measurements of
weak lensing (which samples gravitation directly) and semi-analytic models
(which predict bias) improve this freedom will be reduced.

####  VI.1.4 Absence of anisotropic stress and contribution the cosmological
Poisson equation

A sufficiently small anisotropic stress associated with the vector field may
be implemented by fine tuning the parameter $c_{13}$ to be very small (see
equation (32)). However, as was discussed in Subsection IV, even this will
tend to come at the expense of other desired behavior of the field. An
appreciable time variation of the anisotropic stress over the time from last
scattering to today can result in a very poor fit to the low-$\ell$ CMB
$C_{\ell}$. This problem is likely to be common in theories of modified
gravity.

As we have seen, it is a combination of time variation of the anisotropic
stress and time variation of the field $\Phi$ via the vector field’s effect on
the cosmological Poisson equation that contribute to the ISW effect. Though
both effects are absent in the cold dark matter case, one may imagine both
effects being present in the vector field model but being of equal and
opposite sign. As with the case of the isolated anisotropic shear
contribution, it seems this is not possible whilst maintaining the other
constraints like the existence of a growing mode.

####  VI.1.5 Effective minimal coupling to the gravitational field

Even if the vector field growing mode gives an appropriate (dark matter-like)
contribution to the Poisson equation, the link between the overdensity and the
corresponding $\Phi$ can differ from the CDM case. This difference can come
from curvature terms in the vector field stress energy tensor, and its main
consequence is a time-dependent rescaling of Newton’s constant $G$. Thus
$\Phi$ may gain a time dependence during the matter era even there is a
completely standard dark matter contribution to the Poisson equation. The
converse may also be possible: the time dependence of an incorrect Poisson
contribution could be counteracted by a time dependence of the effective $G$.

###  VI.2 Being Dark Energy is easy

As a model for dark energy the Generalized Einstein-Aether theory is more
successful: we have obtained constraints on its parameters and found that it
generates spectra that fit the data across a reasonable range of its parameter
space. It is clear that modified gravity approaches to explaining late-time
acceleration are viable and can provide motivated explanations for dark energy
(though this theory retains the co-incidence problem in the guise of the
parameter $M$).

####  VI.2.1 Closeness to $\Lambda$

The most interesting constraint on this branch of the theory is on the
parameter $n_{\mathrm{ae}}$ and is shown in figure 6. In some sense this
parameter describes how closely the theory mimics $\Lambda$ (which has
$n_{\mathrm{ae}}=0$). The fact that this parameter is rather free,
$n_{\mathrm{ae}}<0.126$, (95% CL), is consistent with the fact that a wide
variety of other theories can also explain dark energy: present structure
formation data is not very informative about the nature of dark energy, and
deeper require expansion probes like baryon acoustic oscillation and
supernovae.

####  VI.2.2 Sound speed

The other notable constraint on Einstein-Aether dark energy, which may extend
to other modified gravity approaches, is the limits on the sound speed,
illustrated in figure 7. As shown in figure 11, an incorrect sound speed can
lead to large scale oscillations by modifying the other parameters of the
theory and permitting a growing mode excitation at late time.

####  VI.2.3 Other constraints

The other constraints on the theory (which are easily fulfilled by choosing
the $c$ parameters) come from ensuring that no growing mode can disrupt the
power spectra, that the acceleration is close to the $\Lambda$CDM value, and
that the value of effective $G$ remains positive at all times.

###  VI.3 Future issues for modified gravity and structure formation

Because $\Lambda$CDM is such a good fit to current cosmological data, modified
gravity will never be favoured in a model comparison exercise using only
current data about linear structure. It is only in combination with physics on
galactic and smaller scales that it can be be persuasive. This work highlights
a few issues for future model-building in this vein.

The Generalized Einstein Aether model is a member of a class of models in
which the scale $M\sim H_{0}$ associated with Dark Energy is visible to dark
matter. Many of the issues raised here will be relevant to any such models
which try to use a dark matter scale consistent with small-scale modifications
to gravity.

A combination of probes sensitive only to the background (like Type 1A
supernovae) and to the behavior of cosmological perturbations is needed to
fully constrain these theories. For example, a value $n_{\mathrm{ae}}=0.3$ has
$w(z)\sim 1$ at low redshift but is ruled out by our constraints. Similarly
there are models with $n_{\mathrm{ae}}\sim 0.75$ that provide reasonable
spectra, but they are ruled out by $w(z)$ constraints.

There are, of course, a number of extensions to the theory and features of it
that could be change; we could, for example, allow $M$ to vary dynamically, or
add more terms to the kinetic component $F$ in equation (19). There are also
myriad possibilities in more changing more general aspects of the primordial
conditions or cosmological parameters: what happens if we add add tensors? Can
we include an isocurvature mode? Would massive neutrinos help? Or curvature?
This leads us to a key caveat that applies to this and all similar work
constraining new physics with linear structure: a simple constraint from the
data alone is worthless, since any of the numerous other parameters we could
change might conspire to counteract whatever problem it solves. We need a
physical explanation of a constraint’s origin to understand whether it is
robust to the cosmologist’s tinkering.

Acknowledgments: We thank Constantinos Skordis and David Jacobs for useful
discussions. GDS was supported by a grant to the CWRU particle/astrophysics
theory group from the US DOE. JZ is supported by an STFC rolling grant. TGZ is
supported by Perimeter Institute for Theoretical Physics. Research at
Perimeter Institute is supported by the Government of Canada through Industry
Canada and by the Province of Ontario through the Ministry of Research &
Innovation.

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##  Appendix A The MOND regime F(K) cannot yield late-time acceleration

The MOND branch of $F$ applies in the regime $0\leq K\ll 1$; outside this
range MOND does not prescribe its form. Inside that range the MOND value is:

$\displaystyle c_{1}F$ $\displaystyle=$
$\displaystyle-2K+\frac{2}{3}K^{\frac{3}{2}}$ (56) $\displaystyle(0\leq$
$\displaystyle K$ $\displaystyle\ll 1)$

The weak field limit of the Einstein Aether theory is:

$K^{\mathrm{ae}}_{\mathrm{WF}}=-c_{1}\frac{(\nabla\phi)^{2}}{M^{2}},$ (57)

where $\phi$ is the conformal Newtonian potential. The weak field MOND limit
is:

$K^{\mathrm{MOND}}_{\mathrm{WF}}=\frac{(\nabla\phi)^{2}}{a_{0}^{2}}.$ (58)

Clearly we can equate these by setting M${}^{2}=-c_{1}a_{0}^{2}$. Using this
relation with equation (14) gives us this value for the present-day
cosmological $K$:

$K_{\mathrm{FRW}}(t_{0})=-3\frac{\alpha}{c_{1}}\frac{H_{0}^{2}}{a_{0}^{2}}$
(59)

There are now two cases: $\alpha/c_{1}<0$ and $\alpha/c_{1}>0$. In either case
we need the measured values of $H_{0}$ and $a_{0}$; they are suggestively
similar:

$\displaystyle H_{0}$ $\displaystyle\equiv$ $\displaystyle\kappa a_{0}$
$\displaystyle\kappa$ $\displaystyle\approx$ $\displaystyle 6$ (60)

where

If $\alpha/c_{1}<0$ then $K>0$ and we are directly in the MOND regime. Then we
obtain the modified Friedman equation:

$H^{2}\left(1+\frac{\alpha}{c_{1}}+\frac{3}{2}\kappa^{2}\left(\frac{H}{H_{0}}\right)\left(\frac{-3\alpha}{c_{1}}\right)^{\frac{3}{2}}\right)=8\pi
G\rho$ (61)

A self-accelerating solution to this is only possible if the quantity in
brackets is positive definite, so that $H\rightarrow$ _const_ as
$\rho\rightarrow 0$. This could only happen if $\alpha/c1<-1$, but that would
violate our requirements that $K\ll 1$.

In the other case $\alpha/c_{1}>0$ we must extrapolate the MOND form of $F$ in
equation (A) to $K<0$. To make the extension continuous across the $K=0$, we
should set $F(-K)=-F(K)$, so that the MOND form becomes:

$\displaystyle c_{1}F$ $\displaystyle=$
$\displaystyle-2K-\frac{2}{3}(-K)^{\frac{3}{2}}$ (62) $\displaystyle(0\leq$
$\displaystyle-K$ $\displaystyle\ll 1)$

The Friedman equation then becomes:

$H^{2}\left(1+\frac{\alpha}{c_{1}}-\frac{3}{2}\kappa^{2}\left(\frac{H}{H_{0}}\right)\left(\frac{3\alpha}{c_{1}}\right)^{\frac{3}{2}}\right)=8\pi
G\rho$ (63)

which does have an accelerating solution. Unfortunately, solving this equation
at the present day for reasonable values of $\Omega_{m}$ shows that it
requires $K\sim 2$, which again violates our requirement that $|K|\ll 1$.
Having exhausted our other options we are forced to require $K\geq 1$, outside
the true MOND regime. This is consistent with another separate analysis of the
solutions of (16) Cardone and Radicella (2009). It seems likely that if the
vector field is responsible for the late time acceleration then it is a result
of behavior of the function away from the MONDian limit. There is still,
though a role for the near numerical coincidence of $a_{0}$ and $H_{0}$ \- it
lessens the fine tuning of the other parameters in $F$ in order for the
acceleration to happen at suitably late times.

##  Appendix B Vector And Tensor Modes

In this appendix, we provide the perturbed equations of motion for the vector
and tensor modes of the various fields appearing in these models, namely the
metric, the vector field and the Lagrange multiplier. The latter, being
scalar, only have a spin$-0$ mode. The other two fields, $g_{ij}$ and $A^{i}$
are perturbed as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-a^{2}d\tau^{2}+a^{2}B_{i}d\tau
dx^{i}+a^{2}(\gamma_{ij}+h_{ij})dx^{i}dx^{j}$ $\displaystyle h_{ij}$
$\displaystyle=$ $\displaystyle 2\partial_{(i}E_{j)}+2E_{ij}$ $\displaystyle
A^{\mu}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{a},\frac{A^{i}}{a}\right),$ $\displaystyle
A_{\mu}$ $\displaystyle=$ $\displaystyle\left(-a,aV_{i}\right),$ (64)

where the different fields introduced satisfy

$\partial_{i}B^{i}=0=\partial^{j}E_{ij}=\partial_{i}E^{i}=E^{i}_{i}\text{ and
}\partial_{i}A^{i}=0.$ (65)

We also introduce the useful quantity $V^{i}=A^{i}+B^{i}$ and remind that all
the latin indices on the perturbed fields are raised and lowered thanks to the
Kronecker flat metric $\delta_{ij}$.

In the following we use the unperturbed results

$\displaystyle K$ $\displaystyle=$ $\displaystyle
3M^{-2}\mathcal{H}^{2}\alpha,$ $\displaystyle J^{0}_{\ 0}=6c_{2}\mathcal{H}$ ,
$\displaystyle J^{i}_{\ j}=2\mathcal{H}\alpha\delta^{i}_{j},$ $\displaystyle
I^{\sigma}_{00}=6c_{2}\mathcal{H}\delta^{\sigma}_{0}$ , $\displaystyle
I^{\sigma}_{0i}=0,\
I^{\sigma}_{ij}=-2\mathcal{H}\alpha\delta^{\sigma}_{0}\delta_{ij}.$ (66)

The Einstein Equation (5) without matter introduces the stress tensor

$\displaystyle\tilde{T}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\nabla_{\sigma}\left[I^{\sigma}_{\alpha\beta}\right]+\hat{T}_{\alpha\beta}$
$\displaystyle\hat{T}_{\alpha\beta}$ $\displaystyle=$ $\displaystyle-
F_{K}Y_{(\alpha\beta)}+\frac{1}{2}M^{2}g_{\alpha\beta}\mathcal{F}+\lambda
A_{\alpha}A_{\beta},$

where

$I^{\sigma}_{\alpha\beta}=F_{K}\left[J_{(\alpha}^{\ \
\sigma}A_{\beta)}-J^{\sigma}_{\
(\alpha}A_{\beta)}-J_{(\alpha\beta)}A^{\sigma}\right].$ (67)

As intermediate results we have

$\delta K^{\alpha\beta}_{\ \ \ \gamma\delta}=c_{1}\left(\delta
g^{\alpha\beta}g_{\gamma\delta}+g^{\alpha\beta}\delta
g_{\gamma\delta}\right),$ (68)
$\delta\left(\nabla_{\alpha}A^{\gamma}\right)=\delta^{\gamma}_{i}\partial_{\alpha}\left[\frac{A^{i}}{a}\right]+\Gamma^{\gamma}_{\alpha
k}\frac{A^{k}}{a}+\frac{\delta\Gamma^{\gamma}_{\alpha 0}}{a},$ (69) $\delta
Y_{0i}=\delta Y_{i0}=c_{1}\left(\mathcal{H}V_{i}+V_{i}^{\prime}\right),$ (70)
$\displaystyle\delta\hat{T}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}M^{2}Fa^{2}h_{ij},$ $\displaystyle\delta\hat{T}_{0i}$
$\displaystyle=$ $\displaystyle\frac{1}{2}a^{2}B_{i}M^{2}F-\lambda
a^{2}V_{i}-F_{K}\delta Y_{(0i)},$ (71) $\displaystyle\delta J^{0}_{\ i}$
$\displaystyle=$
$\displaystyle\frac{2\mathcal{H}c_{3}V_{i}}{a}-\frac{2c_{1}V_{i}^{\prime}}{a},$
$\displaystyle\delta J^{\ i}_{0}$ $\displaystyle=$ $\displaystyle-\delta
J^{0}_{\ i}+\frac{2\mathcal{H}}{a}(c_{1}+c_{3})B^{i},$ (72)
$\displaystyle\delta J^{\ 0}_{i}$ $\displaystyle=$ $\displaystyle-\delta
J^{j}_{\ 0}\delta_{ij}+\frac{2\mathcal{H}}{a}(c_{1}+c_{3})B_{i},$ (73)
$\displaystyle\delta J^{i}_{\ 0}$ $\displaystyle=$ $\displaystyle
2\frac{c_{13}B^{i}}{a}\mathcal{H}-2\frac{c_{1}\mathcal{H}}{a}V^{i}+\frac{2c_{3}V^{i^{\prime}}}{a};$
(74)

and

$\displaystyle\delta J^{i}_{\ j}$ $\displaystyle=$ $\displaystyle
2c_{1}\left[\frac{\partial^{i}A_{j}}{a}+\frac{h^{i^{\prime}}_{j}}{2a}+\frac{\partial^{[i}B_{j]}}{a}\right],$
(75)
$\displaystyle+2c_{3}\left[\frac{\partial_{j}A^{i}}{a}+\frac{h^{i^{\prime}}_{j}}{2a}+\frac{\partial_{[j}B^{i]}}{a}\right].$

We then obtain,

$\displaystyle\delta I^{\sigma}_{0i}$ $\displaystyle=$ $\displaystyle
F_{K}\left[2\mathcal{H}c_{3}V_{i}-2c_{1}V_{i}^{\prime}-2\mathcal{H}\alpha
B_{i}\right]\delta^{\sigma}_{0}$
$\displaystyle+2F_{K}(c_{1}-c_{3})\partial^{[k}V_{i]}$

with $\alpha=c_{1}+3c_{2}+c_{3}$, and

$\displaystyle\delta I^{i}_{00}$ $\displaystyle=$ $\displaystyle
F_{K}[-2c_{1}V^{i^{\prime}}-2c_{1}\mathcal{H}V^{i}+2\mathcal{H}c_{3}V^{i}$
(77) $\displaystyle+2c_{3}V^{i^{\prime}}+6c_{2}\mathcal{H}A^{i}].$

We also get

$\delta I^{\sigma}_{ij}=-\delta^{\sigma}_{0}F_{K}\delta J_{(ij)}.$ (78)

To obtain the perturbed Einstein equation, we then plug these results into the
relations

$\displaystyle\delta\nabla_{\sigma}I^{\sigma}_{0i}$ $\displaystyle=$
$\displaystyle\partial_{0}\left(\delta
I^{0}_{0i}\right)+\partial_{k}\left(\delta
I^{k}_{0i}\right)+2\mathcal{H}\delta I^{0}_{0i}$
$\displaystyle-(B^{j^{\prime}}+\mathcal{H}B^{j})I^{0}_{ji}-6\mathcal{H}^{2}c_{2}B_{i}F_{K}-\Gamma^{0}_{ji}\delta
I^{j}_{00},$ $\displaystyle\delta\nabla_{\sigma}I^{\sigma}_{ij}$
$\displaystyle=$ $\displaystyle\partial_{0}\left(\delta
I^{0}_{ij}\right)+2\mathcal{H}\delta
I^{0}_{ij}-I^{0}_{kj}\delta\Gamma^{k}_{0i}-\mathcal{H}\delta_{ki}\delta
I^{k}_{0j}$
$\displaystyle-\delta\Gamma^{k}_{0j}I^{0}_{ik}-\mathcal{H}\delta_{kj}\delta
I^{k}_{i0}.$

We decompose these equations into Fourier components:

$\displaystyle X^{i}(t,\overrightarrow{x})$ $\displaystyle=$
$\displaystyle\sum_{\overrightarrow{k}}\sum_{m=0,1}X(t,\overrightarrow{k})Y^{i(\pm
m)}_{\overrightarrow{k}},$ $\displaystyle T^{ij}(t,\overrightarrow{x})$
$\displaystyle=$
$\displaystyle\sum_{\overrightarrow{k}}\sum_{m=0,1,2}T(t,\overrightarrow{k})Y^{ij(\pm
m)}_{\overrightarrow{k}}.$ (79)

where the orthonormal modes

$Y^{(0)},\ Y^{(\pm 1)},\ Y^{(\pm 2)}$ (80)

are eigenmodes of the Laplace-Beltrami operator: $\Delta
Y_{\mathbf{I}}^{(m)}=-k^{2}Y_{\mathbf{I}}^{(m)}$, $\mathbf{I}$ being an
arbitrary set of Lorentz indices. For more information on these functions, see
Lim (2005). The perturbed vector field equation can then be written

$\displaystyle 2\lambda aV$ $\displaystyle=$ $\displaystyle
F_{KK}K^{\prime}\left(2\mathcal{H}c_{3}V-2c_{1}V^{\prime}\right)-2c_{1}F_{K}V^{\prime\prime}$
(81)
$\displaystyle-4F_{K}c_{1}\mathcal{H}V^{\prime}+V\left(2\frac{a^{\prime\prime}}{a}c_{3}F_{K}+\mathcal{H}^{2}(2c_{3}+2c_{1})F_{K}\right)$
$\displaystyle-
F_{K}\left(2c_{1}k^{2}A+(c_{1}-c_{3})k^{2}B\right)+3F_{K}c_{2}k^{2}E^{\prime}.$

The spin$-2$ part of the $ij$ Einstein equations gives

$E^{{}^{\prime\prime}}+2\mathcal{H}E^{{}^{\prime}}+\frac{k^{2}E}{1+F_{K}(c_{1}+c_{3})}+2F_{KK}K^{\prime}(c_{1}+c_{3})E^{{}^{\prime}}=0,$
(82)

while the spin$-1$ part of the ${}^{0}_{i}$ Einstein equations gives the same
equation as (81) and the spin$-1$ component of the $ij$ Einstein equation is,
in space conventions,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{(i}E_{j)}^{{}^{\prime\prime}}+\frac{2(1-2\alpha)\mathcal{H}+2c_{13}\frac{1}{a^{2}}(a^{2}F_{K})^{\prime}+4\alpha\mathcal{H}F_{K}}{1+2F_{K}c_{13}}\partial_{(i}E_{j)}^{{}^{\prime}}$
(83)
$\displaystyle-\frac{\partial_{(i}B_{j)}^{{}^{\prime}}-c_{13}F_{K}\partial_{(i}A_{j)}^{{}^{\prime}}}{1+2F_{K}c_{13}}$
$\displaystyle-\frac{2(2\mathcal{H}^{\prime}+\mathcal{H}^{2})+M^{2}\mathcal{F}a^{2}+4\alpha({\cal
H}F_{K})^{\prime}+8\mathcal{H}^{2}\alpha\mathcal{F}}{1+2F_{K}c_{13}}\partial_{(i}E_{j)}$
$\displaystyle-\frac{2\mathcal{H}\partial_{(i}B_{j)}}{1+2F_{K}c_{13}}$
$\displaystyle+\frac{c_{13}F_{KK}K^{\prime}+2c_{13}\mathcal{H}F_{K}}{1+2F_{K}c_{13}}\partial_{(i}A_{j)}.$

In order to compare with the results of Lim (2005) let us consider the gauge
$E_{j}=0$. The spin$-1$ part of the $ij$ Einstein equations (83) implies

$B_{i}=c_{13}F_{K}A_{i}.$ (84)

After that we can rewrite (81) as

$\displaystyle A^{{}^{\prime\prime}}$ $\displaystyle=$
$\displaystyle-\left(2\mathcal{H}+\frac{F_{KK}}{c_{1}F_{K}}K^{{}^{\prime}}\right)A^{{}^{\prime}}-\frac{F_{K}}{2c_{1}}\frac{2c_{1}+F_{K}(c_{1}^{2}-c_{3}^{2})}{1+F_{K}c_{13}}k^{2}A$
$\displaystyle+\left[\mathcal{H}\frac{c_{3}F_{KK}}{c_{1}F_{K}}K^{{}^{\prime}}+\frac{a^{\prime\prime}}{a}\frac{c_{3}}{c_{1}}+\mathcal{H}^{2}\frac{c_{13}}{c_{1}}-\frac{\lambda
a}{F_{K}c_{1}}\right]A$

and obtaining the exact expression of $\lambda$ in the case $F(K)=K$,

$\lambda=3c_{13}\frac{\mathcal{H}^{2}}{a}-3c_{2}\frac{a^{\prime\prime}}{a^{2}}+6c_{2}\frac{\mathcal{H}^{2}}{a},$
(85)

implies in this latter case

$\displaystyle c_{1}\left(2\mathcal{H}A^{\prime}+A^{\prime\prime}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left[c_{1}-c_{3}+\frac{c_{13}}{1+c_{13}}\right]k^{2}A-$
(86)
$\displaystyle\left[2\alpha\mathcal{H}^{2}-\alpha\frac{a^{\prime\prime}}{a}+c_{1}\frac{a^{\prime\prime}}{a}\right]A.$

It is interesting to find that in the limit where $\mathcal{F}(K)=K$ we
recover results from Lim (2005) for (82,B) and (84).

A quick look at (82) and (B) implies the existence of two different speeds

$\displaystyle C_{T}^{2}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{1+F_{K}c_{13}},$ $\displaystyle C_{V}^{2}$
$\displaystyle\equiv$
$\displaystyle\frac{F_{K}}{2c_{1}}\frac{2c_{1}+F_{K}(c_{1}^{2}-c_{3}^{2})}{1+F_{K}c_{13}},$
(87)

being the sound speeds respectively of the tensor perturbation $E_{ij}$ and
the vector perturbation $A_{i}$. These speeds must be positive, so that it
implies constraints on the parameters of the theory and the function $F(K)$.
In particular, $F_{K}\neq\text{const.}$ will imply non-trivial constraints
between the fields of the theory and the parameters $c_{i}$:

$\left\\{\begin{array}[]{ccc}1+F_{K}c_{13}&>&0\\\
F_{K}\left[1+F_{K}\frac{c_{1}^{2}-c_{3}^{2}}{2c_{1}}\right]&>&0\end{array}\right.$
(88)