arXiv:astro-ph/0703259v3 29 Oct 2007
Unified Dark Matter in Scalar Field Cosmologies
Daniele Bertacca∗and Sabino Matarrese†
Dipartimento di Fisica “G. Galilei,” Universit` a di Padova,
and INFN, Sezione di Padova, via Marzolo 8, Padova I-35131, Italy
INFN, Sezione di Padova, via Marzolo 8, I-35131, Italy
(Dated: February 5, 2008)
Considering the general Lagrangian of k-essence models, we study and classify them through
variables connected to the fluid equation of state parameter wκ.
around which the scalar field describes a mixture of dark matter and cosmological constant-like
dark energy, an example being the purely kinetic model proposed by Scherrer. Making the stronger
assumption that the scalar field Lagrangian is exactly constant along solutions of the equation of
motion, we find a general class of k-essence models whose classical trajectories directly describe a
unified dark matter/dark energy (cosmological constant) fluid. While the simplest case of a scalar
field with canonical kinetic term unavoidably leads to an effective sound speed cs = 1, thereby
inhibiting the growth of matter inhomogeneities, more general non-canonical k-essence models allow
for the possibility that cs ≪ 1 whenever matter dominates.
This allows to find solutions
In the current standard cosmological model, two unknown components govern the dynamics of the Universe: the
dark matter (DM), responsible for structure formation, and a non-zero cosmological constant Λ (see, e.g. ref. ), or
a dynamical dark energy (DE) component, that drives cosmic acceleration [2, 3, 4].
If the DE is given by a Λ term, besides having a non-trivial fine-tuning problem to solve (unless one resorts to an
anthropic argument), one does not know why ΩDM and ΩΛare both of order unity today. In these years alternative
routes have been followed, for example Quintessence [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and k-essence [17, 18, 19]
(a complete list of dark energy models can be found in the recent review ). The k-essence is characterized by a
Lagrangian with non-canonical kinetic term and it is inspired by earlier studies of k-inflation [21, 22].
Some models of k-essence have solutions which tend toward dynamical attractors in the cosmic evolution so that
their late-time behavior becomes insensitive to initial conditions (see, e.g., [17, 23, 24, 25]). Other models, besides
having this property allow to avoid fine-tunning and are able to explain the cosmic coincidence problem [18, 19].
Subsequently, it was realized that the latter models [18, 19] have too small a basin of attraction in the radiation era
 and lead to superluminal propagation of field fluctuations .
An important issue is whether the dark matter clustering is influenced by the dark energy and if, when this happens,
the dark energy can indirectly smooth the cusp profiles of dark matter at small radii. Another hypothesis is to consider
a single fluid that behaves both as dark energy and dark matter. The latter class of models has been dubbed Unified
Dark Matter (UDM). Among several models of k-essence considered in the literature there exist two types of UDM
models: generalized Chaplygin gas [28, 29, 30, 31, 32, 33, 34] model and the purely kinetic model considered by
Scherrer . Alternative approaches to the unification of DM and DE have been proposed in Ref. , in the frame
of supersymmetry, and in Ref. , in connection with the solution of the strong CP problem.
The generalized Chaplygin model can be obtained via a Born-Infeld-type Lagrangian. This “fluid” has the property
of behaving like dark matter at high density and like a cosmological constant at low density.
The kinetic model introduced by Scherrer  can evolve into a fluid which describes at the same time the dark
matter and cosmological constant components. In this case, perturbations do not show instabilities but, at early
times, the fluid evolves like radiation, leading to a possible conflict with the constraints coming from primordial
nucleosynthesis. Moreover, the parameters of the model have to be fine-tuned in order for the model not to exhibit
finite pressure effects in the non-linear stages of structure formation .
∗Electronic address: firstname.lastname@example.org
†Electronic address: email@example.com
‡Electronic address: firstname.lastname@example.org
In this paper we consider the general Lagrangian of k-essence models and classify them through two variables
connected to the fluid equation of state parameter wκ. This allows to find attractor solutions around which the
scalar field is able to describe a mixture of dark matter and cosmological constant-like dark energy, an example
being Scherrer’s  purely kinetic model. Next, we impose that the Lagrangian of the scalar field is constant, i.e.
that pκ= −Λ, where Λ is the cosmological constant, along suitable solutions of the equation of motion, and find a
general class of k-essence models whose attractors directly describe a unified dark matter/dark energy fluid. While the
simplest of such models, based on a neutral scalar field with canonical kinetic term, unavoidably leads to an effective
speed of sound cswhich equals the speed of light, thereby inhibiting the growth of matter perturbations, we find a
more general class of non-canonical (k-essence) models which allow for the possibility that cs≪ 1 whenever matter
The plan of the paper is as follows. In Section 2 we introduce the general class of k-essence models and we propose
a new approach to look for attractor solutions. In Section 3 we apply our formalism to obtain the attractors for the
purely kinetic case. In Section 4 we generalize our model giving general prescriptions [Eqs. (4.12) and (4.14)] which
allow to obtain unified models where the dark matter and a cosmological constant-like dark energy are described by
a single scalar field along its attractor solutions
Section 5 contains our main conclusions. The scaling solutions for a particular case of k-essence are discussed in
Let us consider the following action
S = SG+ Sϕ+ Sm=
X = −1
We use units such that 8πG = c2= 1 and signature (−,+,+,+).
The energy-momentum tensor of the scalar field ϕ is
∇µϕ∇νϕ + L(ϕ,X)gµν.(2.3)
If X is time-like Sϕdescribes a perfect fluid with Tϕ
µν= (ρκ+ pκ)uµuν+ pκgµν, where
L = pκ(ϕ,X), (2.4)
is the pressure,
ρκ= ρκ(ϕ,X) ≡ 2X∂pκ(ϕ,X)
− pκ(ϕ,X) (2.5)
is the energy density and the four-velocity reads
Now let us assume that our scalar field defines a homogeneous background X =
differentiation w.r.t. the cosmic time t) and consider a flat Friedman-Robertson-Walker background metric. In such
a case, the equation of motion for the homogeneous mode ϕ(t) becomes
The k-essence equation of state wκ≡ pκ/ρκis just
2˙ ϕ2(where the dot denotes
¨ ϕ +∂pκ
∂X(3H ˙ ϕ) +
∂ϕ= 0 . (2.7)
while the effective speed of sound, which is the quantity relevant for the growth of perturbations, reads [21, 22]
If we assume that the scalar field Lagrangian depends separately on X and ϕ, i.e. that it can be written in the
pκ(ϕ,X) = f(ϕ)g(X) ,(2.10)
then Eq. (2.5) becomes
≡ f(ϕ)β(X). (2.11)
Notice that the requirement of having a positive energy density imposes a constraint on the function g, namely,
dX> g ,
having assumed f > 0.
Defining now the variables λ = (1/f)df/dN and α = −dlnβ/dN, where N = lna, we can express the energy
with¯K an integration constant. We can also rewrite the energy continuity equation in the form
dN+ λβ + 6Xdf
dX= 0. (2.14)
In terms of α and wκ, or, equivalently, of α and λ, the effective speed of sound, Eq. (2.9), reads
s= −(wκ+ 1)
= −α − λ
Therefore, in purely kinetic models (λ = 0) X can only decrease in time down to its minimum value.
The case α = 0 is analyzed in Appendix A.
III. STUDY OF THE ATTRACTORS FOR PURELY KINETIC SCALAR FIELD LAGRANGIANS
In the λ = 0 case the Lagrangian L (i.e. the pressure pκ) depends only on X, that is we are recovering the equations
that describe the purely kinetic model studied in Ref.  and the Generalized Chaplygin gas [29, 30]. In this section
we want to make a general study of the attractor solutions in this case.
For λ = 0, Eq. (2.14) gives the following nodes,
The general solution of the differential equation (2.14) in the λ → 0 limit is 
with k a positive constant. This solution was previously derived although in a different form in Ref. . As N → ∞,
X or dg/dX (or both) must tend to zero, which shows that, depending on the specific form of the function g(X),
each particular solution will converge toward one of the nodes above.
In what follows we will provide some examples of stable node solutions of the equation of motion, some of which
have been already studied in the literature. The models below are classified on the basis of the stable node to which
they asymptotically converge.
= 0 , 2)X =? X = 0 ,(3.1)
with? X a constant. Both cases correspond to wκ= −1, as one can read from Eq. (2.8).
A. Case 1): Scherrer solution
For the solution of case 1) we want to study the function g around some X =? X ?= 0. In this case one can
g = g0+ g2(X −? X)2.
with g0and g2suitable constants. This solution, with g0< 0 and g2> 0, coincides with the model studied by Scherrer
in Ref. .
If we now impose that today X is close to? X so that ǫ ≡ (X −? X)/? X = (a/a1)−3≪ 1, we obtain
ρκ= −g0+ 4g2? X2
In order for the density to be positive at late times, we need to impose g0< 0. In this case the speed of sound (2.9)
turns out to be
approximate g as a parabola with
(X −? X)
(3X −? X)
We notice also that, for (a/a1)−3≪ 1 we have c2
Eq. (3.4) tells us that the background energy density can be written as ρκ= ρΛ+ ρDM, where ρΛbehaves like a
“dark energy” component (ρΛ= const.) and ρDMbehaves like a “dark matter” component (ρDM∝ a−3). Note that,
from Eq. (3.4),? X must be different from zero in order for the matter term to be there. For this particular case the
It is interesting to notice that an alternative model, proposed in Ref.  in the frame of extended Born-Infeld
dynamics, actually converges to the Scherrer solution in the regime (X −? X)/? X ≪ 1.
radiation. Scherrer  imposed the constraint ǫ0 = ǫ(a0) = −g0/(8g2? X2) ≪ 10−10, requiring that the k-essence
is obtained by Giannakis and Hu , who considered the small-scale constraint that enough low-mass dark matter
halos are produced to reionize the universe. One should also consider the usual constraint imposed by primordial
nucleosynthesis on extra radiation degrees of freedom, which however leads to a weaker constraint.
Hubble parameter H is a function only of the UDM fluid H2= ρκ/3.
It is immediate to verify that in the early universe case (X ≫? X i.e. ρκ ≫ (−g0)) the k-essence behaves like
behaves as a matter component at least from the epoch of matter-radiation equality. The stronger bound ǫ0≤ 10−18
B.Case 1): Generalized Scherrer solution
Starting from the condition that we are near the attractor X =? X ?= 0, we can generalize the definition of g,
pκ= g = g0+ gn(X −? X)n
with n ≥ 2 and g0and gnsuitable constants.
The density reads
extending the Scherrer model in the following way
ρκ= (2n − 1)gn(X −? X)n+ 2? Xngn(X −? X)n−1− g0
If ǫn= [(X −? X)/? X]n≪ 1, Eq. (3.2) reduces to
X =? X 1 +
(where an−1≪ a) and so ρκbecomes
ρκ≃ 2n? Xngn
with (1/an−1)−3= [1/(ngn)](k/? X2n−1)1/2for ǫn≪ 1.
purely kinetic model of Ref. , i.e. a cosmological constant and a matter term. We can therefore extend the
constraint of Ref.  to this case obtaining (ǫ0)n−1= −g0/(4n? Xngn) ≤ 10−10. A stronger constraint would clearly
If we write the general expressions for wκand c2
We have therefore obtained the important result that this attractor leads exactly to the same terms found in the
also apply to our model by considering the small-scale constraint imposed by the universe reionization, as in Ref. .
wκ= − 1 +
(X −? X)n
1 − 2n? X
(X −? X)n−1− (2n − 1)
(X −? X)
(X −? X)n
2(n − 1)? X + (2n − 1)(X −? X).
For ǫ ≪ 1 we obtain a result similar to that of Ref. , namely
wκ≃ −1 + 2n
2(n − 1)ǫ .(3.13)
On the contrary, when X ≫? X we obtain
2n − 1
In this case we can impose a bound on n so that at early times and/or at high density the k-essence evolves like
dark matter. In other words, when n ≫ 1, unlike the purely kinetic case of Ref. , the model is well behaved also
at high densities.
C.Case 2): Generalized Chaplygin gas
An example of case 2) is provided by the Generalized Chaplygin (GC) model (see e.g. Refs. [28, 29, 30, 31, 32, 33,
34]), whose equation of state has the form
where now pGC= pκand ρGC= ρκand ρ∗and p∗are suitable constants.
Through the equation ρκ= 2Xdg(X)
and ρGC as functions of either X or a. When the pressure and the energy density are considered as functions of X
one gets 
−g(X) and the continuity equationρGC
dN+(ρGC+pGC) = 0 we can write pGC
1 − µX
1 − µX
with µ = const..
It is necessary for our scopes to consider the case γ < 0, so that c2
standard “Chaplygin gas” model.
Another model that falls into this class of solution is the one proposed in Ref. , in which g = b√2X −Λ (with b
a suitable constant) which satisfies the constraint that p = −Λ along the attractor solution X =? X = 0. This model,
s> 0. Note that γ = −1 corresponds to the
however is well-known to imply a diverging speed of sound.
IV. UNIFIED DARK MATTER FROM A SCALAR FIELD WITH NON-CANONICAL KINETIC TERM
Starting from the barotropic equation of state p = p(ρ) we can describe the system either through a purely kinetic
k-essence Lagrangian (if the inverse function ρ = ρ(p) exists) or through a Lagrangian with canonical kinetic term,
as in quintessence-like models. The same problem has been solved in Ref. , although with a different procedure
and for a different class of models. In the first case we have to solve the equation
ρ(p(X)) = 2Xdp(X)
− p(X) (4.1)
when X is time-like. In the second case we have to solve the two differential equations
χ − V (φ) = p(φ,χ)
χ + V (φ) = ρ(φ,χ)
where χ =˙φ2/2 is time-like. In particular if we assume that our model describes a unified dark matter/dark energy
fluid we can proceed as follows: starting from ˙ ρ = −3H(p+ρ) = −√3ρ(p+ρ) and 2χ = (p+ρ) = (dφ/dρ)2˙ ρ2we get
φ = ±1
(p(ρ′) + ρ′)1/2, (4.4)
up to an additive constant which can be dropped without any loss of generality. Inverting the Eq. (4.4) i.e. writing
ρ = ρ(φ) we are able to get V (φ) = [ρ(φ) − p(ρ(φ))]/2. If one requires the exact condition that our unified DM fluid
has a constant pressure term p = −Λ and looks for a scalar field model with canonical kinetic term, one arrives at an
exact solution with potential V (φ) = (Λ/2)[cosh2(√3φ/2) + 1] (see also Ref. [43, 44]). Using standard criteria (e.g.
Ref. ) it is immediate to verify that the above trajectory corresponds to a stable node even in the presence of an
extra-fluid (e.g. radiation) with equation of state wfluid≡ pfluid/ρfluid> 0, where pfluidand ρfluidare the fluid pressure
and energy density, respectively. Along the above attractor trajectory our scalar field behaves precisely like a mixture
of pressure-less matter and cosmological constant. Using the expressions for the energy density and the pressure we
immediately find, for the matter energy density
ρm= ρ − Λ = Λsinh2
∝ a−3. (4.5)
A closely related solution was found by Salopek & Stewart , using the Hamiltonian formalism. Like any scalar
field with canonical kinetic term , such a UDM model however predicts c2
inhibits the growth of matter inhomogeneities. Such a “quartessence” model therefore behaves exactly like a mixture
of dark matter and dark energy along the attractor solution, whose matter sector, however is unable to cluster on
sub-horizon scales (at least as long as linear perturbations are considered).
We can then summarize our findings so far by stating that purely kinetic k-essence cannot produce a model which
exactly describes a unified fluid of dark matter and cosmological constant, while scalar field models with canonical
kinetic term, while containing such an exact description, unavoidably lead to c2
structure formation. In order to find an exact UDM model with acceptable speed of sound we consider more general
scalar field Lagrangians.
s= 1, as it is clear from Eq. (2.9), which
s= 1, in conflict with cosmological
A. Lagrangians of the type L(ϕ,X) = g(X) − V (ϕ)
Let us consider Lagrangians with non-canonical kinetic term and a potential term, in the form
L(ϕ,X) = g(X) − V (ϕ) .(4.6)
The energy density then reads
ρ = 2Xdg(X)
− g(X) + V (ϕ) , (4.7)
while the speed of sound keeps the form of Eq. (2.9). The equation of motion for the homogeneous mode reads
One immediately finds
p + ρ = 2Xdg(X)
≡ 2F(X) .(4.9)
We can rewrite the equation of motion, Eq. (4.8), in the form
= 0 .(4.10)
It is easy to see that this equation admits 2 nodes, namely: 1) dg/dX|?
X= 0 and 2)? X = 0. In all cases, for N → ∞,
the potential V should tend to a constant, while the kinetic term can be written around the attractor in the form
g(X) = M4
X −? X
n ≥ 2 , (4.11)
with M a suitable mass-scale and the constant? X can be either zero or non-zero. The trivial case g(X) = X obviously
Following the same procedure adopted in the previous section we impose the constraint p = −Λ, which yields the
general solution ρm= 2F(X).
This allows to define ϕ = ϕ(ρm) as a solution of the differential equation
reduces to the one of Section 4.
As found in the case of k-essence, the most interesting behavior corresponds to the limit of large n and? X = 0 in
Eq. (4.11), for which we obtain
leading to V (ϕ) ≈ ρm/2n − Λ, and c2
s= 1/(2n − 1) ≈ 0. The Lagrangian of this model is similar to that analyzed in
B.Lagrangians of the type L(ϕ,X) = f(ϕ)g(X)
Let us now consider Lagrangians with a non-canonical kinetic term of the form of Eq. (2.10), namely L(ϕ,X) =
Imposing the constraint p = −Λ, we obtain f(ϕ) = −Λ/g(X), which inserted in the equation of motion yields the
The latter equation, together with Eq. (4.12) define our general prescription to get UDM models describing both
DM and cosmological constant-like DE.
As an example of the general law in Eq. (4.14) let us consider an explicit solution. Assuming that the kinetic term
is of Born-Infeld type, as in Refs. [43, 44, 47, 48, 49],
with M a suitable mass-scale, which implies ρ = f(ϕ)/?1 − 2X/M4, we get
g(X) = −
1 − 2X/M4, (4.15)
1 +¯ka−3, (4.16)
where¯k = ρm(a∗)a3
impose that the Universe is dominated by our UDM fluid, i.e. H2= ρ/3. This gives
∗/Λ and a∗is the scale-factor at a generic time t∗. In order to obtain an expression for ϕ(a), we
which, replaced in our initial ansatz p = −Λ allows to obtain the expression (see also Ref. [43, 44])
If we expand f(ϕ) around ϕ = 0, and X/M4≪ 1 we get the approximate Lagrangian
2M4˙ ϕ2− Λ
Note that our Lagrangian depends only on the combination ϕ/M2, so that one is free to reabsorb a change of the
mass-scale in the definition of the filed variable. Without any loss of generality we can then set M = Λ1/4, so that
the kinetic term takes the canonical form in the limit X ≪ 1. We can then rewrite our Lagrangian as
?1 − 2X/Λ
L = −Λ
This model implies that for values of√3ϕ ≈ −π and 2X/Λ ≈ 1,
1 − 2X/Λ ∝ a−3/2, (4.21)
the scalar field mimics a dark matter fluid. In this regime the effective speed of sound is c2
To understand whether our scalar field model gives rise to a cosmologically viable UDM solution, we need to check
if in a Universe filled with a scalar field with Lagrangian (4.20), plus a background fluid of e.g. radiation, the system
displays the desired solution where the scalar field mimics both the DM and DE components. Notice that the model
does not contain any free parameter to specify the present content of the Universe. This implies that the relative
amounts of DM and DE that characterize the present universe are fully determined by the value of ϕ0≡ ϕ(t0). In
other words, to reproduce the present universe, one has to tune the value of f(ϕ) in the early Universe. However, a
numerical analysis shows that, once the initial value of ϕ is fixed, there is still a large basin of attraction in terms of
the initial value of dϕ/dt, which can take any value such that 2X/Λ ≪ 1.
The results of a numerical integration of our system including scalar field and radiation are shown in Figures 1 -
3. Figure 1 shows the density parameter, ΩUDMas a function of redshift, having chosen the initial value of ϕ so that
today the scalar field reproduces the observed values ΩDM≈ 0.268, ΩDE≈ 0.732 . Notice that the time evolution
of the scalar field energy density is practically indistinguishable from that of a standard DM plus Lambda (ΛCDM)
model with the same relative abundances today. Figure 2 shows the evolution equation of state parameter wUDM;
once again the behavior of our model is almost identical to that of a standard ΛCDM model for 1 + z < 104. Notice
that, since c2
s= −wUDM, the effective speed of sound of our model is close to zero, as long as matter dominates, as
required. In Figure 3 we finally show the redshift evolution of the scalar field variables X = ˙ ϕ2/2 and ϕ: one can
easily check that the evolution of both quantities is accurately described by the analytical solutions above, Eqs. (4.16)
and (4.17), respectively (the latter being obviously valid only after the epoch of matter-radiation equality).
s= 1 − 2X/Λ ≈ 0, as
In this paper we have investigated the possibility that the dynamics of a single scalar field can account for a unified
description of the dark matter and dark energy sectors. In particular, we have studied the case of purely kinetic
k-essence, showing that these models have only one late-time attractor with equation of state wκ= −1 (cosmological
constant). Studying all possible solutions near the attractor we have found a generalization of the Scherrer model
, which describes a unified dark matter fluid.
FIG. 1: Evolution of the scalar field density parameter vs. redshift. The continuous line shows the UDM density parameter;
the dashed line is the density parameter of the DM + DE components in a standard ΛCDM model; the dotted line is the
radiation density parameter.
FIG. 2: The redshift evolution of the scalar field equation of state parameter wUDM (continuous line) is compared with that of
the sum of the DM + DE components in a standard ΛCDM model (dashed line).
Generalizing our analysis to the case where the Lagrangian is not purely kinetic, we have given general prescriptions
[Eqs. (4.12) and (4.14)] to obtain unified models where the dark matter and a cosmological constant-like dark energy
are described by a single scalar field along its attractor solutions. Moreover, we have given explicit examples for which
the effective speed of sound is small enough whenever matter dominates, thus allowing for the onset of gravitational
instability. Studying the detailed consequences of such unified dark matter model for cosmic microwave background
anisotropies and for the formation of the large-scale structure of the Universe will be the subject of a subsequent
We thank Nicola Bartolo and Fabio Finelli for useful discussions.
FIG. 3: Redshift evolution of the scalar field of the scalar field variables X = ˙ ϕ2/2 (top) and ϕ (bottom).
APPENDIX A: STUDY OF THE SOLUTIONS FOR α = 0
If X is constant (dX/dN = 0) then, from Eq. (2.14), α = 0. In this situation we have that
dt= λ = −3(wκ+ 1)(A.1)
is constant because wκis function only of X.
Now if we consider the case in which the universe is dominated by a fluid with constant equation of state wBthen
H =˙N ∼ 2/[3(wB+ 1)t] and Eq. (A.1) becomes dlnf/dlnt ∼ −2(wκ+ 1)/(wB+ 1).
Therefore we get f ∼ t−2wκ+1
recovered, although in a more general way, the result of Ref. . These models have been dubbed “scaling k-essence
” (see also Ref. ). In such a case, the equation of state wκcan be written as wκ= β(wB+1)/2−1, where f = ϕ−β.
If wB= wκwe have only the k-essence as background and we get β = 2.
Using the latter approach it is simpler to see that when α → 0 all the viable solutions1converge to the scaling
solution. In fact, if a priori α ?= 0 and f = ϕ−βwe have that
wB+1(because if ˙ ϕ is constant then ϕ ∼
√2Xt). In other words, we have
α = 3(wκ+ 1) −3
Starting from Eq. (2.14), we note that when α → 0 then 2Xdg
must be a constant2. Therefore, provided that g ?= b√2X + const. the solution of the equation of motion converges
to the scaling solution.
dX− g → const. ?= 0. It is then easy to see that X
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