#### Duality between k-essence and Rastall gravity

Eur. Phys. J. C
Duality between k-essence and Rastall gravity
Kirill A. Bronnikov 0 1 2
Júlio C. Fabris 0 3
Oliver F. Piattella 3
Denis C. Rodrigues 3
Edison C. Santos 3
0 National Research Nuclear University “MEPhI” , Kashirskoe sh. 31, Moscow 115409 , Russia
1 Institute of Gravitation and Cosmology, RUDN University , ul. Miklukho-Maklaya 6, Moscow 117198 , Russia
2 VNIIMS , Ozyornaya ul. 46, Moscow 119361 , Russia
3 Universidade Federal do Espírito Santo , Vitória, ES CEP 29075-910 , Brazil
The k-essence theory with a power-law function of (∂φ)2 and Rastall's non-conservative theory of gravity with a scalar field are shown to have the same solutions for the metric under the assumption that both the metric and the scalar fields depend on a single coordinate. This equivalence (called k-R duality) holds for static configurations with various symmetries (spherical, plane, cylindrical, etc.) and all homogeneous cosmologies. In the presence of matter, Rastall's theory requires additional assumptions on how the stress-energy tensor non-conservation is distributed between different contributions. Two versions of such nonconservation are considered in the case of isotropic spatially flat cosmological models with a perfect fluid: one (R1) in which there is no coupling between the scalar field and the fluid, and another (R2) in which the fluid separately obeys the usual conservation law. In version R1 it is shown that k-R duality holds not only for the cosmological models themselves but also for their adiabatic perturbations. In version R2, among other results, a particular model is singled out that reproduces the same cosmological expansion history as the standard CDM model but predicts different behaviors of small fluctuations in the k-essence and Rastall frameworks.
1 Introduction
The century-old general relativity (GR) theory still
successfully passes all local experimental tests. However, there are
many reasons to consider this theory not as an ultimate theory
of gravity but only as a reasonable approximation, working
well in a large but finite range of length and energy scales.
Among such reasons are the old problem of unifying
gravity with other physical interactions and the difficulties in
attempts to quantize GR. Other reasons for dealing with
modifications of GR are the well-known problems experienced
by the theory itself: its prediction of space-time singularities
in the most physically relevant solutions, actually showing
situations where the theory does not work any more, and its
inability to explain the main observable features of the
Universe without introducing so far invisible forms of matter,
dark matter (DM) and dark energy (DE), which add up to as
much as 95% of the energy content of the Universe.
The existing modifications and extensions of classical GR
can be divided into two large classes. The first one changes
the geometric content of the theory and includes, in
particular, f (R) theories, multidimensional theories and
nonRiemannian geometries. The second class introduces new
fundamental, non-geometric fields and includes, in
particular, scalar-tensor theories, Horndeski theory [1] and
vectortensor theories. Of much interest are the cases where it is
possible to establish connections between different
representatives of the same class or even different classes of theories
(possibly the best-known example of such a connection is
the equivalence of f (R) theories with a certain subclass of
scalar-tensor theories; see, e.g., [2,3]). In the present paper
we discuss such an equivalence between large families of
models of k-essence theories and Rastall’s non-conservative
theories with a scalar field as a source of gravity.
The k-essence theories, introducing a non-standard form
of the kinetic term of a scalar field [4,5], evidently belong
to the class of theories with non-standard fundamental fields
coupled to gravity. They proved to be a way of obtaining
both early inflation and the modern accelerated expansion
of the Universe [5–7] driven by a scalar kinetic term instead
of a potential. Notably, a kind of k-essence structure also
appears in string theories, for example, in the Dirac–Born–
Infeld action, where the kinetic term of the scalar field has
a structure similar to that of the Maxwell-like term in Born–
Infeld electrodynamics [8].
configurations with dustlike matter are discussed in Sect. 6.
Our conclusions are presented in Sect. 7.
2 Scalar-vacuum space-times
2.1 k-essence
The k-essence theories can be defined as general relativity
with generalized forms of scalar fields minimally coupled to
gravity. In the absence of matter nonminimally coupled to
gravity, the most general Lagrangian is
L =
with
X = ηφμφμ,
√−g[R + F (X, φ) + Lm ],
Rastall’s theory [9] is one more generalization of GR,
which relaxes the conservation laws expressed by the zero
divergence of the stress-energy tensor (SET) Tμν of matter.
In this theory, the quantity ∇ν Tμν is linked to the gradient
of the Ricci scalar, and in this way Rastall’s theory may
be viewed as a phenomenological implementation of some
quantum effects in a curved background. Rastall’s theory
leads to results of interest in cosmology, e.g., the evolution
of small DM fluctuations is the same as in the CDM model,
but DE is able to cluster. This might potentially provide an
evolution of DM inhomogeneities in the non-linear regime
different from the standard CDM model [10]. The whole
success of the CDM model is reproduced at the background
and linear perturbation levels, but new effects are expected in
the non-linear regime, where the CDM model faces some
difficulties [11,12]. It has also been shown [13] that Rastall’s
theory with the canonical SET of a scalar field, in the context
of cosmological perturbations, is only consistent if matter
is present. An interesting observation in this analysis is that
scalar field coupling with gravity leads to equations very
similar to those in some classes of Galileon theories.
A consideration of static, spherically symmetric solutions
in k-essence theories with a power-law kinetic function [14]
and similar solutions in Rastall’s theory in the presence of
a free or self-interacting scalar field [15] has shown that
some exact solutions of these two theories describe quite the
same geometries, although the properties of the scalar fields
are different. Also, a no-go theorem concerning the
possible emergence of Killing horizons, proved in the k-essence
framework [14], has its counterpart in the Rastall-scalar field
system [15]. These similarities indicate a deeper
relationship between the two theories, to be analyzed in this paper.
We will show that in the absence of other matter than the
(possibly self-interacting) scalar fields, the two theories lead
to completely coinciding geometries (we will call this k–R
duality) under the assumptions that the relevant quantities
depend on a single spatial or temporal coordinate and that
the k-essence theory is specified by a power-law function;
then there emerges a simple relation between the numerical
parameters of the theories. If there are other forms of
matter, the situation is more involved and depends on how the
non-conservation of the SET is distributed between
different matter contributions in Rastall’s theory. We will discuss
two variants of such non-conservation in the case of isotropic
spatially flat cosmological models and show that k–R duality
generically takes place.
The paper is organized as follows. In the next section we
discuss vacuum solutions of k-essence and Rastall theories.
In Sect. 3, isotropic cosmological models are analyzed with
a matter source in the form of a perfect fluid. Some
considerations on the speed of sound are presented in Sect. 4,
while Sect. 5 is devoted to some special values of the
numerical parameters of both theories. Some concrete cosmological
(
1
)
(
2
)
(
3
)
(
4
)
(
5
)
(
6
)
(
7
)
(
8
)
where φμ = ∂μφ, F (X, φ) is an arbitrary function, and η =
±1 is used to make X positive since otherwise in the cases
like general power-law dependence F will be ill defined for
X < 0; Lm is the Lagrangian density of other kinds of matter
having no direct coupling to the curvature or the φ field. We
are using the system of units where c = 8π G = 1
Variation of the Lagrangian (
1
) with respect to the metric
and the scalar field leads to the field equations
Gνμ ≡ Rμν − 21 δμν R = −Tμν [φ] − Tμν [m],
Tμν [φ] ≡ η FX φμφν − 21 δμν F,
η∇α(FX φα) − 21 Fφ = 0,
where Gνμ is the Einstein tensor, FX = ∂ F/∂ X , Fφ =
∂ F/∂φ, and Tμν [m] is the SET of matter due to Lm .
Now, let us make the following assumptions:
(i) The k-essence Lagrangian is
F (X, φ) = F0 X n − 2V (φ),
where n = const = 0 and V (φ) is an arbitrary function
(the potential).
(ii) φ = φ (u), where u is one of the coordinates, which
may be temporal or spatial.
(iii) The metric has the form
ds2 = η e2α(u)du2 + hik dxi dx k ,
where i, k are the numbers of coordinates other than u,
and the determinant of hik has the factorized structure
det(hik ) = e2σ (u)h1(xi ).
In this case, we have X = e−2α(u)φu2 (the index u means
d/du), and the SET of the φ field has the following
nonzero components:
(there is no summing over an underlined index). The
scalar field equation has the form
Rastall’s theory of gravity is characterized by the following
equations [9]:
V (φ) = (3 − 2b)W (ψ ).
e−2nαφu2n−2 (2n − 1)φuu + σu φu
1 d V
−(2n − 1)αu φu = − n F0 dφ .
2.2 Rastall’s theory with a scalar field
Rμν − λ2 δμν R = −Tμν ,
∇ν Tμν =
λ − 1
2
∂μ R,
where λ is a free parameter and Tμν is the SET of matter. At
λ = 1, GR is recovered.
These equations can be rewritten as
Gνμ = − T ν
μ −
b − 1 δν T
2 μ
≡ −Tμν ,
∇ν Tμν = b −21 ∂μT , b := 2λ − 1 , T = Tαα.
3λ − 2
In this parametrization, GR is recovered if b = 1.
Let us consider matter in the form of a minimally coupled
scalar field ψ , so that
Tμν [ψ ] =
ψμψ ν − 21 δμνψαψ α
+ δμνW (ψ ),
where = ±1, indicating an ordinary (+1) or phantom (−1)
nature of the ψ field, ψμ ≡ ∂μψ , and W (ψ ) is a potential.
The scalar field equation follows from (
15
) and has the form
ψ + (b − 1)
ψ μψ ν ∇μψν
ψ αψα
dW
= − (3 − 2b) dψ
.
Let us now, in full similarity with what was done for
kessence theory, assume that ψ = ψ (u) and the metric has
the form (
7
). Then the nonzero components of the modified
scalar field SET in the right-hand side of Eq. (
14
) are
Tuu [ψ ] = 21 bη e−2αψu2 + (3 − 2b)W (ψ ),
Tii [ψ ] = 21 (b − 2)η e−2αψu2 + (3 − 2b)W (ψ ),
(
9
)
(
10
)
(
11
)
(
12
)
(
13
)
(
14
)
(
15
)
(
16
)
(
17
)
(
18
)
(19)
while the scalar field equation (
17
) takes the form
e−2α bψuu + ψu (σu − bαu ) = − η(3 − 2b)Wψ ,
(20)
where Wψ ≡ dW /dψ and, as before, η = sign guu .
2.3 Comparison
We assume that in the k-essence system there is no other
matter than the scalar field φ and in the Rastall system there
is no other matter than the scalar ψ . Let us see under which
conditions the right-hand sides of the Einstein equations (
9
)
and (
10
) coincide with those of the effective Einstein
equations of Rastall’s theory, (
18
) and (19). This will guarantee
that the solutions for the metric are also the same.
To begin with, we identify the potentials:
(21)
(22)
(23)
Next, we equate the ratios of the kinetic parts of Tμν [φ] and
Tμν [ψ ], to obtain
2n − 1
−1
b
= b − 2
⇒ (2 − b)n = 1.
Then, equating the kinetic parts themselves, we find that
2
ψu =
ηn F0φu2n e2(1−n)α.
Under the three conditions (21)–(23), the metric field
equations of the two theories completely coincide, therefore
their sets of solutions are also identical. Substituting (23) to
(20), one can easily verify that under these conditions the
scalar field equations (
11
) and (20) are also equivalent.
This general result covers many static symmetries
(spherical, plane, cylindrical, etc.), homogeneous cosmologies
(FRW, all Bianchi types, Kantowski–Sachs) and even
inhomogeneous ones if their metrics are of the form (
7
), (
8
).
Here and in most of the paper we consider the generic
values of the parameters n and b and exclude from consideration
their special values that require a separate analysis, such as,
for example, b = 0, b = 3/2 and n = 1/2. Some remarks
on these special cases will be made in Sect. 5.
3 Cosmology with matter
When, besides the scalar field, matter is present, it is
better, for evident technical reasons, to restrict ourselves from
the beginning to a certain type of metrics. We will consider
cosmological FLRW spatially flat metrics
ds2 = dt 2 − a(t )2[dx 2 + dy2 + dz2],
(24)
so that in (
7
) and (
8
) we have η = 1, eα = 1, and eσ = a(t )3.
Matter will be taken in the form of a perfect fluid, so that
Let us consider two (of an infinite number of) alternatives
in incorporating matter to Rastall’s theory with a scalar field:
Tμν [m] = diag(ρ , − p, − p, − p),
where ρ is the density and p is the pressure.
3.1 k-essence cosmology
In the FLRW metric (24) and with φ = φ (t ), the field
equations (
3
)–(
5
) with matter (where we denote ρ = ρk , p = pk )
take the form
3H 2 = 21 (2n − 1)F0φ˙ 2n + V (φ) + ρk ,
2H˙ + 3H 2 = − 21 F0φ˙ 2n + V (φ) − pk ,
1
φ˙ 2n−2[(2n − 1)φ¨ + 3H φ˙ ] = − n F0 Vφ ,
where H = a˙ /a is the Hubble parameter and Vφ ≡ dV /dφ.
The SET of matter satisfies the conservation law ∇ν Tμν [m] =
0, whence
ρ˙k + 3H (ρk + pk ) = 0.
3.2 Rastall cosmology with a scalar field and matter
The Rastall equations have the form (
14
) and (
15
), where
now Tμν is the total energy-momentum tensor,
T ν
μ = Tμν [ψ ] + Tμν [m],
with Tμν [ψ ] given by (
16
) and Tμν [m] by (25). In Eqs. (
14
),
the modified energy-momentum tensor Tμν is then a sum of
Tμν [ψ ] given by (
18
) and (19) and Tμν [m] with the
components
1
Ttt [m] = 2 [(3 − b)ρ + 3(b − 1) p] ≡ ρ,
1
Tii [m] = 2 [(1 − b)ρ + (3b − 5) p] ≡ − p
(we preserve the notation ρ and p without indices for matter
in Rastall gravity). Hence the Rastall equations read
3H 2
1
= 2 bψ˙ 2 + (3 − 2b)W + ρ,
1
2H˙ + 3H 2 = 2 (b − 2)ψ˙ 2 + (3 − 2b)W − p,
while the equation for ψ depends on further assumptions on
how the non-conservation of the full SET according to (
15
)
is distributed between ψ and matter. One notices that
ρ + p = ρ + p
and
ρ − ρ = p − p =
1 − b
2 (ρ − 3 p).
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
R1: The SETs of ψ and matter obey (
15
) each separately, so
there is no mixing between the two sources of gravity;
R2: The SET of matter is conservative, so that
ρ˙ + 3H (ρ + p) = 0.
3.3 Case R1: no mixing of scalar field and matter
In this case we have
∇ν Tμν [ψ ] = b −2 1 ∂μT [ψ ], (37)
∇ν Tμν [m] = b −2 1 ∂μT [m]. (38)
The first of these conditions leads to the scalar field equation
(20), which in the present case reads
bψ¨ + 3H ψ˙ = − (3 − 2b)Wψ .
With (34), the condition (38) has the form
ρ˙ + 3H (ρ + p) = 0.
The full set of equations consists of (32), (33), (39), and (40),
with the definitions (31).
From (34) it follows that if matter satisfies the null energy
condition (NEC), then the same is true for the “effective”
density and pressure (ρ and p) in Rastall’s theory. However,
from (31) it can be verified that the positivity of the energy
density (or pressure) is not guaranteed in the k-essence case
if it is imposed in the Rastall theory, and vice versa.
It is easy to see that the right-hand sides of Eqs. (32) and
(33) coincide with those of (26) and (27) if, in addition to
the relationships (21)–(23) for scalar variables, we identify
ρk = ρ,
pk = p.
The correctness of this identification is confirmed by the
identity of the conservation laws (29) and (40). Thus, as in the
vacuum case, the parameters n and b are related by (22), that
is, n(2 − b) = 1, and the scalar fields φ and ψ are related by
Eq. (23) which now reads
3.4 Case R2: conservative matter
We now have ∇ν Tμν [m] = 0. This condition is particularly
important for the structure formation in the Universe for
the case of a pressureless fluid since ordinary matter must
agglomerate.
(36)
(39)
(40)
(41)
(42)
In this case, for the scalar field SET we have
1
∇ν Tμν [ψ ] = 2 (b − 1)(∂μT [ψ ] + ∂μT [m]),
which leads to the scalar field equation
The full set of equations consists of (32), (33), (44), and (29),
with the definitions (31). Note that Eq. (44) mixes the scalar
field and the matter fluid even though the fluid is conserved
as in GR.
This conservation lets us identify the matter SET
components in the k-essence and Rastall theories: ρ = ρk , p = pk .
As a result, identification of the other parts of the total SET
is only partly the same as in the previous case.
Identifying, as before, the potentials according to (21)
(that is, V = (3 − 2b)W ) and comparing the
Friedmannlike equations (32) and (33) with their k-essence counterparts
(26) and (27), we obtain, as before,
The correctness of this identification is verified by
substituting (45) into the scalar field equation (44): indeed, since we
have now, due to (35),
this substitution leads precisely to the scalar field equation
(28) of the k-essence theory.
It is important that in the case of conservative matter, a
comparison between the two theories does not lead to a direct
relationship like (22) between their numerical parameters n
and b. Instead, we have the equality (46), from which (22)
is restored only in the special case ρ = 3 p (zero trace of the
matter SET, radiation).
3.5 Further consequences of matter conservation
Equation (46) creates a connection between the temporal
behavior of ψ and the matter content. Indeed, inserting (46)
to (44) with zero or constant potential, we find
6n
F˙ + 2n − 1 H F = 0,
where F = ψ˙ 2. This leads to
ψ˙ 2 = ψ0a−6n/(2n−1),
where ψ0 is an integration constant.
(49)
(50)
(52)
(53)
(44)
(45)
(46)
(47)
(48)
From Eq. (46) it is clear that the matter density and
pressure must also evolve by a power law as functions of the
scale factor. Hence, only an equation of state (EoS) of the
type p = wρ, with w = const, is possible. In this case,
ρ = ρ0a−3(1+w),
ρ0 = const,
implying the relation between w and n
1
w = 2n − 1 .
We see that a substitution of p = wρ into the scalar field
equation relates the EoS factor w with the k-essence power
n, while the Rastall constant b remains arbitrary. Moreover,
Eqs. (26) and (27) show that the pure k-essence scalar field
φ behaves as a perfect fluid with the same EoS factor (50)
(see also [16]).
In other words, assuming a zero or constant potential V =
(3 − 2b)W and conservative matter in the Rastall framework,
we find that the k–R duality is only possible if matter is a
perfect fluid with the linear EoS p = wρ, coinciding with
the effective EoS of the scalar field φ.
With p = wρ Eq. (46) gives
ψ˙ 2 = kρ , k = n(bbn−−1)2(n1 −+ 31w) . (51)
Inserting this to the Friedmann-like equation (32), we obtain
in terms of n or w
3H 2
ρ
= V + 2n − 1
2nb(b − 1)(n − 2)
bn − 2n + 1
+ 2(2b − 3) + 2n(3 − b)
ρ b(b − 1)(1 + w)(1 − 3w)
= V + 2 (b − 2)(1 + w) + 2w
+ 3 − b + 3w(b − 1) .
The right-hand side must be positive. Therefore, given n and
V (or alternatively w and V ), we obtain a restriction on b.
For example, if V = 0 and w = 0 (dust, n → ∞),1 we
have either b < 3/2 or b > 2 (provided ρ > 0). For w = 1
(stiff matter, n = 1), there is no restriction on b, and we
obtain H 2 = V = const > 0, hence a de Sitter expansion,
a(t ) ∝ eHt . In this case, stiff matter precisely cancels the
contribution from the scalar field ψ or φ.
If we introduce a variable potential or a more complex
equation of state, the situation becomes much more involved.
It must be stressed that the EoS p = wρ with w = const
covers most of the interesting cases in cosmology. Moreover,
1 This relation makes sense even if the Lagrangian formulation becomes
ill defined; see Ref. [16].
we expect that the perturbative behavior may be very different
in the two theories even in this case.
There emerge two more natural questions. First, we have
found that in k-essence theory there are simultaneously a
scalar field and a perfect fluid with the same EoS and hence
the same time evolution of their densities and pressures. Can
we unify them by, for example, redefining the scalar field? A
probable answer is “no” because these two kinds of matter
are expected to behave quite differently at the perturbative
level.
Another question is: how is it possible to have a
completely definite situation in k-essence theory but arbitrariness
in the parameter b in the dual solution of Rastall’s theory?
An answer is that this arbitrariness is compensated by the
corresponding non-conservative behavior of the scalar field
ψ .
4 Perturbations and the speed of sound
A power-law k-essence model with V = 0 is equivalent in
the cosmological framework (such that (∂μφ)2 > 0) to a
perfect fluid with the equation of state p = wρ, where the
constant w is related to the power n:
(54)
(55)
(56)
(57)
,
In a fluid, adiabatic perturbations propagate as a sound with
the speed vs such that
Scalar field perturbations for general Lagrangians of the
form F (X, φ) have been treated in detail in Refs. [6,17]. It
has been shown there that a k-essence theory implies
and this expression is valid even if there is an arbitrary
potential term V (φ). In particular, for the theory (
6
), where
F (X ) = F0 X n − 2V (φ), we find again
2 1
vs = 2n − 1
in full agreement with (55). Thus there is a complete
equivalence between a perfect fluid and k-essence without a
potential not only for a cosmological background but even on the
perturbative level as far as adiabatic perturbations are
concerned. In particular, if w < 0, corresponding to n < 1/2,
the model is perturbatively unstable since it implies vs2 < 0.
(58)
(59)
This is true both for a perfect fluid and for k-essence.
Moreover, although the presence of a potential changes the scalar
field dynamics, the propagation speed of its perturbations,
coinciding with the derived speed of sound [17], is still the
same as with V = 0.
In Rastall’s theory things may be different. The speed of
sound for a scalar field is given by [13,18]
2
vs =
2 − b .
b
In scalar vacuum and in the R1 case (matter obeys the
non-conservation equation (37)), we have the relation (22),
(2 − b)n = 1, which makes (57) and (58) identical.
Furthermore, the fluids in the corresponding models obey different
equations of state; see (41). However, in the Rastall model
we can still characterize the fluid by the “effective” density
and pressure, ρ and p, the SET written in their terms is
conservative, hence the squared speed of sound of the Rastall
fluid is equal to d p/dρ = d pk /dρk . Thus we can conclude,
even without performing a complete perturbation analysis,
that the models belonging to the two theories coincide not
only at the background level but also at the level of adiabatic
perturbations.
In the case R2 (conserved matter), we have another
relation, Eq. (46), between the parameters b and n, without such
a simple connection. As a consequence, in principle it is
possible that an unstable model in k-essence theory may
correspond to a stable model in Rastall’s theory, or vice versa,
since, as shown above, Eq. (22) does not hold, b being now
essentially independent of n up to some possible restrictions
on their range. In fact, in this case, even non-adiabatic
perturbations may appear, due to the coupling between the scalar
field and matter.
5 Some special cases
5.1 n = 1/2
3H = −2F0−1Vφ .
In this case, the k-essence scalar field equation takes the form
Thus if V = const, we have H = 0, hence a = const, and
flat space-time is obtained. One can certainly obtain H = 0
in Rastall gravity under special assumptions, but the question
of k–R duality looks meaningless in this trivial case.
If V = V (φ), the φ field has no dynamics of its own,
but Eq. (59) expresses it in terms of H , and the Friedmann
equations (26) and (27) are meaningful.
The Rastall counterpart in the scalar-vacuum and R1 cases
is then obtained with b = 0, V = 3W (according to (22) and
(21)) and
(60)
6 Examples
In the scalar-vacuum and R1 cases we return to the above
description for n = 1/2.
With conserved matter (case R2), the Rastall-scalar field
takes the form
6.2 Dust and Rastall-R1 models
Suppose that in k-essence theory, in addition to the scalar
field φ, there is pressureless fluid (dust), so that
Let us now consider some specific examples of the
equivalence discussed above, assuming a zero or constant potential
and dust as a possible matter contribution.
6.1 Scalar vacuum
Consider scalar vacuum with zero potential. The k-essence
equations give
where φ1 is a combination of the previous constants, and we
have written w = 1/(2n − 1), thus identifying the k-essence
with a perfect fluid with the EoS p = wρ.
In Rastall’s theory, the dual solution contains the same
a(t ), while the scalar field is given by
ψ˙ ∝ a−3(1+w)/2 = a−3/b ∝ t −1,
where now we should put w = (2 − b)/b. We notice that
while the k-essence scalar field behavior depends on the EoS
parameter w, the Rastall scalar is simply ψ = log t + const.
Addition of a constant potential, equivalent to a
cosmological constant, does not change the scalar field evolution laws
(62) and (66) in terms of a but makes the time dependences
more complex, not to be considered here.
In the presence of matter, as we saw above, the form of
k–R duality depends on how matter couples to the scalar
field.
(62)
(63)
(64)
(65)
(66)
2 ψ˙ 2 = F0φ˙ .
In the case R2 (conserved matter), b remains arbitrary,
but k–R duality still holds. Indeed, if we substitute (60) into
Eq. (44) and use the Friedmann-like equations (32), (33) to
calculate ρ˙ − 3 p˙, we obtain Eq. (59).
The main feature in this case is that nontrivial solutions
with n = 1/2 and k–R duality are only achieved in the
presence of a variable potential.
5.2 b = 3/2
With this value of b, the potential disappears from Rastall’s
gravity, hence the k–R duality implies V = 0, and we
deal with zero potentials. In other respects, the situation is
described as in the general case.
A feature of interest is that with b = 3/2 Eq. (31) leads
to ρ = 3 p. Therefore, in the scalar-vacuum and R1 cases,
the dual k-essence counterpart of this Rastall model contains
matter with ρk = 3 pk (see (41)). Due to (22), in addition,
n = 2, so that the φ field also behaves as radiation.
In the case R2 (conserved matter), Eqs. (41) and (22) are
no more valid, and the general description is applicable.
5.3 b = 2
Equations (54) and (58) give zero values of pressure and
the speed of sound of a scalar field in Rastall’s theory. The
corresponding expression (57) in k-essence theory leads to
n → ∞ according to the general relation (22).
If there is conserved matter (case R2), then (unless this
conserved matter is pure radiation, ρ = 3 p) Eq. (22) is no
more valid, so that the speeds of sound of scalar fields are
different in the two theories. It means that k–R duality does
not exist for perturbations even though it does exist for the
isotropic background.
5.4 b = 0
3H ψ˙ 2 = −3W˙ − 21 (ρ˙ − 3 p˙),
looking like a constraint equation since it contains only the
first-order derivative. However, k–R duality still works, as
before: thus, a substitution of (42) (what is important, with
arbitrary n = 0) and ρ − 3 p from Eqs. (32), (33) into (61)
leads to (28), which is a second-order equation unless n =
1/2.
(61)
pk = 0,
ρk = ρ1a−3,
ρ1 = const.
(67)
Then in the R1 version of Rastall’s theory, according to (41),
we have the conditions
1
2 (3 − b)ρ + 3(b − 1) p
1
2 (b − 1)ρ + (5 − 3b) p
= ρ = ρk ,
= p = 0,
(68)
5 − 3b
ρ = 2(3 − 2b) ρk ,
1 − b 1 − b
p = 5 − 3b ρ = 2(3 − 2b) ρk ,
leading to the following relations for the density and pressure:
both evolving as ρ ∝ p ∝ a−3. Thus in Rastall cosmology
the fluid acquires pressure (except for the GR value b = 1).
The scalar fields in both models satisfy the same relations as
in the vacuum case, valid for any values of n and b such that
n(b − 2) = 1.
Adding a constant potential V = (3 − 2b)W does not
change the relations (68) and (69) and introduces an effective
cosmological constant. We then obtain a three-component
model with matter, a cosmological constant and a scalar field
whose behavior is determined by n or, equivalently, by w =
1/(2n − 1). In the dual Rastall model, we have an effective
pressure even though in the k-essence model pk = 0.
If, on the contrary, we introduce matter with p = 0 in
Rastall’s (R1) theory, then in the dual k-essence model we
have
1
ρk = ρ = 2 (3 − b)ρ ,
1
pk = p = 2 (b − 1)ρ ,
and their evolution law agreeing with Eq. (40) reads
ρk ∝ ρ ∝ a−6/(3−b) = a−3(1+wk ),
where the EoS parameter wk of the fluid in the k-essence
model is
corresponds to the limit n → ∞, which is, however, well
defined. In this way we obtain a ∝ t 2/3 as in the pure dust
model of GR. For the scalar fields it follows in this limit that
(69)
φ˙ = const ⇒ φ ∝ t,
ψ˙ ∝ a−3/2 ∝ 1/t.
(70)
(71)
(72)
(73)
(74)
(75)
(76)
The condition for b is obtained from (73): writing ψ˙ = ψ0/t
and ρ = ρ0/t 2, we arrive at
Thus the value of b is determined by the relative contributions
of matter and the scalar field. Moreover, the speed of sound
of the scalar field now does not follow the adiabatic relation
verified in the R1 case.
A cosmological constant can be easily introduced in the
form of V = (3 − 2b)W = const. The scalar field again
follows the law (73), and the whole configuration reduces to
the CDM model where is given by the constant
potential and the matter component consists of the scalar field
and ordinary matter. All background relations of the CDM
model are preserved in this case, but the degeneracy between
the scalar field and usual matter is broken at the
perturbative level. Due to the fact that the CDM model is subject
to problem at the perturbative level in the non-linear regime
(see, e.g., [11,12]), such a more complex configuration in
k-essence and Rastall models may lead to interesting results,
to be studied in the future.
7 Conclusion
We have studied the conditions of equivalence between the
kessence and Rastall theories of gravity in the presence a scalar
field (k–R duality). These two theories have actually emerged
in very different contexts, the k-essence theory being based
on a generalization of the kinetic term of a scalar field,
suggested by some fundamental theories, while Rastall’s theory
is a non-conservative theory of gravity which can be seen
as a possible phenomenological implementation of quantum
effects in gravitational theories. Such equivalence has been
revealed in the case of static spherically symmetric models
[14,15], and it has been more explicitly stated here for all
cases where the metric and scalar fields essentially depend
on a single coordinate, and the k-essence theory is specified
by a power-law function of the usual kinetic term, to which
a potential term can be added. This generalization covers
diverse static and cosmological models, including all
homogeneous cosmologies.
We have discussed cosmological configurations with
scalar fields and matter in the form of a perfect fluid whose
evolution in Rastall’s theory can follow one of two
possible laws: one (R1) assumes no mixing between matter and
b − 1 n − 1
wk = 3 − b = n + 1
(not to be confused with w = 1/(2n − 1) characterizing the
φ field behavior); as before, the relation n(2 − b) = 1 holds.
The model thus obtained is quite different from the one with
dust introduced in k-essence theory.
6.3 Dust and Rastall-R2 models
Let us again assume Eq. (67) but now consider version R2
of Rastall’s theory, so that now ρ ∝ a−3 and
Combining this with the relation ψ˙ 2 = n F0φ˙ 2n and the
field equation (28) with Vφ = 0, we find that this situation
the scalar field, each of them separately obeying the
nonconservation law (
15
), and the other (R2) ascribes the whole
non-conservation to the scalar field while matter is
conservative (∇ν Tμν [m] = 0). Let us summarize the main results
obtained in this context:
1. k–R duality has been established for version R1 of
Rastall’s theory with an arbitrary EoS of matter. It has
been found that the EoS of matter is different in the
mutually dual k-essence and Rastall models; however, it is
argued that the respective speeds of sound are the same.
Since the speeds of sound characterizing the scalar fields
(φ in k-essence theory and ψ in Rastall’s) also coincide,
we conclude that k–R duality is maintained not only for
the cosmological backgrounds but also for adiabatic
perturbations.
2. For version R2 of Rastall’s theory, it has been found that
k–R duality exists only with fluids having the EoS p =
wρ , w = const, which is the same for k-essence and
Rastall models. Moreover, in the k-essence model the
scalar field obeys the same effective EoS. However, on
the perturbative level the mutually dual models behave,
in general, differently.
3. Some special cases have been discussed, showing how
there emerge some restrictions on the free parameters of
each theory.
4. An example has been considered in which a cosmological
model completely equivalent to the CDM model of GR
is obtained at the background level, but different features
must appear at the perturbative level.
The equivalence between the two theories discussed here
is somewhat surprising because of their basically different
origin. A curious aspect is that the k-essence theory has a
Lagrangian formulation unlike Rastall’s theory. It is
possible that the equivalence studied here may lead to a restricted
Lagrangian formulation of Rastall’s theory in the
minisuperspace in terms of metric functions depending on a single
variable. If this is true, it might suggest how to recover a
complete Lagrangian formulation for Rastall’s theory in a more
general framework.
Acknowledgements We thank CNPq (Brazil) and FAPES (Brazil) for
partial financial support. KB thanks his colleagues from UFES for kind
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