Present accelerated expansion of the universe from new Weyl-Integrable gravity approach

We investigate if a recently introduced formulation of general relativity on a Weyl-integrable geometry, contains cosmological solutions exhibiting acceleration in the present cosmic expansion. We derive the general conditions to have acceleration in the expansion of the universe and obtain a particular solution for the Weyl scalar field describing a cosmological model for the present time in concordance with the data combination Planck + WP + BAO + SN.


I. INTRODUCTION
The present accelerated expansion of the universe has become an interesting topic, not just for the absence of a fully satisfactory explanation of its origin, but for the wide range of gravity theories introduced in the quest for viable answers to the problem. Many have been the attempts to construct viable models. This issue has been address basically in two approaches. In one of them the acceleration is generated by an extra material component in the universe, the dark energy (in which is included the cosmological constant). In the other the acceleration is a purely gravitational effect within the framework of gravitation theories alternative to the Einstein's general relativity. Modifications to Einsitein's gravitational theory going back to the works of Eddington and Schrödinger [1]. Physical scalar fields have been included in some dark energy models, but unfortunately not always with a solid motivation for its introduction beyond to justify the acceleration in the expansion of the universe. However, none of these is free of problems, as for instead some quintessence fields [2]. The majority of quintessence models have been proposed in the light of general relativity. However, within the second approach, the Weyl-Integrable geometry provides a solid tool to incorporate a scalar field as a part of the geometry of the spacetime. A consequence of this is that the scalar field can also describe the gravitational field (in addition to the metric tensor g µν ), in a new version of scalar-tensor theory [3,4].
The Brans-Dicke (BD) theory of gravity is the simplest scalar-tensor theory considered as an alternative to general relativity, in which gravity is described by both a tensor field g µν and a scalar field Φ . In this theory the scalar field, which is not of geometric nature, is not a matter field, instead it determines the inverse of the gravitational coupling parameter and in this sense it is part of gravity. This is the reason why the BD scalar field does not appear in the geodesic equation for both massive and massless particles [5,6]. This aspect may be considered an inconvenient in the sense that even when Φ is a part of the gravitational field, it does not appears as a part of the geometry, which in this case is the Riemannian one.
A more congruent gravitational theory of the BD type would be one in which the BD scalar field would play an active role in the dynamical field equations of the theory, as same as on its geometrical structure, describing together with the metric tensor g µν , the gravitational field. Recently C. Romero and collaborators, by means of the Palatini variational method, found that the scalar field of a BD theory of gravity can turn the space time geometry (assumed Riemannian in DB theories) into a Weyl-Integrable one, generating in this manner a scalar-tensor theory that differs from the original BD one [7]. They also found that the Weyl-Integrable geometry contains a Riemannian structure which allows to formulate general relativity on Weyl-Integrable manifolds [4].
A similar result has been obtained in [8]. In this letter it is postulated a conformal equivalence principle which mathematically is associated with the conformal invariance of the field equations of the regarded theory of gravitation. The conformal equivalence principle is formulated as follows: the laws of gravity look the same, no matter which one of the different conformally related frames is chosen to describe them. Under this point of view is analized an alternative interpretation of conformal transformations of the metric, establishing that they can be viewed as a mapping between Riemann and Weyl-Integrable geometries. Thus, when the conformal equivalence principle is assumed to be valid, the transformations relate complementary geometrical pictures of the same physical reality. As an example, it is shown that in order to have a BD theory of gravity in concordance with the conformal equivalence principle, the background geometry of the BD theory must be the Weyl-Integrable geometry.
Thus, in view of the preview results, the Weyl-Integrable geometry as gravity theory has several interesting aspects to address the problem of the acceleration in the cosmic expansion. In this letter we investigate if the formulation of the general theory of relativity on Weyl-Integrable geometry, contains cosmological solutions compatible with an accelerated expansion of the universe, without the introduction of a dark energy component. For this purpose the letter is organized as follows. In Sect. II we give a review of the proposal of C. Romero and Collaborators to formulate a general theory of relativity on a Weyl-Integrable geometry [4]. In Sect. III we establish the dynamical field equations of the gravity model on cosmological scales. In Sect. IV we derive the general conditions under which the cosmological solutions of the theory exhibit accelerated expansion on cosmological scales, and found a particular solution compatible with the data combination Planck+WP+BAO+SN [9]. Finally, in Sect. V we develope some final comments.

II. A NEW APPROACH OF WEYL-INTEGRABLE GEOMETRY
As it is well-known the Weyl geometry is the simplest extension of Riemann geometry. The general theory of relativity is formulated on the base of Riemannian geometry. Unlike Riemannian geometry, in Weyl geometry the nonmetricity condition has different form. In a coordinate chart the Weyl nonmetricity condition reads [4,8] where (w) ∇ α is the Weyl covariant derivative, σ α is a 1form field also known as gauge vector field, and g µν are the covariant components of the tensor metric. It can be easily shown that the nonmetricity condition (1) is invariant under the Weyl rescaling transformations being Ω(x) a non-vanishing differentiable function. In Weyl geometry the affine connection is assumed torsionless and hence the condition (1) yields where α µ ν denotes the Christoffel symbols.
The presence of σ µ in (1) has its consequences when paralell transport of vectors is implemented. One of them is that the length of a vector: l 2 = g αβ l α l β varies point to point even in a closed path: l = l 0 exp dx µ σ µ /2. This effect is known as " the second clock effect" which basically consists in the broadening of the atomic spectral lines of the electrons immersed in the σ α field. This second clock effect is unobserved as it was pointed out by Einstein, and thus the Weyl geometry was considered not viable [10].
Subsequently, Weyl proposed a particular subclass of his geometry known as Weyl-Integrable (WI) geometry, that does not suffered from the second clock effect. This achievement was due to the fact that Weyl expressed the 1-form field as the gradient of a scalar field: σ µ = ∂ µ ϕ. This scalar field is geometrical in nature and is known as the Weyl scalar field. In WI geometry the nonmetricity condition (1) reads while the torsionless affine connection (3) leads to which is the WI connection. Along a closed path the Stokes's theorem ensures that and thus the length of a vector is preserved when it is parallel transported along a closed path. However, in the case of an open path the length of a vector continue varying point to point. This variation is due to the variation of the scalar product of two vectors g(u(λ), v(λ)) when they are parallel transported along a path C characterized by the parameter λ. If λ 0 corresponds to a point a and λ corresponds to a different point b, the scalar product is given by This expression is interpreted as the Weyl scalar field is responsible for the non invariance of the scalar product along an opened path, and this is the interpretation usually found in the literature.
However, recently C. Romero and collaborators have shown that we can have a novel interpretation of (7), in which a Riemannian structure can be recovered into the Weyl-Integrable geometry. To show it, they rewrite the equation (7) in the form e ϕ(x(λ)) g(v(λ), u(λ)) = e ϕ(x(λ0)) g(v(λ 0 ), u(λ 0 )). (8) This equation can be interpreted as there is an isometry between the tangent spaces of the manifold (spacetime) at the points a = C(λ 0 ) and b = C(λ) only in the effective metricĝ µν = e ϕ g µν . It is easy to see that with this effective metric the nonmetricity condition (4) becomes which corresponds to a Riemannian nonmetricity condition. Moreover, asĝ µν = e ϕ g µν is invariant under Weyl rescaling transformations, then any geometrical object constructed only withĝ µν is also invariant. The isometry acts as a correspondence between a Weyl-frame (M, g, ϕ) and a unique Riemannian frame (M,ĝ = e ϕ g, 0). In this sense the isometry in (8) implies that a new kind of invariance can be established and the same physical phenomena may appear in different representations. For example, the present accelerated expansion of the universe, may be address on both frames, however a possible advantage is that in a Weyl-frame we can find a more satisfactory solution to the problem, taking into account that in WI gravity the gravitational field has an scalar-tensor nature, whereas in the Riemann frame gravity is just described by a tensor field. Thus, it results convenient to study the possibility that the present acceleration in the expansion of the universe could be explained simply as an effect of gravity, without the need of any dark energy. The answer will be investigated along the next sections.

III. THE FIELD EQUATIONS ON COSMOLOGICAL SCALES
Let us start considering the simplest action for a Weyl-Integrable (WI) gravity [4,8] (w) being (w) R the Ricci scalar calculated with the WI connection, κ is a coupling constant and g is the determinant of the metric g αβ . This action respects the conformal equivalence principle. When a Palatini variational principle is applied to this action, the appropiate background geometry results to be the Weyl-Integrable geometry [8]. The action (10) must be invariant under the Weyl-Integrable rescaling transformations:ḡ αβ = e f g αβ , ϕ = ϕ − f , with f being a coordinate depending smooth function. The Riemann frame can be obtained simply when we make the particular choose f = ϕ in the WI rescaling transformations i.e. with this election we pass from an arbitrary Weyl frame (M, g, ϕ) to the Riemann frame (M,ḡ = e ϕ g,φ = 0). In the Riemann frame the action (10) becomes the usual action for the standard theory of general relativity, where the Weyl scalar fieldφ becomes null. The sources of matter are described by the Lagrangian density L m which is considered independent of the Weyl scalar field. Employing the expression and avoiding divergence terms, the field equations derived from the action (10) can be written as where T = g µν T µν is the trace of the energy-momentum tensor, G µν = R µν − (1/2)Rg µν is the Einstein tensor, ∇ α is the covariant derivative calculated with the Christoffel symbols and = g αβ ∇ α ∇ β is the D'Alambertian operator. It is important to stress that in WI gravity the gravitational field is given by the pair (g µν , ϕ) and thus ϕ is a geometrical scalar field that describes gravity, so it is not a matter field.
In order to consider some cosmological implications from this theory, we introduce the Friedmann-Robertson-Walker (FRW) line element being dS 2 3 = δ ij dx i dx j the spatial 3D Minkowskian line element and a(t) is the cosmological scale factor. For observational reasons we will only consider models with 3D spatially flat curvature.
Thus, regarding a perfect fluid with total energy density ρ T and a total pressure p T , for the metric in (14) the field equations (12) yield where H =ȧ/a is the Hubble parameter and the dot is denoting derivative with respect the cosmic time t.
Similarly, the equation (13) now reads which is not an independent equation. Now, in order to implement the cosmological principle on large scales, let us assume that the Weyl scalar field has two contributions, one on cosmological scales and another one on smaller scales. Thus we use the separation formula where φ(t) is the part on cosmological scales that satisfies the cosmological principle and δϕ(t, x i ) is valid on smaller scales. Once we assume (19), it results natural consider also two contributions for the total energy density and the total pressure in the form where ρ(t) = ρ m (t) + ρ r (t) and p(t) = p m (t) + p r (t), being ρ m and ρ r the energy density for matter (baryonic matter and dark matter) and the energy density for radiation, while p m and p r denote the pressure for matter and radiation, respectively.
Under these conditions the equations (15) and (16) read on cosmological scales as 2ä a + H 2 +φ + 4Hφ + 1 where we have regarded that on cosmological scales the condition |δϕ| ≪ |φ| holds, thus given account of the fact that on cosmological scales the astrophysical contributions to the Weyl scalar field are subdominant. On smaller scales (astrophysical scales mainly) the system (15)-(16) yield Notice that the dynamics of the Weyl scalar field on astrophysical scales is depending of its cosmological solution part. The dynamical equations (23)-(24) can be simplified if we assume on astrophysical scales valid the opposite condition: |δϕ| ≫ |φ|, however, this analysis goes out of the scope of this letter.

IV. COSMOLOGICAL SOLUTIONS EXHIBING ACCELERATED EXPANSION
Now we are in position to investigate if there exist solutions of the new Weyl-Integrable gravitational approach, capable to describe the present period of accelerated expansion of the universe. In order to do so, let us to derive the general conditions for accelerated expansion solutions.
The cosmological dynamics is described by the equations (21) and (22). Hence, after straightforward calculations they yield It can be easily seen from this equation that to achievë a > 0 solutions, the condition must holds. For barotropic equations of state for matter and radiation: p m = 0, p r = 1/3ρ r , the condition (26) becomes where Ω m and Ω r are the density parameter for matter and radiation, respectively.
As we have mentioned the Weyl scalar field is geometrical in origin and can not be considered as another component of the cosmic fluid. It is in fact part of the gravitational field. However, in order to have a comparation with observational data we can assume, just for this purpose, that the Weyl scalar field describes a scalar field fluid characterized by an energy density ρ φ and a pressure p φ . If it is the case, it follows from the dynamical equations (21) and (22) that ρ φ = − same physical reality. The main difference is that in a Weyl frame the acceleration in the expansion is just a pure gravitational effect, whereas in the Riemann frame the dark energy component necessarily leads to the question about the origin and dynamics of this exotic component, which is a problem in the majority of the cosmological models.