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Vector-field model with compensated cosmological constant and radiation-dominated FRW phase

A special model of a massless vector-field is presented, which has an extra modified-gravity-type interaction term in the action. The cosmology of the model is studied with standard noninteracting relativistic matter added. It is found that this cosmology can have an early phase where the vector-field starts to compensate a (Planck-scale) cosmological constant and a late Friedmann-Robertson-Walker (FRW) phase where the relativistic-matter energy density dominates the dynamic vacuum energy density.


Introduction
The cosmological constant problem 1,2 is perhaps the most important outstanding question of modern physics. In a nutshell, the problem is to explain why the energy scale corresponding to the measured value of the cosmological constant Λ is negligible compared to the energy scales of elementary particle physics, The cosmological constant Λ can be canceled dynamically and without finetuning in a particular massless-vector-field model. 3,4 But this model runs into two obstacles. First, the local Newtonian dynamics is ruined. 5 Second, the final phase of the model universe expands so rapidly (scale factor a ∝ t) that any standard matter contribution (ρ M ∝ 1/a p with p = 4 for relativistic matter or p = 3 for nonrelativistic matter) becomes irrelevant compared to the decreasing remnant vacuum energy density (ρ V ∝ 1/t 2 ).
Following previous work on the q-theory approach 6,7,8,9,10,11,12 to the main cosmological constant problem, an extended model with two massless vector-fields has been found, 13,14 which evades the first obstacle regarding the local Newtonian dynamics. Here, we present another model with a single vector-field and a modifiedgravity-type interaction term, which circumvents the second obstacle by having a final phase with slower expansion (a ∝ t 1/2 ).

Model and Ansatz
Consider the following model of a massless vector-field A α (x) with an effective action ( = c = 1) where R is the Ricci scalar, Λ the effective cosmological constant, and f (R) a function to be specified later. For the moment, the contribution of other matter fields is set to zero, L M = 0. The q-theory-type scalar 6,7,8 entering the above action is defined as follows: with the spacetime coordinate (x) = (x 1 , x 2 , x 3 , t) as argument and the semicolon standing for covariant derivation. The particular contraction (1b) was mentioned in Vector-field model with compensated Λ and FRW phase 3 Ref. 4 but not studied in detail.
In the following, we consider a spatially-flat Friedmann-Robertson-Walker (FRW) universe with scale factor a(t) for cosmic time t and take the isotropic Ansatz 3,4 for the vector-field: The Ansätze (2) give where the overdot stands for differentiation with respect to the cosmic time t and the Hubble parameter H(t) is defined by H(t) ≡ȧ(t)/a(t).
Note that the variable ( for a brief introduction to the basic ideas of q-theory. This introduction to q-theory may give the reader some background information but is not needed to understand the present article, which is entirely self-contained.

Vector-field equation
The variational principle applied to the vector-field of action (1a) gives the following equation: The Ansätze (2) reduce this field equation to a single ordinary differential equation with ζ ≡ (2 Q 3 ) −1 dǫ/dQ 3 .

Generalized FRW equations
The energy-momentum tensor of the vector-field A α follows from the variation of the action with respect to the metric field g αβ (x). The result is where the round brackets around spacetime indices denote symmetrization and the prime on f stands for differentiation with respect to R. With the vector-field equation (4), the energy-momentum tensor (6) becomes From the gravitational field equations, the Ansätze (2), and the energymomentum tensor (7), the generalized FRW equations are having added a further matter contribution from a nonzero term L M in the ac- the corresponding homogeneous perfect fluid. The vector-field contributions on the right-hand sides of (8) are given by The special model considered in this article has a linear function f (R) in the interaction term of the action (1a), and a quadratic vector-field function ǫ(Q 3 ), for nonzero dimensionless coefficients κ and ζ 0 . In order to be able to cancel a nonzero cosmological constant, these two coefficients must obey the condition in terms of the sign function sgn(x) ≡ x/|x| for x = 0 and sgn(0) ≡ 0. The particular form of condition (10c) will be derived at the end of this subsection. a The Ansatz (10a) reduces the energy density and the isotropic pressure from (9) to the following expressions: Similarly, the Ansatz (10b) reduces the ODE (5) to the following equation: Inserting the function (10a) in the last ODE and using (3a) gives Equation (13) is the core result of this article, as will become clear in Sec. 4.2.
For t → ∞, the asymptotic solution of (13) is given by a Condition (10c) is purely technical and appears because of the simple ǫ-function used in (10b).
For an appropriate, more complicated, ǫ-function (cf. Refs. 6,7,8,13), the cancelation of the cosmological constant Λ can be expected to hold for any sign of Λ.
with a constant θ to be determined by the solution of the combined field equations (see Sec. 4.3). Substituting this asymptotic solution into the energy density ρ(A) and the pressure P (A) from (11), we obtain having added the contributions from the cosmological constant Λ in the action (1a).

Heuristic argument
Returning to the vector-field ODE (13), we can give the following heuristic argument for the appearance of the FRW-like asymptotic solution (14), even with a nonzero cosmological constant Λ present in the action (1a). The physically relevant case has both coefficients κ and ζ 0 nonvanishing, but, for completeness, also the other cases will be briefly mentioned. For κ = 0 and ζ 0 = 0, (13) implies A 0 = 0 (i.e., θ = 0) for generic H(t) and the cosmological constant can no longer be canceled, as exemplified by (15).
For κ = ζ 0 = 0 in the Ansätze (10), the vector-fields have disappeared from the action (1a) altogether and the cosmological constant problem remains unsolved.
Returning to the case of κ = 0 and ζ 0 = 0, the above heuristic argument for obtaining a radiation-dominated FRW universe, thus, relies on having an action employing the Q 3 field [which gives the 3 HȦ 0 + 3Ḣ A 0 combination in (13)] and the Ricci scalar (which vanishes precisely for H = 1 2 t −1 ). An entirely different question is whether or not solution (14) appears dynamically as an attractor. Here, this question will be addressed numerically, leaving a proper mathematical treatment for the future (see Note Added).

Additional relativistic matter and dimensionless ODEs
From now on, we also consider an additional standard-matter component with energy density ρ M and pressure P M . These and other dimensional variables can be replaced by the following dimensionless variables: if appropriate powers of the (Planck-type) energy scale (8π G) −1/2 are used, without further numerical factors. Henceforth, an overdot stands for differentiation with respect to τ , for example, h(τ ) ≡ȧ(τ )/a(τ ).
The previous results (8b) and (13), together with the extra standard-matter component, give the following dimensionless ODEs: v + 3 hv + 3ḣ + 6 (κ/ζ 0 )(ḣ + 2 Observe that the right-hand side of (18), evaluated for the asymptotic solution (14), has canceling O(t 0 ) terms for one particular value of θ 2 , provided condition (10c) holds. Specifically, the two possible θ values are given by in terms of the dimensionless cosmological constant λ ≡ (8π G) 2 Λ. The actual sign of θ will be determined by the initial boundary conditions.

Numerical results
Numerical calculations have been performed for the model defined by (1) and (10) With special boundary conditions to ensure the exact vanishing of the vectorfield, Fig. 1 shows that the cosmological constant λ rapidly dominates the matter component and that exponential expansion sets in. Asymptotically, this model universe approaches de Sitter space havingḣ = r M = 0. The behavior of h(τ ) and r M (τ ) shown in Fig. 1 is completely standard and can also be obtained analytically.
With boundary conditions of the vector-field in an appropriate domain, Fig. 2 shows that the vector-field is able to compensate a Planck-scale cosmological constant λ = O(1) and that Minkowski spacetime is approached asymptotically, h(τ ) → 0 for τ → ∞. The numerical solution v(τ ) asymptotically takes the form (14), with the precise linear coefficient needed for the complete compensation of the  The final phase shown in Fig. 2 corresponds to an FRW universe (h ∼ 1 2 τ −1 ) dominated by relativistic matter. The solid curve in the right panel of Fig. 2 corresponds to an asymptotic universe with r M ∼ 1.09 τ −2 and r V ≡ (3 h 2 − r M ) ∼ −0.34 τ −2 (hence, ratio r M /|r V | ∼ 3) and the dashed curve to an asymptotic universe with r M ∼ 0.687 τ −2 and r V ∼ 0.063 τ −2 (hence, ratio r M /r V ∼ 11). For these numerical solutions, the action density term R A 2 0 ∝ −6 (ḣ + 2 h 2 ) v 2 has also been found to drop to zero faster than τ −1 .

Discussion
The vector-field model defined by (1) and (10)  In this way, there is a more or less realistic physical description of the earliest cosmological phase as a radiation-dominated FRW universe with a dynamically canceled cosmological constant. Possibly, this description needs to be augmented with the effects from inflation. 14 As discussed in Refs. 9, 10, 11, quantum-dissipative processes can be expected to lead to a further (exponential) reduction of |ρ V (t)| in the very early universe (kT ≫ b Most likely, the model can be extended to allow for a similar compensation of a cosmological constant Λ of arbitrary sign. 6,7,8,13 For the case of positive Λ, also a final cosmological phase with H(t) = (2/3) t −1 can be obtained by replacing ǫ(Q 3 ) in (1a) with ǫ(Q 1 , Q 2 ) = ζ 1 (Q 1 ) 2 +ζ 2 (Q 2 ) 2 , for the contractions (Q 1 ) 2 ≡ A α;β A α;β and (Q 2 ) 2 ≡ A α;β A β;α , and by taking the coefficient κ = −(ζ 1 + ζ 2 )/2 in (10a). This type of model may be relevant to inflationary effects in vectorfield theories with a dynamically canceled cosmological constant. 14 At this point, it should be mentioned that certain scalar-tensor theories (with a fundamental scalar field φ) have also been argued to give both a compensation of the cosmological constant and a final FRW-like phase. 15