The Newton constant and gravitational waves in some vector field adjusting mechanisms

TL;DR

This paper investigates how certain Lorentz-breaking vector field models affect gravitational wave propagation and the Newton constant, confirming previous claims and proposing new models with similar desirable properties.

Contribution

The paper explicitly verifies the behavior of the Newton constant and gravitational waves in specific Lorentz-breaking models, and introduces new models with similar features.

Findings

Newton constant approaches correct asymptotic behavior

Gravitational waves remain well-behaved at large times

New models with similar adjustment mechanisms are proposed

Abstract

As is well known, there exist some Lorentz breaking scenarios which explain the smallness of the cosmological constant in the present era. An important aspect to analyze is the propagation of gravitational waves and the screening or enhancement of the Newton constant $G_N$ in these models. The problem is that the Lorentz symmetry breaking terms may induce an unacceptable value of the Newton constant $G_N$ or introduce longitudinal modes in the gravitational waves. Furthermore there may spoil the standard dispersion relation $\omega=ck$. In [21] the authors have presented a model for which they suggest that the behavior of the gravitational constant is the correct one for asymptotic times. In the present work, an explicit checking is made and we finally agree with these claims. Furthermore, it is suggested that the gravitational waves are also well behaved for large times. In the process, some new models with the same behavior are obtained, thus enlarging the list of possible adjustment mechanisms.

Equations

$$L=ϵ(Q_A,Q_B)+L_{EH},Q_A=\sqrt{A_{\mu ;\nu }{A}^{\mu ;\nu }},Q_B=\sqrt{B_{\mu ;\nu }{B}^{\mu ;\nu }}.$$

$${\nabla }^{\alpha }({\xi }_A{\nabla }_{\alpha }{A}_{\beta })=0,{\nabla }^{\alpha }({\xi }_B{\nabla }_{\alpha }{B}_{\beta })=0.$$

$${\xi }_A=\frac{1}{2Q_A}\frac{dϵ}{d{Q}_A},{\xi }_B=\frac{1}{2Q_B}\frac{dϵ}{d{Q}_B},$$

$$d{s}^2=d{t}^2-a^2(t)(d{x}^2+d{y}^2+d{z}^2).$$

$${\Gamma }_{ij}^0={\delta }_{ij}a\dot{a},{\Gamma }_{0j}^i={\delta }_{ij}\frac{\dot{a}}{a}.$$

$$A_{0;0}={\dot{A}}_0,A_{i;j}=-{\delta }_{ij}a\dot{a}{A}_0,A^{0;0}={\dot{A}}_0,A^{i;j}=-{\delta }_{ij}\frac{\dot{a}}{a^3}{A}_0.$$

$${\dot{\dot{A}}}_0+(3H+\frac{{\dot{\xi }}_A}{{\xi }_A}){\dot{A}}_0-3H^2{A}_0=0,$$

$${\dot{\dot{B}}}_0+(3H+\frac{{\dot{\xi }}_B}{{\xi }_B}){\dot{B}}_0-3H^2{B}_0=0.$$

$$T_{\alpha \beta }=ϵ(Q_A,Q_B)g_{\alpha \beta }-2{\xi }_A(A_{\alpha ;\gamma }{A}_{\beta }^{;\gamma }+A_{\gamma ;\alpha }{A}_{;\beta }^{\gamma })-2{\xi }_B(B_{\alpha ;\gamma }{B}_{\beta }^{;\gamma }+B_{\gamma ;\alpha }{B}_{;\beta }^{\gamma })$$

$$+{\nabla }^{\gamma }[{\xi }_A(A_{\alpha }{A}_{\gamma ;\beta }+A_{\beta }{A}_{\gamma ;\alpha }+A_{\alpha }{A}_{\beta ;\gamma }+A_{\beta }{A}_{\alpha ;\gamma }-A_{\gamma }{A}_{\alpha ;\beta }-A_{\gamma }{A}_{\beta ;\alpha })].$$

$$+{\nabla }^{\gamma }[{\xi }_B(B_{\alpha }{B}_{\gamma ;\beta }+B_{\beta }{B}_{\gamma ;\alpha }+B_{\alpha }{B}_{\beta ;\gamma }+B_{\beta }{B}_{\alpha ;\gamma }-B_{\gamma }{B}_{\alpha ;\beta }-B_{\gamma }{B}_{\beta ;\alpha })].$$

$$\rho (A,B)=T_0^0,-P(A,B)=T_1^1=T_2^2=T_3^3,$$

$$\rho =ϵ(Q_A,Q_B)-Q_A\frac{dϵ}{d{Q}_A}-Q_B\frac{dϵ}{d{Q}_B},$$

$$P=-\rho+\frac{d}{dt}\bigg{(}\frac{HA_{0}^{2}}{Q_{A}}\frac{d\epsilon}{dQ_{A}}\bigg{)}+\frac{d}{dt}\bigg{(}\frac{HB_{0}^{2}}{Q_{B}}\frac{d\epsilon}{dQ_{B}}\bigg{)}-\frac{\dot{A}_{0}^{2}}{Q_{A}}\frac{d\epsilon}{dQ_{A}}-\frac{\dot{B}_{0}^{2}}{Q_{B}}\frac{d\epsilon}{dQ_{B}}.$$

$$Q_A^2={\dot{A}}_0^2+3H^2{A}_0^2,Q_B^2={\dot{B}}_0^2+3H^2{B}_0^2,$$

$$H^2=8\pi G_N(\Lambda +\rho (A,B)),$$

$$2\dot{H}+3H^2=8\pi G_N(\Lambda -P(A,B)).$$

$$\delta {\Gamma }_{\mu \nu }^{\alpha }=\frac{g^{\alpha \kappa }}{2}({\nabla }_{\mu }\delta g_{\kappa \nu }+{\nabla }_{\nu }\delta g_{\kappa \mu }-{\nabla }_{\kappa }\delta g_{\mu \nu }).$$

$$\delta L=\delta L_A+\delta L_B+\delta L_{AB}=\frac{1}{2}\frac{d^2ϵ}{d{Q}_A^2}\delta Q_A^2+\frac{1}{2}\frac{d^2ϵ}{d{Q}_B^2}\delta Q_B^2+\frac{d^2ϵ}{d{Q}_{B}d{Q}_A}\delta Q_A\delta Q_B.$$

$$\frac{d^2ϵ}{d{Q}_A^2}=Q_A\frac{d}{d{Q}_A}(\frac{1}{Q_A}\frac{dϵ}{d{Q}_A})+\frac{1}{Q_A}\frac{dϵ}{d{Q}_A}.$$

$$\delta Q_A=\frac{1}{2Q_A}\delta (A_{\alpha ;\beta }{A}^{\alpha ;\beta })=\frac{1}{2Q_A}(A_{\alpha ;\beta }\delta A^{\alpha ;\beta }+A^{\alpha ;\beta }\delta A_{\alpha ;\beta })=\frac{1}{Q_A}{A}_{\alpha ;\beta }\delta A^{\alpha ;\beta }.$$

$$\delta A_{\alpha ;\beta }=-\delta {\Gamma }_{\alpha \beta }^{\gamma }{A}_{\gamma }+(\delta A_{\alpha }{)}_{;\beta }.$$

$$\delta Q_A^2=\frac{1}{Q_A^2}{A}^{\gamma ;\delta }{A}^{\alpha ;\beta }[(\delta A_{\gamma }{)}_{;\delta }(\delta A_{\alpha }{)}_{;\beta }-2\delta {\Gamma }_{\gamma \delta }^{ϵ}{A}_{ϵ}(\delta A_{\alpha }{)}_{;\beta }+\delta {\Gamma }_{\alpha \beta }^{ϵ}\delta {\Gamma }_{\gamma \delta }^{\eta }{A}_{ϵ}{A}_{\eta }].$$

$$\delta L_{A}=\bigg{[}Q_{A}\frac{d}{dQ_{A}}(\frac{1}{Q_{A}}\frac{d\epsilon}{dQ_{A}})+\frac{1}{Q_{A}}\frac{d\epsilon}{dQ_{A}}\bigg{]}\frac{1}{2Q_{A}^{2}}A^{\gamma;\delta}A^{\alpha;\beta}$$

$$\times [(\delta A_{\gamma }{)}_{;\delta }(\delta A_{\alpha }{)}_{;\beta }-2\delta {\Gamma }_{\gamma \delta }^{ϵ}{A}_{ϵ}(\delta A_{\alpha }{)}_{;\beta }+\delta {\Gamma }_{\alpha \beta }^{ϵ}\delta {\Gamma }_{\gamma \delta }^{\eta }{A}_{ϵ}{A}_{\eta }]$$

$$=\frac{1}{2Q_{A}}\bigg{[}\frac{d}{dQ_{A}}(\frac{1}{Q_{A}}\frac{d\epsilon}{dQ_{A}})A^{\gamma;\delta}A^{\alpha;\beta}+\frac{d\epsilon}{dQ_{A}}g^{\alpha\gamma}g^{\beta\delta}\bigg{]}$$

$$\times [(\delta A_{\gamma }{)}_{;\delta }(\delta A_{\alpha }{)}_{;\beta }-2\delta {\Gamma }_{\gamma \delta }^{ϵ}{A}_{ϵ}(\delta A_{\alpha }{)}_{;\beta }+\delta {\Gamma }_{\alpha \beta }^{ϵ}\delta {\Gamma }_{\gamma \delta }^{\eta }{A}_{ϵ}{A}_{\eta }].$$

$$\delta Q_A\delta Q_B=\frac{1}{Q_A{Q}_B}{A}_{\mu ;\nu }{B}^{\gamma ;\delta }\delta A^{\mu ;\nu }\delta B_{\gamma ;\delta }=\frac{1}{Q_A{Q}_B}{A}_{\mu ;\nu }{B}^{\gamma ;\delta }{g}^{\mu \alpha }{g}^{\nu \beta }\delta A_{\alpha ;\beta }\delta B_{\gamma ;\delta }$$

$$=\frac{1}{Q_A{Q}_B}{A}^{\alpha ;\beta }{B}^{\gamma ;\delta }[(\delta B_{\gamma }{)}_{;\delta }(\delta A_{\alpha }{)}_{;\beta }-\delta {\Gamma }_{\alpha \beta }^{ϵ}{A}_{ϵ}(\delta B_{\gamma }{)}_{;\delta }-\delta {\Gamma }_{\gamma \delta }^{\eta }{B}_{\eta }\delta A_{\alpha ;\beta }+\delta {\Gamma }_{\alpha \beta }^{ϵ}\delta {\Gamma }_{\gamma \delta }^{\eta }{A}_{ϵ}{B}_{\eta }].$$

$$\delta L_{AB}=\frac{1}{Q_A{Q}_B}\frac{d^2ϵ}{d{Q}_{B}d{Q}_A}{A}^{\alpha ;\beta }{B}^{\gamma ;\delta }[(\delta B_{\gamma }{)}_{;\delta }(\delta A_{\alpha }{)}_{;\beta }-\delta {\Gamma }_{\alpha \beta }^{ϵ}{A}_{ϵ}(\delta B_{\gamma }{)}_{;\delta }-\delta {\Gamma }_{\gamma \delta }^{\eta }{B}_{\eta }\delta A_{\alpha ;\beta }+\delta {\Gamma }_{\alpha \beta }^{ϵ}\delta {\Gamma }_{\gamma \delta }^{\eta }{A}_{ϵ}{B}_{\eta }].$$

Full text

1. Introduction

One of the most interesting observations of the last century is the discovery of the cosmic acceleration. As gravity is an attractive force, the velocity of the distant galaxies may be expected to slow down. Contrary to this, the astronomical observations support an increasing velocity [1]-[3]. Another crucial phenomenon is the discrepancy between the luminous matter of several objects in the universe and their gravitational effects [4]-[5]. In fact, there is experimental evidence supporting a flat universe, which implies that the actual universe energy density should be of the order of the critical one, $\rho_{c}\sim 10^{-47}$GeV4 [6]. This scenario does not agree with the contributions corresponding to the dynamically measured non relativistic mass density, which is approximately $(0.1-0.3)\rho_{c}$.

Several scenarios have been proposed to explain these results. Some of them postulate the existence of dark matter. This is an unknown matter sector whose contribution to the energy density compensates the difference between the critical and the observed densities [4]-[5]. Furthermore, the acceleration of the universe expansion suggests the presence of a cosmological constant [1]-[3]. If this were to be interpreted as vacuum energy density, then its value would be a considerable fraction of the critical density $\rho_{c}$.

This picture has several theoretical problems. As is well known, the predictions of different Quantum Field Theories of the vacuum energy density are at least 55 orders of magnitude larger than $\rho_{c}$ [17]. Thus, there exist two problems to be explained namely, why the energy density of the universe is so small and why it is so close to the critical one $\rho_{c}$.

One of the approaches for solving the first problem is to assume the existence of an unknown matter component whose evolution screens the QFT energy density at late times. The first scenarios of this type include an scalar component $\varphi$ non minimally coupled to the curvature $R$ of the background space time [7]. The energy density $\rho(\varphi)$ of this field has a sign opposite to the critical energy $\rho_{c}$, in such a way that at larges times an screening of the cosmological constant takes place. However, one of the problems that arise is that the evolution also changes the Newton constant $G_{N}$ to an unacceptable numerical value [7]. There exist a no go theorem due to Weinberg which discourages the possibility of solving the cosmological constant problems in terms of scalar fields [8]. However, this theorem is formulated under certain assumptions which may be avoided in some modified gravity theories. Some examples of adjustment mechanisms in terms of scalar fields in modified gravity theories is given in [9].

There exist alternative scenarios which assume that the QFT vacuum energy is not gravitating. Their underlying idea is that, if GR were the full gravity theory, then the graviton would interact with this vacuum energy. However, if the gravity description available is only an effective one, then the gravitons may be quasi-particles and do not necessarily experience all the degrees of freedom. Quintessence models are examples of these type of scenarios [10]. The cosmological constant in this case is modeled by the potential energy of an unknown scalar field.

Another type of models are the self-tuning vacuum variable scenarios [11]. The idea behind these models is that the vacuum is a self-sustainable medium, that is, it has a definite volume even in an empty environment. These works postulate a new degree of freedom, called the q-variable, whose role is the equilibration of the quantum vacuum. Other thermodynamical scenarios of this type were considered in [12]-[15]. The q-scenarios inspired partially the interest in vector vector fields adjustment mechanisms, or even tensor ones. Some vector models capable to adjust vacuum energy to a very low value were presented several years ago in [16]-[17]. These models spontaneously break the Lorentz symmetry, and can be considered as particular cases of more general models considered by Bjorken [18]-[19]. A first obstacle in the original formulation [16]-[17] is that the effective Newton constant $G_{N}$ obtains an unacceptable numerical value [20]. Furthermore, they strongly modified the dispersion relation of gravitational waves and introduce longitudinal components wide beyond the experimental accuracy [20]. However, there exist new scenarios which apparently overcome this problem [21]-[25].

The new scenarios [21]-[25] are precisely the motivation to write the present note. In these works it is claimed that the Newton constant $G_{N}$ is well behaved for large proper times $t\to\infty$. The problem is that the arguments these authors present are not completely accessible to our knowledge, since they require an strong knowledge of the q-models mentioned above. For this reason it is presented here an independent check about the behavior of the gravitational constant $G_{N}$ and the propagation of gravitational waves in these models. This may be useful for a reader which is not acquainted with the ideas of [11]-[15]. The present work is focused in the models constructed in [21], and the results presented here agree with these authors claims. The gravitational constant stabilizes to a constant value which is identified as the Newton one $G_{N}$. Besides, the longitudinal components of the gravitational waves vanish at $t\to\infty$. In the process of deriving these results, general expressions for the deviation of the Planck mass $\delta M_{p}$ and the longitudinal components of the gravitational waves are found. Further new solutions with the same property are found as well.

The present work is organized as follows. Section 2 contains the basic equations of the model. Section 3 presents the equations for the perturbations of the model. In section 4, the Newtonian limit of these theories is analysed and an expression for the effective Planck mass is derived. Section 5 contains the equation for gravitational waves in these scenarios, and the terms that potentially may introduce longitudinal components are derived explicitly. In section 6 these features are analyzed for the models of [21] and it is strongly suggested that these models work fine for asymptotic times. In Section 7 new models with these properties are presented. Section 8 contains the discussion of the obtained results, and open perspectives to be investigated further are suggested.

2. The basic equations describing the model

The present work is related to the scenarios presented in [21]. The degrees of freedom of these models are two vector fields $A_{\mu}$ and $B_{\mu}$, together with the metric field $g_{\mu\nu}$. The corresponding lagrangian is given by

[TABLE]

Here $L_{EH}$ is the standard Einstein-Hilbert lagrangian and $\epsilon(Q_{A},Q_{B})$ is, at this point, an arbitrary function of both scalars $Q_{A}$ and $Q_{B}$. The action is clearly non gauge invariant. The equations of motion that are obtained by varying the action corresponding to (2.1) with respect to the vector fields $A_{\mu}$ and $B_{\mu}$ are given by

[TABLE]

Here the following derivatives

[TABLE]

have been introduced. In the following, an isotropic and homogeneous universe will be considered, and the signature to be employed is $(+,-,-,-)$. In addition, it will be assumed that the spatial curvature of the universe $k=0$. In this situation, there exists a coordinate system $(t,x,y,z)$ for which the metric takes the following form

[TABLE]

Here $t$ is the proper time. The only non vanishing components of $A_{\mu}$ and $B_{\mu}$ when the space time metric is of the form (2.3) are the components $A_{0}$ and $B_{0}$, and the only non vanishing Christoffel symbols for such distance element are given by

[TABLE]

The unique non zero derivatives of $A_{0}$ are given by

[TABLE]

Here the latin indices are spatial ones. Analogous formulas hold for $B_{0}$.

The explicit dependence of the components $A_{0}$ and $B_{0}$ with respect to the proper time $t$ is found by solving (2.2) taking into account (2.4)-(2.5). The resulting equations are

[TABLE]

[TABLE]

These equations should be supplemented with the Einstein equations. The energy momentum tensor for the vector fields is given by

[TABLE]

[TABLE]

[TABLE]

In an isotropic and homogeneous situation the only non zero components are given by

[TABLE]

and their explicit expression can be found by use of (2.8), the result is

[TABLE]

[TABLE]

In finding these formulas, the equation of motions (2.6)-(2.7) should be taken into account. Furthermore, the following quantities

[TABLE]

have been introduced. In these terms the Einstein equations $R_{\alpha\beta}-g_{\alpha\beta}R=\kappa T_{\alpha\beta}$ become

[TABLE]

[TABLE]

In the following section, it will be assumed that a solution $a(t)$, $A_{0}(t)$ and $B_{0}(t)$ of (2.6)-(2.7) and (2.10) has been found, and perturbations around this vacuum will be characterized. The presence of non zero values $A(t)$ and $B(t)$ breaks the Lorentz symmetry, since it indicates a preferred direction in the space time manifold.

3. Generic perturbations for the model

Consider a given solution $g_{\mu\nu}$, $A_{\mu}$ and $B_{\mu}$ of the previous model. The task in consideration is to analyze the behavior of perturbations of the form $\widetilde{g}_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu}$, $\widetilde{A}_{\mu}=A_{\mu}+c_{\mu}$ and $\widetilde{B}_{\mu}=B_{\mu}+d_{\mu}$ in the given background. Here $h_{\mu\nu}$, $c_{\mu}$ and $d_{\mu}$ are small perturbations around the background $g_{\mu\nu}$, $A_{\mu}$ and $B_{\mu}$. Recall that, in general, given an space time $(M,g_{\mu\nu})$ with a metric $g_{\mu\nu}$ smooth at an small neighbor of a point $p$, the variation of the Christoffel symbols under an infinitesimal but smooth metric change $\widetilde{g}_{\mu\nu}=g_{\mu\nu}+\delta g_{\mu\nu}$ in this neighbor is given by

[TABLE]

A variation of a metric $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$ are not independent, in fact they are related by $\delta g_{\mu\nu}=-g_{\mu\beta}(\delta g^{\beta\delta})g_{\delta\nu}$. Now, given $L=\epsilon(Q_{A},Q_{B})$, its expansion to second order is

[TABLE]

The first term can be worked out by use of

[TABLE]

In addition one has the identity

[TABLE]

Analogous formulas are true for $Q_{B}$. The variation $\delta A_{\alpha;\beta}$ has contributions due to the fluctuations $\delta g_{\mu\nu}$ of the metric and due to the fluctuations $\delta A_{\alpha}$ of the vector field. Thus

[TABLE]

Here the derivative $(\delta A_{\alpha})_{;\beta}$ is taken with respect to the unperturbed metric. In these terms

[TABLE]

By taking into account the identity $A^{\gamma;\delta}A^{\alpha;\beta}=g^{\alpha\gamma}g^{\beta\delta}Q_{A}^{2}$ it is deduced that a

[TABLE]

[TABLE]

[TABLE]

[TABLE]

An analogous expression follows for $\delta L_{B}$. The term $\delta L_{AB}$ is worked out by the identity

[TABLE]

[TABLE]

In these terms, it is found that

[TABLE]

By taking into account that our background is isotropic and homogeneous, and that the only non zero vector components of the vector fields $A_{\nu}$ and $B_{\nu}$ are the time components $A_{0}$ and $B_{0}$, it follows that the relevant part of $\delta\Gamma^{\alpha}_{\beta\gamma}$ is the following

[TABLE]

[TABLE]

These expressions follow directly from (3.11) and (2.4). By further making the definition

[TABLE]

it follows that the covariant derivatives of the field perturbations are

[TABLE]

[TABLE]

The second order variation of the lagrangian (2.1) is then given by [21]

[TABLE]

[TABLE]

[TABLE]

This second order variation should be supplemented with the Einstein-Hilbert lagrangian $\delta L_{EH}$, which is well known and its explicit form is given in any standard book of cosmology.

The importance of the perturbation lagrangian (3.15) is that it describes the propagation of inhomogeneities around the given background $g_{\mu\nu}$, $A_{\mu}$ and $B_{\mu}$. These inhomogeneities always exist, and an stern test for a realistic theory is that these perturbations do not grow beyond the experimental accuracy. The next subsections are devoted to two important aspects of this perturbations, namely the screening or enhancement of the effective Newton constant $G_{N}$ and the propagation of non transversal modes for gravitational waves. These effects should be suppressed for these theories, otherwise these model would not be realistic.

4. The screening or enhancement of the Planck mass $M_{p}$

In the following, the Newton limit of the general class of theories described above will be derived. In other words, it will be assumed that there is a spherically symmetric matter source, such an spherical planet. In the non relativistic limit, the gravitational field is described by the Poisson equation

[TABLE]

Here $\Phi$ is the Newton potential of standard classical mechanics and the Newton constant is identified as $G_{N}\sim M_{p}^{-2}$ in natural units. The energy momentum tensor $T_{00}$ is due to the source, and $T_{00}\sim\delta(x)\delta(y)\delta(z)$, that is, the source is a mass point located at the origin.

Now, in the cosmological setup we are working with, the solutions $A_{0}(t)$, $B_{0}(t)$ and $a(t)$ of the system (2.6)-(2.7) and (2.10) are functions of the proper time $t$. However, in the following, it will be assumed that this time dependence is not relevant at the present times and these quantities may be considered as almost constants. The idea is to include an almost static mass source in this static background, and to calculate the components $h_{\mu\nu}$ due to this perturbation. It is important to remark that in GR, the Newtonian potential $\Phi$ is identified as usual by the relation $g_{00}=1+2\Phi$ from where it follows that

[TABLE]

Thus, special attention should be paid to the component $h_{00}$. The time dependence of these quantities will be assumed to be very slow. Now, the terms in the lagrangian (3.15) which contains the perturbations $c_{\alpha}$ are generically of the form

[TABLE]

[TABLE]

By taking (2.5) into account, it is found the following expression for the coefficients $p$, $q$, $r$ and $s$

[TABLE]

[TABLE]

All these functions are evaluated at the present time $t_{0}$. As stated above, these coefficients will be considered as simple constants, this assumption will be justified later on. Here the latin indices are spatial ones and greek indices are generic, that is, they may denote an spatial or time component. For the Newtonian limit, one may choose the gauge $h_{0g;}^{\;\;\;\;\;g}=0$. In fact, one have that

[TABLE]

From this expression, by taking into account that $H\to 0$ for $t\to\infty$ it follows that this gauge is approximately $h_{0g,g}=0$, which is the Coulomb gauge usually employed for studying perturbations in Minkowski space time. Furthermore, since the source is an static mass, the time dependence of the components $h_{\alpha\beta}$, $c_{\alpha}$ and $d_{\alpha}$ will be neglected. In particular, from the expressions

[TABLE]

and by taking into account that $H\to 0$ for large times, it follows that the covariant derivatives $h_{\alpha\beta;0}$ may be neglected as well. The same follows for the covariant derivatives $c_{\alpha;0}$, as follows from (3.14). These are all the assumptions to be used when deriving the equations of motions

[TABLE]

The equation of motion for the component $c_{0}$ is given by

[TABLE]

The analogous equation follows for $d_{0}$. Thus, one has that

[TABLE]

with $f$ such that $\Delta f=0$. This harmonic function should be zero at infinite. By further assuming that when $h_{00}\to 0$ then $c_{0}\to 0$ and $d_{0}\to 0$, it is reasonable to assume that $f$ should be zero everywhere and thus

[TABLE]

In addition, the corresponding equations for the spatial components $c_{i}$ are given by

[TABLE]

[TABLE]

[TABLE]

which, after simplifying the Kronecker deltas, becomes

[TABLE]

Analogous equations hold for the component $d_{i}$. By taking into account the gauge $h_{0g;}^{\;\;\;\;g}=0$, these equations are solved by

[TABLE]

This can be seen as follows. The condition $h_{0g;}^{\;\;\;\;g}=0$ together with (4.24) imply that $c_{g;}^{\;\;\;g}=d_{g;}^{\;\;\;g}=0$. If this conclusion is introduced into (4.23) the resulting equation is

[TABLE]

It is direct to check that (4.24) is solution of this equation, which is what we wanted to show. The same argument applies for the $d_{i}$ components.

The final task is to consider the part of the lagrangian (3.15) corresponding to the graviton $h_{ij}$. This is explicitly given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here the coefficients $p$, $q$, $r$ and $s$ are given in (4.19) and the remaining ones follows from (3.15) and (2.5). Their explicit expression is

[TABLE]

Note that the deltas $\delta^{ij}$ are related to spatial indices, therefore the only contributions to the equations of motion comes from the terms proportional to $s$ and $v$. By taking into account (4.19), the identification (4.24) and the gauge $h_{0,gg}=0$ the resulting equations corresponding to $h_{00}$ are given by

[TABLE]

Here

[TABLE]

the last identity follows from the fact that the time dependence of the metric $h_{00}$ can be neglected and also that $\Gamma^{0}_{ij}\sim H\to 0$ for $t\to\infty$. By taking into account that $2\Phi=h_{00}$, as discussed above in (4.17), the last equation (4.27) resembles the Poisson equation (4.17) with an effective Planck mass given by

[TABLE]

Thus, one of the conditions for these models to be realistic is that the quantity

[TABLE]

goes to zero or to a constant for large times.

The other point to be checked is that $h_{0i}=0$ and $h_{ij}=0$ are solutions of the model, otherwise an spherically mass may induce anisotropies. The equations for these degrees of freedom derived from (4.25) and are identically zero due to the gauge condition $h_{0,i}^{i}=0$. Thus this type of spherically symmetrical solutions are allowed in the model.

5. Gravitational waves

The next important aspect to study is the propagation of gravitational wave in the given Lorentz breaking background. In this case, the components $h_{\mu\nu}$, $c_{\mu}$ and $d_{\mu}$ are assumed to be time dependent. By taking into account (4.20), it is seen that $h_{\alpha\beta;0}=h_{\alpha\beta,0}$ for large times. But this will be imposed at the end of the calculations. The lagrangian (3.15) in these circumstances is given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The coefficients in the previous expression can be obtained by comparing (2.5) and (3.15). The explicit result is given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

A convenient gauge for analyzing gravitational waves propagation is $d_{0}=h_{i0}=0$. The first condition can be fixed by use of convenient Lorentz transformation while the second is obtained by use of spatial rotations. This gauge was already considered in [20] when studying this type of models. By taking this gauge into account the equations of motion for $c_{0}$ and $d_{0}$ may be expressed as follows

[TABLE]

[TABLE]

By use of the same gauge, the equations for the spatial components $c_{i}$ and $d_{i}$ are respectively given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The equation for $h_{00}$ is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here the term $E_{00}$ is the contribution of $\delta L_{EH}$. For the component $h_{0k}$ one has that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Finally, for $h_{ik}$, it is obtained that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

As stated above, the quantities $E_{\alpha\beta}$ in equations (5.34)-(5.36) are the contributions for the standard perturbed Einstein-Hilbert lagrangian $\delta L_{EH}$. If this terms were only present, then the solution of the model will be the propagation of gravitational waves, which are transversal and has the dispersion relation $\omega=k$. It is the presence of the additional terms the ones which may introduce longitudinal modes or to modify the dispersion relation. Thus, another stern requirement for a realistic theory is that the effect of these additional terms is negligible for large times $t\to\infty$.

6. Tests for known solutions

Having derived the deviation from the mass Planck mass $\delta M_{p}$ in (4.29) and the deviations from the gravitational wave equations in (5.34)-(5.36), the next step is to test the known solutions of the model. In the work [21] the function $\epsilon(Q_{A},Q_{B})$ the authors consider is given by

[TABLE]

with $\delta$ a dimensionless quantity. By introducing the dimensionless time $\tau=M_{p}t$, the following asymptotic behavior for the following dimensionless quantities is derived [21]

[TABLE]

[TABLE]

This behavior follows from a numerical analysis of equations (2.6)-(2.7) and (2.10). Here $p\sim 3.6$ and $n\sim 2.1$ [21]. In terms of these quantities, the deviation from the Planck mass (4.29) is given by

[TABLE]

Now, for the solution (6.38) the function (6.37) is asymptotically

[TABLE]

Thus, the mass deviation (6.39) is given by

[TABLE]

Both terms in the parenthesis have a divergent behavior proportional to $\tau^{2}$. However, it follows from (6.38) directly that

[TABLE]

and this, combined with (6.40) implies that $\delta M_{p}^{2}\to 0$ for $\tau\to\infty$. In other words, this solution does not affect considerably the Newton constant $G_{N}$ at large times.

The argument given above has a problem. In the derivation of (4.29) and (4.23)-(4.27) the time dependence of the coefficients of the perturbation lagrangian have been neglected. However, the coefficients $a_{0}^{2}r$ and $a_{0}^{2}s$ posses an asymptotic behavior of the form $\tau^{2}$, which is reflected in (6.41). So, the approximation just made may be non consistent. However, we have checked that the resulting terms that arise when the time dependence of these coefficients is not neglected are proportional to the time derivative of $\delta M_{p}^{2}$, which also tends to zero for large times. Thus, our conclusions are not modified for $t\to\infty$.

The next step is to analyze the gravitational wave issue for these models, which is a bit more complicated. This analysis requires the knowledge of the time dependence of the coefficients (5.31). In order to see this aspect, it is convenient to study the dependence of the quantities

[TABLE]

with respect to the proper time at $t\to\infty$. From here, by taking into account the asymptotic behavior (6.38), it follows that

[TABLE]

Therefore, the coefficients (5.31) have the following asymptotic behavior

[TABLE]

From here, by taking into account that asymptotically $A_{0}\sim B_{0}\sim\tau^{p}$, the following behavior of the coefficients of the gravitational wave equations (5.32)-(5.36) is inferred

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We have not included the coefficients $\alpha$ and $a$ given by

[TABLE]

since we have seen above that they do not go to zero for $\tau\to\infty$. The same consideration applies for the last three coefficients in (6.43), which go like $\tau^{2}$. By neglecting the other terms, it follows that (5.33) become asymptotically

[TABLE]

[TABLE]

From the two equations (6.45) it is inferred that

[TABLE]

This relation is completely schematic, but it states that the behavior of $c_{i}$ and $d_{i}$ are determined by the same quantity $f(h_{ij})$, which depends on the metric components $h_{ij}$, up to a proportionality factor $A_{0}$ or $B_{0}$. Now, by use of this and the formulas (6.43) it follows that in both equations (5.32) the dominant term for large $\tau$ is the one proportional to $\alpha A_{0}$ or $aA_{0}$. Thus, both equations give asymptotically that \bigg{(}\alpha A_{0}h_{00;\alpha}\bigg{)}^{;\alpha}=0 at very large times. However, it will be useful to consider very large but finite times. By the first formulas (6.43) it follows that the next order dominant term is proportional to $2\sigma A_{0}+\lambda B_{0}$ and $2sB_{0}+\lambda A_{0}$, together with $\sigma c_{0;0}$ and $\lambda c_{0;0}$. In this approximation the equations (5.32) reduce to

[TABLE]

[TABLE]

On the other hand the asymptotic form of the equation (5.34) is

[TABLE]

[TABLE]

Here the term $E_{00}$ is the contribution of $\delta L_{EH}$. Now, it will be shown that several of the terms in (6.48) cancel asymptotically. In order to visualize this, note that the first and the fourth term of the equation (6.48) combine as

[TABLE]

where in the last line the relation (6.46) was taken into account. By bearing in mind the formulas (6.44) it follows that the quantity in parenthesis in (6.49) is proportional to the quantity given in (6.39), that is

[TABLE]

But the right hand side has been shown in (6.39) to vanish for large times $t\to\infty$, since it is proportional to the deviation of the Newton constant $\delta M_{p}^{2}$. Thus the contribution for this pair of terms for asymptotic times is negligible. The same consideration follows for second and the fifth term of (6.48), and the third and the sixth. It remains to see that the seventh and the eight terms of (6.48) cancel each other. To analyze this, consider again the equations (6.47). The first of these equations may be rewritten as

[TABLE]

Equipped with this equation, one would like to analyze the seventh term of (6.48). This term is elaborated, by use of the last identity, as follows

[TABLE]

[TABLE]

[TABLE]

Here, the fact that $A_{0}\sim\tau^{p}$ has been taken into account. By using the integration part formula on the first term of the last equality, one obtains finally

[TABLE]

[TABLE]

In the same fashion, from the second (6.47) it is obtained that the eight term of (6.48) is given by

[TABLE]

[TABLE]

Clearly both (6.51)-(6.52) have the same structure, except for a proportionality factor $\alpha A_{0}^{2}$ and $aB_{0}^{2}$. In other words

[TABLE]

for $\tau\to\infty$. Thus, equation (6.48) reduces to $E_{00}=0$, which is one of the standard equation of gravitational waves in GR.

In addition, for the component $h_{0k}$ the equation (5.35) reduces to

[TABLE]

[TABLE]

By use of (6.50) and (6.46) it is seen again that all the terms cancel when chosen in pairs conveniently, and the equation reduces to $E_{0k}=0$, which is another standard gravitational wave equation in GR. Finally, for $h_{ik}$, it is obtained that

[TABLE]

Again, all the terms cancel and one obtains asymptotically that $E_{kl}=0$. Thus, all the standard gravitational waves equations of GR have been obtained, under the approximations made. Based on this, we suggest that there are no large deviations from transversality or from the standard dispersion relation $\omega=k$ of the gravitational waves for asymptotic times $t\to\infty$ for the solutions presented in [21].

7. New solutions

In the previous section, it was shown that for a generic behavior (6.38) with $p>2$ and $n>2$, the resulting dynamics will not generate neither an unacceptable effective gravitational constant $G_{N}$ nor longitudinal gravitational waves for large times. Thus, there is no reason for restricting the attention to the lagrangian density (6.37). One can consider any other lagrangian density $\epsilon(Q_{A},Q_{B})$ which give rise to any behavior (6.38) with $p>2$ and $n>2$. For instance, consider the following lagrangian density

[TABLE]

Here $m$ is, at this point, an arbitrary power. By assuming that the solutions corresponding to this model posses the asymptotic behavior (6.38), then by plugging this behavior it into the equations of motion (2.6)-(2.7) and (2.10) one should obtain a relation between the exponents in (6.38). The dimensionless version of the equations of motion (2.6)-(2.7) and (2.10) are the following

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here the overdot stands for differentiation with respect to $\tau$. When the asymptotic solution (6.38) is considered, one has that

[TABLE]

[TABLE]

Here, the following coefficients have been introduced

[TABLE]

By taking into account (7.58), it follows after some algebraic operations that the asymptotic behavior (6.38) is compatible with equations (7.54)-(7.55) when the following algebraic relation is satisfied

[TABLE]

On the other hand, the quantity $\widetilde{e}(q_{A},q_{B})$ introduced in (7.57) can be worked out as

[TABLE]

By taking the last equation into account, it follows that the Friedmann equation (7.57) becomes

[TABLE]

By requiring that this equation holds when $\tau\to\infty$, it is obtained that

[TABLE]

where in the last step the definition (7.59) has been taken into account. The solution of this algebraic equation gives

[TABLE]

The final equation to solve is (7.56). For working out this equation one should calculate first that

[TABLE]

[TABLE]

In these terms, equation (7.56) simplifies by noticing that $\lambda+\widetilde{e}$ vanishes at zero order in $\delta$, so only the part of $\widetilde{e}$ to be considered is the first order one. After some elementary calculations, it is found that the terms of this equations which do not contain $\delta$ vanish identically, and therefore

[TABLE]

Since the values of $n$ and $p$ are not known, then one should insure that the right hand of the equation goes to [math] when $\tau\rightarrow\infty$. This can be shown to be, after some algebraic manipulations, the following condition

[TABLE]

This algebraic equation insures that the right hand is zero. Both equations (7.60) and (7.63) are equivalent to the system

[TABLE]

[TABLE]

This system has the compatibility condition

[TABLE]

with coefficients

[TABLE]

[TABLE]

[TABLE]

For instance, for $m=0.89$, it is calculated that $n=2.5$ and $p=2.6$. This solution pass the test of the previous section since $p>2$ and $n>2$. There are several other solutions with this behavior. Thus, we have enlarged the list of possible adjustment mechanisms initiated in [21]. However, the best values for $n$ from the phenomenological point of view are in the interval $[0.5,0.66]$. In view of this, we have to emphasize that the condition $p>2$ and $n>2$ is sufficient, but not necessary. We have considered this case in order to simplify the analysis. For other situations, there may be a cancellation but the analysis is much more involved. We leave this for a future investigation.

8. Discussion and open questions

In the present work some evidence was collected that suggest that the vector adjustment vector fields presented in [21] do not affect the Newtonian dynamics of the planets at present times. Furthermore, it was suggested her that the gravitational waves that these models predict are asymptotically transversal and with the right dispersion relation $\omega=k$. In the process for checking this some new models were found. The models interpolate between an initial de Sitter universe and a Friedmann-Robertson-Walker type. It would be be interesting to study these matters for the models [22]-[25], which seem to be more general than these ones. Another open question is to interpret the lagrangian $\epsilon(Q_{A},Q_{B})$ presented as an effective one, and to identify an underlying theory. Of course this may not be a simple task, but perhaps the Bjorken ideas [18]-[19] may be useful in this context.

There exist several open problems to be investigated further. The present paper address the possibility of making the effective cosmological small, but it does not explain why the actual energy density is so close to the critical one $\rho_{c}$. The value of $\rho_{c}$ has an uncanny relation within the pion mass $m_{\pi}$ and the Planck mass $M_{p}$. There exist some speculations that interpret this as a result of an interaction between the hadronic sector and a hidden one, which can be very weak, of gravitational order [26]-[31]. It may be interesting to find a combined model with includes an adjustment field which cancel the large contribution of the different QFT to the universe energy density and leave the contribution of this extremely weak interaction untouched at large times. We leave this for a future investigation.

Acknowledgments

O. P. S is supported by the CONICET.