Vector field theories in cosmology
aa r X i v : . [ g r- q c ] A ug Vector field theories in cosmology
A. Tartaglia and N. RadicellaDipartimento di Fisica del Politecnico and INFN section of TurinCorso Duca degli Abruzzi 24, I-10129 Torino, Italye-mail: [email protected] and [email protected] 25, 2018
Abstract
Recently proposed theories based on the cosmic presence of a vectorialfield are compared and contrasted. In particular the so called Einsteinaether theory is discussed in parallel with a recent proposal of a strainedspace-time theory (Cosmic Defect theory). We show that the latter fitsreasonably well the cosmic observed data with only one, or at most two,adjustable parameters, whilst other vector theories use much more. TheNewtonian limits are also compared. Finally we show that the CD theorymay be considered as a special case of the aether theories, correspondingto a more compact and consistent paradigm.
I Introduction
The most successful field theories of the XXth century are in general tensortheories on a four-dimensional manifold. This is true for the electromagneticfield as well of course as for the gravitational interaction. In the former case the”root” of the theory is in a four-vector potential, in the latter also the potentialis a rank 2 symmetric tensor (the metric tensor).Although, properly speaking, ”tensor” includes any rank from 0 (a scalar)up to higher values, by ”tensor” theory one normally means a theory based onan at least rank 2 tensor. On this respect we shall here discuss ”vector” theoriesas rank 1 tensor theories.Theories of this sort have not been considered as frequently as scalar andscalar-tensor theories. Of course we need a motivation and ours has been atthe cosmic scale. Since 1998 evidence has been found pointing at an acceleratedexpansion of the universe (see [1]) and theorists have been working since thento find an explanation of the alleged behaviour. In various forms a sort of darkenergy fluid has been envisaged (see [2] and [3]) or modifications of the classicalGeneral Relativity theory have been proposed producing the sought for effects(see [4] and [5]). One of us has put forth the idea that a cosmological vectorfield be responsible for the acceleration: let us call the one based on this vector,”Cosmic Defect” (CD) theory [6]. 1nother cosmic vector field has been discussed in the literature in recentyears in a group of theories that we shall call here, for short, Vector ÆtherTheories (VET), although their authors have used different names or no nameat all. One of these theories has indeed been christened Einstein Æther (Æ) [7].In all cases by ”æther” the cosmic vector field is meant.The initial motivation for the VET was not related to cosmology, but ratherto fundamental quantum field theory, where reasons exist to doubt of an exactLorentz invariance. In fact, the Lorentz group is non-compact and leads todivergences in quantum field theory, associated with states of arbitrary highenergy and momentum. Furthermore, because of the non-compactness of thegroup, it is not possible to experimentally test the invariance at all scales ofenergy.Considering a D -dimensional universe with D − II Vector Æther Theories
Adopting the traditional approach to field theory, we may write the total actionintegral for a space time containing a vector field U as the sum of three parts S = Z d xc √− g (cid:18) κ R + L U + L m (cid:19) . (1)Of course g is the determinant of the metric tensor and R is the scalar curvatureof the manifold. L U is the Lagrangian density of the vector field, and L m is theone of matter; κ = πGc is the coupling constant between matter and geometry.Writing the action in the form (1) we are implicitly assuming that no directcoupling between the vector field U and matter exists; both the vector field andmatter couple with geometry.According to [12] and [13], the most general form for the Æther Lagrangiandensity in the action integral (1) for a vector-tensor theory, including terms upto second order in derivatives and fourth order in the fields, is written as follows: L U = K αβµν ∇ α U µ ∇ β U ν − k ( U µ U µ − M n α n α ) , (2)We can recognize a kinetic and a potential term for the vector field. The coef-ficients of the kinetic term are contained in the rank 4 tensor: K αβµν = K (1) αβµν + K (2) αβµν ,K (1) αβµν = c g αβ g µν + c δ αµ δ βν + c δ αν δ βµ + c U α U β g µν , (3) K (2) αβµν = c δ αν U β U µ + c g αβ U µ U ν + c δ αµ U β U ν + c U α U β U µ U ν , in which the terms in c and c represent directional covariant derivatives alongthe field U µ .The potential term in (2) is the gravitational analogue of the Higgs mecha-nism of gauge theories, so that the vector field acquires a vacuum expectationvalue M n α that breaks the Lorentz invariance; n is a unit four-vector. Theaction integral written with (2) is slightly different from the one that appears3n [12] and [14] because of the different choice made for constants. Assuming U to be dimensionless, one is left with dimensionless c i ’s and M , too. Notice,as already stressed before, that the matter Lagrangian couples only with themetric and not directly with the U µ ’s.The claim of generality on (2) must be taken with some caution, because itdepends on a number of limiting assumptions on space-time. Within a widerframework and in the attempt of constructing a theory in which both GeneralRelativity and the Standard Model are taken into account, Konsteleck´y has de-veloped SME (Standard Model Extension), a theory whose effective Lagrangiancontains the fields of the Standard Model as well as gravity together with addi-tional Lorentz symmetry-violating terms. The most general formulation of SME[15] uses an Einstein-Cartan background, including torsion; in this framework(2) appears as a special subclass of Lagrangians. An earlier version of the SMEin a Minkowski spacetime had already been studied in [16].Konsteleck´y and Samuel in [8], as early as in 1989, considered a Lagrangiandensity which now could be seen as a subclass of the VE theories. It has beenput forth again by Jacobson and Mattingly [10] in 2001, in practice consideringthe only K(1) term of (3) and replacing the potential term by a constraint onthe norm of the vector field, introduced by means of a Lagrange multiplier λ .The Lagrangian density for the vector field is then L U = K (1) αβµν ∇ α U µ ∇ β U ν + λ ( U α U α − . (4)From now on we shall refer to this theory as Einstein Æther (Æ), from thename used in [10].The case analyzed in the earlier formulation corresponds to choosing allparameters to be zero except for c and c , for which c + c = 0 holds.A theory equivalent to this choice for the Æther Lagrangian had alreadybeen studied by Nambu [17] in the case of Minkowski spacetime, who proved itto be equivalent to electrodynamics in a non-linear gauge. Other contributionsin the non-flat background case are found in [10] and [18].A variant of Æ[19] introduces the vector field in the action in the form of afunction F of the scalar K obtained from the K (1) αβµν after choosing c = 0: K = M − K αβµν ∇ α U µ ∇ β U ν K αβµν = c g αβ g µν + c δ αµ δ βν + c δ αν δ βµ . This approach was motivated by its authors, Zlosnik, Ferreira, and Starkman(ZFS, for short) by the quest of a modified Newtonian gravity at galactic scales,as an alternative to dark matter. The Lagrangian for the Æther is now written([19]) L U = M F ( K ) + λ ( U µ U µ −
1) (5)assuming that the Lagrange multiplier λ has the dimension of the inverse of asquared length. Of course (5) coincides with the Æ Lagrangian when F ( K ) ≡ K . We shall use a + − −− signature throughout the paper.
4n the general Lagrange density (2), U µ is neither restricted to have a fixednorm nor to be timelike. In the case of a homogeneous and isotropic universe ,however, the assumed space isotropy implies the vector field to be timelike U µ = ( U ( t ) , , , , and still leaves six free parameters in the equations of motion [12]: c , c , c , c , c , and k .This freedom is somewhat reduced in [7],[10], [19] and in the analysis madeby Carroll [20] and Lim [21]. All these authors constrain the vector field to bea unit vector U µ = (1 , , , , and maintain four (see [7][10][22]), or three (see [19][20][21]) free parameters,like in (4) and (5).Varying (1) with respect to the metric tensor elements, we obtain, as usual,the Einstein equations in the form G αβ = κ (cid:0) T Uαβ + T mαβ (cid:1) , (6)where T mαβ is the stress-energy tensor for matter, while T Uαβ is the one of thevector field. In the case of the Æ theory, the explicit form of T Uαβ is T Uαβ = 12 ∇ σ h F ′ ( J σ ( α U β ) − J σα U β ) − J ( αβ ) U σ ) i (7)+ c F ′ [( ∇ ν U α )( ∇ ν U β ) − ( ∇ α U ν )( ∇ β U ν )] + 12 g αβ M F + λU α U β , where F ′ = d F d K J ασ = ( K αβσγ + K βαγσ ) ∇ β U γ . Varying the action with respect to U µ , under the same hypotheses, one obtainsthe equations of motion for the vector field ∇ α ( F ′ αβ ) = 2 λU β . (8) III The Cosmic Defect theory
The CD theory, as the theories mentioned in the previous section, ascribes thebehaviour of the universe as a whole to the presence of a cosmic four-vectorfield. The difference with respect to the VE theories is mostly in the motivationand interpretation of the vector, then in the choice of the initial Lagrangiandensity for space time. The line element is of the form ds = c dt − a ( t ) δ ij dx i dx j . . A vector field naturally arises from this view:the ”radial” rate of stress γ induced by the defect [6]. Now ”radial” means alongthe cosmic time axis.The mentioned identification of the vector field implies it to be divergence-free, which means γ µ ; µ = 0 (9)In the RW symmetry and adopting a co-moving cosmic reference frame, (9)has the solution γ = Q a (10) γ i = 0where Q is an integration constant and a is the scale factor of the universe. Infact the time component of the vector is also a measure of its norm χ = (cid:0) γ (cid:1) This result introduces a first difference with respect to the Æ theory becausethere the cosmic four-vector is constrained to have unit norm, in fact coincidingwith the four-velocity of an observer co-moving with the cosmic fluid. This isnot the case of the general Lagrangian density (2), where there is no fixed normconstraint.As for the choice of the Lagrangian density, the CD theory uses anotheranalogy based on the remark that the phase space of a RW universe is bidimen-sional and that it formally coincides with the one describing a point particlemoving across a viscous fluid. Starting from this formal correspondence theaction integral for the only space time is assumed to be [6]: S = 12 cκ Z e − g µν γ µ γ ν R √− g d x (11)where γ is again the already mentioned four-vector.In a RW symmetry the action integral, including matter (Friedman Robert-son Walker case), reduces to S = S g + S m = V k (cid:20) − Z κ e − Q /a (cid:0) a ¨ a + a ˙ a (cid:1) dτ + κ Z f a ˙ a dτ + ̟ Z ha dτ (cid:21) (12) In the case of a spatially flat spacetime we should rather refer to a singular surface thanto a singular event, but the logic structure remains the same. τ is the cosmic time (= ct ); dots denote cosmic time derivatives; V k is the part of the Lagrangian which is not affected by any variation withrespect to the metric; κ and ̟ are appropriate coupling constants; f and h arescalar functions of a accounting for anything we could widely speaking dub as”matter”.We remark that the second derivative of a with respect to τ appears linearlyin the Lagrangian (the integrand of (12)). This means that integrating thecorresponding term by parts in the action leads to Z e − Q /a a ¨ adτ = e − Q /a a ˙ a (cid:12)(cid:12)(cid:12) τ τ − Z e − Q /a (cid:18) Q a + a (cid:19) ˙ a dτ. One is then left with a surface term, whose variation is by definition zero, and afirst order derivative term so that in practice the effective Lagrangian becomes L = 3 κ e − Q /a (cid:18) Q a + a (cid:19) ˙ a + κ f a ˙ a + ̟ha . It is now possible to write the Hamiltonian density function for the system.This is: H = ˙ a ∂ L ∂ ˙ a − L = (cid:20) κ f a + 3 κ e − Q /a (cid:18) Q a + a (cid:19)(cid:21) ˙ a − ̟ha and is, as usual, interpreted as the energy density (in the universe). We mayeasily verify that d H dτ = 0 . (13)We may then write (cid:20) κ f a + 3 κ e − Q /a (cid:18) Q a + a (cid:19)(cid:21) ˙ a − ̟ha = W = constant . Then the field equation becomes˙ a = ̟ha + W h κ f a + κ e − Q /a (cid:16) Q a + a (cid:17)i . (14)Actually, if we want to recover the usual meaning of the matter term in aco-moving reference frame we must choose κ = 0so the rate of expansion equation becomes:7 a = κ ̟ha + W e − Q /a (cid:16) Q a + a (cid:17) . (15)In the absence of a defect, it would be (FRW universe)˙ a a = 8 πG c ρc = κ ρc (16)Evaluating (15) with Q = χ = 0, that is looking at the equation that comesfrom the action of the CD theory but in the absence of a defect, we obtain˙ a a = κ ̟ha + W a . (17)Of course the value of the W constant depends on the type of space-time weconsider: in the classical empty case (no defect, no matter) it would be W = 0.We then conclude that in order to recover the classical result, i.e. comparing(16) and (17), it must be ̟ = 1 h = ρc where now ρ represents the usual mass-energy density function.The final expansion rate equation is˙ a = κ ρc a + W e − Q /a (cid:16) Q a + a (cid:17) (18)Introducing the new variable ˜ a = a/Q and using Q also as the unit for time( τ → τ Q ) we may recast (18) in the form˙ a = ˜ κ ρc ˜ a + W a ˜ a e / ˜ a (19)where the coupling constant κ , as well as the volume entering the definition of ρ , have been rescaled on Q :˜ κ = κQ ; ˜ ρ = Q ρ. IV The accelerated expansion
I In the Vector Æther theories
Let us investigate, now, cosmological solutions deduced from the VE theories,i.e. from (6). As we already know, in the case of a homogeneous and isotropicuniverse the constraints of ZFS and Æ theory force the four vector field to be U µ = (1 , , , T mαβ = ρc u α u β + p ( g αβ − u α u β ), where ρc isthe energy density, p is the pressure and u µ is the four velocity of the fluid (i.e. g αβ u α u β = 1). Equation (8) for the vector field can be used to deduce λ andput it in the stress-energy tensor for U µ , eq.(7), so that one is left with the twoEinstein equations: H = κ (cid:18) αH F ′ − F M (cid:19) + κ ρc − H − aa = − κ (cid:20) F ′ α (cid:18) H + ¨ aa (cid:19) − ˙ F ′ αH + 12 F M (cid:21) + κp, (20)where α is a combination of the c i ’s, namely α = c + 3 c + c and H ≡ ˙ aa . Theequations(20) govern the evolution of the universe. An accelerated expansionis indeed obtained choosing, for an appropriate range of K values, [19] F ( K ) = C ( −K ) n (21)where C is a constant and n an integer. K is found to be 3 αH M .Restrictions on the c i ’s are obtained when studying the consistency of thetheory in the perturbative regime, that is performing classical perturbationsin flat space time, and at a quantum level, when the Hamiltonian has to bepositive definite. The analysis has been performed by Lim in [21] in the caseof the Æ theory; the Lagrangian for the Æther turns out to have a scalar-typeand a vector-type perturbation (i.e. a spin-0 mode and a spin-1 mode). Theconstraints obtained by Lim are the following: • c <
0, in order to insure that spin-0 states have positive norm, i.e. arenon-ghost-like; • ≤ c + c + c c ≤
1, in order to have a well-behaved propagation of thespin-0 mode; • c + c ≥ c < c ≤ c + c + c ≤ , i.e., α ≤
0; this is the reason why K appears with a minus sign in F in (21). In[19] it is also shown that one can rewrite the Einstein equations to obtain, withthe particular choice of F written above in (21): " ǫ (cid:18) HM (cid:19) n − H = κ ρc , There is a slight difference between the equations written here and those found in [19],because of the definition of the Lagrangian for the Æther: L U here is 16 πG/c times the onefound in the cited article. ǫ = − (1 − n ) C ( − α ) n /
6. This solution introduces two more completelyfree parameters, besides the ones already present in the Lagrangian. It is thenpossible to choose them so that ǫ < H tends to the attractor:˜ H = M ( − ǫ ) / n (1 − n ) . Besides the c i ’s, C and n , the mass scale M is alsopresent; the authors (ZFS) relate it to the acceleration scale a of Milgrom’sMOND theory [24], in order to have the right limit at galactic scale.Restricting to Æ case (cfr. [10]) the equations reduce to H = κ αH + κ ρc − (cid:18) H + 2 ¨ aa (cid:19) = κ (cid:20) − α (cid:18) H + 2 ¨ aa (cid:19) + p (cid:21) . (22)The analysis of these equations has been performed by Carroll and Lim in [20],but their α parameter is opposite in sign with respect to the one used here,because we are following the notations of [19]; furthermore in their case κα is adimensionless quantity.Inspecting the first equation in (22), which is the 00-th component of the Ein-stein equations, one can easily see that the contribution from the stress-energytensor of the vector field is proportional to the square of the Hubble parameter.In practice the equations can be rewritten as the usual Friedmann equationsjust rescaling the gravitational constant G : H = κ − κα ) ρc ¨ aa = − πG c c ( ρc + 3 p ) , The effective gravitational constant G c is G c ≡ G − πGα/c . (23)Since α ≤ G is not directly measurable. In order to obtain constraints on α values, one has to consider other situations, first of all the Newtonian limit.We note that in Æ theory, i.e. in the last analysis we have outlined, there is noaccelerated expansion, since it retraces the GR solution. II In the CD theory
The interesting feature of eq. (18) is that it does indeed contain an acceleratedexpansion phase in the history of the universe. Studying the properties of (18)we see that the expansion rate starts with an infinite value at the origin andtends to 0 at infinity. The initial expansion is exponential, i.e. inflationary; atthe other end, for any reasonable behaviour of matter, the expansion continuesfor ever at a rate asymptotically tending to 0.If the defect is a property of space-time the expansion (which is our wayto describe what actually is a static state in four dimensions) is present even10 a da / d τ Accelerated expansion
Figure 1: Expansion rate of the universe versus the scale factor a according tothe CD theory. The graph is valid both for empty space-time and for a universefilled with an incoherent dust.in the absence of matter, and, remarkably, one has a sequence of decelerated-accelerated-decelerated expansion. In fact ˜ a from (19) with ˜ ρ = 0 has twoextrema corresponding to ˜ a = (cid:16) ± √ (cid:17) / (24)The same result is obtained when matter is present in the form of dust. In thatcase mass conservation implies ρ = ρ ˜ a ˜ a leading to a renormalization of constants not modifying (24). Fig.1 shows thebehaviour of the expansion rate as a function of the cosmic scale factor.The situation is different if we allow for more general forms of matter. Fora simple barotropic fluid with an equation of state ρc = wp, where 0 ≤ w ≤ /
3, the conservation laws of thermodynamics imply that ρ = ρ ˜ a w )0 ˜ a w ) .
11n this case the equation for the extrema, from (19), is˜ a w (cid:0) − a + ˜ a (cid:1) W − a (3 w − − ˜ a (1 + 3 w ) −
36 = 0We are left with two free parameters, w and W = W / (cid:16) c ˜ ρ ˜ a w )0 (cid:17) , tobe determined in order to recover both the observed onset of the acceleratedexpansion and the age of the universe. V The Newtonian limit
Since General Relativity satisfies all the Solar system tests, any extension ormodification of GR must possess a correct Newtonian limit. In this section wewant to compare the theories we have been discussing until now, also on thisrespect. In practice we expect that, given any material source, the field equa-tions for gravity, in weak field approximation, reduce to the Poisson equationfor the potential.
I In Vector Æther Theories
Let us consider the field equations (6) in the static, weak field limit. The waychosen both in [19] and in [20] is to expand both the metric and the vector fieldaround a Minkowski background. At the lowest non-trivial order the approxi-mated line element may be written as follows: ds = (1 + 2Φ( x, y, z )) dτ − (1 − x, y, z ))( dx + dy + dz ) , where Φ and Ψ are suitable potentials. Since we are in the weak field limit, weshall neglect terms beyond the first order in the potentials; under this assump-tion the space components of the Æther stress-energy tensor disappear. Thespatial components of the Einstein equations reduce then to( δ ij ∇ − ∂ i ∂ j )(Φ − Ψ) = 0 , (25)where i and j are space indices ranging from 1 to 3. Assuming that both Ψand Φ vanish at space infinity, (25) implies that Ψ = Φ. Using this result, whilecombining the linearized 00-th component of the Einstein equations and thevector field equation, we obtain ~ ∇ · (cid:20) (2 + 16 πGc c F ′ ) ~ ∇ Φ (cid:21) = 8 πGc ρ, (26)where ρ is the mass density of the matter distribution. In the framework of Ætheory, it is simply F ′ = 1 and equation (26) becomes ∇ Φ(1 + 8 πGc c ) = 4 πGc ρ, G N = G c πGc . (27)The results (27) and (23) can be used to obtain one more constraint on theparameters of the theory, as analyzed in [20] and [21].In the ZFS version of the theory [19] , the authors are first led to identifythe mass scale M with something of the order of a , as we have seen before, inorder to recover the MOND limit of the theory. For them it actually is K = − c ( ~ ∇ Φ) M and c < F ′ contribution must be small. For these reasons, theauthors assume that in the Solar system (cid:16) ~ ∇ Φ (cid:17) ≫ M and expand F ′ as aseries of inverse powers of K / . At this point, however, the non-linearity due to F ′ makes the equations very difficult, thus making very hard to draw clear-cutconclusions, as remarked in [25]. II In the CD theory
In the case of the CD theory, as in the previous section, we have to considerthe weak-field limit of the theory, expanding both the metric and the vectorfield around a background configuration, but now we choose the FRW ratherthan Minkowski spacetime, because we want to maintain a link between thecosmological and the local solution. The source of the perturbation is assumedto be some local, static matter distribution, superposed to the cosmic one. Thedetails of the whole procedure may be found in [6]; the essentials are outlinedin the following.The perturbed line element is now written as: ds = (1 + h ( x, y, z )) c dt − a ( t )(1 + h s ( x, y, z ))( dx + dy + dz ) , (28)with h , h s << = χ (1 + f ( x, y, z ))Υ i = χf is ( x, y, z ) , (29) Note that, again, the definition of the Lagrangian for the Æther L U here is πGc timesthe one found in the cited article. χ , and we assume that f , f is << h ’s), and depend on the space coordinates.The time dependence is contained in the scale factor a ( t ) only. The divergence-lessness condition (9) applied to Υ must still hold, because it is broken only atthe site of a space-time defect and in this respect nothing has changed, in thesense that no other singularities have been introduced, besides the cosmic one.Eq. (9), at first order in the perturbations, becomes ~ ∇ · ~f s = 0 . (30)The invariance of the norm of Υ produces the condition f = − h . (31)The next steps may be summarized as follows: a) introduce the metric (28)in the CD action integral (11) plus matter, then linearize it in the perturbations;b) deduce the field equations for the geometry (the equivalent of the Einsteinequations); c) consider that the zero order of the equations is automaticallysatisfied with the cosmic fluid energy momentum tensor; d) write down the firstorder equations with the local matter energy momentum tensor T µν assumed tobe isotropic in space around any given point. You will get: − e − χ (cid:2) ∇ h s + χ ( ∇ h + 2 ∇ h s ) (cid:3) = 4 πGc T , (32)where ∇ = a ( ∂ x + ∂ y + ∂ z ) and T is the energy density of the local source.As we know, there is a freedom for the choice of the coordinates, so that theLorentz gauge can be imposed, leading to h s = − h . The final equation is then ∇ h = 4 πc Ge χ χ T . (33)This equation is the Poisson equation with a renormalized gravitational ”con-stant” slowly changing with time: G ∗ = Ge χ χ . (34)The cosmic vector field γ does indeed affect the local gravitational field, throughits norm χ . This influence is not perceivable on the usual time scales, in thesense that the Newtonian behaviour is fully recovered; however in cosmic timesthe effective coupling ”constant” of gravity, in the Newtonian formalism, slowlychanges. Had we started from a Minkowski background, this adiabatic effectwould not have been visible, as it happens in the Æ theory where two formallydifferent renormalizations of G are obtained at the cosmic and at the local scale.14 I Correspondence between the theories
VET, as well as the more general SME, and CD are apparently rather differentfrom each other, however, as we shall show here, it is possible to recast thelatter in a form which will make it emerge as a special case of the former. Thecomparison will then be made at the level of the effective action integrals.Considering the CD action (11) we remark that it could be thought of asbeing the result of a conformal transformation from some previous appropriatemetric. To evidence this interpretation in the present section, we shall mark theentities used in the CD theory with a ∼ assuming that ˜ g µν = e ω g µν , being ω aconformal factor. Let us rewrite (11) accordingly: S = 12 κc Z e − χ ˜ R p − ˜ g d x. (35)Consistently with the approach we are describing here, the curvature and thesquare root of the determinant of the metric tensor may be written as˜ R = e − ω [ R − g µν ∇ µ ∇ ν ω − g µν ( ∇ µ ω ) ( ∇ ν ω )] (36) p − ˜ g = e ω √− g (37)˜ χ = e ω χ (38)If we now choose the conformal factor so that χ = 2 ωe − ω , (39)since of course e ω e − χ e ω = 1, the effective Lagrangian density before thetransformation turns out to be[ R − g µν ∇ µ ∇ ν ω − g µν ( ∇ µ ω ) ( ∇ ν ω )] √− g (40)Recalling that χ is the norm of the vector field γ µ , we can rewrite the secondand third terms in the square brackets of (40) as explicitly depending on γ .The solution to this trascendental equation is the Lambert function, in par-ticular 2 ω = − W k ( − χ ) . The Lambert function W k ( z ) is a multivalued function of the complex vari-able z and k is an integer that represents the branch we are looking at. Inour case the variable is the norm of the vector field that is time-like: we mustrestrict to the case in which z is real, let us say x , and x >
0. Furthermore, wewant the conformal factor, as well as the Lambert function, to be real, so thatwe consider only the case in which the argument is greater than − /e . But inour solution the argument of the Lambert function is − χ , that is, the norm ofthe vector filed can change only between 0 and 1 /e . In order to enhance thisrange we can solve the subsequent equation, instead of 39: e ω e − χ e ω = const ⇒ χ e ω − ω = c. (41)15his simply means that we are changing the value of the dimensional con-stant in front of the action integral. The solution, now, is2 ω = c − W k ( − e c χ )and the range of the variable for which the Lambert function is real. To enhanceit we have to choose c < W ( x ) ≥ − W ( x ), or simply W ( x ), andis called the principal branch; for W ( x ) ≤ − W − ( x ).If we look for the principal branch, a Taylor series can be found but due tothe singularity at x = − /e the series converges for | x | < /e . The series is W ( x ) = ∞ X n =1 ( − n ) n − n ! x n = x − x + 32 x + . . . Rewriting the solution in terms of this series, and remebering that the actionintegral of VET is fourth order in the fields, we can drop all terms beyond secondorder, since the variable x corresponds to the norm of the vector field, that issecond order in the field.Now, we are left only with the terms that appear in ˜ R . Apart from the firstterm, that is the Ricci scalar in terms of the old metric, there are two additionalterms. Let us look at the first: g ab ∇ a ∇ b ω ∼ g ab ∇ a ∇ b ( c + e c χ − e c χ χ )= 2 e c g ab ∇ a ( γ c ∇ b γ c ) − e c g ab ∇ a ∇ b ( χ χ )= 2 e c g ab ∇ a ( γ c ∇ b γ c ) − e c g ab ∇ a ( χ ∇ b χ ) (42)where both terms are divergence, that is they reduce to surface terms whenintegrating. We are left only with the second term in (36), in which, in orderto obtain terms up to fourth order, reduce to the following one: g ab ( ∇ a ω )( ∇ b ω ) ∼ g ab ∇ a ( e c χ ) ∇ b ( e c χ )= e c g ab (2 γ c ∇ a γ c ) (cid:0) γ d ∇ b γ d (cid:1) = 4 e c g ab γ c γ d ∇ a γ c ∇ b γ d . (43)Our effective Lagrangian density is now: (cid:0) R − e c g µν γ α γ β ∇ µ γ α ∇ ν γ β (cid:1) √− g (44)What here is called γ , in the VE theories is the U vector, so that comparing(44) with the Lagrangian density in (2) we see that the CD theory correspondsto a VE theory with the only c coefficient differing from 0. Actually it is c = − e c .As we wrote in section (II), VET may be considered as special cases ofthe SME theory so that a direct comparison to that theory is appropriate. Asimple example of a cosmic vector field is the so called ”bumblebee” vectorfield illustrated in appendix B of ref.[15]. The bumblebee field is indeed a16imelike vector B dynamically depending on a suitable potential in a torsionlessspacetime. Writing B µν = ( ∇ µ B ν − ∇ ν B µ ), the Lagrangian density is assumedto be L B = ξ κ B µ B ν R µν − B µν B µν − V (cid:0) B µ B µ ± b (cid:1) (45)where ξ is a parameter and V ( x ) is a scalar potential; b sets the position of theminimum of the potential. Comparing (45) with (2) and (3) we see that theformer corresponds to the latter for peculiar combinations of the c i ’s (of courseto see this one must express R µν in terms of the g µν ’s and their derivatives, thenmaking some integrals by parts in the action). In particular the ”bumblebee”model is recovered when c = − / , c = 1 / − c . VII Conclusion
We have been analyzing in parallel the CD theory, on one side, and the VEtheories (and especially the Einstein Æther theory) on the other. Both (groupsof) theories are based on the presence of a cosmic timelike vector field, and maybe thought of as special cases of the SME theory.The VET contain a big number of adjustable parameters, and, in the ZFScase, also a free function. Various ways to exploit this wide freedom allow fordifferent approaches and lead to different conclusions. In fact, rather than aglobal scenario, a number of specific, not completely consistent pictures emerge.For instance the accelerated expansion is present in one version, and not inothers; the gravitational coupling constant has formally different limits in thecosmic and in the Newtonian limit. Furthermore the physical meaning of themajority of the free parameters of the theory is unclear.In the CD theory only one free parameter exists in the description of spacetime: a sort of global scale constant, to be determined on the basis of theobserved behaviour of the universe. One more parameter appears when con-sidering the coupling between matter and spacetime. The theory accounts forthe accelerated expansion and possesses a Newtonian limit with a renormalizedgravitational ”constant” slowly changing in time.Apart from the details, an important difference between VET and CD isin the embedding paradigm. In the case of VET we are in the mathematicalframework of vector-tensor field theories, and the hypothesized vector field, aswell as many parameters, lack a physical motivation other than the final result.In the case of CD the paradigm is based on some analogies with problems of thephysics we already know, and the vector field is thought of as the consequenceof the strain induced in a four-dimensional medium (spacetime) by the presenceof a defect, in the sense of the elasticity theory. This paradigm makes the theoryvery compact, minimizing the number of adjustable coefficients and making thecomparison with observation easier or, at least, the conclusions sharper.The fact that the real main difference is in the interpretation paradigmsappears clearly when, as we did in Sec. (VI), we show that CD may be lookedat as to a special case of VET. However, had we gone from the Vector Æther17heory to the Cosmic Defect, the corresponding peculiar choice of the parametersof VET would have appeared to be completely arbitrary. On the contrary theapproach used for CD provides a consistent interpretation scheme, which in theend is shown to be mathematically equivalent to one specific Æther theory.For these reasons we think that the CD paradigm can be fruitfully exploitedagain for a deeper understanding of the evolution of our universe.
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