Does the Cosmological Expansion Change Local Dynamics?

It is well a known fact that the Newtonian description of dynamics within Galaxies for its known matter content is in disagreement with the observations as the acceleration approaches $a_0 \approx 1.2 \times 10^{-10}m/s^2$ (slighter larger for clusters). Both the Dark Matter scenario and Modified Gravity Theories (MGT) fails to explain the existence of such an acceleration scale. Motivated by the closeness of this acceleration scale and $c H_0\approx 10^{-9} h$ $ m/s^2$, we analyse whether this coincidence might have a Cosmological origin for scalar-tensor and spinor-tensor theories, performing detailed calculations for perturbations that represent the local matter distribution on the top of the cosmological background. Then, we solve the field equations for these perturbations in a power series in the present value of the Hubble constant. As we shall see, for both theories the power expansion contains only even powers in the Hubble constant, a fact that renders the cosmological expansion irrelevant for the local dynamics. At last, we show what a difference a theory predicting linear terms in H makes in the local dynamics.

The discrepancy between the Newtonian prediction that orbital velocities within spiral Galaxies fall off as v ∼ (M G/r) 0.5 away from the bulk of the galactic mass distribution and observations that reveal that in every spiral galaxy the velocity distribution reaches a plateau as the accelerations approach the value a 0 ≈ 1.2 × 10 −10 m/s 2 [1] led to two diametrically distinct approaches to the conundrum: (i) the Dark Matter Scenario [2] where putative non-barionic dark matter with a spherical distribution involving the disk galaxy provides the needed mass deficit to conform to the observed flat rotation curves and still adhere to the Newtonian paradigm -in this case, the Newtonian potential has a logarithmic dependence on r which is what is needed to provide the flat rotation curves; (ii) Mond Scenario [3] , [4] in which the relation between the acceleration and Newtonian gravitational potential is given by where µ(x) is a function such that µ(x) → 1 as x >> 1 to recover the Newtonian limit and µ(x) → x as x << 1 to reproduce the flat rotation curves of galaxies. One of the immediate consequences of this approach is the automatic reproduction of the Tully-Fisher Law that states that the galaxy luminosity of the galaxy scales as L ∼ v 4 , where v is the orbital velocity away from the mass distribution, provided that Luminosity tracks the Mass. The defenders of Mond claim that in order to the dark matter paradigm to conform to the Tully's -Fisher law , a very precise (and quite unreasonable ) fine-tuning between the hallo distribution and the observed mass distribution in the galactic disk is required [5]. The MOND paradigm evolved into a relativistic equation TeVeS [6] involving the metric, a scalar and a vector field phrased in terms of a Lagrangian principle. The theory is very successful in reproducing the rotation curves in spiral Galaxies but is at odds with observed background radiation anisotropies [7]. Furthemore it is in blatant disagreement with weak lensing observations. The latter is made particularly transparent by the Bullet Cluster lensing observations [8], [9].
While the dark matter paradigm cannot explain the existence of the transition acceleration scale a 0 , in TeVeS it enters as a God-Given parameter in the Lagrangian. Neither one of these possibilities is theoretically acceptable. Intriguingly, a 0 comes very close to cH 0 ≈ h10 −9 m/s 2 and raises the question whether the change on the dynamical behaviour has a cosmological origin. This avenue was exploited to some degree in the past [10] , [11].
According to Birkhoff's theorem, in pure Einstein's theory the gravitational field of a spherical symmetric mass configuration is determined by the mass within a sphere of the radius of the observed point alone. Therefore we do not expect the Universe to play any role in the local dynamics. A gauge vector field is likewise of no avail; by Gauss' theorem it also depends upon the internal configuration. Thus if the Cosmological expansion is to "leak" into the Galactic dynamics, scalar, spinors or non-gauge vector fields must be called for.
In this paper we deal with a Brans-Dicke theory and carefully write down the field equations for linearised perturbations on the top of the cosmological background. In the next section we shall write down these equations in terms of one scalar field and 3D scalar, vector and tensor fields. These equations are corrections of the dynamical equations and contain correction terms in powers of H 0 . The exact field equations are then solved perturbatively in powers of H 0 . The gravitational potential contains only even powers of H 0 and we expand it up to H 4 0 . It turns out that all corrections are way too small to play any role in the local dynamics. Then, in the following section we study a massless spinor field and show that also in this case are no linear corrections in H 0 . Since there is no a priori reason for the absence of odd powers in the Hubble constant, we discuss the prospects of a linear term in H 0 and show that it brings about noticeable changes the local dynamics .

BRANS-DICKE THEORY
Brans-Dicke theory is defined by the equations of motion and where and the matter and vacuum energy distributions are represented by where p M = 0 and p Λ = −ρ Λ , for the present state of the Universe. Consequently T M = −(ρ M + 4ρ Λ ). For future reference, we recall that We wish to construct the field perturbations on the top of a cosmological background for the Brans Dicke Theory; they represent the local matter distribution. First things first, we start by solving the equations for the background fields. In the absence of any dimensional parameter we assume that for a short time interval (the observation time )ȧ for some dimensionless η ∼ O(1). Then, with this parametrization and We identify the energy density and the pressure exerted by the field as Defining as usual ρ c = 3H 2 φ/8π and Ω X = ρ X /ρ c , from Friedmann's equations The field equation for the Brans-Dicke field yieldṡ Aiming solving the perturbed equations, we display Einstein's equations in a more convenient form and There are two relevant coordinate systems, the r-frame (r a coordinates) locally attached to the local mass distribution and the x-frame (x a coordinates) which is the cosmological comoving frame, with r α = a(t)x α . The r-frame is the physically meaningful frame for local dynamics but the x-frame turns out to be much more convenient for performing calculations. Accordingly, we construct static local disturbances in the r-frame (we are not interested in galactic evolution), make a coordinate transformation to the x-frame and perform calculations, obtaining the perturbed fields. Then, we transform them back to the r-frame. Let h ab ( r) represent the static metric perturbations in the r-frame, then the line element is where dr 0 = dt , g ab is the cosmological smooth background. Under a 'r' to 'x' coordinate transformation the line element perturbation looks where H =ȧ/a and we recall that h ab ( r) = h ab (a x).
Inspecting this form, we express the perturbed metric in the x-frameh ab in the form: ψ, W α and f αβ where ψ, W α and f αβ are to be regarded as scalar, vector and tensor fields of a flat three dimensional space . It is reasonable to assume that the global space curvature is unimportant on a local scale, thus locally we take g We represent the local mass distribution as a disturbance of the global smooth distribution. In this case, δp stands for the pressure and δρ the mass density of the local matter distribution. Locally δp = 0 and δρ = ρ G , the local Galactic mass distribution. There is still one missing field u a , the difference between the velocity of locally static observer in the r-frame with respect to a cosmological comoving observer. For a static observer in the local frame x α = a −1 r α with constant r α . Thus the corresponding velocity in the x-frame is: Recalling that V b is the velocity of the cosmological comoving observer Clearly Preparing the ground for calculating the perturbations of the field equations we first evaluate, and consequently with γ c = g ab γ c ab . We adopt the Lorentz gauge condition, in which case and simply drop the last term in eq. (27). We can express this gauge condition in terms of the effective 3D-fields: The field equations governing the local scalar field is We translate back our equations in terms of r-frame variables. In contrast to the comoving derivative ξ ,α = ∂ξ/∂x α we define the local derivative ∂ α ξ = ∂ξ/∂r α . Then and as the rule of the thumb we automatically replace everywhere ∂/∂x α → a∂/∂r α . Furthermore Then With the the replacemenṫ the scalar field equation (eq. (34)) looks in its final form The field equations for the gravitational field are given by the linear perturbations of Einstein's equations: Let me start with the lhs. We borrow from MTW [12]: and rewrite the divergence of the gauge condition [eq. (28)] in the form With the rule for the commutation of derivates for (1, 1) tensors it follows that and then (46) This expression is quite general. For a homogenous and isotropic background the Weyl tensor vanishes, and the Riemann tensor is entirely described by the Ricci curvature : (48) In that case Furthermore, The linear variation of eqs. (17) and (18) provide the source terms of the gravitational field equations: together with Working out the components We shall put all the pieces together ,(51)-(53) with eqs. (54)-(56) and (59)). We use the gauge conditions (eqs.(30)) and the replacements (36), (38). The 'scalar equation' that arises from the 00 component is while the vector equation that arises from the 0α component is −H 2 f αβ +H 2 P ψ + Qξ + r · ∂ξ δ αβ +H 2 − η 2 where we defined the numerical coefficients coefficients :

Solving the equations by Perturbation
At this stage a remark of caution is in order. Albeit the perturbation fields ψ, W α and f αβ stand forh αβ , are functions of the local coordinate r, they are still metric perturbations in the x-frame [see eqs. (20),(21)]: (73) Transforming back to the r-frame: Clearly We shall consider spherically symmetric configurations alone. In this case where A, B and W are 'scalar fields'. Then and also Next we introduce these expressions into the their corresponding equations (64) , (64)-(66) and solve them per-tubatively in powers of H. The zeroth order satisfying the gauge conditions is The easiest way of getting W is by substituting the previous results into the gauge condition (30) . From now on we drop numerical coefficients, then and by virtue of (79) it follows that W ∼ M G and no r-dependence and then To the second order we have and (Ω M + 4Ω Λ )(ξ (0) + ψ (0) ) − η(f (0) + 3ψ (0) ) + 2(η + 2) r · ∂ξ (0) + r α r β ∂ α ∂ β ξ (0) (88) whose solution is αβ ∼ M Gr(δ αβ +r αrβ ) at higher orders Acccordingly, The fourth order equations for ψ (4) and ξ (4) are identical to (87),(88) and therefore Thus by virtue of eq. (75), The term H 2 r 2 arrives from the coordinate transformation from the x frame to r-frame [see eq (75)]. Comparison with the Newtonian potential term GM/r tells that it becomes relevant as r 3 ∼ M GH −2 or r ∼ 400kpc for a typical galaxy. On the same grounds, he correction H 2 M Gr becomes relevant only at the Hubble distance r ∼ H . Notice that there are no linear terms on H that could bring about relevant corrections to the local dynamics.

SPINOR FIELD
In Brans-Dicke theory the lowest order in H corrections of the field equations are quadratic in the Hubble constant. We wonder if a spinor field, whose energy momentum tensor contains first derivatives of the spinor field could remedy the problem and yield larger contributions. Since we agreed upon not to settle the scale of a 0 through external given parameters, we concentrate on a massless particle. All non-zero momentum modes can be swept into the energy momentum tensor of the matter distribution and the discussion is similar to that of the previous section. Nevertheless, the zero mode has no particle content and must be dealt separately. We think this mode as being a cosmological substrate that is deformed in the presence of a mass distribution and calculate its contribution to the energy-momentum tensor.
The energy momentum tensor is where the swapping m ↔ n of indexes is carried for symmetrisation. The tetrads of the Robertson-Walker metric are diagonal: where Greek indexes run over the spatial components and a = a(t) is the cosmological radius scale . In this case the only non-vanishing components of spin-connection are after some algebra the Dirac Equation reads As in the previous section, W α starts at the order∼ O(H) (it is related to T 0α equation and it vanishes for a static configuration). Then to the lowest order in H Applying γ · ∂ on both sides whose solution is and primed functions means they are expressed in terms of r ′′ . Expanding he spinor equation (112) to the first order in H reads The energy momentum tensor corresponding to disturbance of the cosmological substrate is We are mainly interested in the δT 00 component. Recalling that C 0ab = 0 , Ψ(t) = (a 0 /a) 3/2 Ψ 0 we get NowΨγ a Ψ is real and the currentΨ 0 γ α Ψ 0 = 0 since there is no preferred cosmological direction. Furthermore, for a spherical symmetrical configuration σ 0αβ = 0 [ see eq.(110)], thus Clearly, in a spherical symmetrical configuration F αβ is symmetric, thus where F = α F αα . Accordingly, a spinor cannot induce a first order in H correction to the Newtonian potential.
Unforseenably, none of the field theories studied in this paper can produce odd corrections in H to the local gravitational fields and therefore, cannot bring about substantial corrections to the local dynamics.
In the lack of a general principle forbidding odd powers in H, it is conceivable that some field theory could bring about odd powers in the H-expansion. Should such a theory exist, the lowest order corrections are linear in H and on dimensional grounds ψ ∼ − M G r + Hr + HM G ln(r) + . . .
Accordingly, the velocity profile, away from the mass distribution would be v 2 ∼ M G r + Hr + M GH + · · · (125) The last term yields flat rotation curves, but comparing to the Newtonian term reveals that it becomes relevant only at scales r 0 ∼ H −1 , thus meaningless. The second term gives a linearly growing velocity curve at a very much small slope such that could be mistakenly taken for a flat rotation curve at galactic scales. Furthermore, comparison with the Newtonian potential reveals that it becomes relevant at scales r 0 ∼ (M G/H) 0.5 ∼ 5kpc for a typical galaxy. At the r ∼ r 0 region where there is dynamical transition from the Newtonian behaviour to the Hr term the velocity scales is v 4 0 ∼ M 2 G 2 /r 2 0 ∼ M GH, which is nothing but Tully-Fisher's Law ! Furthermore, the corresponding acceleration scale in this region a 0 ∼ v 2 0 /r 0 ∼ H. Needless to say the utmost importance of scrutinizing field theories that could bring about linear corrections in H to the gravitational potential or either showing that odd term corrections are forbidden. acknowledgements I am grateful to Prof. J. D. Bekenstein for enlightening discussions.