Cosmological screening and the phantom braneworld model

The scalar and vector cosmological perturbations at all length scales of our Universe are studied in the framework of the phantom braneworld model. The model is characterized by the parameter $\Omega_M\equiv M^3/2m^2H_0$, with $M$ and $m$ the 5- and 4-dimensional Planck scales, respectively, and $H_0$ the Hubble parameter today, while $\Omega_M\rightarrow 0$ recovers the $\Lambda\rm CDM$ model. Ignoring the backreaction due to the peculiar velocities and also the bulk cosmological constant, allows the explicit computation of the gravitational potentials, $\Phi$ and $\Psi$. They exhibit exponentially decreasing screening behaviour characterized by a screening length which is a function of the quasidensity parameter $ \Omega_M$.


Introduction
In the braneworld (BW) model the 3 + 1− dimensional Universe we live in is a timelike hypersurface (the brane) of codimension one or more, embedded in a higher dimensional spacetime (the world), see [1,2] for a vast review and also references therein. Unlike the higher dimensional theories such as Gauss-Bonnet gravity, e.g. [3], in the BW model all standard model matter fields are confined on the brane whereas only gravity can propagate in the extra dimension(s).
The existence of the extra dimension implies departure from General Relativity. For example in the Randall-Sundrum model with a single extra dimension, the modification occurs at the small scales [4,5]. The extra dimension needs to neither be small nor compact and can even be infinite. Compact extra dimensions, on the other hand, imply an infinite and discrete Kaluza-Klein spectrum on the brane, see e.g. [6]. We further refer our reader to [7]- [12] for a description of fitting the galaxy rotation curves and we can ignore at small scales, lead to modifications of the gravitational potential at large scales [46]- [56]. By approximating the inhomogeneities of our universe as delta function sources, a first order analytical formalism for the cosmological scalar and vector perturbations for the ΛCDM model was developed recently in [46], where a Yukawa-like fall-off of the gravitational potential was derived at large scales. Various extensions of this work, including the case of interacting fluid sources, can be found in [47,48,49,50,51,52]. Discussions on the N -body simulations in the context of cosmic screening can be seen in [53,54]. We further refer our reader to [55,56] for second order computations on the scalar perturbation pertaining respectively to the ΛCDM and the Einstein de Sitter models. The extra dimensional scenario is certainly not included in the above examples. Motivated by this, we shall study in this work the first order cosmological screening in the phantom braneworld model. Our chief goal would be, apart from casting the perturbation equations in a suitable form and solving them, to point out differences of this model from ΛCDM, that can arise at very large scales.
The paper is organized as follows. In the next section we briefly review the phantom braneworld model. In Section 3 we develop the first order equations pertaining to the scalar and the vector perturbations with no bulk cosmological constant. In Section 4 we solve for the scalar perturbation ignoring the peculiar velocities, and compare it both analytically and numerically with the ΛCDM model. We conclude with a discussion Section 5.
We shall use mostly negative signature for the metric and will set c = 1 throughout.

The phantom braneworld model
Let us first briefly review the basic features of the phantom braneworld model, details of which can be seen in e.g. [40] and references therein. The relevant action is given by, where R and R are the Ricci scalars corresponding to five (the bulk) and four dimensions (the brane) and M and m are the respective Planck masses. The quantity Λ 5D is the cosmological constant in the bulk and σ is the brane tension, related to the brane cosmological constant Λ by Λ = σ/m 2 . K is the trace of the extrinsic curvature of the brane. L(g µν , φ) stands collectively for all matter fields, φ, confined to the brane and g µν is the induced metric on it. For our current purpose, φ would correspond only to the cold dark matter. Being interested in the 3+1-dimensional physics, we choose to measure energies in units of the 4dimensional Planck mass m. So, we set m = 1 throughout.
Using the Gauss-Codacci relations, the Einstein equations on the brane become are convenient parameters, and The tensor C µν is traceless, coming from the projection of the five-dimensional Weyl tensor onto the brane. Taking the divergence of Eq. (2) yields the constraint equation, The spatially homogeneous Einstein equation reads (in conformal time, η) with the cold dark matter as the source, where a ≡ a(η) is the scale factor, H = a −1 da/dη is the Hubble rate andρ the time independent background homogeneous cold dark matter density in co-moving coordinates. The constant C is due to the existence of the Weyl tensor in the bulk. Due to the radiation like behavior of the term containing C, it is often named "Weyl radiation". We shall ignore its backreaction effects onto the cosmological background, though we shall take into account the inhomogeneous perturbations of the projection of the Weyl tensor. We will also ignore the backreaction effects of Λ 5D . Taking Ω M → 0 in the above equation one recovers the ΛCDM limit. Notice that Eq. (6) in the absence of Λ 5D and C may be conveniently expressed as We will also need the derivative of this equation with respect to conformal time η , Consequently, Eq. (6) takes the simple form in everything that follows we have replacedρ in favor of 3H 2 Ω m a, M 3 in favor of 2Ω M H/a and Λ in favor of 3H 2 Ω σ . It is very convenient since everything we derive may be expressed as functions of Ω m and Ω M , only (Ω σ is solved for from Eq. (10)). The advantage of this procedure is twofold, firstly these parameters are dimensionless and we claim rather intuitive to handle, secondly these will make comparison to ΛCDM trivial by simply taking Ω M to zero.

Derivation of scalar and vector perturbation equations
We shall extend below the linear perturbation scheme developed for the ΛCDM model in [46] to the phantom braneworld model described in the preceeding section. We start with the ansatz for the first order McVittie metric on the brane in the Cartesian coordinates, 1 We would like to mention that the quantity Ω M is often defined as √ Ω l in the related literature where Φ, Ψ and B i 's are respectively the scalar and vector perturbations and the bold font is used to indicate a vector, which determines the position in space where the potentials are evaluated at. Note that unlike the ΛCDM, Φ = Ψ here, owing to the anisotropic stresses originating from the bulk, e.g. [40]. We shall consider the backreaction effects due to N self gravitating moving point masses. Following [46], we define the proper interval for the n-th mass, The peculiar velocities appearing above can be evaluated by subtracting from the observed velocity of the mass, the velocity due to the Hubble flow, e.g. [38]. The energy momentum tensor for these point masses is then given by where x n is the value of the x coordinate (as defined in the metric Eq. (11)) where the n th particle is located at. Existing data shows that the peculiar velocities are in general rather small or non-relativistic, at most of the order of 10 6 ms −1 [58]. Putting these all in together, we find from Eq. (13) the energy momentum tensor up to the first order, where each ρ n corresponds to a delta function point mass located at r n , We decompose the total energy density ρ in Eq. (14) as, where δρ(η, x) stands for the contribution of the inhomogeneities. The index n runs over all N particles in the Universe. Note here that δρ is not treated as a perturbation, due to the fact that it is dominant at small scales (see [60]). Since we must have |Ψ|, |Φ| 1 in Eq. (11), we write from Eq. (14) at first order, the geodesic equation for the n th particle in Eq. (12) also reads, where the 'prime' denotes differentiation once with respect to the conformal time η and the variations δT µν , δC µν and δQ µν depend on both space and time. Since we wish to build a perturbation scheme valid all the way to superhorizon scales, we cannot assume that the perturbations' spatial variations dominate over the temporal ones, unlike the case of the study of cosmic structures, e.g. [38].
Finally, we come to the perturbation of the Weyl tensor's projection onto the brane, δC µν . Its most generic form is given by, e.g. [40], where δπ ij = (∇ i ∇ j − g ij ∆/3)δπ C (∆ stands for the Euclidean 3-Laplacian) is trace free and δρ C , v C and δπ C are scalars. In particular, v C can be regarded as a momentum potential, whose backreaction effects will also be ignored, while considering its time evolution also negligible. The Einstein equations on the brane Eq. (2), at first order read, after using Eq. (17), Eq. (19), which is the 00 component, and for We also have for the vector perturbation, where ∆ as earlier is the Laplacian on the Euclidean 3-space and also the function m eff ≡ m eff (η) has been introduced m 2 eff ≡ 1 + Ω M The ΛCDM limit in the above equations is obtained by letting Ω M → 0 in which case we recover the results of [46]. At small length scales relevant to cosmic structures, the spatial derivatives of the potential in Eq. (20) dominate over its temporal derivatives and the other effective mass-like terms appearing on the left hand side. Accordingly, at such small scales, Eq. (20) reduces to the Poisson equation, yielding a gravitational potential falling off as 1/r, along with a modified Newton's constant [41]. For ΛCDM in particular, we have δρ C = 0, yielding Newton's potential. However, at length scales much larger than those of cosmic structures, the temporal derivative and the effective mass terms can be comparable and, as we will show in Section 4, this leads to a significant modification in the behavior of the solution of Eq. (20), as is expected due to the presence of the mass-like term on its left-hand side.
The divergence of Eq.
Eq. (22) has the same form as the corresponding solved in [46]. In this work, we are chiefly interested in distinguishing the phantom braneworld model from ΛCDM with respect to the cosmological screening, which is certainly impossible unless we go to very large length scales. Note that at such scales, the backreaction effects due to the peculiar velocities, which are essentially non-relativistic, would be negligible, e.g. [58]. Thus for our current purpose, we shall from now on ignore the peculiar velocities (and hence the vector perturbation) throughout.

Solutions ignoring peculiar velocities
On the other hand we can write Eq. (21) and recall that in a marginally closed universe with a vanishing bulk cosmological constant, one has [39], Combining Eq. (28) and Eq. (29) to eliminate δπ C we get In order to solve for Φ and Ψ we want one more equation. This comes from the spatial component of Eq. (5), after using Eq. (18) and Eq. (27), we obtain We can substitute Eq. (30) into Eq. (27) and Eq. (31) to obtain a system of two equations with only two unknowns, the perturbations Φ and Ψ. The constant in Eq. (28) and Eq. (31) has to be zero in order for the potential to be vanishing at infinity. The solution of the system is straightforward and where Eq. (33) is identical to the one obtained for ΛCDM derived in [46] if we drop the eff subscripts, it is trivial to solve our equation by comparing with [46], and using eff subscripts wherever appropriate. The solution is where For a single particle -a single central over-density -the solution for the potential Ψ, valid for all length scales is where m 0 is the mass of the central overdensity and r = |x|. We have dropped the 1 3 which is generated by the existence of an infinite number of point particles. We will prove that this occurs naturally when considering an isolated sub-region of the Universe (e.g. the observable Universe) at the end of this section. The exponential appearing above clearly indicates the suppression of Newton's potential at large scales, originating from the term present in the perturbation equation behaving as an effective mass. Thus the length scale, λ, should be interpreted as a screening length.
In every case we can find Φ solving Eq. (32) Fig. 1 − Fig. 3 elucidate various properties of the gravitational potentials and the screening length. We use the values for cosmological parameters Ω m = 0.3089 and H 0 = 67.74 km s −1 M pc −1 as specified in [59] and we examine the gravitational potentials for one particle with mass M = 1.989 · 10 30 kg. Fig. 1 depicts the behavior of the effective mass density parameter and the screening length versus Ω M . The ΛCDM limit is obtained by letting Ω M → 0.
We also note that since the screening length is typically of the order of O(10 3 ) Mpc (Fig. 1), at length scales comparable of the size of a typical cosmic structure i.e. O(100) Mpc, Eq. (37) recovers the 1/r falloff of the gravitational potentials. However, the 1/m 2 eff,Φ ≡ I/m 2 eff,Ψ and 1/m 2 eff,Ψ terms present modify Newton's 'constant' in Φ and Ψ respectively and make it time dependent, as discussed in [41]. Fig. 2 depicts the behavior of the effective Newton's constant for Ψ and Φ. In the Ω M → 0 limit both of them aproach 1 recovering the ΛCDM limit. Note also that in this limit setting furtherρ → 0 (Ω m → 0) removes the exponential fall off since then λ → ∞ (cf., Eq. (36),Eq. (34)), yielding Newton's potential for a point mass located in a de Sitter universe. It is easy to verify that, as expected, this is the linearized approximation of the Schwarzschild-de Sitter metric in the McVittie coordinate frame. Similar conclusions hold for the potential Φ one particle . Finally, we depict the potentials in Fig. 3.
We would now like to show that with respect to the universe visible to an observer located at some point x, we can actually get rid of the first term in Eq. (35). Indeed, let N be the total number of point sources in Eq. (35) and let N be the number located within the Hubble horizon radius of an observer located at  x. Clearly, we may expect that only these N particles would contribute significantly into Eq. (35). On the other hand, since we should have N → ∞ in order to obtain a non-vanishingρ, it is natural to consider N N . We next split the summations in Eq. (35) into two parts Since the second summation gets contributions from all particles outside the Hubble horizon of the observer, we can average the potential of this part following [46] and usingρ = ∞ 1 m n /V ≈ ∞ N +1 m n /V . It is easy to see that this average cancels-out the constant (1/3) term in Eq. (35), leading to Ψ many particle; average = − 1 8πm 2 eff,Ψ a N n=1 m n |x − x n | e − |x−xn | λ where "average" in the subscript refers to the aforementioned averaging over sources located outside the observer's Hubble horizon. Note that the above formula has a smooth one particle ( N = 1) limit, recovering Eq. (37).

Discussion
At very large length scales of our universe not decoupled from the cosmic expansion, one might expect the gravitational potential to be modified from that of Newton's, as has explicitly been demonstrated for the ΛCDM model in [46]. It is an interesting task to investigate the same for other viable gravity models as well. Being motivated by this, we have investigated the cosmological screening at such large length scales for the phantom braneworld model described in Section 2, with the expectation that the qualitative differences of this model compared to ΛCDM should be maximum at the (super-)horizon scales of our universe. We have presented the equations governing the first order scalar and vector perturbations in Section 3. Finally, by ignoring the backreaction effects due to the bulk cosmological constant and the vector perturbation, we have demonstrated analytically and numerically, the behaviour of the two potentials up to the superhorizon length scale in Section 4. It seems to be an interesting task to investigate the tensor perturbation for this model in an early universe scenario. We hope to address this issue in future work.