Analysis on the evolution of the effective Geff from Linear Nash-Greene fluctuations

From the linear Nash-Greene fluctuations of background metric, we present the perturbation equations in an embedded four space-time. In the context of a five-dimensional bulk, we show that the perturbations are only propagated by the gravitational tensorial field equation. In a Newtonian conformal gauge, we study the matter density evolution in sub-horizon regime and on how such scale may be affected by the extrinsic curvature. We use the “extended Gold 2018” growth-rate dataset with 25 datapoints and the best fit Planck2018/ΛCDM parameters of TT,TE,EE+lowE spectra on 68% interval. Hence, we determine the evolution equation for the density growth δ(a) as a function of the expansion factor a as a result from the embedded equations of the background geometry. By using solar constraints, we analyse the evolution of the effective Newtonian constant Geff and showing that applying Taylor expansion to theGeff (a) function under the constraint of time-derivative of Geff,0 = G, G is the bare Newtonian constant, at a = 1 in matter domination era, we get an agreement with Big Bang Nucleosynthesis (BBN) and also a reduction of the 3-σ tension to 1-σ contour between (σ8-Ωm) of the observations from Cosmic Microwave Background (CMB) and Large Scale Structure (LSS) probes. Corresponding author. ar X iv :2 00 9. 02 16 7v 1 [ gr -q c] 3 S ep 2 02 0


Introduction
The Occam's razor is one of the cornerstone philosophical principles in science that states that the simplest solution of a problem must be followed in detriment of the complex ones. In this realization, the ΛCDM model has been the simplest and successful solution to deal with the accelerated expansion of the universe as corroborated for several independent observations in the last two decades [1][2][3][4][5][6][7][8][9][10][11]. Despite its success, the ΛCDM model has important drawbacks that must be taken into account. For instance, the main components of ΛCDM model lack a fundamental explanation about the nature of the cosmological constant Λ and also the (Cold) Dark Matter (CDM) problem [12][13][14][15][16][17][18].
In this paper, we focus on studying the 3-σ tension in the σ 8 -contours revealed by the mismatch of the data inferred from Planck Cosmic Microwave Background (CMB) radiation and Large Scale Structure (LSS) observations considering the concordance ΛCDM model as a background. The σ 8 is the r.m.s amplitude of matter density at a scale of a radius R ∼ 8h.Mpc −1 within a enclosed mass of a sphere [19]. The main problem apparently resides in the fuzzy origin of such mismatch, which could be a result of systematics or due to deviations of gravity. The situation still resists in both early and late universe landscapes even if one does not consider the Planck CMB data [20] and evidences that similar tensions may also occur in the matter distributions around 2-σ [11,[21][22][23]. In particular, to avoid a biased dependence of σ the quantity f σ 8 (z) is a good model-independent discriminator for mapping the growth rate of matter. This alleged tension opens an interesting arena for testing gravitational models, once the possibility to alleviate it may come from modified gravity [24][25][26][27][28][29][30].
In the context that gravity may be modified departed from Einstein gravity or other fundamental principle, we explore the embedding of geometries (or hypersurfaces) to elaborate a model independent based on seminal works on the subject [31][32][33] in order to tackle the aforementioned σ 8 -issue in the problem of explanation of the accelerating expansion of the universe. In hindsight, the seminal problem of embedding theories lies in the hierarchy problem of fundamental interactions. The possibility that gravity may access extra-dimensions is taken as a principle for solving the huge ratio of the Planck masses (M P l ) to the electroweak energy scale M EW in such M P l /M EW ∼ 10 16 . This option has been explored more vigorously in the last two decades as a solution to the dark energy paradigm. Most of these models have been Kaluza-Klein or/and string inspired, such as, for instance, the works of the Arkani-Hamed, Dvali and Dimopolous (ADD) model [34], the Randall-Sundrum model [35,36] and the Dvali-Gabadadze-Porrati model (DPG) [37]. Differently from these models with specific conditions, and apart from the braneworld standards and variants, we have explored the embedding as a fundamental guidance for elaboration of a gravitational physical model. Until then, several authors explored embedding of geometries and its physical consequences as a mathematical structure to apply to gravitational problems [31][32][33][38][39][40][41][42][43][44][45][46][47][48][49][50][51].
The plan of the paper is organized in sections. In the second section we revise the embedding of geometries and how it may be used to construct a physical framework. In this context, the Nash-Greene theorem is discussed. The third and forth sections verse on the background Friedmann-Lemaître-Robertson-Walker (FLRW) metric, transformations and gauge variables involving the extrinsic curvature, respectively. The fifth section shows the resulting conformal Newtonian gauge equations. In the sixth section, we show the contrast matter density δ m (a) as a result of the Nash fluctuations and an effective Newtonian constant G ef f is also determined that carries a resulting signature of extrinsic curvature. In the seventh section, we compare our model with the ΛCDM model in f σ 8 (z) measurements and use the "extended Gold 2018" compilation on growth-rate data and the Planck 2018 TT, TE, EE+lowE spetra within the 68% interval for the best fit related parameters. We perform our analysis on the growth factor and on G ef f using the data points of SDSS [52][53][54], 6dFGS [55], IRAS [56,57], 2MASS [56,58], 2dFGRS [59], GAMA [60], BOSS [61], WiggleZ [62], Vipers [63], FastSound [64], BOSS Q [65] and additional points from the 2018 SSSD-IV [26,[66][67][68]. In the final section, we present our remarks and prospects.
It is noteworthy to point out that we adopt the Landau time-like convention (−−−+) for the signature of the four dimensional embedded metric and speed of light c = 1. Concerning notation, capital Latin indices run from 1 to 5. Small case Latin indices refer to the only one extra dimension considered. All Greek indices refer to the embedded space-time counting from 1 to 4. Hereon we indicate the non-perturbed (background) quantities by the upper-script symbol "0".

The induced dimensional equations in a four embedded space-time
Although embedding can be made in an arbitrary number of dimensions (see [31-33, 41, 42, 45-48, 50, 51]), the current alternative models of gravitation are normally stated in five dimensions at most. Then, we start with a model defined by a gravitational action functional in the presence of confined matter field on a four-dimensional embedded space-time embedded in a five-dimensional larger space that has the form where κ 2 5 is a fundamental energy scale on the embedded space, 5 R denotes the five dimensional Ricci scalar of the bulk and L * m denotes the confined matter Lagrangian in such the matter energy momentum tensor fulfills a finite hypervolume with constant radius l along the fifth-dimension. Thus, the variation of Einstein-Hilbert action in Eq.(2.1) with respect to the bulk metric G AB leads to the Einstein equations where α is the energy scale parameter and T AB is the energy-momentum tensor for the bulk [31][32][33]42]. In accordance with the Nash-Greene theorem [69,70], that verses on orthogonal perturbations of the metric in which induce the appearance of the second curvature in that direction, an embedded space-time results from the background fluctuations. To our purposes, we are restricted to the four-dimensionality of the space-time embedded in a five dimensional bulk following the confinement hypothesis [71,72] such dimensionality will suffice based on experimentally high-energy tests [73]. This model can be regarded as a four-dimensional hypersurface dynamically evolving in a five-dimensional bulk with constant curvature whose related Riemann tensor is where G AB denotes the bulk metric components in arbitrary coordinates and the constant curvature K * is either zero (flat bulk) or it can have positive (deSitter) or negative (anti-deSitter) constant curvatures.
In accordance with recent observations [11], with a very small value of the cosmological constant Λ, we do not consider any dynamical contribution of such quantity. Then, we chose K * = 0, although our results also hold for any other choice of K * . The bulk geometry is actually defined by the Einstein-Hilbert principle in Eq.(2.1), which leads to Einstein's equations as shown in Eq.(2.2). The confinement condition implies that K * = Λ * /6 = 0 and the confined components of T AB are proportional to the energy-momentum tensor of General Relativity (GR): α * T µν = 8πGT µν , where G is the bare gravitational Newtonian constant. On the other hand, since only gravity propagates in the bulk we have T µa = 0 and T ab = 0. In this sense, it is possible to search a more general physical theory based on the geometries of embedding. Although it is not explicitly showed here, depending on the type of the embedding (e.g., local or global, isometric, analytic or differentiable, etc.), brane-world models may be an example of this framework [32]. Another important aspect of the original Nash embedding is that it was applied to a flat D-dimensional Euclidean space and explored by J. Rosen [74] with an analysis on pseudo-Euclidean spaces. Its generalization to pseudo-Riemannian manifolds to non-positive signatures and the result that the embedding of the space-times may need a larger number of dimensions was made only two decades later by Greene [70]. Hereon, we simply call the Nash-Greene theorem.
In a nutshell, the smoothness of the embedding is the cornerstone of the Nash-Greene theorem, once this embedding results from a differentiable mapping of functions of the manifolds. On the other hand, it is not capable of telling us about the physical dynamic equations or evolution of the gravitational field. Thus, a natural choice for the bulk is that its metric satisfies the Einstein-Hilbert principle. By design, it represents the variation of the Ricci scalar and the related curvature must be "smoother" as possible [41]. It warrants that the embedded geometry and their deformations will be differentiable too.
Let be a Riemannian manifold V 4 endowed with a non-perturbed metric (0) g µν being locally and isometrically embedded in a five-dimensional Riemannian manifold V 5 . Given a differentiable and regular map X : V 4 → V 5 , one imposes the embedding equations where we have denoted X A the non-perturbed embedding coordinate, G AB the metric components of V 5 in arbitrary coordinates, and 0 η denotes the non-perturbed unit vector field orthogonal to V 4 . This mechanism avoids possible coordinate gauges that may drive to false perturbations. The colons denote ordinary derivatives. The meaning of those former set of equations is that Eq.(2.3) represents the isometry condition between the bulk and the embedded space-time. The orthogonality between the embedding coordinates X and 0 η is represented in Eq. The main concern of our work is provide a complement to the Einstein gravity by adding a second curvature, i.e., the extrinsic curvature and to study its implications to a physical theory. As defined in traditional textbooks [75], the extrinsic curvature of the embedded space-time V 4 is the projection of the variation of the vector 0 η onto the tangent plane A geometric objectΩ always can be constructed in V 4 in any direction 0 η by the Lie transport along the flow for a certain small distance δy. It is worth noting that it is irrelevant if the distance δy is time-like or not, nor it is positive or negative. Then, the Lie transport is given by Ω =Ω + δy£0 ηΩ , where £0 η denotes the Lie derivative with respect to the normal vector 0 η. In this sense, the Lie transport of the Gaussian coordinates vielbein {X A µ , 0 η A a }, defined on V 4 , can be written as Interestingly, from Eq.(2.8), it is straightforward the derivative of 0 η is affected by perturbations in a sense η ,µ = 0 η ,µ . Concerning perturbations of the embedded geometry V 4 , there is a set of perturbed coordinates Z A to satisfy the embedding equations likewise Eqs.(2.3), (2.4) and (2.5), as As seen in the non-perturbed case, the perturbed coordinate Z defines a coordinate chart between the bulk and the embedded space-time. Replacing Eqs.(2.7) and (2.8) in Eqs.(2.9) and (2.6), for instance, we obtain the fundamentals objects of the new manifold in linear perturbation It is important to note that the Nash-Greene fluctuations on the perturbed metric g µν = g (0) µν + δg µν + δ 2 g µν + .... are continuously smooth and naturally go on adding small increments δg µν of the background metric.
In addition, Nash's deformation formula can be obtained by the derivative of Eq.(2.10) with respect to y coordinate given by It is noteworthy to point out that the ADM formulation gives a similar expression later discovered by Choquet-Bruhat and J. York [76]. In a physical context, the interpretation of Eq.(2.12) reinforces the confinement of matter as a consequence of the well established experimental structure of special relativity, particle physics and quantum field theory, using only the observables which interact with the standard gauge fields and their dual properties. It imposes a geometric constraint that it localizes the matter in the embedded space-time [31,32].
To avoid redundances, the parameter y does not appear explicitly in the line elements once the perturbation process is triggered. It also holds true for any perturbations resulting from a n-parameter families of embedded submanifolds extended to a larger set of y a . The resulting perturbed geometry can be bent and/or stretch without ripping the embedded spacetime. This feature is exclusive to embedding geometries with dynamical embeddings which it is not possible to do in the Riemannian realm as acknowledged by Riemann himself [77].
The integrability conditions for these equations are given by the non-trivial components of the Riemann tensor of the embedding space given by where 5 R ABCD is the five-dimensional Riemann tensor. The semicolon denotes covariant derivative with respect to the metric. The brackets apply the covariant derivatives to the adjoining indices only. The first equation is called Gauss equation that shows that Riemann curvature of the embedding space acts as a reference for the Riemann curvature of the embedded space-time. The second equation (Codazzi equation) evinces the projection of the Riemann tensor of the embedding space along the normal direction that is given by the tangent variation of the extrinsic curvature. This guarantees to reconstruct the five-dimensional geometry and to understand its properties from the dynamics of the four-dimensional embedded space-time V 4 . As a result, we can write in embedded vielbein {Z A ,µ , η A } for the metric of the bulk in the vicinity of V 4

Background FLRW metric
The basic familiar line element of FLRW four-dimensional metric is given by where the expansion factor is denoted by a ≡ a(t). The coordinate t denotes the physical time. In the Newtonian frame, the former equations turns out to be 3.1 Non perturbed field equation in a embedded space-time Using Eqs. (2.2), (2.15) and (3.2), we can obtain the induced field equations in a fivedimensional bulk where the energy-momentum tensor of the confined perfect fluid is denoted by T µν are given as usual [78,79]: where the Hubble parameter is defined in the standard way by H ≡ H(t) =ȧ a .
The non-perturbed extrinsic term Q where we denote the mean curvature h 2 = h·h and h = 0 g µν 0 k µν . The term is the Gaussian curvature. The equation (3.5) is readily conserved in the sense that Since the extrinsic curvature is diagonal in FLRW space-time, one can find the components of extrinsic curvature using Eq.(3.4) that can be split into spatial and time parts: In the Newtonian frame, the spatial components are also symmetric and using the former relation one can obtain k and the following objects can be determined: where we define the function B = B(t) ≡˙b b in analogy with the Hubble parameter.
Since the conservation equation of the deformation tensor in Eq.(3.6) is identically satisfied in its covariant form, the only new information gained comes from calculating the components of Q µ µ;i = 0 with respect to spatial coordinates in the FLRW Newtonian frame, likewise the procedure of obtaining Euler equations in GR. Using the relations in Eqs.(3.8), (3.9), (3.10) and (3.11), we obtain B H = α 0 , where α 0 is an integration constant. Without any further ado, it can be easily solved by quadrature and gives the relation It is worthy noting that the former relation completes univocally the set of the components of the extrinsic curvature in Eq.(3.7), as shown in previous works [32,42].

Hydrodynamical equations
The stress energy tensor in a non-perturbed co-moving fluid is given by The conservation of T (0) µν;µ = 0 leads to 13) and the resulting Friedmann equation turns where ρ (0) is the present value of the non-perturbed matter density (hereon . For a pressureless fluid, it is a standard way to write the matter density in terms of redshift as and we can rewrite Eq.(3.14)in terms of redshift as Using the definition of the cosmological parameter Ω i = 8πG where Ω m(0) is the current cosmological parameter for matter content and for a flat universe Ω ext(0) = 1 − Ω m(0) and H 0 is the current value of Hubble constant in units of km.s −1 Mpc −1 .
It is worth noting that Eq.(3.16) with the α 0 -parameter nearly resembles the wCDM model in terms of comparison with their Friedmann equations at background level, where w is a dimensionless parameter of the fluid equation of state w = p ρ [80]. As we are going to show, the strikingly differences will appear at perturbation level. In this particular study, we do not consider the radiation term since it can be neglected for late times. Likewise, in conformal time η such that dt = a(η)dη and H ≡ aH, we can write the Friedmann equation in this frame as where k 0 ≡ 8 3 πG and the conformal Hubble parameter is H = a a . The prime symbol represents the conformal time derivative. Hence, the conformal time derivative of Hubble parameter is given by and completes the set of equations for a non-perturbed fluid in a conformal Newtonian frame.

Transformations and gauge variables
Using the standard line element of FLRW metric in Euclidean coordinates in Eq.(3.2), one finds where a = a(η) is the expansion parameter in conformal time. We start with the standard process as known in GR [78,79]. The novelty of this approach is the inclusion of the extrinsic curvature in the theoretical framework. Thus, let be the coordinate transformation x α → x α = x α + ξ α such as ξ α 1, then we have for a second order tensor . Henceforth, we can write the perturbed metric tensor in the new coordinates δg αβ as where the infinitesimally vector function ξ α = ξ (4) + ξ i can be split into two parts in which ξ i ⊥ is the orthogonal part decomposition and ζ is a scalar function. The prime symbol denotes the derivative with respect to conformal time η. As a result, we can obtain

4)
δg 44 = δg 44 − 2a(aξ (4) ) . (4.5) Using Eq.(4.2), we obtain a similar transformation for k µν as And taking into account the Nash-Greene theorem where we denote g ∂y , and y is the coordinate of direction of perturbations from the background to the extra-dimensions (in this case, just one extra-dimension). Thus, we can rewrite Eq.(4.6) as and we get straightforwardly Taking the previous expressions and to avoid the implications of the ambiguity of two "times" coordinates, likewise the Rosen bi-metric theory [2,81] that led to erroneous results such as a dipole gravitational waves, we adopt y as a set of space-like coordinates. Then, one obtains For scalar perturbations the metric takes the form where φ = φ( x, η), ψ = ψ( x, η), B = B( x, η) and E = E( x, η) are scalar functions. For the tensors G µν , T µν and Q µν , one can use the same set of transformations. In this sense, for small perturbations, we can write the Einstein tensor in a coordinate systemx as where δG µν denotes linear perturbations in the new coordinate system Immediately, we have a similar expression for T µν T µν = T (0) µν + δT µν , that leads to 14) and also, for the deformation tensorQ And using Eq.(4.12), we obtain for δG µν : For the perturbed stress energy tensor δT µν , one obtains the set of equations Likewise, for the perturbed extrinsic part δQ µν we have

Scalar perturbations in newtonian gauge
In longitudinal conformal Newtonian gauge, the main condition resides in the vanishing functions of B = B( x, η) and E = E( x, η), as well as the quantities ξ (4) , ξ , ζ. Hence, the metric in Eq.(4.12) turns to be where Φ = Φ( x, η) and Ψ = Ψ( x, η) denotes the Newtonian potential and the Newtonian curvature, respectively. In addition, we obtain a simplification of the previous transformations of the curvature-related quantities and the set of following outcomes:

Perturbed gravitational equations
Taking into account all the former results of Eqs.(5.2), we can write the perturbed field equations as µν + δg µν generated by Nash's fluctuations are given byg (5.5) and the related perturbed extrinsic curvaturẽ where we can identify δk µν = (0) g σρ k and we have respectively the components in the conformal frame, Finally, the perturbation of the deformation tensor Q µν can be made from its background form in Eq.(3.5) and the resulting k µν perturbations from the Nash fluctuations of Eq.(5.7) such as The quantity δQ µν is also independently conserved in a sense that δQ µν;ν = 0. Moreover, using the background relations of Eqs. (3.8), (3.9), (3.10), (3.11), we can determine the components of δQ µν and we get the basic gauge invariant field equations in the conformal Newtonian frame: where we denote γ 0 = α 0 b 2 0 .

Hydrodynamical gravitational perturbed equations
For a perturbed fluid with pressure p and density ρ, one can write the perturbed components of the related stress-tensor where δu i denotes the tangent velocity potential and ρ 0 and p 0 denote the non-perturbed components of density and pressure, respectively. Hence, we can rewrite Eqs.(5.15), (5.16) and (5.17) as Those set of equations can be better understood in the Fourier k -space wave modes. Taking the Fourier transform of each main quantity (with subscript "k "), we obtain a new set of equations:

Matter density evolution under subhorizon regime
In order to obtain a bare response of an influence of the extrinsic terms, we do not consider anisotropic stresses and pressure for Eq.(5.25), Eq.(5.25) and Eq.(5.26), we obtain the following equation It is important to notice that when γ 0 → 0 in Eq.(6.1), the standard GR equations are obtained and we can recover the subhorizon approximation with k 2 H 2 or k 2 a 2 H 2 which means Φ k , HΦ k ∼ 0. To determine the gravitational potential Φ, we need to work with the continuity and Euler equations from calculating the components δT µ µ;4 and δT µ µ;i to give, respectively where the former expressions can be also written in terms of the fluid parameter w = p 0 ρ 0 . Taking Eq.(6.2) under a Fourier transform, we obtain the following equation in the k -space: which in subhorizon approximation gives For the pressureless form of Eq.(6.3), we have and using the background formula from conservation equation of Eq.(3.13), we have Performing the definition of the "contrast" matter density δ m ≡ δρ ρ 0 , and using Eqs. (6.4), (6.5) and (6.1), we obtain a relation with Φ k and δ m as where G ef f is the effective Newtonian constant and is given by Taking the conformal time derivative of Eq.(6.4) and writing Eq.(6.5) in terms of δ m and using Eq.(6.6), we obtain the equation of evolution of the contrast matter density δ m (η) in conformal longitudinal Newtonian frame To express Eq.(6.8) in terms of the physical time, we use the notationδ m ≡ δ m a and obtain the useful relation δ m = a 2δ m + a 2 Hδ m . We are leading tö Finally, we can obtain an alternatively way to express the former equation in terms of the expansion factor a(t). We use the notation dδm which solutions are possible only numerically. For instance, in the context of GR, where G ef f = G that turns δ m (a) independent of the scale k, with the fluid parameter w, one has the following solution where 2F 1 (a, b; c; z) is a hypergeometric function.
7 Analysis on evolution of G ef f and f σ 8 The form for the effective Newtonian constant is given by Eq.(6.7) as a result of the linear Nash fluctuations of the metric and the extrinsic curvature as shown in Eq.(2.11). To alleviate the tension of f σ 8 , one possible way is to modify gravity in some sense [24][25][26][27][28][29][30] and hence, the G ef f function plays a paramount role. Firstly, we adopt the minimum value of expansion parameter a min = 0.001 and for sub-horizon scales we fix the value of k = 300 H 0 ∼ 0.1hMpc −1 with a numerical analysis from a modified available publicly code [25]. The former consideration holds our analysis in the linear matter growth landscape. Since gravity has been tested in several experiments at solar scale, the G ef f function must be constrained by G ef f a=0 = 1, G ef f a=1 = 1 and at solar scale by the ordinary time-derivative dG ef f dt a=1 = 0. The latter relation is satisfied when γ 0 → 0 for b 0 → 0, once γ 0 = α 0 b 2 0 that means that the extrinsic curvature must vanish at solar scale and only at cosmological range the extrinsic curvature may induce topological changing of the universe [31,32,42,47,48]. As familiar in most modified gravity models, we also assume that G ef f is scale independent and does not depend on the matter density [25,26]. Thus, we expand Eq.(6.7) around a = 1 in a Taylor series and once we have fixed the wave-number k, we can blend all constants (b 0 is an arbitrary constant) and simplify G ef f in terms of two non-dimensional parameters (α 0 , β 0 ) as Accordingly, the related graphic behavior is shown in Fig.(1). We show five curves with the values of α 0 varying from 0.5 to 4, where we have summarized the results of minimizing data in Table 1. Error estimates were calculated using Fisher Matrices around the related best-fit values for each model. In the course of this study, we realized that the values of α 0 parameter seem to obey a narrower window of validity on the range α 0 = [1,3], and outside of this range, the G ef f constraints are not satisfied and the related models are ruled out. Also, this set of models defined by α 0 matches the Big Bang Nucleosynthesis (BBN) expectations  Table 1.
Another test consist to use the σ 8 parameter that measures the amplitude of growth of r.m.s fluctuations on the scale of 8h −1 Mpc and is an important reference for selection of dark energy and/or modified gravity models. A biased-free analysis can be performed by the measure of the quantity f σ 8 (a) ≡ f (a).σ 8 (a) .
where f (a) = ln δ ln a is the growth rate and the growth factor δ(a) is given by Eq.(6.9). In the χ 2 -statistics, one must consider the observed growth parameter f (a obs ) in minimization due to the Alcock-Paczynski effect to take into account redshift-space distortions. We use the "extended Gold-2018" growth-rate compilation as shown in Table 2 on the data points of SDSS [52][53][54], 6dFGS [55], IRAS [56,57], 2MASS [56,58], 2dFGRS [59], GAMA [60], BOSS [61], WiggleZ [62], Vipers [63], FastSound [64], BOSS Q [65] and an additional points from the 2018 SSSD-IV [26,[66][67][68]. These additional data points provide the growth-rate at relatively higher redshifts. Moreover, as pointed out in Refs. [25,26], to compatibilize the data dependence from the fiducial cosmology and another cosmological surveys, it is necessary to rescale the growth-rate data by the ratio r(z) of the Hubble parameter H(z) and the angular distance D A (z) by where the subscript "f " corresponds a quantity of fiducial cosmology. Similarly, the compatibilization of the related χ 2 statistics is also necessary. It can be done using the expression denotes a set of vectors that go up to ithdatapoints at redshift z i for each i = 1...N . N is the total number of datapoints of a related collection of a data. The set of f σ 8,i datapoints come from theoretical predictions [25]. The set of C −1 ij denotes the inverse covariance matrix. A final important correction concerns the necessity to disentangle the datapoints related to WiggleZ dark energy survey which are at first correlated. Then, the covariant matrix C ij [62] is given by where the set of σ 2 's denote the N -variances. In Fig.(2), we present the σ 8 -contours with 68.3%, 95.4% and 99.7% confidence levels (C.L.) in the (σ 8 − Ω m ) plane. In the left panel, as expected, we identify the 3-σ tension for the model α 0 = 0 that means that the extrinsic curvature vanishes (and Einstein equations of GR are restored) and thus G ef f = G reproduces Planck2018/ΛCDM discrepancy. The black points denote the best-fit point of the models for the values of the σ 8 parameter. The right panel shows the model for α 0 = 2, β 0 = 0.71 ± 0.265 and χ 2 = 18.64 and exhibits the 3-σ tension ceases falling well within the 1-σ level. In both panels, the red points pinpoint the best-fit of Planck2018/ΛCDM values of TT, TE, EE+lowE spectra for the (σ 8 − Ω m ) plane.
In addition, we present in Fig.(3) the resulting plot of the f σ 8 from the datapoints of Table 2   where Inverf(x) is the inverse of the error function Erff (x) and Γ(1, ∆χ 2 2 ) is the incomplete gamma function, with ∆χ 2 = χ 2 model (2) − χ 2 model (1) . The resulting σ-distance D σ = 1.79 results from the distance of the best-fit values of our model and Planck 2018 results within the 1-σ contour. On the other hand, we obtain D σ = 2.898 for ΛCDM and Planck18/ΛCDM which reinforces that the σ 8 tension persists in the Planck2018 dataset per se.
The three additional lines represent the best-fit of G ef f evolution for the models defined by the value of α 0 = (1, 2, 3) as presented in Table 1. In Fig.(03), they are represented by the blue thick, black dot-dashed and red thin lines, respectively. In particular, the growth rate fluctuations represented by the blue thick line α 0 = 2 is the best accommodated curve considering the constraints from various redshift surveys [11] and reinforces the previous results as shown in Fig.(1) and (2) consisting a promising result for further studies.

Remarks
In this paper we studied cosmic perturbations of matter in a search of understanding if the contribution of the extrinsic curvature to complement Einsteinian gravity is rather than semantics and relies on physical reality. It may pinpoint a renewed consideration of the concept of curvature, a paramount element of contemporary physics, as a fundamental physical agent itself. From the linear Nash-Greene perturbations of metric, we have shown how to transpose the initial process in the background metric of the embedding of geometries to trigger the perturbations. In this sense, we have shown the perturbed field equations. An interesting fact resides that in five dimensions the gravitational tensor equation is indeed a perturbed equation, once the perturbation of the Codazzi equation does not propagate cosmological perturbations being hampered by linear Nash's fluctuations. On the other hand, this landscape can be dramatically different in dim ≥ 6 with appearance of new geometric objects, such as the third fundamental form A µνa that is associated to gauge fields. We also calculated the longitudinal Newtonian gauge of this framework in the simplest case that the gravitational potentials coincide Ψ = Φ. Moreover, we have obtained in the subhorizon scale the contrast matter density ignited by the embedding equations. The finding of the matter overdensity equation δ m is a paramount quantity for latter studies to identify any signature of modifications of gravity due to cosmic acceleration. We also have shown the determination of effective Newtonian constant G ef f that matches the Big Bang Nucleosynthesis (BBN) constraint. Furthermore, the 3-σ tension paradigm may be solved as a result from a modification of gravity with a narrow set of values for the parameters of the model and a consistent behaviour of f σ 8 growth descriptor. These results pose an interesting scenario since the model seems to provide a necessary gravitational strength to correct the σ 8 discrepancy. This was obtained by the inclusion of the extrinsic curvature as a pivot element to modify standard Einstein's gravity. As prospects, the integrated Sachs-Wolfe (ISW) effect will be analysed for the G ef f model in the light of large surveys on dark energy and to study the impact of the model on CMB power spectrum.