A space dependent Cosmological Constant

In a specific adiabatic perfect fluid, intrinsic entropy density perturbations are the source of a space-dependent cosmological constant responsible for local void inhomogeneity. Assuming an anisotropic Locally Rotationally Symmetric space time, using the 1+1+2 covariant approach and a Lemaître space time metric, we study the cosmological implication of such a scenario giving a proper solution to the Hubble constant tension and providing, locally, also an effective equation of state with w ≤ -1.


Introduction
The presence of a Dark Fluid (DF) component in the composition of the Universe is posing challenging problems for his interpretation both to particle physicists than to cosmologists.The ΛCDM model is the simplest model describing the main feature of the present observations [1] where the Dark side of the Universe is given by a Dark Matter (DM) component (with zero equation of state w and sound velocity c 2 s ) and a Dark Energy (DE) component (with equation of state w = −1) consisting of a Cosmological Constant (CC).
Here we study a minimal DF model where the late period of the universe, dominated by DM and DE, is described as a unique adiabatic perfect fluid.For convenience we name such a fluid Next to Minimal ΛCDM: NΛCDM, due to the fact that the background dynamics on a Friedman-Robertson-Walker (FRW) space time (without perturbations) is identical to the ΛCDM model.The NΛCDM is described both with a field theory approach, using an EFT where a single Lagrangian is describing the system and the dof are the goldstone modes of the spontaneously broken symmetries, that a fluid description where only the hydrodynamical eqs take places (conservation eqs of the energy momentum tensor and of some currents).The first approach is very powerful to build specific Lagrangians and to check if any symmetry principle is involved into the system (chapter 2).The eqs of motion of the goldstone mode, can be translated in the fluid description that results much easier to be solved and to be compared with physical predictions (chapters (3,4)).In particular the NΛCDM model results an adiabatic perfect fluid where the ratio of entropy density s(t, ⃗ x) over number density n(t, ⃗ x) (the entropy per particle σ = s(t, ⃗ x)/n(t, ⃗ x)) is conserved in time but it results 3-d space dependent σ = σ(⃗ x) (see chapter (3)).For a particular class of Lagrangians, the entropy per particle is dynamically connected to the CC so that, the main dynamical implication is the presence of a space dependent CC.Contrary to the large amount of literature dedicate to a time dependent CC (see [2] for a seminal paper and [3] for a recent paper), a space dependent CC results not to have been explored in depth in the literature (see for example [4] for spherically symmetric static solutions).Analysing the cosmological perturbations of the NΛCDM model around FRW it was shown the presence of peculiar growing modes for the comoving curvature perturbations at late time [5], [6]. 1 We show as the acceleration (a vector field) of the fundamental observer is the source of the intrinsic instability of the FRW background (we used the 1+3 covariant approach [7]).To evade such a behaviour we included the acceleration field into a new background utilising a Locally Rotationally Invariant (LRS) formalism (we used the 1+1+2 covariant approach [8]).We show as such a DF, also once is dominated by his CC component, doesn't evolve to a de Sitter phase due to the presence of pressure gradients.The adiabatic fluid contains frozen initial entropic fluctuations that behave effectively as a space-dependent CC.In chapter (4.1) we give the eqs needed to study the propagation of photons in such a space time.In particular we stress the presence of two Hubble rates: a radial one (along the line of sight) and the orthogonal one (transverse).In chapter (5) we formulate the evolution equations of the system using a Lema ître metric.We perturbatively solve the eqs of motion in two different setting: in chapter (6.1), we start from a FRW background and we add a small r-dependent CC as a perturbation while in chapter (6.2) we obtain an exact small r expansion solution.In chapter (7) we get the null geodesic solutions and then we perform some comparison with the existing nearby cosmological data.The price to pay for such an unusual spherically symmetric background come from the constraints from the observed homogeneity of the space.The observer must be positioned close to the centre of a spherical structure.In fact, from CMB constraint, the observer can be at most few tens of M pc from the origin in a Lema ître Tolman Bond (LTB void) [10]. 2 Effective field theory for Perfect Fluids and Thermodynamics Perfect fluids, and media in general, can be described, by using an effective field theory formalism, in terms of four scalar fields Φ A (A = 0, 1, 2, 3) [12].3 The mechanical and thermodynamical properties of the medium can be encoded in a set of symmetries of the scalar field action selecting order by order in a derivative expansion a finite number of operators.Following [13] we require the Lagrangian to be invariant under:

Global shift symmetries: Φ
(2.1) The global shift symmetry requires the scalars to enter the action only through their derivatives, while field dependent symmetries plus internal rotational invariance select, at the leading order derivative, the following operators As a result, our starting point is the action The corresponding EMT can be easily obtained using the formulas that gives the conserved energy momentum tensor (EMT) that has a perfect fluid form 4 with energy density ρ and pressure p given by The gradient of the velocity field u µ can be decomposed as (see appendix A for details) where the tensor h ν µ = δ ν µ + u µ u ν is a projector on the orthogonal surface to u α , θ is the expansion rate, σ αβ the shear tensor, ω αβ the rotation tensor and A α = u β ∇ β u α the acceleration vector.We introduce also the covariant derivative of a scalar function f along and orthogonal to u µ : two approaches, in any case, result complementary.The first one is much more powerful once we want stress some symmetry principle, couple exactly the media with gravity or electromagnetism or try some quantum treatment.The second one is more physical and intuitive having physical quantities as dynamical variables.
For example, the relativistic Kelvin Helmholtz theorem (obtained from the eqs of motion for perfect fluid) related to the vorticity conservation, in the field formulation is generated by the Noether's theorem for the local currents of the volume preserving symmetry. 4We have introduced the notation The model features the presence of two conserved currents (the last one related to the shift symmetry (2.2) of the Φ 0 field) It also follows from (2.9 -2.10) that the ratio σ ≡ U Y b is conserved, indeed EMT conservation and number density current conservation J µ = n u µ generate the dynamical equation of motion of the fluid with n representing the number particle density.Indeed projecting the EMT conservation equations along and orthogonal to u µ we have that with eq (2.9) constitute the equation of motion of the fluid.To stress the interconnection in between the field formulation and the fluid formulation is the fact that the equation of motion of the scalar fields are given exactly by the conservation laws of the EMT T µν and the conserved current J µ as it happens in hydrodynamics.Hydrodynamical constitutive eqs that give the structure of T µν and J µ as a function of energy, pressure, entropy, etc. are, in the field theory approaches, condensed inside the function U.All of this is similar to the Lagrangian approach of classical fields where the eqs of motion for many variables can be compressed inside a single functional, the Lagrangian.The thermodynamical dictionary that relates composite operators to thermodynamical quantities was already studied in ref. [14] pressure where n 0 and T 0 are constants normalising factors.Finally we can identify the potential U to be proportional to the Free energy and are exactly satisfied with our identifications (2.14,2.15,2.16).The differential structure of the pressure was studied in ref. [19] and is given by (we also define Γ ≡ c 2 ρ dσ) and factor out an adiabatic contribution, proportional to δρ with c 2 s the adiabatic sound speed and the non adiabatic contribution Γ [15] can be further factorised for time and space derivatives as specifically we have (for ρ + p ̸ = 0) (2.21) A final important parameter (mainly to estimate the cosmological impact of the system on the Universe evolution) is the effective equation of state given by the ratio Perfect fluids with w = const are generate by the following potentials: with U a generic function.We see that a DM fluid with w = 0 is generated by a potential of the form U = b U (Y ) while for a DE fluid with w = −1 we have U = U (b Y ).Once the perfect fluid potential is given U(b, Y ), one can compute c 2 s , c 2 b , c 2 ρ and w, then Γ is known in a fully non perturbative way.In appendix (B) we revisit the thermodynamical stability conditions applied to such a general formalism.Stability analysis for the perturbations of a media can be developed around a Minkowski space where the pressure and the density of the system are constant in space and time.In fact analysing also the perturbations around a FLRW background in the short wave length limit, the same stability conditions are recovered for a medium on Minkowski spacetime [16].We consider the goldstone perturbations π A where Φ A = x A + π A (A=0,...,4) (we can further decompose the spatial components as π a = ∂ a π L + V a with ∂ a V a = 0, a = 1, 2, 3) so that we have two scalar dof (π 0 and π L ) and two divergence-less vector dof V a .In general the goldstone quadratic Lagrangian can be parametrised using four mass parameters Mi , i = 0, 1, 2, 3, 4 defined in terms of the derivatives of the medium Lagrangian (2.3), their explicitly expression is given in appendix of ref. [16].From the mechanical point of view we have: perfect fluids when M1,2 = 0, solids for M1 = 0, superfluids for M2 = 0 and supersolids when M1,2 ̸ = 0. So, in our case, a perfect fluid gives the following quadratic Lagrangian5 The dispersion relation for transverse modes V a is ω V (k) = 0 while for the two scalar dof (note that in the scalar sector this lagrangian has the same structure of a gyroscopic system see [17]) we have ω 0 (k) = 0 and ω L (k) = cs k where c2 is the sound speed.So, in general, only longitudinal waves propagates with a finite sound speed.The transverse fluctuations correspond to an infinitesimal version of a fluid vortex.The fact that, in the EFT approach to perfect fluids, some field excitations lack of gradient energy results an open problem [18], in particular once we try to quantise the system and it results strictly related to the volume diff symmetries of the world space Φ a .For scalar perturbation the bridge in between the π 0,L fields and the fluid approaches with observables like the energy density perturbation δρ, the pressure perturbation δp, the entropy per particle perturbation δσ, or the velocity ⃗ v is given by the following Fourier transformed relationships (x = x + δx and x = ρ, p, σ with x constants). ) with c2 b = − M0 / M4 .The equations of motion for the density and the entropy per particle fluctuations describe the evolution of the system Unacceptable exponential instabilities are present for c2 s < 0. While for c2 s ≥ 0 we have the following solutions with c1,2 the initial conditions for the energy fluctuations.Note that for zero sound speed c2 s = 0 there are power law growing mode in time.We stress such a specific case because the system we are interested in is characterised exactly by the configuration with c2 s = 0 and c2 b = −1, see next chapter.
3 The Next to Minimal ΛCDM Model (NΛCDM ) In order to describe the Matter-Dark energy dominated period of the Universe evolution, we introduce a single Perfect fluid where both DM and DE take place.Thanks to the structure of potential with w constant in (2.23) we can add two term, one corresponding to w = −1 (DE) and one to w = 0 (DM) to get a system describing DE and DM all together.The dynamics of such a system is described by the following action, already studied in ref. [19] where the CC (w = −1) and the Matter (w = 0) content is explicitly shown.Energy density, pressure and thermodynamical parameters are given by The EMT conservation is providing the exact eqs of motion6 where the first eq.results from the projection along the u µ direction (u µ ∇ µ Λ ≡ Λ and u µ A µ = 0, A µ being the acceleration field (2.7)), the second eq.instead is given by the orthogonal projection obtained multiplying by h µ ν = δ µ ν + u ν u µ and D ν ≡ h µ ν ∇ µ (see also Appendix A).If we want to describe the present DM/DE transition in a FRW framework, we have just to tune the parameter to the present energy abundance where H 0 is the present Hubble constant, Ω m and Ω Λ the fraction of DM and DE at present time (Ω m + Ω Λ = 1).So that and it corresponds exactly to the ΛCDM predictions for the last period of DM/DE domination.Now that we set the background behaviour we can look to the structure of the perturbations.In ref. [19] it was shown that, for a FRW background, the comoving curvature perturbation R at large scales grows up as a 3 at late times (3.16).To show directly the instability of the FRW background we can write the full evolution equation for a specific gauge invariant operator: the acceleration vector A µ .The time evolution of the acceleration A µ , for a generic perfect fluid is given by the eq.(see appendix A)7 Note that the left handed side of eq.(3.12) is first order in perturbations around FRW while the right handed side is at least second order.Then for the NΛCDM fluid c 2 s = 0 (3.3) and that at first order gives A ∝ a 2 so that A µ ∝ a 3 ∼ 1 n , i.e. the acceleration growths as the third power of the scale factor, at any scales and in any DM-DE dominated period, entering soon or later in a non perturbative regime and destabilising the FRW background.As discussed in ref. [22], one of the necessary conditions for a FRW limit there is a zero acceleration. 8

1+1+2 formalism
FRW background is a space geometrically isotropic about all the fundamental world lines implying zero shear, vorticity and acceleration.A non zero value for these quantities would pick out preferred directions in the 3-d space orthogonal to the vector u α .To describe relativistic cosmology around a FRW the use of the covariant 1+3 approach (see [7]) results quite a powerful tool, especially for the definition of the gauge-invariant and covariant perturbation formalism.The space time is splitted in time and space relative to the fundamental observer represented by the timelike unit vector field u α (u 2 = u α u α = −1), representing the observer's 4-velocity.In this way the covariant 1+3 threading irreducibly splits any 4-vector into a scalar part parallel to u α and a 3-vector part orthogonal to u α .Furthermore, any second rank tensor is covariantly and irreducibly split into scalar, 3-vector and projected symmetric trace-free 3-tensor parts.The previous FRW analysis showed the presence of a gauge invariant vector field A µ , the acceleration, that growth very fast in time at all scales.In this section we change the background hypothesis going from an homogeneous and isotropic model (FRW) with a preferred time-like vector field u α to a Locally Rotationally Symmetric (LRS) spacetime where, in addition to the time-like vector field u α , it exists also a covariantly defined unique preferred spatial direction, v α , that in our case is the direction of the acceleration field A α .To describe the structure of such a kind of spaces the 1+1+2 formalism, developed in [8] (see also [23] for developments), is therefore ideally suited for a covariant description in terms of invariant scalar quantities that have physical or direct geometrical meaning.The preferred spatial direction in the LRS spacetimes constitutes a local axis of 8 Following [19], the Master equation for the Bardeen potential Φ = Φ(a) on FRW (3.10) is (Φ ′ = ∂Φ/∂a etc.) whose non homogenous solution in Matter and in Λ dominated periods behaves as we get so that at late time, a → ∞ during the DE domination era, also if the Bardeen potential remains constant, the comoving scalar curvature (due to the presence of the scalar vector component v) grows as a 3 (see also (3.12)), soon overwhelming the perturbative limit.
symmetry and is just a vector pointing along the axis of symmetry and is thus called a radial vector.Since LRS spacetimes are defined to be isotropic, this allows for the vanishing of all 1+1+2 vectors and tensors, such that there are no preferred directions in the sheet (the 2-d space orthogonal both to u α and v α ).Thus, all the non-zero 1+1+2 variables are covariantly defined scalars.The variables needed to describe the LRS space form an irreducible 1+1+2 set (see Appendix B for the mathematical details): • From a generic EMT we have density ρ, pressure p, the projected energy flow Q and the projected anisotropic scalar Π variable.
• The split of the (1+3) kinematical variables (related to the gradient of ∇ α u β ) gives the projected acceleration A, shear Σ and vorticity Ω.
• The decomposition of the gradient of ∇ α v β gives the structure of the 2-d sheet space with the sheet expansion ϕ and the twist ξ.
• Finally the various projections of the Weyl tensor generate two scalars: the electric E and the magnetic H one.
So, the geometrical scalar variables that fully describe LRS spacetimes are A subclass of the LRS spacetimes, called LRS-II, contains all the LRS spacetimes that are rotation free.As consequence in LRS-II spacetimes the variables Ω, ζ and H are identically zero and the rest of the variables in D 2 fully characterise the kinematics.
• LRS Class II: Ω = ξ = 0 then also the magnetic part of the Weyl curvature tensor H is vanishing.These models contain spherical, hyper-spherical and plane symmetric (cylindrical) solutions.
Singh et al. [24] found a new class of LRS spacetimes that in presence of non zero heat flux have nonvanishing rotation and spatial twist.For the analysis of our model we choose the space like unit vector v α as the spatial direction of the acceleration field: 9 Here some useful identities Table 1.Background parameters needed to describe various LRS space times.For the column "Matter" we have the non zero background scalars relative to the EMT and the particle density current.Into the column "Geometry: ∇u" we have the non zero background kinematic form factors relative to the gradient of the time like u α vector.Into the column "∇v" we have the non zero background kinematic form factors relative to the gradient of the space like v α vector.Into the column "Weyl" we have the non zero background form factors relative to the Weyl tensor.KS=Kantowski-Sachs models, LTB=Lema ître Tolman Bondi models, FRW=Friedmann Robertson Walker models.The κ factor is related to the topology of the sheet 2-d space.
Due to the presence of two special vectors, one time-like u α and the other space-like v α we have two directional derivatives for a generic scalar function f The EMT conservation eqs for a PF and the differential structure of the pressure (2.19) give The mixed derivatives f or ḟ for the density and pressure reads (see eq.(A.16) and app.(D)) Deriving eq.(2.19) we get (for ρ + p ̸ = 0) the general expression for the evolution of A 10 10 Interestingly, for c 2 s ̸ = 0, we can rewrite eq.(4.14) as function of σ (replacing ρ with σ) and in the limit of constant c 2 s and c 2 ρ (always For the NΛCDM fluid we have c 2 s = 0 and c 2 ρ = −1 so that the acceleration evolves as supplemented by the eqs (3.3,3.8) Note that a positive acceleration is related to a positive gradient of the cosmological constant (negative for the pressure).The general evolution/propagation equations for the rest of the scalar functions can be classified depending on the kind of covariant derivatives (ˆor ˙) and are obtained from the Bianchi and Ricci identities respectively: • Evolution eqs Being the vorticity zero, we have that the vector u α is hypersurphace orthogonal to space like 3-d surfaces with 3-curvature (3) R given by with K the Gaussian curvature of the 2-d sheet given by characterised by the following evolution/propagation equations The K variable and his evolution eqs (4.22) can be used to replace one of the variable in subset D 2 , (4.1).It is interesting to rewrite a subset of coupled evolution equations using the Rearranging the above eqs we can get whose non perturbative solutions, as a function of two boundaries f 0,1 functions, are 11 As soon as f 0,1 ̸ = 0 we can rewrite the full system of eqs (4.23) as a single evolution equation for A12 : ...
and using K = −ϕ K, we can write the eq.for the spatial gradient of A as while the eqs for the other variables result From eqs (4.15) and (4.22) we see that A and K evolve in time in opposite way, so that, if A is growing, K is decreasing or vice versa.Analogous behaviour happens for the thermodynamical quantities, density and temperature such that: ṅ = −θ n and Ṫ = θ T .
11 Strictly speaking we have to distinguish the positive/negative acceleration cases with the corresponding solutions ϕ = 2 3 A + f 0 √ ±A .In order to have compact notations we wrote only √ A assuming that f0 will be pure complex when A < 0.

Null Geodesics
Due to the fact that cosmological observations rely on the detection of the light emitted from far away sources, a non homogeneous background makes thinks more subtle for the mixing in between temporal and spatial variations.Moreover while the Einstein eqs are traced by the matter geodesic vector u α , the geometry of light propagation is dictated by the tangent vector k α = dx α /dν (with ν the affine parameter).In the framework of geometrical optics, null geodesics eqs are given by The photon momentum can be decomposed along the u α vector as where e α is a space like vector (e α e α = 1, e α u α = 0) that defines the direction of the photon and E = −k α u α is the energy of the photon relative to the observer defined by the vector u α .The energy evolution along the null geodesic is given by the derivative of E along ν where we applied the decomposition (2.7).The redshift z of a source is defined as the observed photon wavelength divided by the wavelength at the source minus one, and being the wave length inversely proportional to E we get where u α is the four-velocity of the perfect fluid evaluated for the galaxy (G) and for the observer (O).Then we normalised the null affine parameter with ν = 0 at the observational point setting dz = dE.With (4.35) we get the evolution of the redshift along the light cone (specialised to a LRS space time) : Where we see the various cosmological contributions to the redshift of the photon coming from the expansion rate , the acceleration of the observer and the shear.
It is clear that redshift measurements are observations on null cones that sample the radial direction.Also the two dimensional orthogonal space directions (the screen space) experience the expansion of the space time, but in a non homogeneous space it is important to distinguish such effects.So we introduce the observed radial Hubble rate (line of sight expansion) as13 versus the orthogonal Hubble rate (transverse expansion rate) σ αβ e α e β (4.39) Finally the volume expansion rate that gives the usual definition of the Hubble parameter, i.e. the expansion of the volume parameter θ, is given by where l is a representative length given by dV = η µαβγ dx µ dx α dx β dx γ = det| (3) and l 3 = det| (3) The presence of different expansion rates in different directions is at the basis of of the Alcock-Paczynski [26] test in a general spacetime.For an object that is known to be spherically symmetric, the ratio between his observed angular size over the radial extent in redshift space is a function of the redshift and the space time geometry.An isotropic Hubble rate clearly implies H ⊥ = H ∥ that is the case in a FRW model.

Coordinate approach, Lema ître metric
To obtain explicit solutions we need to connect the above formalism directly to the metric components.As described in detail in [8] (see also [9] for applications to shell crossing), for a LRS space we can use the following local metric in (t, r, x, y) coordinates: where D(x, κ) = (sin x, x, sinh x) for κ = (1 spherical, 0 euclidean, −1 hyperbolic), that describes the geometry of the 2-d sheet and R(t, r) is the area radius coordinate.For a spherically symmetric inhomogeneous fluid (κ = 1) we have the so called Lema ître metric [28] while in the special case of dust (p = 0) with a cosmological constant, the above metric reproduces the Lema ître-Tolman (LT) model 14 .Within the Lema ître models the coordinates are comoving with matter flow and for a perfect fluid both pressure and energy density are functions of t, r variables.The matter flow has four velocity u α = (1/F, 0, 0, 0) while the special space four vector is v α = (0, 1/X, 0, 0).The centre of the space is given by the solutions of R(t, r) = 0.The covariant derivatives for a scalar function f (t, r) become (4.6) Note also that in many articles the X(t, r) function is written as 14 It is important to stress the difference with the huge literature present for the LT models.At background level, as already show in Tab 4, LT models have null acceleration A = 0 → ∂rp = 0 → F (0,1) (t, r) = 0 → Λ ′ (r) = 0, so their metric can be written as with E(r) an arbitrary function of integration proportional to the curvature of space at each r value (to be compared with (5.4)).The time independence of E allows an explicit solution for eq (5.13).
where the E function (> −1 ) corresponds to the curvature parameter [31].The geometrical quantities in D 2 , (4.1), are given by 15 and the two Hubble rates (4.38, 4.39) result Note that the scalar shear Σ is proportional to the difference between the radial and azimuthal expansion rates.

The NΛCDM model
Applying the above eqs to the NΛCDM model we can integrate the density and the EMT conservation equations F , Λ ′ ̸ = 0 (5.9) where Λ ≡ Λ(r) and n ≡ n(r) are effectively introduced as spatial boundary conditions.Then we introduce the Misner-Sharp mass (see also [27], [28], [29]) that, in the Newtonian limit, represents the mass inside the shell of radial coordinate r.
Using the M variable the Einstein eqs read 16 15 In a conformal metric of a flat FRW we have that θ = − 2 3 S = 3 ȧ a = 3 a ′ a 2 , ϕ = 2 r a . 16From eq.(5.11) the infinite density limit is obtained when R = 0 with R ̸ = 0 (5.14).These singularities correspond to shell-crossing singularities or caustics (i.e.cusps for the dark matter distribution).It is a generic prediction that the evolution for an absolutely cold dark matter system from near uniform initial conditions under the effect of gravity, at non linear level, bring to generation of caustics.In general it is expected that, on the smallest scales, the fluid matter model, that results a macroscopic approximation for the smooth behaviour of matter fields, is not appropriate for the study of highly non linear gravitational phenomenon.For realistic dark matter candidates it is expected that such a divergences can be tamed for example by thermal effects and it results as a motivation for alternatives to CDM (warm dark matter, fuzzy dark matter, dust replaced by the Vlasov model, inviscid fluid replaced by viscous).In Tolman models (where p = 0) the necessary and sufficient conditions which ensure no shell crossings are described in [30].
The pressure equation can be integrated giving where m ≡ m(r) is another spatial boundary conditions.Inserting (5.12) in (5.10) we get while the radial derivative of R, using the expression for the energy density (5.11) and the density (5.9), can be written as The same expression can be used also to relate n and m as To have a flat FRW space time we need the following conditions: • R = r a(t), F = a(t), X = a(t) so that H ⊥ = ȧ a and R = 1 • Boundary conditions for the matter content implies m = 3 H 2 0 Ω m r 3 and (5.15 While the matter conditions can be easily implemented, the constancy of Λ is contrary to our assumptions.Following [31], we see that the general structure of the Local Hubble parameter H ⊥ (t, r) is quite similar to the FRW Hubble equation and allows to identify various contributions 17 ) (5.17) 17 We note as doesn't show such a similarity with the FRW Hubble equation.
with the constraint Ω Λ (r) + Ω M (t 0 , r) + Ω c (t 0 , r) = 1 and where the current Hubble rate is H 0 (r) = H ⊥ (t 0 , r).The structure of Ω c contains a term, proportional to m ′ n 2 − 9 , that depends only on dark matter structure while the second term, proportional to Λ ′ , disappears for a true constant CC where Λ ′ → 0. In order to have a close differential eq for only one component of the metric we can rewrite eq.( 5.15) as so that X (from (5.9)) is completely determined from the R dynamics.Eq.( 5.18) and eq.( 5.13) represent a close system of partial differential equations governing the dynamics of the F and R functions.In Appendix we give the self contained evolution equation for the R function (E.2).The rest of kinematical quantities are here specified (for θ and Σ we don't use eq (E. 2) to have a simplest expression) From the non perturbative eqs (4.25) we can relate the various space dependent integration constant Note that for n = m ′ 3 we have

Analytical Approximations
We assume a spatial background distribution of DM compatible with the FRW conditions n = m ′ 3 .Then eqs (5.18) and (5.13) give the closed system that we approximatively solve in different contexts.In chapter (6.1) we perturbatively solve the above eqs expanding around an homogeneous FRW space time with a small, space dependent, correction to the CC.Instead in chapter (6.2) we solve the above system expanded for small r values.Finally in chapter (7) we study the light ray propagation with the Lema ître metric using the above solutions.

Solutions around an approximated FRW model
The exact FRW solution for the eqs (6.1) are obtained for giving F = 1, R = a(t) r.The scale factor a(t), in a De Sitter-Matter dominated universe, is We perturb such a configuration with a small space-dependent correction λ(r) to the CC and we expand the metric coefficient at order O(λ n ) as For F (t, r) we get, at second order, 18 where x .For the components of R(t, r) and X(t, r) we get while for the other geometrical quantities we have The convergence region of our series expansion can be inferred looking to eq (6.6) 12) 18 Where we use 2F1 − Then analysing the solution for R (6.7) and X (6.8) we need also the present perturbativity constraints or, in the future (a > 1), we can perturbatively reach the region with Note that (6.13) must be valid for all r values so, for example, in the limit r → 0 we need an expansion of the form λ(r) ∼ λ 2 r 2 + ... without the linear term and the constant term that can be absorbed inside Ω Λ (the structure of the above expansion is confirmed also in the next chapter).At present time a(t 0 ) = 1 and at the center of the vacuum bubble r = 0 from (6.10) we get Conversely, far away r → ∞, we impose that λ(r) is approaching a constant λ 0 (that can be zero) with λ ′ (r), λ ′′ (r) ∼ 0 (the space gradients die earlier) so that matching the FRW background solution (6.3).We stress that the above eqs ( are perturbative solutions of the background equations (6.1) and not perturbations of the Lema ître metric (5.1).In this approximation, from the eqs (6.15,6.16),we identify H 0 with the Hubble Planck data, see (7.17).

Solution in the small r expansion limit
Explicit results from the equations (6.1) can be obtained from the solutions around our neighbour universe.Following ref. [32], we perform a small r expansion for all the form factors of the metric and associated quantities Note that the coefficients of bold quantities f n are just numbers.Matching order by order the continuity eqs and the Einstein equations (6.1), for spherically symmetric solutions (κ = 1), we get the following relations: 19  At order O(r 0 ) the Hubble function is given by Table 2.The leading coefficients from the series (6.17) that solve the eqs of motion for the different variables.
after the identifications: In such approximation for the background Hubble constant we use the symbol H 0 that refers to the nearby Hubble constant measurement (7.42) contrary to the previous case (6.3)where it was used H 0 (Planck value).Note the presence of an unusual Ω χ component whose eqs of state result w χ = − 4 3 ≃ −1.33.The M (t, r) function (5.10) start his expansion at O(r 3 ) From (5.11) we get m i = 3 n i−1 corresponding to n(r) = m ′ (r) 3 .Finally there are two functions R 2,3 (t) that satisfy a coupled system of evolution equations (Λ i ≡ 3 H 2 0 ℓ i ) (F.1,F.2).The leading terms for the kinematical variables result The above expansion result reliable as soon as and at present time 7 Null Geodesics on a Lema ître universe Because the observation are done along the light cone, first of all we have to study the null geodesic eqs as a functions of the redshift.For a source directed towards an observer located at the symmetry centre of the model, null geodesic are given by [32] ( where r = r(z) and t = t(z).Concerning the photon geodesic eq.(4.37), being dz dν = E dz dτ = (1+z) F dz dt and v α e α = cos(θ) on a LRS space we get to be compared with the result of eq.( 7.1) So that exactly we have cos(θ) = −1.This corresponds to have the observer exactly at the centre of the space time (r = 0) so that congruence are isotropic.For off centre observers, where cosθ ̸ = ±1 it appears a dipole correction induced by the acceleration and a quadrupolar correction by the shear.There are many "distance" definitions between two points in cosmology, measuring the separation between events on radial null trajectories we have: Following [38] we can work out some formulas inspired by the more familiar FRW models that can help understanding the dynamics of the system.For example we can build an effective equation of state along the radial (∥) or the angular (⊥) directions as also the effective acceleration can be a useful function In the next chapters we will solve eqs (7.1) in the almost FRW approximation (7.1) and in the small r limit (7.2).

Geodesics for small r expansion
In the small redshift expansion from eqs (7.1, 7.3) we get (where we used Ω Λ = 1 − Ω m − Ω χ ).We can parametrize the various corrections as functions of the expansion coefficients of the acceleration and shear functions, see eq. (6.21): .33)where A i and Σ i are constants depending from the initial conditions and free parameters: To disentangle the degeneracy A 1 − Σ 1 we need an extra independent cosmographic measure that can be the redshift drift, as suggested in [32].

Conclusions
We analysed an adiabatic perfect fluid cosmological model characterised by two terms into the Lagrangian (3.1), one featuring a CC and another a DM component, both terms are builded with the same field content (goldstone modes of the spontaneously broken space and time) (2.1).In this sense we have a single fluid mimicking both DE and DM.The presence of a space dependent CC results thermodynamically related to the conserved (in time) entropy density (3.8).Such a DE component breaks the homogeneity of the space time and induces a non trivial space dependence for the pressure (3.3) such that the comoving observers are not anymore geodesic A µ ̸ = 0 (3.8).
We study the equation of motion of such a system using a 1+1+2 formalism where we use two background vectors: one is time-like u µ (the geodesic of the fluid) and the other is space-like v µ (proportional to the non zero acceleration) (4.5).
In chapter (4.1) we give the eqs for the propagation of light bundles and we stress the difference in between the radial H ∥ and transverse H ⊥ expansion rates (4.38, 4.39).
In chapter (5) we study the eqs of motion in a comoving coordinate system using a rotationally invariant Lema ître metric (5.1) characterized by three form factors functions of the t, r variables (F (t, r), X(t, r), R(t, r)).The eqs of motion (6.1) are solved in two limiting cases.
In chapter (6.1) we perturb a FRW background with a small r-dependent cosmological constant (6.4) while in chapter (7.1) we give the corresponding solutions of the null geodesics.
In chapter (6.2) instead we study the solutions to (6.1) in the small r H 0 ≪ 1 expansion regime and in chapter (7.2) we give the corresponding null geodesic paths.As observable, to check the outcome of our model, we study the luminosity distance d L (7.4), the effective eqs of state along the radial and transverse direction w ∥,⊥ (7.5) computed along the light of sight.In chapter (7.1) we analyse two kinds of CC r-profiles (7.15,7.16)and we report our main results in fig. 2 and 3.
In chapter (7.2) the small r expansion is translated in a small z redshift expansion where we can extract the deceleration parameter Q 0 (7.36) as a function of constants describing the background evolution.It is interesting to note as, in such a limit, the background Hubble law (6.19)shows the presence of an extra matter component Ω χ (proportional to the local acceleration parameter (7.34)) whose equation of state result w χ = −4/3.
To simplify our computations we put the observer at the center of the void region violating the Copernican (Cosmological) Principle. 22Our model represent an alternative to the more popular spherically symmetric LTB model where the matter component of the universe is the responsible of the inhomogeneity.Assuming the ΛCDM model, at present, the discrepancy between the Hubble values measured around local distance ladders and from CMB Planck data (see [39], [40] for a full reference list) has reached the 4-6 σ level.This model represents a minimal modification to the ΛCDM scenario (for this reason we named next to minimal ΛCDM model (NΛCDM)) able to modify the null geodesic paths in agreement with present data.This paper is dedicated to the theoretical analysis of the model therefore we have not carried out a statistical analysis for the best possible parameter space of the model by just analysed two generic possible space dependent shapes for the CC shown in fig. 1 The first kind of shape is featuring a bump of Λ(r), while the second one a smooth increase of Λ(r) till a constant value.Choosing appropriate parameters we first adjust the discrepancy for the Hubble parameter nearby and far away (Planck), then we cheque the implications for the effective equations of state.For example we note that data favours the second model where an effective nearby equation of state w ≤ −1 is obtained mimicking a phantom dark energy component [41] [42].Following the classification mechanism solutions for the H 0 -crisis [39], we see that this kind of models can be inserted in the class of the alternative proposals operating at late time: Local Inhomogeneity.It is clear that to probe this kind of radial inhomogeneity around us we further have to do many other test: CMB spectral distortions, BAO, type Ia supernovae, cosmic chronometers, ect.as is in progress for the ΛCDM model [43].
• Coefficient of adiabatic compressibility All of them satisfy the above relations where C p,V ≡ V c p,V .Working with densities, thermodynamical stability requires the |s ij | matrix to be negative defined so that and also c p > c V > 0 or T κ T c V > 0 bringing to the following constraint for the potential Note that such a conditions are non perturbative and background independent. 24

C Cosmological Constant and Λ-Media
The largest symmetry inside the internal scalar space is a full unitary Diffeomorphism transformation whose feature is to select one specific invariant operator Z that results the building block for a thermodynamical non trivial CC: where e αβγδ and e ABCD are the permutation tensors in the 4-Dim space-time and the 4-Dim scalar space.The action generating an effective CC perfect fluid results (that we named Λ-Media in [5,13]) The eqs of motion for the Φ A scalar fields (for U ′′ ̸ = 0) are given by ) and imply the exact non perturbative solution ∇ µ Z = 0 → Z = Z 0 constant in space and time (C.5) The corresponding Energy Momentum Tensor (EMT) can be easily obtained 25T µν = (U − Z U ′ ) g µν ≡ −Λ g µν (C.6) so that we have an effective CC: Λ = (−U + Z U ′ ).For such a model we can write the energy density ρ, the pressure p and the entropy per particle σ as [5] Using the thermodynamical analysis of the previous chapter, for the specific potentials U = U (b Y ), we get that Perfect Fluid Λ-media are thermodynamically unstable.In fact we get violating eq.(B.11).Note the paradox that dynamically the system is frozen to and exact de Sitter phase with no runaway solutions for the energy/entropy observables.

D Mixed derivatives for thermodynamical quantities
We give the mixed derivatives (A.

E Evolution equation for the radial form factor in almost FRW
In order to get a single evolution equation for the radial form factor of the metric R(t, r) we can insert eq.(5.14) in (5.13) so we get F as a function of R Finally to have a self contained eq. for R we insert (E.1) inside (5.18) F Evolution equation for the radial form factors in small r regime Evolution for the form factor R 2,3 (t) where R(t, r) = a(t) r + R 2 (t) r 2 2 + R 3 (t) r 3 6 + ....We prefer to shift the derivation, from the variable t to a(t) (6.19) such that f ′ (t) = a H(a) f ′ (a).where we used Λ i ≡ 3 H 2 0 ℓ i .

1 ) 2 U
where m is a mass parameter multiplying the density field b (we will use the notation m b = n) and the potential U (b Y ) corresponds to a Λ perfect fluid (whose details are given in appendix (C)).In Minkowski space the perturbations of such a system are characterised by M0,3,4 = b Y (b Y ) (where b = Y = 1) that gives c2 b = −1 and c2 s = 0. So, no dof is propagating being ω V,0,L (k) = 0 and no ghost like pathologies are present.On a generic space time background the corresponding EMT is