Large scales space-time waves from inflation with time dependent cosmological parameter
LLarge scales space-time waves from inflation with time dependent cosmologicalparameter Juan Ignacio Musmarra ∗ , , Mauricio Bellini. † Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,Universidad Nacional de Mar del Plata, Funes 3350, C.P. 7600, Mar del Plata, Argentina. Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR),Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Mar del Plata, Argentina.
We study the emission of large-scales wavelength space-time waves during the inflationary expan-sion of the universe, produced by back-reaction effects. As an example, we study an inflationarymodel with variable time scale, where the scale factor of the universe grows as a power of time. Thecoarse-grained field to describe space-time waves is defined by using the Levy distribution, on thewavenumber space. The evolution for the norm of these waves on cosmological scales is calculated,and it is shown that decreases with time.
I. INTRODUCTION
The study of the General Relativistic dynamics is a very important topic that has been subject of research sincemany years ago[1, 2]. However, during many time, the study of this problem has not evolved too much. The strategyused in [2], generalized in [3] for null surfaces, consisted of adding an appropriate term to the original action, sothat, when the action was changed, the boundary terms become null. The original problem resides in that, whenwe consider the Einstein-Hilbert (EH) action I , which describes gravitation and matter (for κ = 8 π G — we shallconsider (cid:126) = c = 1 throughout the work): I = (cid:90) V d x √− g (cid:20) R κ + L m (cid:21) , (1)the variation of the EH action δ I = (cid:90) d x √− g (cid:2) δg αβ ( G αβ + κT αβ ) + g αβ δR αβ (cid:3) = 0 , (2)includes some boundary terms that cannot be arbitrarily neglected to obtain the Einstein’s equations. These termsare the last inside the brackets and must be studied in detail. In some earlier works, we have studied some physicalconsequences and applications of this terms[4–9]. One of the interesting applications which deserve study is theemission of space-time waves during the inflationary evolution of the universe. Cosmic inflation describes a primordialera in which the universe growth quasi-exponentially and provides a solution to some cosmological problems thatcannot be explained otherwise[10–15]. During inflation the equation of state ω = P/ρ remained close to a vacuumexpansion: ω (cid:39) −
1, so that the universe became at cosmological scales spatially flat, isotropic, homogeneous, andthe energy density was very close to the critical one: ρ (cid:39) H π G = λ π G . This primordial expansion was governed by aunknown kind of energy, named dark energy (a possible explanation about its origin was done in[16]).In this work we are aimed to study the space-time waves which came from the local geometrical inhomogeneitiesduring inflation. Back-reaction effects[17, 18], which are described by a massless field σ , are the geometrical responseto the local fluctuations of the inflaton field ϕ ( x ) − (cid:104) ϕ ( x ) (cid:105) = δφ ( x ). In other words, back-reaction effects are thelocal geometrical space-time distortion due to the local fluctuations of the scalar massive inflaton field. Since thesefluctuations are local, on cosmological scales they can be seen as uniformly distributed sources of space-time wavesemitted during inflation. This is a very interesting topic to be studied.The manuscript is organized as follows: In Sect. II we revise and extend Relativistic Quantum Geometry (RQG)by emphasizing the role of the local flux δ Φ in the boundary conditions, when we minimize the Einstein-Hilbertaction. We obtain the equation of motion for the trace of these space-time waves produced by this flux, and wemake a preliminary description of the quantum space-time. In Sect. III we make a description of inflation with avariable time scale. In Sect. IV we study solutions in this inflationary model for back-reaction effects and the traceof space-time waves. In particular, we study solutions on cosmological scales with large wavelengths using a Levydistributions on the momentum space. Finally, in Sect. V we develop some final comments and conclusions. ∗ [email protected] † Corresponding author : [email protected] a r X i v : . [ g r- q c ] J un II. RELATIVISTIC QUANTUM GEOMETRY WITH NONZERO FLUX
Now we consider the 4-vector δW α , given in terms of the varied symmetric connections: δ Γ αβ(cid:15) [19] δW α = δ Γ (cid:15)β(cid:15) g βα − δ Γ αβγ g βγ . (3)Since we shall consider that the extended manifold given by a displacement of connections, does not preserve the nullnon metricity on the extended manifold: ( g (cid:15)ν ) | α (cid:54) = 0, will be useful, when we variate an Einstein-Hilbert (EH) action,to consider a flux δ Φ of δW α , as g αβ δR αβ − δ Φ = [ δW α ] | α − (cid:0) g αβ (cid:1) | β δ Γ (cid:15)α(cid:15) + ( g (cid:15)ν ) | α δ Γ α(cid:15)ν , (4)where ” | ” denotes the covariant derivative on the extended manifold. Here, δR αβ is δR αβγα = δR βγ = (cid:0) δ Γ αβα (cid:1) | γ − (cid:0) δ Γ αβγ (cid:1) | α , (5)were we have used an extension of the Palatini identity[23]. A. Minimum action with δR αβ = λ ( t ) δg αβ In this work we shall consider the case where δR αβ is related to the variation of the metric tensor δR αβ = λ ( t ) δg αβ . (6)Here, λ ( t ) is the called cosmological parameter, which is a decaying function of time[21, 22]. This parameter, takes intoaccount, in an effective manner, the geometrical contribution of back-reaction effects on the background metric. Sincethese effects have a geometrical origin, they are incorporated in the redefined Einstein tensor: ¯ G αβ = G αβ − λ ( t ) g αβ in such manner that the varied action (2), will can be written as δ I = (cid:90) d x √− g (cid:2) δg αβ ( G αβ − λ ( t ) g αβ + κT αβ ) (cid:3) = 0 , (7)where we have made use of the fact that δg αβ g αβ = − δg αβ g αβ . (8)Notice that λ ≡ λ ( t ) is not a constant. This is because the extended manifold on which we describe back-reactioneffects is not a Riemann manifold and the nonmetricity is not null: g αβ | (cid:15) (cid:54) = 0. Therefore, vectors and tensors defined onthis manifold have a nonconservative norm. Intuitively, we can say that we are dealing with an elastic geometry on thisextended manifold. This is necessary to can define gauge invariance on a geometry with ”roughness”, when we makean effective description on a metric which is isotropic and homogeneous (or, ”smooth”). As in previous works[7, 20],we shall consider the connections given by Levi-Civita symbols plus a variation that drives the displacement withrespect to the Riemann manifold Γ αβγ = (cid:26) αβ γ (cid:27) + δ Γ αβγ = (cid:26) αβ γ (cid:27) + b σ α g βγ . (9)Here, σ α ≡ σ ,α is the ordinary partial derivative of σ with respect to x α . On the Riemman manifold it is requiredthat non-mectricity to be null: ∆ g αβ = g αβ ; γ dx γ = 0. However, on the extended manifold the variation of the metrictensor is δg αβ = g αβ | γ dx γ = − b ( σ β g αγ + σ α g βγ ) dx γ , (10)where g αβ | γ is the covariant derivative on the extended manifold given by (9).For b = 1 /
3, it is obtained in (3) that δW α = − σ α , and hence the expression (4) can be written as g αβ δR αβ − δ Φ = [ δW α ] | α − (cid:0) g αβ (cid:1) | β δ Γ (cid:15)α(cid:15) + ( g (cid:15)ν ) | α δ Γ α(cid:15)ν = ∇ α δW α = g αβ [ (cid:3) δ Ψ αβ − λ ( t ) δg αβ ] = −∇ α σ α ≡ − (cid:3) σ = 0 , (11)where δ Ψ βγ , can be interpreted as the components of gravitational waves in a more general sense than the standard one.They comply with a tensor-wave equation with a source λ ( t ) δg αβ , but its trace is nonzero: g αβ δ Ψ αβ (cid:54) = 0. Furthermore,in the standard formalism for gravitational waves, the equation of motion is obtained in a linear perturbative expansionwith respect to the background and is valid only as a weak field approximation. However, in our work we are dealingwith a non-perturbative formalism which is valid for arbitrary gravitational fields. Finally, in the standard formalismthe gravitational waves come from a non-conservative quadrupolar momentum, but in our case this is not necessarythe case. Notice that eq. (11) is true only in the case with b = 1 /
3, where the flux can be written in terms of a4-divergence for δW α defined in terms of covariant derivatives in the Riemann manifold. To calculate the flux δ Φ, wemust know δR αβ . The variation of the Ricci tensor on the extended manifold is δR αβ = ( δ Γ (cid:15)α(cid:15) ) | β − ( δ Γ (cid:15)αβ ) | (cid:15) = 13 (cid:20) ∇ β σ α + 13 ( σ α σ β + σ β σ α ) − g αβ (cid:18) ∇ (cid:15) σ (cid:15) + 23 σ ν σ ν (cid:19)(cid:21) , (12)so that, in agreement with the equation (6), it is possible to write the left side of (11) as: g αβ [ δR αβ − λ ( t ) δg αβ ] = 0,and therefore we obtain the most restrictive equation13 (cid:20) ∇ β σ α + 13 ( σ α σ β + σ β σ α ) − g αβ (cid:18) ∇ (cid:15) σ (cid:15) + 23 σ ν σ ν (cid:19)(cid:21) − λ ( t ) δg αβ = 0 , (13)where the last term in (13) is due to the flux that cross the closed 3D-hypersurface: δ Φ = λ ( t ) g αβ δg αβ .In this work we shall consider an expanding universe that is isotropic and homogenous. Due to this fact, it ispossible to define the redefined background Einstein equations¯ G αβ = G αβ − λ ( t ) g αβ = − κ T αβ , (14)with the redefined boundary conditions (13). The set of equations, given by (13) and (14), comply with the minimumaction principle given in the equation (7), because the transformation (14) preserves the EH action, and the flux thatcross the 3D-gaussian hypersurface, δ Φ, is related to the cosmological parameter λ ( t ), and the variation of the scalarfield, δσ δ Φ = − λ ( t ) δσ. (15)The field χ ( x (cid:15) ) ≡ g µν χ µν is a classical scalar field, such that χ µν = δ Ψ µν δS describes the space-time waves produced bythe source through the 3D-Gaussian hypersurface (cid:3) χ = δ Φ δS , (16)where δS = U α dx α . Here, U α = dx α dS are the components of the 4-velocity, given as a solution of the geodesic equationon the Riemann manifold: dU α dS + (cid:26) αβ γ (cid:27) U β U γ = 0 , (17)with unity squared norm: U α U α = 1. The differential operator (cid:3) that acts on χ in (16), is written in terms of thecovariant derivatives defined on the background metric: (cid:3) ≡ g αβ ∇ α ∇ β , such that ∇ α give us the covariant derivativeon the Riemann background manifold. B. Quantum space-time
If we deal with a space-time which is quantum in nature, we can describe it as a Fourier expansion in terms of themodes δ ˆ x α ( t, (cid:126)x ) = 1(2 π ) / (cid:90) d k ˇ e α (cid:104) b k ˆ x k ( t, (cid:126)x ) + b † k ˆ x ∗ k ( t, (cid:126)x ) (cid:105) , where b † k and b k are the creation and annihilation operators of space-time, that comply with the algebra (cid:68) B (cid:12)(cid:12)(cid:12)(cid:104) b k , b † k (cid:48) (cid:105)(cid:12)(cid:12)(cid:12) B (cid:69) = δ (3) ( (cid:126)k − (cid:126)k (cid:48) ) and ˇ e α = (cid:15) αβγδ ˇ e β ˇ e γ ˇ e δ . The operators of creation δ ˆ x α ( x β ), applied to a backgroundstate | B (cid:105) , return an eigenvalue dx α dx α | B (cid:105) = U α dS | B (cid:105) = δ ˆ x α ( x β ) | B (cid:105) , (18)so that the background line element results to be dS δ BB (cid:48) = ( U α U α ) dS δ BB (cid:48) = (cid:104) B | δ ˆ x α δ ˆ x α | B (cid:48) (cid:105) , (19)and the quantum states are described by a Fock space. III. INFLATIONARY UNIVERSE WITH TIME DEPENDENT COSMOLOGICAL PARAMETER
As we have demonstrated in a previous work[8], variable time scale can have played an important role in theevolution of the universe. In order to study an inflationary model where the time scale is variable, we shall considerthe line element dS = e − (cid:82) Γ( t ) dt dt − a e (cid:82) H ( t ) dt δ ij dx i dx j . (20)Here, the rate of co-moving events is described by the physical time τ . From the point of view of a co-movingrelativistic observer, its clock evolves as dτ = U dx = √ g dx . If the expansion of the universe is driven by theinflaton field (cid:104) ϕ (cid:105) ≡ φ ( t ) on an isotropic and homogeneous background metric (20), the action will be I = (cid:90) d x √− g (cid:32) R πG − (cid:34) ˙ φ e (cid:82) Γ( t ) dt − V ( φ ) (cid:35)(cid:33) . (21)The dynamics for the background inflaton field, that drives the expansion of the universe, is¨ φ + [3 H ( t ) + Γ( t )] ˙ φ + δ ¯ Vδφ = 0 , (22)and the relevant Einstein equations with the time dependent cosmological parameter included, are3 H ( t ) + λ ( t ) = 8 πGρ, (23a) − [3 H ( t ) + 2 ˙ H ( t ) + 2Γ( t ) H ( t ) + λ ( t )] = 8 πGP, (23b)where P and ρ are respectively the pressure and energy density. The equation of state describes the ration betweenboth scalar quantities ω = Pρ = − (cid:32) H ( t ) + Γ( t ) H ( t ))3 H ( t ) + λ ( t ) (cid:33) , (24)such that ω must remain close to a vacuum dominated state during the inflationary expansion of the universe.Furthermore, from the Einstein equations, we obtain that the redefined scalar potential ¯ V ( φ ) and ˙ φ , are respectivelygiven by ¯ V = e − (cid:82) Γ( t ) πG (cid:16) H ( t ) + ˙ H ( t ) + Γ( t ) H ( t ) + λ ( t ) (cid:17) , (25a)˙ φ = (cid:115) − ( ˙ H + Γ H )4 πG e − (cid:82) Γ( t ) dt = (cid:114) p (1 − q )4 πG t q . (25b)Using this in (22), we obtain that the following equation must be fulfilled:˙ λt − λqt − p q − pq + 2 pq = 0 . (26)The solution for λ ( t ) is λ ( t ) = C t q + pq (1 − q − p ) q + 1 1 t , (27)where C is a constant. If q = 0, λ ( t ) = C and we can recover a traditional power-law expansion[7, 24]. In this workwe shall consider C = 0 and p = (1 − q ) q q +1) . Then we obtain λ ( t ) = 3 H ( t ) = p t . On the other hand, for p > q <
1, which is consistent with a real ˙ φ in (25b). IV. SPACE-TIME WAVES FROM BACK-REACTION EFFECTS
With the choice b = 1 /
3, by using the expression (11), we obtain that g αβ δR αβ = − (cid:20) (cid:3) σ + 23 σ ν σ ν − δ Φ (cid:21) = 0 , (28)and, on the another hand, we have that ∇ α δW α ≡ − (cid:3) σ = 0 . (29)In order to make resoluble the system of equations, we shall consider the gauge σ ν σ ν = λ ( t ) δσ , where for a co-movingobserver U = (cid:112) g , is fulfilled δσ = U α σ α = U σ . (30)With this choice, and using the equations (4) and (16), the dynamics for σ and χ , results to be (cid:3) σ = 0 , (31a) (cid:3) χ = − U λ ( t ) ˙ σ, (31b)where Γ( t ) = qt and H ( t ) = pt . Therefore, in order to solve the dynamics we must first find the solution of (31a), tothen solve the equation (31b), with (30).In order to describe the fields χ and σ , we can expand these fields as Fourier series χ ( x α ) = 1(2 π ) (cid:90) d k (cid:104) A k e i(cid:126)k.(cid:126)r Θ k ( t ) + c.c. (cid:105) , (32a) σ ( x α ) = 1(2 π ) (cid:90) d k (cid:104) B k e i(cid:126)k.(cid:126)r ξ k ( t ) + c.c. (cid:105) , (32b)where ξ k are the time dependent modes of the field σ , which once normalised, are[8]: ξ k ( t ) = (cid:114) π p + q − t − ( q +3 p − H (2) ν [ y ( t )] . (33)Here, H (2) ν [ y ( t )] is the second kind Hankel function, with ν = q + 3 p − p + q − , (34a) y ( t ) = k t p + q t − ( p + q − a ( p + q − . (34b)The solution can be obtained using the expressions (32a), (32b) and (33) in (31b), with the general solution for Θ k ( t ):Θ k ( t ) = Θ ( h ) k ( t ) + Θ ( p ) k ( t ) . (35)Here, the homogeneous part of the solution for the modes of χ , isΘ ( h ) k ( t ) = C t − ( q +3 p − J − ν [ y ( t )] + C t − ( q +3 p − Y − ν [ y ( t )] . (36)The general solution finally results to beΘ ( p ) k ( t ) = 1 k (cid:114) π p + q − (cid:34) h ,k ( t ) (cid:90) H (2) ν [ y ( t )] t q λ ( t ) f ,k ( t ) g k ( t ) dt + h ,k ( t ) (cid:90) H (2) ν [ y ( t )] t q λ ( t ) f ,k ( t ) g k ( t ) dt (cid:35) . (37)where J − ν and Y − ν are the first and second kind Bessel functions with parameter − ν . Furthermore, h ,k ( t ), h ,k ( t ), f ,k ( t ), f ,k ( t ) and g k ( t ) are functions given by the expressions h ,k ( t ) = − t − ( p − q +1) a t p + q ( p − q + 1) Y ν [ y ( t )] − t − ( q +3 p − k Y ν [ y ( t )] , (38a) f ,k ( t ) = a t p + q t p + q − ( p − q + 1) J ν [ y ( t )] + k J ν [ y ( t )] , (38b) h ,k ( t ) = t − ( p − q +1) a t p + q ( p − q + 1) J ν [ y ( t )] + t − ( q +3 p − k J ν [ y ( t )] , (38c) f ,k ( t ) = a t p + q t p + q − ( p − q + 1) Y ν [ y ( t )] + k Y ν [ y ( t )] , (38d) g k ( t ) = Y ν [ y ( t )] J ν [ y ( t )] − Y ν [ y ( t )] J ν [ y ( t )] , (38e)with parameters ν = q − p − p + q − and ν = p +3 q − p + q − , such that they are related by the expression ν = ν + 1.In order to obtain a solution to the physical problem in which the waves are produced by the source, the homogeneoussolution must be null, so that we impose C = C = 0. For an analytical expression of the particular solution, wemust approach in the limit case where y ( t ) (cid:28)
1, corresponding to long wavelengths. So we obtain:Θ ( p ) k ( t ) | y ( t ) (cid:28) (cid:39) i Γ( ν ) (cid:114) π p + q − (cid:20) ( p + q − ν (cid:16) a k (cid:17) ν − (cid:18) A t p − − A t p +2 q (cid:19) + ( p − q + 1) ν (cid:16) a k (cid:17) ν +1 (cid:18) B t p − q − B t p +2 (cid:19)(cid:21) , (39)where the constants A , A , B and B are given by the expressions A = C ( p − q + 1) (cid:32) t − ( ν − p + q )0 π (3 q + p − q + 3 p − p − q ) + 2 ν t p + q )0 ( q + 2 p − p − ( q − ] (cid:33) , (40a) A = pq (6 p + q − (cid:32) t − ( ν − p + q )0 ( p − q + 1)3 π (3 q + p − q + 3 p − p + 2) + 2 ν t p + q )0 (2 p + q − q + 2 p − q + 2 p − p + q − (cid:33) , (40b) B = C (cid:32) t − ν ( p + q )0 ( p − q + 1) ν ( p + q − π (3 q + p − p − q )( q + 1) + t p + q (3 q + p − p − ( q + 1) ]( q + 1)[ p − ( q − ] (cid:33) , (40c) B = pq (6 p + q − (cid:32) t − ν ( p + q )0 ( p + q )( p − q + 1) ν +1 ( p + q − π (3 q + p − p + 2) + t p + q (3 q + p − p − ( q + 1) ]( q + 1)[ p − ( q − ] (cid:33) . (40d) A. Redefined fields
The dynamics of σ and χ fields are described respectively by the equations¨ σ + [3 H ( t ) + Γ( t )] ˙ σ − e − (cid:82) ( H ( t )+Γ( t )) dt a ∇ σ = 0 , (41a)¨ χ + [3 H ( t ) + Γ( t )] ˙ χ − e − (cid:82) ( H ( t )+Γ( t )) dt a ∇ χ = − U λ ( t ) ˙ σ e (cid:82) t ) dt . (41b)Notice that the right side of the equation (41b) is originated in the flux δ Φ of δW α ≡ σ α , that cross the 3D-Gaussianhypersurface. In our case, because the relativistic observer is in a co-moving frame the unique nonzero relativisticvelocity is U , so that only contributes ˙ σ in (41b). In order to simplify the structure of the equations (41a) and (41b),we make the changes of variables σ = e − (cid:82) (3 H ( t )+Γ( t )) dt u and χ = e − (cid:82) (3 H ( t )+Γ( t )) dt v , and we obtain:¨ u + [ ∇ − k ( t )] a e (cid:82) ( H ( t )+Γ( t )) dt u = 0 , (42a)¨ v + [ ∇ − k ( t )] a e (cid:82) ( H ( t )+Γ( t )) dt v = − U λ ( t ) ˙ σ e − (cid:82) ( H ( t ) − Γ( t )) . (42b) B. Coarse-grained
The coarse-grained approach provides a description of the dynamics for a desirable part of the spectrum. In ourcase this part is the long-wavelength sector of the spectrum, which is described by wavelengths much bigger than thesize of the Hubble horizon. This wavelengths variate with time, because the horizon is expanding. The wavenumberrelated to the horizon wavelength in a co-moving frame is k ( t ), so that we shall be interested in wavenumbers k , whichare smaller than k in order to describe the cosmological sector (infrared sector), of the spectrum during inflation k (cid:28) k ( t ) ≡ a e (cid:82) ( H ( t )+Γ( t )) dt (cid:34) [3 H ( t ) + Γ( t )] H ( t ) + ˙Γ( t )2 (cid:35) / . (43)At this point, we will define the coarse-grained fields[15], by using a suppression factor f ( k, t ) to select the desirablewavenumbers of the infrared sector of the spectrum: u cg = 1(2 π ) (cid:90) d k f ( k, t ) (cid:104) A k e i(cid:126)k(cid:126)r ξ k ( t ) + c.c. (cid:105) , (44a) v cg = 1(2 π ) (cid:90) d k f ( k, t ) (cid:104) B k e i(cid:126)k(cid:126)r ˜ ξ k ( t ) + c.c. (cid:105) , (44b)where the suppression factor f ( k, t ) is given by a Levy distribution, f ( k, t ) = (cid:114) (cid:15)k ( t )2 π e − (cid:15)k t )2( k − (cid:15)k t )) ( k − (cid:15)k ( t )) . (45)The square fluctuations for the coarse-grained fields are[13] (cid:10) B | u cg | B (cid:11) = (cid:90) ∞ dkk P u cg ( k ) = 12 π (cid:90) k dk k | ξ k ( t ) | f ( k, t ) , (46a) (cid:10) B | v cg | B (cid:11) = (cid:90) ∞ dkk P v cg ( k ) = 12 π (cid:90) k dk k | ˜ ξ k ( t ) | f ( k, t ) . (46b)In the figures (1) and (2) we show the power spectrums of P σ cg ( k ) and P χ cg ( k ), for different times (during theinflationary era — time scale is in Planckian times), with q = 0 . p = 1 .
5. In both cases the peaks move towardhigher k -values, while their intensities decrease with time. Notice that the intensities of P χ cg ( k ) are weakest than the P σ cg ( k ) ones. V. FINAL COMMENTS
We have shown that back-reaction effects in the primordial universe act as sources of space-time waves that prop-agates in all directions. These sources would be homogeneously and isotropically distributed in cosmological scales,which is the scale that concern us. They do not be the standard gravitational waves, but rather space-time wavesoriginated by local space-time fluctuations which have a quantum origin. It is expected that these waves would camefrom all directions, as cosmic background radiation, but its intensity is so low to be detected with the instrumenta-tion available today. We have calculated the spectrums for the squared fluctuations of σ cg and χ cg . In both cases,it is shown that the amplitudes are decreasing with time and the distributions are dispersed along the large-scale k -spectrum. In this work we have supposed that these sources are scalar fluctuations. All these sources can be viewedon the background metric as a decaying cosmological parameter λ ( t ), due to the fact we have supposed the universe asglobally isotropic and homogenous in the distribution of the sources. However, if we loss homogeneity, this parameterwould be a function of r and t [ i.e., a λ ( r, t )]. This topic will be studied in a future work. Acknowledgements
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Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton , Rend. Circ. Mat. Palermo :203-212 (1919). [English translation by R.Hojman and C. Mukku in P.G. Bergmann and V. De Sabbata (eds.) Cosmologyand Gravitation, Plenum Press, New York (1980)].[24] S. Kumar, Mon. Not. R. Astron. Soc. : 2532 (2012). FIG. 1: P u cg ( k ) for p = 1 . q = 0 .
9, with parameters t = G / , a = 0 . G / and (cid:15) = 10 − . The red, blue and greenlines correspond respectively to the cases t = 1 . × G / , t = 1 . × G / and t = 1 . × G / .FIG. 2: P v cg ( k ) for p = 1 . q = 0 .
9, with parameters C = 0, t = G / , a = 0 . G / and (cid:15) = 10 − . The red, blue andgreen lines correspond respectively to the cases t = 1 . × G / , t = 1 . × G / and t = 1 . × G /2