On the anomalous large-scale flows in the Universe
aa r X i v : . [ a s t r o - ph . C O ] S e p EPJ manuscript No. (will be inserted by the editor)
On the anomalous large-scale flows in the Universe
Davor Palle
Zavod za teorijsku fiziku, Institut Rugjer Boˇskovi´cBijeniˇcka cesta 54, 10002 Zagreb, CroatiaRevised: 19 June 2010
Abstract.
Recent combined analyses of the CMB and galaxy cluster data reveal unexpectedly large andanisotropic peculiar velocity fields at large scales. We study cosmic models with included vorticity, acceler-ation and total angular momentum of the Universe in order to understand the phenomenon. The Zel’dovichmodel is used to mimic the low redshift evolution of the angular momentum. Solving coupled evolutionequations of the second order for density contrast in corrected Ellis-Bruni covariant and gauge-invariantformalism one can properly normalize and evaluate integrated Sachs-Wolfe effect and peculiar velocityfield. The theoretical results compared to the observations favor a much larger matter content of the Uni-verse than that of the concordance model. Large-scale flows appear anisotropic with dominant componentsplaced in the plane perpendicular to the axis of vorticity (rotation). The integrated Sachs-Wolfe term hasa negative contribution to the CMB fluctuations for the negative cosmological constant and it can explainthe observed small power of the CMB TT spectrum at large scales. The rate of the expansion of theUniverse may be substantially affected by the angular momentum if its magnitude is large enough.
PACS.
PACS-98.80.Es Observational cosmology(including Hubble constant, distance scale, cosmologicalconstant, early Universe, etc.) – PACS-12.10.Dm Unified theories and models of strong and electroweakinteractions – PACS-04.60.-m Quantum gravity
Modern cosmology relies heavily on two paradigms in order to fit vast astrophysical datasets: 1. the existence of darkmatter and 2. the existence of dark energy. It is assumed that dark matter consists of the neutral weakly interactingfermion decoupled from primordial plasma as a nonrelativistic species (cold dark matter=CDM). The CDM-mass-density behavior on redshift is the standard one of nonrelativistic matter. The origin and the redshift behavior of darkenergy is rather obscure. The concordance
ΛCDM model presumes that dark energy is positive cosmological constant.Dark matter is thus a problem of particle physics, while dark energy is a common problem of field theory, particlephysics and the theory of gravity.The common mathematical difficulties of the zero distance singularity and causality motivate the proposal for anew symmetry-breaking mechanism in particle physics (BY theory of ref. [1]) and new studies in the Einstein-Cartantheory of gravity. Heavy Majorana neutrinos within the nonsingular and causal SU(3) electroweak-strong unification[1] represent the main building block of the Universe as a cold dark matter (DM) particles. They are cosmologicallystable τ N i >> τ U with large annihilation cross sections [2]. The H.E.S.S. source J1745-290 [3] in the center of ourgalaxy and WMAP haze [4] are possible indications of the heavy DM particle annihilations.Light Majorana neutrinos trigger vorticity of the Universe [5,6] through the spin-torsion relations in the Einstein-Cartan cosmology at early times of the evolution. The formation of large-scale structures in the form of galaxies andclusters, as well as, the present anisotropy of spacetime, not predicted by inflation of the concordance ΛCDM model,indicate the existence of nonvanishing total angular momentum of the Universe.The current measurements of the CMB by WMAP, the large catalogues of SDSS, the new cluster and the peculiarvelocity catalogues motivate us to undertake considerations to explain features that are not expected in the concordance
ΛCDM model-anisotropic anomalous large-scale flows. The low power of large-scale TT CMB fluctuations observedby COBE and WMAP, discrepant with the
ΛCDM model, should be explained with new cosmologies.In the next chapter we derive the evolution equations of density contrast for models with expansion, vorticity,acceleration and angular momentum within the corrected Ellis-Bruni formalism. A comparison is made with thestandard formulas of isotropic and homogeneous spacetime. One can then evaluate integrated Sachs-Wolfe effect andRMS peculiar velocity field for any model.
Davor Palle: On the anomalous large-scale flows in the Universe
The last chapter is devoted for discussions, conclusions and suggestions for future work. Appendices contain detailedframework for Einstein-Cartan cosmology with field equations, propagation equations and a comparison of the correctedand standard Ellis-Bruni fluid-flow approach for the density-contrast evolution.
Our interest is the evolution of the Universe during the matter dominated epoch. The standard lore for the evolutionof the density contrast, peculiar acceleration and peculiar velocity field gives the following equations [7,8,9]: δ ( a, k ) ≡ δ ( a ) δ k , δ ( x ) = Z d ke ı k · x δ k , v k = ı k k a ˙ aδ k dδ ( a ) da , a = R ( t ) /R = 1 / (1 + z ) . (1)The root mean square (RMS) values of the mass-density contrast and the peculiar velocity field at a certain scale S and the redshift z using Gaussian window functions [7,8,9] are defined as:( δM/M ) RMS ( a, S ) ≡ h ( δM/M ) i ( a, S ) = N V − W Z d kW ( k , S ) δ ( a ) | δ k | , (2) v RMS ( a, S ) ≡ h v i ( a, S ) = N V − W Z d kW ( k , S ) 1 k ( a ˙ a dδ ( a ) da ) | δ k | , (3) V W = 4 π S , W ( k , S ) = (2 π ) / S e − S k . The aim is to study cosmological models with vorticity, acceleration and nonvanishing total angular momentumthrough torsion effects. We use the standard CDM power spectrum P ( k ) = | δ k | defined in [8].The growth function δ ( a ) must be studied carefully. Our choice of covariant spacelike vectors within a fluid-flowformalism differs from that of Ellis and Bruni [10]: δ = ( −D µ D µ ) / , D µ ≡ R ( t ) ρ − h νµ ˜ ∇ ν ρ, (4) L µ ≡ R ( t ) h νµ ˜ ∇ ν Θ. These vectors fulfil the Stewart-Walker lemma [11] and their evolution equations result in a correct solution for adensity contrast formed from their scalar invariants. This is not the case for the standard Ellis-Bruni covariant vectors.The detailed derivation of the equations one can find in Appendix A while a comparison between two fluid-flowapproaches is in Appendix B.The resulting second order coupled equations in our corrected scheme are given by (matter dominated regime):¨ D µ −
13 ˙ Θ D µ − Θ ˙ D µ + a µ a ν D ν + u µ ˙ a ν D ν + u µ a ν ˙ D ν − ( 13 Θδ λµ + u µ a λ + ˜ ω λ.µ + σ λ.µ )( 13 Θ D λ − u λ a ν D ν − ˙ D λ − D ρ ( σ ρ.λ + ˜ ω ρ.λ ) + ΘR a λ ) − a µ R ˙ Θ − R ( (3) ˜ ∇ µ N ) − κρ D µ + ˙ D λ ( σ λ.µ + ˜ ω λ.µ )+ D λ ( ˙ σ λ.µ + ˙˜ ω λ.µ ) − R Θ a µ − ΘR ˙ a µ = 0 , (5) N = 2 σ − ω − ˜ ∇ µ a µ . We use the Zel’dovich model to describe the evolution of the total angular momentum of the Universe at smallredshifts [9] assuming the surplus of right-handed over left-handed galaxies and clusters [6,12] avor Palle: On the anomalous large-scale flows in the Universe 3 L ( t ) ∝ a ˙ a ¯ ρ Z V L d q ( q − ¯ q ) × [ ∇ Φ ( q ) − ∇ Φ (¯ q )] , L ( U niverse ) ∝ [ n ( right ) − n ( lef t )] L ( t ) ,Q = torsion ∝ L ( U niverse ) ⇒ Q ( a ) = Q a − / f or z = a − − < z cr = 4 , (6) where z cr = 4 is put arbitrarily, otherwise Q ( z > z cr ) = 0 . The Einstein-Cartan equations remain unaltered with respect to the functional form of the time dependence oftorsion (see Appendix A).We have to factorize density contrast on the space and time dependent parts. One can achieve this goal transformingthe evolution equations for covariant vectors to the local Lorentzian frame by tetrads: D a = v µa D µ , g µν = v µa v νb η ab , δ = ( −D a D a ) / ,η ab = diag (+1 , − , − , − , µ, ν = 0 , , , , a, b = ˆ0 , ˆ1 , ˆ2 , ˆ3 . In the Appendix A one can find evaluated coefficients for the following spacetime metric: ds = dt − R ( t )[ dx + (1 − λ ( t )) e mx dy ] − R ( t ) dz − R ( t ) λ ( t ) e mx dydt, (7) m = const. It is easy to verify that in the Friedmann limit one recovers the standard form of the density contrast [13] (seeAppendix A): δ ( a ) = H ( a ) H Z a daa − [ H H ( a ) ] , (8) H ( a ) = H ( Ω m a − + Ω Λ ) / . One can evaluate properly normalized peculiar velocities and integrated Sachs-Wolfe effect for various cosmologicalmodels [13]: a ISWlm = 12 πı l Z d kY m ∗ l (ˆ k ) δ k ( H k ) Z daj l ( kr ) χ ISW , (9) χ ISW = − Ω m dda ( δ ( a ) /a ) , r = Z a daa − H − ( a ) , δ ( a = 1) = 1 . Supplied with all the necessary equations we can evaluate properly normalized peculiar velocities and integratedSachs-Wolfe effect. We use the same normalization for all cosmological models:( δM/M ) RMS ( a = 1 , S = 10 h − M pc ) = 1 . This is the standard normalization suitable for the study of peculiar velocities (see p.262 of ref.[9]). We aim tocompare cosmic models with respect to the concordance
ΛCDM model, not to perform a precision fit to data.Let us fix relevant parameters of the models in Table I (unit H = 1 is used for parameters m, λ and Q ;EdS=Einstein-de Sitter, EC=Einstein-Cartan, ΛCDM =concordance model, Ω Λ = 1 − Ω m , λ = λ R − , Q = Q R − / , R = H − ).The formation of small-scale structures and the age of the Universe can be explained with a larger mass-densityand smaller Hubble constant (see Table II). This statement is valid if we assume that the total angular momentum ofthe Universe at low redshifts, acting through torsion terms, is much smaller than mass-density terms. The evolutionwith large torsion terms must contain feedback from matter to the background geometry, changing substantially its Davor Palle: On the anomalous large-scale flows in the Universe
Table 1.
Model parameters Ω m h m λ Q Λ CDM 0.3 0.7 0 0 0EdS 1 0.5 0 0 0EC1 2 0.4 0.03 0.067 -0.2EC2 2 0.4 0 0 0EC3 2 0.4 0.3 0.67 0
Table 2.
Age of the Universe Λ CDM EdS EC1 EC2 τ U (Gyr) 13.77 13.33 13.14 13.09 Table 3. v rms ( km/s ) for the Λ CDM modelz \ S(Mpc) 50 187.5 325 462.5 6000 511.67 184.73 111.75 79.95 62.210.25 515.49 186.11 112.58 80.55 62.671 467.16 168.66 102.03 73.00 56.80
Table 4. v rms ( km/s ) for the EdS modelz \ S(Mpc) 50 187.5 325 462.5 6000 766.27 245.82 145.55 103.27 80.000.25 685.38 218.87 130.18 92.37 71.561 541.84 173.82 102.92 73.03 56.57
Table 5. v rms ( km/s ) for the EC1 modelz \ S(Mpc) 50 187.5 325 462.5 6000 1090.94 335.81 197.54 139.83 108.190.25 901.75 277.58 163.28 115.58 89.421 678.04 208.71 122.78 86.91 67.24
Table 6. v rms ( km/s ) for the EC2 modelz \ S(Mpc) 50 187.5 325 462.5 6000 1091.69 336.04 197.68 139.92 108.260.25 902.28 277.74 163.38 115.65 89.481 678.45 208.84 122.85 86.96 67.28 expansion and vorticity. It is possible to utilize this approach within N-body simulations. Thus, we limit numericalevaluations in this paper to small torsion contributions.From Tables III-VI it is clear that only models with large mass-density can enhance large-scale peculiar velocitiesobserved in the analyses with combined cluster and WMAP data [14,15,16]. Rather small amounts of vorticity,acceleration or torsion do not essentially influence RMS velocities. However, components of a density contrast D ˆ i , i =1 , , λ ( a ) = λ a − ⇒ ω ∝ a − (see Apendix fordefinitions).The density contrasts normalized at zero-redshift do not depend on the initial cosmic scale factor, but a differencebetween components does depend. One can estimate the resulting angle between the axis of vorticity (z-axis) andthe anisotropic bulk velocity. The angle depends on the initial redshift and the magnitude of the vorticity ( ω ( t ) = mλ ( t ) R ( t ) − ): a ( initial ) = 10 − , a ( f inal ) = 1 , model = EC , ω H = 12 mλ = 10 − ⇒ (ˆ n ( f low ) , ˆ n ( axis )) = arctg ( D + D ) / |D ˆ3 | = 55 . ◦ ,a ( initial ) = 10 − , a ( f inal ) = 1 , model = EC , but m = 0 .
15 : (ˆ n ( f low ) , ˆ n ( axis )) = 57 . ◦ ,a ( initial ) = 10 − , a ( f inal ) = 1 , model = EC (ˆ n ( f low ) , ˆ n ( axis )) = 72 . ◦ . avor Palle: On the anomalous large-scale flows in the Universe 5 den s i t y - c on t r a s t s redshiftDensity-contrasts "LCDM""EC1" 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 den s i t y - c on t r a s t s redshiftDensity-contrasts "LCDM""EC1" Fig. 1.
Density contrasts for the Λ CDM and EC1 models.
Since the metric describes rotation around z-axis, the dominant components of over(under)densities, the peculiaraccelerations and the velocities are placed in the plain perpendicular to the axis of rotation (vorticity). We estimatethe angle between the measured directions of the axis of vorticity [17] and the large-scale flows [15]:ˆ n ( f low ) = ( l = 287 ◦ , b = 8 ◦ ) , ˆ n ( axis ) = ( l = 260 ◦ , b = 60 ◦ ) ⇒ (ˆ n ( f low ) , ˆ n ( axis )) = 53 ◦ . The reader can compare and visualize density contrasts and their derivatives for two crucial models ( Λ CDM andEC) in Figs. 1 and 2.The integrated Sachs-Wolfe effect is negative for large mass-density models (EC) with negative cosmological con-stant, while it is positive for Λ CDM, as can be seen in Fig. 3. The negative contribution of the ISW decreases the totalamplitude of the CMB fluctuation, while the positive ISW of the Λ CDM increases it [18]. The observations point tovery small power at large scales, in contradiction with the Λ CDM model.It seems that the introduction of rotational degrees of freedom (torsion, spin, vorticity, angular momentum) isinevitable in order to understand and fit all observational data. Two scenarios emerge as viable resolutions: (1) smallHubble constant with small amount of the total angular momentum of the Universe at present or possibly: (2) largerHubble constant if the total angular momentum appears much larger. Torsion terms (linear and quadratic) alwaysgive a negative contribution to the effective mass-density, as it can be seen from Einstein-Cartan field equations (seeAppendix A).Recent measurements of the Hubble constant [19] give a large value, thus EC1 type models with small contributionof torsion (angular momentum) at low redshifts are ruled out. The concordance model cannot accommodate to thelow power of density fluctuations at large scales because of the positive contribution of the ISW effect for the posi-tive cosmological constant. It has unsurmountable difficulties to explain large peculiar velocities, while the observedanisotropies of the CMB fluctuations and peculiar velocities are complete surprise for the astrophysical communityviolating fundamental cosmological principle of the isotropic Universe.
Davor Palle: On the anomalous large-scale flows in the Universe g r ad i en t s o f den s i t y - c on t r a s t s redshiftGradients of density-contrasast "LCDM""EC1" 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 g r ad i en t s o f den s i t y - c on t r a s t s redshiftGradients of density-contrasast "LCDM""EC1" Fig. 2.
Gradients of the density contrasts dδda for the Λ CDM and EC1 models.
Let us briefly comment some results of the analyses of cluster data by various projects and groups. Some groupsconclude that the mass density of the Universe is low like in the concordance model [20], while other groups draw aconclusion for a large mass density [21]. Large uncertainty in the L-T relations is a probable cause of the discrepantconclusions.There is a general belief that the CMB measurements are the most reliable tool to constrain cosmic parameters.One cannot ignore disturbing objections for possibly erroneous analyses of time ordered data [22] or problems withbeam profile sensitivity [23] in WMAP papers. Any error can affect the estimate of cosmic parameters.The equation of state for dark energy is a target of an extensive research by supernovae [24] or by combined data[25]. Even tests of the concordance model by studying local group dynamics provoke more questions than answers [26].It should be mentioned that the unusual time dilation in quasar light curves [27] should be eventually explained bylensing within different cosmic models.The second scenario with large torsion content at low redshifts is the most plausible model capable to sur-mount difficulties of the concordance model and EC1 type models. From the work in ref.[5] one can conclude thatlim R →∞ ρ M /ρ Λ = −
2, but lim R →∞ ρ M = 0 and then ρ Λ = 0. It follows that the torsion contribution plays a role ofthe negative dark energy (see [5] and Eq.(16)) ρ M, /ρ torsion, ≃ −
2, making possible the introduction of the largeHubble parameter necessary for more accurate cosmic clocks and the age of the Universe (see Eq.(17)). Considerationswith large angular momentum (torsion) of the Universe must include N-body numerical simulations. Dark energy,described by a torsion, should be a clustered physical quantity [28] dependent on redshift [29].To conclude, it is of great importance to search for an independent method to fix ρ M , not only the total density ρ tot . The idea of Zwicky [30] to study galaxy and cluster catalogues (SDSS etc.) to estimate directly the mass densityof matter could be advantageous. avor Palle: On the anomalous large-scale flows in the Universe 7 -1-0.8-0.6-0.4-0.2 0 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 c h i - f un c t i on s o f t he I S W e ff e c t redshiftIntegrated Sachs-Wolfe effect "LCDM""EC1"-1-0.8-0.6-0.4-0.2 0 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 c h i - f un c t i on s o f t he I S W e ff e c t redshiftIntegrated Sachs-Wolfe effect "LCDM""EC1" Fig. 3.
Integrated Sachs-Wolfe χ ISW function for the Λ CDM and EC1 models.
We provide here a complete set of conventions, identities and equations for Einstein-Cartan theory. g µν ia a metrictensor defined in Eq.(7), R(t) is a cosmic scale factor, λ (t) function is proportional to the acceleration vector a µ inEq.(11), m parameter of g µν defines vorticity in Eq.(11), Q α. βγ is a torsion tensor, S α. βγ is a spin tensor, v µa are tetradfields, ρ denotes mass density, p denotes pressure of the fluid, Λ is a cosmological constant, u µ is a velocity four-vector, Q = 1 / Q µν Q µν , Q α. µν = u α Q µν , a = R ( t ) /R in Eq.(17) is a dimensionless cosmic scale factor, h in Eq.(17) is aHubble parameter, A in Eq.(18) is a normalization constant.The metric is defined as: g µν = v µa v νb η ab , η ab = diag (+1 , − , − , − ,µ, ν = 0 , , , , a, b = ˆ0 , ˆ1 , ˆ2 , ˆ3 , ˜ Γ αβµ = Γ αβµ + Q α.βµ + Q .. αβµ. + Q .. αµβ. , ˜ R λ.σµν = ∂ µ ˜ Γ λσν − ∂ ν ˜ Γ λσµ + ˜ Γ λβµ ˜ Γ βσν − ˜ Γ λβν ˜ Γ βσµ . Field equations and Ricci identities look thus [31]:˜ R µν − g µν ˜ R = κ ˜ T µν , ˜ R µν = ˜ R λ.µλν , ˜ R = ˜ R µ.µ ,Q µ.ab + 2 v µ [ a Q b ] = κS µ.ab , κ = 8 πG N c − ,Q a = v µa Q µ , Q µ = Q µ.µν , [ ab ] = 1 / ab − ba ) , ( ab ) = 1 / ab + ba ) Davor Palle: On the anomalous large-scale flows in the Universe ( ˜ ∇ µ ˜ ∇ ν − ˜ ∇ ν ˜ ∇ µ ) u λ = − ˜ R σ.λµν u σ − Q σ.νµ ˜ ∇ σ u λ , ˜ ∇ α u β = ∂ α u β − ˜ Γ νβα u ν . Conformal or Weyl tensor ˜ C σλµν is defined as:˜ R σλµν = 12 ( g σµ ˜ R λν − g σν ˜ R λµ − g λµ ˜ R σν + g λν ˜ R σµ ) −
16 ˜ R ( g σµ g λν − g σν g λµ ) + ˜ C σλµν . The energy-momentum tensor of the Weyssenhoff fluid is derived by Obukhov and Korotky [32]: T µν = − ( p − Λ ) g µν + u µ [ u ν ( ρ + p ) + 2 u α ˜ ∇ β S β.αν ] . (10)The Ehlers-decomposition of the velocity-gradient can be written as˜ ∇ µ u ν = ˜ ω νµ + σ µν + 13 Θh µν + u µ a ν , (11) u µ u µ = 1 , h µν = g µν − u µ u ν , a µ = u ν ˜ ∇ ν u µ , Θ = ˜ ∇ ν u ν , ˜ ω µν = h αµ h βν ˜ ∇ [ β u α ] , σ µν = h αµ h βν ˜ ∇ ( α u β ) − Θh µν . The vorticity is uniquely defined by a variational principle (see Eq. (3.11) of [32]):˜ ω ij = v µi ( ˜ ∇ α v νj ) u α g µν , i, j = ˆ1 , ˆ2 , ˆ3 . (12)The above two formulas for vorticity agree, but a formula for vorticity in (5.5) of [32] has a wrong sign, as well asdefinitions of vorticity in [33] and [34]. Eventually, this confusion caused some wrong terms in the derivation of theevolution equations in [35], as it is pointed out in [34] and later in [36].The standard procedure leads to the evolution equations (Frenkel condition employed u µ Q κ.µν = 0):˙ F ≡ u µ ˜ ∇ µ F, ˜ ω = 12 ˜ ω µν ˜ ω µν , σ = 12 σ µν σ µν , ˙ Θ = ˜ ∇ µ a µ + 2˜ ω − σ − Θ − ˜ R σν u σ u ν ,h [ να h λ ] β ˙˜ ω νλ = − Θ ˜ ω αβ − σ γ [ α. ˜ ω γ | β ] − h [ να h λ ] β ˜ ∇ ν a λ + h [ να h λ ] β u µ u κ ˜ R κλµν ,h αν h βµ ˙ σ αβ = h αν h βµ ˜ ∇ ( α a β ) − a ν a µ − ˜ ω ( ν | ρ ˜ ω ρ. | µ ) − σ να σ αµ. − Θσ νµ − h νµ [2(˜ ω − σ ) + ˜ ∇ α a α ] − E ( νµ ) ,E αβ = ˜ C σαµβ u σ u µ . We use the following identity: (3) ˜ ∇ µ ( ˙ f ) − h νµ ( (3) ˜ ∇ ν f ) . = a µ ˙ f + (˜ ω λ.µ + σ λµ. + 13 Θh λµ ) (3) ˜ ∇ λ f, (13)and the continuity equation in matter dominated epoch: u µ ˜ ∇ µ ρ + ρ ˜ ∇ µ u µ = 0 (14)to derive the evolution equation for density contrast Eq.(5). avor Palle: On the anomalous large-scale flows in the Universe 9 We evaluate the coefficients of the coupled evolution equations for components in the local Lorentzian frame D a with the metric of Eq.(7) (time component D ˆ0 can be set to zero because of the relation u a D a = 0):¨ D ˆ i + b ij ˙ D ˆ j + a ik D ˆ k + d i = 0 , i, j, k = 1 , , , (15) a = −
13 ˙ λ − κΛ − Q − κρ −
53 ˙
RR λ ˙ λ −
13 ¨ λλ + 53 mR λQ − λ ˙ R R −
512 ( λ mR ) −
13 ¨ RR ( − λ ) ,a = 12 mR ˙ λ + 14 ˙ RR Q + 12 λ mR ˙ RR , a = 0 ,b = 2 ˙ RR , b = − Q + λ mR , b = 0 ,a = − mR ˙ λ −
14 ˙
RR Q − λ mR ˙ RR ,a = −
43 ˙ λ − κΛ − Q − κρ −
113 ˙
RR λ ˙ λ −
13 ¨ λλ + 53 mR Qλ − λ ( ˙ RR ) −
512 ( λmR ) −
13 ¨ RR ( − λ ) ,a = 0 , b = 2 Q − λ mR , b = 2 ˙ RR , b = 0 ,a = 0 , a = 0 ,a = −
13 ˙ λ − κΛ − Q − κρ −
53 ˙
RR λ ˙ λ −
13 ¨ λλ + 23 mR λQ −
23 ˙ R R λ − λ m R −
13 ¨ RR ( − λ ) ,b = 0 , b = 0 , b = 2 ˙ RR ,d = d = 0 ,d = R [ − λ ( ˙ RR ) + ˙ λ ˙ R R ( − λ ) + 2 ˙ λ + λ ( ... λ + λ ... RR )+ ˙ λ (4 Q + κ (2 Λ − ρ ) − mR Qλ + λ (5¨ λ + 2 λ m R + 9 λ ¨ RR ))+ ˙ RR (¨ λ (3 + 7 λ ) + λ (17 ˙ λ + 2 κΛ − Q − κρ + mR λQ )+ λ ¨ RR (3 + 5 λ ))] ,f or i = j : a ij = − a ji , b ij = − b ji . The symmetric parts of Einstein-Cartan equations are given by:(ˆ0ˆ0) : − λ ¨ RR + ˙ R R (3 − λ ) + m R ( − λ ) − RR ˙ λλ = κ ( ρ + Λ ) + 2 mλQR − Q , (ˆ1ˆ1) : 2(1 − λ ) ¨ RR + (1 − λ ) ˙ R R − m λ R − ˙ λ − RR ˙ λλ − ¨ λλ = κΛ + Q − mλQR , (ˆ2ˆ2) : 2 ¨ RR + (1 − λ ) ˙ R R − m λ R − RR ˙ λλ = κΛ + Q − mλQR , (ˆ3ˆ3) : (1 − λ )(2 ¨ RR + ˙ R R ) − (4 − λ ) m R − RR ˙ λλ − ˙ λ − ¨ λλ = κΛ + Q , (ˆ0ˆ1) : mλR ( λ ˙ RR + 32 ˙ λ ) = ˙ Qλ + 2 ˙ λQ + 3 λQ ˙ RR , (ˆ0ˆ2) : 2 λ ( ˙ R R − ¨ RR ) = 0 , (ˆ1ˆ2) : m R ( ˙ λ + 2 λ ˙ RR ) = 0 . In the limit of small λ << R R = 13 ( κ ( ρ + Λ ) − Q + 2 mλQR + ( mR ) ) , ¨ RR = 12 ( 23 κΛ − κρ + 43 Q − mλQR −
13 ( mR ) ) . (16)The age of the Universe follows immediately: τ U ( Gyr ) = 10 h Z − daa [ Ω Λ + Ω m a − − Q + 23 mλQa + m a ] − / , (17) λ = λ a − , Q = Q a − / ; m, λ , Q evaluated in the unit H = 1 . Note that even a term linear in torsion Q is negative because [6] mλ > mλ <
0) implies
Q < Q > P ( k ) = | δ k | = Ak (1 + βk + αk . + γk ) , (18) k = | k | , β = 1 . Ω m h ) − M pc, α = 9 . Ω m h ) − . M pc . , γ = 1 . Ω m h ) − M pc . One can introduce the following general vector variable fulfilling the Stewart-Walker lemma D µ ( k ) ≡ R k ( t ) ρ − h νµ ˜ ∇ ν ρ .It is easy to verify that the correct Friedmann limes of density contrast can be achieved by rescaling the scalar invariantof the vector variable δ ∝ R − k ( t )[ −D µ ( k ) D µ ( k )] / . We show below that our choice ( k = 2) and rescaled Ellis-Bruni choice ( k = 1) [10] give different results forgeometries beyond that of Friedmann. Thus, only our choice of the variable ( k = 2) gives a correct density contrastwithout any ad hoc posterior rescaling.Ellis-Bruni covariant vector variables are defined as [10]:¯ D µ ≡ R ( t ) ρ − h νµ ˜ ∇ ν ρ, (19)¯ L µ ≡ R ( t ) h νµ ˜ ∇ ν Θ. The same procedure as in Appendix A results in the following equation for a density contrast in the matterdominated epoch: avor Palle: On the anomalous large-scale flows in the Universe 11 ¨¯ D µ + a µ a λ ¯ D λ + u µ ˙ a λ ¯ D λ + u µ a λ ˙¯ D λ − κρ ¯ D µ +( ˙¯ D λ + ¯ D ν ( σ ν.λ + ˜ ω ν.λ ))( 23 Θδ λµ + u µ a λ + ˜ ω λ.µ + σ λ.µ )+ 23 Θu µ a ν ¯ D ν + ˙¯ D λ (˜ ω λ.µ + σ λ.µ ) + ¯ D λ ( ˙ σ λ.µ + ˙˜ ω λ.µ ) − R [2 a µ ˙ Θ + (3) ˜ ∇ µ N + Θa λ ( 23 Θδ λµ + u µ a λ + ˜ ω λ.µ + σ λ.µ )+ 13 Θ a µ + Θ ˙ a µ ] = 0 . (20)The corresponding equations in the local Lorentzian frame are:¨¯ D ˆ i + ¯ b ij ˙¯ D ˆ j + ¯ a ik ¯ D ˆ k + ¯ d i = 0 , i, j, k = 1 , , , (21)¯ a = 2 ˙ R R + ¨ RR − Q − κρ + mR λQ − ( λm R ) , ¯ a = 12 mR ˙ λ −
74 ˙
RR Q + 32 λ mR ˙ RR , ¯ a = 0 , ¯ b = 4 ˙ RR , ¯ b = − Q + λ mR , ¯ b = 0 , ¯ a = −
12 ˙ λ mR + 74 ˙
RR Q − λ ˙ RR mR , ¯ a = ¨ RR − ˙ λ − Q − κρ − λ ˙ λ ˙ RR + mR Qλ − ( λ m R ) − ( ˙ RR ) ( − λ ) , ¯ a = 0 , ¯ b = 2 Q − λ mR , ¯ b = 4 ˙ RR , ¯ b = 0 , ¯ a = 0 , ¯ a = 0 , ¯ a = 2 ˙ R R + ¨ RR − κρ, ¯ b = 0 , ¯ b = 0 , ¯ b = 4 ˙ RR , ¯ d = ¯ d = 0 , ¯ d = R [2 ˙ λ + λ ( ... λ + ... RR ) + ( ˙ RR ) ˙ λ (6 + 13 λ )+ ˙ λ (4 Q + κ (2 Λ − ρ ) − mR λQ + λ (5¨ λ + 2 λ m R + 9 λ ¨ RR ))+ ˙ RR (¨ λ (3 + 7 λ ) + λ (17 ˙ λ + 2 κΛ − Q − κρ + mR λQ + λ ¨ RR (3 + 5 λ ))] ,f or i = j : ¯ a ij = − ¯ a ji , ¯ b ij = − ¯ b ji . The authors in [10] equalize components of their variables ¯ D µ with a scalar density contrast δ . This is possiblein the Friedmann limes when all components are equal. However even then, the scalar quantity formed from theirvariables must be ad hoc multiplied by the cosmic scale factor to achieve the correct result: -2.5-2-1.5-1-0.5 0 0 2 4 6 8 10 d i ff e r en c e i n pe r c en t age redshiftComparison of the two fluid-flow approaches "difference"-2.5-2-1.5-1-0.5 0 0 2 4 6 8 10 d i ff e r en c e i n pe r c en t age redshiftComparison of the two fluid-flow approaches "difference" Fig. 4.
Comparison between the two fluid-flow approaches for the model EC3: δ ≡ [ −D µ D µ ] / and ¯ δ ≡ R ( t )[ − ¯ D µ ¯ D µ ] / ,difference ≡ ( δ − ¯ δ ) /δ , δ ( z = 0) = ¯ δ ( z = 0) = 1. δ ∝ R ( t )[ − ¯ D µ ¯ D µ ] / . Our corrected variables D µ , on the contrary, give immediately good and correct Friedmann limes: δ ∝ [ −D µ D µ ] / = [ −D a D a ] / (22)Let us stress that beyond Friedmannian geometry two quantities are not equal: R ( t )[ − ¯ D µ ¯ D µ ] / = [ −D µ D µ ] / , hence we use throughout our paper corrected variables D µ . In Fig. 4 the reader can find comparison between twoformulas when the vorticity and the acceleration do not vanish. References
1. D. Palle, Nuovo Cim.
A 109 , 1535 (1996)2. D. Palle, Nuovo Cim.
B 115 , 445 (2000); ibidem
B 118 , 747 (2003)3. F. Aharonian et al.(H.E.S.S. Collaboration), Phys. Rev. Lett. , 221102 (2006)4. D. P. Finkbeiner, Astrophys. J. , 186 (2004)5. D. Palle, Nuovo Cim. B 111 , 671 (1996)6. D. Palle, preprint arXiv:0802.2060v2 (2009)avor Palle: On the anomalous large-scale flows in the Universe 137. P. J. E. Peebles,
The Large-Scale Structure of the Universe , (Princeton University Press, New Jersey 1980)8. E. W. Kolb, M. S. Turner,
The Early Universe (Addison-Wesley, Redwood City 1990)9. T- Padmanabhan,
Structure formation in the Universe (Cambridge University Press, Cambridge 1995)10. G. F. R. Ellis, M. Bruni, Phys. Rev.
D 40 , 1804 (1989); G. F. R. Ellis, J. Hwang, M. Bruni, ibidem
D 40 , 1819 (1989);G. F. R. Ellis, M. Bruni, J. Hwang, ibidem
D 42 , 1035 (1990)11. J. M. Stewart, M. Walker, Proc. R. Soc. Lond.
A 341 , 49 (1974)12. M. J. Longo, preprint arXiv:0812.3437 (2008)13. A. Cooray, Phys. Rev.
D 65 , 103510 (2002)14. A. Kashlinsky, F. Atrio-Barandela, D. Kocevski, H. Ebeling, Astrophys. J. , L49 (2008); A. Kashlinsky, F. Atrio-Barandela, H. Ebeling, A. Edge, D. Kocevski, Astrophys. J. , L81 (2010)15. R. Watkins, H. A. Feldman, M. J.Hudson, M.N.R.A.S. , 743 (2009); H. A. Feldman, R. Watkins, M. J.Hudson, ibidem , 756 (2009)16. G. Lavaux, R. B. Tully, R. Mohayaee, S. Colombi, Astrophys. J. , 483 (2010)17. K. Land, J. Magueijo, Phys. Rev. Lett. , 071301 (2005)18. M. R. Nolta et al. (WMAP Collaboration), Astrophys. J. Suppl. , 296 (2009)19. A. G. Riess et al., Astrophys. J. , 539 (2009)20. J. A. S. Lima, J. V. Cunha, J. S. Alcaniz, Phys. Rev. D 68 , 023510 (2003); D. Rapetti, S. W. Allen, A. Mantz, M.N.R.A.S. , 1265 (2008); A. Vikhlinin et al., Astrophys. J. , 1060 (2009)21. S. C. Vauclair et al., Astron. and Astrophys. , L37 (2003); A. Blanchard, M. Douspis, M. Rowan-Robinson, S. Sarkar,Astron. and Astrophys. , 925 (2006); L. D. Ferramacho, A. Blanchard, Astron. and Astrophys. , 423 (2007)22. H. Liu, T.-P. Li, preprint arXiv:0907.2731 (2009); H. Liu, T.-P. Li, preprint arXiv:1003.1073 (2010); B. F. Roukema,preprint arXiv:1004.4506 (2010)23. U. Sawangwit, T. Shanks, preprint arXiv:1006.1270 (2010)24. R. Kessler et al., Astrophys. J. Suppl. , 32 (2009)25. S. Basilakos, M. Plionis, J. A. S. Lima, preprint arXiv:1006.3418 (2010)26. P. Kroupa et al., preprint arXiv:1006.1647 (2010), Astron. and Astrophys. in press27. M. R. S. Hawkins, M.N.R.A.S. , 1940 (2010)28. S. Basilakos, J. C. Bueno Sanchez, L. Perivolaropoulos, Phys. Rev.
D 80 , 043530 (2009)29. R. Bean, M. Tangmatitham, Phys. Rev.
D 81 , 083534 (2010)30. F. Zwicky, Helv. Phys. Acta , 110 (1933)31. J. A. Schouten, Ricci-Calculus (Springer Verlag, Berlin 1954)32. Yu. N. Obukhov, V. A. Korotky, Class. Quantum Grav. , 1633 (1987)33. Th. Chrobok, Yu. N. Obukhov, M. Scherfner, Phys. Rev. D 63 , 104014 (2001)34. Th. Chrobok, H. Herrmann, G. Rueckner, Technische Mechanik , 1 (2002)35. D. Palle, Nuovo Cim. B 114 , 853 (1999)36. S. D. Brechet, M. P. Hobson, A. N. Lasenby, Class. Quant. Grav.24