Delta gravity and the accelerated expansion of the universe

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(1)December 14, S0218301311040086. 2011 15:34 WSPC/INSTRUCTION. FILE. International Journal of Modern Physics E Vol. 20, Supplement 1 (2011) 65–72 c World Scientific Publishing Company DOI: 10.1142/S0218301311040086. DELTA GRAVITY AND THE ACCELERATED EXPANSION OF THE UNIVERSE. JORGE ALFARO Facultad de Fı́sica, Pontificia Universidad Católica de Chile Avda. Vicuña Mackenna 4860, Santiago, Chile [email protected]. We study a model of the gravitational field based on two symmetric tensors. The equations of motion of test particles are derived. We explain how the Equivalence principle is recovered. Outside matter, the predictions of the model coincide exactly with General Relativity, so all classical tests are satisfied. In Cosmology, we get accelerated expansion without a cosmological constant. Keywords: General Relativity; Quantum Field Theories; Delta Gravity.. 1. Introduction General Relativity describes gravitation with enormous precision at very different scales.1 However it fails at the quantum level because it is not a renormalizable theory. The strongest candidate to the quantization of gravity is String Theory.2 But, due to the plethora of consistent vacua, it is far from phenomenological predictions. General Relativity without matter and without a cosmological constant is finite at one loop.7 Several years ago8,9 we introduced a particular type of gauge theories(semiclassical or delta). They are fully fledged Quantum Field Theories that have the property of living at one loop only. All higher loops are automatically zero. In 10 we apply a generalization of the methods employed in 8,9 to produce a Quantum Theory of Gravity (Delta Gravity) that lives at one loop.12 The main property of Delta Gravity is that Classical equations of motion are satisfied in the full Quantum theory. They are obtained through the extension of the former symmetry of the model, introducing an extra symmetry(δ), which is formally obtained as the variation of the original symmetry. In this work, we review some of the results of ref. 10 .. 2. Definition of Delta Gravity We use the metric convention (− + ++); κ = 65. 8πG c4. and κ2 is an arbitrary constant..

(2) December 14, S0218301311040086. 66. 2011 15:34 WSPC/INSTRUCTION. FILE. J. Alfaro. The action of delta-gravity is: Z √ 1 S(g, g̃) = dd x −g − R + LM 2κ Z √ 1 + κ2 Rµν − gµν R + κTµν −g g̃ µν dd x . 2. (1). Tµν is the matter energy-momentum tensor. In this work, we treat matter as an external source, subjected to the conservation law Tµν;ν = 0. 2.1. Symmetries Action (1) is invariant under the following transformations(δ), if Tµν;ν = 0: ρ ρ δgµν = gµρ ξ0,ν + gνρ ξ0,µ + gµν,ρ ξ0ρ = ξ0µ;ν + ξ0ν;µ ρ ρ δg̃µν (x) = ξ1µ;ν + ξ1ν;µ + g̃µρ ξ0,ν + g̃νρ ξ0,µ + g̃µν,ρ ξ0ρ .. (2). 2.2. Equations of motion The equation of motion for gµν is: δTµν µν 1 g̃ , S γσ + (Rg̃ γσ − gµν Rσγ g̃ µν ) = κ 2 δgγσ. (3). S σγ = F (σγ)(αβ)ρλ g̃αβ;λρ ;. (4). where. F (µν)(αβ)ρλ = P ((ρµ)(αβ)) g νλ + P ((ρν)(αβ)) g µλ − P ((αβ)(µν)). P ((µν)(αβ)) g ρλ − P ((ρλ)(αβ)) g µν ; = 41 g αµ g βν + g αν g βµ − g αβ g µν .. The equation of motion for g̃ µν is Einstein equation: 1 Rµν − gµν R + κTµν = 0 . 2. (5). (6). Covariant derivatives as well as raising and lowering of indices are defined using gµν . Notice that outside the sources(Tµν = 0), a solution of (3) is g̃ µν = λg µν , for µν a constant λ, since g;α = 0. We will have g̃ µν = g µν , assuming that both fields satisfy the same boundary conditions. 3. Energy Momentum Tensor for a Perfect Fluid We start from Tµν = pgµν + (p + ρ)Uµ Uν , g µν Uµ Uν = −1 ;. p is the pressure and ρ is the fluid density. Then: δTµν µν 1 g̃ = gµν (T γµ g̃ σν + T σµ g̃ γν ) . δgγσ 2. (7). (8).

(3) December 14, S0218301311040086. 2011 15:34 WSPC/INSTRUCTION. FILE. Delta Gravity and the Accelerated Expansion of the Universe. 67. 4. Particle Motion in the Gravitational Field The action of a particle in a gravitational field is Z Z p √ 1 dn y −gR − m dt −gµν ẋµ ẋν . 2κ. The variation in xµ produces the geodesic equation. Instead in δ gravity we have: Z Z p 1 n √ d y −gR − m dt −gµν ẋµ ẋν 2κ Z √ +κ2 dn y −g(Gµν + κTµν )g̃ µν , where: m Tµν (y) = √ −g. Z. (9). ẋµ ẋν dt p δ(y − x) . −gαβ ẋα ẋβ. Notice that Tµν is t-parametrization invariant. Call κ′2 = κ2 κ. It is dimensionless. The new action has a term: Z dt Sp = m p ẋµ ẋν (g µν + κ′2 g̃ µν ) = −gαβ ẋα ẋβ Z dt gµν ẋµ ẋν . (10) m p −gαβ ẋα ẋβ Since far from the sources, we must have free particles in Minkowski space, it follows that we are describing the motion of a particle of mass m′ : m′ = m(1 + κ′2 ) . And since outside the sources g̃ µν = g µν , the geodesic equation outside the sources is the same as Einstein’s equation. In order to include massless particles, we prefer to use the action: R 1 K 2 −1 µ ν L = dt vm µ ν − v ẋ ẋ gµν ; K = ẋµ ẋν gµν , 2 ẋ ẋ gµν. which, using the equation of motion of v reduces to (10) This action is invariant under re-parameterizations:. x′ (t′ ) = x(t) ; dt′ v ′ (t′ ) = dtv(t) ; t′ = t − ε(t) . Using this gauge freedom, we can fix v = 1. The equation of motion of v is: m2. K ẋµ ẋν g. µν. + v −2 K = 0 ⇒. p −ẋµ ẋν gµν v=− , m 6= 0 . m. m2 −ẋµ ẋν gµν. = v −2 (K 6= 0) ;.

(4) December 14, S0218301311040086. 68. 2011 15:34 WSPC/INSTRUCTION. FILE. J. Alfaro. For massive particles, introducing mdt = dτ , the lagrangian reduces to Z 1 ẋµ ẋν gµν µ ν L1 = m2 dτ − ẋ ẋ g ; ẋµ ẋν gµν = −1 . µν 2 ẋµ ẋν gµν. From L1 the equation of the geodesic for massive particles is derived:. d(ẋµ ẋν gµν ẋβ gαβ + 2ẋβ gαβ ) 1 µ ν − ẋ ẋ gµν ẋβ ẋγ gβγ,α − ẋµ ẋν gµν,α = 0 . dτ 2 We will discuss the motion of massive particles elsewhere. For massless particles, we get R K = ẋµ ẋν gµν = 0 ; L0 = − dt 21 (ẋµ ẋν gµν ) .. (11). So, the massless particle moves in a null geodesic of gµν . 5. Local Lorentz Frames. Using general coordinate transformations(gct) and δ gct, we can choose at each point x0 : gµν (x0 ) = ηµν , gµν,λ (x0 ) = 0 ; g̃µν (x0 ) = ηµν , g̃µν,λ (x0 ) = 0 . In these local Lorentz frames, the equation of motion for massive particles implies: d2 xµ = 0. dτ 2 That is Local Lorentz frames are inertial frames. Therefore the Equivalence Principle is satisfied. 6. Distances and Time Intervals Eq.(11) satisfies the important property of preserving the form of the proper time in a particle in free fall. Notice that in our case the quantity that is constant using (11) is ẋµ ẋν gµν . Therefore we define proper time using the original metric gµν , √ dτ = −g00 dx0 . (12) Following Landau citeLandau, the change in x0 for a roundtrip of a light ray from a point with coordinates xα to a neighboring point with coordinates xα + dxα is q 2 ∆x0 = (g0α g0β − gαβ g00 ) ; g00 r r g00 g0α g0β dl = gαβ − ; g00 g00 dl2 = γij dxi dxj ; g0i g0j g00 (gij − ). γij = g00 g00.

(5) December 14, S0218301311040086. 2011 15:34 WSPC/INSTRUCTION. FILE. Delta Gravity and the Accelerated Expansion of the Universe. 69. 7. Friedman-Robertson-Walker Metric In this case, assuming flat three dimensional metric: −ds2 = dt2 − R(t)2 dr2 + r2 dθ2 + r2 sin2 θdφ2 ; −ds̃2 = Ã(t)dt2 − B̃(t) dr2 + r2 dθ2 + r2 sin2 θdφ2 . Equations (3, 8) give:. 1 1 1 3 − ṘB̃˙ − pRB̃ + R−1 Ṙ2 B̃ − ρ R3 Ã + RṘ2 Ã = 0 ; 2 2 6 2 ¨ − R−2 Ṙ2 B̃ + 2R−1 R̈B̃ + 2R−1 ṘB̃˙ − pB̃ − 2B̃ +ρR2 Ã + Ṙ2 Ã + 2RṘÃ˙ + 2RÃR̈ = 0 .. (13). (14). Einstein’s equations are: 2 2 2 3 ddt R d d = κρ ; 2 R R + R = −κR2 p . R2 d t2 dt We use the state equation: p = wρ , to get, for w 6= −1 :. 2. w−1. R = R0 t 3(1+w) ; Ã = 3wl2′ t( w+1 ) ; 4 w−1 B̃ = R02 l2′ tb , b = + . 3w + 3 w + 1. (15). 7.1. Redshift Photons moves on a null geodesic of g:. Introduce:. e 2 + (R2 + κ′ B̃)(dr2 + r2 dθ2 + r2 sin2 θdφ2 ) . 0 = −(1 + κ′2 A)dt 2 R̃(t) =. s. (16). R2 + κ′2 B̃ (t) . 1 + κ′2 Ã. Then: z=. R̃(t0 ) −1. R̃(t1 ). (17). 7.2. Luminosity Distance The luminosity distance is: dL =. R̃(t0 ) R̃(t1 ). Z. z 0. dz = (1 + z) H(z). Z. 0. z. dz ′ R̃˙ , H̃ = . H̃(z ′ ) R̃. (18).

(6) December 14, S0218301311040086. 70. 2011 15:34 WSPC/INSTRUCTION. FILE. J. Alfaro. 8. Supernova Ia data The supernova Ia data gives, m (apparent or effective magnitude) as a function of z. This is related to distance dL by: dL m = M + 5log . 10pc Here M is common to all supernova and m changes with dL alone. We compare δ gravity to General Relativity(GR) with a cosmological constant: H 2 = H02 (Ωm (1 + z)3 + (1 − Ωm )) ; ΩΛ = 1 − Ωm . Notice that Ã = 0 for w = 0 in (15). So, it seems that we cannot fit the supernova data. However w = 0 is not the only component of the Universe. The massless particles that decoupled earlier still remain. It means that true 0 6 w < 13 , but very close to w = 0. So, we will fit the data with w = 0.1, 0.01, 0.001 and see how sensitive the predictions are to the value of w. Using data from Essence, we notice that R2 test does not change much with w. a Moreover l2 scales with w, l2 ∼ 3w . As an approximation to the limit w = 0, we get: √ a ; (19) R̃(t) = R(t) √ a−t q 1 3w renormalizes the derivative of R̃ at t = 0. It is not divergent, because for. t → 0, w → 13 . a is a free parameter. Of course, the complete model must include the contribution of normal matter(w = 0) plus relativistic matter (w = 13 ). But, at later times, the data should tend to (19). Let us fit the data to the simple scaling model (19). We get: Ωm = 0.22 ± 0.03, M = 43.29 ± 0.03 , χ2 (perpoint) = 1.0328, General Relativity a = 2.21 ± 0.12, M = 43.45 ± 0.06, χ2 (perpoint) = 1.0327, Delta Gravity δ-gravity with non-relativistic(NR) matter alone give a fit to the data as good as GR with NR matter plus a cosmological constant. According to the fit to data, a Big Rip will happen at t = 2.21 in unities of t0 (today). It is a similar scenario as in 11 . Finally, we want to point out that since for t → 0, we have w → 13 , then R̃(t) = R(t). Therefore the accelerated expansion is slower than (19) when we include both matter and radiation in the model. Below, we have the graphs for general relativity and delta gravity versus the data in Essence. 9. Conclusions and Open Problems Delta Gravity agrees with General Relativity when Tµν = 0, imposing same boundary conditions for both tensor fields..

(7) December 14, S0218301311040086. 2011 15:34 WSPC/INSTRUCTION. FILE. Delta Gravity and the Accelerated Expansion of the Universe. 71. Fig. 1. General Relativity with NR matter plus a cosmological constant and Essence data. On the vertical axis, we have m. On the horizontal axis we have z.. Fig. 2. Delta Gravity with NR matter without a cosmological constant and Essence data. On the vertical axis, we have m. On the horizontal axis we have z.. Local Lorentz frames exist; they are also local inertial frames. In this way the Equivalence Principle is implemented, since in them gravity (represented by both gµν and g̃µν ) is cancelled out: massive particles are in free fall and photons move on null geodesics of ηµν locally. We recover the Newtonian approximation. See 10 . In a homogeneous and isotropic universe, we get accelerated expansion without a cosmological constant or additional scalar fields. Notice that equation (15) implies that R̃ = R at the beginning of the Universe, when w = 13 , corresponding to ultra-relativistic matter. That is, the accelerated expansion started at a later time, which is needed if we want to recover the observational data of density perturbations and growth of structures in the Universe. An earlier acceleration of the expansion would prevent the growth of density perturbations. Work is in progress to compute the PPN(Postnewtonian) parameters,the growth of density perturbations, the anisotropies in the CMBR,BAO, WL and the evolution of massive stars..

(8) December 14, S0218301311040086. 72. 2011 15:34 WSPC/INSTRUCTION. FILE. J. Alfaro. Acknowledgements The author wants to thank the organizers of the First Caribbean Symposium on Nuclear and Astroparticle Physics - STARS2011 La Habana, Cuba, 1-4 May 2011,for a pleasant and stimulating atmosphere. The work of JA is partially supported by VRAID/DID/46/2010 and Fondecyt 1110378. He wants to thank R. Avila and P. González for several useful remarks; The author acknowledges interesting conversations with L. Infante, G. Palma, M. Bañados and A. Clocchiatti. In memoriam of my father, Juan de Dios Alfaro Jorquera who departed on May 1, 2011. References 1. C.M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Relativity 9 (2006), http://www.livingreviews.org/lrr-2006-3; S.G. Turyshev, Annual Review of Nuclear and Particle Science 58, 207 (2008). 2. For a modern review of string theory see: M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, vols. 1, 2, Cambridge University Press (1987); J. Polchinski, String Theory, vols. 1,2 (Cambridge University Press, Cambridge, 1998). 3. For a review of Dark Matter and its detection , see S. Weinberg, Cosmology (Oxford University Press, Oxford, 2008); D. Hooper and E.A. Baltz, Annu. Rev. Nucl. Part. Sci. 58 (2008) 293314. 4. A.G. Riess et al., (Supernova Search Team), Astron. J. 116 (1998) 1009; S. Perlmutter et al., (Supernova Cosmology Project), Astrophys. J. 517 (1999) 565 ; For a recent review, see R. R. Caldwell and M. Kamionkowski, The Physics of Cosmic Acceleration, astro-ph 0903.0866. 5. J.A. Frieman, M.S. Turner, and D. Huterer, Dark Energy and the Accelerating Universe, Annu. Rev. Astron. Astrophys. 46 (2008) 385432 . 6. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972). 7. G.’t Hooft and M. Veltman, Ann. Inst. Henri Poincaré 20 (1974) 69. 8. J. Alfaro, BV Gauge Theories, hep-th 9702060. 9. J. Alfaro and P. Labraña, Phys. Rev. D 65 (2002) 045002 . 10. J. Alfaro, Delta Gravity and Dark Energy, gr-qc 1006.5765. 11. R.R. Caldwell, M. Kamionkowski, and N.N. Weinberg, Phys. Rev. Lett. 91 (2003) 071301. 12. J. Alfaro, R. Avila and P. González, Quantization of Delta-Gravity,gr-qc 1009.2800. 13. W. Siegel, http://arxiv.org/abs/hep-th/9912205v3 14. L. Landau and L.M. Lifshitz, The Classical Theory of Fields, (Pergamon Press, Oxford, 1980). 15. R.F.C. Vessot et al., Phys. Rev. Lett. 45 (1980) 2081. 16. P.J. Mohr, B.N. Taylor, and D.B. Newell in Rev. Mod. Phys. 80 (2008) 633. 17. W.M. Wood-Vasey et al., Astrophys.J. 666 (2007) 694..

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