Sachs-Wolfe effect in Spacetimes with Torsion

Cartan torsion contribution to Sachs-Wolfe effect in the inflationary phase of the Universe is discussed.From the COBE data of the microwave anisotropy is possible to compute the spin-density in the Universe as 10 mks units.The spin-density fluctuations at the hadron era is shown to coincide with the anisotropy temperature fluctuations. PACS number(s) : 0420,0450 Departamento de F́ısica Teórica Instituto de F́ısica UERJ Rua São Fco. Xavier 524, Rio de Janeiro, RJ Maracanã, CEP:20550-003 , Brasil. E-mail : [email protected] The Sachs-Wolfe effect [1, 2, 3] is given by δT T = −Φ (1) where Φ is the Newtonian gravitational potential.Since there vis no restriction on the origin of the gravitational potential in this expression one may extend this expression to allow for alternative theories of gravity.Thus one is able to compute the anisotropy fluctuations in the CMB temperatures.In this letter we shall be concerned with the computation of the temperature fluctuations in the context of weak field limit approximation of the Einstein-Cartan gravity.In Newtonian approximation the Einstein-Cartan field equation may be expressed as 1 c ∂Φ ∂t −∇ Φ = −4πG(ρ−Gσ) (2) where σ is the spin-torsion energy density and ρ is the usual matter density.Since the cosmological spacetime used is of the form ds = dt − (1− 2Φ)(dx + dy + dz) (3) is homogeneous ∇Φ = 0 equation (2) reduces to Φ̈ = −4πGc(ρ−Gσ) (4) Solution of this expression can be easily obtained obtained by considering the RHS of (4) is constant in the average, thus Φ(t) = −4πGc(ρ−Gσ)t +Bt+ C (5) where B and C are integration constants.Therefore spin-torsion effects contribute to the temperature fluctuations through the term δT T torsion = 4πGcσt (6) Since at the inflation era t = 10s and from COBE data δT T < 10 one may obtain an expression for the spin-torsion density at the inflation era as σ = 1 4π (Gct) δT T torsion (7)