Big bounce from spin and torsion
aa r X i v : . [ a s t r o - ph . C O ] A ug General Relativity and Gravitation
Vol. , No. 4 (2012) pp. 1007–1014c (cid:13) Springer Science+Business Media, LLC
BIG BOUNCE FROM SPIN AND TORSION
Nikodem J. Pop lawski
Department of Physics, Indiana University, Swain Hall West,727 East Third Street, Bloomington, Indiana 47405, USA ∗ The Einstein-Cartan-Sciama-Kibble theory of gravity naturally extends general relativity to ac-count for the intrinsic spin of matter. Spacetime torsion, generated by spin of Dirac fields, inducesgravitational repulsion in fermionic matter at extremely high densities and prevents the formationof singularities. Accordingly, the big bang is replaced by a bounce that occurred when the energydensity ǫ ∝ gT was on the order of n /m (in natural units), where n ∝ gT is the fermionnumber density and g is the number of thermal degrees of freedom. If the early Universe containedonly the known standard-model particles ( g ≈ times smaller than its present value, giving ≈ µ m. If more fermionsexisted in the early Universe, then the spin-torsion coupling causes a bounce at a lower energy andlarger scale factor. Recent observations of high-energy photons from gamma-ray bursts indicatethat spacetime may behave classically even at scales below the Planck length, supporting the clas-sical spin-torsion mechanism of the big bounce. Such a classical bounce prevents the matter in thecontracting Universe from reaching the conditions at which a quantum bounce could possibly occur. Keywords: torsion, spin fluid, bouncing cosmology, nonsingular universe.
The Einstein-Cartan-Sciama-Kibble (ECSK) theory of gravity naturally extends Einstein’s general relativity (GR)to account for the quantum-mechanical, intrinsic angular momentum (spin) of elementary particles that composegravitating matter [1–12]. The ECSK gravity is based on the Lagrangian density for the gravitational field that isproportional to the curvature scalar, as in GR [13]. It removes, however, the constraint of GR that the torsion tensor(the antisymmetric part of the affine connection) be zero by promoting this tensor to a dynamical variable, as themetric tensor [1–12]. The torsion tensor is then given by the principle of least action and in many physical situations itturns out to be zero. But in the presence of fermions, which compose all stars in the Universe, spacetime torsion doesnot vanish because Dirac fields couple minimally to the torsion tensor [1–12]. At macroscopic scales, such particlescan be averaged and described as a spin fluid [14–16]. It has been shown in [17] that the spin-fluid form of the spintensor results from the conservation law for this tensor [5–12].The field equations of the ECSK gravity can be written as the general-relativistic Einstein equations with themodified energy-momentum tensor [1–12]. Such a tensor has terms which are quadratic in the spin tensor and thus donot vanish after averaging [16, 18]. These terms are significant only at densities of matter that are much larger than thedensity of nuclear matter; otherwise the ECSK gravity effectively reduces to GR. The ECSK gravity therefore passesall current tests of GR. These terms generate gravitational repulsion in spin-fluid fermionic matter, which becomessignificant in the early Universe and inside black holes. Such a repulsion prevents the formation of singularities fromfermionic matter [16, 18–22]. It replaces the singular big bang by a nonsingular state of minimum but finite radius[18, 23–25]. This extremely hot and dense state is a (big) bounce that follows a contracting phase of the Universeand initiates its rapid expansion [26–34].In [24, 25], we considered the dynamics of a closed universe immediately after such a bounce. We showed thata negative and extremely small (in magnitude) spin-torsion density parameter naturally explains why the presentUniverse appears spatially flat, homogeneous, and isotropic. The ECSK gravity therefore not only eliminates anunphysical cosmological singularity but also provides an alternative to cosmic inflation without requiring additionalfields and specific assumptions on their potentials. Another advantage of the ECSK theory is that it has no freeparameters. We also suggested that the coupling between spin and torsion may be the mechanism that allows for ascenario in which every black hole produces a new universe inside, instead of a singularity [24, 25]. The contraction ∗ Electronic address: [email protected] of our Universe before the bounce at the minimum radius may thus correspond to the dynamics of matter inside acollapsing black hole existing in another universe [35–43]. A scenario in which the Universe was born in a black holeseems more reasonable than its contraction from infinity in the past [44] because the latter does not explain whatcaused such a contraction. If our Universe was born in a black hole that has formed in a parent universe, then itwould interact with the parent universe. Recent measurements of the large-scale bulk flows of galaxy clusters [45],which cannot be explained within the standard theoretical framework, suggest that our Universe may be interactingwith other parts of spacetime. Torsion may therefore provide a natural scenario for what existed before the Universebegan to expand [24, 25, 35–43].In [24, 25], we estimated the torsion density parameter and the conditions at the big bounce from the currentnumber density of neutrinos which are the most abundant fermions in the Universe. We found that the energy densityof matter at the bounce is a few orders of magnitude larger than the Planck energy density. The scenario of a bigbounce within the classical ECSK theory may therefore be inadequate because the Planck regime is expected to bedescribed by a quantum theory of gravity. Interestingly, loop quantum gravity (LQG), which assumes that spacetimeis discrete at the Planck scale, also predicts a cosmic bounce at the Planck energy [46–48]. In this paper, we refinethe results of [24, 25] by including the thermal degrees of freedom arising not only from photons and neutrinos, butalso from other standard-model particles that are ultrarelativistic in the early Universe.We consider a closed ( k = 1), homogeneous, and isotropic universe filled with fermionic matter macroscopicallyaveraged as a spin fluid [16]. The Einstein-Cartan field equations for the Friedman-Lemaˆıtre-Robertson-Walker(FLRW) metric describing such a universe are given by the Friedman equations for the scale factor a ( t ) [18, 23–25]:1 c (cid:16) dadt (cid:17) + k = 13 κ (cid:16) ǫ + ǫ S (cid:17) a + 13 Λ a , (1) ddt (cid:0) ( ǫ + ǫ S ) a (cid:1) + ( p + p S ) ddt ( a ) = 0 , (2)where ǫ S = − κs , p S = ǫ S (3)are the contributions to the energy density of matter ǫ and pressure p from the spin-torsion coupling [16]. The quantity s in (3) is the square of the dispersion of the spin density distribution around its average value and it is equal, forunpolarized spins, to [49] s = 18 ( ~ cn ) , (4)where n is the fermion number density. In the early Universe, the matter is ultrarelativistic: p ≈ ǫ/
3. We can alsoneglect in (1) the terms with k and the cosmological constant Λ. The big bounce occurs when da/dt = 0, which gives ǫ = 14 κs = 132 κ ( ~ cn ) . (5)In natural units ( ~ = c = 1), this condition is ǫ = π n m , (6)where m Pl is the Planck mass.The energy density of ultrarelativistic matter in kinetic equilibrium is given by [50, 51] ǫ ( T ) = π g ⋆ ( T ) ( k B T ) ( ~ c ) , (7)where T is the temperature of the early Universe. The effective number of thermal degrees of freedom g ⋆ ( T ) is givenby [50, 51] g ⋆ ( T ) = g b ( T ) + 78 g f ( T ) , (8)where g b = P i g i is summed over relativistic bosons and g f = P i g i is summed over relativistic fermions. For eachparticle species, g i is the number of its spin states. We include in g ⋆ ( T ) only relativistic species (whose rest masses m i satisfy m i c < k B T ) because the energy density of relativistic particles in the early Universe is much larger thanthat of nonrelativistic particles. The fermion number density is given by [50, 51] n ( T ) = ζ (3) π g n ( T ) ( k B T ) ( ~ c ) , (9)where ζ (3) ≈ .
202 is the Riemann zeta function of 3, and g n ( T ) = 34 g f ( T ) . (10)Substituting the energy density (7) and the number density (9) into the condition for a torsion-driven bounce (5)gives the temperature at the big bounce: T bb = (cid:18) (cid:19) / π / ζ (3) ( g b + 7 g f / / g f T Pl , (11)where T Pl = k − ( ~ c /G ) / is the Planck temperature. Substituting the temperature (11) into (7) gives the energydensity of matter at the bounce: ǫ bb = ξ ( g b + 7 g f / g ǫ Pl , (12)where ǫ Pl = c / ( ~ G ) = 5 . × kg / m · c is the Planck energy density and ξ = 130 (cid:18) π ζ (3) (cid:19) . (13)For T > m t , where m t ≈
175 GeV is the rest mass of a t quark, all known particles are relativistic. If the earlyUniverse contained only all known standard-model particles (with equal temperatures), then g b = 28 and g f = 90[50, 51] at the big bounce, which gives ǫ bb = 1 . × ǫ Pl . (14)In this case, the energy density at the bounce is at the Planck scale, where the classical ECSK theory should bereplaced by a quantum theory of gravity. LQG predicts a quantum cosmic bounce at the same scale [46–48]. If,however, much more fermionic degrees of freedom existed at extremely high energies, then the spin-torsion couplingcauses a bounce below the Planck energy. Such a scenario would be possible, for example, if standard-model fermionswere composed of more elementary particles [52–57]. If, for example, g f = 10 at the bounce and the ratio g b /g f isconstant, then ǫ bb = 1 . × − ǫ Pl . In this case, the classical description of gravity is sufficient. Such a classicalbounce would prevent the matter in the contracting Universe from reaching densities at which a quantum bouncewould happen (if LQG is correct). Consequently, LQG would not be able to provide cosmological signatures in thepresence of torsion.Recent observations of high-energy photons from gamma-ray bursts [58, 59] suggest, however, that spacetime maybe continuous at the Planck length and lower scales [60]. Spacetime may therefore behave classically even above thePlanck energy. As a result, the classical spin-torsion mechanism of the big bounce should be valid without additionalfermionic degrees of freedom.The energy density of ultrarelativistic matter scales according to ǫ R ∼ a − , so that its present-day value is givenby ǫ R0 = ǫ bb (cid:18) a bb a (cid:19) ( g b + 7 g f / | ( g b + 7 g f / | bb , (15)where a bb is the scale factor at the bounce and a is the current scale factor. Subscripts 0 denote quantities evaluatedat the present time. The second ratio on the right-hand side of (15) represents the decrease of the number of relativisticdegrees of freedom: their present-day values are g b = 2 (photon) and g f = 6 (neutrinos and antineutrinos) [50, 51].Using the current density parameter for radiation, Ω R = ǫ R0 /ǫ cr , where ǫ cr = 9 . × − kg / m · c is the currentcritical energy density [24, 25, 61], we obtain a bb a = (cid:18) Ω R ǫ cr ǫ bb ( g b + 7 g f / | bb ( g b + 7 g f / | (cid:19) / . (16)Putting Ω R = 8 . × − [24, 25, 61] gives a bb a = 2 × − . (17)The spin-torsion contribution to the energy density, ǫ S ∝ n , scales according to ǫ S ∼ a − [18, 23–25], so that itspresent-day value is given by ǫ S0 = ǫ S | bb (cid:18) a bb a (cid:19) (cid:18) g f | g f | bb (cid:19) . (18)The second ratio on the right-hand side of (18) represents the decrease of the number of relativistic degrees of freedom.The condition (5) reads ǫ S | bb = − ǫ bb . (19)Using the current density parameter for the spin-torsion coupling, Ω S = ǫ S0 /ǫ cr [24, 25], we obtain a bb a = (cid:18) − Ω S ǫ cr ǫ bb g | bb g | (cid:19) / . (20)Putting Ω S = − . × − [24, 25] gives a bb a = 1 . × − . (21)This value is smaller than (17) because Ω R , which appears in (16) in the (1 / ǫ cr /ǫ bb from (16) and (20) yields a bb a = r − Ω S Ω R g f | bb g f | (cid:18) ( g b + 7 g f / | ( g b + 7 g f / | bb (cid:19) / . (22)This relation refines the formula for the minimum normalized scale factor a bb /a = p − Ω S / Ω R found in [24, 25].Putting the values of Ω R and Ω S [24, 25] gives a bb a = 1 . × − . (23)This value is smaller than (21) because Ω R appears in (22) in the ( − / a = 2 . × m [24, 25], derived from the WMAP data [61], therefore gives the value of the scale factor of theUniverse at its minimum size (at the bounce): a bb = 4 . × − m . (24)Eliminating a bb /a from (16) and (20), and using (12) leads toΩ S = − s Ω Ω Pl (cid:18) g | ξ ( g b + 7 g f / | (cid:19) / , (25)where Ω Pl = ǫ Pl /ǫ cr . This relation shows why the spin-torsion density parameter Ω S is extremely small in magnitude,which can explain the flatness and horizon problems without inflation [24, 25]. | Ω S | ≪ Pl = 5 . × . Acknowledgments
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