Analysis of Big Bounce in Einstein--Cartan Cosmology
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Analysis of big bounce in Einstein{Cartan cosmology Jordan L. Cubero∗ and Nikodem J. Poplawski y Department of Mathematics and Physics, University of New Haven, 300 Boston Post Road, West Haven, CT 06516, USA We analyze the dynamics of a homogeneous and isotropic universe in the Einstein{Cartan theory of gravity. The spin of fermions produces spacetime torsion that prevents gravitational singularities and replaces the big bang with a nonsingular big bounce. We show that a closed universe exists only when a particular function of its scale factor and temperature is higher than some threshold value, whereas an open and a flat universes do not have such a restriction. We also show that a bounce of the scale factor is double: as the temperature increases and then decreases, the scale factor decreases, increases, decreases, and then increases. Introduction. The Einstein{Cartan (EC) theory of a size where the cosmological constant starts cosmic ac- gravity provides the simplest mechanism generating a celeration. This expansion also predicts the cosmic mi- nonsingular big bounce, without unknown parameters crowave background radiation parameters that are con- or hypothetical fields [1]. EC is the simplest and most sistent with the Planck 2015 observations [11], as was natural theory of gravity with torsion, in which the La- shown in [12]. grangian density for the gravitational field is proportional The spin-fluid description of matter is the particle ap- to the Ricci scalar, as in general relativity [2, 3]. The con- proximation of the multipole expansion of the integrated sistency of the conservation law for the total (orbital plus conservation laws in EC. The particle approximation for spin) angular momentum of fermions in curved spacetime Dirac fields, however, is not self-consistent [13]. The spin- with the Dirac equation allowing the spin-orbit interac- fluid description also violates the cosmological principle tion requires that the antisymmetric part of the affine [14]. On the other hand, the Dirac form of the spin ten- connection, the torsion tensor [4], is not constrained to sor for fermionic matter, which follows directly from the zero [5]. Instead, torsion is determined by the field equa- Dirac Lagrangian using the Sciama-Kibble variation with tions obtained from the stationarity of the action un- respect to the torsion tensor [2, 15], is consistent with the der variations of the torsion tensor. In EC, the spin of cosmological principle [16]. The minimal coupling be- fermions is the source of torsion. tween the torsion tensor and Dirac fermions also averts The multipole expansion [6] of the conservation law for the big-bang singularity [17, 18]. Other forms of the tor- the spin tensor in EC gives a spin tensor which describes sion tensor in cosmology are investigated in [19]. fermionic matter as a spin fluid (ideal fluid with spin) The spin-torsion coupling modifies the Dirac equa- [7]. Once the torsion is integrated out, EC reduces to tion, adding a term that is cubic in spinor fields [20]. general relativity with an effective spin fluid as a matter As a result, fermions must be spatially extended [13], source [3]. The effective energy density and pressure of which could eliminate infinities arising in Feynman dia- 2 2 a spin fluid are given by ~ = − αnf andp ~ = p − αnf , grams involving fermion loops. In the presence of torsion, where and p are the thermodynamic energy density the four-momentum operator components do not com- and pressure, nf is the number density of fermions, and mute and thus the integration in the momentum space α = κ(~c)2=32 with κ = 8πG=c4 [8, 9]. The negative in Feynman diagrams must be replaced with the sum- corrections from the spin-torsion coupling generate grav- mation over the discrete momentum eigenvalues. The itational repulsion which prevents the formation of grav- resulting sums are finite: torsion naturally regularizes arXiv:1906.11824v1 [gr-qc] 27 Jun 2019 itational singularities and replaces the big bang with a ultraviolet-divergent integrals in quantum electrodynam- nonsingular bounce, at which the universe transitions ics [21]. Torsion may also explain the matter-antimatter from contraction to expansion. These corrections lead asymmetry [22] and the cosmological constant [23]. to a violation of the strong energy condition by the spin The analysis in [17] considered a closed, homogeneous, fluid when + 3p − 4αn2 drops below 0, thus evading f and isotropic universe in EC. However, the calculations of the singularity theorems [8]. Accordingly, this violation the maximum temperature and the minimum scale factor could be thought of as the cause of the bounce. at a bounce neglected the factor k = 1 in the Friedmann The dynamics of the universe filled with a spin fluid in equations (which is justified during and after inflation EC has been studied in [9]. The expansion of the closed but not at a bounce before inflation), de facto consider- universe with torsion and quantum particle production ing a flat universe. In addition, the time-dependence of shortly after a bounce is almost exponential for a finite the scale factor appeared to have a cusp-like behavior at period of time, explaining inflation [10]. Depending on the bounce. In this article, we refine those calculations by the particle production rate, the universe may undergo taking k into account and analyzing the turning points of several bounces until it produces enough matter to reach the universe for all three cases: k = 1 (closed universe), 2 k = 0 (flat universe), and k = −1 (open universe). We bosons and fermions, respectively. For standard-model discover that a closed universe exists only when some particles, gb = 29 and gf = 90. In the presence of spin function of the scale factor and temperature is higher and torsion, the first Friedmann equation is therefore [17] than a particular threshold, whereas an open and flat a_ 2 1 universes are not restricted by this condition. Accord- + k = κ(h T 4 − αh2 T 6)a2: (8) ingly, a closed universe forms in a region of space within c2 3 ? nf a trapped null surface when this threshold is reached. The first law of thermodynamics (7) gives Such a region could be the interior of a black hole [24]. Torsion therefore may explain the origin of our universe dT 3αh2 da [10]. − nf T dT + = 0 (9) T 2h a Dynamics of scale factor and temperature. If ? we assume that the universe is homogeneous and or isotropic, then it is described by the Friedmann- dT T dT da Lema^ıtre-Robertson-Walker metric, which is given in the − 2 + = 0; (10) isotropic spherical coordinates by T Tcr a a2(t) where ds2 = c2dt2 − (dr2 + r2dθ2 + r2 sin2 θ dφ2); (1 + kr2=4)2 2h 1=2 T = ? = 2:218 × 1031 K (11) (1) cr 3αh2 where a(t) is the scalar factor as a function of the cosmic nf time t [25]. The Einstein field equations for this metric is the critical temperature. Equation (10) determines the become the Friedmann equations: scale factor a as a function of temperature T and can be integrated to a_ 2 1 + k = κa~ 2 (2) 2 ∗ T 2 c 3 C 2 a(T ) = e 2Tcr ; (12) and T ∗ a_ 2 + 2aa¨ where C is an integration constant. This constant is + k = −κpa~ 2; (3) c2 positive because the scale factor and temperature are positive. where a dot denotes the derivative with respect to t. The We define nondimensional quantities: effective energy density and pressure for a Dirac field are given by [15, 17]. T x = ; (13) Tcr ~ = − αn2; p~ = p + αn2; (4) a f f y = ; (14) a where cr ct 9 τ = ; (15) α = κ( c)2: (5) acr 16 ~ where Multiplying the first Friedmann equation by a and differ- 4 1=2 entiating over time, and subtracting from it the second 27 αhnf −35 acr = ~c 3 = 6:661 × 10 m: (16) Friedmann equation multiplied bya _ gives an equation 8 h? that has the form of the first law of thermodynamics for an adiabatic universe: Henceforth, we will use a dot to denote the derivative with respect to the new time coordinate τ. Equation (8) d d ( − αn2)a3 + (p + αn2) (a3) = 0; (6) can be written, using these quantities, as dt f f dt 2 4 6 2 which gives y_ + k = (3x − 2x )y : (17) 3 3 3 Equation (12) can be written as a d − 2αa nf dnf + ( + p)d(a ) = 0: (7) C 2 The matter in the early universe is ultrarelativistic. If y(x) = ex =2; (18) 4 we assume kinetic equilibrium, then we can use = h?T , x 3 p = /3, and nf = hnfT , where T is the temperature of ∗ 2 4 3 where C is a positive constant, related to C by the universe, h? = (π =30)(gb + (7=8)gf)kB=(~c) , and 2 3 3 ∗ hnf = (ζ(3)/π )(3=4)gfkB=(~c) [26]. The quantities gb C C = : (19) and gf are the numbers of spin states for all elementary acrTcr 3 FIG. 1: The nondimensionalized scale factor y as a function of the nondimensionalized temperature x for C = 1. The minimum value of y is positive: the universe is nonsingular. FIG. 2: A curve representing the function on the left-hand side of Eq. (23) has two points of intersection with a horizon- tal line 1=C2 if C > e−1=2. Substitution of y(x) in Eq. (18) into Eq. (17) gives −1=2 2 If C < e , Eq. (20) has no turning points and the y_2 + k = C2(3x2 − 2x4)ex : (20) universe would not exist. The function y(x), given by Eq.