UNIVERSE IN A BLACK HOLE IN EINSTEIN–CARTAN GRAVITY

Relativistic quantum mechanics predicts that elementary particles that are fermions, described by the Dirac equation, have the intrinsic angular momentum (spin). The conservation law for the total angular momentum (orbital plus intrinsic) that admits the exchange between its orbital and intrinsic components (spin–orbit interaction) in curved spacetime requires an asymmetric affine connection (Hehl & McCrea 1986; Hammond 2010; Popławski 2013a). The Einstein–Cartan or Einstein–Cartan–Sciama–Kibble (ECSK) theory of gravity naturally extends the metric general relativity by removing its constraint of the symmetry of the connection (Kibble 1961; Sciama 1962, 1964a, 1964b). The antisymmetric part of the affine connection, the torsion tensor, becomes a dynamical variable related to the spin density of matter (Kibble 1961; Sciama 1962, 1964a, 1964b; Hehl 1971, 1973, 1974, 2014; Hehl & Datta 1971; Hehl et al. 1976; Lord 1976; de Sabbata & Gasperini 1985; de Sabbata & Sivaram 1994; Hammond 2002; Shapiro 2002; Popławski 2009). Since Dirac fields couple to the connection, the spin of fermions acts like a source of torsion. The ECSK theory also passes all tests of general relativity, because even at nuclear densities the contribution from torsion to the Einstein equations is negligibly small and both theories give indistinguishable predictions at these densities.

At extremely high densities existing in black holes and in the very early universe, the minimal spinor–torsion coupling manifests itself as gravitational repulsion, which avoids the formation of singularities from fermionic matter (Kopczyński 1972, 1973; Trautman 1973; Hehl et al. 1974; Gasperini 1986; Brechet et al. 2008; Popławski 2010b, 2012a, 2013c; Magueijo et al. 2013; Alexander et al. 2014; Bambi et al. 2014). Accordingly, the singular big bang is replaced by a nonsingular big bounce, before which the universe was contracting (Kuchowicz 1978; Garcia de Andrade 2001; Popławski 2010c, 2011a). In addition to eliminating the initial singularity, this scenario solves the flatness and horizon problems in cosmology (Popławski 2010c, 2011a, 2012a, 2012b, 2013b; Magueijo et al. 2013; Alexander et al. 2014; Bambi et al. 2014). Cosmic inflation, which is supported by the measurements of the inhomogeneities of the cosmic microwave background radiation (Rich 2001), also solves the flatness and horizon problems, but it requires additional matter fields with specific conditions on their form and does not address the big bang singularity (Kazanas 1980; Guth 1981; Linde 1982). Torsion therefore provides the simplest and most natural mechanism that solves these three major problems of the standard big bang cosmology. The ECSK theory may also solve the problem of divergent integrals in quantum field theory by providing fermions with spatial extension and thus introducing an effective ultraviolet cutoff for their propagators (Popławski 2010b, 2013c).

The contraction of the universe before the big bounce must be caused by a physical initial condition. The general features of nonsingular universes and bouncing cosmologies have been reviewed in the literature (Stachowiak & Szydłowski 2007; Novello & Perez Bergliaffa 2008; Battefeld & Peter 2015; Brandenberger & Peter 2016). Such a contraction could correspond to gravitational collapse of matter inside a newly formed black hole existing in another universe (Novikov 1966; Pathria 1972; Frolov et al. 1989, 1990; Smolin 1992; Stuckey 1994; Easson & Brandenberger 2001; Smoller & Temple 2003; Popławski 2010a, 2010c, 2011a). To describe gravitational collapse of a dustlike medium (Tolman–Bondi collapse) in general relativity, one can choose a reference system that is both synchronous and comoving (Landau & Lifshitz 1975). Accordingly, each spatial point in the interior of a black hole locally evolves toward the singularity as an independent, spatially homogeneous, and isotropic universe (Landau & Lifshitz 1975; Lord 1976). Numerical analysis of generic gravitational collapse in general relativity shows that spatial derivatives of the metric and curvature tensors become negligible in the dynamics, and each spatial point in the interior of a black hole locally evolves toward the singularity as an independent, spatially homogeneous universe (Berger et al. 1998; Garfinkle 2004). This evolution looks like the Belinskii–Khalatnikov–Lifshitz oscillatory dynamics (Belinskii et al. 1971).

In general relativity, the singularity forms before the collapse has completed (Landau & Lifshitz 1975; Hochberg & Visser 1998). The infalling matter and fields make the interior of the black hole a very dynamical spacetime where the inhomogeneities and anisotropies are very large. During the collapse, multiple trapped null surfaces with a very complicated structure can form in the interior and fold in on themselves (Hochberg & Visser 1998). In the ECSK theory, we expect that each spatial point evolves toward a state of an extremely high but finite density and curvature. At such a state, the local contraction ends, the matter undergoes a bounce, and the local expansion begins. Accordingly, multiple (at least two) dynamical wormhole throats form (Hochberg & Visser 1998). Quantum effects in the presence of an extremely strong gravitational field cause an intense particle production, which creates an enormous amount of mass (Parker 1968, 1969; Zel'dovich 1970; Parker 1971a, 1971b; Zel'dovich & Starobinskii 1977) without changing the total energy (matter plus gravitational field) in the black hole (Popławski 2014). This creation may increase the existing inhomogeneities and anisotropies in the interior (Chrobok et al. 2002).

The dynamics of spacetime in a black hole is very complex, but we conjecture that eventually the wormholes will merge into one wormhole and the outermost trapped surface can be regarded as the event horizon. Asymptotically, the throat of this wormhole and the event horizon would coincide. The entire interior of a black hole would then become a new, closed universe whose expansion is not visible from the outside of the black hole because of an infinite redshift at its event horizon. Such a universe would be initially inhomogeneous and anisotropic. Its expansion would eventually stop because of the closed geometry (Landau & Lifshitz 1975). The universe in a black hole would then contract until it reaches a minimum radius and maximum density, undergoes a bounce, and expands again. Subsequent contracting phases and bounces could make it more homogeneous and isotropic. The spin tensor could also contribute to the evolution of the shear and twist tensors in the Raychaudhuri equation on a timescale to allow a homogeneous and isotropic universe to form (Chrobok et al. 2002). This universe can be thought of as a three-dimensional analog of the two-dimensional surface of a sphere (Popławski 2010c, 2011a).

When the universe in a black hole creates sufficient amounts of mass, it reaches the size at which the cosmological constant becomes dominant and the expansion proceeds to infinity (Lord 1976). The boundary of the black hole (event horizon) becomes an Einstein–Rosen bridge (wormhole) connecting this universe with the outer universe in which the black hole exists. Every astrophysical black hole candidate may thus be a wormhole to a new universe on the other side of its event horizon. Accordingly, our universe may be the interior of a black hole existing in another universe and a part of a multiverse. This scenario could also explain the arrow of time and the black hole information paradox (Popławski 2010c, 2011a, 2014).

This paper is structured as follows. Section 2 briefly describes the ECSK theory of gravity. Section 3 briefly describes a spin fluid. Section 4 presents the cosmology of the early universe in the presence of spin and torsion. The next sections constitute the novel part of the paper. Section 5 analyzes the dynamics of a relativistic universe in a black hole without particle production. For simplicity, we consider a homogeneous and isotropic universe. Section 6 adds quantum particle production caused by an extremely high curvature and analyzes its effects on the dynamics of a universe in a black hole. Section 7 shows that inflationary expansion can be derived as a special case of the dynamics described in Section 6. Cosmic inflation can thus originate from the ECSK theory of gravity with particle production, without requiring additional matter fields. The results are briefly discussed in Section 8.

The affine connection ${{\rm{\Gamma }}}_{i\,j}^{\,k}$ in the ECSK theory of gravity is not symmetric and has the antisymmetric part, the torsion tensor ${S}_{\,{ij}}^{k}=({{\rm{\Gamma }}}_{i\,j}^{\,k}-{{\rm{\Gamma }}}_{j\,i}^{\,k})/2$ (Schrödinger 1954; Kibble 1961; Sciama 1962, p. 415; 1964a, 1964b; Hehl 1971, 1973, 1974, 2014; Hehl & Datta 1971; Hehl et al. 1974, 1976; Lord 1976; de Sabbata & Gasperini 1985; de Sabbata & Sivaram 1994; Hammond 2002; Shapiro 2002; Hehl & Obukhov 2007; Popławski 2009, 2010b, 2013c). We use the notation of Popławski (2009). The curvature tensor is given by ${R}_{\,{mjk}}^{i}={\partial }_{j}{{\rm{\Gamma }}}_{m\,k}^{\,i}-{\partial }_{k}{{\rm{\Gamma }}}_{m\,j}^{\,i}+{{\rm{\Gamma }}}_{l\,j}^{\,i}{{\rm{\Gamma }}}_{m\,k}^{\,l}-{{\rm{\Gamma }}}_{l\,k}^{\,i}{{\rm{\Gamma }}}_{m\,j}^{\,l}$, and its contraction gives the Ricci tensor ${R}_{{ik}}={R}_{\,{ijk}}^{j}$. The metricity condition ${g}_{{ij};k}=0$, where a semicolon denotes the covariant derivative with respect to ${{\rm{\Gamma }}}_{i\,j}^{\,k}$, gives ${{\rm{\Gamma }}}_{i\,j}^{\,k}=\{{}_{i\,j}^{\,k}\}+{C}_{\,{ij}}^{k}$, where $\{{}_{i\,j}^{\,k}\}=(1/2){g}^{{km}}({g}_{{mi},j}+{g}_{{mj},i}-{g}_{{ij},m})$ are the Christoffel symbols of the metric tensor g_ik and ${C}_{\,{jk}}^{i}={S}_{\,{jk}}^{i}+2{S}_{({jk})}^{{i}}$ is the contortion tensor. The curvature tensor can be decomposed as ${R}_{{klm}}^{i}={P}_{{klm}}^{i}+{C}_{{km}:l}^{i}-{C}_{{kl}:m}^{i}+{C}_{{km}}^{j}{C}_{{jl}}^{i}-{C}_{{kl}}^{j}{C}_{{jm}}^{i}$, where ${P}_{{klm}}^{i}$ is the Riemann tensor (the curvature tensor constructed from the Levi–Civita connection $\{{}_{i\,j}^{\,k}\}$) and a colon denotes the covariant derivative with respect to $\{{}_{i\,j}^{\,k}\}$.

The ECSK theory of gravity is based on the Lagrangian density of the gravitational field, which is proportional to the Ricci curvature scalar $R={R}_{{ik}}{g}^{{ik}}$, similarly to the metric general relativity. The field equations are obtained from the total action for the gravitational field and matter, $I=\tfrac{1}{c}\int (-R\sqrt{-g}/(2\kappa )+{{\mathfrak{L}}}_{{\rm{m}}}){d}^{4}x$, where ${{\mathfrak{L}}}_{{\rm{m}}}$ is the Lagrangian density of matter, $g={\rm{\det }}({g}_{{ik}})$, and $\kappa =8\pi G/{c}^{4}$, with respect to the metric and torsion tensors. Varying the action with respect to the torsion gives the Cartan field equations that relate algebraically the torsion of spacetime to the canonical spin tensor of matter ${s}^{{ijk}}=2(\delta {{\mathfrak{L}}}_{{\rm{m}}}/\delta {C}_{{ijk}})/\sqrt{-g}$ (Kibble 1961; Sciama 1962, p. 415; 1964a, 1964b; Hehl 1971, 1973, 1974; Hehl & Datta 1971; Hehl et al. 1974, 1976; Lord 1976; de Sabbata & Gasperini 1985; de Sabbata & Sivaram 1994; Hammond 2002; Shapiro 2002; Popławski 2009, 2010b, 2013c; Hehl 2014):

$$\begin{eqnarray}&&{S}_{{jik}}-{S}_{i}{g}_{{jk}}+{S}_{k}{g}_{{ji}}=-\displaystyle \frac{1}{2}\kappa {s}_{{ikj}},\end{eqnarray} \tag{ 1 }$$

where ${S}_{{i}}={S}_{{ik}}^{k}$. Varying the action with respect to the metric and using Equation (1) gives the Einstein field equations that relate the curvature of spacetime to the canonical energy-momentum tensor of matter ${\sigma }_{{ij}}$,

$$\begin{eqnarray}&&{R}_{{ik}}-\displaystyle \frac{1}{2}{{Rg}}_{{ik}}=\kappa {\sigma }_{{ki}}.\end{eqnarray} \tag{ 2 }$$

The symmetric, dynamical energy-momentum tensor ${T}_{{ik}}=2(\delta {{\mathfrak{L}}}_{{\rm{m}}}/\delta {g}^{{ik}})/\sqrt{-g}$ (Schrödinger 1954; Landau & Lifshitz 1975) is related to the canonical energy-momentum tensor by ${T}_{{ik}}={\sigma }_{{ik}}-(1/2)({{\rm{\nabla }}}_{j}-{2S}_{{j}})({s}_{{ik}}^{\,j}-{S}_{k\,i}^{j}+{s}_{\,{ik}}^{j})$, where ∇_k denotes the covariant derivative with respect to ${{\rm{\Gamma }}}_{i\,j}^{\,k}$.

The Einstein and Cartan equations can be combined to give (Hehl et al. 1974, 1976; Popławski 2009, 2010b, 2013c)

$$\begin{eqnarray}{G}^{{ik}} & = & \kappa {T}^{{ik}}+\displaystyle \frac{1}{2}{\kappa }^{2}\left(\phantom{\rule{}{3ex}}{s}_{\phantom{{ij}}j}^{{ij}}{s}_{\phantom{{kl}}l}^{{kl}}-{s}_{\phantom{{ij}}l}^{{ij}}{s}_{\phantom{{kl}}j}^{{kl}}-{s}^{{ijl}}{s}_{\phantom{k}{jl}}^{k}\right.\\ & & +\left.\displaystyle \frac{1}{2}{s}^{{jli}}{s}_{{jl}}^{\phantom{{jl}}k}+\displaystyle \frac{1}{4}{g}^{{ik}}(2{s}_{j\phantom{l}m}^{\phantom{j}l}{s}_{\phantom{{jm}}l}^{{jm}}-2{s}_{j\phantom{l}l}^{\phantom{j}l}{s}_{\phantom{{jm}}m}^{{jm}}+{s}^{{jlm}}{s}_{{jlm}})\right),\end{eqnarray} \tag{ 3 }$$

where ${G}_{{ik}}={P}_{{ik}}-(1/2){{Pg}}_{{ik}}$ is the Einstein tensor of general relativity constructed from the contractions of the Riemann tensor, ${P}_{{ik}}={P}_{\,{ijk}}^{j}$, and $P={P}_{{ik}}{g}^{{ik}}$. The second term on the right-hand side of Equation (3) is the correction to the curvature of spacetime from the spin. The spin tensor also appears in T_ik because ${{\mathfrak{L}}}_{{\rm{m}}}$ depends on torsion. The contributions from the spin tensor to the right-hand side of the Einstein equations are significant only at extremely high densities, on the order of the Cartan density (Popławski 2010b, 2013c). Below this density, the predictions of the ECSK theory do not differ from the predictions of the metric general relativity and reduce to them in vacuum, where torsion vanishes.

Quarks and leptons, which compose all stars, are fermions described in relativistic quantum mechanics by the Dirac equation. Since Dirac fields couple minimally to the torsion tensor, the torsion of spacetime at microscopic scales does not vanish in the presence of fermions (Kibble 1961; Sciama 1962, 1964a, 1964b; Popławski 2009, 2010b, 2013c). A correct coupling of fermions to torsion should treat spinors as anticommuting fields (Chang 1990). At macroscopic scales, such particles can be averaged and described as a spin fluid (Kopczyński 1972, 1973; Trautman 1973; Hehl et al. 1974; Kuchowicz 1978; Gasperini 1986; Brechet et al. 2008; Popławski 2009, 2010c, 2011a). Even if the spin orientation of particles is random, the terms quadratic in the spin tensor in Equation (3) do not vanish. These terms are significant only at densities of matter much higher than the density of nuclear matter because of the factor κ².

The Bianchi identities, ${R}_{\,n[{jk};l]}^{i}=2{R}_{\,{nm}[j}^{i}{S}_{\,{kl}]}^{m}$ and ${R}_{\,[{jkl}]}^{m}=-2{S}_{\,[{jk};l]}^{m}+4{S}_{\,n[j}^{m}{S}_{\,{kl}]}^{n}$, together with the Einstein and Cartan field equations, give the conservation laws for the canonical energy-momentum and spin tensors: ${T}_{\,:j}^{{ij}}={C}_{{jk}}^{\,i}{T}^{{jk}}+(1/2){s}_{{klj}}{R}^{{klji}}$ and ${s}_{{ij}\,;k}^{\,k}-2{S}_{{k}}{S}_{{ij}}^{\,{k}}={T}_{{ij}}-{T}_{{ji}}$ (Kibble 1961; Sciama 1962, 1964a, 1964b; Hehl 1971, 1973, 1974, 2014; Hehl & Datta 1971; Hehl et al. 1974, 1976; Lord 1976; de Sabbata & Gasperini 1985; de Sabbata & Sivaram 1994; Hammond 2002; Shapiro 2002; Popławski 2009). Using the Papapetrou method of multipole expansion for these laws (Mathisson 1937; Papapetrou 1951; Nomura et al. 1991) leads to the formulas for the macroscopic canonical energy-momentum and spin tensor of a spin fluid in the point-particle approximation (Popławski 2009, 2010b, 2013c). The canonical energy-momentum tensor of a spin fluid is given by (Hehl et al. 1974)

$$\begin{eqnarray}&&{\sigma }_{{ij}}=c{{\rm{\Pi }}}_{i}{u}_{j}-p({g}_{{ij}}-{u}_{i}{u}_{j}),\end{eqnarray} \tag{ 4 }$$

and its spin tensor by

$$\begin{eqnarray}&&{s}_{{ij}}^{\,k}={s}_{{ij}}{u}^{k},\,{s}_{{ij}}{u}^{j}=0,\end{eqnarray} \tag{ 5 }$$

where Π_i is the four-momentum density of the fluid, uⁱ its four-velocity, s_ij its spin density, and p its pressure. Substituting the macroscopic tensors into Equation (3) gives (Hehl et al. 1974; Kuchowicz 1978; Gasperini 1986; Brechet et al. 2008; Popławski 2010c, 2011a)

$$\begin{eqnarray}{G}^{{ij}} & = & \kappa \left(\epsilon -\displaystyle \frac{1}{4}\kappa {s}^{2}\right){u}^{i}{u}^{j}-\kappa \left(p-\displaystyle \frac{1}{4}\kappa {s}^{2}\right)({g}^{{ij}}-{u}^{i}{u}^{j})\\ & & -\displaystyle \frac{1}{2}\kappa ({\delta }_{k}^{l}+{u}_{k}{u}^{l}){({s}^{{ki}}{u}^{j}+{s}^{{kj}}{u}^{i})}_{:l},\end{eqnarray} \tag{ 6 }$$

where $\epsilon =c{{\rm{\Pi }}}_{i}{u}^{i}$ is the rest energy density of the fluid, and

$$\begin{eqnarray}&&{s}^{2}=\displaystyle \frac{1}{2}{s}_{{ij}}{s}^{{ij}}\gt 0\end{eqnarray} \tag{ 7 }$$

is the square of the spin density.

If the spin orientation of particles in a spin fluid is random, then the last term on the right-hand side of Equation (6) vanishes after averaging. Thus, the Einstein–Cartan equations for such a spin fluid are equivalent to the Einstein equations for a perfect fluid with the effective energy density $\epsilon -\kappa {s}^{2}/4$ and the effective pressure $p-\kappa {s}^{2}/4$ (Hehl et al. 1974; Kuchowicz 1978; Gasperini 1986; Brechet et al. 2008; Popławski 2010c, 2011a). The square of the spin density for a fluid consisting of fermions with no spin polarization is given by (Nurgaliev & Ponomariev 1983; Gasperini 1986; Brechet et al. 2008)

$$\begin{eqnarray}&&{s}^{2}=\displaystyle \frac{1}{8}{({\hslash }{{cn}}_{{\rm{f}}})}^{2}.\end{eqnarray} \tag{ 8 }$$

where n_f is the number density of fermions. The effective energy density and pressure of a spin fluid are thus given by

$$\begin{eqnarray}&&\tilde{\epsilon }=\epsilon -\alpha {n}_{{\rm{f}}}^{2},\quad \tilde{p}=p-\alpha {n}_{{\rm{f}}}^{2},\end{eqnarray} \tag{ 9 }$$

where $\alpha =\kappa {({\hslash }c)}^{2}/32$.

A closed, homogeneous, and isotropic universe is described by the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, which, in the isotropic spherical coordinates, is given by ${{ds}}^{2}={c}^{2}{{dt}}^{2}-{a}^{2}(t){(1+{{kr}}^{2}/4)}^{-2}({{dr}}^{2}+{r}^{2}d{\theta }^{2}$ $+{r}^{2}{\sin }^{2}\theta d{\phi }^{2})$, where a(t) is the scale factor and k = 1 (Landau & Lifshitz 1975; Lord 1976). In the comoving frame of reference, in which the four-velocity uⁱ of the cosmological spin fluid satisfies u⁰ = 1 and u^α = 0 (α denotes spatial indices), the Einstein field Equation (6) for this metric becomes the Friedmann equations (the cosmological constant is negligible in the early universe; Kuchowicz 1978; Gasperini 1986; Brechet et al. 2008; Popławski 2010c, 2011a):

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{a}}^{2}}{{c}^{2}}+k=\displaystyle \frac{1}{3}\kappa (\epsilon -\alpha {n}_{{\rm{f}}}^{2}){a}^{2},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{a}}^{2}+2a\ddot{a}}{{c}^{2}}+k=-\kappa (p-\alpha {n}_{{\rm{f}}}^{2}){a}^{2},\end{eqnarray} \tag{ 11 }$$

where a dot denotes the differentiation with respect to the cosmic time t. Differentiating Equation (10) multiplied by a with respect to t and combining it with Equation (11) multiplied by $\dot{a}$ gives a conservation law

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}\{(\epsilon -\alpha {n}_{{\rm{f}}}^{2}){a}^{3}\}+(p-\alpha {n}_{{\rm{f}}}^{2})\displaystyle \frac{d}{{dt}}({a}^{3})=0,\end{eqnarray} \tag{ 12 }$$

The spin fluid in the early universe is formed by ultrarelativistic matter in kinetic equilibrium, for which $\epsilon (T)={h}_{\star }{T}^{4}$, $p(T)=\epsilon (T)/3$, and $n(T)={h}_{n}{T}^{3}$, where T is the temperature of the universe and ζ is the Riemann zeta function, and we defined ${h}_{\star }=({\pi }^{2}/30){g}_{\star }{k}_{{\rm{B}}}^{4}/{({\hslash }c)}^{3}$ and ${h}_{n}=(\zeta (3)/{\pi }^{2}){g}_{n}{k}_{{\rm{B}}}^{3}/{({\hslash }c)}^{3}$ (Rich 2001). The effective numbers of thermal degrees of freedom are ${g}_{\star }(T)={g}_{{\rm{b}}}(T)+(7/8){g}_{{\rm{f}}}(T)$ and ${g}_{n}(T)={g}_{{\rm{b}}}(T)+(3/4){g}_{{\rm{f}}}(T)$, where ${g}_{{\rm{b}}}={\sum }_{i}{g}_{i}$ is summed over relativistic bosons, ${g}_{{\rm{f}}}={\sum }_{i}{g}_{i}$ is summed over relativistic fermions, and g_i is the number of the spin states for each particle species i. Accordingly, ${n}_{{\rm{f}}}(T)={h}_{n{\rm{f}}}{T}^{3}$ and ${h}_{n{\rm{f}}}=(\zeta (3)/{\pi }^{2})(3/4){g}_{{\rm{f}}}(T){k}_{{\rm{B}}}^{3}/{({\hslash }c)}^{3}$. Substituting these functions into Equation (12) gives

$$\begin{eqnarray}&&\left(\displaystyle \frac{\dot{a}}{a}+\displaystyle \frac{\dot{T}}{T}\right)\left(1-\displaystyle \frac{3\alpha {h}_{n{\rm{f}}}^{2}}{2{h}_{\star }}{T}^{2}\right)=0,\end{eqnarray} \tag{ 13 }$$

since g_b and g_f are constant in the range of T where torsion is significant. Relation (13) is satisfied for all values of T if

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}({aT})=0.\end{eqnarray} \tag{ 14 }$$

Integrating Equation (14) gives

$$\begin{eqnarray}&&a=\displaystyle \frac{{a}_{{\rm{r}}}{T}_{{\rm{r}}}}{T},\end{eqnarray} \tag{ 15 }$$

where a_r is the scale factor at a reference temperature T_r. Relation (15) between a and T does not depend on the presence of spin (Popławski 2010c, 2011a). The energy density of particles in kinetic equilibrium scales like  ∼ a⁻⁴, and their number density scales like n ∼ a⁻³. Accordingly, the number of particles N ∝ na³ in the universe does not change if Equation (14) is satisfied.

When a trapped null surface forms in a black hole (Hochberg & Visser 1998), the region within this surface is causally connected and mathematically equivalent to a new, closed universe (Berger et al. 1998; Garfinkle 2004; Popławski 2010c, 2011a). We will also assume, for simplicity, that such a universe is already homogeneous and isotropic (Hochberg & Visser 1998). In a black hole, fermions composing the spin fluid have energies much greater than their rest energies. These particles are thus described by an ultrarelativistic barotropic equation of state, p = /3, as for radiation. If we assume that the universe begins to contract when a = a_i and $\dot{a}=0$, where $\alpha {n}_{{\rm{f}}}^{2}\ll \epsilon$, then Equation (10) gives $\kappa \epsilon {a}_{i}^{2}/3=1$. Accordingly, the initial temperature is given by

$$\begin{eqnarray}&&{T}_{i}={\left(\displaystyle \frac{3}{\kappa {h}_{\star }{a}_{i}^{2}}\right)}^{1/4}.\end{eqnarray} \tag{ 16 }$$

If no particles are produced, then we can use Equation (14). The reference values in Equation (15) can be a_r = a_i and T_r = T_i. The dynamics of the new universe is described by Equations (15) and (10), which can be written as

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{a}}^{2}}{{c}^{2}}+1=\displaystyle \frac{1}{3}\kappa ({h}_{\star }{T}^{4}-\alpha {h}_{n{\rm{f}}}^{2}{T}^{6}){a}^{2}\end{eqnarray} \tag{ 17 }$$

or

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{a}}^{2}}{{c}^{2}}+1=\displaystyle \frac{1}{3}\kappa \left({h}_{\star }\displaystyle \frac{{T}_{i}^{4}{a}_{i}^{4}}{{a}^{2}}-\alpha {h}_{n{\rm{f}}}^{2}\displaystyle \frac{{T}_{i}^{6}{a}_{i}^{6}}{{a}^{4}}\right).\end{eqnarray} \tag{ 18 }$$

The Friedmann Equations (10) and (11) also give

$$\begin{eqnarray}&&\displaystyle \frac{\ddot{a}}{{c}^{2}a}=-\displaystyle \frac{1}{3}\kappa \epsilon +\displaystyle \frac{2}{3}\kappa \alpha {n}_{{\rm{f}}}^{2}=-\displaystyle \frac{\kappa }{3}({h}_{\star }{T}^{4}-2\alpha {h}_{n{\rm{f}}}^{2}{T}^{6}).\end{eqnarray} \tag{ 19 }$$

As the scale factor a decreases, the temperature T increases. The spin–torsion contribution to the energy density is negative and generates gravitational repulsion, which becomes stronger as T increases. Since this contribution scales with T faster (∼ T⁶) than  ∼ T⁴, $\dot{a}$ eventually reaches zero and the universe undergoes a nonsingular bounce. At a bounce, where we can neglect k, Equation (10) gives $\epsilon -\alpha {n}_{{\rm{f}}}^{2}=0$. Accordingly, the maximum temperature is given by

$$\begin{eqnarray}&&{T}_{{\rm{\max }}}={\left(\displaystyle \frac{{h}_{\star }}{\alpha {h}_{n{\rm{f}}}^{2}}\right)}^{1/2}=\sqrt{\displaystyle \frac{2{\pi }^{5}}{15}}\displaystyle \frac{{g}_{\star }^{1/2}}{\zeta (3)\tfrac{3}{4}{g}_{{\rm{f}}}}{T}_{{\rm{P}}},\end{eqnarray} \tag{ 20 }$$

where T_P is the Planck temperature. The temperature given by Equation (20) depends only on the numbers of thermal degrees of freedom. Relation (15) determines the minimum scale factor at a bounce of the universe in a black hole:

$$\begin{eqnarray}&&{a}_{{\rm{\min }}}=\displaystyle \frac{{a}_{i}{T}_{i}}{{T}_{{\rm{\max }}}}.\end{eqnarray} \tag{ 21 }$$

The universe is accelerating if $\ddot{a}\gt 0$, which is equivalent, using Equation (19), to $T\gt {T}_{{\rm{\max }}}/\sqrt{2}$ ($a\lt \sqrt{2}{a}_{{\rm{\min }}}$). After a bounce, the universe accelerates until $T={T}_{{\rm{\max }}}/\sqrt{2}$ ($a=\sqrt{2}{a}_{{\rm{\min }}}$). The universe continues to expands with $\ddot{a}\lt 0$ until it reaches its maximum scale factor (a crunch) at a = a_i, after which another contraction begins. This expansion looks like the time reversal of the contraction from a = a_i to a = a_min. The universe is therefore oscillatory: its scale factor oscillates between the minimum and maximum scale factors (Kuchowicz 1978). The period of the oscillation from a bounce to a crunch and back is equal to

$$\begin{eqnarray}&&{\rm{\Delta }}t=2{\int }_{{a}_{{\rm{\min }}}}^{{a}_{i}}\displaystyle \frac{{da}}{\dot{a}}=\displaystyle \frac{2}{c}{\int }_{{a}_{{\rm{\min }}}}^{{a}_{i}}\displaystyle \frac{{da}}{{(\kappa \tilde{\epsilon }{a}^{2}/3-1)}^{1/2}},\end{eqnarray} \tag{ 22 }$$

where $\tilde{\epsilon }={h}_{\star }{({a}_{i}{T}_{i}/a)}^{4}-\alpha {h}_{n{\rm{f}}}^{2}{({a}_{i}{T}_{i}/a)}^{6}$.

The scale factor depends on the time according to

$$\begin{eqnarray}&&{\left({a}^{2}-\displaystyle \frac{\alpha {h}_{n{\rm{f}}}^{2}{a}_{i}^{2}{T}_{i}^{2}}{{h}_{\star }}-\displaystyle \frac{3{{ka}}^{4}}{\kappa {h}_{\star }{a}_{i}^{4}{T}_{i}^{4}}\right)}^{-1/2}{a}^{2}{da}\\ &&\quad ={\left(\displaystyle \frac{\kappa {h}_{\star }}{3}\right)}^{1/2}{({a}_{i}{T}_{i})}^{2}{cdt}.\end{eqnarray} \tag{ 23 }$$

For $a\ll {a}_{i}$, the term with k in Equation (23) can be neglected. If we define x = a/a_min and choose t = 0 at a = a_min, then integrating Equation (23) gives (Popławski 2010c, 2011a)

$$\begin{eqnarray}\displaystyle \frac{t}{\tau } & = & {\displaystyle \int }_{1}^{x}\displaystyle \frac{{x}^{2}{dx}}{{({x}^{2}-1)}^{1/2}}=\displaystyle \frac{x}{2}\sqrt{{x}^{2}-1}\\ & & +\displaystyle \frac{1}{2}{\rm{ln}}| x+\sqrt{{x}^{2}-1}| ,\end{eqnarray} \tag{ 24 }$$

where

$$\begin{eqnarray}&&\tau =\displaystyle \frac{\alpha {h}_{n{\rm{f}}}^{2}}{c}{\left(\displaystyle \frac{3}{\kappa {h}_{\star }^{3}}\right)}^{1/2}=\displaystyle \frac{45}{4}\sqrt{\displaystyle \frac{5}{{\pi }^{13}}}\displaystyle \frac{{\zeta }^{2}(3){\left(\tfrac{3}{4}{g}_{{\rm{f}}}\right)}^{2}}{{g}_{\star }^{3/2}}{t}_{{\rm{P}}}\end{eqnarray} \tag{ 25 }$$

is the characteristic time of the torsion-dominated phase of the expansion. This time is on the order of the Planck time t_P and depends only on the numbers of thermal degrees of freedom.

The Hubble parameter $H=\dot{a}/a$ is determined by

$$\begin{eqnarray}&&H=\pm c{\left[\displaystyle \frac{1}{3}\kappa ({h}_{\star }{T}^{4}-\alpha {h}_{n{\rm{f}}}^{2}{T}^{6})-\displaystyle \frac{k}{{a}^{2}}\right]}^{1/2},\end{eqnarray} \tag{ 26 }$$

where a plus sign corresponds to expansion and a minus sign corresponds to contraction. The magnitude of H has a maximum $| H{| }_{{\rm{\max }}}=\sqrt{4/27}{\tau }^{-1}$ at $T=\sqrt{2/3}{T}_{{\rm{\max }}}$ (at $a=\sqrt{3/2}{a}_{{\rm{\min }}})$. The total density parameter ${\rm{\Omega }}=\tilde{\epsilon }/({\rho }_{c}{c}^{2})$, where ${\rho }_{c}=3{H}^{2}/(\kappa {c}^{4})$ is the critical density, is determined by Equation (10): ${\rm{\Omega }}=1+{{kc}}^{2}/{\dot{a}}^{2}$, which leads to

$$\begin{eqnarray}{\rm{\Omega }} & = & 1+\displaystyle \frac{3}{\kappa ({h}_{\star }{T}^{4}-\alpha {h}_{n{\rm{f}}}^{2}{T}^{6}){a}^{2}}\\ & = & 1+\displaystyle \frac{3}{\kappa ({h}_{\star }{T}^{2}-\alpha {h}_{n{\rm{f}}}^{2}{T}^{4}){a}_{i}^{2}{T}_{i}^{2}}.\end{eqnarray} \tag{ 27 }$$

The density parameter has a minimum ${{\rm{\Omega }}}_{{\rm{\min }}}=1+4c\tau /{a}_{i}$ at $T=\sqrt{1/2}{T}_{{\rm{\max }}}$ (at $a=\sqrt{2}{a}_{{\rm{\min }}})$. A closed universe contains $N\approx {(\dot{a}/c)}^{3}={({\rm{\Omega }}-1)}^{-3}$ causally disconnected regions. Their maximum number is thus ${N}_{{\rm{\max }}}\approx {({a}_{i}/{l}_{{\rm{P}}})}^{3}$, where l_P is the Planck length. As the universe expands from a_min to $\sqrt{2}{a}_{{\rm{\min }}}$, Ω rapidly decreases from infinity to Ω_min, which appears to be tuned to 1 to a high precision since $c\tau \ll {a}_{i}$ (Popławski 2010c, 2011a). As the universe expands from $\sqrt{2}{a}_{{\rm{\min }}}$ to a_i, Ω increases to infinity.

If we assume that only the known standard-model particles exist, then g_b = 28 and g_f = 90 (Rich 2001). For these values, we find

$$\begin{eqnarray}{T}_{{\rm{\max }}} & = & 1.15\times {10}^{32}\,{\rm{K}},\quad \tau =4.75\times {10}^{-45}\,{\rm{s}},\\ | H{| }_{{\rm{\max }}} & = & 8.1\times {10}^{43}\,{{\rm{s}}}^{-1},\quad {{\rm{\Omega }}}_{{\rm{\min }}}-1=5.7\times {10}^{-36},\\ {N}_{{\rm{\max }}} & \approx & {10}^{52}.\end{eqnarray} \tag{ 28 }$$

An extremely small magnitude of ${{\rm{\Omega }}}_{{\rm{\min }}}-1$ solves the flatness problem. An extremely large N solves the horizon problem. A typical stellar black hole has the Schwarzschild radius a_i = 10⁴ m. For this value, we find

$$\begin{eqnarray}&&{T}_{i}=1.38\times {10}^{12}\,{\rm{K}},\quad {a}_{{\rm{\min }}}=1.19\times {10}^{-16}\,{\rm{m}}.\end{eqnarray} \tag{ 29 }$$

The volume of a closed universe is given by $V=2{\pi }^{2}{a}^{3}$ (Landau & Lifshitz 1975). The conservation law given by Equation (12) can thus be written in the form of the first law of thermodynamics (Garcia de Andrade 2001):

$$\begin{eqnarray}&&{dE}+{pdV}={TdS}=12{\pi }^{2}\alpha {h}_{n{\rm{f}}}^{2}{a}^{2}{T}^{5}d({aT}),\end{eqnarray} \tag{ 30 }$$

where E = V is the thermal energy of the universe and S its entropy. This law also leads to a lower limit for the scale factor of the universe (Garcia de Andrade 2001). Relations (14) and (30) show that the entropy of a cyclic universe in a black hole without particle production is constant.

A closed universe in a black hole without particle production is oscillatory. It does not reach the size and mass of the observed universe. To identify our universe with a universe in a black hole, we must include quantum effects in curved spacetime, which are responsible for particle production by the gravitational field (Parker 1968, 1969; Zel'dovich 1970; Parker 1971a, 1971b; Zel'dovich & Starobinskii 1977).

An initial, accelerated expansion of a closed universe after the big bounce defines the torsion-dominated era. As the universe expands, the spin–torsion term $\alpha {n}_{{\rm{f}}}^{2}$ in Equation (10) decreases faster than . The universe begins to decelerate and enters the radiation-dominated era. Eventually, the matter in the universe becomes nonrelativistic. At the matter–radiation equality, the energy density of nonrelativistic matter exceeds the energy density of radiation and the universe enters the matter-dominated era. The universe cannot expand to infinity unless it enters another phase of acceleration. The simplest source of such an acceleration is a cosmological constant Λ, which modifies the Lagrangian density of the gravitational field: ${{\mathfrak{L}}}_{{\rm{g}}}=-(R+2{\rm{\Lambda }})\sqrt{-g}/(2\kappa )$. The cosmological constant does not change the Cartan equations but adds a term to the Einstein Equations (2): ${R}_{{ik}}-\tfrac{1}{2}{{Rg}}_{{ik}}=\kappa {\sigma }_{{ki}}+{\rm{\Lambda }}{g}_{{ik}}$. The Friedmann Equation (10) in the radiation-dominated era becomes

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{a}}^{2}}{{c}^{2}}+1=\displaystyle \frac{1}{3}\kappa \epsilon {a}^{2}+\displaystyle \frac{1}{3}{\rm{\Lambda }}{a}^{2}.\end{eqnarray} \tag{ 31 }$$

In the matter-dominated era, the energy density is dominated by the mass density of nonrelativistic matter ρ:  ≈ ρc². The mass of the nonrelativistic matter in the universe is equal to M_univ = ρV. Accordingly, the Friedmann Equation (10) in the matter-dominated era is

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{a}}^{2}}{{c}^{2}}+1=\displaystyle \frac{D}{a}+\displaystyle \frac{1}{3}{\rm{\Lambda }}{a}^{2},\end{eqnarray} \tag{ 32 }$$

where $D=4{{GM}}_{{\rm{univ}}}/(3\pi {c}^{2})$.

A closed universe in a black hole expands to infinity if (Lord 1976)

$$\begin{eqnarray}&&D\gt \displaystyle \frac{2}{3\sqrt{{\rm{\Lambda }}}}.\end{eqnarray} \tag{ 33 }$$

At the matter–radiation equality, where a = a_eq and T = T_eq, we have $(1/3)\kappa {\epsilon }_{{\rm{eq}}}{a}_{{\rm{eq}}}^{2}\,=\,D/{a}_{{\rm{eq}}}$ or

$$\begin{eqnarray}&&D=\displaystyle \frac{1}{3}\kappa {h}_{\star }{T}_{{\rm{eq}}}^{4}{a}_{{\rm{eq}}}^{3}.\end{eqnarray} \tag{ 34 }$$

Using Equation (15), this constant can be written as

$$\begin{eqnarray}&&D=\displaystyle \frac{1}{3}\kappa {h}_{\star }{T}_{{\rm{eq}}}{\tilde{a}}_{i}^{3}{T}_{i}^{3}=\displaystyle \frac{1}{3}\kappa {h}_{\star }{T}_{{\rm{eq}}}{\left(\displaystyle \frac{{\tilde{a}}_{i}}{{a}_{i}}\right)}^{3}{({a}_{i}{T}_{i})}^{3},\end{eqnarray} \tag{ 35 }$$

where ${\tilde{a}}_{i}$ is the scale factor at T = T_i in the expanding phase. Without particle production, ${\tilde{a}}_{i}={a}_{i}$. With particle production, ${\tilde{a}}_{i}\gt {a}_{i}$. Since ${\tilde{a}}_{i}$ and a_i correspond to the same temperature, the ratio ${\tilde{a}}_{i}/{a}_{i}$ represents the expansion of the universe due to particle production. Relation (33) shows that

$$\begin{eqnarray}&&\displaystyle \frac{{\tilde{a}}_{i}}{{a}_{i}}\gt {\left(\displaystyle \frac{2}{\kappa {h}_{\star }{T}_{{\rm{eq}}}{({a}_{i}{T}_{i})}^{3}\sqrt{{\rm{\Lambda }}}}\right)}^{1/3}.\end{eqnarray} \tag{ 36 }$$

The value of T_eq depends on the number ratio of photons to massive particles and on the masses of the particles. In our universe, T_eq = 8820 K and Λ/κ = 5.24 × 10⁻¹⁰ Pa (Rich 2001), which gives

$$\begin{eqnarray}&&\displaystyle \frac{{\tilde{a}}_{i}}{{a}_{i}}\gt {10}^{10}.\end{eqnarray} \tag{ 37 }$$

Accordingly, Equations (27) and (28) lead to

$$\begin{eqnarray}&&{\tilde{{\rm{\Omega }}}}_{{\rm{\min }}}-1=({{\rm{\Omega }}}_{{\rm{\min }}}-1){\left(\displaystyle \frac{{a}_{i}}{{\tilde{a}}_{i}}\right)}^{2}\lt {10}^{-55}.\end{eqnarray} \tag{ 38 }$$

This value solves both the flatness and horizon problems (Popławski 2010c, 2011a).

If Equation (33) is not satisfied, then the universe in a black hole eventually reaches a crunch at which it stops expanding and starts contracting. The contraction lasts until the universe reaches the next bounce at T = T_max and a > a_min. After this bounce, the next expanding phase begins. The universe is more isotropic and has a larger scale factor at T = T_eq than in the previous expanding phase. Accordingly, it has a larger value of D. If this new value satisfies Equation (33), then the universe expands to infinity. If not, the universe has another cycle of contraction and expansion. After a finite number of cycles, D satisfies Equation (33) and the universe expands indefinitely. The last bounce can be regarded as the big bang, or rather the big bounce, of this universe.

In an isotropic universe, described by the FLRW metric, an alternating gravitational field causes production of particle–antiparticle pairs (Beilin et al. 1980). For a strong field in general relativity, the local rate of production of massive particles with spin j = 1 is equal to

$$\begin{eqnarray}&&\displaystyle \frac{1}{\sqrt{-g}}\displaystyle \frac{d}{{dt}}(\sqrt{-g}{n}_{1})=\displaystyle \frac{c}{288\pi }{P}^{2},\end{eqnarray} \tag{ 39 }$$

where n₁ is their number density. The local rates of production of massless particles with spin j = 1 and particles with spins j = 0 and j = 1/2 are proportional to P and can be neglected. In the ECSK theory, the torsion tensor may modify Equation (39). We can thus write

$$\begin{eqnarray}&&\displaystyle \frac{1}{\sqrt{-g}}\displaystyle \frac{d}{{dt}}(\sqrt{-g}{n}_{1})={cK},\end{eqnarray} \tag{ 40 }$$

where K > 0 is a scalar of dimension m⁻⁴, constructed from the curvature, torsion, and metric tensors. For the FLRW metric, we have

$$\begin{eqnarray}&&\displaystyle \frac{1}{{a}^{3}}\displaystyle \frac{d}{{dt}}({a}^{3}{n}_{1})={cK}.\end{eqnarray} \tag{ 41 }$$

Using ${n}_{1}(T)={h}_{n1}{T}^{3}$, where ${h}_{n1}=(\zeta (3)/{\pi }^{2}){g}_{n1}{k}_{{\rm{B}}}^{3}/{({\hslash }c)}^{3}$ and ${g}_{n1}={g}_{{\rm{b}}1}={\sum }_{i}{g}_{i}$ is summed over relativistic, massive bosons with spin j = 1, we find

$$\begin{eqnarray}&&\displaystyle \frac{\dot{a}}{a}+\displaystyle \frac{\dot{T}}{T}=\displaystyle \frac{{cK}}{3{h}_{n1}{T}^{3}}.\end{eqnarray} \tag{ 42 }$$

If we assume that Equation (10) is satisfied, then Equation (11) must be modified. Relation (42) generalizes Equation (14) and describes, together with Equation (17), the dynamics of the universe.

Particle production does not change the total energy and momentum of the matter and gravitational field in the universe (Popławski 2014). However, it increases the entropy in the universe. Combining Equations (30) and (42) gives

$$\begin{eqnarray}&&\displaystyle \frac{{dS}}{{dt}}=\displaystyle \frac{2\alpha {{ch}}_{n{\rm{f}}}^{2}}{{h}_{n1}}{a}^{3}{T}^{2}K.\end{eqnarray} \tag{ 43 }$$

The motion of matter through the event horizon is unidirectional, and thus it defines the arrow of time in the universe in a black hole. This arrow is also entropic: although black holes are states of maximum entropy in the frame of reference of outside observers, new universes expanding inside black holes continue to increase entropy.

For the FLRW metric, the Weyl tensor vanishes and the two independent invariants of dimension m⁻⁴ constructed from the Riemann and metric tensors are P² and ${P}_{{ik}}{P}^{{ik}}$. Using Equation (6) without the last term on the right-hand side and Equation (9) gives

$$\begin{eqnarray}P & = & -\,\kappa (\tilde{\epsilon }-3\tilde{p})=-2\kappa \alpha {n}_{{\rm{f}}}^{2},\\ {P}_{{ik}}{P}^{{ik}} & = & {\kappa }^{2}({\tilde{\epsilon }}^{2}+3{\tilde{p}}^{2}).\end{eqnarray} \tag{ 44 }$$

At a bounce, where $\dot{a}=\tilde{\epsilon }=0$ and T = T_max, the rate of production of particles (and thus K) should vanish to avoid $\dot{T}\gt 0$ in Equation (42). Otherwise, the increase of T would lead to T > T_max and ${\dot{a}}^{2}\lt 0$ in Equation (17). A scalar, vanishing for $\tilde{\epsilon }=0$ and constructed from the Riemann and metric tensors, is proportional to $K={P}^{2}-3{P}_{{ik}}{P}^{{ik}}$. Including the torsion tensor can give other forms for K. Ultimately, K should be derived from quantum field theory in the Riemann–Cartan spacetime of the ECSK theory of gravity.

The simplest form of K that vanishes at a bounce and has the same dimension as P² is

$$\begin{eqnarray}&&K=\beta {(\kappa \tilde{\epsilon })}^{2},\end{eqnarray} \tag{ 45 }$$

where β > 0 is a nondimensional constant. Near a bounce, where we can neglect k, Equation (10) can be written as

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\dot{a}}{{ca}}\right)}^{2}=\displaystyle \frac{1}{3}\kappa \tilde{\epsilon }.\end{eqnarray} \tag{ 46 }$$

Combining Equations (42), (45), and (46) gives

$$\begin{eqnarray}&&\displaystyle \frac{\dot{a}}{a}\left[1-\displaystyle \frac{3\beta }{{c}^{3}{h}_{n1}{T}^{3}}{\left(\displaystyle \frac{\dot{a}}{a}\right)}^{3}\right]=-\displaystyle \frac{\dot{T}}{T}.\end{eqnarray} \tag{ 47 }$$

The signs of $\dot{a}$ and $\dot{T}$ must be opposite to avoid an indefinitely long, exponential increase of the scale factor. This condition is satisfied during a contracting phase ($\dot{a}\lt 0$). During an expanding phase ($\dot{a}\gt 0$), we must have

$$\begin{eqnarray}&&\displaystyle \frac{3\beta }{{c}^{3}{h}_{n1}}{\left(\displaystyle \frac{\dot{a}}{{aT}}\right)}^{3}=\displaystyle \frac{3\beta }{{h}_{n1}}{\left[\displaystyle \frac{\kappa }{3}({h}_{\star }{T}^{2}-\alpha {h}_{n{\rm{f}}}^{2}{T}^{4})\right]}^{3/2}\lt 1.\end{eqnarray} \tag{ 48 }$$

The above function has a maximum at $T={T}_{{\rm{\max }}}/\sqrt{2}$. We thus find

$$\begin{eqnarray}&&\beta \lt {\beta }_{{\rm{cr}}}=\displaystyle \frac{\sqrt{6}}{32}\displaystyle \frac{{h}_{n1}{h}_{n{\rm{f}}}^{3}{({\hslash }c)}^{3}}{{h}_{\star }^{3}}=\displaystyle \frac{{15}^{3}\sqrt{6}}{4{\pi }^{14}}\displaystyle \frac{{\zeta }^{4}(3){g}_{{\rm{b}}1}{\left(\tfrac{3}{4}{g}_{{\rm{f}}}\right)}^{3}}{{g}_{\star }^{3}}.\end{eqnarray} \tag{ 49 }$$

If we assume that only the known standard-model particles exist, then g_b1 = 9 (Rich 2001). For this value, we find

$$\begin{eqnarray}&&{\beta }_{{\rm{cr}}}\approx \displaystyle \frac{1}{929},\end{eqnarray} \tag{ 50 }$$

which is on the order of $1/(288\pi )$ in Equation (39).

The exponential expansion of space in the early universe, known as cosmic inflation, also solves the flatness and horizon problems (Kazanas 1980; Guth 1981; Linde 1982). During inflation, the scale factor increases according to a ∝ e^Ht with constant values of H and T. In addition, inflation predicts the observed spectrum of primordial density fluctuations (Bardeen 1980; Mukhanov et al. 1992). Standard inflation, which assumes the existence of a fundamental scalar field with a specific potential, requires fine-tuning and is geodesically incomplete in the past (Borde et al. 2003).

If β is slightly lesser than β_cr, then at $T={T}_{{\rm{\max }}}/\sqrt{2}$,

$$\begin{eqnarray}&&\left|\displaystyle \frac{3\beta }{{c}^{3}{h}_{n1}}{\left(\displaystyle \frac{\dot{a}}{{aT}}\right)}^{3}\right|\lesssim 1.\end{eqnarray} \tag{ 51 }$$

At this temperature, Equation (47) therefore gives

$$\begin{eqnarray}&&\displaystyle \frac{\dot{T}}{T}\approx -\displaystyle \frac{2\dot{a}}{a}\end{eqnarray} \tag{ 52 }$$

in a contracting phase and

$$\begin{eqnarray}&&\dot{T}\approx 0\end{eqnarray} \tag{ 53 }$$

in an expanding phase. Accordingly, Equation (42) in an expanding phase gives an exponential expansion,

$$\begin{eqnarray}&&\displaystyle \frac{\dot{a}}{a}\approx \displaystyle \frac{c\beta {(\kappa \tilde{\epsilon })}^{2}}{3{h}_{n1}{T}^{3}}\approx c{\left(\displaystyle \frac{1}{3}\kappa \tilde{\epsilon }\right)}^{1/2},\end{eqnarray} \tag{ 54 }$$

at a nearly constant energy density

$$\begin{eqnarray}&&\tilde{\epsilon }\approx \displaystyle \frac{{h}_{\star }^{3}}{8{\alpha }^{2}{h}_{n{\rm{f}}}^{4}}.\end{eqnarray} \tag{ 55 }$$

As a increases after a bounce, the term on the left-hand side of Equation (51) increases from 0 to nearly 1 and T decreases from T = T_max to $T={T}_{{\rm{\max }}}/\sqrt{2}$. When $T\approx {T}_{{\rm{\max }}}/\sqrt{2}$, the term in Equation (51) is nearly 1 and the universe begins to expand exponentially at a nearly constant T (inflation). An exponential expansion lasts until the term in Equation (51) decreases significantly below 1 and T continues to decrease. The universe then starts expanding according to Equation (42) and moves from the torsion-dominated era to the radiation-dominated era. In a scenario with one bounce, relations (37) and (54) constrain the time interval t_infl of the exponential expansion: Ht_infl ≳ 23. This interval depends on $\tau {({\beta }_{{\rm{cr}}}-\beta )}^{-1}$.

For β = 0, the universe in a black hole is oscillatory with an infinite number of bounces and crunches (cycles). If 0 < β < β_cr, then the universe is cyclic with a finite number of cycles. In this case, the scale factor at a crunch is greater than the scale factor at the preceding crunch. As β increases, the number of cycles decreases and the accelerated expansion of the universe in each cycle is closer to exponential. If β ≲ β_cr, then the universe has only one bounce and expands through the torsion-dominated era with a period of an inflationary phase, radiation-dominated era, matter-dominated era, and then indefinitely in the cosmological-constant era. If $\beta \geqslant {\beta }_{{\rm{cr}}}$, then an exponential expansion of the universe would last indefinitely (eternal inflation). This behavior as a function of β was confirmed numerically in Desai & Popławski (2016).

Gravitational repulsion induced by spin and torsion, which becomes significant at extremely high densities, prevents singularities in black holes and at the big bang. Because of this repulsion and particle production, every black hole may create a new universe on the other side of its event horizon. The formation of trapped surfaces and wormhole throats, described in Section 1, could be explored numerically in the ECSK theory. This scenario simultaneously can explain the future of black hole interiors and the origin of the universe (Popławski 2010c, 2011a). Moreover, it naturally solves the flatness and horizon problems and can generate an inflationary phase that lasts for a finite amount of time without requiring finely tuned scalar fields, replacing R in the gravitational Lagrangian density by more complicated functions, or adding other hypothetical objects. Our universe could be a stable part of the multiverse and may contain inflating regions of spacetime (new universes) only in black holes.

Numerical analysis of this scenario (Desai & Popławski 2016) showed that the dynamics of the universe is insensitive to the value of a_i, and thus to that of a_min, and effectively depends on the particle production coefficient β only. Therefore, this scenario is insensitive to the detailed circumstances leading to the formation of a closed universe in a black hole. The same behavior is obtained if we consider the dynamics of a universe with a given value of a_min and T_max given by Equation (20) as the initial conditions for expansion (Desai & Popławski 2016). Equations (17) and (42) can describe the dynamics of the early universe even if we do not assume a black hole as its origin. However, the interior of a black hole can naturally provide the conditions to create a closed universe that can reach temperatures on the order of T_max. After such a universe is formed, spin and torsion provide the mechanism to avoid a singularity and to bounce (when the universe reaches T_max), and particle production provides the mechanism to generate inflation in the subsequent expansion. The only problem, which is more philosophical and cannot be solved, is this: if the black hole that created our universe exists in a larger universe and that universe was created inside a black hole, then what started the entire process?

The scenario presented in this paper would be more complete if we use the spin–torsion coupling beyond the point-particle approximation of a spin fluid (Popławski 2010b, 2013c) and combine it with the Dirac-spinor description of fermions. We showed in Popławski (2012a) that a semiclassical spin–spin interaction, generated by the minimal coupling between the torsion tensor and Dirac spinors, averts the unphysical big bang singularity and replaces it with a cusp-like big bounce at a finite minimum scale factor, before which the universe was contracting. In that scenario, the effective energy density and pressure were given by $\tilde{\epsilon }=\epsilon -\alpha {n}_{{\rm{f}}}^{2},\tilde{p}=p+\alpha {n}_{{\rm{f}}}^{2}$, where $\alpha =(9/16)\kappa {({\hslash }c)}^{2}$. A similar spin–spin interaction was analyzed in Khriplovich & Rudenko (2013), obtaining the same dynamics, but the cusp-like bounce was not noted; instead, it was assumed that the compression would not be followed by the expansion. To find whether the cusp can be smoothed out, Dirac spinors should be treated not semiclassically but as anticommuting fields (Chang 1990).

The observed matter–antimatter asymmetry in the universe may be caused by the nonlinearity of the Dirac equation in the presence of torsion (Popławski 2011b). Particle production by quantum effects in strong gravitational fields may also be affected by torsion (Rumpf 1979a, 1979b, 1979c). Primordial density fluctuations in the early universe formed in a black hole should be generated before the big bounce. If the big bounce is followed by a period of an exponential expansion, generated by torsion and particle production, then such fluctuations should be consistent with the observed spectrum. The consistency, for a particular range of the production rate, was recently found in Desai & Popławski (2016). The nature of dark matter and dark energy should be related to later epochs in the universe. The presented scenario predicts phenomena that are similar to those in some other cosmological models such as inflationary bubbles in the multiverse (Kazanas 1980; Guth 1981; Linde 1982), cyclic universe (Steinhardt & Turok 2002), or holographic world (Susskind 1995). However, contrary to the other models, this scenario does not need new, hypothetical fields. Finally, we note that the high-energy observations of photons originating from gamma-ray bursts indicate that specatime is continuous at the Planck temperature (Vasileiou et al. 2015), which is on the order of the temperature near a bounce. Therefore, the classical description of the gravitational field used in this scenario may be sufficient. I am grateful to my parents, Bożenna Popławska and Janusz Popławski. This research was supported by the University Research Scholar program at the University of New Haven.