arXiv:1808.08327v2 [gr-qc] 1 Mar 2019

The Astrophysical JournalVol. 870, No. 2, 78 (2019)c©The American Astronomical Society

BIG BOUNCE AND CLOSED UNIVERSE FROM SPIN AND TORSION

Gabriel Unger1,2∗ and Nikodem Pop lawski1†1Department of Mathematics and Physics, University of New Haven,

300 Boston Post Road, West Haven, CT 06516, USA and2Department of Mechanical Engineering and Applied Mechanics,

University of Pennsylvania, 220 South 33rd Street, Philadelphia, PA 19104, USA

We analyze the dynamics of a homogeneous and isotropic universe in the Einstein–Cartan theory ofgravity. The coupling between the spin and torsion prevents gravitational singularities and replacesthe Big Bang with a nonsingular big bounce, at which the universe transitions from contraction toexpansion. We show that a closed universe exists only when the product of the scale factor andtemperature is higher than a particular threshold, contrary to a flat universe and an open universe,which are not restricted. During inflation, this product must increase to another threshold, so thatthe universe can reach dark-energy acceleration.

Keywords: cosmology: theory, early universe, gravitation.

I. COSMOLOGY IN EINSTEIN–CARTAN (EC) GRAVITY

The simplest mechanism generating a nonsingular big bounce and inflation, involving only one unknown parameterand no hypothetical fields, arises in the EC theory of gravity [1]. EC is the simplest and most natural theory of gravitywith torsion, with the Lagrangian density for the gravitational field proportional to the Ricci scalar, as in generalrelativity. The conservation law for the total (orbital plus spin) angular momentum of fermions in curved spacetime,consistent with the Dirac equation, requires that the antisymmetric part of the affine connection, the torsion tensor[2], is not constrained to zero [3, 4]. Instead, torsion is determined by the field equations obtained from varying theaction with respect to the torsion tensor [5–14]. In EC, the spin of fermions is the source of torsion. The multipoleexpansion [15] of the conservation law for the spin tensor in EC gives a spin tensor that describes fermionic matter asa spin fluid (ideal fluid with spin) [16]. Once the torsion is integrated out, EC reduces to general relativity with aneffective spin fluid as a matter source [10–14]. The effective energy density and pressure of a spin fluid are given by

ε = ε− αn2f , p = p− αn2f , (1)

where ε and p are the thermodynamic energy density and pressure, nf is the number density of fermions, and α =κ(~c)2/32 [17].

The negative corrections from the spin-torsion coupling in (1) generate gravitational repulsion, which prevents theformation of gravitational singularities and replaces the Big Bang with a nonsingular bounce, at which the universetransitions from contraction to expansion [18–20]. These corrections lead to a violation of the strong energy conditionby the spin fluid when ε + 3p − 4αn2f drops below 0, thus evading the singularity theorems [17]. Accordingly, thisviolation could be thought of as the cause of the bounce. The dynamics of the EC universe filled with a spin fluid(1) has been studied in [21–25], with a parity-violating extension in [26], and with torsion coupled to the spinor fieldin [27]. The expansion of the closed universe with torsion and quantum particle production shortly after a bounce isalmost exponential for a finite period of time, explaining inflation [1]. Depending on the particle production rate, theuniverse may undergo several bounces until it produces enough matter to reach a size where the cosmological constantstarts cosmic acceleration. This expansion also predicts the cosmic microwave background radiation parameters thatare consistent with the Planck 2015 observations [28, 29], as was shown in [30].

∗[email protected]†[email protected]

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The avoidance of singularities can also occur in cosmological models based on Riemann–Cartan geometries withoutspin density: Poincare gauge theories with quadratic terms in curvature and torsion [31, 32], scalar-tensor theories withtorsion [33, 34], and higher-dimensional geometries with torsion [35]. Therefore, it seems to be a generic feature of theRiemann–Cartan spacetime, rather than a particular feature of EC. We consider EC because it has other interestingconsequences. The spin-torsion coupling modifies the Dirac equation, adding a term that is cubic in spinor fields [36].As a result, fermions must be spatially extended [37, 38], which could eliminate infinities arising in Feynman diagramsinvolving fermion loops. In the presence of torsion, the four-momentum operator components do not commute andthus the integration in the momentum space in Feynman diagrams must be replaced with the summation over thediscrete momentum eigenvalues. The resulting sums are finite: torsion naturally regularizes ultraviolet-divergentintegrals in quantum electrodynamics [39, 40]. Torsion may also explain the matter-antimatter asymmetry and darkmatter [41], and the cosmological constant [42].

The analysis in [1] considered a closed, homogeneous, and isotropic universe in EC. However, the calculations ofthe maximum temperature and the minimum scale factor at a bounce neglected the factor k = 1 in the Friedmannequations (which is justified during and after inflation but not at a bounce before inflation), de facto considering aflat universe. In this article, we refine these calculations by taking k into account and analyzing the expansion of theuniverse for all three cases: k = 1 (closed universe), k = 0 (flat universe), and k = −1 (open universe). We discoverthat a closed universe exists only when the product of the scale factor and temperature is higher than a particularthreshold, whereas open and flat universes are not restricted by such a condition. Accordingly, a closed universe formsin a region of space within a trapped null surface [1] when this threshold is reached.

II. DYNAMICS OF SCALE FACTOR AND TEMPERATURE

If we assume that the universe is homogeneous and isotropic, then it is described by the Friedmann–Lemaıtre–Robertson–Walker metric in the isotropic spherical coordinates [43]:

ds2 = c2dt2 − a2(t)

(1 + kr2/4)2(dr2 + r2dθ2 + r2 sin2 θ dφ2), (2)

where a(t) is the scale factor as a function of the cosmic time t. The Einstein field equations for this metric becomethe Friedmann equations [43]:

a2

c2+ k =

1

3κεa2 (3)

and

a2 + 2aa

c2+ k = −κpa2, (4)

where a dot denotes the derivative with respect to t and κ = 8πG/c4. Multiplying the first Friedmann equationby a and differentiating over time, and subtracting from it the second Friedmann equation multiplied by a gives anequation that has the form of the first law of thermodynamics for an adiabatic universe:

d

dt(εa3) + p

d

dt(a3) = 0. (5)

For EC, the Friedmann equations have the same form but the energy density and pressure are replaced by ε and p[21, 22, 24, 25]:

a2

c2+ k =

1

3κ(ε− αn2f )a2 (6)

and

d

dt[(ε− αn2f )a3] + (p− αn2f )

d

dt(a3) = 0. (7)

The spin fluid in the early universe is formed by an ultrarelativistic matter in kinetic equilibrium, for whichε = h?T

4, p = ε/3, and nf = hnfT3, where T is the temperature of the universe, h? = (π2/30)(gb + (7/8)gf)k

4B/(~c)3,

and hnf = (ζ(3)/π2)(3/4)gfk3B/(~c)3 [44]. The quantities gb and gf are the numbers of spin states for all elementary

3

bosons and fermions, respectively. For standard-model particles, gb = 29 and gf = 90. In the presence of spin andtorsion, the first Friedmann equation is therefore [1, 45]

a2

c2+ k =

1

3κ(h?T

4 − αh2nfT 6)a2. (8)

The first law of thermodynamics (7) gives [1]( aa

+T

T

)(1− 3αh2nf

2h?T 2)

= 0, (9)

which yields

a

a+T

T= 0. (10)

Remarkably, the last equation is the same as that for the relativistic universe without spin and torsion.

III. ANALYSIS OF SOLUTIONS FOR A CLOSED UNIVERSE

Let us consider a closed relativistic universe. We define nondimensional quantities:

x =T

Tcr, (11)

y =a

acr, (12)

τ =ct

acr, (13)

where

Tcr =( 2h?

3αh2nf

)1/2= 9.410× 1031 K (14)

and

acr =9~c8√

2

(αh4nfh3?

)1/2= 3.701× 10−36 m. (15)

Henceforth, we will use the dot to denote the derivative with respect to the new time coordinate τ . Equation (8) canbe written as

y2 + 1 = (3x4 − 2x6)y2. (16)

Equation (10) can be integrated to

xy = C, (17)

where C is a positive constant (because the scale factor and temperature are greater than 0). Substitution of thisrelation into Equation (8) gives

y2 + 1 =3C4

y2− 2C6

y4. (18)

Since the left-hand side of this equation is positive, y cannot reach zero because the right-hand side of this equationwould have to become negative. Consequently, a cosmological singularity is never produced for any value of C.

The big bounce and big crunch of a closed universe are turning points (there is no expansion or contraction at thesepoints), so therefore they are determined by a condition

y = 0. (19)

4

Using this relation, Equation (18) can be resolved for a quadratic, in terms of y and C. The resulting quadraticequation for y2 is

y4 − 3y2C4 + 2C6 = 0. (20)

The solutions of this equation are:

y2± =3C4 ±

√9C8 − 8C6

2. (21)

At the big bounce, y = y−, and at the big crunch, y = y+. These turning points of a closed universe exist if theexpression under the square root in Equation (21) is positive or zero. In order for that expression to remain positive

or zero, C must be greater than or equal to√

8/9. Consequently, an inequality

C ≥√

8/9, (22)

equivalent to

aT ≥√

8/9acrTcr, (23)

is a necessary condition for creating a closed universe in a region of space with the local values of a and T .If C =

√8/9, then the turning points coincide, y− = y+, and the universe is stationary (no expansion or contraction)

with the constant value of the scale factor of y =√

32/27. Since the values of y and C are now both known, the

corresponding constant value of x can be found using Equation (17) to be√

3/2. Such a stationary universe has a

constant scale factor a =√

32/27acr and temperature T = (√

3/2)Tcr.

If C is greater than√

8/9, then the universe has two turning points and both the big bounce and big crunch occur.The value of y2 is nonnegative in the range from y = y− to y = y+, so therefore the universe oscillates between y = y−and y = y+. If C >

√8/9, then Equation (18) can be rearranged to give

y2± = 3C4[1±

√1− 8

9C2

2

]. (24)

In the limit C � 1, using the formula (1− x)n ≈ 1 + nx for |x| � 1 gives

y2 = 3C4[1± (1− 4

9C2 )

2

]. (25)

Accordingly, y− ≈√

2/3C and y+ ≈√

3C2.The absolute minimum value for y among all possible values of C can be determined from a condition dy2−/dC = 0,

giving

C = 1. (26)

For this value of C, the nondimensionalized minimum scale factor and the corresponding maximum temperature are

x = 1, ymin = 1. (27)

Accordingly, the constant acr is equal to the least possible value of the scale factor of a closed universe in EC. A closeduniverse with spin and torsion is nonsingular (y ≥ 1 and thus y > 0).

The squared values of ymin, ymax, xmin, and xmax for different values of C are shown in Table I. The greatest possiblevalue of the temperature in a closed universe in EC is

√3/2Tcr. When C is much greater than 1, the universe can

expand by a factor of 9C2/2 and its temperature can decrease by the same factor, reaching the value at which thetransition from the radiation (relativistic) domination to the matter (nonrelativistic) domination occurs.

Equation (18) can be solved analytically. We substitute

y = [E − F cos(2θ)]1/2, y = [E − F cos(2θ)]−1/2F sin(2θ)θ, (28)

where E and F are positive constants such that E > F , obtaining

[F 2 sin2(2θ)θ2][E − F cos(2θ)] + [E − F cos(2θ)]2 − 3C4[E − F cos(2θ)] + 2C6 = 0. (29)

5

C y2min y2max x2max x2min√8/9 32

273227

34

34

1 1 2 1 12

� 1 2C2

33C4 3

21

3C2

TABLE I: The minima and maxima of the nondimensionalized temperature x and scale factor y for different values of theintegration constant C in a closed universe. The domain of C is [

√8/9,∞).

FIG. 1: Nondimensionalized scale factor y as a function of the nondimensionalized time τ near a bounce with C = 1. The timeτ = 0 is set at the bounce.

Putting 2E = 3C4 and E2 − F 2 = 2C6 reduces this equation into

θ2[E − F cos(2θ)] = 1, (30)

which integrates to

τ =

∫ θ

0

[E − F cos(2θ)]1/2dθ = (E − F )1/2∫ θ

0

[1 + ξ2 sin2 θ]1/2dθ, (31)

where ξ2 = 2F/(E − F ), and θ = 0 at t = 0. The integral in this equation is the elliptic integral of the second kindwith an imaginary ξ. Therefore, the time dependence y(τ) of the scale factor is given in the parametric form by thefirst equation in (28) and Equation (31) with

E = 3C4/2, F =√

9C8/4− 2C6. (32)

The parameter θ runs from 0 (bounce) through π/2 (crunch) to π (next bounce) for one cycle. The minimum andmaximum scale factors are given by y± =

√E ± F , in agreement with (24). The value of F is real, giving the

condition (22). Figures 1, 2, and 3 show the nondimensionalized scale factor y as a function of the nondimensionalizedtime τ near a nonsingular, smooth bounce with C = 1, C = 10, and C = 100, respectively. Figure 4 shows thenondimensionalized scale factor y as a function of the nondimensionalized time τ for one cycle with C = 1, 10, 100using a logarithmic scale.

The results of this section show that a necessary condition for the creation of an expanding universe in a givenvolume of space is

C >√

8/9 (33)

for all points in that volume. For all allowed values of C, a closed universe in nonsingular.

6

FIG. 2: Nondimensionalized scale factor y as a function of the nondimensionalized time τ near a bounce with C = 10. Thetime τ = 0 is set at the bounce.

FIG. 3: Nondimensionalized scale factor y as a function of the nondimensionalized time τ near a bounce with C = 100. Thetime τ = 0 is set at the bounce.

IV. ANALYSIS OF SOLUTIONS FOR A FLAT AND AN OPEN UNIVERSE

For a flat relativistic universe, Equation (18) becomes

y2 =3C4

y2− 2C6

y4. (34)

The corresponding quadratic equation for y2 at the turning points is

3y2 − 2C2 = 0. (35)

7

FIG. 4: Nondimensionalized scale factor y as a function of the nondimensionalized time τ for one cycle with C = 1, 10, 100using a logarithmic scale.

The physical solution of this equation is

y =√

2/3C, (36)

which is the minimum value of the nondimensionalized scale factor. This value is equal to the nondimensionalizedscale factor at a bounce for the case of a closed universe in the limit C � 1. Consequently, a flat universe in EC isalso nonsingular, because C is positive. Since a flat universe has only one turning point (at a bounce), it expands toinfinity. Contrary to a closed universe, there are no further restrictions on the value of C.

For an open relativistic universe, Equation (18) becomes

y2 − 1 =3C4

y2− 2C6

y4. (37)

The corresponding quadratic equation for y2 at the turning points is

y4 + 3y2C4 − 2C6 = 0. (38)

The physical solution of this equation is

y2 =−3C4 +

√9C8 + 8C6

2, (39)

with y > 0. This solution is the minimum value of the nondimensionalized scale factor. As for a flat universe, anopen universe is also nonsingular and has only one turning point (at a bounce), so it expands to infinity. There areno further restrictions on the value of C.

V. ANALYSIS WITH COSMOLOGICAL CONSTANT

A flat and an open universe expand to infinity. Without further considerations, a closed expanding universe hastwo turning points (provided that C >

√8/9) and therefore it reaches the maximum value of the scale factor, after

which it contracts. To avoid the big crunch and the subsequent contraction, another term in the first Friedmannequation is needed that can cause the acceleration of a late universe. The simplest term is given by a cosmologicalconstant, which enters the first Friedmann equation (for relativistic matter) according to

y2 + 1 =3C4

y2− 2C6

y4+ λy2, (40)

8

where λ > 0 is the nondimensionalized cosmological constant:

λ =1

3Λa2cr = 5.0× 10−124. (41)

This small value results from the small cosmological constant Λ = 1.1× 10−52 m−2.For a late universe, where the values of y are large, the y−4 component of this equation can be ignored, reducing

this equation to

y2 + 1 =3C4

y2+ λy2. (42)

Putting y = 0 to find the turning points, the equation can then be rewritten as the quadratic given below:

λz2 − z + 3C4 = 0, (43)

where z = y2. Solving the quadratic shows that in order for no turning point to exist (in a late universe), λ must begreater than 1/(12C4).

This condition can also be written as

C > (12λ)−1/4. (44)

A closed universe expands to infinity if the cosmological constant is sufficiently high. Such expansion is asymptoticallyexponential: y2 ≈ λy2 in a late universe gives y ∼ exp(

√λτ). According to the results of Section III and the condition

(33), a closed universe forms in a given region of space when the local value of C is equal to√

8/9. When C >√

8/9,the universe expands. To expand to infinity, the universe must have a mechanism to increase the value of C from√

8/9 to (12λ)−1/4. If (12λ)−1/4 <√

8/9, then the universe expands to infinity regardless. A natural and physicalmechanism for the growth of C is provided by quantum particle-pair production in strong gravitational fields [46–52].This mechanism also generates a brief period of exponential expansion of a very early universe [1], thus naturallyderiving inflation [53–55].

This analysis would be valid for a relativistic universe. However, a relativistic universe transitions from the radiation-dominated era to the matter-dominated era and becomes nonrelativistic. Equation (42) becomes

y2 + 1 =B

y+ λy2, (45)

where B is a positive constant. This transition occurs when

3C4

y2=B

y(46)

and at temperature Teq = 8.8× 103 K. Using (11) and (17), we obtain

B = 3C3x3eq, (47)

where

xeq =TeqTcr

= 9.4× 10−29. (48)

A nonrelativistic universe expands to infinity if the cosmological constant is sufficiently high. The turning pointsof Equation (45) are given by a cubic equation

y3 − y

λ+B

λ= 0. (49)

This equation has no real positive solutions if [9]

B >2

3√

3λ. (50)

Consequently, (47) gives the following condition for the absence of turning points in a late universe:

C >( 2

9√

3λ

)1/3 1

xeq= 1.9× 1048. (51)

9

If this condition is satisfied, the universe expands to infinity.Numerical analysis of generic gravitational collapse in general relativity shows that each spatial point in the interior

of a black hole locally evolves toward the singularity as an independent, spatially homogeneous, closed universe [56, 57].In EC, this evolution does not reach a singularity, but instead undergoes a nonsingular bounce after which such auniverse expands [1]. Accordingly, if our Universe is closed, its contraction before the big bounce could correspond togravitational collapse of matter inside a newly formed black hole existing in another universe [58–67]. In this scenario,the formation of our Universe corresponds to the moment when the quantity C, representing the product of the scalefactor and temperature, begins to satisfy the inequality (33) in a given volume of space in the black hole. Duringinflation, this quantity increases because of quantum particle-pair production in strong gravitational fields, and mustreach the threshold (51) so that the Universe could start the observed current acceleration. If this threshold is notreached, the closed universe contracts to another bounce and starts another cycle of expansion. The last bouncebefore reaching the threshold can be regarded as the Big Bang of the Universe.

This work was funded by the University Research Scholar program at the University of New Haven.

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