A big bounce, slow-roll inflation and dark energy from conformal gravity

Jack Gegenberg,∗ Shohreh Rahmati,† and Sanjeev S. Seahra‡ Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada E3B 5A3 (Dated: October 1, 2018) We examine the cosmological sector of a gauge theory of gravity based on the SO(4,2) conformal group of Minkowski space. We allow for conventional matter coupled to the spacetime metric as well as matter coupled to the field that gauges special conformal transformations. An effective vacuum energy appears as an integration constant, and this allows us to recover the late time acceleration of the universe. Furthermore, gravitational fields sourced by ordinary cosmological matter (i.e. dust and radiation) are significantly weakened in the very early universe, which has the effect of replacing the big bang with a big bounce. Finally, we find that this bounce is followed by a period of nearly- exponential slow roll inflation that can last long enough to explain the large scale homogeneity of the cosmic microwave background.

I. INTRODUCTION galaxies and weak gravitational lensing [1]. Alternative gravity theories have also been proposed to explain the In order to reconcile Einstein’s general relativity with late time acceleration of the universe [2,3], as well as observational facts, modern cosmology incorporates a early time inflationary acceleration [4]. Quantum correc- number of elements that are difficult to justify from fun- tions to general relativity have been used to tame the big damental physics: For example, dark matter is required bang singularity; for example, in loop quantum cosmol- to describe the clustering of stars and galaxies, dark en- ogy semiclassical equations of motion yield a big bounce ergy is required to explain the late time acceleration of instead of a big bang [5]. the universe, and an inflationary mechanism in the early In this paper, we study the cosmological implications of universe is needed to explain both the large scale homo- a modified gravity model that simultaneously addresses geneity of the cosmic microwave background as well as the issues of the initial singularity, the mechanism driv- the origin of primordial fluctuations. ing inflation, and the late time acceleration of the uni- The simplest framework that incorporates all of these verse. Our model belongs to the class of gauge theories elements has become known as the concordance model of gravity [6–13] in which the central object is a gauge of cosmology: namely, ΛCDM with single field inflation. potential analogous to the gauge potentials of particle In this paradigm, one assumes the existence of cold dark physics. The action functional is taken to be quadratic matter (CDM) that is not part of the standard model of in the field strength of the gauge potential, just as in con- particle physics, a cosmological constant Λ that is added ventional Yang-Mills theory. Geometric quantities, such by hand to the Einstein-Hilbert action of general relativ- as the metric and connection, are defined as functions of ity, and a scalar inflaton field with an appropriate po- the gauge potential. This ultimately leads to a metric tential to drive nearly de Sitter (dS) inflation for a finite theory of gravity. We take the gauge group to be the period in the early universe. (We note that none of these conformal group of Minkowski spacetime SO(4,2), and axioms of standard cosmology are inconsistent with gen- the resulting theory is invariant under local conformal eral relativity.) Even after observational evidence is ac- (Weyl) transformations. Non-cosmological aspects of the counted for via these mechanisms, theoretical challenges SO(4,2) gauge gravity model have been studied in [10– remain for concordance cosmology. For example, if infla- 13]. tion is finite the classical equations of motion for general A ubiquitous feature of gauge-gravity theories is man- relativity imply that the universe started with a big bang. ifolds with non-vanishing torsion. When models based arXiv:1604.06369v3 [gr-qc] 1 May 2017 (It should be mentioned that while this singular initial on the Poincar´e[14] or de Sitter groups [15, 16] were state is conceptually unappealing, it is not actually in applied in cosmology, it was found that nonzero tor- direct conflict with observations.) sion can drive late time acceleration. Actually, in the In an attempt to avoid some of the more ad hoc el- de Sitter case nonzero torsion is a necessary condition ements of concordance cosmology, many authors have for the existence of non-radiation cosmological matter. considered the possibility that general relativity is not It should also be noted that there are other non-gauge the correct theory of gravitation. Some of the oldest gravity models where torsion is responsible for singular- modified gravity theories have attempted to provide a ity avoidance and inflation in the early universe [17–20]. non-particle explanation of dark matter, but such mod- However, while the SO(4,2) theory considered here allows els can have difficulty accounting for the clustering of for nonzero torsion, in this work we find that it is not re- quired to reconcile the model with observations, explain dark energy, or to alter early universe dynamics. How- ∗ [email protected] ever, we do require a population of matter that couples † [email protected] directly to the fields gauging the generators of special ‡ [email protected] conformal transformations. 2

In sectionII we present the action of our model, and Note that in this model, it is not necessary to assume α α write down the equations of motion assuming vanishing T βγ = 2Γ [βγ] = 0; however, we will concentrate on the torsion and matter which couples to the metric as well vanishing torsion case in this paper. as fields gauging special conformal transformations. In The action functional of the model is sectionIIIA, we specialize to homogeneous and isotropic 1 Z √ spacetimes and write down the Friedmann equation for 4 αµ βν A B S = − 2 d x −gg g hABFαβFµν + Sm, (6) the model. In sectionIIIB, we discuss solutions of the 2gYM

Friedmann equation and demonstrate the existence of a M N bounce, slow-roll inflation and late time acceleration. In where hAB = f AN f BM is the Cartan-Killing metric sectionIV, we summarize and discuss our results. on so(4,2). The non-trivial components of hAB are:

ha¯b = hab¯ = −2ηab, h14,14 = 2, II. GRAVITATIONAL YANG-MILLS THEORY h[ab][cd] = h[cd][ab] = −4ηa[cηd]b. (7) The notation here is that a, a¯ = 0, 1, 2, 3 denote com- We consider the gravity theory described in [13]. The ponents in the direction of translations Pa and special reader can find a more complete description of the model conformal transformations Ka, respectively. The six in- in that paper; here, we focus on a subsector of the the- dices [ab] consist of [12], [23], [31], [01], [02], [03] and de- ory obtained by making a number of simplifying assump- note directions along the distinct non-zero generators Jab tions. of Lorentz transformations. Finally, the index 14 denotes Our model is based on the SO(4,2) conformal group the component in the direction of the generator D of di- of Minkowski spacetime, which is the largest group of latations. We view (6) and (7) as the defining relation- transformations that leaves null geodesics invariant. We ships for our model. begin with the so(4,2)-Lie algebra-valued vector potential The action (6) is manifestly diffeomorphism invariant, and as demonstrated in [13], it is invariant under local A a a ab Aα = Aα JA = eαPa + lαKa + ωα Jab + qαD, (1) gauge transformations described by an eleven parameter subgroup of SO(4,2) with generators {Ka, Jab, D}. The where the JA = {Pa, Ka, Jab, D} are the generators of behaviour of the gauge potential under these infinitesimal the Lie algebra and α = 0 ... 3 is a spacetime index. gauge transformations is: The components of the associated field strength Fαβ = F A J are given by A A A A B C αβ A Aα 7→ Aα + ∂α + f BC Aα  , A a ab A A A A B C  JA = λ Ka + Λ Jab + ΩD. (8) Fαβ = ∂αAβ − ∂βAα + f BC Aα Aβ , (2) In particular, the component of the gauge potential in with the structure constants defined by [JA, JB] = C the direction of D transforms as: f ABJC . We identify various components of Aα in the JA basis 1 δqα = ∂αΩ + 2 λα. (9) with geometric quantities in a 4-dimensional manifold M α with Lorentzian metric gαβ and affine connection Γ βδ. It is obvious that we can impose the gauge condition a In particular, we take eα as the components of an or- qα = 0 via a simple series of gauge transformations of the ab A a thonormal frame fields on M, with ωα as the associated form  JA = λ Ka. This gauge condition is preserved connection one-forms. Hence, the metric and connection under the gauge transformation generated by: are given by: A aα ab  JA = −2e ∂αΩ Ka + Λ Jab + ΩD. (10) a b γ γ a ac gαβ = ηabeαeβ, Γ αβ = ea(∂αeβ + ωα ecβ). (3) Under this class of restricted gauge transformations the In these expressions, lowercase Greek and Latin indices metric transforms as: are raised and lowered with gαβ and ηab, respectively. The curvature one-forms are anti-symmetric in their δgαβ = Ωgαβ. (11) (ab) frame indices ωα = 0, from which it follows that the That is, the model is invariant under local conformal affine connection is metric-compatible [21]: (Weyl) transformations. For the rest of this paper we will work in the qα = 0 gauge. 0 = ∇αgβγ , (4) While qα = 0 is a gauge choice and may be imposed without loss of generality, we will also enforce a number of where ∇ is the derivative operator defined by Γα . The α βδ additional conditions that are actually physically restric- Riemann curvature and torsion tensors of M are given by: tive. In general the affine connection Γ on the spacetime µν µ ν ab ac b manifold M has non-vanishing torsion. It might be the R αβ = ea eb (dω + ω ∧ ωc )αβ, α α a ac case that torsion plays an important role in cosmology, T βγ = ea (de + ω ∧ ec)βγ . (5) but in this work we concentrate on the case where torsion 3

a is not present; i.e., we impose Tµν = 0. It was demon- III. FRIEDMANN-ROBERTSON-WALKER strated in [13] that the torsion-free condition is preserved COSMOLOGY under the gauge transformations (10). Another assump- tion concerns the dependence of the matter action on the A. The Friedmann equation gauge potential. Specifically, we assume that the matter a action is a functional of the metric gαβ, the field lα gaug- The goal of this section is to study the evolution of ing special conformal transformations, and matter fields a spatially homogenous and isotropic spacetime in our (generically denoted by ψ) only: model. We therefore assume the Friedmann-Robertson- a Walker (FRW) line element: Sm = Sm[gαβ, lα, ψ]. (12) More general types of matter-gauge potential coupling  dr2  ds2 = −dt2 + A2 + r2dθ2 + r2 sin2 θ dφ2 , are discussed in [13]. Finally, in the full theory derived 2 2 1 − kr0/r from (6) there is an antisymmetric tensor, (18) 1 a b where A = A(t) is the scale factor, r is a constant Fαβ = ηabe l , (13) 0 2 [α β] with the dimension of length, and k = 0, +1, −1 for flat, appearing in the field equations that satisfies Maxwell- 3-sphere and 3-hyperboloid spatial geometries, respec- like equations for the electromagnetic field strength. tively. Since our primary interest is cosmology below, we ex- Due to the isotropy and homogeneity of the spacetime, pect such a tensor would be ruled out by isotropy and the symmetric matter source aµν must take the form: homogeneity and hence we set Fαβ = 0. We note that it µν µ ν µν is easily confirmed that the vanishing of Fαβ is a gauge a = (ξ1 + ξ2) u u + ξ2g , (19) invariant condition: Under the transformations (10), we α have: where u ∂α = ∂t. This is algebraically identical to the stress-energy tensor of a perfect fluid, but we caution that δ = −2∇ ∇ Ω = 0. (14) Fαβ [α β] aµν should not be interpreted in this way: As seen in (16), Variation of the action (6) with respect to the gauge this tensor arises from the variation of the matter action a potential under these assumptions yields the equations with respect to lα, not from the variation with respect of motion: the metric. Also, the parameters ξ1 and ξ2 appearing in (19) do not have the dimensions of density and pressure; αν 1 2 αν [ν µ]α αν 2 0 = B + 16 gYMT − ∇µ∇ a¯ − Q , (15a) rather, they have dimensions of (mass) . Substituting α 0 = ∇ aαβ, (15b) the trace of (19) into (15b) and (15c) and solving yields 0 = ∇βa. (15c) Π Π ξ = − − Λ, ξ = − + Λ, (20) Here and below, the vanishing of the torsion implies that 1 A4 2 3A4 ∇ is the ordinary covariant derivative operator as de- α where Λ and Π are constants of integration. fined from the Levi-Civita connection. Also, a de- αβ For any homogeneous and isotropic spacetime the Bach scribes matter coupling to la while is the ordinary α Tαβ and Weyl tensors vanish identically, and therefore (15a) stress-energy tensor: √ reduces to g2 δ( −gL ) µν √YM m bµ a = b e , 1 2 αν [ν µ]α 1 4 −g δl gYMT = ∇µ∇ a¯ − (2Sλµ − a¯λµ) √ν 16 2 λ[µ ν]α α[µ ν]λ 2 δ( −gLm) × (g a¯ − g a¯ ). (21) Tµν = −√ . (16) −g δgµν We fix the “ordinary” matter content of the universe to The other quantities appearing in (15) are given by: be non-interacting pressureless dust and radiation as in αν 1 αλµν 1 αν 1 ΛCDM: Q = 2 aλµC − 8 τ − 2 (2Sλµ − a¯λµ) λ[µ ν]α α[µ ν]λ µν (m) (r) × (g a¯ − g a¯ ), (17a) T = Tµν + Tµν , (22) α α αβ Bµν = −∇ ∇αSµν + ∇ ∇µSαν + CµανβS , (17b) with S = 1 (R − 1 Rg ), (17c) αβ 2 αβ 6 αβ (m) (r) ρσ αβγρ σ ρσ αβγδ Tµν = ρmuµuν , Tµν = (ρr + pr)uµuν + prgµν , τ = −4C Cαβγ + g C Cαβγδ, (17d) 1 pr = ρr/3. (23) a¯αβ = aαβ − 6 gαβa, (17e) α a = a α. (17f) Demanding that each matter source is separately con- µ (m) µ (r) served (∇ Tµν = ∇ Tµν = 0) yields: Here, Cαβγδ is the Weyl tensor, while Sαβ and Bµν are the Schouten and Bach tensors. Finally, by taking the −3 ρ˙m + 3Hρm = 0 ⇒ ρm = ρm,0A , (24a) divergence of (15a), we find that the stress-energy tensor −4 α ρ˙ + 4Hρ = 0 ⇒ ρ = ρ A . (24b) is conserved as usual: ∇ Tαβ = 0. r r r r,0 4 where ρm,0 and ρr,0 are constants, we use an overdot to which when substituted into equation (21) yields: denote d/dt, and we have defined the Hubble parameter 2 ˙ 3gYM H = A/A. The (00) component of (21) then yields the Gαβ + Λgαβ = Tαβ. (32) Friedmann equation 8Λ Here, Gαβ is the Einstein tensor, so this is equivalent to 2 ! the field equations of general relativity with a cosmolog- g ρm + ρr k Λ Π H2 = YM − + − , (25) ical constant provided we identify the Planck mass as in 8 Λ + Π r2A2 3 3A4 A4 0 (27). where we have used (19) and (20).1 In the late-time limit 4 A  |Π/Λ|, we obtain B. Cosmological dynamics

ρm + ρr k Λ H2 ≈ − + , (26) It is useful to write the Friedmann equation in terms of 3M 2 r2A2 3 Pl 0 the same density parameters used to describe the ΛCDM where we have identified the Planck mass as: model: ρ ρ Ω = m,0 , Ω = r,0 , 8 Λ m 2 2 r 2 2 2 3MPlH0 3MPlH0 MPl = 2 . (27) 3 gYM Λ k Ω = , Ω = − , (33) Λ 3H2 k r2H2 Equation (26) is the same as the Friedmann equation in 0 0 0 ΛCDM provided that we interpret the constant of inte- where we have assumed that A = 1 and H = H0 at the gration Λ as the cosmological constant. Probes of the present epoch. We also define a dimensionless density expansion history of the late time universe gives us the parameter for the “dark radiation” (i.e. the integration order of magnitude of Λ: constant Π): Π (10−3 eV)4 −33 2 ΩΠ = 2 . (34) Λ ∼ 2 ∼ (10 eV) . (28) 3H0 MPl 2 Note that since observations imply that Λ ∼ H0 , we have This in turn fixes the size of Yang-Mills coupling constant that ΩΠ ∼ Π/Λ; i.e., it is roughly the ratio of the two 2 −120 to be gYM ∼ 10 . constants appearing in our solution for aαβ. In terms of Before moving on, we make a few remarks about the in- these, the Friedmann equation (25) becomes terpretation of the Friedmann equation (25). This equa- 2 tion can be rewritten in a form more familiar from gen- H ΩΛΩm ΩΛΩr Ωk ΩΠ = + + + ΩΛ − , H2 3 ΩΠ 4 ΩΠ A2 A4 eral relativity if one introduces a time-varying Newton’s 0 A (ΩΛ + A4 ) A (ΩΛ + A4 ) constant and a “dark radiation” field with density ∝ −Π: (35) where we have made use of (27). Evaluating this at the 8πG (A) k Λ Π 2 eff present epoch (when A = 1 and H = H0) yields a con- H = (ρm + ρr) − 2 2 + − 4 , (29) 3 r0A 3 3A straint amongst the density parameters: where ΩΛ(Ωm + Ωr) 1 = + Ωk + ΩΛ − ΩΠ. (36) ΩΛ + ΩΠ 3g2  A4  1  A4  YM Note that if |Ω |  1 we recover the standard ΛCDM Geff(A) = 4 = 2 4 . Π 64π ΛA + Π 8πMPl A + Π/Λ relation (30) We see that the effective Newton constant decreases with 1 = Ωm + Ωr + Ωk + ΩΛ. (37) decreasing A; i.e., the force of gravity is weaker in the In order to qualitatively analyze the cosmological dy- past. As seen in sectionIIIB, this screening of the grav- namics, it is useful to rewrite the Friedmann equation as itational field sourced by ordinary matter will have im- the equation of motion of a zero-energy particle moving portant consequences for early universe dynamics. in a one-dimensional effective potential: We also note that if Π = 0, we recover the Friedmann equation of general relativity exactly. Indeed, if Π = 0 1 dA2 + V (A) = 0, (38) we have 2 dτ eff where we have defined τ = H t and aαβ = Λgαβ, (31) 0

ΩΛΩm ΩΛΩr Veff(A) = − − ΩΠ 2 ΩΠ 2A(ΩΛ + A4 ) 2A (ΩΛ + A4 ) 1 ˙ Ω Ω A2 Ω The spatial components of (21) yield an equation for H that can − k − Λ + Π . (39) be derived from the formula already presented. 2 2 2A2 5

We note that (36) can be used to eliminate Ωk in either As mentioned above, in order to recover an acceptable (35) or (39). The utility of the Friedmann equation writ- late-time cosmology, we must have that |ΩΠ|  1. Let ten as (38) is that we can immediately conclude that all us assume that 0 < ΩΠ  ΩΛ, and hence obtain the values of the scale factor with Veff(A) > 0 are classically following approximate form of the potential: forbidden, and we can obtain the acceleration of the uni- verse in a given epoch from A¨ = −V 0 (A). It is also of    −1 eff 1 Ωm Ωr ΩΠ interest to define the “slow-roll” parameter Veff(A) ≈ − + 2 1 + 4 2 A A ΩΛA ˙ ¨ 0 2 H A A Veff(A) Ω − 1 ΩΠ ΩΛA H = − 2 = 1 − 2 = 1 − . (40) + + 2 − , (43) H H A 2 Veff(A) 2 2A 2 This is a direct measure of the rate of change of the Hub- where we have defined ble parameter. Using these quantities, we obtain three equivalent conditions for the Universe to be accelerating: Ω = Ωm + Ωr + ΩΛ, (44) ¨ 0 A > 0 ⇔ H < 1 ⇔ Veff(A) < 0. (41) as in standard ΛCDM cosmology. By making further as- sumptions on the size of A and preforming some straight- Now, in order to be consistent with late times probes forward analysis, we can write V in various epochs: of cosmological expansion (such as Supernovae of Type eff IA), we demand that the Friedman equation (35) reduce Ω Ω Ω−1A2 A  A  A , down to the ΛCDM form when A 1. This implies that  r Λ Π 1 2 &  −2 Ω − 1 1 ΩrA ,A2  A  A3, |ΩΠ|  1. Furthermore, to avoid a singularity in the Veff(A) ≈ − −1 the Friedmann equation for finite A > 0, we will assume 2 2 ΩmA ,A3  A  A4, 2  2 that ΩΠ > 0. Given that we recover ΛCDM for A & 1, ΩΛA ,A4  A, we expect that the other density parameters will take on (45) their concordance values [22]: where we have defined

Ω = 0.27 ± 0.04, Ω = 0.73 ± 0.04, Ω ' 8.24 × 10−5. 1/2 m Λ r " r 2 !# ΩΠ Ω − 1 (Ω − 1) (42) A1 = + + ΩΛΩr , In Figure1, we plot the Hubble parameter, effective ΩΛΩr 2 4 potential and slow-roll parameter as functions of the scale  1/4  1/3 ΩΠ Ωr Ωm factor assuming the central values of cosmological param- A2 = ,A3 = ,A4 = . (46) eters in (42). Since values of the scale factor for which ΩΛ Ωm ΩΛ Veff(A) > 0 are classically forbidden, there will be an To obtain (45), we have further assumed that early Universe “big bounce” that occurs when Veff(A) = 0 and there is no big bang singularity in our model. The replacement of the big bang with a big bounce in this Ω = O(1), ΩΛ = O(1), Ωm = O(1), 2 model is a direct consequence of the weakening of the Ωr  Ωm, ΩΠ  Ωr  Ωr, (47) gravitational field sourced by ordinary matter in the early universe (c.f. equation 25). Essentially, strong gravita- in order to ensure the hierarchy A1  A2  A3  A4. tional forces implied by high densities are mitigated by (These assumptions are all consistent with the observa- the reduction of Geff in the distant past, which allows the tional values quoted above.) We can name the various universe to escape an initial singularity. epochs (45) in analogy to the behaviour of Veff in stan- Immediately after this cosmological bounce there is a dard cosmology: phase of nearly dS early-time acceleration. The universe 0 early quasi-dS acceleration: A1  A  A2 undergoes two further transitions where Veff(A) = 0: The first transition marks when the acceleration in the early radiation domination: A2  A  A3 Universe ends and the radiation dominated epoch starts, matter domination: A3  A  A4 and the second transition occurs in the late Universe late quasi-dS acceleration: A4  A. when matter domination ends and the second acceler- ation epoch starts. The latter is consistent with the ob- In particular, if we assume Ω ≈ 1 then we find that in served late-time acceleration of the Universe. We give an the “early quasi-dS acceleration” phase example of numerical solutions of (38) for the scale factor 1/2 1/2 −1/2 in Figure2, which clearly demonstrates the existence of A ≈ exp[Ωr ΩΛ ΩΠ H0(t − t0)],A1  A  A2; a bounce in the early Universe. (48) i.e., we have exponential expansion (t0 is an integration constant). From figure1, we expect the early acceleration phase 2 This is not a necessary assumption, and it would be interesting to be preceeded by a cosmological bounce that occurs to consider the Ω < 0 case in future work. Π when Veff(A) = 0. By performing a 3-term Taylor series 6

FIG. 1. Hubble parameter H (left), effective potential Veff (centre), and slow-roll parameter H (right) as functions of scale −5 factor. Here, we have taken (Ωm, Ωr, ΩΛ) = (0.27, 8.24×10 , 0.73). There is a cosmological bounce at early times when H = 0, 2 Veff = 0, and H → −∞. This bounce is followed by a period of quasi de-Sitter acceleration when H ≈ constant, V ∝ −A , and H ≈ 0. The early time acceleration ends when Veff switches from decreasing to increasing and is followed by epochs of radiation, matter, and late time acceleration similar to ΛCDM.

FIG. 2. Numeric solutions for the scale factor A assuming −5 0 (Ωm, Ωr, ΩΛ) = (0.27, 8.24 × 10 , 0.73). All simulations show FIG. 3. Curves Veff(A) = 0, Veff(A) = 0, A = A1 and A = a bounce at time t = t0. The scale factor at the bounce A2 in the (ΩΠ,A) plane with (Ωm, Ωr, ΩΛ) = (0.27, 8.24 × −5 increases with increasing ΩΠ. 10 , 0.73). We see that for 0 < ΩΠ  1 the Veff(A) = 0 0 and A = A1 curves coincide, while the Veff(A) = 0 and A = A2 curves coincide. This is an explicit confirmation that the expansion of (39) about A = 0 and working to leading bounce occurs at A ≈ A1 and the quasi-dS inflation ends order in ΩΠ, we find that when A ≈ A2 for these parameters and 0 < ΩΠ  1.

Veff(A1) ≈ 0, 0 < ΩΠ  1, (49) where A1 is given by (46). That is, the bounce occurs where A2 is given by (46). To test these approximations, at A ≈ A1 when ΩΠ is small and positive. On the other 0 we can plot the curves Veff(A) = 0, V (A) = 0, A = A1 hand, the transition from early time acceleration to ra- eff and A = A2 in the (ΩΠ, a) plane with (Ωm, Ωr, ΩΛ) held diation domination at A ≈ A2 occurs when the potential constant. An example of such a plot is given in Figure3. switches from a decreasing to increasing function of A. Therefore, we also expect Given formulae for A1 and A2, we can estimate how 0 Veff(A2) ≈ 0, 0 < ΩΠ  1, (50) many e-folds N of exponential expansion occur after the 7 bounce: The metric and connection of the spacetime manifold are identified from various components of the gauge poten- A2 1 1 1 tial. The ensuing gravitational theory is in general rather N = ln = − ln ΩΠ + ln ΩΛ + ln Ωr A1 4 4 2 rich and complex, but we made a number of simplifying r ! 1 Ω − 1 (Ω − 1)2 assumptions to aid our analysis. For example, the full − ln + + Ω Ω . (51) theory admits manifolds with torsion and matter which 2 2 4 Λ r couples to the gauge potential in exotic ways. In this paper, we considered torsion-free solutions and retained We see from this that we can make N arbitrarily large only matter directly coupled to the metric (as in general by selecting Ω to be very small. If we take cosmological Π relativity) and matter coupled to fields la gauging special parameters as their central values in (42), then we have α conformal transformations. Our model is invariant under local conformal (Weyl) transformations. 1 ΩΠ 1 ΩΠ N ∼ 60 − ln ∼ 66 − ln . (52) When we specialized to isotropic and homogeneous 4 10−109 4 g2 YM spacetimes, we found that the contribution to the field a We also note that the Hubble scale during this early “in- equations of matter coupled to lα is highly constrained. Indeed, the tensor a encoding this contribution is com- flationary” period is also fixed by ΩΠ: αβ pletely determined by two integration constants: Λ and 1/2 1/2 −1/2 Π. We derived the Friedman equation governing the dy- Hinf ≈ Ωr ΩΛ ΩΠ H0. (53) namics and deduced that at late times the model reduces This is commonly characterized by the energy scale dur- down to the standard ΛCDM cosmology with Λ playing ing inflation: the role of the cosmological constant. It is worth not- ing that the cosmological constant in our model was not 2 2 1/4 1/4 1/4 1/4 −1/4 1/2 1/2 Einf = (3MPlHinf) ≈ 3 Ωr ΩΛ ΩΠ MPl H0 . put into the action by hand, as in ΛCDM, rather it is a (54) generated dynamically from the matter coupled to lα. Again taking central values for the usual density param- If the other constant Π in the solution for aαβ is set to −33 eters and H0 ∼ 10 eV, we find zero, we recover ΛCDM exactly for all times. However, if it is not zero there are fascinating repercussions in the  Ω −1/4  Ω −1/4 early universe. If Π > 0, the big bang of general relativ- E ∼ 1015 GeV Π ∼ 5×1017 GeV Π . inf 10−109 g2 ity is replaced by a cosmological bounce. Furthermore YM 2 (55) if 0 < Π  H0 , then the bounce is followed a period of We note the relationship between N and Einf in this quasi-dS acceleration. That is, there exists a period of model slow-roll inflation in the early universe. This inflation- ary period can be made arbitrary long by selecting Π to  E  N ∼ 60 + ln inf , (56) be arbitrarily small. The physical reason for these ef- 1015 GeV fects is that the effective Newton constant mediating the gravitational force exerted by ordinary matter (i.e. dust again assuming central values for (Ωm, Ωr, ΩΛ). and radiation) becomes small in the past, allowing for a Finally, for this model to accurately reproduce ob- bounce. served light element abundances, we require that the To summarize, we have presented a theory of gravity cosmological expansion history from big bang nucleosyn- whose cosmological solutions are free of singularities, and thesis onwards be not significantly different than that which incorporate quasi-dS epochs of acceleration in the of standard ΛCDM. This can be guaranteed if we have early and late universe. One may be concerned about the A2  ABBN, where ABBN is the scale factor at big bang naturalness of such a theory. A priori, our cosmological nucleosynthesis. Using standard formulae, this condition solutions involve one dimensionless constant appearing in can be re-written as 2 the action gYM, and two dimensionful constants of inte-  −4 gration Λ and Π. We fixed Λ by comparing to observa- −35 TBBN ΩΠ  2 × 10 , (57) tions of late time acceleration. The Yang-Mills coupling 100 keV was then fixed by requiring the late time Friedmann equa- tion have the correct dependence on the Planck mass. where T is the radiation temperature at big bang BBN Since there is a large hierarchy between the Planck and nucleosynthesis. dark energy scales, this yielded a small Yang-Mills cou- 2 −120 pling gYM ∼ 10 . In order to recover acceptable late IV. DISCUSSION time cosmology, we required Π  Λ, which implies the most “natural” nonzero value for Π is In this paper we have considered cosmological solutions 2 2 Π ∼ gYMΛ ⇒ ΩΠ ∼ gYM. (58) of a gauge theory of gravity. The action of our model re- sembles that of a Yang-Mills theory with gauge group With this choice, the early time dS-phase involves ∼ 66 SO(4,2); i.e. the conformal group of Minkowski space. e-folds of exponential expansion (which is sufficient to 8 explain the homogeneity of the cosmic microwave back- forces. This means that the dynamics of cosmological ground) at an energy scale of ∼ 5 × 1017 GeV (which perturbations may be significantly different from general implies high temperature inflation). Furthermore, this relativity, which could lead to definitive observational value of Π will yield an expansion history consistent with tests of the model; both in the late universe via observa- big bang nucleosynthesis. Therefore, just as in ΛCDM, tions of large scale structure and in the early universe via our model does involve one unnaturally small number the quantum generation of fluctuations during inflation. forced upon us by the observed hierarchy between the (Primordial perturbations in models with a cosmological Planck mass and cosmological constant; the other con- bounce have been considered in [23–27].) Finally, the role stant Π can take on a natural value and still generate an of torsion in this model is interesting at both the back- acceptable cosmological model. ground and perturbative level, and needs to be explored in greater detail. In the future, this model needs to be rigorously com- pared with observations. By comparing the predictions of the modified Friedmann equation (35) with probes ACKNOWLEDGMENTS of the expansion history (such as type IA supernove), we can obtain direct bounds on ΩΠ ∼ Π/Λ. Perhaps SR would like to thank Jonathan Ziprick for useful more importantly, as shown in [13], matter perturba- discussions. We are supported by National Sciences and tions in this model can source long-range gravitational Engineering Research Council of Canada (NSERC).

[1] B. Famaey and S. S. McGaugh, Living Reviews in Rela- 010 (2008), arXiv:0801.0905 [gr-qc]. tivity 15 (2012), 10.1007/lrr-2012-10. [16] C.-G. Huang and M.-S. Ma, Phys. Rev. D80, 084033 [2] G. Dvali, in Proceedings, Nobel Symposium 127 on String (2009), arXiv:0906.2837 [gr-qc]. theory and cosmology (2004) arXiv:hep-th/0402130 [hep- [17] N. J. Poplawski, Phys. Lett. B694, 181 (2010), [Er- th]. ratum: Phys. Lett.B701,672(2011)], arXiv:1007.0587 [3] T. Clifton, P. G. Ferreira, A. Padilla, and C. Sko- [astro-ph.CO]. rdis, Phys. Rept. 513, 1 (2012), arXiv:1106.2476 [astro- [18] N. J. Poplawski, Gen. Rel. Grav. 44, 1007 (2012), ph.CO]. arXiv:1105.6127 [astro-ph.CO]. [4] A. A. Starobinsky, Phys. Lett. B91, 99 (1980). [19] N. J. Poplawski, Phys. Rev. D85, 107502 (2012), [5] M. Bojowald, Phys. Rev. Lett. 86, 5227 (2001), arXiv:gr- arXiv:1111.4595 [gr-qc]. qc/0102069 [gr-qc]. [20] S. Desai and N. J. Pop lawski, Phys. Lett. B755, 183 [6] K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 64, (2016), arXiv:1510.08834 [astro-ph.CO]. 866A882ˆ (1980). [21] S. M. Carroll, Spacetime and geometry: An introduc- [7] E. A. Ivanov and J. Niederle, Phys. Rev. D 25, 976 tion to general relativity (Addison-Wesley, San Francisco, (1982). USA, 2004). [8] R. Utiyama, Phys.Rev. 101, 1597 (1956). [22] B. Carroll and D. Ostlie, An Introduction to Modern As- [9] T. W. B. Kibble, J. Math. Phys. 2, 212 (1961). trophysics (Pearson Addison-Wesley, 2007). [10] J. T. Wheeler, Phys.Rev. D44, 1769 (1991). [23] Y.-F. Cai and X. Zhang, JCAP 0906, 003 (2009), [11] J. S. Hazboun and J. T. Wheeler, Class.Quant.Grav. 31, arXiv:0808.2551 [astro-ph]. 215001 (2014), arXiv:1305.6972 [gr-qc]. [24] Y.-F. Cai, T.-t. Qiu, J.-Q. Xia, and X. Zhang, Phys. [12] J. T. Wheeler, Phys.Rev. D90, 025027 (2014), Rev. D79, 021303 (2009), arXiv:0808.0819 [astro-ph]. arXiv:1310.0526 [gr-qc]. [25] Y.-F. Cai, D. A. Easson, and R. Brandenberger, JCAP [13] J. Gegenberg, S. Rahmati, and S. S. Seahra, Phys. Rev. 1208, 020 (2012), arXiv:1206.2382 [hep-th]. D93, 064025 (2016), arXiv:1505.06058 [gr-qc]. [26] J.-Q. Xia, Y.-F. Cai, H. Li, and X. Zhang, Phys. Rev. [14] K.-F. Shie, J. M. Nester, and H.-J. Yo, Phys. Rev. D78, Lett. 112, 251301 (2014), arXiv:1403.7623 [astro-ph.CO]. 023522 (2008), arXiv:0805.3834 [gr-qc]. [27] Y.-F. Cai, Sci. China Phys. Mech. Astron. 57, 1414 [15] C.-G. Huang, H.-Q. Zhang, and H.-Y. Guo, JCAP 0810, (2014), arXiv:1405.1369 [hep-th].