We construct an effective action for gravity in which all homogeneous solutions are

arXiv:gr-qc/9210014v1 24 Oct 1992Brown-HET-876September 1992A NONSINGULAR UNIVERSE⋆Robert H. BrandenbergerPhysics DepartmentBrown UniversityProvidence, RI 02912, USAABSTRACTWe construct an effective action for gravity in which all homogeneous solutions arenonsingular1).In particular, there is neither a big bang nor a big crunch.The actionis a higher derivative modification of Einstein’s theory constructed in analogy to how theaction for point particle motion of particles in special relativity is obtained from Newtonianmechanics.⋆Invited talk at International School of Astrophysics “D. Chalonge”, 2nd course, 6-13 September 1992,Erice, Italy, to be published in the proceedings (World Scientific, Singapore, 1992).

1. IntroductionThe singularity theorems of general relativity2) prove that space–time manifolds in gen-eral relativity are geodesically incomplete.

The key assumptions which are used in the proofsof these theorems are that the action of gravity is an unmodified Einstein action, and thatmattter satisfies the energy dominance condition ǫ > 0 and ǫ + 3p ≥0, where ǫ and p areenergy density and pressure respectively.Whereas the singularity theorems do not provide any general information about thenature of the singularity, well known examples of space–time show that at a singularitytypically some of the curvature invariants R , Rµν Rµν , Rα β γ δ Rα β γ δ , . .

. blow up.

Thesesingularities should be viewed as a breakdown of general relativity at high curvatures.Quantum gravity and alternative fundamental theories of all four forces, such as stringtheory, have been invoked as ways towards a solution of the singularity problems. As yetthese avenues have not been developed far enough to provide a solution.

We believe, however,that a new fundamental theory which includes gravity will solve the singularity problem. Oneway towards realizing a singularity free theory is to guess an effective action for gravity whichcontains no singularities.

Constructing such a theory is what we attempt in this work.As a preliminary, recall that the well known and successful theories of special relativity(SR) and quantum mechanics are based on inequalities v < c and ∆x∆p ≥¯h respectively.Hence, the hope arises that it might be possible to construct a new theory of gravity based onan inequality involving Newton’s constant G. For example, in a theory with a fundamentallength ℓf (i.e. ℓ≥ℓf for all lengths ℓ), all curvature invariants are automatically bounded(R ≤ℓ−2f, Rµν Rµν ≤ℓ−4f, etc.).

However, a fundamental length is incompatible with acontinuum theory of space and time, and thus we will attempt to realize the constraints onthe curvature invariants directly..Our goal is to construct a theory in which all curvature invariants are bounded andin which space–time is geodesically complete.This formidable problem can be reducedsubstantially by invoking the “Limiting Curvature Hypothesis”3), according to which onei) finds a theory in which a small number of invariants is explicitly bounded, andii) when these invariants take on their limiting values, a definite nonsingular solution(namely de Sitter) is taken on.2

As a consequence of the limiting curvature hypothesis, automatically all invariants arebounded, and space–time is geodesically complete in its asymptotic regions.The limiting curvature hypothesis has interesting consequences for Friedmann modelsand for spherically symmetric vacuum space–times4). A collapsing Universe will not reach abig crunch, but will end up as a contracting de Sitter Universe (k = 0) or a de Sitter bounce(k = 1) followed by re-expansion (see Fig.

1). For a spherically symmetric vacuum solution,there would be no singularity inside the Schwarzschild horizon; instead, a de Sitter Universewill be reached when falling through the horizon towards large curvature (see Fig.

1.).Fig. 1: Penrose diagrams for a collapsing Universe (left) and for a black hole (right) inEinstein’s theory and after implementing the limiting curvature hypothesis (bottom).Wavy lines denote a singularity (in the case of the collapsing Universe the big crunch),the symbols C, DS and E stand for collapsing phase, de Sitter phase and expandingphase respectively, and H denotes the Schwarzschild horizon.3

2.ConstructionIn order to realize the limiting curvature hypothesis, we must abandon at least one ofthe assumptions of the Penrose–Hawking theorems. Unlike in inflationary cosmology5) wedo not invoke “strange” matter which violates the energy dominance condition.

Instead, wedrop the assumption that gravity is described by a pure Einstein action.The theory discussed here is a higher derivative modification of Einstein gravity. It isreasonable to consider such modifications since the Einstein theory is known to break downat high curvatures – based on perturbative quantum gravity calculations, quantum fieldtheory effects in curved space–time, and on taking low energy limits of fundamental theoriesof all forces such as string theory.Most higher derivative gravity models have much worse singularity properties than theEinstein theory.

Hence, it is a nontrivial task to construct a model which has better prop-erties. As an added bonus, the construction which leads to our nonsingular Universe is wellmotivated in analogy to how the action for particle motion in special relativity emerges fromthe point particle action in Newtonian mechanics.Special relativity is a theory in which point particle velocities v are bounded.

Startingfrom Newtonian mechanics (in which v is unbounded) for which the point particle action isS = mZdt 12 ˙x2 ,(2.1)m being the particle mass, the action for special relativity can be obtained6) using a Lagrangemultiplier constructionS = mZdt12 ˙x2 + ϕ ˙x2 −V (ϕ). (2.2)Provided that V (ϕ) increases no faster than ϕ at large ϕ, the quantity which couples to ϕ,namely ˙x2, is automatically bounded, as follows from the variational equation with respectto ϕ˙x2 = ∂V∂ϕ .

(2.3)In order to recover the correct Newtonian limit at low velocities, V (ϕ) must be proportional4

to ϕ2 as ϕ →0. Thus, the conditions on V (ϕ) areV (ϕ) ∼ ϕ|ϕ| →∞ϕ2ϕ →0 .

(2.4)Up to factors of 2, the simplest potential which satisfies (2.4) isV (ϕ) =2ϕ21 + 2ϕ(2.5)Eliminating the Lagrange multiplier ϕ via (2.3) and substituting into (2.2), the action of apoint particle in special relativityS =Zdtp1 −˙x2(2.6)results.Our idea1) is to imitate the above construction in gravity. Starting with Einstein’s theoryof general relativity with actionS =Zd4x √−g R(2.7)and unbounded Ricci scalar curvature, we construct a new gravity theory by introducing aLagrange multiplier ϕ1, with potential V1(ϕ1) which couples to R, the quantity we wish tobound:S =Zd4x √−g [R + ϕ1R + V1(ϕ1)] .

(2.8)The potential V1(ϕ1) must satify the same asymptotic properties as given in (2.4).However, the action (2.8) is not sufficient. In order to obtain a nonsingular Universe,we must implement the limiting curvature hypothesis.

This is achieved once again by usingthe Lagrange multiplier technique. At this point we restrict our attention for the momentto homogeneous and isotropic space–times.Consider the invariantI2 = 4Rµν Rµν −R2 .

(2.9)This invariant is positive semidefinite, and vanishes only if space–time is de Sitter. Hence,we will implement the limiting curvature hypothesis by forcing I2 to zero at high curvatures.5

We chose the actionS =Zd4x √−ghR + ϕ1R + ϕ2pI2 + V1(ϕ1) + V2(ϕ2)i. (2.10)Provided thatV2(ϕ2) ∼(const|ϕ2| →∞ϕ22ϕ2 →0(2.11)then for |ϕ2| →∞space–time becomes de Sitter, and the low curvature limit of the theoryagrees with general relativity.By construction, a theory with action (2.10) becomes de Sitter at large ϕ2.

It remainsto be shown that there are no singularities for finite values in the ϕ1/ϕ2 phase space. Toshow this, we need a specific model.3.

Specific ModelAs the most simple realization of a nonsingular Universe we consider the action1)S =Zd4x √−g(1 + ϕ1)R −ϕ2 +6√12 ϕ1I1/22+ V1(ϕ1) + V2(ϕ2)(3.1)withV1(ϕ1) = 12H20ϕ211 + ϕ11 −ℓn(1 + ϕ1)1 + ϕ1V2(ϕ2) = −√12 H20ϕ221 + ϕ22(3.2)Apart from the logarithmic term in V1, the above potentials are the most simple ones whichsatisfy the asymptotic conditions (2.4) and (2.11). It was necessary1) to add the next leading(logarithmic) term in V1 in order to prevent trajectories from reaching ϕ1 →∞for |ϕ2| < 1.The general variational equations which follow from (3.1) are rather complicated (seeRef.

7). However, when applied to a collapsing Universe with metricds2 = −dt2 + a2(t)dx2(3.3)6

and Hubble parameterH = ˙aa < 0 ,(3.4)the variational equations become simple1):H2 = 112 V ′1 ,(3.5)˙H = −1√12 V ′2(3.6)and3(1 −2ϕ1) H2 + 12 (V1 + V2) =6√12 H( ˙ϕ2 + 3Hϕ2) . (3.7)From (3.5) it follows that ϕ1 > 0, from (3.6) that |ϕ2| →∞is equivalent to de Sitter space,and (3.7) can be combined with the time derivative of (3.5) and with (3.6) to yielddϕ2dϕ1=V ′′1V ′1 V ′2−14(1 −2ϕ1) V ′1 + 12 (V1 + V2) +32√12 V ′1 ϕ2,(3.8)an equation from which the trajectories of this dynamical system in ϕ1/ϕ2 phase space canbe read off.The system of equations (3.5, 3.6 & 3.8) must be analyzed to show that there are nosingular solutions.

The asymptotic regions |ϕ1| , |ϕ2| ≪1 and |ϕ1| , |ϕ2| ≫1 can be analyzedanalytically1). It can be seen that there are two types of solutions: periodic solutions aboutMinkowski space (ϕ1 = ϕ2 = 0) and solutions which start and end at |ϕ2| = ∞, i.e.

in deSitter space (see Fig. 2).

It can be shown numerically7) that there are indeed no singularpoints for finite values of phase space and that the trajectories connect in a way which can beguessed from the analytical analysis of the asymptotic regions. Thus, we have demonstratedthat all solutions are nonsingular.So far, only vacuum solutions of our new gravitational theory have been discussed.

It iseasy to include matter in the analysis by considering the actionSfull = S + Sm ,(3.9)where S is the gravitational action of (3.1), and Sm is the action for matter in the presenceof the metric gµν. We have investigated7) the model obtained by adding hydrodynamicalmatter with an equation of state p = wρ and w = 0 (cold matter) or w = 1/3 (radiation).7

The interesting result of our analysis7) is that for |ϕ2| →∞the trajectories are un-changed when adding matter, even though for a contracting spatially flat Universe the energydensity is increasing exponentially.The only change for a spatially closed Universe is that the contracting de Sitter phase isreplaced by a de Sitter bounce.In conclusion, we have presented a model in which all homogeneous and isotropic solu-tions are nonsingular.8

4.Wild SpeculationsSince matter does not change the evolution of space–time at large curvatures, the grav-itational interactions are asymptotically free, i.e. the effective coupling Geffof matter togravity tends to zero.

This is a first very nice property of our model.Secondly, when applied to an expanding Universe, our theory implies that it has emergedfrom an initial de Sitter phase. Thus, an inflationary period is obtained without assumingthe presence of matter violating the energy dominance condition.

This result, however, isno surprise, since it is well known8) that higher derivative gravity models lead to inflation.Let us now combine the first two results and consider the quantum generation of densityperturbations in the initial de Sitter phase. These perturbations are streched by inflationand may become the seeds for structures in the Universe.

In scalar field driven inflationarymodels, the magnitude of the scalar metric fluctuations is too large without requiring that aparticle physics parameter (coupling constant of a λϕ4 interaction term or a mass scale m ina theory of chaotic inflation with potential 12m2 ϕ2) be artificially small. However, since themagnitude of these perturbations is proportional to Geff, it is conceivable that in our modelthere will be no fine tuning for inflation.Next, let us consider an application to black holes.

For black holes in Einstein’s theoryof general relativity, Hawking radiation leads to its evaporation with ever increasing speed.However, the strength of Hawking radiation is proportional to Geff. Hence, in our theoryHawking radiation may automatically shut offas the black hole mass decreases towards itscritical value Mcrit, which is in turn determined by when curvature invariants like C2 reachtheir limiting values (H40).A consequence of the above is that black hole remnants will remain.

Hence, there willbe no loss of quantum coherence in the presence of black holes (when calculated in thesemiclassical approximation). Neither will there be global charge violation by black holes.5.

Extension to an Anisotropic UniverseHopefully the reader is at this point persuaded that it is worth while to explore ourtheory further and see if the wild speculations mentioned in the previous section can indeedbe realized.9

Fig. 2: Phase diagram for the solutions of (3.8), arrows pointing in the direction ofincreasing time.

As can be shown using (3.6), all asymptotic solutions are de Sitter.10

As a first step, we have explored9) whether our theory can damp out anisotropy at highcurvatures, such that asymptotically also an anisotropic Universe will lead to de Sitter space.Obviously, the action (3.1) with invariant I2 given by (2.9) is insufficient, since I2 doesnot depend on the anisotropy. However, we can easily improve the prospects by changing I2toI2 = 4Rµν Rµν −R2 + C2(5.1)where C2 = Cµνρσ Cµνρσ and Cµνρσ is the Weyl tensor.

We maintain the form of the action(3.1).Based on our previous investigations, we should expect to be able to achieve our goal.As ϕ2 →∞, the invariant I2 is again driven to zero. This will imply (in cases when C2 ≥0) bothC2 = 0(5.2)and4Rµν Rµν −R2 = 0 .

(5.3)The condition (5.2) implies decrease in anisotropy, and then (5.3) tells us that the asymptoticsolution (which is homogeneous) will be de Sitter space.To verify the above claims, we have considered9) the simplest anisotropic Universe withmetricgµν =−1a2eβ(t)a2eβ(t)a2e−2β(t). (5.4)The variational equations can be derived using a convenient trick: we replace the time–timecomponent g00 by −α(t)2, insert the metric into (3.1) and vary with respect to α(t) , a(t) , β(t) , ϕ1(t)and ϕ2(t).

Still, the resulting equations are rather complicated.It must be shown that for |ϕ2| →∞the anisotropy tends to zero, i.e. ˙β →0.

This canbe done by picking out the terms which dominate in the equations of motion in the limit|ϕ2| →∞. As demonstrated in Ref.

9, this is indeed the case.11

ConclusionsWe have presented an effective action for gravity based on a higher derivative mod-ification of Einstein’s theory of general relativity in which all homogeneous solutions arenonsingular. All corresponding space–time manifolds are geodesically complete and eitherapproach de Sitter space asymptotically or oscillate about Minkowski space.

We have spec-ulated that in our theory also singularities inside the black hole horizon might be avoided.AcknowledgementsThe results and ideas presented in this article are based on key ideas by and joint workwith V. Mukhanov. I am also grateful to my collaborators M. Mohazzab, A. Sornborger andM.

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