Universal scattering laws for quiescent bouncing cosmology
CCERN-TH-2020-101
Universal scattering laws for bouncing cosmology
Bruno Le Floch, ∗ Philippe G. LeFloch, † and Gabriele Veneziano ‡ Philippe Meyer Institute, Physics Department, École Normale Supérieure, PSL Research University, Paris, France. Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Sorbonne Université, Paris, France. CERN, Theory Department, Geneva, Switzerland & Collège de France, Paris, France. (Dated: August 2020)Bouncing cosmologies can arise from various gravity theories. We model them through a singularityscattering map, as we call it, relating large scale geometries before and after the bounce. Byclassifying all suitably local maps we uncover universal laws (scaling of Kasner exponents, canonicaltransformation of matter). We study the singularity scattering map for Bianchi I bounces in stringtheory, loop quantum cosmology and modified matter models: our classification then determineshow general spatial inhomogeneities and anisotropies (without BKL oscillations) are transmittedthrough bounces.
INTRODUCTION
Toward a unification of bouncing scenarios.
An impor-tant class of proposals to resolve the initial singularityproblem in Cosmology are bouncing scenarios where theBig Bang is replaced by a Big Bounce, and in which theUniverse undergoes a contracting phase followed by theexpanding phase that includes the present time. Suchscenarios have been constructed through various modifiedgravity theories, matter violating energy conditions, orquantum gravity effects in string theory and loop quan-tum cosmology. In this Letter, we revisit this old problemby abstracting away all microscopic details of the model.
Formulation and classification.
We propose the newnotion of singularity scattering map , which extractsthe essence of the bounce by focusing on the Einstein equa-tions themselves, imposed before and after the bounce.The relevant scattering map in a given scenario is deter-mined, in a second stage only, from an extended model ofgravity. Our main contribution is a characterization of allpossible junction conditions that may arise from such aphysical model. For proofs in d = 3 space dimensions andapplications to plane-symmetric cyclic spacetimes, see ourpaper [17] and Figure 1 below. Interestingly, our stand-point naturally leads us to distinguish between universaland model-dependent aspects of junction relations. Universality.
Our classification uncovers universalscattering laws for the geometry and matter across thebounce. We prove that Kasner exponents before and afterthe bounce obey the universal scaling relation( | g | / ˚ K ) after = γ ( | g | / ˚ K ) before (1)(for a constant γ ∈ R ), with g the spatial metric insynchronous gauge, | g | / its volume factor, and ˚ K thetraceless part of the extrinsic curvature. We also provethat matter, modeled away from the bounce as a mini-mally coupled and massless scalar field φ , undergoes acanonical transformation. Essentially, as explicited in (9),Φ : ( π φ , φ ) before ( π φ , φ ) after preserves dπ φ ∧ dφ (2) where π φ is the momentum conjugate to φ . In addition,the metric after the bounce is entirely determined from Φ. Model-dependence.
We then study the singularity scat-tering maps associated with the pre Big Bang scenario,loop quantum cosmology, and some modified matter mod-els. The map for a given model encapsulates microscopicphysics constrained by the universal scattering laws (1)and (2). It is determined by calculations in homoge-neous (but anisotropic) Bianchi I universes, yet describesbounces with arbitrary spatial inhomogeneities. Through-out, our focus is on quiescent cosmological singularities, aclass first identified by Barrow [4] (see also [1, 7]) so thatthe BKL oscillating behavior [5] is suppressed.
Perspectives.
We do not attempt here to review thevast subject of bouncing cosmologies and we focus on theissues mentioned above only. For further material we referto the review papers by Ashtekar [2], Brandenberger andPeter [6], and Gasperini and Veneziano [10].
SINGULARITY SCATTERING MAPS
Bounce hypersurface.
We introduce here our notion ofsingularity scattering maps for quiescent singularities in a( d + 1)-dimensional spacetime ( d (cid:62)
2) with a scalar field.We focus on bouncing scenarios in which corrections toEinstein gravity are negligible away from the bounce locus,which we model as a spacelike singularity hypersurface H (cf. [17] for timelike singularities), but are essential at(small) time scales t b around H . Provided spatial inho-mogeneities are mild (see below) the spacetime, on eachside, is well described at larger time scales by a solutionof Einstein equations which is singular along H , and thesetwo solutions are connected using a suitable junction. ADM formalism.
We work with a Gaussian (or syn-chronous gauge) foliation in which the metric reads g ( d +1) = − dt + g ( t, x ), the bounce hypersurface being atthe proper time t = 0. Each constant-time hypersurfaceis endowed with a Riemannian metric g = g ab and anextrinsic curvature K = K ba such that K ac = K ba g bc issymmetric. Here, a, b, . . . are local coordinate indices on a r X i v : . [ g r- q c ] A ug each time slice. We consider the ADM formulation of theEinstein equations for the evolution of a self-gravitatingmassless scalar field φ : − ∂ t φ + Tr( K ) ∂ t φ = − ∇ a ∇ a φ,∂ t g ab + 2 K ab = 0 ,∂ t K ba − (Tr K ) K ba = R ba − ∂ a φ∂ b φ, (Tr K ) − Tr( K ) − ( ∂ t φ ) = − R + ∂ a φ∂ a φ, ∇ a K ab − ∂ b (Tr K ) + ∂ t φ∂ b φ = 0 , (3)which consists of three evolution and two constraint equa-tions. (We normalize the speed of light to c = 1 andNewton constant to 8 πG = 1.) Data on a bounce hypersurface.
Near the singularity,asymptotic profiles describing the main behavior of a solu-tion are found by neglecting spatial derivatives comparedto t derivatives, namely neglecting right-hand sides of (3),and then solving the resulting equations. This family of asymptotic profiles (denoted by a ∗ subscript) reads g ±∗ ( t ) = e | t/t ∗ | ) k ± g ± , K ±∗ ( t ) = − (1 /t ) k ± ,φ ±∗ ( t ) = φ ± ln | t/t ∗ | + φ ± , (4)parametrized by singularity data ( g ± , k ± , φ ± , φ ± ) pre-scribed on each side ± = sgn( t ) of the bounce. Theasymptotic metric is expressed in terms of the matrixexponential of ( k ± ba ), and t ∗ > k ± = 1 together with an asymptotic form ofthe Hamiltonian and momentum constraints, that is,1 − k ± ab k ± ba = ( φ ± ) , ∇ ± a k ± ab = φ ± ∂ b φ ± , (5)where ∇ ± is the connection associated with g ± . In ourcontext, a data set ( g ± , k ± , φ ± , φ ± ) is called quiescent if Kasner exponents k ± i (eigenvalues of k ± ) are positive.The trace and Hamiltonian constraints read P i k ± i = 1and 1 − P i ( k ± i ) = ( φ ± ) , respectively. Asymptoticprofiles are generally not exact solutions. Validity of asymptotic profiles.
Asymptotic profilesare defined for all times ± t ∈ (0 , ∞ ), but are only goodapproximations in some range t b (cid:28) | t | (cid:28) t s : indeed, cur-vature generically blows up as t → ± so that corrections(e.g. higher-curvature corrections) become important at asmall time | t | ’ t b , while the spatial derivatives neglectedin (3) stop being negligible at some large time scale t s since they decay slower than time derivatives at | t | → ∞ .Our assumption of mild spatial inhomogeneities isthat t b (cid:28) t s . Equivalently, we require that at time t b thespatial derivative terms in (3) such as ∇ a ∇ a φ/φ , ∂ a φ∂ a φ or R are parametrically smaller than the typical scale 1 /t of the left-hand sides, so that they remain smaller on sometime interval ( t b , t s ). Under this assumption we retrievethe data for the asymptotic profile as the (approximately FIG. 1. Cyclic spacetime arising from colliding plane gravi-tational waves [17], depicted in null coordinates (the futureis up). Left: the height of spacetime “bubbles” is the areaelement A in plane-symmetry orbits. Right: singular locus A = 0 across which we apply a simple junction condition( g + , k + , φ +0 , φ +1 ) = ( e k − − /d ) g − , k − , φ − , φ − + φ − ). constant for t b (cid:28) | t | (cid:28) t s ) values( g ± , k ± , φ ± , φ ± ) (6):= (cid:0) | t/t ∗ | tK g, − tK, t∂ t φ, φ − t ln | t/t ∗ | ∂ t φ (cid:1) t b (cid:28)| t |(cid:28) t s . In the idealized cases t b = 0 (singular bounce) or t s = ∞ (spatially homogeneous case) the singularity data can bedefined as t → ± or t → ±∞ limits of (6), respectively. The new notion.
By construction, all slices of the foli-ation are diffeomorphic to the t = 0 slice H . We denoteby I ( H ) the set of all singularity data ( g ± , k ± , φ ± , φ ± ),obeying the conditions (5). By definition, a singularityscattering map is a local diffeomorphism-covariant map S : I ( H ) → I ( H ). General covariance and locality ensurethat S is characterized by its effect on any small ball, sothe notion of singularity scattering map does not dependon the bounce hypersurface H . We introduce the junc-tion conditions ( g + , k + , φ +0 , φ +1 ) = S ( g − , k − , φ − , φ − )associated with a given singularity scattering map S andrelating asymptotic data (6). An example of map is givenas part of Figure 1. Mathematical advances.
The existence of solutions tothe Einstein equations asymptotic to quiescent profiles (4)and satisfying the junction conditions is proven in thecompanion paper [17] based on the earlier work [1, 7].We also refer to [13–16] for recent progress on the theoryof weak solutions with singularities. Our definition isa generalization to singularity hypersurfaces of Israel’sjunction conditions [11] for hypersurfaces across whichthe metric remains regular. Our junction conditions arereminiscent of kinetic relations for phase boundariesin fluid dynamics and material science [19, 15].
SELECTED EXAMPLES OF BOUNCES
Reduction to Bianchi I.
We now exhibit the two fea-tures (1) and (2) for singularity scattering maps of severalmodels (pre Big Bang, modified matter, etc.) in spatiallyhomogeneous bounces. As we argue in the next section,these laws can also be derived model-independently froman ultralocality assumption, without spatial homogeneity.For now, though, we work with a ( d (cid:62)
2) Bianchi I metric g ( d +1) = − dt + ω ( t ) /d P di =1 e α i ( t ) dx i dx i , (7)with anisotropic stress parameters α i summing to zeroand volume factor ω := | g | / . Asymptotic profiles.
As explained before (6), spatialhomogeneity means that t s = ∞ namely the bounce iswell-described for all | t | (cid:29) t b by the asymptotic profiles (4)(from here on we normalize t ∗ = 1), which are exactBianchi I solutions to Einstein equations with a free scalarfield. Explicitly, in the notation (7) we consider bouncesthat are asymptotic (at t → ±∞ ) to ω = ± ω ± ( t − t ± ) , φ = φ ± ln | t − t ± | + φ ± , (8) α i = ( k ± i − d ) ln | t − t ± | + ν ± i , ( φ ± ) + | k ± | = 1for some constants ( t ± , ω ± , k ± i , ν ± i , φ ± , φ ± ) such that P i ν ± i = 0, the Kasner exponents k ± i (eigenvalues of k ± )sum to 1, and | k ± | = Tr( k ± ) . We define Kasner radii r ± = r ( φ ± ) = q − dd − ( φ ± ) = q dd − Tr(˚ k ± ) ∈ [0 , k ± = k ± − d is the traceless extrinsic curvature.We are interested in the map that relates parametersdescribing the two limits. Invariance under time transla-tions and coordinate redefinitions of each x i ensures that t − , ln ω − , ν − i appear precisely as shifts of t +0 , ln ω +0 , ν + i , re-spectively, so ( t +0 − t − , ω +0 /ω − , ν + i − ν − i , k + i , φ +0 , φ +1 ) onlydepend on ( k − i , φ − , φ − ). For simplicity, we focus here on( ω +0 /ω − , k + i , φ +0 , φ +1 ) and do not discuss the proper timeoffset and metric scale factors. Universal scattering laws.
The first scattering law (1)that we will exhibit in concrete bounces translates, inBianchi I notations, to ω +0 ( k + i − d ) = γω − ( k − i − d ) forsome γ ∈ R . In particular, ω +0 r ( φ +0 ) = | γ | ω − r ( φ − ).Consider next the map Φ : ( k − , φ − , φ − ) ( φ +0 , φ +1 )describing scalar fields. The second scattering law (2)states that Φ is a canonical transformation at fixed ˚ k − /r − ,in that it preserves the volume form d (cid:0) φ /r ( φ ) (cid:1) ∧ dφ = dφ ∧ dφ /r ( φ ) up to the sign (cid:15) = sgn γ :det (cid:18) ∂ φ − ( φ +0 /r + ) ∂ φ − φ +1 ∂ φ − ( φ +0 /r + ) ∂ φ − φ +1 (cid:19) = (cid:15)∂ φ − (cid:16) φ − r − (cid:17) = (cid:15) ( r − ) . (9) Pre Big Bang scenario.
Our first concrete model isa singular bounce inspired from string theory [22, 9, 10],described by string frame fields φ SF , g ( d +1)SF obeying suit-ably truncated metric-dilaton equations. Bianchi I so-lutions related by scale-factor duality are glued along t SF = 0, assuming that higher derivative and/orhigher loop corrections resolve the singularity: they are φ SF = ln | g ( d +1)SF | / − ln | t SF | and g ( d +1)SF = − dt + - - - φ +0 /r + φ +1 FIG. 2. Pre Big Bang canonical transformation Φ ( d = 3, β + = − β − , u + = u − ). It preserves volume d ( φ ± /r ± ) dφ ± , asseen in this plot of equally-spaced constant-( φ − /r − ) verticallines and constant- φ − curved lines. P di =1 e u i ± | t SF | β i ± dx i dx i on both sides ± t SF >
0. Theconstants u i ± , β i ± obey P i β i ± = 1 and each β i + = β i − .Only β ± and differences u i + − u i − are coordinate-invariant;they depend on β − and how the singularity is resolved.The Einstein frame metric g ( d +1) = e − φ SF / ( d − g ( d +1)SF ,proper time t , and canonically normalized scalar φ = φ SF / √ d − ω ± ( k ± i − d ) = β i ± − d P j β j ± . Among the 2 d choicesof β + allowed by scale-factor duality, β + = − β − gives aninteresting junction, with ω +0 ( k + i − d ) = − ω − ( k − i − d ), φ +0 = − (cid:0) √ d − d + 1) φ − (cid:1)(cid:14)(cid:0) d + 1 + d √ d − φ − (cid:1) , (10)and φ +1 = ( r + /r − ) φ − + f ( β − ) with a function f thatdepends on how the singularity is resolved. Interestingly,regardless of f both laws (1) and (2) are obeyed. Thecanonical transformation Φ is depicted in Figure 2.All other sign choices (except the trivial β + = β − )violate these laws. Our general results in the next sec-tion prove that the corresponding junction conditionsdo not extend to inhomogeneous spacetimes (specifically,applying the transformation pointwise would violate themomentum constraint). Modified gravity and loop quantum cosmology.
Bothin loop quantum cosmology [3, 23] and in quite generalmodified gravities [8] (Brans–Dicke theory, kinetic gravitybraiding, mimetic gravity, etc.), the densitized shear ˚ K √ g is continuous (up to a sign) across Bianchi I bounces. Toderive this, the authors of [8] assumed that modificationsof gravity are encapsulated in an effective stress-tensor,preserve spatial rotation invariance, and are strong enoughto lead to a bounce but are negligible away from it.This is precisely our first universal scattering law (1)(with γ = − withoutany symmetry assumption. It would be very inter-esting to determine the precise scattering maps for thesemodels and check our second scattering law (2) directly.
Bounces with modified matter.
Consider now Einsteingravity coupled to a scalar field with Lagrangian L ( φ, X )where X = −|∇ φ | = ˙ φ . It is beyond the scope of this - -
10 0 10 201020 ω/ω − t/t b - - - ln( ω/ω − ) φ FIG. 3. Bianchi I symmetric modified matter bounces withLagrangian L = ˙ φ − | ˙ φ | e − φ / /t b + e − φ /t for fixed t − , φ − ,and ω − (normalized to 1). Each color corresponds to one valueof φ − , which affects t → + ∞ asymptotics ω ’ ω +0 ( t − t +0 ) and φ ’ φ +0 ln ω + ( φ +1 − φ +0 ln ω +0 ) manifest in the two plots. Letter to analyse which specific models lead to bouncingsolutions (exemplified in Figure 3); such bounces arisewith ekpyrotic matter [12], ghost condensates, Brans–Dicke theory in Einstein frame, etc. For our purposes,Bianchi I solutions should asymptote to free scalar ones (8)at t → ±∞ , where X → | φ | → ∞ . We thus demand L ’ X/ S = R (cid:0) L ( φ, ˙ φ ) − d − d ˙ ω /ω + P i ˙ α i (cid:1) ω dt .As observed in [8], the equation of motion ∂ t ( ω ˙ α i ) = 0for α i states that λ i = ω ˙ α i are constants so their t → ±∞ limits ± ω ± ( k ± i − d ) coincide. This proves in this settingour first scattering law (1) with γ = −
1, for any modifiedmatter Lagrangian L that exhibits bounces.Next, we switch to the Hamiltonian formalism withmomenta π φ = 2 ω ˙ φ∂ X L , π ω = − d − d ˙ ω/ω , π i = ω ˙ α i con-jugate to φ, ω, α i . By Liouville’s theorem, the symplecticform $ = dπ φ ∧ dφ + dπ ω ∧ dω + dπ i ∧ dα i is time-invariantso its t → ±∞ limits coincide. The asymptotics L ’ X/ $ t →±∞ = ± d (cid:0)(cid:0) dd − | λ | (cid:1) / φ ± /r ( φ ± ) (cid:1) ∧ dφ ± + dλ i ∧ dν ± i , and these limits must coincide. At fixed λ , this means ± d ( φ ± /r ± ) ∧ dφ ± are equal, so the map Φ is a canonicaltransformation as stated in (9) with (cid:15) = sgn γ = −
1. Thisestablishes the second scattering law (2) for modified-matter bounces. One can check that φ +0 , φ +1 only dependon φ − , φ − , and the scattering map takes the explicitform (12) given below in our model-independent analysis. UNIVERSALITY AND MODEL-DEPENDENCE
Classification of singularity scattering maps.
As ob-served in the analysis of quiescent cosmological singu-larities in [5, 1, 7], spatial derivatives can be neglectednear a singularity, so that each spatial point undergoes an(almost) independent evolution in time. When describ-ing a bounce as the junction of two singular solutions toEinstein equations along the bounce hypersurface H , it isnatural to assume that the same “ultralocality” property holds through the bounce. Namely, we focus on ultralo-cal scattering maps, as we call them, for which thevalue of ( g + , k + , φ +0 , φ +1 ) at a point x of H depends on( g − , k − , φ − , φ − ) at the same point but is independent of(spatial) derivatives thereof.This simple postulate has far reaching consequences,leading to a model-independent classification of singularityscattering maps: any ultralocal scattering map iseither an anisotropic map S aniΦ ,γ (12) or an isotropicmap S iso∆ ,ϕ,(cid:15) (13). We present here the classification in d (cid:62) d = 3 proof [17]. Proof sketch.
By general covariance the scalars read( φ +0 , φ +1 ) = Φ( φ − , φ − , χ m ) in terms of scalar invariants χ m := Tr(˚ k − /r − ) m , (cid:54) m (cid:54) d . Likewise the tensors˚ k + and ln( g + ( g − ) − ) are linear combinations of (˚ k − ) n ,0 (cid:54) n < d . A calculation then gives ∇ + a ˚ k + ab = Ω − ∇ − a (cid:0) Ω˚ k + ab (cid:1) − X b / , (11)where X b is a sum of (scalar) ∂ b (scalar) terms andΩ = p | g + | / | g − | . We want the momentum constraint ∇ ± a ˚ k ± ab = φ ± ∂ b φ ± on the “ − ” side to imply the “+” one.This requires (1) to hold, namely Ω˚ k + = γ ˚ k − , so that theright-hand side of (11) reduces to (scalar) ∂ b (scalar) terms:any other choice would involve ∇ − a ((˚ k − ) n ) ab , n >
1, henceall derivatives of ˚ k − . Tracking precisely (scalar) ∂ b (scalar)terms yields solvable differential equations for the scalarcoefficients σ n in ln( g + ( g − ) − ) = P n σ n (˚ k − /r − ) n , whichimply our second scattering law (2) in the form (9). Anisotropic ultralocal scattering.
Scattering maps forwhich γ = 0 are characterized by the canonical trans-formation Φ obeying (9) with (cid:15) = sgn γ . Explicitly, S aniΦ ,γ : ( g − , k − , φ − , φ − ) ( g + , k + , φ +0 , φ +1 ) reads( φ +0 , φ +1 ) = Φ( χ m , φ − , φ − ) , ˚ k + = (cid:15) ( r + /r − )˚ k − , (12) g + = (cid:12)(cid:12)(cid:12) γr − r + (cid:12)(cid:12)(cid:12) d exp (cid:18) ξ ˚ k − χ r − + d − X n =2 σ n (cid:18)(cid:16) ˚ k − r − (cid:17) n − χ n +1 ˚ k − χ r − (cid:19)(cid:19) g − , where σ n = ( ∂ χ n +1 ξ + 2 (cid:15) ( φ +0 /r + ) ∂ χ n +1 φ +1 )( n + 1), ξ vani-shes at φ − = ±√ ( d − /d , and ∂ φ − ξ = − (cid:15) ( φ +0 /r + ) ∂ φ − φ +1 .Remarkably, (i) S aniΦ ,γ depends on a single canonicaltransformation Φ : ( φ − , φ − ) ( φ +0 , φ +1 ) parametrized bythe scalar invariants χ m ; (ii) the densitized trace-free partof the extrinsic curvature ( k ± − d δ ) p g ± is unchanged upto a constant factor γ , and in particular its eigenvectors(Kasner frame) are preserved; (iii) the metric is scaledanisotropically in each eigenvector direction of k ± . Isotropic ultralocal scattering.
The isotropic map isgiven by taking γ = 0 in (1), namely ˚ k + = 0. Theconstraints (5) then fix | φ +0 | and make φ +1 constant, butleave the spatial metric arbitrary: S iso∆ ,ϕ,(cid:15) : ( g − , k − , φ − , φ − ) ( g + , k + , φ +0 , φ +1 )= (cid:0) ∆(˚ k − , φ − , φ − ) g − , δ/d, (cid:15) p ( d − /d, ϕ (cid:1) (13)for any constant ϕ ∈ R , sign (cid:15) = ±
1, and function ∆ = P d − n =0 ∆ n ( φ − , φ − , χ m )(˚ k − ) n with positive eigenvalues.While the constant map Φ = ( φ +0 , φ +1 ) is not strictlyspeaking a canonical transformation since it sits at asingular point of the symplectic form (9), it can be realizedas a limit of canonical transformations. However, theisotropic map (13) is not a limiting case of the anisotropicmap (12) since (13) allows a much more general metric g + .The isotropic scattering map S iso∆ ,ϕ,(cid:15) physically describesan irreversible bouncing scenario in which almost all infor-mation is lost: (i) Since k + = δ/d , the bounces producean isotropic and homogeneous evolution . (ii) Thematter field is constant in space. (iii) However, the metricis scaled differently along different eigenvectors of k − . Outlook on universality and model-dependence.
Ournotion of singularity scattering map extracts the macro-scopic effects induced by a microscopic model. Remark-ably, by our ultralocality postulate we establish a fullclassification which proves universal laws while leavingroom for model-dependence to affect the Universe after thebounce. The universal laws are obeyed by models rangingfrom string theory to loop quantum cosmology and modi-fied gravity: (1) continuity of densitized shears ˚ K ± p g ± and (2) canonical transformation of matter.In the pre Big Bang scenario our approach selects thenatural choice of signs β + = − β − and leads us to ananisotropic scattering map characterized by an explicit Φ;cf. (10). For modified matter models, the map depends onthe Lagrangian yet obeys the universal scattering laws (inhomogeneous cases at least) and fits in our classification.We treat inhomogeneous bounces in [18].The keys for our classification were the constraint equa-tions and the fact that space derivatives are negligiblenear the singularity. Our method should generalize toother matter fields, a cosmological constant, bounces thatdo not asymptote to general relativity, and Penrose’s con-formal cyclic cosmology [20, 21]. For compressible fluidswe find in [15] an interesting interplay between geometricsingularities, fluid shock waves, and phase transitions. ∗ bruno@le-floch.fr † contact@philippelefloch.org ‡ [email protected][1] L. Andersson and A.D. Rendall,
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