IImperial/TP/2019/KSS/02
A Safe Beginning for the Universe?
Jean-Luc Lehners ∗ and K. S. Stelle † Max–Planck–Institute for Gravitational Physics(Albert–Einstein–Institute), 14476 Potsdam, Germany Blackett Laboratory, Imperial College London,Prince Consort Road, London SW7 2AZ, U.K.
When general relativity is augmented by quadratic gravity terms, it becomes a renor-malisable theory of gravity. This theory may admit a non-Gaussian fixed point asenvisaged in the asymptotic safety program, rendering the theory trustworthy toenergies up to the Planck scale and even beyond. We show that requiring physicalsolutions to have a finite action imposes a strong selection on big-bang-type universes.More precisely we find that, in the approach to zero volume, both anisotropies andinhomogeneities are suppressed while the scale factor is required to undergo accel-erated expansion. This provides initial conditions which are favourable to the onsetof an inflationary phase while also providing a suitable starting point for the secondlaw of thermodynamics in the spirit of the Weyl curvature hypothesis. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - t h ] O c t Contents
I. Introduction II. Quadratic gravity III. Anisotropies IV. Inhomogeneities V. Discussion Acknowledgments References I. INTRODUCTION
From observations of the cosmic background radiation we know that at an early epoch,the universe was highly isotropic and homogeneous, spatially flat to a good approximation,only containing small nearly scale-invariant and Gaussian density fluctuations [1]. It is clearthat this configuration of the universe was very special. But one may wonder whether itwas the outcome of a special state in the very early universe, or whether it arose from ageneric early state. That is to say, could the dynamics alone have been sufficient to explainthe state of the universe at recombination, maybe because of a strong attractor? Or wasthere rather a selection rule that determined the initial state? A debate between these twoviewpoints has been at the heart of cosmology for as long as people have thought about thebeginning of the universe.With the advent of inflation [2–4], many people started believing that dynamics alonewould be sufficient . More precisely, the idea was that with “generic” initial conditions (forinstance with some sort of “equipartition” of energies at the Planck scale, as in chaoticmodels [6]) inflation would occur somewhere, and in that region the universe would expandso much that it quickly came to dominate over any surrounding region where inflation did Though, interestingly, Starobinsky’s early model was based on the premise that a de Sitter geometry is aspecial solution [5]. not take hold. In all parts where inflation came to an end, the universe would then automat-ically be flat, isotropic and with the required density perturbations amplified from quantumfluctuations during inflation. However, observations put an upper limit on the Hubble rateduring inflation, with the upper bound being 5 orders of magnitude below the Planck scale[1]. Since kinetic, gradient and potential energies scale very differently with the expansionof the universe, any kind of equipartition at the Planck scale would evolve to something farfrom the conditions required for inflation at the scale of the inflationary potential [7], im-plying that inflation itself requires special initial conditions to get underway. Semi-classicalquantisation of inflationary universes points to the same need for a suitable pre-inflationaryphase [8, 9]. Meanwhile, one should bear in mind that we do not actually know whetheran inflaton field, with a viable potential, actually exists. Dynamical mechanisms explain-ing the state of the universe are also central to cyclic models [10, 11], where an ekpyroticcontracting phase plays a role similar to that of inflation [12], while the bounce can act asan additional “filter” as envisaged in the phoenix universe [13, 14]. These models remainequally speculative, since we do not know if ekpyrotic matter exists and since the physics ofthe bounce remains work in progress [15–19].The opposite point of view is to find an explanation for a special initial state of the uni-verse. This has been the approach pursued in the no-boundary proposal [20], the tunnelingproposal [21] and more recent ideas such as the CPT symmetric universe [22, 23]. So far theno-boundary and tunneling proposals have been considered in conjunction with inflation orekpyrosis [24, 25], and a debate is still unfolding as to how to implement these theories inthe best way [26–32]. These proposals certainly remain promising candidate explanationsfor the state of the early universe, and besides the question of a precise mathematical im-plementation it will be interesting to see if they can lead to observationally distinguishablepredictions. In the same camp of theories for the initial state resides Penrose’s Weyl cur-vature hypothesis [33, 34]. Penrose noted that around the time of recombination the Weylcurvature was extremely low, while during gravitational collapse it increases (and divergesnear the centres of black holes). Thus originated his suggestion that the Weyl curvaturewas zero (or as small as the uncertainty principle allows) at the big bang. If gravitationalentropy is related to the Weyl curvature, this has the further implication of explaining thelow initial entropy in the universe, thereby providing a suitable starting point for the secondlaw of thermodynamics. But to date no concrete implementation of this proposal has beenfound.In the present paper we want to investigate possible implications of quadratic gravity onthe question of initial conditions . It seems clear that general relativity (GR) will not bethe final theory of gravity, as it must be quantised, but it happens to be non-renormalisable.However, upon inclusion of terms quadratic in curvature tensors, gravity does become renor-malisable [36], and for this reason we will focus our attention on this theory. There is someevidence that quadratic gravity is in fact asymptotically safe [37–42], implying that it can betrusted up to Planck scale energies and even higher . This makes it particularly appealingin terms of exploring the consequences for the big bang. It is here that we find a crucialdifference between pure general relativity and the quadratic curvature terms. There existmany solutions of general relativity that contain an inhomogeneous, anisotropic spacelikesingularity in their past. In fact, the singularity theorems of Penrose and Hawking tell usthat this is generically the case [43]. Despite the singularity many such solutions never-theless have finite GR action and would thus not be questionable from the quantum pointof view. However, we find that the quadratic gravity terms modify this conclusion: theygenerically cause the action to blow up if there are anisotropies and inhomogeneities presentat the big bang. Since infinite action solutions cannot be part of a well-defined quantumtheory of gravity, such spacetimes are filtered out. In this manner quadratic gravity selectsa homogeneous and isotropic big bang, thus potentially explaining some of the most basicpuzzles of the early universe. What is more, finiteness of the action requires the scale factorto undergo accelerated expansion, so that conditions favourable for a subsequent inflationaryphase are obtained.Quantum quadratic gravity thus acts as a strong selection principle, as it requires ahomogeneous and isotropic big bang. This implements the Weyl curvature hypothesis, andcan therefore also explain the low entropy present at early times. Let us emphasise that thestructure of the theory is already enough by itself to select a class of very special possibleinitial states, differing only in their early expansion rates. Moreover, it is conceivable that Quantum cosmological aspects of quadratic gravity have previously been studied in [35]. We do not take a definite position here as to whether quadratic gravity is fully acceptable as an ultimatetheory of gravity. It might be considered as an intermediate effective theory of gravity, valid to energiesfar above those for which Einstein’s theory alone is applicable. A deeper theory may still be required tounderstand the microscopic nature of spacetime and gravitons. future work will show that even this rate is determined, e.g. by the relevant renormalisationgroup behaviour of the cosmological constant. This would constitute a resolution of thedebate mentioned at the beginning of this introduction, but without having to introducefurther ad hoc principles: quadratic gravity combined with basic quantum principles mayalready be enough to explain why our universe had a safe beginning.
II. QUADRATIC GRAVITY
The action that we will consider is that of (4-dimensional) general relativity, includ-ing a cosmological constant Λ , augmented by terms that are quadratic in the Riemanntensor R µνρσ . As is well known, the Gauss-Bonnet identity implies that the particularcombination √− g ( R µνρσ R µνρσ − R µν R µν + R ) forms a total derivative, so that we mayrestrict our attention to two independent quadratic gravity terms. We will take these tobe the Ricci scalar squared, and the Weyl tensor C µνρσ squared, which satisfies the relation (cid:82) √− g C µνρσ C µνρσ = (cid:82) √− g (cid:0) R µν R µν − R (cid:1) . Consequently our action can be written as S = (cid:90) d x √− g (cid:20) κ R − Λ − σ C µνρσ C µνρσ + ω σ R (cid:21) . (1)The gravitational coupling is given in terms of Newton’s constant G as κ = 16 πG. In termsof dimensionless couplings we can write κ = µ g N and Λ = λµ , where µ is an energy scaleand g N , λ are the dimensionless Newton coupling and cosmological constant respectively.The couplings of the quadratic terms, σ and ω, are already dimensionless by definition.Pure general relativity is not renormalisable [44–46], in the sense that an infinite numberof counterterms, up to arbitrarily high powers in the curvature tensors, would be requiredto specify the high energy behaviour upon including loop corrections. This situation is dra-matically improved if instead of pure general relativity one takes the action of quadraticgravity (1) as the starting point [36]. The 1 /k momentum dependence of the propagatorat high frequencies then renders this theory renormalisable, and does not require the in-clusion of counterterms of yet higher orders in the curvature tensors. The evolution of the(dimensionless) couplings under a change of energy scale µ are given at one loop order bythe following beta functions (calculated in a Euclidean setting [37–42]), µ ddµ g N = f g ( g N , λ, σ, ω ) , µ ddµ λ = f λ ( g N , λ, σ, ω ) ,µ ddµ σ = − π σ , µ ddµ ω = −
25 + 1098 ω + 200 ω π σ . (2)Here f g and f λ are functions of all the couplings, but whose precise form we will not require.The beta functions for the higher-derivative couplings form a closed subsystem. Interestingly,there is evidence for a non-trivial fixed point at σ (cid:63) = 0 , ω (cid:63) ≈ − . g (cid:63)N , λ (cid:63) , with σ approaching zeroin inverse proportion to the logarithm of the energy scale, σ ∼ / ln µ [39–42] . The precisevalues of the asymptotic couplings will change once matter contributions are included, butour arguments will be independent of such refinements. The existence of a non-trivial fixedpoint is in line with the goals of the asymptotic safety programme initiated by Weinberg[48]. Note that at high energies only a specific linear combination of the quadratic terms,namely C µνρσ C µνρσ − ω (cid:63) R , remains relevant to leading order in σ . (Most of the work onasymptotic safety has focussed on the full space of couplings of diffeomorphism invariantgravitational theories, i.e. on general relativity extended by the infinite series of terms ofhigher orders in the Riemann curvature, see [49–51] and references therein. Here we restrictto quadratic gravity because of its special renormalisability properties.)To further understand the implications of these running couplings we can consider a met-ric fluctuation h µν around a reference metric ¯ g µν . Since we are taking quadratic gravity as ourstarting point, we must rescale the metric fluctuations h such that the kinetic term ln µ ( ∂ h ) becomes canonical, i.e. we have to define (ln µ ) / h ≡ ˜ h . Then the higher-derivative kineticterm will be of the form ( ∂ ˜ h ) . Under such a re-scaling, the cubic interaction term con-tained in the Einstein-Hilbert term will be of the form µ / (ln µ ) / ˜ h ∂ ˜ h, demonstratingthat the effective (dimensionful) gravitational coupling µ / (ln µ ) / blows up, with the con-sequence that gravity is strongly coupled at large energies (while the dimensionless coupling1 / (ln µ ) / tends to zero). The cubic interactions contained in the quadratic gravity termsgo as (ln µ ) / (ln µ ) / = 1 / (ln µ ) / and thus go to zero. In other words, the quadratic gravity We should mention that other works have found evidence for another non-trivial fixed point at which thequadratic couplings would both take non-zero values [47]. terms become asymptotically free [37, 38].The quadratic gravity terms have the effect of introducing additional degrees of freedominto the theory [52]. Beyond massless gravitons, there are negative energy spin 2 excitationsof mass squared σ/κ ∼ µ / ln µ and spin 0 excitations of mass squared σ κ ω ∼ µ / ln µ. Notethat the scalar degree of freedom is non-ghost [52] but becomes tachyonic in the approachof the fixed point ω (cid:63) < µ → ∞ , although simultaneously the masses of the new degreesof freedom are pushed to infinity. The extent to which these additional excitations lead toinstabilities remains a matter of debate, to which we have nothing new to add here. Here,we will consider quadratic gravity as an effective quantum theory of gravity, and we willexplore consequences for the early universe that will be independent of this issue.The renormalisability and asymptotic safety of quadratic gravity imply that we can trustthe theory up to arbitrarily high energies. If we now consider a transition amplitude, cal-culated for instance as a sum over paths weighted by the action, then we only obtain arelevant contribution to this sum if the action is well-defined and not divergent. This leadsus to impose the following condition, motivated by basic quantum mechanics: all physicalsolutions of the theory should have finite action . This criterion will be especially relevantnear the big bang. Note that usually one is not bothered by a divergence in the action nearthe big bang, as one would usually relegate this problem to new physics at the Planck scale.But here, since we can take the theory seriously as written down, we can also take solutionsseriously up to arbitrarily high energies. Imposing the condition of finite action will turnout to be a strong selection principle.We should note that the relation between renormalisation scale and physical scales isnot completely clear: often one can identify the energy scale with the ambient temperature,but in the early universe there may not have been any radiation at early stages, and evenif so it may not have been in any sort of equilibrium. It may thus make more sense toidentify the energy scale with the curvature scale stemming from the Ricci scalar, but morework is required in general to elucidate the relationship between renormalisation scales andcosmological evolution. What we will do in the present paper is the following: we will assumethat the metric is such that we approach zero volume as the time coordinate t → . Thenat fixed (and arbitrarily large) scale µ we will impose the requirement that the action isconvergent, in particular that the time integral in the action does not diverge as the initialtime is taken to zero. In the same vein, we will assume that the spatial volume of theuniverse is finite. We are now in a position to explore the consequences of this criterion forearly anisotropies and early inhomogeneities, which we will do in turn. III. ANISOTROPIES
A useful way to analyse anisotropies is to consider the Bianchi IX metric, which can alsobe regarded as a non-linear completion of a gravitational wave. This analysis will serve as agood indication for the fate of general anisotropies. The Bianchi IX metric can be writtenas ds IX = − dt + (cid:88) m (cid:18) l m (cid:19) σ m , (3)where σ = sin ψ dθ − cos ψ sin θ dϕ , σ = cos ψ dθ + sin ψ sin θ dϕ , and σ = − dψ + cos θ dϕ are differential forms on the three sphere with coordinate ranges 0 ≤ ψ ≤ π , 0 ≤ θ ≤ π ,and 0 ≤ φ ≤ π. We may then rescale l = a e ( β + + √ β − ) , l = a e ( β + −√ β − ) , l = a e − β + , (4)such that a represents the spatial volume while the β s quantify a change in the shape ofspatial slices. When β − = β + = 0 one recovers the isotropic case. The Einstein-Hilbertaction in these coordinates is given by S EH = (cid:90) d x √− g R π (cid:90) dta (cid:18) − a + 34 a ( ˙ β + ˙ β − ) − U ( β + , β − ) (cid:19) , (5)where the anisotropy parameters evolve in the effective potential U ( β + , β − ) = − (cid:16) e β + + e − β + −√ β − + e − β + + √ β − (cid:17) + (cid:16) e − β + + e β + − √ β − + e β + +2 √ β − (cid:17) . (6)The Friedmann equation resulting from this action alone is given by (with H = ˙ a/a )3 H = 34 ˙ β + 34 ˙ β − + 1 a U ( β + , β − ) , (7)and it can be used to simplify the on-shell action which is then given by the compactexpression S on − shellEH = (cid:90) d x √− g R − π (cid:90) dt aU ( β + , β − ) . (8)The full action that we will consider includes the contributions due to quadratic gravity.The Weyl squared part is given by (up to total derivative terms) − (cid:90) d x √− g C µνρσ C µνρσ = 2 π (cid:90) dt (cid:110) a (cid:20)(cid:18) ¨ aa + H (cid:19) ( ˙ β − + ˙ β ) − ¨ β − − ¨ β − ( ˙ β − + ˙ β ) (cid:21) + 4 a (cid:20) − ( ¨ β − + 3 H ˙ β − ) U ,β − − ( ¨ β + + 3 H ˙ β + ) U ,β + − (cid:18) aa + ˙ β − + ˙ β (cid:19) U (cid:21) + 643 a (cid:16) − e − β + + e − β + −√ β − + e − β + + √ β − − e − β + + e β + − √ β − + e β + +3 √ β − − e β + −√ β − − e β + + √ β − − e β + − √ β − − e β + +4 √ β − + e β + − √ β − + e β + +2 √ β − (cid:17) (cid:111) . (9)The anisotropy potential U was defined in (6). Meanwhile, the R part is (cid:90) d x √− gR = 2 π (cid:90) dta (cid:20) aa + 6 ˙ a a + 32 ( ˙ β + ˙ β − ) − a U ( β + , β − ) (cid:21) . (10)The solutions to the two-derivative equations of motion, starting from “generic” condi-tions at some time and evolving back towards the big bang, correspond to the Belinsky-Khalatnikov-Lifshitz (BKL)/mixmaster chaotic solutions [53, 54], in which the universeevolves increasingly locally, with time derivatives being much more important than spa-tial derivatives, and contracting anisotropically. The solution then jumps between Kasnerepochs, during which the anisotropy parameters evolve logarithmically, β ± ∼ ln( t ) and theuniverse contracts as a ( t ) ∼ t / . We may wonder if this behaviour is significantly alteredwhen the four-derivative terms are added to the action. To investigate this question, wemay start by analysing the equations of motion for small anisotropies, here retaining onlythe leading terms arising from the Weyl squared action. These are then given by0 = 1 κ (cid:18) ¨ β + + 3 H ˙ β + + 8 β + a (cid:19) + 1 σ (cid:32) β (4)+ + 6 Hβ (3)+ + 4¨ a ¨ β + a + 7 ˙ a ¨ β + a + 20 ¨ β + a + 4 ˙ a ¨ a ˙ β + a + a (3) ˙ β + a + ˙ a ˙ β + a + 20 ˙ a ˙ β + a + 64 β + a (cid:33) , (11)0 = 1 κ (cid:18) aa + H + 34 ˙ β + 1 a (cid:19) + 1 σ (cid:32) β (3)+ ˙ β + −
14 ¨ β + H ˙ β + ¨ β + + ¨ a ˙ β a + ˙ a ˙ β a + 5 ˙ β a (cid:33) , (12)while the constraint reads0 = 1 κ (cid:18) H + 3 a − β a −
34 ˙ β (cid:19) + 1 σ (cid:32) − β (3)+ ˙ β + + 34 ¨ β − H ˙ β + ¨ β + − a ˙ β a − a ˙ β a −
15 ˙ β a − β a (cid:33) . (13)At the level of the two-derivative theory we know that the relevant solution is the one with a ∝ t / and ˙ β + ∝ /t. So first we can try to solve around this solution, i.e. we assumethat a ∝ t / and solve for the time dependence of β + using only the dominant terms in(11). This yields four solutions β + ∝ t b with b ∈ { , , , } . All of these are decayingsolutions as t → , except for b = 0 which actually corresponds to the original ˙ β + ∝ /t solution with β + growing logarithmically. The subleading terms will induce corrections.However, this shows that the higher-derivative terms do not drastically change the expectedBKL behaviour in the approach to a spacelike singularity (there may however exist isolatedislands of stability, as suggested by [55]). Numerical studies of the full equations of motionconfirm this expectation, and show that typically the solutions crunch faster when thehigher-derivative terms are added – an example is shown in Fig. 1.As we just saw, when the higher derivative terms are added the anisotropies remainlarge, and the behaviour remains chaotic, near the big bang. In this sense the higher orderterms do not help in understanding the early state of the universe. However a theory ofinitial conditions must restrict the allowed solutions – otherwise it has no explanatory power.The way this is achieved here is via a consideration of the action: our premise, based onquantum theory, will be that a solution with infinite action does not contribute to physical0 a ( t ) t σ = ∞ , ω = σ = / ω =
30 10 20 30 40 50 60020406080100 β + ( t ) t10 20 30 40 50 60 - - Figure 1: Plots of the scale factor (left) and the anisotropy (right) as a function of time, for a typicalexample of the approach to a singularity. The dashed curves show the solution in the absence ofquadratic gravity terms, where one can see that after one reflection of the anisotropy parameter, asingularity is reached at a finite time. When quadratic curvature terms are added, the only effect isto accelerate the occurrence of the crunch, as seen in the continuous curves. The initial conditionsused here for the two-derivative theory are: a ( t = 0) = 100 , ˙ a (0) = − / , β + (0) = 1 . For the casewith higher-derivative terms added, we fixed the additional initial data by using the two-derivativesolution, and we checked that the constraint remains satisfied to sufficient accuracy. transition amplitudes. In this respect there is an interesting difference between the pureEinstein-Hilbert theory, and the theory with the quadratic curvature terms added in.A special but illustrative example (without reflections off the anisotropy walls) is thesolution along one of the symmetry axes of the Bianchi IX spacetime, i.e. a solution with β − = 0 . The anisotropy potential reduces to U ( β + ,
0) = e − β + − e − β + , and the asymptoticsolution is given by a ( t ) = a t / e β + = b + t − / (14)As t → i.e. β + → ∞ . The on-shell Einstein-Hilbert action (8) canthen be approximated by S on − shellEH = − π (cid:90) t t dt aU ( β + , ≈ π a b + (cid:90) t t dt t , (15)where the approximate expression holds for small t. We can see that the action will be finiteas the lower bound of integration tends to zero, t → . Thus, within the pure Einstein-Hilbert theory, there is no obstruction to considering a highly anisotropic solution of thiskind. But now we can also evaluate the action when the quadratic curvature terms are1included. We may, for instance, look at the Weyl squared term. With β − = 0 , it reduces to − (cid:90) d x √− g C µνρσ C µνρσ ( β − = 0)= 2 π (cid:90) dt (cid:110) a (cid:20)(cid:18) ¨ aa + H (cid:19) ˙ β − ¨ β − ˙ β (cid:21) + 4 a (cid:20) − ( ¨ β + + 3 H ˙ β + ) U ,β + − (cid:18) aa + ˙ β (cid:19) U (cid:21) + 643 a (cid:0) − e − β + + 2 e − β + − e − β + (cid:1) (cid:111) . (16)Near t = 0 , for our example this action is approximately given by − (cid:90) d x √− g C µνρσ C µνρσ ≈ π a (cid:90) t t dt t + · · · , (17)where we have dropped subdominant terms. This clearly diverges as t → , hence on thisbasis we would exclude this solution. Thus we have an example of a solution which hasfinite Einstein-Hilbert action, but infinite Weyl squared action and diverging Weyl tensor.The question now is how general this behaviour is.Based on the BKL analysis, we know that close to a spacelike singularity the genericsolution is locally of Bianchi IX form, and for short periods of time this is well approximatedby Kasner solutions. Thus we may try an ansatz a ∝ t s e √ β − ∝ t m e β + ∝ t p . (18)The chaotic mixmaster solutions imply that most of the time, i.e. in between reflections offthe potential walls, we will have s = 1 / m and p will be related to each other. Butin our context it makes sense to keep these coefficients general. As explained in the previoussection, the Einstein-Hilbert term remains important at high energies. Near t = 0 , thevarious terms in the Einstein-Hilbert action then scale as powers of time with the exponents3 s − ,s + 2 p + 1 , s − p + 1 ,s − p − m + 1 , s − p + m + 1 , s + 2 p − m + 1 , s + 2 p + 2 m + 1 (19)where we are showing the integrated version ( i.e. we have performed the integral over t –for instance, the first term (cid:82) a ˙ a leads to (cid:82) t s − ∝ t s − ). In the absence of cancellations2between terms (which we generically cannot expect to arise), we would want all of theseexponents to be positive, implying the conditions s > , −
12 (1 + s ) < p <
14 (1 + s ) , −
12 (1 + s ) < p ± m < s . (20)These conditions can easily be satisfied – note that they allow for negative values of p and m, and consequently to a blowing up of anisotropy (and of Weyl curvature) near t = 0.Convergence of the terms involving the scale factor alone simply leads to the requirement s > / . Meanwhile, with the above ansatz, the Weyl squared and R actions lead to the scalings(again barring cancellations between terms)3 s − , s − − p − m , s − − p + m , s − p + 2 m , s − p − ms − p , s − − p , − s − p , − s − p − m , − s − p + m , − s − p − s + p + 3 m , − s + p − m , − s + p + m , − s + p − m , − s + 4 p + 4 m , − s + 4 p − m , − s + 4 p + 2 m , − s + 4 p − m (21)Requiring these exponents to be positive leads to the conditions s > , −
12 ( s − < p <
14 ( s − , −
12 (1 − s ) < p <
18 (1 − s ) p + m > s − , p + m < s − , p − m < s − , p − m > s − , (22)which are clearly mutually exclusive. We can only obtain a finite action if the anisotropyparameters β ± become zero near t = 0 . As the universe expands the anisotropies will re-main small at the level of the background (if an inflationary phase ensues, fluctuations willbe produced by further quantum effects). Thus the imposition of finite action suppressesanisotropies quite generally at the earliest times.
IV. INHOMOGENEITIES
What about spatially dependent (inhomogeneous) perturbations? Are they also sup-pressed, or can there be large variations in space? To answer this question it is instructive3to consider a metric of the form ds = − dt + A (cid:48) F dr + A ( dθ + sin θdφ ) , (23)which belongs to the Lemaˆıtre-Tolman-Bondi class [56, 57]. Here the scale factor dependsnot only on time, but also on the radial direction r , that is to say A = A ( t, r ) . A primedenotes a derivative w.r.t. r. Meanwhile F = F ( r ) is a function of r only that describesthe inhomogeneity in the r direction. A spherical symmetry in the two remaining spa-tial directions is retained for simplicity. Relaxing this assumption would only strengthenthe arguments below. When F ( r ) = 1 and A factorises into a time-dependent and an r -dependent function, then after a field redefinition this metric reduces to the standard flatRobertson-Walker form.The Einstein-Hilbert action is given by (cid:90) d x √− g R π (cid:90) dt dr A (cid:48) F (cid:32) − F + ˙ A − AF F (cid:48) A (cid:48) + 2 A ˙ A ˙ A (cid:48) A (cid:48) + 2 A ¨ A + A A (cid:48) ¨ A (cid:48) (cid:33) . (24)As for the quadratic gravity terms, the R action is given by (cid:90) d x √− g R = 16 π (cid:90) dt dr A (cid:48) A F (cid:32) − F + ˙ A − AF F (cid:48) A (cid:48) + 2 A ˙ A ˙ A (cid:48) A (cid:48) + 2 A ¨ A + A A (cid:48) ¨ A (cid:48) (cid:33) , (25)while the Weyl squared action is (cid:90) d x √− g C µνρσ C µνρσ = 16 π (cid:90) dt dr A (cid:48) A F (cid:32) − F + ˙ A + AF F (cid:48) A (cid:48) − A ˙ A ˙ A (cid:48) A (cid:48) − A ¨ A + A A (cid:48) ¨ A (cid:48) (cid:33) . (26)We are interested in the conditions under which the action is finite when integrated fromzero spatial volume onwards, i.e. we would like to see under what conditions the actionintegral diverges in the approach to the big bang. For this we will again look at the scalingsof the various terms contributing to the action. As above, we will make the ansatz thatthe scale factor has a power-law time dependence in the approach to A = 0 , i.e. we willposit A ( t, r ) ∼ t s near t = 0, with s being a positive number. The Einstein-Hilbert action4then has two different types of terms: those related to the time evolution of the scale factor,such as (cid:82) dt ˙ A A (cid:48) F ∼ (cid:82) dt t s − ∼ t s − and which require s > / (cid:82) dt A (cid:48) F ∼ (cid:82) dt t s ∼ t s +1 . These latter terms arealways convergent, since s > R and the Weyl squared actions lead to temporal scalingsof the form (cid:90) dt t s − , (cid:90) dt t s − , (cid:90) dt t − s , (27)where the last term comes from the pure inhomogeneity terms like (cid:82) A (cid:48) F A F . Convergence near t = 0 then requires s > , s < . (28)The second requirement, which comes from the inhomogeneity contributions, is in clearconflict with the first one. This shows that near t = 0 the inhomogeneity must be damped, F ( r ) → , in order for the action to remain finite. In this manner only initially homogeneousuniverses are allowed, with the scale factor undergoing accelerated expansion, A ∼ t s with s > . V. DISCUSSION
Quadratic gravity is not only in complete agreement with gravity experiments, but itis a renormalisable theory, which may in fact be asymptotically safe and thus trustworthyup to high energies. Taking this theory seriously then has the immediate and surprisingconsequence of explaining the special nature of our universe at the earliest times: the bigbang is required to have been homogeneous and isotropic. Thus basic aspects of the largescale properties of our universe would be explained by the quantum requirement that physicalamplitudes must have finite action. At the same time, such a safe beginning of the universe5has vanishingly small Weyl curvature, and can thus explain the low initial entropy requiredfor the unfolding of the second law of thermodynamics.Our arguments are simple and direct, but make use of a number of assumptions thatdeserve clarification in future work. Especially the relation between energy scale and cur-vature needs to be better understood, not just in the present context but more generallyin early universe cosmology. An improvement on this front would also allow one to includethe running of couplings into the equations of motion, and thus determine more preciselythe evolution after the big bang: how long would a subsequent cosmological-constant-driveninflationary phase last (see also [58])? And could the energy scale be such that the observedamplitude of fluctuations in the cosmic background could be explained? Is the running of the R term such that it can grow in relative importance compared to the Einstein-Hilbert term(and obtain the correct sign in its coupling) such that an inflationary phase of Starobinskytype could take place? Moreover, if the Higgs scalar is added, what are the properties ofits potential at high energies, and would it influence the inflationary dynamics implied bythe cosmological constant or R terms? Most importantly: does the scenario described herelead to any distinguishable observational consequences? This will of course be crucial inassessing the ultimate validity of our proposal. Acknowledgments
We would like to thank Andrei Barvinsky, Steffen Gielen, Caroline Jonas and RobertoPercacci for helpful discussions and correspondence. JLL gratefully acknowledges the sup-port of the European Research Council in the form of the ERC Consolidator Grant CoG772295 “Qosmology”. JLL would like to thank Imperial College for hospitality. The workof KSS was supported in part by the STFC under Consolidated Grant ST/P000762/1. KSSwould like to thank the Albert Einstein Institute for hospitality on several occasions duringthe course of the work. [1]
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