sign in sign up
<<
Home , Flatness problem, Horizon problem

PHYSICAL REVIEW D 97, 043512 (2018)

Solving the with an anisotropic instanton in Horava-Lifshitzˇ

Sebastian F. Bramberger,1 Andrew Coates,2 João Magueijo,3 Shinji Mukohyama,4,5 Ryo Namba,6 and Yota Watanabe5,4 1Max Planck Institute for Gravitational Physics ( Institute), 14476 Potsdam-Golm, Germany 2School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 3Theoretical Physics, Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom 4Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502 Kyoto, Japan 5Kavli Institute for the Physics and Mathematics of the (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 6Department of Physics, McGill University, Montr´eal, Qu´ebec H3A 2T8, Canada

(Received 5 October 2017; published 12 February 2018)

In Horava-Lifshitzˇ gravity a scaling isotropic in space but anisotropic in , often called “anisotropic scaling,” with the dynamical critical exponent z ¼ 3, lies at the base of its renormalizability. This scaling also leads to a novel mechanism of generating scale-invariant cosmological perturbations, solving the without . In this paper we propose a possible solution to the flatness problem, in which we assume that the initial condition of the Universe is set by a small instanton respecting the same scaling. We argue that the mechanism may be more general than the concrete model presented here. We rely simply on the deformed dispersion relations of the theory, and on equipartition of the various forms of energy at the starting point.

DOI: 10.1103/PhysRevD.97.043512

I. INTRODUCTION the problem and we simply assume that Λ has the observed value. The slowest decaying In a homogeneous and isotropic component on the right-hand side of the Friedmann Universe is described by the Friedmann equation equation is the spatial curvature term −3K/a2 and is the 3 source of the flatness problem in the standard . 3 2 8π ρ − K Λ ρ H ¼ G 2 þ ; ð1Þ Inflation, once it occurs, makes almost constant for an a extended period in the early Universe so that even the curvature term decays faster than ρ. The initial condition of where H is the Hubble expansion rate; G is Newton’s the standard cosmology is thus set at the end of inflation in constant; ρ is the energy density; K ¼ 0; 1; −1 is the such a way that the curvature term is sufficiently smaller curvature constant of a maximally symmetric 3-space; a than 8πGρ. Subsequently, the ratio of the curvature term to Λ is the ; and is the cosmological constant. 8πGρ grows but the initial value of the ratio at the end of ρ The asymptotic value of at late times can be set to zero inflation is so small that the Universe reaches the current Λ ρ by redefinition of . In the standard cosmology, then epoch before the ratio becomes order unity. This is how 4 includes energy densities of radiation (∝ 1/a ) and pres- inflation solves the flatness problem. sureless (∝ 1/a3). The fact that all but Λ decay as the If a theory of quantum gravity predicts that the ratio Universe expands is the source of the cosmological con- ð3K/a2Þ/ð8πGρÞ were sufficiently small at the beginning of stant problem. The present paper does not intend to solve the Universe then this could be an alternative solution to the flatness problem. The purpose of the present paper is to propose such a solution based on the projectable version of Published by the American Physical Society under the terms of Horava-Lifshitzˇ (HL) gravity [1,2], which has recently been the Creative Commons Attribution 4.0 International license. proved to be renormalizable [3,4] and thus is a good Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, candidate for a quantum gravity theory. Since our proposal and DOI. Funded by SCOAP3. is solely based on a fundamental principle called “anisotropic

2470-0010=2018=97(4)=043512(10) 043512-1 Published by the American Physical Society SEBASTIAN F. BRAMBERGER et al. PHYS. REV. D 97, 043512 (2018) scaling,” which is respected by all versions of the HL theory, anisotropic scaling and the semiclassical evolution of it is expected that the same idea can be implemented in HL theory, the curvature is sufficiently suppressed to other versions of HL gravity. solve the flatness problem without the need for inflation. One of the fundamental principles of HL gravity is the The solution may be more general than the concrete model so-called anisotropic scaling, or Lifshitz scaling, presented here, as argued in Sec. IV, where we show that on dimensional grounds we can always predict the modifica- t → bzt; x⃗→ bx;⃗ ð2Þ tions to (1) from the modified dispersion relations of the theory. Together with equipartition of energy at the initial where t is the time coordinate, x⃗are the spatial coordinates point, evolution in this regime enforces the necessary and z is a number called the “dynamical critical exponent.” suppression of the curvature. In Appendixes A and B we In 3 þ 1 dimensions the anisotropic scaling with z ¼ 3 in discuss the generation of scale-invariant perturbations and the ultraviolet (UV) regime is the essential reason for evolution after the instanton based on the concrete setup of renormalizability. The anisotropic scaling with z ¼ 3 also Sec. III. Appendix C then discusses further generalization leads to a novel mechanism of generating scale-invariant of the already general scenarios of Sec. IV. cosmological perturbations, solving the horizon problem without inflation [5]. II. PROJECTABLE HL GRAVITY In the context of quantum cosmology, the initial conditions The basic variables of the projectable version of HL of the Universe are typically set by quantum tunneling gravity are described by an instanton, i.e. a classical solution to some Euclidean equations of motion with suitable boundary lapse∶ NðtÞ; shift∶ Niðt; x⃗Þ; 3d metric∶ g ðt; x⃗Þ: conditions. In relativistic theories, where z ¼ 1, quantum ij tunneling is thought to be dominated by an Oð4Þ symmetric ð4Þ instanton, implying that T ¼ L, where T and L are the Euclidean time and length scales, respectively. After analytic The theory respects the so-called “foliation preserving continuation to the real time evolution, this causes the diffeomorphisms,” flatness problem unless inflation follows. Setting z ¼ 3, however, the story is completely different. t → t0ðtÞ; x⃗→ x⃗0ðt; x⃗Þ: ð5Þ An instanton should lead to T ∝ L3 and thus Adopting the notation of [6], the action of the gravity sector T ≃ M2L3; ð3Þ is then given by Z 2 ffiffiffi where T and L are again the Euclidean time and length MPl p 3 ij 2 I ¼ Ndt gd x⃗ðK K − λK − 2Λ þ R þ L 1Þ; scales, respectively, and M is the scale above which the g 2 ij z> 3 anisotropic scaling (2) with z ¼ becomes important. If ð6Þ the theory is UV complete then the scaling (3) is expected to apply to any kind of instanton deep in the UV regime, where i.e. for L ≪ 1/M. If the size of the instanton L is indeed much smaller than 1/M then this implies that T ≪ L and 2 MPl i jk i j k i L 1 c1D R D R c2D RD R c3R R R thus the instanton has a highly anisotropic shape. We thus 2 z> ¼ð i jk þ i þ i j k “ ” call this kind of instanton an anisotropic instanton. If the j i 3 j i 2 c4RR R c5R c6R R c7R : 7 creation of the Universe is dominated by a small anisotropic þ i j þ Þþð i j þ Þ ð Þ instanton then in the real time Universe after analytic Here, Kij ¼ð∂tgij − DiNj − DjNiÞ/ð2NÞ is the extrinsic continuation, the spatial curvature length scale will be ij ik jl much greater than the cosmological time scale. In this way curvature of the constant t hypersurfaces, K ¼ g g Kkl, ij j ij j the anisotropic instanton may solve the flatness problem K ¼ g Kij, Ni ¼ gijN , g is the inverse of gij, Di and Ri are the covariant derivative and the Ricci tensor constructed without inflation. pffiffiffiffiffiffiffiffiffi i 1 π The rest of the present paper is organized as follows. from gij, R ¼ Ri is the Ricci scalar of gij, MPl ¼ / 8 G is In Sec. II we review projectable HL theory, obtaining the the Planck scale, and λ and cn (n ¼ 1; …; 7) are constants. equivalent of Friedmann’s equation (1) in this theory. New In HL gravity, as already stated in (4), a spacetime curvature-dependent terms are found, which will be essen- geometry is described by a family of spatial metrics tial for the solution to the flatness problem proposed here. parametrized by the time coordinate t, together with the In Sec. III we examine a quantum state inspired by the no- lapse function and the shift vector. The 3D space at each t boundary proposal: the idea that the Universe nucleated can have nontrivial topology and may consist of several from nothing, as represented by Euclidean evolution connected pieces, Σα (α ¼ 1; ), each of which is dis- replacing the singularity. We find that under connected from the others. In this situation, we have a

043512-2 SOLVING THE FLATNESS PROBLEM WITH AN … PHYS. REV. D 97, 043512 (2018)   common lapse function and a set of shift vectors and a set 3λ − 1 ∂ α 3 α 2 2 tHα 3 2 3Kα 2Kα − Kα Λ þ Hα ¼ 6 þ 4 2 þ : of spatial metrics parametrized by not only (continuous) t 2 N aα aα aα but also (discrete) α,as ð14Þ i i ⃗ α ⃗ ⃗∈ Σ N ¼ Nαðt; xÞ;gij ¼ gijðt; xÞ; ðx αÞ: ð8Þ Integrating this equation once, we obtain

3 3λ − 1 α 3 3α 2 3 The equation of motion for NðtÞ then leads to a global ð Þ 2 Cα − 3Kα − 2Kα − Kα Λ Hα ¼ 3 6 4 2 þ ; ð15Þ Hamiltonian constraint of the form 2 aα aα aα aα X Z where Cα is an integration constant. The first term on the 3 ⃗H 0 d x g⊥ ¼ ; ð9Þ right-hand side behaves like a pressureless dust and thus α Σα is called “dark matter as integration constant” [7,8].The equation of motion for NðtÞ then leads to the global where Hamiltonian constraint of the form (9). For example, if ∀ Kα ¼ 1 for α then the global Hamiltonian constraint is 2 ffiffiffi MPl p ij 2 simply H ⊥ ¼ gðK K − λK þ 2Λ − R − L 1Þ: ð10Þ g 2 ij z> X Cα ¼ 0: ð16Þ Because of the summation over mutually disconnected α pieces of the space fΣαg in (9), For the reason already explained in the previous paragraph, Z we do not need to consider this equation, if we are interested 3 Σ d x⃗Hg⊥ ≠ 0 ð11Þ in a Universe in one of f αg. Σα III. ANISOTROPIC INSTANTON is possible, provided that the sum of them over all α is zero. Therefore, if we are interested in a Universe in one of fΣαg As we have shown in the previous section, a homo- then there is neither a local nor a global Hamiltonian geneous and isotropic Universe in the projectable HL constraint that needs to be taken into account. On the other gravity is described by i ⃗ ⃗ hand, the equations of motion for N ðt; xÞ and gijðt; xÞ are 3 3λ − 1 α 3 3α 2 3 ð Þ 2 C − 3K − 2K − K Λ local and thus must be imposed everywhere. The absence of 2 H ¼ 3 6 4 2 þ : ð17Þ a Hamiltonian constraint introduces an extra component a a a a that behaves like dark matter [7,8], as we shall see below Here, the subscript α has been suppressed. For simplicity, explicitly for a homogeneous and isotropic Universe. we set α2 ¼ 0 and Λ ¼ 0, giving We now consider a homogeneous and isotropic Universe 3 3λ − 1 α 3 3 in each connected piece of the space Σα (α ¼ 1; ), ð Þ 2 C 3K K H ¼ − − : ð18Þ described by 2 a3 a6 a2 We assume that there is a UV fixed point of the renorm- i 0 α 2Ω Nα ¼ ;gij ¼ aαðtÞ ij; ð12Þ alization group (RG) flow with a finite value of λ larger than 1, as in the case of 2 þ 1 dimensions [9]. Since we are Ωα λ where ij is the metric of the maximally symmetric three- interested in quantum tunneling in the UV, it is ideal to set dimensional space with the curvature constant Kα ¼ 0; 1; −1 to a constant value (>1) at the UV fixed point. However, ij α i j i j 3 1 and the Riemann curvature R ½Ω ¼Kαðδ δ − δ δ Þ. since the RG flow in þ dimensions has not yet been kl k l l k λ 1 The action is then investigated, we shall consider as a free parameter (> ). 1 We shall adopt units in which MPl ¼ . Z X Z Hereafter in this section, we consider the creation of a 3 2 3 ⃗ 3L L 1 Ig ¼ MPl Ndt d xaα α; α closedR (K ¼ ) Universe. Switching to Euclidean time α Σα τ ¼ i t Nðt0Þdt0 þ const, we obtain 3 2 1 − 3λ α3Kα α2Kα Kα Λ 2 − 2 ¼ Hα þ 6 þ 4 þ 2 ; ð13Þ 3ð3λ − 1Þ ð∂τaÞ C α3 3 2 3aα aα aα 3 ¼ − þ þ : ð19Þ 2 a2 a3 a6 a2 ∂ α 4 3 2 → 0 τ → 0 where Hα ¼ð taαÞ/ðNaαÞ, 2 ¼ ðc6 þ c7Þ/MPl and Supposing that a þ as þ , the leading behavior of α 24 3 9 2 τ ≃ τ1/3 3 ¼ ðc3 þ c4 þ c5Þ/MPl. The variation of the action a for small is a a1 , where a1 is a constant. Hence, with respect to aα leads to the dynamical equation, expanding a around τ ¼ 0 as

043512-3 SEBASTIAN F. BRAMBERGER et al. PHYS. REV. D 97, 043512 (2018)

τ τ λ 2 α 1 FIG. 1. Loglog plots of a vs. / in in solid blue with the analytic solution (26) superimposed in dashed red. We have ¼ , 3 ¼ , α2 ¼ 0 for both plots; however on the left we have C ¼ 6 while on the right C ¼ 50. This confirms the validity of the analytic solution in the large C limit. The figures were obtained by solving Eq. (17) numerically from τ ¼ 10−4 using the small τ expansion until −22 ∂τa ≈ 10 . jτ¼τin

1/3 2/3 a ¼ a1τ þ a2τ þ a3τ þ; ð20Þ For large positive C, we expect a to be small in the 0 ≤ τ ≤ τ whole interval in. Hence in this limit we can safely and plugging this into the Euclidean equation of motion ignore the last term on the right-hand side of (19): (19), we obtain 2 3ð3λ − 1Þ ð∂τaÞ C α3   ≃− : 24 6α 1/6 2 2 3 þ 6 ð Þ 3 0 a a a a1 ¼ 3λ − 1 ;a2 ¼ ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We then have an approximate analytic solution given by 3α2 6 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! a3 ¼ : ð21Þ pffiffiffiffiffi 10 α3ð3λ − 1Þ 2 2 α3 C τ ≃ 1 − 1 − a3 ; ð25Þ 3ð3λ − 1Þ 3C α3 By using this formula, it is easy to solve (19) numerically τ ϵ τ ϵ from ¼ towards larger , where is a small positive or equivalently number. The solution is unique for a given value of the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi integration constant C as we have already fixed another   ffiffiffiffiffi 9 1/3 2 τ p 2 integration constant corresponding to a constant shift of . a ≃ 3 α3T − CT ; T ¼ τ: ð26Þ Some numerical solutions are shown in Fig. 1. For a 4 3ð3λ − 1Þ positive α3 and a large enough positive C, one finds that ∂ τ τ τa vanishes at a finite value of , which we call in, i.e. As a result, we have ffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ∂ 0 p 1 τajτ τ ¼ : ð22Þ 2 α3 3ð3λ − 1Þ α3 /3 ¼ in τ ≃ ;a≃ ; ð27Þ in 3C 2 in C The Lorentzian evolution of the Universe after the quantum ≡ τ tunneling is then obtained by WickR rotating the Euclidean where ain að inÞ. This implies that τ τ τ τ t 0 0 solution at ¼ in as ¼ in þ i Nðt Þdt , meaning that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the instanton is represented by the solution in the range 3 a 3α3 ϵ ≤ τ ≤ τ ϵ → 0 in ≃ in with þ . The contribution of the connected τ 2 3λ − 1 ¼ const: ð28Þ in ð Þ piece of the space of interest to the Euclidean action iIg is then Since we have set K ¼ 1, the scale factor a has the Z   τ 1 − 3λ α dimension of length. For a positive α3 and a large positive 6π2 in τ ∂ 2 − 3 − ≡ τ SE ¼ lim d að τaÞ 3 a value of C, ain að inÞ is small as seen in (27).As ϵ→þ0 ϵ 2 3a Z   expected from the scaling argument (3) in the τ 2α 6π2 in τ C − 3 − 2 Introduction and as confirmed numerically in Fig. 2,we ¼ lim d 3 a ; ð23Þ ϵ→þ0 ϵ 3 3a have the scaling relation (28). These results support the claim that a small anisotropic instanton may solve the where we have used the equation of motion (19). flatness problem in HL gravity.

043512-4 SOLVING THE FLATNESS PROBLEM WITH AN … PHYS. REV. D 97, 043512 (2018)

−1/2 ∼ _ τ −1/2 ∼ K T jHð inÞj ;L 2 : ð29Þ ain Since the equation of motion implies that

1 3 _ τ ∼ K jHð inÞj 4 2 ; ð30Þ M ain it follows that

T ∼ M2L3; ð31Þ

−1/4 3 τ where M ≡ α3 is the momentum scale above which the FIG. 2. The plot shows ain/ in as a function of C and confirms the expected analytic scaling behavior in the large C limit shown anisotropic scaling with z ¼ 3 and thus the curvature cubic in dashed red. To obtain the plot, we kept λ ¼ 2, α3 ¼ 1, α2 ¼ 0 term in (17) become important. This is exactly what we have and integrated the Euclidean equation of motion from τ ¼ 0 to expected in (3) from general arguments. Because of the τ τ ¼ in for various values of the integration constant C. uniqueness of the Euclidean solution, this scaling holds for L ≪ 1/M, independently from the boundary condition and 0 To see if the small instanton dominates the creation of the any physical conditions near a ¼ . Universe, we need to estimate the tunneling rate, which in the regime of validity of the semiclassical approximation IV. GENERAL ARGUMENT is given by the exponential of the Euclidean action (23). This however turns out to be a difficult task. First, both Although we have proposed a concrete framework for i i solving the flatness problem within HL gravity, the argu- the (Euclidean) extrinsic curvature K ¼ δ ∂τ ln a and the Ej j ments presented are more general and may be valid on purely spatial curvature Ri ¼ 2δi /a2 diverges in the limit τ → þ0, j j dimensional grounds for any UV complete theory with an indicating that the semiclassical description should break τ 0 anisotropic scaling of spacetime. This can be suspected from down near ¼ . We are thus unable to rely on the the simple argument presented in Sec. I, but we now take the semiclassical formula for the tunneling rate. Indeed, the dimensional argument further. All that we shall need from the τ dominant termpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the integrand of (23) for small is concrete model presented are its dispersion relations (as in ∝ α 3 ∝ α λ − τ τ 3/a 3ð3 1Þ/ , whose integralp overffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the small HL theory) and equipartition at the starting point (as imposed region exhibits a divergence of order α3ð3λ − 1Þ ln ϵ. by the anisotropic instanton). Thus the quantum state employed in this paper, while Let a general UV complete theory have modified inspired by the no-boundary proposal, does not have a dispersion relations for its massless particles (including regular beginning. Quantum effects such as the RG flow of gravitons) of the form coupling constants might somehow ameliorate the loga- rithmic divergence but this is beyond the scope of the E2 ¼ M2fðp2/M2Þ; ð32Þ present paper. Second, based on a formulation of the Lorentzian path integral for quantum cosmology, it was where f is a smooth function with the following asymptotic recently suggested that the semiclassical formula for the behavior, tunneling rate may have to be drastically modified [10–12].  This may place some doubts on the no-boundary proposal x; ð0 ≤ x ≪ 1Þ fðxÞ¼ ; ð33Þ [13–17] in general relativity. It is certainly worthwhile to xz; ðx ≫ 1Þ investigate whether a similar argument applies to HL gravity or not. and the momentum scale M may be taken to be of the order The Euclidean solution that we found is unique up to an of the Planck scale or not. This is a Hamiltonian constraint integration constant C and a physically irrelevant, arbitrary for particles, so we may expect that in a Friedmann– shift of the origin of the Euclidean time coordinate τ,aslong Lemaître–Robertson–Walker (FLRW) setting a corre- as the homogeneous and isotropic ansatz with a positive sponding Hamiltonian constraint for vacuum solutions will three-dimensional curvature is adopted. Therefore one can result from replacing E2 → H2 and p2 → jKj/a2.Even easily show that the scaling (3) holds for large C, independ- when such a constraint does not strictly exist (as is the case ently from the boundary condition near a ¼ 0.Thisis with the HL model), an effective one may be present, because the (Euclidean) time scale T and the length scale resulting in a Friedmann-like equation. On dimensional τ τ L at ¼ in can be defined locally, without referring to the grounds we expect the corresponding Friedmann equation τ τ behavior of the solution away from ¼ in,as in vacuum to read

043512-5 SEBASTIAN F. BRAMBERGER et al. PHYS. REV. D 97, 043512 (2018)

H2 ¼M2fðjKj/a2M2Þ: ð34Þ matter (with a general equation of state w) and curvature. Curvature can be seen as a fluid with energy density, The sign on the right-hand side may be either positive or ρ 3 2 2 2 2 negative and the following argument does not rely on the K ¼ MPlM fðjKj/a M Þ; ð38Þ choice of the sign. Addition of matter energy density ρ [or some component that stems from gravity but that behaves and we can tweak this formula as appropriate, to contain the like matter, such as the term C/a3 in (17)] then leads to concrete model. Equipartition, then, implies 1 ρ 2 2 2 2 ρ ≈ ρ ∼ ρ ; ð39Þ H ¼ 3 2 M fðjKj/a M Þ; ð35Þ K in MPl which is equivalent to the suppression of curvature K/a2 where we have assumed that the ratio between the effective derived from the anisotropic instanton presented in Sec. III. for the homogeneous and isotropic ρ 2 −1 1 However, defined in terms of K there is no suppression. cosmology and ð8πM Þ is (approximately) constant ρ ∼ ρ ∼ ρ Pl Indeed K in initially and the subsequent evolution and we have absorbed such a ratio into the definition of takes care of the suppression. Whether we phrase things in ρ. To complete the system we have to specify the second 2 terms of K/a or ρK the final result is the same. Friedmann equation (which indeed was the starting point Let the curvature be measured by for our concrete model), or alternatively, the conservation equation for ρ. Let us first assume conservation (this is ρ Ω ¼ K : ð40Þ in fact not needed and violations of energy conservation K ρ þ ρ only refine and reinforce the argument; see Appendix C K ρ for details). With a general equation of state w ¼ p/ we Using (37) and (38) with (33) we see that for M2 ≪ then have 2 ≪ 2 ρ 2 2 1/z ρ ≪ jKj/a M ð in/MPlM Þ or equivalently for z→1 ρ ≪ ρ , where ρ_ þ 3Hð1 þ wÞρ ¼ 0; ð36Þ in   2 2 3ð1þwÞ M M 2z integrating into ρ ∼ ρ Pl z→1 in ρ ; ð41Þ in 1 ρ ∝ : ð37Þ a3ð1þwÞ we have

In our concrete model we have z ¼ 3 and w ¼ 0, but this 3ð1þwÞ−2z ΩK ∝ a ; ð42Þ setup is more general. Let us now assume that at some time, deep in the UV 2 2 whereas for jKj/a ≪ M or equivalently for ρ ≪ ρ →1 we M z regime far beyond the scale , the Lorentzian signature have the standard flatness problem instability: Universe is created, after which it is subject to (semi) classical evolution. We assume that the theory we are Ω ∝ 3ð1þwÞ−2 considering is UV complete, so there is no need to fear K a : ð43Þ going beyond the scale M. This “initial time” of creation Ω can be seen as the result of tunneling from vacuum, via an So that K may be suppressed in the first stage of evolution instanton, similar to our concrete model, or it can be the we see that a necessary condition for solving the flatness result of any other process, e.g. a phase transition from a problem in an expanding Universe is disordered quantum geometry. The point is that the 3ð1 þ wÞ Universe undergoes a transition into (semi)classical evo- z> : ρ 2 ð44Þ lution in the UV complete theory at a density in, assumed to be ρ ≫ M2 M2. in Pl 3 Let us now also assume that an equipartition principle In our concrete model this is satisfied since z ¼ and 0 3 is in action, that is, we assume roughly equal amounts of w ¼ , but in fact for the z ¼ HL theory this would work 1 1 energy for different types of contributions that enter the with any w< . With standard gravity (i.e. z ¼ )we −1 Hamiltonian. In our setting there are just two contributions: would need w< /3, i.e. inflation. The above is a necessary but not sufficient condition. ρ 1 The exact condition will involve M and in as well as z and In the concrete example of the previous section, this ratio w. Assuming for simplicity the Universe exits the UV phase depends on λ and thus is in general subject to running under the 2 2 RG flow. However, as we have assumed the existence of a UV around jKj/a ∼ M to enter a standard hot big bang model, fixed point with finite λ, this assumption is justified. then curvature must be suppressed at this time by

043512-6 SOLVING THE FLATNESS PROBLEM WITH AN … PHYS. REV. D 97, 043512 (2018)     2 4 1 T M 2 initial condition is expected to satisfy this scaling property Ω ≪ Ω ≡ CMB Pl K sup zeq ρ ; ð45Þ with L ≪ 1/M, meaning that the curvature length scale of MPl z→1 the Universe is much longer than the expansion time scale. This is exactly what is needed to solve the flatness problem. where TCMB is the present temperature of the cosmic microwave background, we have used (43) with w ¼ 1/3 Based on the projectable version of the HL theory for concreteness, we have found a family of instanton solutions and 0 before and after matter radiation equality, and zeq is ρ ∼ 4 parametrized by an integration constant C. This family of the of matter radiation equality. If z→1 MPl, with standard assumptions we have roughly Ω ∼ 10−60,asis solutions is unique under the FLRW ansatz for the pure sup gravity system, i.e. without any matter fields, for a given set well known. of parameters in the action. For positive and large enough In order to obtain this suppression while ρ →1 < ρ < ρ z in C, the spatial size a and the (Euclidean) temporal size τ we should therefore impose the condition in in of the instanton are decreasing functions of C. We con- 3 3 1 τ ≃ ρ − ð þwÞ firmed the scaling relation ain/ in const. in the large C in 2z−3ð1þwÞ ≫ Ωsup ; ð46Þ limit, both numerically and analytically. Moreover, by ρ →1 z τ τ _ 2 defining T and L locally at ¼ in through H and K/a 2 3 where we have used (42) in conjunction with ρ conserva- as in (29), we have seen that the scaling T ∼ M L holds tion [and solution (37)], even though the latter is not strictly independently from the boundary condition and any physi- ρ Ω ρ 0 necessary. Expressing z→1 and sup in terms of in and M, cal conditions near a ¼ . We call those instantons with this translates to in four-dimensional Euclidean spacetime (but   with isotropy in three-dimensional space) anisotropic 4z ρ 1 M M 2z−3ð1þwÞ instantons. The anisotropic instanton provides a concrete in ≫ Pl : ð47Þ M2 M2 z T2 example of a physical system that realizes the scaling Pl eq CMB property T ≃ M2L3 and thus may solve the flatness Equation (47) is the general condition for solving the problem in cosmology. flatness problem in the vast class of models considered We have also given a more general argument for the here. For the concrete model proposed in this paper (z ¼ 3, solution of the flatness problem presented here, based on 0 the assumption of equipartition among different contribu- w ¼ ), if we set M ¼ MPl for concreteness then we have   tions of energy density to the Hamiltonian of the system. ρ1/4 1 M 2 The equipartition between the highest time derivative term in ≫ Pl ∼ 1058: ð48Þ M z T and the highest spatial gradient term can be considered as a Pl eq CMB restatement of the anisotropic scaling and thus is expected Equation (47) establishes the general condition for a to be universally applicable to many physical systems in solution of the flatness problem in general UV complete any possible UV complete theories with anisotropic scal- theories with anisotropic scaling. In summary, they must ing. We showed that a large class of theories and cosmo- start operating sufficiently above the Planck scale and logical models satisfying this property will be free from the satisfy equipartition in some form at this initial point. This flatness problem. The flatness of the Universe is then an applies to our concrete model with a starting point defined expression of the fact that the Universe started deep in the by an anisotropic instanton. However, the formal mecha- UV regime and of this scaling property of quantum gravity. nism is more general. ACKNOWLEDGMENTS V. SUMMARY AND DISCUSSIONS The work of S. F. B. is supported in part by a grant from We have proposed a possible solution to the cosmologi- the Studienstiftung des Deutschen Volkes. S. F. B. and cal flatness problem without relying on inflation. To do so A. C. are grateful for the hospitality of the Yukawa we have made use of the renormalizable theory of gravity Institute of Theoretical Physics where some of this work called Horava-Lifshitzˇ (HL) gravity. We further assumed was completed. The work of A. C. was undertaken in part that the initial condition of the Universe respects the so- as an overseas researcher under a Short-Term Fellowship of called anisotropic scaling (2), with z ¼ 3 which is the the Japan Society for the Promotion of Sciences. The work minimal value that guarantees renormalizability of HL of J. M. was supported by a STFC consolidated grant and gravity. Because of this scaling, any physical system in the John Templeton Foundation. The work of S. M. was the deep ultraviolet (UV) regime tends to possess the supported by Japan Society for the Promotion of Science scaling property T ≃ M2L3, where T and L are the time (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) scale and the length scale of the system and M is the No. 17H02890, No. 17H06359, and No. 17H06357. The momentum scale characterizing the anisotropic scaling. work of S. M. and Y. W. was supported by World Premier If the Universe started in the deep UV regime then the International Research Center Initiative (WPI), MEXT,

043512-7 SEBASTIAN F. BRAMBERGER et al. PHYS. REV. D 97, 043512 (2018)

3 3λ − 1 3 1 Japan. The work of R. N. was supported by the Natural ð Þ 2 C − K ρ ρ Λ Sciences and Engineering Research Council of Canada and 2 H ¼ 3 2 þ 2 ð mat þ radÞþ ; ðB1Þ a a MPl by the Lorne Trottier Chair in Astrophysics and Cosmology at McGill. The work of Y. W. was supported by JSPS ρ ρ Grant-in-Aid for Scientific Research No. 16J06266 and by where mat and rad are the energy densities of radiation the Program for Leading Graduate Schools, MEXT, Japan. and matter, respectively, and we have absorbed the ratio between the effective gravitational constant for the homo- 8π 2 −1 geneous and isotropic cosmology and ð MPlÞ into the APPENDIX A: SCALE-INVARIANT ρ ρ definition of matter and rad. We have recovered the PERTURBATION cosmological constant Λ to account for the late-time In the projectable Horava-Lifshitzˇ gravity, there are three accelerated expansion. We further assume an instantaneous physical degrees of freedom: two from the tensor graviton reheating by the decay of C for simplicity, and that the ρ ρ and one from the scalar graviton. Actually, one can values of C, rad and mat shift before and after reheating as consider the scalar graviton as a perturbation of the dark matter as integration constant, i.e. the C/a3 term in (17).In t

043512-8 SOLVING THE FLATNESS PROBLEM WITH AN … PHYS. REV. D 97, 043512 (2018)

2/3 2/9 1/9 τ α Ω t0 ρ ρ propagation for massless particles. This falls under the in ≃ 3 j Kð Þj eq 0 28/93 Ω8/9 1 2/3 2/3 general umbrella term of “varying ” (see [19] ain s m ð þ zrehÞ M   Pl  for an early review). In the guise of MDRs, such theories M 2/3 3λ − 1 1/2 ≃ 1.80 × 10−47 Pl lead to several astrophysical and cosmological implications M 2   (e.g. [20,21]). The phenomenon may be quantified by the 10 2/3 10 jΩ ðt0Þj phase speed cp ¼ E/p or the group speed cg ¼ dE/dp. × K ; ðB7Þ 1 þ z 0.005 In the case of (32) and (33), in the UV we have reh   p z−1 ≡ α−1/4 where M 3 as defined in (31), and in the last approxi- cp ∝ cg ∝ : ðC1Þ mate equality we have used the observed values to plug in M ρ ≃ 5 67 10−28 4 ρ ≃ 1 01 10−30 4 eq ð . × MPlÞ , 0 ð . × MPlÞ and In view of this, it is tempting to map the Friedmann Ω ≃ 0 308 1 m . . Since we have set K ¼ , the scale factor a equation (35) into the standard-looking Friedmann equation: has the dimension of length. As we learn from (B7), we need an anisotropic instanton 1 2 2 Kch ≲ −47 H ¼ ρ − ; ðC2Þ with the level of anisotropy of order T/L 10 in order 3 a2 to respect the observational upper bound jΩKðt0Þj ≲ 0.005 [18], provided that the reheating occurs before Big Bang also with a time dependent c, and where we have reinstated K ≳ 1010 λ nucleosynthesis (zreh ), that at the time of tunneling as the culprit for the sign ambiguity of the curvature term λ 1 ≠ 0 does not deviate much from its (expected) IR value IR ¼ , (relevant in what follows). Assuming K we have ∼ O 2 2 2 and that M/MPl ð1Þ. This small value is to account ch ¼ a M f, so that in the UV for the present flatness of the Universe by the proposed   −1 mechanism in Sec. III, which in the inflationary cosmology 1 z c ≈ : ðC3Þ would be compensated by the duration of inflation ∼50–60 h Ma e-foldings. This also sets the lower bound on the energy ∝ ∝ scale that the instanton tunneling has to occur. At the time We see that in the deep UV we have ch cg cp (with the of this transition, (17) gives understanding that comparisons assume the replacements 2 2 2   E → H and p → jKj/a2). Thus in the deep UV thevarious 3 3ð3λ − 1Þ C α3 α3 a c may be used interchangeably. The transition from UV to IR H2 ∼ ∼ ≃ in 2 in 3 6 3 τ may be different, but this is not important here. ain ain s in   This interpretation at once connects the solution of the M4 1 þ z 2 ≃ ð1.28 × 1035Þ4 reh flatness problem presented here to that in [22]. This is M2 1010   Pl particularly relevant if we wish to consider the implication 0.005 3 of nonconservation of energy mentioned above. As shown × ; ðB8Þ in [22] such violations actually help to solve the flatness jΩ ðt0Þj K problem, reinforcing the argument. As is well known, violations of Lorentz invariance may where Hin is the value of the Hubble parameter at the time of instanton transition (in Planck units), and this bring about nonconservation. This depends on how we close the system started by (C2). In the concrete model correspondsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi to the energy scale at the transition to be ≡ p ≳ 1035 presented in this paper, conservation of ρ is assumed (or Ein MPlHin M. If we set M ¼ MPl and if reheating occurs at the time K/a2 ∼ ð1/M4ÞðK/a2Þ3, rather, one starts from the second Friedmann equation and ρ ∼ ∼ 1056 Ω −3/2 ∼ then integrates it into the first, building conservation into i.e. C zrehj Kðt0Þj , then one finds zreh 1060 Ω −1/2 2 ∼ the model). An alternative is to assume no modifications j Kðt0Þj and therefore from (B8), Hin 58 4 −7/2 to the second Friedmann equation: ð10 Þ jΩKðt0Þj , which is consistent with the general results in Sec. IV. ä 1 ¼ − ρð1 þ 3wÞ: ðC4Þ a 6 APPENDIX C: A MORE GENERAL SOLUTION TO THE FLATNESS PROBLEM This implies violations of the Bianchi identities and energy In Sec. IV we showed how the concrete model presented conservation. Specifically, in combination with (C2) we in this paper may be part of a more general class of find solutions. In this appendix we expand further on this a_ 6Kc2 c_ argument, both in scope and in terms of interpretation. ρ_ þ 3 ρð1 þ wÞ¼ : ðC5Þ a a2 c There is a simple interpretation of the general argument presented in Sec. IV. It is known that modified dispersion Merely looking at the sign of the rhs is very informative. ρ 3 2 _ relations (MDR) may lead to an energy dependent speed of Defining in ¼ H we see at once that if c/c<0 the

043512-9 SEBASTIAN F. BRAMBERGER et al. PHYS. REV. D 97, 043512 (2018) violations of energy conservation act so as to push the leading to (42), which was obtained by ignoring violations Universe towards flatness. If the Universe is closed (K ¼ 1 of energy conservation. Thus these violations are not very ρ ρ and thus supercritical, > in) then energy is removed important in the solution to the flatness problem, as long as from the Universe; if the Universe is open (K ¼ −1 and curvature is already sufficiently suppressed. ρ ρ < in) then energy is inserted into the Universe. Where these violations may be interesting is in situations No violations occur for a flat model. Thus ρ is pushed in which the Universe does not start from exact equiparti- ρ to in. tion. Let us consider an extreme case. Suppose that initially This does not mean that these violations are needed, or ρ ¼ 0 and K ¼ −1, that is, a Milne Universe beginning. indeed relevant in all regimes. As in [22] we can combine Then the Universe starts with ΩK ¼ 1 and no matter. This (C2) and (C4) to obtain would be hopeless if energy were conserved (the Universe would simply remain empty). However inserting this a_ c_ condition into (C6), we see that the first term initially Ω_ ¼ð1 − Ω ÞΩ ð1 þ 3wÞþ2 Ω : ðC6Þ Ω ∝ 2 K K K a c K vanishes, but the second term leads to K c . Hence curvature is still suppressed (at this rate) while matter is Ω ≪ 1 being dumped into the Universe. ρ ¼ 0 is also pushed to If K this integrates to ρ ρ Ω ≪ 1 ¼ in. Eventually K , after which violations of energy conservation become irrelevant, and suppression Ω ∝ 1þ3w 2 K a c ðC7Þ of curvature proceeds according to (C7).

[1] P. Horava, Phys. Rev. D 79, 084008 (2009). [12] J. Feldbrugge, J. L. Lehners, and N. Turok, Phys. Rev. D 97, [2] T. P. Sotiriou, M. Visser, and S. Weinfurtner, J. High Energy 023509 (2018). Phys. 10 (2009) 033. [13] S. W. Hawking, Pontif. Acad. Sci. Scr. Varia 48, 563 [3] A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. (1982). Sibiryakov, and C. F. Steinwachs, Phys. Rev. D 93, [14] S. W. Hawking, Nucl. Phys. B239, 257 (1984). 064022 (2016). [15] J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 [4] A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. (1983). Sibiryakov, and C. F. Steinwachs, arXiv:1705.03480. [16] J. B. Hartle, S. W. Hawking, and T. Hertog, Phys. Rev. D 77, [5] S. Mukohyama, J. Cosmol. Astropart. Phys. 06 (2009) 001. 123537 (2008). [6] S. Mukohyama, Classical Quantum Gravity 27, 223101 [17] J. D. Dorronsoro, J. J. Halliwell, J. B. Hartle, T. Hertog, and (2010). O. Janssen, Phys. Rev. D 96, 043505 (2017). [7] S. Mukohyama, Phys. Rev. D 80, 064005 (2009). [18] P. A. R. Ade et al. (Planck Collaboration), Astron. [8] S. Mukohyama, J. Cosmol. Astropart. Phys. 09 (2009) 005. Astrophys. 594, A13 (2016). [9] A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. [19] J. Magueijo, Rep. Prog. Phys. 66, 2025 (2003). Sibiryakov, and C. F. Steinwachs, Phys. Rev. Lett. 119, [20] G. Amelino-Camelia, Int. J. Mod. Phys. D 11,35 211301 (2017). (2002). [10] J. Feldbrugge, J. L. Lehners, and N. Turok, Phys. Rev. D 95, [21] G. Amelino-Camelia, M. Arzano, G. Gubitosi, and J. 103508 (2017). Magueijo, Phys. Rev. D 87, 123532 (2013). [11] J. Feldbrugge, J. L. Lehners, and N. Turok, Phys. Rev. Lett. [22] A. Albrecht and J. Magueijo, Phys. Rev. D 59, 043516 119, 171301 (2017). (1999).

043512-10