PHYSICAL REVIEW D 97, 043512 (2018) Solving the flatness problem with an anisotropic instanton in Horava-Lifshitzˇ gravity 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 (Albert Einstein 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 Universe (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 spacetime, 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 horizon problem without inflation. 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 cosmological constant problem and we simply assume that Λ has the observed value. The slowest decaying In general relativity 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 cosmology. 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 scale factor; 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 matter (∝ 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; ÁÁÁ), each of which is dis- replacing the big bang 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.