PHYSICAL REVIEW D 98, 123512 (2018) Oscillons in higher-derivative effective field theories † Jeremy Sakstein* and Mark Trodden Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, 209 S. 33rd St., Philadelphia, Pennsylvania 19104, USA (Received 26 September 2018; published 13 December 2018) We investigate the existence and behavior of oscillons in theories in which higher derivative terms are present in the Lagrangian, such as Galileons. Such theories have emerged in a broad range of settings, from higher-dimensional models, to massive gravity, to models for late-time cosmological acceleration. By focusing on the simplest example—massive Galileon effective field theories—we demonstrate that higher derivative terms can lead to the existence of completely new oscillons (quasibreathers). We illustrate our techniques in the artificially simple case of 1 þ 1 dimensions, and then present the complete analysis valid in 2 þ 1 and 3 þ 1 dimensions, exploring precisely how these new solutions are supported entirely by the nonlinearities of the quartic Galileon. These objects have the novel peculiarity that they are of the differentiability class C1. DOI: 10.1103/PhysRevD.98.123512 I. INTRODUCTION conserved. When the conserved charge results from a global U(1) symmetry [or an unbroken U(1) of some Essentially all of modern physics is described by non- higher gauge group], these objects are typically referred linear partial differential equations (PDEs). In many cases, to as Q-balls [1]. it is sufficient to study highly symmetric configurations Another class of interesting nonlinear solitary objects such that these reduce to ordinary differential equations or are oscillons, sometimes called breathers. Oscillons are to study some linearized regimes in which the equations are stable, extended, quasiperiodic (in time) particlelike exci- easily solved. More generally, of course, multidimensional tations. These are objects with no conserved charges at all and nonlinear processes are crucial to the understanding of and for this reason are typically found in real scalar field a host of phenomena, from weather patterns, to black hole theories. Their stability is due to nonlinearities that result physics, to the early universe. Among the rich and varied in some of their comprising harmonics being localized and phenomena exhibited by nonlinear PDEs, one class of unable to propagate to infinity. In some cases, breathing particularly interesting objects not present in the linear objects may not be exact solutions, and so may be long- regime is solitons, localized particlelike excitations that can lived rather than absolutely stable. It is these objects, be stable or extremely long-lived. Some examples of referred to as quasibreathers, that this paper is concerned solitons include solitary water waves, domain walls, kinks, with. Oscillons can occur in Bose-Einstein condensates and Skyrmions, and vortices. Typically, solitons can be classi- may be relevant in the early universe. Indeed, the end state fied into two classes: topological and nontopological of many models of scalar field inflation is a universe solitons. The former are localized and stable due to the dominated by oscillons that form after the scalar conden- presence of nontrivial homotopy groups of the vacuum sate fractures due to some instability or through parametric manifold of the field equations. These lead to defects that reheating [2,3]. Classical oscillons are supported by non- cannot relax to zero energy configurations due to some linear potentials that are shallower than quadratic away conserved topological quantity. The latter are localized from the minimum [4], but oscillons have also been found and carry some conserved Noether charge Q that requires in Pðϕ;XÞ theories [where X ¼ −ð∂ϕÞ2=2 is the canonical an energetically unfavorable decay in order to remain kinetic term] [5], and, in particular, the Dirac-Born-Infeld action can give rise to such objects. Oscillons can also exist *[email protected] in multifield theories and may form after multifield † [email protected] inflation driven by string moduli [6]. The purpose of this paper is to determine the existence Published by the American Physical Society under the terms of of oscillons in more general classes of higher derivative the Creative Commons Attribution 4.0 International license. – Further distribution of this work must maintain attribution to effective field theories [7 17]. We focus in particular on a the author(s) and the published article’s title, journal citation, simple representative of such theories, known as Galileons and DOI. Funded by SCOAP3. [8], and perform a preliminary investigation of their 2470-0010=2018=98(12)=123512(13) 123512-1 Published by the American Physical Society JEREMY SAKSTEIN and MARK TRODDEN PHYS. REV. D 98, 123512 (2018) properties. Galileon theories are a class of higher-derivative their contribution to the equations of motion enters at the effective field theories whose equations of motion are same order in ε as the canonical kinetic term and mass. precisely second order and therefore are not plagued by (Another way of saying this is that they enter at a lower Ostrogradski ghosts. Rather than being a fully covariant order in ε than one would näively expect.) This approach theory, they are defined on Minkowski space and as such is akin to the construction of so-called flat-top oscillons [4], they arise in the decoupling or low-energy limits of many where one guarantees that higher-order operators are different physical theories, including the Dvali-Gabadaze- important by taking some dimensionless parameter to be Porrati (DGP) braneworld model [18], ghost-free Lorentz- ≫1. While this is necessary to find solutions analytically, invariant massive gravity [19,20], and higher-derivative one typically finds similar large-amplitude objects numeri- covariant theories constructed to avoid Ostrogradski ghosts cally for generic parameter choices [4], and so this [9,10]. Our motivations for searching for breather solutions assumption can ultimately be relaxed. in Galileon theories are manyfold. First, Galileons may Our main results are as follows: 1 play a role in the early universe. Indeed, Galileon inflation (1) 1 þ 1 dimensions: Only the cubic Galileon exists, has been extensively studied (see [28,29] for example), but and it is unable to support quasibreathers without the end point of this process is not clear. Finding Galileon the aid of other non-Galileon operators. We give an oscillons is one step toward determining if Galileon breath- example of this by including shift-symmetric, but ers could be formed after inflation. Similarly, alternatives to not Galileon symmetric, operators and find novel inflation, in particular Galilean genesis, utilizes Galileon breather solutions. This example serves as a warm- field theories [30,31]. Another motivation is that massive up exercise designed to gain an analytic handle on Galileons are a proxy for ghost-free massive gravity [32] the problem, and to glean insight into the construc- and oscillons may act as a potential signature of these tion of Galileon oscillons before moving to higher theories. Finally, the Vainshtein mechanism and nonre- dimensions, where the equations are less analytically normalization theorems that Galileon theories enjoy [8,33] tractable. We construct the profiles for these objects mean that the higher-derivative operators are within the numerically and calculate their amplitude as a regime of validity of the effective field theory (EFT) in function of the model parameters. contrast to Pðϕ;XÞ theories. One example where this is (2) 3 þ 1 dimensions: We find novel oscillons/quasi- problematic for solitons is Skyrmion theories where higher- breathers supported by the nonlinearity of the quartic derivative operators are required for stability so that such Galileon and the canonical kinetic term. These objects objects are outside the regime of validity of the EFT. For are solutions of a highly nonlinear second-order this reason, finding solitons in Galileon theories is inter- ordinary differential equation. Surprisingly, and in esting in its own right. There is a no-go theorem for static contrast to other oscillons relevant for cosmology, solitons [34] but the existence and stability of time- these objects are C1 functions and inhabit the space dependent solitons is still unexplored. W1;1.2 In the next section we give a brief introduction to A. Summary of results and plan of the paper oscillons/quasibreathers in theories with nonlinear poten- tials and Pðπ;XÞ [with X ¼ −ð∂πÞ2=2] operators. In the In this paper we perform a first investigation of oscillon subsequent section we introduce the massive Galileon EFT solutions in massive Galileon EFTs. Because the equations and discuss some salient theoretical features. In Sec. IV we of motion resulting from higher-derivative theories are derive and present the main results of our work outlined rather complicated, it is instructive to find the simplest above. We begin with the warm-up example of constructing system that encapsulates the relevant and new physics; Galileon breathers in 1 þ 1 dimensions before moving as we shall see, Galileons satisfy this requirement. The on to study the quartic Galileon in 3 þ 1 dimensions. We existence of Galileon operators is dimension and geometry conclude and discuss the implications of our findings as dependent. For this reason, we will analyze Galileon well as future directions in Sec. V. Our metric convention breathers in d 1 dimensions where d 1, 2, 3 separately. þ ¼ is the mostly positive one, and we work in units where We determine the existence of oscillons by looking for ℏ ¼ c ¼ 1. small-amplitude solutions controlled by some parameter ε < 1, which is used as an expansion parameter to construct an asymptotic series. Furthermore, we choose parameters 2We remind the reader that a function belongs to the differ- such that the Galileon operators are sufficiently large that entiability class Ck if its derivatives f; f0; …;fðkÞ exist and are continuous.