Oscillons in Higher-Derivative Effective Field Theories

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 potentials 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. A function belongs to the Sobolev space Wk;p if its 1 weak derivative up to order k has a finite Lp norm. The existence Galileons were also a dark energy candidate [21], although 1 their phenomenological viability is more tenuous after of C oscillons is not unreasonable since the solution space GW170817 [22–26] and cosmic microwave background of PDEs is indeed Wp;k rather than C∞ (or even C2). Indeed, the measurements [27]. Hilbert space for the Schrödinger equation is Hk ¼ Wk;2.

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II. OSCILLONS AND QUASIBREATHERS which we will assume from the outset. Note that, by construction, the frequency of the fundamental harmonic Oscillons are localized quasiperiodic excitations of always gets shifted to values smaller than the natural scalar field theories whose existence is not due to a frequency so that the oscillon is primarily supported by conserved Noether charge nor to nontrivial homotopy harmonics with frequencies smaller than the field’s mass groups of the vacuum manifold [35,36]. In some cases, (recall that higher harmonics are suppressed by powers they are exact solutions and are therefore absolutely stable. of ε). This ensures that the oscillon is stable on a long In others, they may only be an asymptotic series labeled by timescale since these modes are unable to propagate. The some small parameter ε with a finite radius of convergence, higher harmonics (with frequencies greater than the field’s say jεj ≤ ε0. It is common that the radius of convergence is mass) can propagate and ultimately lead to radiation at zero and in these cases the objects are referred to as infinity [38–41].3 Had we instead chosen conditions such quasibreathers, since they eventually decay through radi- that the frequency is shifted to higher values, the object ation but are often long-lived. We will use both terms would be unstable and would rapidly decay to free interchangeably in what follows. As an example, consider radiation. the theory in D ¼ d þ 1 dimensions [5,37] Z Given the above considerations, we make the ansatz 1 λ3 S ¼ ddþ1x X − ξX2 þ− m2π2 − π3 X∞ 2 3 π τ ρ n ð ; Þ¼ ε ϕn: ð7Þ λ 1 − 4 π4 n¼ 4 þ ; ð1Þ The equation of motion resulting from (2) up to Oðε3Þ in where the ellipsis denotes higher derivative and higher the new coordinates is order terms in the EFT. This can be brought into dimen- d−1 =2 μ μ ̈ 2 ̈ 2 3 ̈ sionless form by rescaling π → mð Þ π, x → x =m, εðϕ1 þ ϕ1Þþε ðϕ2 þ ϕ2 þ λ3ϕ1Þþε ðϕ3 þ ϕ3 dþ1 ð5−dÞ=2 d−3 ξ → ξ=m , λ3 → λ3m , and λ4 → λ4m , to yield 3 _ 2 ̈ þ 2λ3ϕ1ϕ2 þ λ4ϕ1 þ 3ξϕ1ϕ1Þ¼0; ð8Þ Z 1 λ λ dþ1 − ξ 2 − π2 − 3 π3 − 4 π4 where a dot denotes a derivative with respect to τ and ∇2 is S ¼ d x X X þ 2 3 4 þ ; the Laplacian operator defined with respect to ζi. Equating ð2Þ the OðεÞ term to zero one finds that ϕ1 obeys the equation of a simple harmonic oscillator with unit frequency where all fields and coupling constants are now dimension- ̈ less. We are looking for quasiperiodic localized solutions, ϕ1 þ ϕ1 ¼ 0; ð9Þ and so we will assume a solution in the form of the asymptotic series whose radius of convergence is governed so that with a suitable choice of initial conditions one has by some small parameter ε ≪ 1. We anticipate objects whose characteristic size decreases with increasing ε, and ϕ1ðτ; ρÞ¼Φ1ðρÞ cosðτÞ; ð10Þ so we define the new spatial coordinates for an (as yet) undetermined function Φ1ðρÞ. Substituting ζi ε i 1 … 2 ¼ x ;i¼ ; ;d: ð3Þ this into the Oðε Þ equation one finds We will be interested in homogeneous solutions with an 1 internal SOðdÞ symmetry, and so we also define ϕ̈ ϕ − Φ2 1 2τ 2 þ 2 ¼ 2 1½ þ cosð Þ: ð11Þ 2 ij 2 ij ρ ¼ η ζiζj ¼ ε η xixj: ð4Þ The unique solution of this equation, imposing the initial _ Because we expect that the nonlinearities will shift the conditions ϕ2ð0; ρÞ¼ϕ2ð0; ρÞ¼0 (imposing these con- characteristic frequency away from 1 (m if one puts the ditions is tantamount to shifting the τ origin), is dimensions back in), we define a new time coordinate X∞ 3Mathematically, the oscillon will radiate because the asymp- j τ ¼ ωðεÞt; ωðεÞ¼1 þ ε ωj; ð5Þ totic expansion does not converge for any value of ε, and so an j¼1 oscillon formed from some initial data will radiate as it relaxes to the true solution. (Exceptions to this are integrable theories where for constants ωj. It is convenient to choose initial con- the oscillon is an exact solution such as the sine-Gordon model.) ditions and use the freedom of shifting the time coordinate The radiation occurs with a highly suppressed rate because the to set frequencies of the Fourier modes that comprise the oscillon are pffiffiffiffiffiffiffiffiffiffiffiffi OðεÞ while those of the outgoing radiation modes are Oð1Þ [42]. ω 1 − ε2 Oscillons may also radiate due to quantum effects [42] or the ¼ ; ð6Þ effects of cosmological backgrounds [43].

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Z 5 λ3 2 X gal ϕ Φ −3 τ 2τ 4 On ðπÞ 2 2 2 2 ¼ 6 1½ þ cosð Þþcosð Þ: ð12Þ S ¼ d x c þ L ð∂ π; ð∂ πÞ ; …Þ ; W i Λ3ðn−2Þ HD n¼2 ε At third order in we then find ð20Þ − 1 ̈ 2 ðd Þ 3 where L represents Galileon-invariant higher-derivative ϕ3 þ ϕ3 ¼ ∂ρΦ1 þ ∂ρΦ1 − Φ1 þ ΔΦ1 HD ρ operators acting on the field. Importantly, there is a × cosðτÞþ½cosð2τÞ þ ½ cosð3τÞ; ð13Þ powerful nonrenormalization theorem that protects the Galileon operators in the action (20) [8,33,44]. In particu- with lar, the coefficients ci are not corrected by loops but rather L the coefficients of the operators appearing in HD receive 10 O 1 Δ ≡ ξ − λ λ2 ð Þ corrections. It is therefore consistent to choose to take 4 þ 9 3: ð14Þ the Galileon operators alone as an effective field theory, since one can maintain a parametrically large separation of τ L Now, the term proportional to cosð Þ is a resonance term; scales between these operators and those appearing in HD. i.e., it oscillates at the natural frequency. We therefore have Another nice feature of the Galileon operators is the a secular growth in ϕ3 unless the coefficient of this term Vainshtein mechanism. Fluctuations of the field about δπ vanishes. This gives us a nonlinear equation for Φ1ðρÞ in some background π0ðxμÞ can be described by the effective terms of Δ. Quasibreathers can exist if this equation has a action nontrivial solution. One can show [5,37] that this is the case Z provided that Δ > 0. (We do not give the proof here since it 4 μν∂ δπ∂ δπ is not relevant in what follows; we refer the interested S ¼ d xZ μ ν ; ð21Þ reader to Refs. [5,37].) This procedure of equating each term at order εn to zero can be repeated ad infinitum to build where the kinetic matrix is up the oscillon profile order by order by demanding that all 0 □π0 2 − ∇ ∇ 2 secular resonance terms vanish [37]. μν μν □π ð Þ ð μ νÞ Z η b3 b4 : ¼ þ Λ3 þ Λ6 þ ð22Þ III. MASSIVE GALILEON EFFECTIVE FIELD THEORY This allows for three regimes: the linear regime where □π0=Λ3 ≪ 1; i.e., the fluctuations behave as if they are Galileon-invariant scalar field theories are those whose free; the nonlinear or Vainshtein regime where □π0=Λ3 ≳ 1 actions are invariant under the Galileon symmetry but ∂2=Λ2 ≪ 1 so that nonlinearities are important but μ μ μ quantum corrections are not; and the quantum regime πðx Þ → πðx Þþvμx þ c; ð15Þ 2 2 where ∂ =Λ ≫ 1. This hierarchy is reminiscent of the one appearing in general relativity where nonlinearities are with c and vμ constant. In four spacetime dimensions there important below some scale (the Schwarzchild radius) but are four operators that respect this symmetry [8]: quantum corrections are only important at a much smaller

gal μ1 μn−1 scale, the Planck length. Indeed, one can define an On ðπÞ¼πΠ μ Πμ ; n ¼ 2; 3; 4; 5; ð16Þ 0 3 ½ 1 n−1 □π ∼ Λ analogous Vainshtein radius by ðrVÞ . The Vainshtein mechanism plays an important role in the where Πμν ∂μ∂νπ and square brackets denote the trace of ¼ phenomenology of Galileon theories. In particular, new η a tensor with respect to the Minkowski metric μν. Since forces mediated by Galileons are highly suppressed inside these operators contain two, three, four, and five powers of the Vainshtein radius, allowing them to evade solar system the field, they are referred to as the quadratic, cubic, quartic, tests of gravity [45]. Galileon fluctuations move on the light and quintic Galileons, respectively. Individually, one has cones of Zμν, and one ubiquitously finds superluminal phase and group velocities [46]. This, among other issues, Ogal π π□π 2 ð Þ¼ ; ð17Þ has raised the question of whether Galileons can be embedded into a Lorentz-invariant UV completion gal 2 2 O3 ðπÞ¼πð½Π − ½Π Þ; ð18Þ [47–50], a question which we will not discuss further here. As discussed by [32], the technical obstructions to embed- gal 3 2 3 O4 ðπÞ¼πð½Π − 3½Π½Π þ2½Π Þ: ð19Þ ding the Galileons into a Lorentz-invariant UV completion can be ameliorated if the Galileon has a mass. Furthermore, We will not work with the quintic Galileon in this work for the mass term only breaks the Galileon symmetry softly reasons that we will discuss later. The Wilsonian effective (provided that the field only couples to other fields in a action for a general Galileon theory is then Galileon-invariant manner) so that the nonrenormalization

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gal theorem is not spoiled [33]. The Galileons as defined above Furthermore, the equation of motion for On ðπÞ vanishes arise in certain decoupling limits of other theories, most identically for configurations in which the Galileon field notably DGP braneworld gravity [18] (and its generaliza- depends on n − 1 or fewer coordinates [56]. For example, tions [51]) and ghost-free massive gravity [19,20]. In the the cubic Galileon vanishes in 1 þ 1 dimensions for static former case, the scalar plays the role of the brane-bending configurations and the quartic vanishes for cylindrical mode, while in the latter case the scalar is the helicity-0 configuration in 2 þ 1 dimensions. We are interested in mode of the massive graviton. Thus, massive Galileons are Sd−1-symmetric configurations in D ¼ d þ 1 dimensions, naturally embedded in interacting massive spin-2 theories. and therefore we expect the cubic to contribute in 1 þ 1 Allowing a mass term for the Galileon opens up the dimensions, the quartic and the cubic to contribute in 2 þ 1 possibility of finding periodic quasibreathers, and so the dimensions, and all Galileon operators to contribute in action we will consider in this work is 3 þ 1 dimensions (although we will not study the quintic). Z It is useful to work with dimensionless quantities in what X4 gal On ðπÞ 1 follows, and so we rescale the coordinates and fields as S ¼ d4x c − m2π2 ; ð23Þ i 3ðn−2Þ 2 2 Λ n¼ μ μ d−1 x˜ ¼ mx ; π ¼ m 2 π˜ ð25Þ with c2 ¼ −1=2, and where Λ and m are constants with units of mass. We will consider the case c4 > 0 so that and define the theory admits the Vainshtein mechanism [8] (we also pffiffiffiffiffiffiffiffiffiffiffiffi 1 dþ3 m dþ m 2 require c3 > − 3c4=4). Such actions are theoretically ξ ≡ ≡ M ;g3 c3 Λ ; and well motivated and have been considered in a number of works [32,52,53]. In order to make connections with the dþ3 ≡ m existing oscillon literature it will sometimes be necessary to g4 c4 Λ ; ð26Þ consider extending the action (23) to include a term, Z where we are now working in an arbitrary number of X2 ΔS ¼ d4x ; ð24Þ dimensions. The action we work with is then (dropping the M4 tildes) Z which must enter with a positive sign to ensure that the g3 dþ1 π□π ξ 2 π Π 2 − Π2 theory has a Lorentz-invariant UV completion [54]. S ¼ d x þ X þ 2 ð½ ½ Þ This action is not Galileon invariant (although it is shift g4 symmetric) and is under less control as an EFT compared þ πð½Π3 − 3½Π½Π2þ2½Π3Þ − π2 ; ð27Þ with the action (23), although such terms can appear 24 alongside some Galileon operators when integrating out the heavy radial mode of a U(1) complex scalar [55]. The with all quantities dimensionless. We will now look for presence of these terms is constrained by positivity bounds oscillon solutions of (27) following the procedure outlined [52], but we will not impose these here since we will only in Sec. II. use the action (24) for illustrative purposes and will not draw physical conclusions from the results. We will only A. Warm up: 1 + 1 dimensions use (24) in 1 þ 1 dimensions because in that case, as we As remarked above, only the cubic Galileon contributes will see presently, Galileon oscillons cannot exist without in 1 þ 1 dimensions. We will make the ansatz (7) for π and it. We will view this case as an instructional exercise in (6) for ω and work in the ρð¼ εxÞ and τ coordinates defined order to learn how to construct Galileon oscillons rather in (4) and (5), respectively. By construction, ϕ1 satisfies than as a theory to be taken seriously in its own right. Eq. (9) so that In 2 þ 1 and 3 þ 1 dimensions the action (23) is sufficient to support oscillons, and we will have no need for the ϕ1 ¼ Φ1ðρÞ cosðτÞ: ð28Þ action (24). Now let us examine the equation of motion:

IV. HIGHER DERIVATIVE-SUPPORTED 2 2 2 2 2 2 2 2 2 OSCILLONS ω π̈− ε π00 þ π þ ξ½3ω π_ π̈− ε ω π0 π̈− 4ω ε ππ_ 0π_ 0 2 2 2 00 4 02 00 2 2 00 02 One major difference between Galileon operators and − ω ε π_ π þ 3ε π π þg3ε ω ðππ̈ − π_ Þ¼0: others that are known to support oscillons is that they are ð29Þ highly sensitive to the number of spacetime dimensions. In particular, in D dimensions there exist D Galileon operators In order to obtain a nontrivial profile for Φ1 we need the Ogal π 2 … 1 (ignoring the tadpole) i ð Þ with i ¼ ; ;Dþ [8]. cubic Galileon terms (proportional to g3) to contribute to

123512-5 JEREMY SAKSTEIN and MARK TRODDEN PHYS. REV. D 98, 123512 (2018) τ ε3 3ξ 2 5 the resonance term [proportional to cosð Þ] at order . 00 3 2 2 00 002 Φ1 − Φ1 þ Φ1 þ gˆ Φ1 Φ1 þ Φ1 Φ1 Now, the cubic Galileon contribution to the equation of 4 3 3 4 motion is quadratic in π, and so the solution for ϕ1 will not 5 5 ϕ2 ∼ 2 τ Φ ð3ÞΦ 0Φ Φ ð4ÞΦ 2 0 contribute to the resonance term since 1 cos ð Þ can be þ 3 1 1 1 þ 12 1 1 ¼ : ð33Þ expanded in terms of even harmonics only; i.e., the expansion contains cosð2τÞ but not cosðτÞ. On the other This equation has two parameters,ffiffiffi ξ and gˆ3, butffiffiffi we can hand, the product ϕ1ϕ2 ∼ cosðτÞ cosð2τÞ ∼ cosðτÞþ p p remove ξ by scaling Φ1 → Φ1= ξ and gˆ3 → gˆ3 ξ, and so and will therefore contribute to the resonance term. we can use this scaling to fix ξ ¼ 1. We will do this Following the discussion in Sec. II, we therefore need this ξ 3 presently, but for now it is instructive to keep free. term to contribute to the Oðε Þ equation of motion. This Φ 0 −2 Multiplying by 1 one finds that this equation has a first can be achieved if we take g3 ∼ ε ; i.e., we define our pffiffiffiffiffi integral or conserved energy density given by expansion parameter ε ∼ 1= g3. We therefore define 2 g3 ¼ gˆ3=ε , which is tantamount to considering the region 1 1 3ξ 5 1 E Φ 02 − Φ 2 Φ 4 ˆ2 Φ 00Φ 02Φ − Φ 04 of parameter space where m3=Λ3 ≫ 1. This is analogous to ¼ 2 1 2 1 þ 16 1 þ g3 6 1 1 1 24 1 the procedure used to find analytic solutions to potential- 5 5 supported theories, in which oscillon solutions are stabi- − Φ 002Φ 2 Φ ð3ÞΦ 0Φ 2 24 1 1 þ 12 1 1 1 : ð34Þ lized by gπ6 terms in the scalar potential, resulting in flat-top oscillons [4]. In that scenario, the contribution 5 3 Now, we are looking for objects that satisfy the free (linear) that would typically enter at Oðε Þ enters at Oðε Þ after 00 equation of motion (Φ1 − Φ1 ¼ 0) at large distances, i.e., one makes the choice g ∼ ε−2 ≫ 1. While this choice is necessary to find solutions analytically and to determine the ρ lim Φ1ðρÞ¼B1e ; ð35Þ existence of such objects with small amplitudes, numerical ρ→∞ simulations reveal that large-amplitude objects with B g ∼ Oð1Þ persist but cannot be found analytically. We (the constant 1 must be solved by matching onto the expect similar features in Galileon theories, and so we boundary conditions at the origin) but are localized near the do not treat m3=Λ3 ≫ 1 as being a necessary condition for origin due to nonlinear self-interactions. This configuration has zero conserved energy, and we are therefore looking for the existence of oscillons, but rather as a useful limit in E 0 which to find small-amplitude objects analytically. solutions with ¼ . One can find a further condition by Returning to the calculation at hand, the second-order demanding that the profile is symmetric about the origin Φ 0 0 Φ ð3Þ 0 0 equation of motion is then [ 1 ð Þ¼ 1 ð Þ¼ ]: sffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ̈ ϕ ˆ ϕ̈ϕ00 − ϕ_ 02 0 3ξ 2 þ 2 þ g3ð 1 1 1 Þ¼ ð30Þ 00 2 Φ1 ð0Þ¼− 3ξΦ1ð0Þ − 8: ð36Þ 10g2 with solution ˆ ˆ This gives us a relation between the amplitude and the g3 00 02 g3 00 02 ϕ2 ¼ ðΦ1 Φ1 þ Φ1 Þ − ðΦ1 Φ1 þ 2Φ1 Þ cosðτÞ second derivative at the origin. Note that a second solution 2 3 00 with Φ1 ð0Þ > 0 exists, but we have discarded it since it ˆ g3 02 00 would give rise to an increasing function away from the þ ðΦ1 − Φ1 Φ1Þ cosð2τÞ; ð31Þ 6 origin. Such solutions are typically higher energy and are unstable. One can see from Eq. (36) that the X2 term in the where we have imposed the initial condition ϕ2ð0; ρÞ¼ _ 3 action is necessary in order to have a Galileon supported ϕ2 0; ρ 0 ε ð Þ¼ . The order equation is then oscillon. Indeed, had it been absent then the second 3 2 derivative would be imaginary so that no oscillon solutions ϕ̈ ϕ Φ 00 − Φ Φ 3 ˆ2 Φ 2Φ 00 3 þ 3 ¼þ 1 1 þ 4 1 þ g3 3 1 1 could exist. Including it allows for oscillon solutions provided that 5 5 5 002 ð3Þ 0 ð4Þ 2 þ Φ1 Φ1 þ Φ1 Φ1 Φ1 þ Φ1 Φ1 8 4 3 12 Φ 0 2 1ð Þ > 3ξ : ð37Þ × cosðτÞþ½cosð3τÞ; ð32Þ Φ 0 2 8 where the ellipsis corresponds to expressions that will not If one takes 1ð Þ ¼ 3ξ, then the profile is flat. require the detailed form of in what follows. In order to Interestingly,p theffiffiffiffiffiffiffiffiffiffi amplitude for Pðπ;XÞ-supported oscillons have periodic solutions we must demand that the coefficient is precisely 8=3ξ [5] so that Galileon oscillons neces- of the resonance term [cosðτÞ] vanishes, which gives us an sarily have larger amplitudes than Pðπ;XÞ oscillons for equation for the oscillon profile Φ1ðρÞ, fixed parameters.

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3.5 − 4.0 3.0 − 3.5 2.5 3.0 − 2.0

2.5 1.5 − 1.0 2.0

0.5 1.5

1.0 –10 –5 0 5 10 0.5 1.0 1.5 2.0 2.5 3.0 pffiffiffi pffiffiffi pffiffiffi FIG. 1. Left: The oscillon profile for gˆ3= ξ 1 (black dashed line), gˆ3= ξ 2 (red dotted line), and gˆ3= ξ 3 (blue dot-dashed ¼ ¼ ¼ pffiffiffi line). The black line corresponds to the Pðπ;XÞ-oscillon profile. Right: The amplitude Φ1ð0Þ as a function of gˆ3= ξ. The Pðπ;XÞ pffiffiffi pffiffiffiffiffiffiffiffi prediction ξΦ1ð0Þ¼ 8=3 corresponds to the blue point.

Unlike the case of potential or Pðπ;XÞ-supported oscil- B. 3 + 1 dimensions E 0 lons [4,5,37], the equation governing the profile [ ¼ in In three spatial dimensions there are contributions from the Eq. (34)] does not have an analytic solution, and so we must cubic, quartic, and quintic Galileon operators. As remarked proceed numerically. We are looking for solutions that are above, we will show presently that the combination of the localized near the origin, so that nonlinear terms are kinetic term and the quartic Galileon admits quasibreather −ρ important, but that tend to the linear solution expð Þ at solutions, and so we will set ξ ¼ 0 from here on, having no large distances [see Eq. (35)]. The task at hand is then to other justification for incorporating it into the massive solve Eq. (34) (with E ¼ 0) given the boundary conditions Galileon EFT. Furthermore, the lessons we have learned ð3Þ Φ1ð0Þ¼Φ1 ð0Þ¼0 and (36). This leaves the value of from our warm-up exercise in 1 þ 1 dimensions give us good Φ1ð0Þ undetermined, and so, in the current formulation, cause to neglect the cubic and quintic terms, too. Recall that 2 2 the correct solution must be found by solving the equation the cubic Galileon contributed terms of order ε g3ϕ for different values of Φ1ð0Þ such that limρ→∞Φ1ðρÞ ∼ (ignoring time and space derivatives) to the equation of exp ð−ρÞ. This is a time-intensive process, but it can be motion, which forced us to make a suitable choice of scaling −2 simplified dramatically by reformulating the problem in for g3 (g3 ∼ ε ) to ensure that this term contributed to the 0 2 terms of the phase space variables fΦ1ðρÞ; Φ1 ðρÞg.We Oðε Þ equation of motion and therefore that terms such as 3 give the technical details of this process for the interested ϕ1ϕ2 (again, suppressing derivatives) appeared in the Oðε Þ reader in Appendix. Some examples of the oscillon profiles pffiffiffi equation. This was necessary to ensure that the cubic for varying gˆ3= ξ (the only free combination of param- Galileon contributed to the resonance term [proportional eters) are given in the left panel of Fig. 1; one can see that to cosðτÞ]. Had we not made this choice and had instead Galileons produce oscillons with similar shapes and withffiffiffi chosen g3 such that the cubic operator contributed at third p ϕ2 larger amplitudes that are increasing functionsffiffiffi of gˆ3= ξ. (and higher) order, the quadratic nature (g3 1) of the equation p of motion would mean that no odd harmonics were present. The right panel of Fig.ffiffiffi 1 shows the amplitude ξΦ1ð0Þ as a p In fact, this is completely analogous to the cubic potential function of gˆ3= ξ. One can see that it is indeed an increasing function. In the case of Pðπ;XÞ oscillons, the discussed in Sec. II, the contribution to the equation of profile was calculated analytically in [5] as motion is quadratic in the field, and therefore the above process is necessary. The only difference is that we had to sffiffiffiffiffi −2 take g3 ∼ Oðε Þ owing to the higher-derivative nature of the 8 λ ∼ O 1 Φ1ðρÞ¼ sechðρÞ; ð38Þ Galileons, whereas one can take 3 ð Þ for potential- 3ξ supported oscillons. Now, the contribution of the quartic Galileon to the which is also shown in both figures. In the case of the right equation of motion is cubic in the fields, and hence it is panel, the amplitude is shown using the blue point. Evidently,ffiffiffi sufficient to choose g4 to scale with ε in such a way that it p 3 3 the amplitude tends to this value for small gˆ3= ξ.One first contributes at order ε because terms such as ϕ1 interesting difference between Galileon oscillons and oscil- (suppressing derivatives) contribute odd harmonics, includ- lons in Pðπ;XÞ theories is that the boundary conditions do ing the resonance term cosðτÞ. This is analogous to the not impose any limit on the amplitude. One cannot have quartic potential discussed in Sec. II, the difference being arbitrarily large amplitudes while simultaneously satisfying that g4 must be chosen to cancel the effects of higher the boundary conditions in Pðπ;XÞ theories [4,5]. derivatives, whereas λ4 ∼ Oð1Þ.

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Let us now briefly discuss the quintic Galileon. This where we have once again given only the coefficient of contributes quartically to the equation of motion, and so the resonance term. This must be identically zero in order to pffiffiffiffiffi the situation is akin to the cubic rather than the quartic: avoid secular growth. Scaling Φ1 → Φ1= gˆ4 one then one must choose g5 to scale with ε such that the quintic finds the equation governing the oscillon profile: O ε2 O ε3 contributes to both the ð Þ and the ð Þ equations in order for the resonance term to be affected by its presence. 3Φ 0Φ Φ 03 3Φ Φ 02 2 1 1 00 1 1 1 0 1 Φ1 Φ1 − Φ1 0: Given the above considerations, we will not include the þ 2ρ þ 2ρ þ 4ρ2 þ ρ ¼ cubic Galileon in what follows since it greatly complicates the equations in 3 þ 1 (and 2 þ 1) and all of the new and ð43Þ salient features are captured by including the quartic solely. Similarly, we will not discuss the quintic The task of finding oscillon solutions is then to solve this 4 0 Galileon at all in this paper. One can forbid these terms equation given the boundary condition Φ1 ð0Þ¼0 with either by imposing a Z2 symmetry or the symmetry of the Φ1ð0Þ chosen such that special Galileon [57,58]. The equation of motion in the τ − ρ coordinate system is e−ρ lim Φ1ðρÞ ∼ B3 ð44Þ ρ→∞ ρ 2 π0 ω2ππ̈ 0 π00π0 2 2 00 0 4 2 ω π̈− ε π þ π þ π þ ε g4 − ε 5 ρ ρ ρ ρ for some constant B3; i.e., Φ1 is the spherically symmetric 2 solution of the linear equation ∇ Φ1 − Φ1 ¼ 0 (in three þ 2ω2ππ̈ 00 − ω2π_ 02 ¼ 0: ð39Þ spatial dimensions) at large distances. Let us recall how this is accomplished for Pðπ;XÞ and potential-supported oscillons in d þ 1 dimensions with d>1. Unlike in 1 þ 1 As per our discussion above, we must choose g4 such that dimensions, there is no conserved first integral,6 and so one 3 the quartic contributes at order ε , and so we choose writes the equivalent of Eq. (43) in the form 4 g4 ¼ gˆ4=ε . (One can equivalently view this as the defi- ε ∼ −1=4 nition of our small number g4 .) Once again we need dE 2 2 ¼ − ð∂ρΦ1Þ ; ð45Þ to employ a procedure similar to the flattop oscillon dρ ρ construction, whereby we push the contributions of ε higher-order operators to lower-order in by taking a and uses the phase space approach. In particular, since the ≫ 1 dimensionless parameter to be . In this case, this energy E 6 6 would be conserved if not for the right-hand side, implies we are in the regime where m =Λ ≫ 1.As one can deduce that there is a series of discrete solutions discussed above, we do not take this as a necessary with Eðρ ¼ 0Þ > 0 such that the phase space trajectories 0 0 condition, but rather as a tool that allows us to construct move from ðΦ1; Φ1 Þ¼ðΦ1ð0Þ; 0Þ to ðΦ1; Φ1 Þ¼ð0; 0Þ solutions analytically. We expect similar objects to exist for corresponding to limρ→∞E ¼ 0, which is necessary to other parameter choices, the difference being that they must ensure that the solution tends to the linear one [i.e., the be found numerically. profile tends to the one given in Eq. (44)] at large distances. ε ε2 Following the procedure in Sec. II, the order and (We refer the reader to [59] for the technical analysis of equations of motion are the phase space.) These solutions are characterized by the number of nodes in the profile, the lowest energy solution ϕ̈ ϕ 0 ⇒ ϕ Φ ρ τ 1 þ 1 ¼ 1 ¼ 1ð Þ cosð Þ; ð40Þ having zero nodes and higher energy solutions having an increasing number. Equation (43) is not amenable to such ̈ ϕ2 þ ϕ2 ¼ 0 ⇒ ϕ2 ¼ 0; ð41Þ 4Imposing this, Eq. (43) gives ϕ 0 where we have set 2 ¼ using appropriate boundary qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 conditions. Using Eq. (40) in Eq. (39) one finds the third- 00 2 Φ1 ð0Þ¼− 1 1 þ Φ1ð0Þ ; order equation of motion 3Φ1ð0Þ

00 2 so there is no restriction on Φ1ð0Þ or Φ1 ð0Þ as there was in 1 þ 1 ̈ 00 0 ϕ3 þ ϕ3 ¼þ Φ1 þ Φ1 − Φ1 dimensions. ρ 5This constant is not arbitrary because the full Eq. (43) is B ˆ 3Φ Φ 02 nonlinear. The value of 3 is a global property of the solution; i.e., g4 03 0 00 1 1 Φ 0 þ Φ1 þ 3Φ1Φ1 Φ1 þ cosðτÞ it depends on 1ð Þ. 2ρ 2ρ 6Of course, we still have energy and momentum conservation 3τ resulting from the Poincar´e symmetries of the action, but it is not þ½cosð Þ; ð42Þ necessarily the case that these should hold order by order in ε.

123512-8 OSCILLONS IN HIGHER-DERIVATIVE EFFECTIVE … PHYS. REV. D 98, 123512 (2018)

1.0 0.5 1.0 1.5 2.0 2.5 3.0

0.8

–0.5 0.6

0.4 –1.0

0.2

–1.5 1 2 3 4 5 6

FIG. 2. Left panel: Three profiles for different amplitude cubic Galileon oscillons in 3 þ 1 dimensions. Right panel: The first 0 derivatives Φ1 ðρÞ of the profiles in the left panel. The colors in each figure correspond to the same solution.

0 0 an analysis for several reasons. First, although one can Then, Eq. (43) with Φ1 ðρsÞ¼2ρs=ð3Φ1 ðρsÞÞ gives make judicious integrations by parts to reformulate it in sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the form of Eq. (45), this is not useful because the energy 1 16ρ2 depends on ρ, greatly complicating the phase space Φ ρ ffiffiffi −1 9 − s 1ð sÞ¼p 3 ; ð48Þ analysis. Furthermore, the right-hand side contains a term 2 of indefinite sign so that one cannot deduce anything when A ¼ 0, so that there are no solutions. If instead one meaningful about the energy along oscillon trajectories 7 00 takes A ≠ 0, then real solutions do exist, although we do in phase space. Finally, the coefficient of Φ1 can vanish not give them here since they are solutions of a more identically at a point ρs that depends on the boundary conditions, either at the origin or infinity depending on the complicated quartic equation and their expressions are direction from which ρ is approached. At this point, the long and cumbersome. The same conclusions are reached s if one begins with a solution satisfying Eq. (44) at ρ → ∞ second derivative is not determined from the equation. This and lets ρ → ρ þ. implies that solutions may not be smooth since one could s We seek solutions over the entire positive real interval. construct functions that are discontinuous at ρ . For these s Such solutions must therefore be of the smoothness class reasons, it is instructive to proceed by analyzing the C1. This may be surprising at first, but we remind the reader equation directly. that the solutions of partial differential equations naturally We begin by showing that solutions with nodes cannot ∞ 00 lie in Sobolev spaces rather than the space of C (or even exist. Solving Eq. (43) for Φ1 one finds C2) functions. In fact, it is not uncommon for such 4ρ2Φ − 8ρΦ 0 − 4Φ Φ 02 − 2ρΦ 03 solutions to arise in nonlinear wave equations; see e.g., 00 1 1 1 1 1 1 Φ1 : 46 C ¼ 2ρ 2ρ 3Φ Φ 0 ð Þ [60]. solutions of Eq. (43) that span the positive real ð þ 1 1 Þ 1;1 þ interval, Φ1ðρÞ ∈ W ðR Þ, can be constructed numeri- 0 cally. One finds a continuous spectrum of solutions dis- Since Φ1 ðρÞ¼0 at any stationary point where 00 tinguished by their amplitudes Φ1ð0Þ. Some examples are ρ ≠ f0; ∞g, one has Φ1 ðρÞ¼Φ1ðρÞ at any potential node. This equation implies that the stationary point is shown in Fig. 2. The left panel shows oscillons with different amplitudes. Evidently, these are very flat objects necessarily a minimum if Φ1ðρÞ > 0 and a maximum if and are completely smooth. The flatness was anticipated, Φ1ðρÞ < 0. Clearly this precludes the possibility that such a since we chose g4 ≫ 1, putting us in the flattop regime as point is a node. In practice, we have not been able to find 0 discussed above. The right panel shows Φ1 ρ . One can any smooth zero node solutions for reasons that we now ð Þ see that there is indeed a point ρ where the derivative is discuss. s continuous but not smooth, indicating the C1 nature of Consider the point ρs mentioned above where the 00 coefficient of Φ1 ðρÞ in Eq. (43) vanishes. This point is 0 7 defined implicitly by 3Φ1 ðρsÞΦ1ðρsÞ¼2ρs,andthe It is tempting to conclude from the present discussion that it 00 second derivative Φ1 ðρÞ is undetermined there. To see therefore follows that this, focus on a solution that satisfies the boundary Φ 00 ρ ∞ Φ 0 0 0 lim− 1 ð Þ¼ ; conditions at the origin, i.e., 1 ð Þ¼ for some ρ→ρs Φ1ð0Þ, and assume that but such a conclusion would be erroneous. Rather, the solution to 0 Eq. (43) should be viewed as a weak solution and is therefore 3Φ1 ðρÞΦ1ðρÞ lim 1 þ ¼ A: ð47Þ locally integrable. One can only make meaningful statements ρ→ρ − 2ρ s about this function when integrated against test functions.

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0 2 2 the solution. We have verified numerically that 3Φ1 ðρsÞ the operator X where X ∼ ð∂πÞ . We analyzed the proper- Φ1ðρsÞ¼2ρs at this point, as per our analytic prediction ties of these new objects and derived their amplitude as a above. function of the cubic Galileon coupling. In 3 þ 1 dimensions we found new quasibreathers that are supported V. CONCLUSIONS AND OUTLOOK entirely by the nonlinearities of the quartic galileon. 1 The study of higher-derivative effective field theories is These objects have the novel feature that they are C important for a multitude of reasons. The leading-order functions. derivative corrections to general relativity are higher Having found these objects, several questions about derivative in nature. Similarly, many infrared modifications their nature naturally arise. First, are they stable? Oscillons supported by nonlinear scalar potentials are of gravity make use of higher-derivative structures, an 1 1 important example being Horndeski gravity [7,10] (and stable in þ dimensions, but broad, small-amplitude oscillons are unstable to long wavelength perturbations in beyond [13,17]) which can be viewed as a framework for 3 1 constructing ghost-free dark energy or inflation scenarios. þ dimensions [4] (large-amplitude objects are robust 3 1 Similarly, the decoupling limit of many IR modifications of in þ dimensions). Similarly, small-amplitude oscillons π gravity such as ghost-free massive gravity or braneworld supported by Pð ;XÞ terms exhibit a long wavelength ≥ 3 models are described by higher-derivative EFTs, in par- instability in d . In both of these cases, one can apply ticular massless Galileons [8,19,20,61]. the Vakhitov-Kolokolov stability criterion [69] to study Galileons have two remarkable features. First, they enjoy the dynamics of long wavelength perturbations. The a nonrenormalization theorem whereby the coefficients of criterion assumes that the spatial kinetic term for pertur- the Galileon operators are not corrected by self-loops or bations is the d-dimensional Laplacian, which is not the matter loops (provided that the coupling is Galileon case for Galileon theories. One could attempt to generalize invariant) [8,33,44]. Second, the Vainshtein mechanism the criterion, but that is beyond the scope of the present suppresses fifth forces compared with the gravitational paper. Given the complexity of the field equations, it may force and ensures that the range of validity of the EFT be simpler to study the stability numerically, although this extends to scales far smaller than one would naively expect. too is difficult given the higher degree of nonlinearity in In particular, there is a Vainshtein radius, inside which the equations [63,70]. Galileon nonlinearities can be important while other Another related question is the radius of convergence for higher-derivative operators are still suppressed. The range the asymptotic expansion. Quasibreathers radiate at wave 2 3 4 … of validity is then lowered to smaller distances where these modes k ¼ nm with n ¼ ; ; ; as a result of the radius are comparable to the Galileon terms. Interestingly, these of convergence (in terms of ε) being zero on the real line features are not disrupted by the inclusion of a mass term [38,40,41]. Said another way, oscillons radiate since they for the Galileon, even though it breaks the Galileon are not exact solutions of the field equations. Their long- symmetry [32,33,53]. Massive Galileon theories are more lived nature is due to the fact that they are composed of naturally embedded in ghost-free massive gravity than their wave modes that are far smaller than those into which they massless counterparts and may shed some light on the radiate, and so the radiation process takes a long time [42]. nature of any UV completion [53]. The answer to this question, and indeed the preceding one, Phenomenologically, the Vainshtein mechanism is would be useful in searching for signatures of these objects incredibly efficient and ensures that Galileon theories are in the early universe, observations of which may act as difficult to test experimentally [45,62,63] and one must smoking guns for Galileon inflation or Galilean genesis. 1 resort to exotic and nontraditional tests of gravity [64–68]. Given the C nature of the solutions, there is also the A powerful no-go theorem [34] prohibits the existence of question of how ubiquitous these objects are. Do they form static topological soliton configurations within the regime from generic initial data or do they require special tunings of validity of the Galileon EFT (this is an extension of to be realized? From the point of view of PDEs, these Derrick’s famous no-go theorem for potential-supported features are not new and it is often the case that shock fronts 0 objects), but the fact that a small mass for the Galileon is or C -peaked traveling waves form spontaneously in fluids both theoretically well motivated and interesting opens up and other nonlinear systems. This question is particularly the possibility of finding quasiperiodic localized solutions: pertinent because it also relates to the feasibility of forming oscillons (or quasibreathers). such objects in the early universe. Indeed, the formation of In this paper we have shown that massive Galileon oscillons from the fragmentation of the homogeneous effective field theories can indeed result in quasibreathers/ condensate can dominate the equation of state in the early oscillons. In 1 þ 1 dimensions the cubic Galileon, which is universe [2,71]. the only Galileon operator that can exist, can give rise to a Another important consideration is how these objects fit new type of quasibreather, provided that the EFT contains into the effective field theory of massive Galileons. In order shift-symmetric (but not Galileon-invariant) operators. In to find small-amplitude solutions analytically it was nec- particular, the solution we found required the presence of essary to take m ≫ Λ (g3, g4 ≫ 1), which is certainly

123512-10 OSCILLONS IN HIGHER-DERIVATIVE EFFECTIVE … PHYS. REV. D 98, 123512 (2018) peculiar from an EFT perspective. Indeed, a mass larger these objects and analyzed their basic properties. A future than the cutoff implies that new light states may be present investigation into their stability and formation dynamics that we have not accounted for. Similarly, the higher-order could reveal a plethora of observational tests. Such inves- operators in the Wilsonian effective action [Eq. (20)] scale tigations would likely be heavily numerical. as ð∂=ΛÞpð∂2π=Λ3Þq ∼ ðm=ΛÞpþ3q and are therefore more relevant than the Galileon operators, and so they should ACKNOWLEDGMENTS have been included. This issue also arises when considering the Vainshtein mechanism, but in a slightly different We are extremely grateful for conversations with context. In particular, operators with very large p and q Mustafa Amin, Mark Hertzberg, and Austin Joyce. J. S. become increasingly important near the Vainshtein radius is supported by funds provided by the Center for Particle [48]. One possible resolution is that the Galileons do not Cosmology. M. T. is supported in part by U.S. Department need additional UV physics to preserve unitarity, so that the of Energy (HEP) Award No. DE-SC0013528 and by NASA IR theory does not contain any higher-derivative terms [47]. ATP Grant No. NNH17ZDA001N. Another is that the dimensionless coefficients of the higher- derivative terms are ≪1, contrary to what one would expect, due to a reordering of the counterterms to preserve APPENDIX: PHASE SPACE ANALYSIS FOR THE an approximate symmetry that is present in the limit where CUBIC GALILEON IN 1 + 1 DIMENSIONS the Galileons dominate over the canonical kinetic term In this appendix we provide the technical details of how [48,72]. Said another way, the coefficients are such that the localized solutions of Eq. (34) (with E ¼ 0) can be operators resum into some simple object dictated by the constructed numerically. The strategy will be to work in 0 approximate symmetry. Regardless, we view the condition the phase space fΦ1; Φ1 g rather than the configuration m ≫ Λ not as being necessary for the existence of oscillons space fρ; Φ1ðρÞg. In particular, performing the change of but rather as a convenient limit that allows for their analytic variable construction in a systematic manner in order to demonstrate 0 their existence, at least in this limit. As discussed a number Φ1 ðρÞ¼wðΦ1Þ; ðA1Þ of times above, this is analogous to the procedure used to construct small-amplitude flattop oscillons analytically, and in that case one typically finds oscillons ubiquitously for the entire parameter space; the difference being that they must be found numerically. It would be interesting to solve the Galileon equation numerically for this purpose, but such a study lies outside the scope of this work. Throughout this work we have restricted ourselves to Galileons defined on Minkowski space, our motivation being to find breather solutions in the simplest possible setting. The study of oscillons on de Sitter space is particularly interesting because it leads to radiating tails [43], and it would be interesting to investigate how higher- derivative oscillons behave on curved backgrounds for this reason. Unlike potential and PðXÞ-supported oscillons, Galileons as defined on Minkowski space are not a FIG. 3. An example of the phase space of Eq. (A2). The covariant theory (the exception being the cubic), and the solution at ρ ¼∞ is mapped onto the origin wðΦ1Þ¼ quartic Galileon that was essential for supporting the 0 w ðΦ1Þ¼0, which follows from the relation (35). The same oscillons we found in this work does not have a unique relation gives wðΦ1Þ¼−Φ1 for ρ → ∞, so beginning the covariantization. One must choose from several such as integration at the origin (in Φ1 − w space) and taking this covariant Galileons [9], beyond Horndeski Galileons condition gives the solution for large positive ρ. In particular, 0 [12,13], and massive gravity. We have not done so in this since localized solutions have Φ1 ðρÞ < 0 for ρ > 0, the curve 0 work since this would have required us to investigate a moves into the lower half-plane where wðΦ1Þ¼Φ1 ðρÞ < 0.As ρ 0 more complicated theory. Another approach would be to one moves closer to ¼ , the red curve moves toward the axis. The point where it crosses, i.e., w ¼ 0, Φ1 ¼ Φ1ð0Þ corresponds use different Galileons defined on these spaces, e.g., de 0 to ρ ¼ 0, since the boundary condition Φ1 ¼ 0 (w ¼ 0)is Sitter Galileons [73] or more general theories [74]. satisfied. The upper-half plane of phase space corresponds to 0 Finally, the connection of massive Galileon EFTs to ρ < 0, since w ¼ Φ1 ðρÞ > 0 for localized solutions, and so at massive gravity raises the tantalizing prospect that such this point the red curve moves back toward the origin, arriving 0 objects may exist within massive gravity theories them- with a gradient w ðΦ1Þ ∼ þ1 corresponding to the ρ → −∞ selves. In this preliminary investigation we have discovered solution in Eq. (35).

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0 the equation for the oscillon profile becomes have wðΦ1Þ¼0 ⇒ Φ1 ðρÞ¼0 with Φ1 ≠ 0. The situation is exemplified in Fig. 3. 5 dw 5 dw 2 5 dw Φ 2 3 Φ 2 2 Φ 3 Setting w ¼ 0 in Eq. (A2) one finds two zeros: Φ1 ¼ 0 1 w 2 þ 1 w 2 þ 1w 6 dΦ 12 dΦ1 3 dΦ1 2 8 1 corresponding to the origin and Φ1ð0Þ ¼ 3ξ, which cor- 4 4 00 2 w 2 3Φ1 responds to Φ1 ð0Þ¼0 so that the profile is flat. Since þ w − − Φ1 þ ¼ 0: ðA2Þ 12 8 neither of these corresponds to a physically acceptable solution, we conclude that oscillon solutions exist only if Note that we are now treating w as a function of Φ1. Φ Φ the curvepwffiffiffiffiffiffiffiffið 1Þ crosses the 1 axis at some value (In this section we will use primes to denote derivatives of Φ 0 8 3 functions with respect to their arguments.) Not only have 1ð Þ < = . Whether this is the case or not requires 8 Φ 00 ρ wereducedtheequationfromthirdtosecondorder,but,as a numerical solution. Furthermore, 1 ð Þ is finite and we will now show, we have mapped the domain ρ ∈ nonzero by virtue of Eq. (36), and taking derivatives of 0 Φ 0 Φ 0 ½−∞; ∞ to Φ1 ∈ ½0; Φ1ð0Þ. This makes finding the Eq. (A1) implies that the product w ð 1ð ÞÞwð 1ð ÞÞ is ð3Þ 00 profile far easier because the boundary condition at finite. [The condition Φ1 ð0Þ¼0 also implies w ðΦ1Þ¼ 0 2 ρ ¼ ∞ is now mapped onto the origin. In particular, as −w ðΦ1Þ =w.] Since w ¼ 0 at this point, the curve must 0 ρ → ∞ we have Eq. (35); i.e., Φ1 → 0 implying that the approach it with w ðΦ1Þ → ∞. The oscillon profile can be origin Φ1 ¼ 0 corresponds to the points ρ ¼∞. found by integrating Eq. (A1). Furthermore, differentiating Eq. (35) near ρ ¼∞ we have w ¼Φ1 near the origin. This implies that the 8 boundary conditions for Eq. (A2) are wðΦ1 ¼ 0Þ¼0 and One can try to take limits of equation (A2) where combina- 0 0 Φ w ðΦ1 ¼ 0Þ¼−1. The boundary conditions at ρ ¼ 0 now tions of w ð 1Þ and its derivatives are fixed to satisfy (36) but imply that an oscillon profile exists if one can integrate only the trivial solutions can be found. Intuitively, this is because Φ1ð0Þ is determined by the requirement that the solution obeys Φ1 0 Eq. (A2) from the origin ( ¼ ) given these boundary the boundary conditions at Φ1 ¼ 0 (ρ → ∞) so its value is conditions and find a second point where wðΦ1Þ¼0.This dependent on both the boundary conditions and the differential point corresponds to the amplitude Φ1ð0Þ because here we equation so that it cannot be determined from the latter alone.

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