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K-Oscillons: Oscillons with Non-Canonical Kinetic Terms

Oscillons are long-lived, localized, oscillatory scalar field configurations. In this work we derive a condition for the existence of small-amplitude oscillons (and provide solutions) in scalar field theories with non-canonical kinetic terms. While oscillons have been studied extensively in the canonical case, this is the first example of oscillons in scalar field theories with non-canonical kinetic terms. In particular, we demonstrate the existence of oscillons supported solely by the non-canonical kinetic terms, without any need for nonlinear terms in the potential. In the small-amplitude limit, we provide an explicit condition for their stability in d+1 dimensions against long-wavelength perturbations. We show that for d>2, there exists a long-wavelength instability which can lead to radial collapse of small-amplitude oscillons.


Introduction
Scalar fields with non-canonical kinetic terms are used ubiquitously in cosmology. They are especially prevalent in modeling of the inflaton (e.g. [1,2]), dark energy and modifications of gravity (e.g. [3,4]). A significant amount of work has been done on their homogeneous evolution in an expanding universe and the evolution of linearized fluctuations about this homogeneous background. However, less attention has been paid to their spatially varying, nonlinear dynamics (e.g. [5,6,7,8,9]). In particular, localized, time-dependent, solitonlike solutions in theories with non-canonical kinetic terms have been rarely discussed. If such configurations exist, they would be novel objects from a mathematical standpoint. If they form a significant component of the energy fraction of the universe, they might have cosmological consequences.
In this paper we show that in a general class of scalar field theories with non-canonical kinetic terms and/or nonlinear potentials, there exist extremely long-lived, spatially localized, oscillatory field configurations called oscillons [10,11] * . While oscillons in scalar field theories have been studied extensively in the literature (for e.g. [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]), every instance so far uses a nonlinear term in the potential and a canonical kinetic term. To the best of our knowledge, this is the first time their existence is being demonstrated in the presence of non-canonical kinetic terms. When the non-canonical kinetic terms are significant, we refer to the oscillons as "k-oscillons" † . We derive the condition for their existence, including effects from the non-canonical kinetic term as well as the nonlinearity in the potential. Our results are general enough to include oscillons supported by nonlinear potentials, oscillons supported purely by the non-canonical kinetic terms as well as oscillons supported by a combination of both. The inclusion of non-canonical kinetic terms significantly expands the space of theories where oscillons can exist.
Our analysis is done in a small-amplitude approximation, but is otherwise quite general. We consider scalar field Lagrangians of the form in d + 1 space-time dimensions where η µν is a Minkowski space metric with the 'mostly +' signature. The only restriction of T and X is that they can be written as T (X, ϕ) = X + ξ 2 X 2 + ξ 3 ϕX 2 + . . .
where all the field variables, space-time co-ordinates and coefficients have been made dimensionless using appropriate scalings. In the next section we motivate this Lagrangian and discuss scalings of the parameters and fields in terms of a mass and a cutoff scale. Anticipating a small-amplitude expansion, we have organized the series in terms of powers of the field and kept terms up to fourth order in the fields. Note that terms of the form ϕX and ϕ 2 X which should be included in the above expression can always be absorbed using a field re-definition. Furthermore, note that the X 2 term cannot be eliminated using a field re-definition. Our choice of T and V also ensures that we recover a free, canonical scalar field theory when X, ϕ → 0. This form is general enough to cover a large class of scalar field theories of interest in cosmology including axions [37], DBI inflation [1], monodromy inflation [38,39], k-essence [4] and scalar-tensor theories [3].
For the Lagrangian discussed above, and in the small-amplitude, spherically symmetric case, we provide • a condition for the existence of oscillons in terms of a relationship between the first few coefficients in the series for T and V , • explicit, controlled, analytic solutions in 1 + 1 dimensions and approximate solutions in 3 + 1 and higher dimensions, • a condition for their stability against long wavelength perturbations. The condition shows the presence of a long wavelength instability for d ≥ 3 (in the small-amplitude approximation).
We also calculate the energy loss from these general oscillons due to an expanding background.
For canonical kinetic terms, the authors of [28,25,40,29,30] show that oscillons are produced copiously at the end of inflation (as well as in phase transitions and bubble collisions, for e.g., [41,42,43]) and can dominate the energy density of the universe at that time (e.g. [30]). A similar situation is possible with a somewhat contrived model of dark energy as well [44]. Models with non-canonical kinetic terms and/or nonlinear potentials are well suited for amplifying fluctuations around a homogeneous oscillatory background field. This provides a natural mechanism to amplify quantum fluctuations existing at the end of inflation, possibly leading to the formation of large-amplitude oscillons. We will pursue this possibility in inflationary models with non-canonical kinetic terms in future work. Indeed our original motivation for studying oscillons in non-canonical theories arose while trying to explore self-resonance and preheating in the DBI scenario (see for e.g. [45,46,47]) The rest of this paper is organized as follows. In the section 2 we motivate the Lagrangian discussed in the introduction, in section 3 we derive an oscillon solution in d + 1 dimensions, in section 4 we discuss the stability of the oscillon solutions and in section 5 we summarize the main results and discuss directions for future work.

The effective Lagrangian
In this section we discuss the motivation for the Lagrangian in the introduction. For the reader interested in oscillons from a mathematical standpoint, this section can be omitted without affecting the rest of the paper. From an effective field theory perspective, a general Lagrangian with non-canonical kinetic terms has the form where X φ = −(1/2)∂μφ∂μφ. We are assuming that only first derivatives of the field appear in the Lagrangian. Let U(φ) = (1/2)m 2 φ 2 + . . . and Λ ≫ m is the cutoff scale. Furthermore, we assume that a Taylor expansion exists for all a n (φ) and a 1 (0) = 1. A field re-definition enacted via dφ/dφ = a 1 (φ) yields where Xφ = −(1/2)∂μφ∂μφ. We Taylor expand L around (φ, Xφ) = (0, 0) and assume that as (Xφ,φ) → (0, 0) we recover a canonical free-field Lagrangian. This yields: Let us redefine the fields and space-time variables as follows With these redefinitions we get the Lagrangian we will use for the rest of the paper: where anticipating a small-amplitude expansion, we have only kept terms up to 4th order in the field ϕ. Note that the next order terms in the first and second brackets above would have the form ξ 3 ϕX 2 and (1/5)λ 5 ϕ 5 respectively.
Another way of arriving at the Lagrangian above is as follows (see for e.g. [48]). Consider a Lagrangian with two fields in 3 + 1 dimensions (the generalization to d + 1 dimensions is straightforward): with Λ ≫ m. The heavy field will sit at the minimum of its effective potential, with the corresponding field value given by Substituting into the original Lagrangian and setting the kinetic term of the heavy field to zero, we have With appropriate scalings of space-time and fields with mass m and Λ defined in equation (7), we recover the effective Lagrangian discussed above and in the introduction (see equation (3)).
Note that ifξ 2 andλ n are order one, we expect ξ 2 ≪ λ n and ξ 2 ≪ 1 if Λ ≫ m. However, this need not always be the case. For example, in the DBI case [1] yields ξ 2 = 1/2. For the rest of the paper we do not make any particular assumptions about the sizes of ξ 2 and λ n , apart from assuming that they are not much larger than unity.

Oscillons in d + 1 dimensions
Let us begin with the equations of motion associated with the Lagrangian presented in the introduction and discussed in the previous section: We are interested in small-amplitude, radially symmetric, spatially localized, oscillatory (in time) solutions. First, we rescale the time and space variables by a small parameter ǫ as follows: and expand the solution as a series in ǫ (a possibly asymptotic one): Plugging the above scalings and form of the solution into the equation of motion (12), and collecting terms to lowest order in ǫ we get where we assumed that ∂ τ φ 1 (0, ρ) = 0. At second order in ǫ we get where we assumed that φ 2 (0, ρ) = ∂ τ φ 2 (0, ρ) = 0. At the next order in ǫ, we get If the coefficient of the cos τ term is non-zero, φ 3 would grow linearly with time, inconsistent with the time-periodic solution we are looking for. ‡ To avoid linear resonance, we need to set the coefficient of cos τ to zero. This in turn yields the equation for the profile f (ρ): where for future convenience we have defined whose sign will turn out to determine whether oscillons exist or not.
This profile equation (18) is valid in any dimension d, but can be solved exactly for d = 1. Let us consider the d = 1 case first.

1+1 dimensional oscillons
For d = 1 the profile equation becomes One can think of the above equation as describing the motion of a particle in a potential The energy associated with this motion is conserved, and given by where we have used ∂ ρ f (0) = 0 and f (0) ≡ f 0 . Now since the solutions are localized, the energy E = U(f 0 ) = 0. This immediately, yields For a localized solution to exist, we need This is one of our main results. Before moving on to solving the profile equation, we pause to discuss ∆ in a bit more detail. If the non-canonical terms are absent (ξ 2 = 0), we need the usual "opening up of the potential" condition: −λ 4 + (10/9)λ 2 3 > 0 to get oscillons. More importantly, note that for a quadratic potential (i.e. λ n = 0), the non-canonical terms are sufficient to yield oscillons. For example if λ n = 0, then ξ 2 > 0 is sufficient. It is also worth noting that in models with non-canonical kinetic terms, the sound speed differs from 1. For the model under consideration, the sound speed is Thus, for λ n = 0, the condition for having k-oscillons is the same as the sound speed being less than 1.
Now, let us get back to solving the equation for the profile f (ρ). Using f (0) = f 0 derived above and integrating ( Explicitly in terms of the original variables: where ∆ = ξ 2 − λ 4 + 10 9 λ 2 3 . This is our second main result: a small-amplitude oscillon in 1 + 1 dimensions in theories with non-canonical kinetic terms. This solution has the same functional form as the canonical small-amplitude oscillon, apart from the appearance of ξ 2 in ∆. One can go beyond the leading order as well: where ρ = ǫr and τ = √ 1 − ǫ 2 t. Note that at second order, only the odd term contributes. If the Lagrangian had ϕ → −ϕ symmetry, the the correction is higher order.

3 + 1 dimensional oscillons
For d = 3, the profile equation becomes We can view the above equation as describing the motion of particle in the presence of a potential U(f ) as in the 1 + 1 D case, but now we have a 'friction term' of the form (2/ρ)∂ ρ f (ρ). As a result, the energy is no longer conserved. It changes with ρ as With the requirement that the solution is localized, we need E → 0 as ρ → ∞. Requiring that the solution is smooth at ρ = 0 implies ∂ ρ f (0) = 0. This implies that for a localized solution we must have E(0) = U(f 0 ) ≥ 0. Numerically, one finds a localized solution for § Thus the solution will have the form: The profile F (ρ) looks 'sech' like with F (0) = 1. An excellent approximation (at the few % level in the 'core' region, deteriorating in the tails) is given by An identical analysis can be carried out in d = 1, 3 and we do not repeat the derivation here.

Stability
We now turn to the question of stability of the oscillons against perturbations. To this end, we linearize the equation of motion (12) around the small-amplitude oscillons solutions ϕ osc as follows: We have used the following to arrive at the above equation: and kept terms up to order ǫ 2 . The ξ 2 ϕ 2 osc ∇ 2 δϕ term has to be kept since we have not (yet) restricted ourselves to long wavelength perturbations. Let us remove the term with the linear derivative by redefining the δϕ as follows: With this redefinition, the equation of motion becomes The solutions ϕ osc are periodic in time. Hence stability can be determined via a Floquet analysis. It is tempting to Fourier transform the above equation and try to determine the Floquet exponents (growth-rate) mode by mode. However, since the background varies in space, the Fourier modes do not decouple. A stability analysis in Fourier space, while possible (see [27]), is numerically intensive, especially in higher dimensions. We do not follow this approach here. Instead we will carry out a stability analysis in position space, but restrict ourselves to perturbations which vary on length scales comparable to the size of the oscillon. Even without considering short wavelengths, we will show that there exists an important instability in dimensions ≥ 3.

Long-wavelength stability analysis
When considering perturbations with wavelengths comparable to the size of the oscillon, we can drop the ξ 2φ 2 osc ∇ 2 δϕ ∼ ǫ 4 term in equation (39) since it is higher order in ǫ. This yields At this point, the equation of motion for linearized fluctuations is identical to that in the canonical case, apart from the coefficient ξ 2 . From now onwards, our long wavelength stability calculation closely follows the one by Amin and Shirokoff [26], where we related oscillon stability to the stability criterion derived by Vakhitov and Kolokolov [50] in the context of light focusing in a nonlinear medium. Here, apart from showing this relationship in the context of non-canonical oscillons, we also provide a pedagogical proof of the stability criterion itself in an Appendix (not provided in [26]).
We are interested in determining the stability of the above oscillons to perturbations with wavelengths comparable to the size of the oscillons. With this in mind, let us define a scaled spatial co-ordinateρ =ǫr whereǫ = ǫ/ √ α with α being an order 1 parameter. We expect the most unstable perturbations to oscillate at the oscillon frequency: √ 1 − ǫ 2 with a 'slowly-varying', time-dependent envelope driven by the oscillating background. To capture the time dependence of the envelope we define a slow time T =ǫ 2 t which is to be treated independently from τ . Note that although O[ǫ] ∼ O[ǫ], they are independent. ǫ plays the role of determining the oscillon solution whereasǫ is used for analyzing the stability about this solution. This distinction is made explicit via the introduction of the α parameter. With these definitions we have Furthermore, let us expand the perturbation χ in powers of ǫ as follows: χ = χ 0 +ǫχ 1 +ǫ 2 χ 2 + . . .
We now substitute the space-time scalings, the ǫ expansion of the perturbation and the oscillon solution into the EOM of the perturbation (40). Collecting terms order by order in ǫ we have (up to order ǫ 2 ) Note that treating O[∇ 2 χ 0 ] ∼ O[χ 0 ] we are restricting ourselves to perturbations that vary spatially on the scale of ǫ. The first equation of (44) yields Substituting χ 0 into the second equation and solving for χ 1 we get where we have ignored the homogeneous solution of χ 1 . Finally, substituting χ 0 and χ 1 into the third equation we get where ∆ = ξ 2 − λ 4 + (10/9)λ 2 3 (the combination which appears in the oscillon solution). Avoiding secular growth requires where We are interested in the eigenvalues of the above linear system. Let v(T,ρ) = e ΩT v e (ρ) and u(y, T ) = e ΩT u e (ρ) where (u e , v e ) is an eigenvector of equation (49). Substituting (u, v) = e ΩT (u e , v e ) into our linear system, we get Equivalently Since H 1 and H 2 are real operators, the eigenvalues −Ω 2 are real, that is Ω is purely real or imaginary. There exist exponentially growing modes if and only if min[−Ω 2 ] < 0. Hence, we now try to determine min[−Ω 2 ].
In the above form, our problem becomes similar to that of light focusing in a nonlinear medium as analyzed by Vakhitov and Kolokolov [50]. Following their techniques, we will show in the Appendix that where The proof of the relationship between the sign of dN dα and sign [min(−Ω 2 )] that we used in the first line above is somewhat involved, which is why we have moved it to an Appendix. Here, we discuss the important and interesting consequences of the result.
Recall that α is merely a scaling of ǫ in the oscillon solution. In terms of ǫ, the stability condition is as follows: Oscillons are stable if and only if where where f is the oscillon profile, that is ϕ osc = ǫf (ǫr) cos √ 1 − ǫ 2 t. We have numerically verified the existence of a long wavelength instability for d ≥ 3.
We stress that the stability criterion (55) is applicable in the small-amplitude limit. More precisely it is applicable when a single frequency solution is a good approximation to the true solution. We now connect the above result to some related work in the literature.
In the example of [26] with canonical kinetic terms, unlike our discussion here, the coefficient of the ϕ 6 term was assumed to be unusually large. This allowed for an (approximate) single frequency solution for the entire allowed amplitude range, which in turn allowed for the derivation of the same stability criterion derived above. However, unlike the above case, in the large ϕ 6 case, N was a non-monotonic function of ǫ in 3 + 1 dimensions, allowing stable solutions to exist at large amplitudes. We also note that a similar stability criterion in term of the oscillon energy was also conjectured in [23] based on numerical results in a massless dilaton + scalar field oscillon. While no general stability condition exists for the general, large-amplitude case, stability and lifetime of large amplitude oscillons is often investigated using Gaussian initial profiles with varying widths and amplitudes. For a flavor of such investigations see for example [24,31].

Radiating Tails in an Expanding Universe
While we have discussed the stability of our oscillons against 'external' long-wavelength perturbations, even without external perturbations, oscillons are not exactly stable. Similar to the canonical case, our more general oscillons possess a radiating tail which we expect to be highly suppressed [51,22], with a decay rate ∼ e −1/ǫ ¶ . Nevertheless, in an expanding universe, this tail can be significantly enhanced [52,20,26]. We briefly sketch out the energy loss due to this radiating tail in an expanding universe below.
For simplicity we will only consider the case in 1 + 1 dimensions. In local co-ordinates, the metric for a homogeneous and isotropic expanding space can be written as (space and time are measured in units of m −1 )  [54]. We thank an anonymous referee for pointing this out. A quantum treatment of the radiation will also increase the decay rate, with the decay rate becoming a power law in ǫ [27].
the following form (following the technique used in [20]): and which leads to an energy loss (averaged over time) given by Our analysis appropriately generalizes the result of [20]. The ∆ appearing above contains ξ 2 from the non-canonical kinetic term along with λ 3 and λ 4 whereas in [20], ∆ = −λ 4 . Note that this analysis in only valid when m ≫ H. The energy loss while enhanced compared to the Minkowski case, can still lead to lifetimes ≫ H −1 .

Discussion
In this paper we have shown that oscillons can exist in a significantly larger class of scalar field theories than previously shown. For a rather general class of scalar field Lagrangians of the form (1) and (3), we have demonstrated the following: • For small-amplitude oscillons to exist, ∆ = ξ 2 − λ 4 + (10/9)λ 2 3 > 0 where ξ 2 is the coefficient of the non-canonical part of the kinetic term.
• The solutions are stable against long wavelength perturbations if and only if dN/dǫ > 0, where We have also calculated the energy loss from oscillons due to an expanding background.
There are a number of natural extensions of our results. The stability criterion above is related to long-wavelength instabilities, which we believe to be the most dangerous instabilities. However, as discussed in the stability section, a calculation of the 'Floquet' instability rates at shorter wavelengths while numerically intensive, is also possible. A further detailed investigation of the suppressed radiating tail, effects of expansion on lifetimes, 'Floquet instabilities' as well as a quantum treatment for these non-canonical oscillons would be interesting.
We have concentrated on the small-amplitude regime in this paper. However, largeamplitude oscillons (with large energies) in 3 + 1 dimensions are interesting due to their possible relevance in cosmology (see for e.g [28,25,40,29,30]). In addition, in 3 + 1 dimensions, single field, small-amplitude oscillons can collapse due to perturbations with wavelengths comparable to the size of the oscillons. As we move to larger amplitudes this instability can disappear (e.g. [26]) . Thus an analysis of the large-amplitude case is certainly worth pursuing. However, moving to large amplitudes also requires a larger number of terms in T and V , which in turn requires a case by case analysis of the solutions and their stability. It is of course possible to analyze them numerically. Although we have not presented the results here, we have analysed large-amplitude oscillons in the DBI Lagrangian: , with rather intriguing dynamics appearing at large amplitudes [53]. To keep our analysis as general as possible, and in an effort to present analytic rather than numerical results, we have restricted ourselves to the small-amplitude case in this paper. An analysis of large-amplitude k-oscillons and their implications in a cosmological context is in progress [53]. We will now try to determine the sign of E min . As mentioned earlier, one can also think of H 2 as the Hamiltonian of a non-relativistic particle in a finite potential well V 2 (ρ) = (α/2) 1 − (9/4)sech 2 ( √ αρ) . Let {ψ γ } be the eigenstates of H 2 with energies E γ . Let us expand u e and Φ in terms of these eigenstates as Φ = γ=0 a γ ψ γ and u e = γ=0 b γ ψ γ . Plugging these into (65) and using Φ|u e = 0 we get where a γ = ψ γ |Φ and we have defined Now either β = 0 or g(E) = 0. If β = 0, then from equation (65) we see that min[E] = E min is obtained if u e = ψ 0 : the ground state of H 2 , which is radially symmetric and has no nodes. This contradicts Φ|u e = 0. Hence β = 0 and we have g(E) = 0. We need to find the smallest root of g(E) = 0 We will now make use of the technical properties of H 2 specified at the beginning of this appendix to analyze the minimum value of E that satisfies g(E) = 0. For the lowest radially symmetric eigenstate (ground state of H 2 without any nodes), a 0 = ψ 0 |Φ = 0. Moreover, from (b), E 0 < 0. For any radially asymmetric (angular) eigenstate a γ = 0 since that eigenstate will be orthogonal to Φ. In particular if ψ 1 is the lowest angular eigenstate, then a 1 = ψ 1 |Φ = 0 and by our assumption (b), has E 1 = 0. If ψ 2 is the next radial eigenstate, then by (b), E 2 > 0. Now consider the behavior of g(E) for E < E 0 . In this domain, g(E) > 0. For E 0 < E < E 2 , g(E) varies monotonically from −∞ to +∞ and crosses 0 for the 'first' time. Hence if g(E) = 0 in this domain, the root E is the smallest root E min . † † Moreover, since g(E) varies monotonically from −∞ to +∞ in this domain, the sign of g(0) determines the sign of E min . Explicitly g(0) > 0 ⇐⇒ E min < 0. Hence from (62) Finally, let us now relate g(0) to d Φ|Φ /dα as follows: (70) † † g(E) varies monotonically between every consecutive pair of distinct E γ .
Multiplying both sides by Φ and integrating, we get Thus using the above result and equation (69) Let us now turn to the justification of the property (a) of H 1 assumed in the proof. We will show that u e |H −1 1 |u e is positive definite. Note that the eigenvalue problem H 1 Ψ β = E β Ψ β is the time independent Schrodinger equation for a particle of mass m = 1 in a radial potential well V 1 (ρ) = (α/2) 1 − (3/4)sech 2 ( √ αρ) . Using the profile equation (18) we get H 1 Φ = 0 where Φ has no nodes. This implies that Ψ 0 = Φ is the unique ground state of H 1 (up to a normalization) with energy E 0 = 0 and all other eigenvalues must be greater than 0. Moreover one has the orthonormal set of excited states {Ψ β } with β = 0 which satisfies Ψ β |Φ = 0. For any state which belongs to this subspace spanned by Ψ β , the operator H 1 is positive definite. Hence H −1 1 exists on this subspace and is also positive definite. This follows from Ψ β |H −1 1 |Ψ γ = E −1 β δ βγ . Now, from equation (52), note that −Ω 2 Φ|u e = H 1 Φ|H 2 u e = 0. Hence for Ω = 0, u e lies in the space spanned by {Ψ β }. Thus u e |H −1 1 |u e is positive definite. We still need to show that H 2 has only one bounded eigenmode with a negative eigenvalue, and the lowest angular eigenfunction has 0 eigenvalue. This is somewhat involved, and we refer the reader to [50] where this is discussed further.