Oscillons and other defects

Non-linear field theories may have classical solutions that are localized in space, but are not elementary excitations (particles) of the theory. We speak of topological or non-topological defects or solitons, reflecting whether the stability to collapse is due to non-trivial topology (such as for doman walls, cosmic strings, monopoles and textures) or not (oscillons, Q-balls).

Defects are abundant in condensed matter systems, where they may appear as actual defects in a crystal grid, or in connection with interfaces between liquids [review of condmat defects]. In these systems, the “field” in which the defect can be considered to exist is not microscopic, but an effective order parameter or condensate variable. These system are also typically non-relativistic.

In particle physics, the fields are in principle actual fundamental relativistic quantum fields, and identifying possible defects becomes a mathematical exercise in classical field theory.

The separate question of whether such defects exist in the Universe ties in closely with the symmetries of (extensions of ) the Standard Model. In this way, identifying defects associated with theoretical models and ruling them in or out observationally, provides information about a potential unified theory of particle physics.

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Topological defects do not occur in the Standard Model, although non-localized textures exist in the electroweak sector [1]. Observationally, no topological defects have been seen, for instance ruling out copious production of domain walls in the early Universe [2].

Monopoles are notoriously hard to observe [3], since they are diluted during cosmological expansion.

Strings-like defects (cosmic strings) are the best candidates for neither over- nor under-populating the Universe, and over many years much effort has gone into predicting their observational signatures, as cosmic lenses [4] and in the cosmic microwave background [5]. Because defects are inherently non-perturbative phenomena, large-scale numerical simulations are often used [6].

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Non-topological defects fall into two categories. Some (like Q-balls and breathers) are absolutely stable, due to the conservation of some charge or angular momentum. These are also time-dependent defects, in that they oscillate in time, but in a truly periodic way. They may decay by interactions with other fields, but left on their own, they will continue to oscillate for ever.

The other type is also oscillating and time-dependent, but only semi-periodic. They eventually decay, on a timescale of 100’s or millions of periods. These are known as oscillons [7], and they are stabilized by dynamical effects, essentially the fact that large amplitude oscillations in some non-linear field theories have frequencies smaller than the particle frequency. These oscillations are therefore not able to excite particle degrees of freedom and thereby “decay”.

The study of oscillons is partly analytic [8], but mostly numerical in nature [9]. The current understanding is that for a broad class of non-linear field theories, so-called “quasi-breather” solutions exist, whose time evolution maps out a trajectory in field space. These quasi-breathers are attractors, in the sense of nearby configurations evolving towards them; they are probably unique [13]; and they can be defined as the localized solutions which emit the least amount of “radiation” [10].

Oscillons exist in d+1 space dimensions for d=0 to at least 7. Analytic work limiting the spatial dimensionality to <7 [11] is likely relying too much on a Gaussian anzats [13].

Lifetimes vary hugely for different d, at least for the simplest models; d=0 is stable, d=1 maybe also. In d=2, simulations have been performed up to millions of periods, and no decay has been observed. For d=3 the lifetime is tens of thousands, d=4 thousands and for d>4 the lifetime increases again. Whether there is an upper limit to d is unknown.

In principle, if initial conditions could be tuned arbitrarily close to the true quasi-breather, lifetimes could be even larger. Numerical simulations typically assume Gaussian initial conditions, parameterized by a width and an amplitude. It turns out that one can tune these to overlap with the quasi-breather, giving a spike or band-like structure of the lifetime [12,13].