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Formation, gravitational clustering, and interactions of nonrelativistic solitons in an expanding universe

Mustafa A. Amin and Philip Mocz
Phys. Rev. D 100, 063507 – Published 9 September 2019

Abstract

We investigate the formation, gravitational clustering, and interactions of solitons in a self-interacting, nonrelativistic scalar field in an expanding universe. Rapid formation of a large number of solitons is driven by attractive self-interactions of the field, whereas the slower clustering of solitons is driven by gravitational forces. Driven closer together by gravity, we see a rich plethora of dynamics in the soliton “gas” including mergers, scatterings, and formation of soliton binaries. The numerical simulations are complemented by analytic calculations and estimates of (i) the relevant instability length scales and timescales, (ii) individual soliton profiles and their stability, (iii) number density of produced solitons, and (iv) the two-point correlation function of soliton positions as evidence for gravitational clustering.

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  • Received 5 March 2019

DOI:https://doi.org/10.1103/PhysRevD.100.063507

© 2019 American Physical Society

Physics Subject Headings (PhySH)

Gravitation, Cosmology & AstrophysicsNonlinear DynamicsParticles & Fields

Authors & Affiliations

Mustafa A. Amin1,* and Philip Mocz2,†

  • 1Physics & Astronomy Department, Rice University, Houston, Texas 77005, USA
  • 2Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, New Jersey 08544, USA

  • *mustafa.a.amin@gmail.com
  • philip.mocz@gmail.com

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Vol. 100, Iss. 6 — 15 September 2019

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Images

  • Figure 1
    Figure 1

    Projected comoving “densities” a3|ψ|2 (averaged along the line of sight) at several scale factors (a=1 to a=20) in our 3+1 dimensional lattice simulations, with βM/mpl=0.03, and local gravitational interactions switched on (top panels) and off (bottom panels). The early instability due to self-interactions gives rise to the formation of solitons from an almost homogeneous initial state. A statistical analysis of the locations of solitons at late times shows evidence for clustering only in the case where gravitational interactions are included. Note that inside solitons, |ψ|2=const.; that is, their core density does not redshift, whereas the background |ψ¯|2a3. Moreover, solitons maintain a fixed physical size; hence, the illusion of them shrinking in size in a comoving volume. The initial size of the box is the size of the horizon at the beginning of the simulation LHin1. The solitons contain a dominant fraction of the mass in the simulation volume. On a technical aside, note that the projected comoving density even in the densest (lightest in color) regions in the above plot will be smaller than the density inside the cores because of the small volume occupied by the solitons.

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  • Figure 2
    Figure 2

    Power spectrum of the field ψ (scaled by |ψ¯|2a3). The initial conditions are consistent with vacuum fluctuations, with a cutoff removing relativistic scales. A self-interaction driven instability on the wave number k/a2|ψ¯|2Unl′′(|ψ¯|2) drives the initial growth of the perturbations. These perturbations backreact on the homogeneous condensate around anl2.1 on the physical scale knl/a0.35 first. After this time, solitons soon begin to form, separated by a comoving distance of 2π/knl. Note that in this figure, since we have divided the power spectrum |ψ¯|2, the backreaction takes place when the spectrum is roughly of order unity.

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  • Figure 3
    Figure 3

    The comoving number density of solitons a3nsol in our simulations with (solid) and without (dotted) gravitational interactions. Proper solitons begin to form around a4, with O[103] solitons per Hubble volume H3 at this time. At late times, the number density of solitons is lower in the case when gravity is included due to mergers or disruptions made possible by gravitational clustering. The curves are obtained by averaging over 6 runs.

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  • Figure 4
    Figure 4

    The relationship between the central amplitude and 1/e width of the solitons. The points are extracted from our simulations, whereas the curve is calculated semianalytically. Note that at late times, only solitons that are stable according to the Vakhitov-Kolokolov stability criterion (on the right of the gray line) remain. For our parameters, gravity remains weak and does not significantly alter individual soliton profiles. The gravitational potential at the center of the solitons is plotted on the top axis.

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  • Figure 5
    Figure 5

    The two-point correlation function of soliton locations with and without the inclusion of gravitational interactions. At early times, the correlation functions with and without gravity agree with each other. However, at late times gravitational clustering ξLS(r)r2 is clearly visible for the a=16 and a=20 cases in the above figure.

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  • Figure 6
    Figure 6

    Gravitational clustering facilitates close encounters at late times between solitons. Such close encounters lead to mergers, strong scattering, and formation of soliton binaries. Nongravitational interactions can play a dominant role in the close encounters, with the phase of the scalar field also playing an important role. This richness in the close-encounter dynamics makes the soliton gas distinct from a gravitationally interacting gas of particles. Shown in this figure are projected densities in zoom-ins (box size L/4), around 3 interactions (bounce, merge, and orbit), at 5 times, each separated by time interval corresponding to Δlog(a)=1.16.

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  • Figure 7
    Figure 7

    Left: Colors show the growth rate μk as a function of k and ψ¯. The dark red regions are stable. The color bar indicates the magnitude of the μk and μ˜k. The dotted lines indicate the flow of k and ψ¯ as the universe expands. To compare this plot with the corresponding Floquet chart (right) from the relativistic case, we set ψ¯=ϕ¯/2. The factor of 2 can be seen from ϕ=2[ψeit]. The magnitude of the growth rate of the instability and the boundary of the nonrelativistic instability band (solid black line) deviate from the relativistic one at large amplitudes. The same is true (to a larger extent) for the magnitude of the Floquet exponent. Also notice that the higher order instability bands are absent in the nonrelativistic treatment. We use Vnl(ϕ)=(1/2)m2M2tanh2(ϕ/M)(1/2)m2ϕ2 and Unl(|ψ|2)=Vnl(ϕ)|ψ|4/2(1+|ψ|2) for ϕ/M<π/2, and the comparison at large ϕ, ψ is not justified.

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  • Figure 8
    Figure 8

    Probability density function of density of the field (left panel) and the gravitational potential (right panel). The PDF is shown for the case with and without gravitational interactions included. In the PDF for the gravitational potential, at each time slice the spatial average of the gravitational potential is zero in the simulation volume. Note that the gravitational potential remains small throughout our simulation. The behavior of the density PDF here can be compared to simulations which involve the relativistic Klein-Gordon equation in an expanding universe but with a “passively” calculated gravitational potential (Fig. 3 of [13]).

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