Gravitational collapse in the postinflationary Universe

Benedikt Eggemeier, Bodo Schwabe, Jens C. Niemeyer, and Richard Easther

Phys. Rev. D 105, 023516 – Published 12 January 2022

Abstract

The Universe may pass through an effectively matter-dominated epoch between inflation and big bang nucleosynthesis during which gravitationally bound structures can form on subhorizon scales. In particular, the inflaton field can collapse into inflaton halos, forming “large scale” structure in the very early universe. We combine N-body simulations with high-resolution zoom-in regions in which the nonrelativistic Schrödinger-Poisson equations are used to resolve the detailed, wave-like structure of inflaton halos. Solitonic cores form inside them, matching structure formation simulations with axion-like particles in the late-time universe. We denote these objects inflaton stars, by analogy with boson stars. Based on a semianalytic formalism we compute their overall mass distribution which shows that some regions will reach overdensities of $10^{15}$ if the early matter-dominated epoch lasts for $20 e$ -folds. The radii of the most massive inflaton stars can shrink below the Schwarzschild radius, suggesting that they could form primordial black holes prior to thermalization.

Received 1 November 2021
Accepted 16 December 2021

DOI:https://doi.org/10.1103/PhysRevD.105.023516

Physics Subject Headings (PhySH)

Cosmology Evolution of the Universe Inflation

Astrophysical & cosmological simulations Numerical simulations in gravitation & astrophysics

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Benedikt Eggemeier ^1,*, Bodo Schwabe ^1,2,†, Jens C. Niemeyer^1,‡, and Richard Easther ^3,§

¹Institut für Astrophysik, Georg-August-Universität Göttingen, D-37077 Göttingen, Germany
²CAPA & Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain
³Department of Physics, University of Auckland, Private Bag, 92019 Auckland, New Zealand

^*benedikt.eggemeier@phys.uni-goettingen.de
^†bschwabe@unizar.es
^‡jens.niemeyer@phys.uni-goettingen.de
^§r.easther@auckland.ac.nz

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Vol. 105, Iss. 2 — 15 January 2022

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Images

Figure 1
Scaled sequence of the inflaton density $N = 17$ $e$ -folds after the end of inflation. The rectangular region shows the projected inflaton density of 15% of the full simulation box centered on a selected halo. A slice through the maximum density of the halo illustrates the setup of several refinement levels in our simulation. Note that the white space in the last N-body level correspond to cells without any particles. Compared to the root grid the spatial resolution of the finest finite-difference (FD) level, which is displayed in an enlargement in the upper right panel, is increased by a factor of $2^{8} = 256$ . One can clearly recognize the interference patterns in the filaments, the granular structure inside the collapsed inflaton halo and the solitonic core in its center.
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Figure 2
Radial density profiles of five simulated halos at different number of $e$ -folds $N$ after the end of inflation. The orange curves represent the profiles from solving the Schrödinger-Poisson equations with the finite-difference method while the blue curves are given by the theoretical soliton density profiles from Eq. (11). For comparison, the halo profiles from a pure N-body simulation are shown in green.
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Figure 3
Velocity spectra of three halos inside their respective virial radii for the wave function (solid) and for N-body particles (dotted) in the same region. Fits of Maxwellian distribution functions [cf. Eq. (14)] are displayed in gray. The vertical dashed lines show the soliton velocities $v_{*}$ . The crosses mark $v_{vir}$ of the respective halos. They closely align with the peaks of the Maxwellian distribution.
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Figure 4
Mass increase of three solitonic cores relative to their mass $M_{*, 0}$ at formation. Due to the strong oscillations of the solitons the simulation data was smoothed with a Gaussian kernel with a standard deviation of $σ = 2 \times 10^{- 5} t_{u}$ . The mean and its $1 σ$ deviation band are displayed by the solid lines and the shaded regions, respectively. As shown by the dashed lines, the mass growth obeys Eq. (19) with timescale $τ$ given by Eq. (18).
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Figure 5
Top: central soliton density as a function of time. Bottom: its corresponding power spectrum. Boundaries of the shaded region are determined by the quasi-normal frequencies [see Eq. (20)] using the maximum and minimum $ρ_{*, 0}$ . The spectrum peaks at the frequency $f_{*} = 2.9 \times 10^{4} t_{u}$ corresponding to ${\bar{ρ}}_{*, 0}$ .
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Figure 6
Inflaton star mass function at different number of $e$ -folds $N$ after the end of inflation for an inflaton mass of $m = 6.35 \times 10^{- 6} M_{Pl}$ . Left: solid lines represent the ISMF derived from the numerically obtained IHMF in Ref. [18] while dashed lines display the corresponding mass functions from the Press-Schechter formalism. The upper axis shows the inflaton star overdensity which can be calculated from their mass. Right: ISMF from Press-Schechter theory until thermalization at $T = 1 TeV$ ( $N = 36.5$ ) with overdensities relative to the mean density ${\bar{ρ}}_{therm}$ at thermalization. The gray shaded area highlights masses larger than $M_{*, \max}$ [see Eq. (21)].
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