Spherical vacuum and scalar collapse for the Starobinsky ${R}^{2}$ model

Spherical vacuum and scalar collapse for the Starobinsky $R^{2}$ model

Jun-Qi Guo and Pankaj S. Joshi

Phys. Rev. D 94, 044063 – Published 31 August 2016

Abstract

Spherical vacuum and scalar collapse for the Starobinsky $R^{2}$ model is simulated. Obtained by considering the quantum-gravitational effects, this model would admit some cases of singularity-free cosmological spacetimes. It is found, however, that in vacuum and scalar collapse, when $f^{'}$ or the physical scalar field is strong enough, a black hole including a central singularity can be formed. In addition, near the central singularity, gravity dominates the repulsion from the potential, so that in some circumstances the Ricci scalar is pushed to infinity by gravity. Therefore, the semiclassical effects as included here do not avoid the singularity problem in general relativity. A strong physical scalar field can prevent the Ricci scalar from growing to infinity. Vacuum collapse for the $R \ln R$ model is explored, and it is observed that for this model the Ricci scalar can also go to infinity as the central singularity is approached. Therefore, this feature seems universal in vacuum and scalar collapse in $f (R)$ gravity.

Received 27 July 2015

DOI:https://doi.org/10.1103/PhysRevD.94.044063

Physics Subject Headings (PhySH)

Alternative gravity theories Quantum fields in curved spacetime

Numerical relativity

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Jun-Qi Guo^* and Pankaj S. Joshi^†

Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India

^*junqi.guo@tifr.res.in
^†psj@tifr.res.in

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Issue

Vol. 94, Iss. 4 — 15 August 2016

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Images

Figure 1
Potential for the Starobinsky $R^{2}$ model (5), $f (R) = R + α R^{2}$ , with $α = 1$ . (a) Potential in the Jordan frame (10). (b) Potential in the Einstein frame (12).
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Figure 2
Evolutions for vacuum collapse for the Starobinsky $R^{2}$ model. (a) and (b): Evolutions of $r$ and $f^{'}$ , respectively. The time interval between two consecutive slices is $4 Δ t = 0.002$ . (c) and (d): Apparent horizon and singularity curve of the formed black hole. As shown in (a), in later evolutions, the central singularity is approached $(r \to 0)$ . As a result, the equation of motion for $ϕ$ (16) is reduced to $ϕ_{, t t} \approx - 2 r_{, t} ϕ_{, t} / r$ , which describes a positive feedback system since $- 2 r_{, t} / r$ is positive. Consequently, as shown in (b), around $x \approx 0.35$ where $ϕ_{, t}$ is negative, $ϕ$ , $f^{'}$ , and $R$ are pushed to $- \infty$ , 0, and $- 1 / (2 α)$ , respectively, while around $x \approx 0.1$ where $ϕ_{, t}$ is positive, $ϕ$ , $f^{'}$ , and $R$ are all pushed to $+ \infty$ . Therefore, the $R^{2}$ term in Eq. (5) does not prevent the singularity formation. The discontinuities of $f^{'}$ in (b) at $x \approx 0.35$ in later evolutions come from the fact that the spacelike singularity ( $r = 0$ ) is met in simulations [also see (a) and (c)].
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Figure 3
Dynamics for $ϕ$ in the vicinity of the singularity curve for vacuum collapse. Here, $ξ = t_{0} - t$ and $t_{0}$ is the coordinate time on the singularity curve. (a)–(b) and (c)–(d) are for the slices of $x = 0.1$ and $x = 0.35$ , respectively. (a) and (c) show that near the singularity there is $ϕ_{, t t} \approx - 2 r_{, t} ϕ_{, t} / r$ , which is a positive feedback system of $ϕ_{, t t}$ and $ϕ_{, t}$ . Then, as shown in (d), on the slice of $x = 0.35$ , $ϕ_{, t}$ is negative, and $ϕ$ is pushed to $- \infty$ as the central singularity is approached. However, as shown in (b), on the slice of $x = 0.1$ , $ϕ_{, t}$ is positive, and $ϕ$ is pushed to $+ \infty$ in later evolutions. (b) Here, $ϕ = a \ln ξ + b$ , $a = - 0.18883 \pm 0.00001$ , $b = - 0.2152 \pm 0.0001$ . (d) Here, $ϕ = a \ln ξ + b$ , $a = 0.033650 \pm 0.000004$ , $b = 0.08561 \pm 0.00006$ .
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Figure 4
Evolutions of $f^{'}$ for vacuum collapse for the Starobinsky $R^{2}$ model with $α = 0.01$ . The time interval between two consecutive slices is $2 Δ t = 4 \times 10^{- 4}$ .
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Figure 5
Scalar collapse for the $R^{2}$ model. The results are for the slice $x = 2.8$ . (a), (c), and (e): The terms in the equation of motion for $ϕ$ (16) for different ranges of $r$ . (b), (d), and (f): The corresponding evolutions of $ϕ$ . A strong $ψ$ can make the evolution of $ϕ$ slow down, make a turn, and approach to $- \infty$ .
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Figure 6
Vacuum collapse for the $R \ln R$ model. (a) Potential in the Einstein frame. (b) The terms in the equation of motion for $ϕ$ (16) for the slice $x = 0.35$ . (c) Evolution of $ϕ$ for the slice $x = 0.35$ .
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