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S-brane and bubble-of-nothing cosmological channels

Physical scope

A source-reported Euclidean five-dimensional Kaluza-Klein bubble instanton with a locally checked smooth cap and asymptotic circle, alongside a distinct spacelike S-brane channel. It gives no nucleation rate, universal vacuum-decay conclusion, or global string solution.

Example

On a time-oriented Lorentzian manifold $(X, g_{L})$ , an S-brane here is a smooth embedding $ι : W ↪ X$ of a positive-dimensional worldvolume whose induced metric $ι^{*} g_{L}$ is positive definite; it is spacelike and has no timelike worldvolume direction. A bubble-of-nothing instanton here is instead the Euclidean five-manifold $M_{E} ≅ R^{2} \times S^{3}$ with metric $d s_{E}^{2} = d r^{2} / (1 - R^{2} / r^{2}) + r^{2} d Ω_{3}^{2} + R^{2} (1 - R^{2} / r^{2}) d ϕ^{2}$ , where $d Ω_{3}^{2} = d θ^{2} + \cos^{2} θ d Ω_{2}^{2}$ and $d Ω_{2}^{2}$ is the unit round metric, $r \geq R > 0$ , and $ϕ \sim ϕ + 2 π$ . Here $R, r$ have units of length and $θ, ϕ$ and the $S^{2}$ coordinates are dimensionless. The circle closes smoothly at $r = R$ ; a Lorentzian continuation is a separate prescription. These distinct spacelike-source and end-of-spacetime channels require model-specific boundary conditions, source actions, stability and semiclassical control; neither is a universal vacuum-decay result.

Two distinct source-backed channels illustrate why the background, boundary data, and continuation must be stated separately. In the first, the S-brane is a localized topological defect with the spacelike embedded worldvolume defined above; for a codimension-one example this means the induced metric is positive definite and the defect is localized in time. In the second, $(M_{E}, g_{E})$ is the Euclidean instanton manifold in a five-dimensional Kaluza-Klein model with one compact circle.

For the bubble take $R > 0$ , $r \geq R$ , $ϕ \sim ϕ + 2 π$ , $θ \in [- π / 2, π / 2]$ , and $d Ω_{2}^{2}$ the unit round metric on $S^{2}$ , so $d Ω_{3}^{2} : = d θ^{2} + \cos^{2} θ d Ω_{2}^{2}$ is the unit round metric on $S^{3}$ . The quantities $r, R, ρ$ have units of length; $θ, ϕ$ and the coordinates on $S^{2}$ are dimensionless. Adopt the source instanton metric [A1]. For this source record the S-brane source action, Lorentzian continuation, stability and semiclassical control as model-specific missing data; the metric supplies the Euclidean boundary and cap geometry only. Neither channel is a universal vacuum-decay law.

Example references

Alternative string-cosmology model data

M. Cicoli, J. P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo and I. Zavala, String Cosmology: from the Early Universe to Today, arXiv:2303.04819v2

Facts & Assumptions

[F1]

Alternative string-cosmology model data: Each cosmological datum must state its source, boundary and transition prescriptions, perturbation transfer, and control regime; an absent continuation is not inferred.

[A1]

Adopt the source-reported five-dimensional Euclidean Kaluza-Klein instanton $d s_{E}^{2} = \frac{d r^{2}}{1 - R^{2} / r^{2}} + r^{2} (d θ^{2} + \cos^{2} θ d Ω_{2}^{2}) + R^{2} (1 - R^{2} / r^{2}) d ϕ^{2}$ on the smooth manifold obtained from $[R, \infty) \times S^{3} \times S^{1}$ by collapsing the $ϕ$ -circle at $r = R$ . Here $R, r$ are lengths, $ϕ$ has period $2 π$ , $θ$ and the $S^{2}$ coordinates are dimensionless, and $d Ω_{2}^{2}$ is the unit round metric. The circle cap is smooth with this period; the configuration is not by itself a decay probability or a full string compactification.

Verification

Given: [A1] and the source's distinct S-brane and bubble-of-nothing interpretations.

1.1givenA1algebra

Near $r = R$ , set $ρ^{2} = 2 R (r - R)$ , so $ρ$ has units of length. Then $1 - R^{2} / r^{2} = ρ^{2} / R^{2} + O (ρ^{4} / R^{4})$ and $d r^{2} / (1 - R^{2} / r^{2}) = d ρ^{2} + O (ρ^{2} / R^{2}) d ρ^{2}$ . The last metric term is $ρ^{2} d ϕ^{2} + O (ρ^{4} / R^{2}) d ϕ^{2}$ . Thus the $(r, ϕ)$ part extends as a smooth polar disk at $ρ = 0$ precisely with the stated $2 π$ period; the remaining factor tends to $R^{2} d Ω_{3}^{2}$ .

2.1A1step 1.1algebra

As $r \to \infty$ , $1 - R^{2} / r^{2} \to 1$ , so the metric approaches $d r^{2} + r^{2} d Ω_{3}^{2} + R^{2} d ϕ^{2}$ , the Euclidean Kaluza-Klein vacuum with circle radius $R$ . The instanton therefore has the source's stated asymptotic boundary geometry and a smooth circle cap.

2.2F1A1step 1.1

The S-brane datum in this example is a spacelike defect worldvolume; the bubble datum in [A1] is a Euclidean instanton with a compact circle that degenerates at a smooth cap. Their geometric supports and continuation data differ, so identifying either one with the other would add an unstated premise.

3.1F1A1step 2.1step 2.2∎

An analytic continuation of the Euclidean saddle is a separately chosen prescription for obtaining a Lorentzian expanding bubble. Steps 1.1–2.1 establish only the cap and asymptotic geometry; they do not compute a nucleation rate, prove stability, specify a universal path-integral measure, or establish a global string solution.

Depends on

Alternative string-cosmology model data

Dependency tree · 2 levels

Sources

M. Cicoli, J. P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo and I. Zavala, String Cosmology: from the Early Universe to Today, arXiv:2303.04819v2