Completing the finite-matter alternative string-landscape cosmology: Einstein-equation consistency with nonzero pressure, observational constraints on the inhomogeneity scale, and the big-crunch and light-trapping fate

Clavelli and Goldstein (2013) proposed an alternative to standard string-landscape eternal-inflation cosmology in which the multiverse contains a finite total mass M whose matter density is a spatial delta function at the big bang and diffuses outward thereafter, avoiding the infinite cloning of observers and the measure problem. The authors themselves listed what remains unresolved: a general-relativistic metric formulation incorporating nonzero pressure, observational constraints on the free parameters (M, the inhomogeneity scale R, and the Coleman-De Luccia thin-wall parameters A and S), whether R comparable to or smaller than the Hubble length is viable, whether transitions to negative vacuum energy produce a big crunch, and the future lifetime before light trapping once pressure is included. Two follow-up papers by the authors themselves (2015) address adjacent points — constructing exploratory metrics and entropy-bound constraints on R — but leave the core questions open.

Background

In the conventional picture of string-theory cosmology, the string landscape — the enormous collection of metastable vacua of string theory — combined with eternal inflation (a regime in which quantum fluctuations keep inflation going forever in some regions) produces infinitely many causally separated bubble universes. This entails several conceptually troubling features: the measure problem (there is no agreed prescription for assigning probabilities to events in an infinite spacetime), the production of Boltzmann brains (randomly assembled observers arising from thermal fluctuations), and what Clavelli and Goldstein call infinite cloning — every individual and civilization existing infinitely many times in causally disconnected regions. Clavelli and Goldstein proposed an alternative that keeps a string-landscape explanation of the small observed vacuum energy but abandons two standard presuppositions: the absence of a preferred rest frame and the homogeneity of matter. Their model postulates (i) a finite total rest energy $Mc^{2}$ in the multiverse, (ii) a matter density that at the big bang is a spatial delta function at a preferred origin and thereafter diffuses outward with conserved total mass — for example a Gaussian profile $\rho_m(r,t) = M/(R\sqrt{\pi}\,a(t))^{3}\,e^{-r^{2}/(Ra(t))^{2}}$, where $a(t)$ is the cosmological scale factor (normalized to 1 today) and $R$ sets the inhomogeneity length — and (iii) a cascade of transitions down in vacuum energy governed by the Coleman–De Luccia thin-wall formalism (the standard semiclassical description of vacuum decay through bubble nucleation), with nucleation probability per unit time and volume $dP/(dt\,d^{3}r) = A e^{-B}$ and bounce exponent $B = 27\pi^{2}S^{4}/[2\hbar c(c^{2}\Delta\rho)^{3}]$, where $S$ is the bubble-wall energy per unit area and $\Delta\rho$ the vacuum-energy difference. Here $A$ and $S$ are free parameters fixed by an unknown effective potential. For any finite $R$ the model has finitely many clones, no measure problem (finite total matter), and a possible resolution of the baryon-asymmetry puzzle (the observed excess of matter over antimatter); as $R \to \infty$ it approaches the standard landscape cosmology. The default assumption is $R$ much larger than the Hubble length $L_H = c/H_0 \approx 4.2$ Gpc (about 14 billion light-years), which makes the matter distribution indistinguishable from homogeneous inside the observable universe; the model is then anti-Copernican (our bubble sits near the preferred origin) but observationally hidden. A light trapping region is a region inside which the enclosed mass is so large that light cannot escape to infinity, as at a black hole horizon; the authors argue a future transition to zero vacuum energy protects us from being engulfed by one, and estimate (neglecting pressure) that the visible universe saturates the trapping condition at $t_{max} = 1/H_0 \approx 13.6$ Gyr after the big bang, requiring the present matter-density fraction $\Omega_m(0) > 0.2645$ for the universe not to be already light trapping. The authors flag four clusters of unresolved issues: the metric formulation in general relativity (their kinematics uses a Newtonian continuity equation and a Friedmann–Robertson–Walker-type scale factor — the standard scale factor of homogeneous, isotropic cosmology — without a genuine Einstein-equation solution), the effect of nonzero pressure, the observational status of the free parameters and of small $R$, and the terminal fate (a possible transition to negative vacuum energy producing a big crunch, i.e. a collapsing phase — against which they cite the quantum-gravity arguments of T. Banks — and the pressure-corrected lifetime before light trapping).

Problem Statement

The single overarching objective is to decide whether the finite-matter, delta-function-initial-density cosmology of Clavelli and Goldstein is internally consistent and observationally viable, by completing the four linked tasks the source paper itself leaves open. (1) Metric formulation with pressure: construct space-time metrics that solve Einstein's equations and support a matter density that is a spatial delta function at the big bang, diffuses outward with time-independent total mass $\int d^{3}r\,\rho_m = M$, and incorporates the effect of nonzero pressure — that is, redo the model's kinematic and dynamical results (Hubble-law velocity field, scale-factor evolution, mass-trapping bounds) as genuine general-relativistic results rather than Newtonian approximations. The pressure is modelled as a barotropic component $p = w\rho$; an admissible equation of state is any barotropic law with $-1 \le w \le 1$ that satisfies the null energy condition and keeps the pressure dynamically significant, $\lvert w \rvert \ge 1/3$ (the radiation-like case $w = 1/3$ is the physically motivated member, and the pressureless limit $w = 0$ is excluded as degenerate). A metric construction for this component may adopt any single reported admissible $w$, which fixes the pressure-corrected lifetime of component (4) for that equation of state; a refutation, by contrast, must show that no such construction exists for every admissible $w$. (2) Parameter constraints: determine the observationally viable ranges of the total mass $M$, the inhomogeneity length $R$, and the thin-wall nucleation parameters $A$ and $S$; in particular, state whether values of $R$ comparable to or smaller than the Hubble length $L_H \approx 4.2$ Gpc are consistent with observations — including whether the predicted systematic matter inhomogeneities at large scales do or do not exist — and whether the solar system's proximity to the origin (if $R$ is in the Gpc range) requires an anthropic selection effect. The range determinations are to be stated against the homogeneity-scale and large-scale-structure observations cited below, together with the assumptions and data on which they rest, with the small-$R$ and large-scale-inhomogeneity verdicts given as special cases of the range of $R$; and the thin-wall parameters $A$ and $S$, which the model leaves fixed by an unknown effective potential, are to be reported as observationally constrained or unconstrained according to whether they can be bounded by observation. (3) Terminal fate: determine whether transitions of the vacuum to negative vacuum energy — leading to a big crunch — occur in this model, and whether the arguments of Banks against such transitions apply. (4) Lifetime before light trapping: compute the future lifetime of the current universe before it becomes a light trapping region with the effect of pressure included — generalizing both the pressureless trapping-time estimate $t_{max} = 1/H_0$ and the pressureless condition $\Omega_m(0) > 0.2645$ (the lower bound on the present matter-density fraction for the universe not to be already light trapping). The outcome of component (1) bears on the others: if a construction exists, the questions of components (2)-(4) remain open; if instead no such construction exists for every admissible equation of state, the overarching consistency objective is settled negatively — the finite-matter cosmology is excluded — so the questions of components (2)-(4) do not arise; and an existence obstruction shown for a single chosen equation of state rules out only that equation of state, leaving components (2)-(4) open.

References

1. L. Clavelli and Gary R. Goldstein, "An Alternative String Landscape Cosmology: Eliminating Bizarreness", International Journal of Modern Physics A 28, 1350148 (2013). DOI: 10.1142/S0217751X13501480. arXiv:1205.3138. https://arxiv.org/abs/1205.3138
2. L. Clavelli and Gary R. Goldstein, "Einstein's Equations and a Cosmology with Finite Matter", International Journal of Modern Physics A 30, 1550068 (2015). DOI: 10.1142/S0217751X15500682. arXiv:1411.5966. https://arxiv.org/abs/1411.5966
3. L. Clavelli, "Finite Matter Resolution of the Cosmological Entropy Problem", arXiv:1511.02948 (2015). https://arxiv.org/abs/1511.02948
4. T. Banks, "The Top 10^500 Reasons Not to Believe in the Landscape", arXiv:1208.5715 (2012). https://arxiv.org/abs/1208.5715
5. T. Banks, "TASI Lectures on Holographic Space-Time, SUSY and Gravitational Effective Field Theory", in "String Theory and Its Applications" (World Scientific, 2011). DOI: 10.1142/9789814350525_0002. arXiv:1007.4001. https://arxiv.org/abs/1007.4001
6. Jaswant K. Yadav, J. S. Bagla, and Nishikanta Khandai, "Fractal Dimension as a measure of the scale of Homogeneity", Monthly Notices of the Royal Astronomical Society 405, 2009 (2010). DOI: 10.1111/j.1365-2966.2010.16612.x. arXiv:1001.0617. https://arxiv.org/abs/1001.0617
7. Roger G. Clowes, Kathryn A. Harris, Srinivasan Raghunathan, Luis E. Campusano, Ilona K. Soechting, and Matthew J. Graham, "A structure in the early universe at z ~ 1.3 that exceeds the homogeneity scale of the R-W concordance cosmology", Monthly Notices of the Royal Astronomical Society 429, 2910-2916 (2013). DOI: 10.1093/mnras/sts497. arXiv:1211.6256. https://arxiv.org/abs/1211.6256