Dynamical realization of the smooth maximal-entropy initial state in weak-early-gravity Higgs-dilaton cosmology

Penrose observed that the initial smoothness of the universe represents an extraordinary fine-tuning: generic initial conditions would be dominated by black holes whose Bekenstein-Hawking entropy exceeds that of the smooth hot Big Bang by a factor of order $10^{123}$. Sloan and Ellis (2019) proposed to remove this tuning by making the Standard Model Higgs field play the role of a dilaton whose non-minimal curvature coupling $F(\phi)$ makes Newton's constant effectively dynamical and extremely weak at early times, so that a smooth matter configuration is the maximal-entropy state rather than an improbable one. Their proposal, however, still assumes rather than derives the required initial condition: no model is known in which the early Hubble rate is small enough, for long enough, that thermal processes actually smooth matter before inflation - and the paper proves that the viable large-field models are necessarily past-singular ($H$ tending to infinity), which removes the long weak-gravity era that thermalisation needs. This record asks for a dynamical realization of the smooth maximal-entropy initial state in weak-early-gravity Higgs-dilaton cosmology (or a proof of obstruction within the model class), together with the quantitative determination of what finite suppression of $G$ suffices to remove Penrose's entropy count. Whether such a realization exists remains open.

Background

A foundational problem of standard cosmology is the observed initial smoothness of the universe. As Penrose emphasized (Fashion, Faith, and Fantasy in the New Physics of the Universe, 2016), the maximum-entropy state of a given amount of matter is a black hole - its Bekenstein-Hawking entropy is $S_{BH} = c^3 A/(4G\hbar)$, where $A$ is the horizon area, $G$ Newton's constant, $c$ the speed of light and $\hbar$ the reduced Planck constant - so a generic initial state of the material now visible would be dominated by black holes whose entropy exceeds the smooth Big Bang by a factor of order $10^{123}$; Penrose quantifies the required suppression of early gravitational degrees of freedom as a fine-tuning of order one part in $10^{10^{123}}$. Studies of inflation that perturb Robertson-Walker geometries (the standard spatially homogeneous and isotropic cosmologies) presuppose an already-smooth background and so do not explain it. Penrose's own response is Conformal Cyclic Cosmology, the hypothesis that the universe is a sequence of aeons related by conformal rescaling with entropy effectively reset at each crossover. Greene, Hinterbichler, Judes and Parikh (2011) proposed an alternative: if gravity were much weaker at early times, black-hole formation would be suppressed and the maximum-entropy state of matter would be its ordinary thermal equilibrium, which is smooth, making smooth initial conditions natural rather than fine-tuned; the mechanism's dynamics was left largely undeveloped.

Sloan and Ellis implemented this idea using tested particle physics. They take the Standard Model Higgs field $\phi$ to play the role of a dilaton (the scalar field that sets the local strength of gravity) in a scalar-tensor theory whose gravitational term is $F(\phi)/6\,R$, where $F$ is a non-minimal coupling function and $R$ the Ricci scalar (a standard measure of spacetime curvature), so that the effective Newton constant behaves as $G \sim 1/F(\phi)$ - a Jordan-Brans-Dicke-type construction (scalar-tensor gravity in which Newton's constant is set by a scalar field rather than being a true constant). (The 'Jordan frame' is the variable-Planck-mass formulation; the 'Einstein Frame' is the conformally rescaled one in which gravity takes its standard form, at the price of induced couplings between the transformed field and ordinary matter.) They establish: (i) small-field initial conditions cannot work - avoiding a spurious de-Sitter-like stationary point (an almost-constant-curvature, exponentially expanding solution such as a cosmological constant produces) between the origin and the Higgs vacuum (vacuum expectation value $\Phi$) bounds the change of $F$ across the interval to about $10^{37}$, far short of the $10^{123}$ change in $G$ needed to counter Penrose's count; (ii) large-field models, for example $F = A^2\phi^2/(\phi + \xi^2)$ with $\xi \gg \Phi$, work phenomenologically: at large $\phi$ the effective $G$ is tiny, the dynamics reproduces the Higgs-inflation plateau (Higgs inflation being the scenario in which the Standard Model Higgs with a large non-minimal curvature coupling drives a period of accelerated expansion), fits the Planck satellite constraints on the spectral index (the scale-dependence of the primordial density-perturbation amplitude) and the tensor-to-scalar ratio (the ratio of primordial gravitational-wave to density-perturbation amplitudes), and ends at the observed vacuum with the correct Planck mass, while in models where $F$ is bounded above a parameter $A \sim 10^{123}$ is required; (iii) nevertheless, a universality result holds: any large-field model in this class is necessarily singular in the backward evolution, with Hubble rate $H \to \infty$ as $\phi \to \infty$, and the alternative of turning gravity on only below a fixed scale (the Higgs-mechanism switch-on of Alexander, Barrow and Magueijo) fails for related reasons.

The load-bearing gap is the initial state. Thermalisation of matter before inflation requires an early era that is simultaneously weak-gravity (so black holes cannot form and dominate the entropy) and long-lived with a very small Hubble rate (so interaction rates exceed the expansion rate and matter can equilibrate). Sloan and Ellis note that in principle thermalisation can be achieved if the Hubble parameter were very low to begin with, but that in practice models do not exist where $H$ is very small initially; their solution to Penrose's problem therefore ultimately still assumes that the universe begins in a maximal-entropy - hence, in the weak-gravity era, smooth - state, 'instead of one chock full of black holes'. Whether this assumption can be replaced by dynamics is the open problem, and it is tied by the authors to the quantitative question of what finite value of $1/G$ suffices to remove the entropy problem.

Problem Statement

Construct, or prove unachievable within the analyzed model class, a Higgs-dilaton cosmology in which the smooth pre-inflationary initial state is dynamically produced rather than assumed. The analyzed model class is the Jordan-frame single-field class of Sloan and Ellis, defined by the action

$S = \int \sqrt{g}\left(\frac{F(\phi)}{6}R + \frac{1}{2}\partial_a\phi\,\partial^a\phi - V(\phi)\right)$

with a canonical kinetic term, a non-minimal coupling function $F(\phi)$ multiplying the Ricci scalar $R$, and a Higgs-type potential $V(\phi) = (\Phi^2 - \phi^2)^2$ with vacuum expectation value $\Phi$, on a Robertson-Walker background together with minimally coupled matter, so that the effective Newton constant behaves as $G \sim 1/F(\phi)$.

A complete answer must: (1) specify the non-minimal coupling function $F(\phi)$, the Higgs potential, and parameter values, and solve the background cosmological dynamics, exhibiting an early weak-gravity era in which the effective Newton constant $G \sim 1/F(\phi)$ is suppressed by a finite factor sufficient to remove Penrose's gravitational-entropy count (a count of order $10^{123}$ in entropy; the source notes that what finite value of $1/G$ suffices depends on the detailed model employed), that factor being determined quantitatively by the accompanying entropy accounting; (2) show that during this era the Hubble rate $H$ is small enough, and remains so for long enough, that matter interaction rates exceed the expansion rate and thermal processes smooth out inhomogeneities, so that the universe reaches the maximal-entropy state compatible with weak gravity (the answer must state explicitly the criteria it uses - the matter content and interaction rates considered, the threshold for $H$ being 'small enough' and for the era being 'long enough', and the measure by which dominance of configurations is judged - and the accounting must establish which configurations dominate the phase space when $G$ varies, and that the equilibrium so reached is indeed smooth rather than black-hole-dominated); and (3) show that this era is followed by observationally viable slow-roll inflation (slow roll: the inflaton descends its potential gently enough that the expansion stays close to exponential and the spectral index and tensor-to-scalar ratio are well-defined observables) and by exit to the Standard-Model Higgs vacuum with the observed present-day Planck mass. Alternatively, establish an obstruction theorem - extending Sloan and Ellis's results that small-field models cannot change $G$ by more than roughly $10^{37}$ and that large-field models are necessarily past-singular with $H \to \infty$ - proving that no model in the analyzed class admits such a thermalisation era, so that the maximal-entropy initial state must remain an independent postulate. The obstruction must cover the class defined above, including its viable large-field exemplars such as $F = A^2\phi^2/(\phi + \xi^2)$ with $\xi^2 \gg \Phi^2$; a precisely characterized extension (for example by additional fields, non-canonical kinetic terms (field-kinetic terms more general than a simple quadratic), or specified quantum-gravity effects) may be treated in addition, but cannot be substituted for the analyzed class. The source problem is the assumption-removal: the answer must make explicit which of these outcomes it establishes and within exactly which class of models.

References

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