Solving the cosmological entropy issue with a Higgs dilaton David Sloan∗ Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom and Beecroft Institute of Particle Astrophysics and Cosmology, Department of Physics, University of Oxford, Denys Wilkinson Building, 1 Keble Road, Oxford, OX1 3RH, United Kingdom George Ellisy Mathematics Department, University of Cape Town, Rondebosch, Cape Town 7701, South Africa Abstract: Current cosmological models require the universe to be in a very smooth initial state before the onset of inflation, a situation to which Penrose ascribes a vanishingly small probability, leading to his proposal of a Conformal Cyclic Cosmology. We present an alter- native paradigm, in which the Higgs plays the role of dilaton and resolves this problem by weakening gravity at very early times, thus providing a form of inflation that is compatible with observations and in which the inflaton is solidly related to tested particle physics. PACS numbers: 04.20.Dw,04.60.Kz,04.60Pp,98.80Qc,04.20Fy I. INTRODUCTION A. The nature of the inflaton The theory of inflation is a huge success, be- This paper proposes a unified solution to two ing able to explain cosmological observations to issues of major import in current cosmology. high precision [1, 2]. However it has a major de- First, what is the nature of the inflaton? A ma- fect: we have no solid basis for stating what the jor problem at present is a lack of a proposal that inflaton is. A great many models have been put is bound in to established particle physics in a forward [33], with some fitting the observations solid way. Second, why is the early universe in a better than others; but most have no sound basis very special state that allows inflation to start, in established physics. The inflaton fields may when that is a highly improbable situation when have their roots in some underlying theory such one takes the entropy associated with possible as string theory or supergravity, be an effective primordial black holes into account [35]? description of a higher order theory of gravity We propose solving both issues simultane- (e.g. Starobinsky inflation [38]), or simply be ously by using a Higgs dilaton as the inflaton, designed for purpose. but coupled in such a way that gravity is essen- `Designer' inflaton potentials are particularly tially turned off in the very early universe. Our problematic as they have been shown to be an purpose in this paper is to show that through implementation of Synge's g-method [17]: one suitable couplings the Higgs field can be used to can choose from a very broad range of geometric answer both issues. This unifies proposals made features of a model as desired, and then run the by others into a coherent whole that is well worth Einstein equations Gab = κTab from left to right exploring further. to find the matter content that would give rise to this behaviour, thereby giving an exaat solution ∗Electronic address: [email protected] of the field equations with the desired properties yElectronic address: [email protected] [39]. In cosmology one can pick any behaviour 2 of the scale factor required to produce the ob- B. The fine tuned initial state servations, and then (subject to a few caveats) find a suitable potential V (φ) to give rise to the The second point is that while it is often required evolution [18]. As such, ad hoc inflaton stated that inflation solves the flatness and hori- models are devoid of meaningful physical con- zon problems in the early universe, that is not in tent; had we observed any other behaviour of our fact the case, as Penrose has pointed out in vari- universe, we could derive a model that would be ous writings that are summarised in Chapter 3 of compatible with those observations to `explain' Fashion, Faith, and Fantasy [35]. The essential them in this way. point is that the maximum gravitational entropy The successes of inflationary models are pri- of a given amount of matter is attained by col- marily induced by the geometries of the space- lapsing it into a black hole. By contrast, \the times they bring about. The spectral tilt is Big Bang was an event of extraordinarily low brought about by the slow reduction of the ex- entropy ... the gravitational degrees of freedom trinsic curvature, and the solutions to the hori- were completely suppressed" ([35]:258). Penrose zon and flatness problems have their roots in estimates the extraordinary precision that was this extended, almost-de Sitter phase. There- involved in setting the initial state of the uni- −10123 fore the question of inflation should not be `is verse as 10 ([35]:275). there a scalar field that can bring about this be- Penrose's argument is simple and persuasive. haviour?' to which the answer in any universe is Consider the space of all possible universes that `yes' (as just explained), but rather `does known could have been created by the big bang sub- physics produce inflation with this behaviour?' ject to the condition that they all have the same At present most scalar field models are not based number of baryons as in our observable uni- on tested, observable physics. verse. The entropy per baryon in the cosmic microwave background has been found to be There is however one exception: Higgs infla- around 109 [41]. However, had the same baryons tion [10] is possible [9]25 and fits the observa- been packed into stellar mass black holes one tions very well [33]. For reviews see: [7][37] This would find an entropy per baryon on the order is the one and only case in which the inflaton is of 1020, and for supermassive black holes such related to tried and tested physics [20], because as the one found at the center of our galaxy the Higgs has indeed been observed at the LHC. this becomes around 1026. Thus it is appar- Because its properties underlie key features of ent that in order to increase the entropy per the standard model of particle physics, if the baryon, baryons should be concentrated in enor- Higgs were to be the inflaton, one would have mous black holes, and to maximize it the entire one of the most awesome unifications imaginable mass of the universe should be in a single black in physics: the same particle is responsible both hole, which would have an approximate entropy for mass at the microscale, and for the dynam- of 10123 which completely dwarfs the observed ics of the very early universe, and hence controls 1089. From Boltzmann's law, we know that the the seeds of structure formation at macro scales. entropy of a configuration is the logarithm of This is therefore a proposal that should be seri- the phase-space volume it occupies. Hence if we ously pursued to see if it might work [20]. It is consider the phase-space of all possible configu- the one possibility for an inflaton solidly tied in rations, a universe such as ours occupies a frac- to the standard model of particle physics. tion P of phase space given as the ratio of the In this paper we depart from the aforemen- exponents of the related entropies; tioned models in which new scalar fields not S 1089 e 1 e 123 known previously are introduced, by using a con- P = ≈ ≈ e−10 (1.1) S 10123 struction that is in a sense minimal: it only re- e 2 e quires known standard model (SM) physics cou- Note that in his argument, Penrose points out pled (in a non-minimal way) to gravity hrough that in such numbers the difference between e a single field that already occurs in the SM. and 10 is insignificant given the enormity of the 3 exponents. This argument invokes a uniform This situation is not made clear by standard measure on phase space; that is assuming the inflationary studies because they consider only big bang had an equal probability of distributing perturbed Robertson-Walker geometries. baryons in all possible configurations. Whilst this may be modified somewhat by various An argument due to Kleban - citing a series probability measures, to overcome the large of studies of inhomogeneous initial conditions for factor obtained would require a very special inflation [31] [16] [15] [14] - states that one should measure. consider the deSitter horizon in calculating en- tropy. Take the space of Schwarzschild-de Sit- An objection to Penrose's reasoning along an- ter space-times and maximize the total horizon thropic lines may be raised: If all baryons in the area (the sum of the black hole horizon and the universe are inside black holes then observers de Sitter horizon) holding fixed the cosmologi- such as ourselves would not exist to see them. cal constant. Then this area is maximized by However, these are easily overcome by a minor empty de Sitter space, and therefore this should perturbation to the reasoning; if we restrict our- be considered the highest entropy state. We ob- selves to conditions under which a galaxy similar ject to this line of reasoning on three grounds: to ours formed we still find that the overwhelm- The first is that a de Sitter horizon is not equiv- ing majority of such systems (as measured frac- alent to a trapping surface. The latter have been tionally on phase space as above) would have shown to obey the laws of thermodynamics with most of the matter in a super-massive black area replacing entropy, whereas the former do hole. We similarly note that the overall num- not interact with one another.