Bootstrapped Newtonian cosmology and the cosmological constant problem

Bootstrapped Newtonian gravity was developed with the purpose of estimating the impact of quantum physics in the non-linear regime of the gravitational interaction, akin to corpuscular models of black holes and inflation. In this work, we set the ground for extending the bootstrapped Newtonian picture to cosmological spaces. We further discuss how such models of quantum cosmology can lead to a natural solution of the cosmological constant problem.


Introduction
reproducing Newton's gravity, one should be able to recover General Relativity by reconstructing its (leading-order) non-linearities out of the coupling of each constituent graviton with the collective state (see, e.g. Refs. [4,13,14] for details). This procedure represents, to some extent, a quantum analogue of Deser's works [18,19], although it mainly focuses on the interplay between non-linearities and quantum physics in order to determine possible quantum corrections to the predictions of General Relativity for compact self-gravitating systems in the strong field regime.
In this work, we lay the basis for an extension of the bootstrapped Newtonian picture to cosmology, where the much simpler homogenous and isotropic spaces allow us to analyse also higher order terms. Furthermore, we show how this novel paradigm provides a natural solution to the cosmological constant problem.

Corpuscular models
Let us start from the assumption that matter and the corpuscular state of gravitons together must reproduce the Einstein equations. In particular, we will focus on the Hamiltonian constraint where H M is the matter energy and H G the analogue quantity for the graviton state. We recall that local gravity being attractive in general implies that H G ≤ 0, although this is not true for the graviton self-interaction [6,13].

Corpuscular De Sitter and black holes
The corpuscular models of gravity of Refs. [2][3][4] were first introduced for describing black holes but are perhaps easier to obtain in cosmology, as the two systems share a fundamental "critical" condition. In order to have the de Sitter space-time in General Relativity, one must assume the existence of a cosmological constant term, or vacuum energy density ρ, so that the Friedman equation (for spatially flat Universe) reads Upon integrating on the volume inside the Hubble radius we obtain This relation looks exactly like the expression of the horizon radius for a black hole of ADM mass M and is the reason it was conjectured that the de Sitter space-time could likewise be described as a condensate of gravitons [3].
One can roughly describe the corpuscular model on assuming that the graviton selfinteraction gives rise to a condensate of N (soft off-shell) gravitons of typical Compton length equal to L, so that 5) and the usual consistency conditions N ≃ M 2 /m 2 p for the graviton condensate immediately follows from Eq. (2.4). Equivalently, one finds where ξ is a constant of order one. This shows that for a macroscopic black hole or universe one needs a huge number N ≫ 1.

Post-Newtonian corpuscular models
Let us refine the above corpuscular description. In Refs. [6,13], it was shown that the maximal packing condition which yields the scaling relations (2.6) for a black hole actually follow from the energy balance (2.1) when matter becomes totally negligible. If H M = 0, one is left with where H G is given by the sum of the negative Newtonian energy [6] One therefore immediately recovers the scaling relation (2.6) from Eq. (2.7), regardless of whether L is the horizon radius of the de Sitter universe or of a black hole.

Bootstrapped Newtonian series
Let us now note that where ξ was introduced in Eq. (2.6), and which leads us to conjecture that the complete form of the gravitational Hamiltonian to all orders in the ξ expansion is given by where we remark that L can be the horizon radius of the de Sitter universe or of a black hole. This sort of expansions are strictly related to the argument of resurgence in QFT. In particular, the values of g n should arise directly from this argument (see, e.g. Ref. [20]). For a black hole, a matter contribution is needed and we cannot have H G = 0. In particular, assuming the scaling relation (2.6), from Eq. (3.3) withg n ≃ e ξ −n /n!, we obtain and, from Eq. (2.1), one finds which remarkably reproduces the Newtonian expectation. 2 One more naive observation is that and one could therefore view the exponential formula (3.3) as the realisation of the black hole quantum state in infinite derivative gravity [21]. More precisely, one can see that the Schwarzschild metric, and perhaps General Relativity as a whole, should be recovered as an infinite derivative theory in flat space-time (where the corpuscular picture is naturally formulated). Again, this is in line with bootstrapped Newtonian gravity [13,14] and Deser's works [18,19]. From now on we shall focus on pure gravity and cosmology, for which Eq. (3.3) is more suitable. Let us first set g n = 1, assuming the absolute convergence of the series in Eq. (3.3). By truncating the series at the k th order in ξ, we obtain If we set H G ≃ 0, we obtain from which we can infer that, if k = 2 m + 1, with m ∈ N, then H G ≃ 0 has no real solutions.
On the other hand, if k = 2 m, with m ∈ N, then (3.8) implies that the only acceptable real solution is the corpuscular relation in Eq. (2.6), and this is recovered only by considering an even number of terms in the expansion. However, convergence of Eq. (3.3) for g n = 1 requires ξ < 1, or and the actual size of the cosmological horizon should be (at least slightly) bigger than the corpuscular scaling L N = √ N ℓ p . Let us then try to keep the analysis of Eq. (3.3) as general as possible. We only require g n > 0 in the series (3.3) and the "on-shell" condition with ξ > 0 (and of order one) in the scaling relation (2.6). First of all, the series (3.3) converges provided the coefficients g n satisfy the inequality Note that, for our purpose, convergence is strictly required only for L =L. Assuming however the series converges for L at least in a neighbourhood ofL, the Hamiltonian H G must be a smooth function of L aroundL, that is for some integers 1 ≤ k ≤ K. We then expect to obtain (slightly) more restrictive conditions for the coefficients g n . At this point, it is more interesting to consider other quantities of physical interest, and employ their relations with H G , in order to obtain further conditions on the coefficients g n . For instance, we know that the pressure p = −ρ in pure de Sitter space-time. From where we used M = √ N m p . On considering the first two terms only (n = 1, 2) and H G = 0 up to the same order, that is g 1 = ξ g 2 , one finds (3.14) From Eq. (2.4), we find so that p = −ρ implies (3. 16) Upon replacing this solution into Eq. (3.3), we obtain the off-shell Hamiltonian and we can conjecture that g n ≃ ξ −n . We can easily generalise the above picture for any equation of state of the form where 0 < ω ≤ 1 is a constant. This yields We can then conjecture that g n ≃ ω ξ −n . Note that the pure dust case p = 0 turns out to be rather peculiar in this scenario. Indeed, there are two possible configurations that allow Eq. (3.13) to reduce to p = 0: i) g n = 0, ∀n ≥ 2, i.e. the purely Newtonian case, in agreement with the Newtonian derivation of the Friedmann equation (2.2); ii) ξ = 0, which implies N → 0, thus reducing to Minkowski space. We can finally notice that, upon employing the present value for the size of the Universe, that isL ≃ 10 27 m, Eq. (3.15) yields the correct order of magnitude for the present total energy density without any further assumptions.

Concluding remarks
Inspired by the corpuscular theory of gravity, we have shown that one can naively reconstruct the gravitational Hamiltonian H G for cosmological spaces to all post-Newtonian orders by multiplying the Newtonian energy of the leading order U N for a power series of the quantity ξ = √ Nℓ p /L. In other words, similarly to Deser's classical argument [18,19], one can reconstruct the full quantum Hamiltonian of cosmology by supplying ("bootstrapping") the Newtonian potential energy with the energies due to particle self-interactions within the marginally bound state. At the fundamental level, this further suggests that General Relativity might be recovered as an infinite derivative field theory on a flat space-time.
Moreover, the Hamiltonian constraint of cosmology forces the size of the horizon to satisfy the condition ξ L = √ N ℓ p , with ξ 1, in agreement with the corpuscular predictions, and suggests a functional behaviour for the coefficients of the bootstrapped Newtonian expansion of H G . If one then sets the radius L of the cosmological horizon to the present size of the Universe, the corpuscular Hamiltonian constraint of cosmology yields an energy density that matches the observed one associated to the cosmological constant. Hence, this prediction for the vacuum energy density coming solely from the observed size of the Universe and the corpuscular quantum description of cosmological spaces can arguably be regarded as a natural solution of the cosmological constant problem.