Cosmological constant: relaxation vs multiverse Alessandro Strumiaa, Daniele Teresia;b a Dipartimento di Fisica dell’Universit`adi Pisa b INFN, Sezione di Pisa, Italy We consider a scalar field with a bottom-less potential, such as g3φ, finding that cosmologies unavoidably end up with a crunch, late enough < 2=3 1=3 to be compatible with observations if g ∼ 1:2H0 MPl . If rebounces avoid singularities, the multiverse acquires new features; in particular probabilities avoid some of the usual ambiguities. If rebounces change the vacuum energy by a small enough amount, this dynamics selects a small vacuum energy and becomes the most likely source of universes with anthropically small cosmological constant. Its probability distri- bution could avoid the gap by 2 orders of magnitude that seems left by standard anthropic selection. Contents 1 Introduction2 2 Rolling: a bottom-less scalar in cosmology3 2.1 Cosmology with a falling scalar and thermal friction . .6 arXiv:1904.07876v2 [gr-qc] 29 Aug 2019 3 Rebouncing: a temporal multiverse7 4 Rolling and rebouncing: the hiccupping multiverse8 4.1 Possible hiccuping dynamics . .9 4.2 Probability distribution of the cosmological constant . 10 5 Conclusions 12 1 Introduction The vacuum energy V that controls the cosmological constant receives power-divergent quantum 4 corrections as well as physical corrections of order Mmax, where Mmax is the mass of the heaviest particle. In models with new physics at the Planck scale (e.g. string theory) one thereby expects Planckian vacuum energies, and the observed cosmological constant (corresponding 4 to the vacuum energy V0 ≈ (2:3 meV) ) can be obtained from a cancellation by one part in 4 120 MPl=V0 ∼ 10 . In tentative models of dimensionless gravity the heaviest particle might be the 4 60 top quark (Mmax ∼ Mt, see e.g. [1]), still needing a cancellation by one part in Mmax=V0 ∼ 10 . A plausible interpretation of this huge cancellation is provided by theories with enough vacua such that at least one vacuum accidentally has the small observed cosmological constant. Then, assuming that the vacua get populated e.g. by eternal inflation, observers can only develop < 3 in those vacua with V ∼ 10 V0 [2]. More quantitative attempts of understanding anthropic selection find that the most likely vacuum energy measured by a random observer is about 100 times larger that the vacuum energy V0 we observe [2{5] (unless some special measure is adopted, for instance as in [6{9]). This mild remaining discrepancy might signal some missing piece of the puzzle. Recently [10] (see also [11]) proposed a cosmological model that could make the cosmological constant partially smaller and negative. It needs two main ingredients: a) `Rolling': a scalar field φ with a quasi-flat potential and no bottom (at least in the field 3 < 2=3 1=3 space probed cosmologically), such as Vφ = −g φ with small g ∼ H0 MPl where H0 is the present Hubble constant. Then, a cosmological phase during which the energy density is dominated by Vφ (with a value such that φ classically rolls down its potential) ends up when Vφ crosses zero and becomes slightly negative, starting contraction. During the contraction phase the kinetic energy of φ rapidly blue-shifts and, assuming some interaction with extra states, gets converted into a radiation bath, thus reheating the Universe and maybe triggering the following dynamics. b) `Rebouncing': a mechanism that turns a contracting universe into an expanding universe. Furthermore, to get a small positive (rather than negative) cosmological constant, the authors of [10] assume multiple minima and a `hiccupping' mechanism that populates vacua up to some energy density Vrebounce. Hence, at this stage the Universe appears as hot, expanding and with a small positive cosmolog- ical constant, i.e. with standard hot Big-Bang cosmology. In this way, the cancellation needed to get the observed cosmological constant gets partially reduced by some tens of orders of mag- 1 nitude, such that theories with Mmax ∼ MeV no longer need accidental cancellations [11,10]. 1Notice that the mechanism of [11] can relax vacuum energies up to ∼ TeV4, while having a cutoff in the MeV range or lower. 2 However particles almost 106 heavier than the electron exist in nature. The authors of [10] restricted the parameter space of their model in order to avoid eternal inflation. However other features of the Standard Model, in particular light fermion masses, suggest that anthropic selection is playing a role [12]. The weak scale too might be anthropically constrained [13]. Taking the point of view that a multiverse remains needed, we explore the role that the above ingredients a) and b), assumed to be generic enough, might play in a multiverse context. Is an anthropically acceptable vacuum more easily found by random chance or through the mechanism of [10]? In section2 we consider in isolation the ingredient a), finding that all observers eventually end up in an anti-de-Sitter crunch, that can be late enough to be compatible with cosmological data. In section3 we consider in isolation the ingredient b), finding that it modifies the multiverse structure, in particular leading to multiple cycles of a \temporal multiverse". Adding both ingredients a) and b), in section4 we show that the mechanism of [10] can have a dominant multiverse probability of forming universes with an anthropically acceptable vacuum energy. In such a case, the small discrepancy left by usual anthropic selection (the measured vacuum energy V0 is 100 times below its most likely value) can be alleviated or avoided. Conclusions are given in section5. 2 Rolling: a bottom-less scalar in cosmology A scalar potential with a small slope but no bottom is one of the ingredients of [10]. We here study its cosmology irrespectively of the other ingredients. We consider a scalar field φ with Lagrangian 2 (@µφ) L = − V (φ); (1) φ 2 φ 3 where the quasi-flat potential can be approximated as Vφ(φ) ' −g φ with small g. We consider a flat homogeneous universe with scale-factor a(t) (with present value a0) in the presence of 3 3 φ and of non-relativistic matter with density ρm(a) = ρm(a0)a0=a , as in our universe at late times. Its cosmological evolution is described by the following equations a¨ 4πG = − (ρ + 3p) (2a) a 3 a_ φ¨ = −3 φ_ − V 0 (2b) a φ 2 where G = 1=MPl is the Newton constant; ρ = ρφ + ρm and p = pφ are the total energy density and pressure with φ_2 φ_2 ρ = + V ; p = − V : (3) φ 2 φ φ 2 φ In an inflationary phase with negligible radiation and matter density ρm the scale factor grows N _ 0 0 2 as a / e and φ undergoes classical slow-roll φ ' −Vφ=3H i.e. dφ/dN ' −Vφ=3H as well as 3 ρ in meV4 2/3 ΛCDM Λ g/MPl = (H0/MPl) 100 -54 0 100 0 0 now now t -44 14 t a a = = t t / 10 / 10 a -34 2.24 a 1 a=a0 1 a=a0 -24 34 0.10 -14 0.10 Scale factor Scale factor 0.01 0.01 0.1 0.5 1 5 10 50100 0.1 0.5 1 5 10 50100 Timet in Gyr Timet in Gyr Figure 1: We consider a flat universe with matter fixed to its observed density. Left: evolution of the scale factor (inverse of the temperature) for different cosmological constants. Right: evolution of the scale factor in the presence of a scalar φ with bottom-less potential gφ3, initially fixed at different cosmological constants. 2 2 quantum fluctuations δφ ∼ H=2π per e-fold, where H = 8πV=3MPl. We assume that all other scalars eventually settle to their minimum, such that we can assume V = Vφ, up to a constant that can be reabsorbed in a shift in φ. Classical motion of φ dominates over its quantum fluctuations for field values such that 0 3 2 jVφj H . The critical point is φclass ∼ −MPl=g which corresponds to vacuum energy 2 2 _2 Vclass ∼ g MPl. Classical slow-roll ends when Vφ ∼ φ : this happens at φ ∼ φend ∼ MPl 3 which corresponds to Vφ ∼ Vend ∼ −g MPl. Such a small Vφ ≈ 0 is a special point of the cosmological evolution when Vφ dominates the energy density [11, 10]. The scale factor of an 2 2 universe dominated by Vφ expands by N ∼ MPl=g e-folds while transiting the classical slow-roll region. > Eternal inflation occurs for field values such that Vφ ∼ Vclass: starting from any given point 2 3 φ < φclass the field eventually fluctuates down to φclass after N ∼ jφjMPl=g e-folds. The Fokker-Planck equation for the probability density P (φ, N) in comoving coordinates of finding the scalar field at the value φ has the form of a leaky box [14] @P @ M 2 @H H3=2 @ = Pl P + (H3=2P ) : (4) @t @φ 4π @φ 8π2 @φ This equation admits stationary solutions where P decreases going deeper into the quantum region (while being non-normalizable), and leaks into the classical region. A large density ρ of radiation and/or matter is present during the early big-bang phase. The scalar φ, similarly to a cosmological constant, is irrelevant during this phase. The variation 4 Vacuum energy densityV=V 0 Dark energy parameterw 2/3 0 (g/MPl)/(H0/MPl) 3 -0.6 3 10 Γ=0f ϕ/gϕ=300 GeV -0.7 0.6 0.6 500 -0.95 0.8 0.8 300 now -0.8 t -0.97 = t w 1 1 in GeV -0.99 3 ϕ -0.93 g -0.9 / ϕ 1.2 1.2 f 100 -1.0 1.4 1.4 50 -1.1 1.6 1.6 0.5 1 5 10 50 100 30 0.6 0.8 1 2 3 4 2/3 Timet in Gyr Couplingg in units ofM Pl(H0/MPl) Figure 2: Left: We consider cosmologies that reproduce, at early times, the measured vacuum energy density V0, for different values of the slope parameter g.