Cosmological Constant: Relaxation Vs Multiverse ∗ Alessandro Strumia A, Daniele Teresi A,B, a Dipartimento Di Fisica “E
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Physics Letters B 797 (2019) 134901 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Cosmological constant: Relaxation vs multiverse ∗ Alessandro Strumia a, Daniele Teresi a,b, a Dipartimento di Fisica “E. Fermi”, Università di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy b INFN, Sezione di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy a r t i c l e i n f o a b s t r a c t 3 Article history: We consider a scalar ﬁeld with a bottom-less potential, such as g φ, ﬁnding that cosmologies unavoidably Received 18 June 2019 2/3 1/3 end up with a crunch, late enough to be compatible with observations if g 1.2H M . If rebounces Received in revised form 21 August 2019 0 Pl avoid singularities, the multiverse acquires new features; in particular probabilities avoid some of the Accepted 27 August 2019 usual ambiguities. If rebounces change the vacuum energy by a small enough amount, this dynamics Available online 30 August 2019 selects a small vacuum energy and becomes the most likely source of universes with anthropically small Editor: G.F. Giudice cosmological constant. Its probability distribution could avoid the gap by 2 orders of magnitude that Keywords: seems left by standard anthropic selection. Cosmological constant © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 3 Relaxation (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . Multiverse 1. Introduction Recently [11](see also [12]) proposed a cosmological model that could make the cosmological constant partially smaller and The vacuum energy V that controls the cosmological constant negative. It needs two main ingredients: receives power-divergent quantum corrections as well as physi- 4 cal corrections of order Mmax, where Mmax is the mass of the a) ‘Rolling’: a scalar ﬁeld φ with a quasi-ﬂat potential and no bot- heaviest particle. In models with new physics at the Planck scale tom (at least in the ﬁeld space probed cosmologically), such as (e.g. string theory) one thereby expects Planckian vacuum ener- =− 3 2/3 1/3 V φ g φ with small g H0 MPl where H0 is the present gies, and the observed cosmological constant (corresponding to Hubble constant. 4 the vacuum energy V 0 ≈ (2.3 meV) ) can be obtained from a can- 4 ∼ 120 cellation by one part in MPl/V 0 10 . In tentative models of Then, a cosmological phase during which the energy density is dimensionless gravity the heaviest particle might be the top quark dominated by V φ (with a value such that φ classically rolls down (M ∼ M , see e.g. [1]), still needing a cancellation by one part max t its potential) ends up when V φ crosses zero and becomes slightly 4 ∼ 60 in Mmax/V 0 10 . negative, starting contraction. During the contraction phase the A plausible interpretation of this huge cancellation is provided kinetic energy of φ rapidly blue-shifts and, assuming some inter- by theories with enough vacua such that at least one vacuum action with extra states, gets converted into a radiation bath, thus accidentally has the small observed cosmological constant. Then, reheating the Universe and maybe triggering the following dynam- assuming that the vacua get populated e.g. by eternal inﬂation, ob- ics. 3 servers can only develop in those vacua with V 10 V 0 [2](see also [3]). More quantitative attempts of understanding anthropic b) ‘Rebouncing’: a mechanism that turns a contracting universe selection ﬁnd that the most likely vacuum energy measured by a into an expanding universe. Furthermore, to get a small posi- random observer is about 100 times larger that the vacuum energy tive (rather than negative) cosmological constant, the authors V we observe [2,4–6](unless some special measure is adopted, 0 of [11] assume multiple minima and a ‘hiccupping’ mechanism for instance as in [7–10]). This mild remaining discrepancy might that populates vacua up to some energy density V . signal some missing piece of the puzzle. rebounce Hence, at this stage the Universe appears as hot, expanding and with a small positive cosmological constant, i.e. with standard hot * Corresponding author. E-mail addresses: [email protected] (A. Strumia), Big-Bang cosmology. In this way, the cancellation needed to get [email protected] (D. Teresi). the observed cosmological constant gets partially reduced by some https://doi.org/10.1016/j.physletb.2019.134901 0370-2693/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 A. Strumia, D. Teresi / Physics Letters B 797 (2019) 134901 tens of orders of magnitude, such that theories with Mmax ∼ MeV Classical motion of φ dominates over its quantum ﬂuctuations 1 | | 3 ∼ no longer need accidental cancellations [11,12]. However particles for ﬁeld values such that V φ H . The critical point is φclass almost 106 heavier than the electron exist in nature. − 2 ∼ 2 2 MPl/g which corresponds to vacuum energy V class g MPl. Clas- The authors of [11]restricted the parameter space of their ˙ 2 sical slow-roll ends when V φ ∼ φ : this happens at φ ∼ φend ∼ MPl model in order to avoid eternal inﬂation. However other features 3 which corresponds to V φ ∼ V end ∼−g MPl. Such a small V φ ≈ 0is of the Standard Model, in particular light fermion masses, suggest a special point of the cosmological evolution when V φ dominates that anthropic selection is playing a role [13–16]. The weak scale the energy density [11,12]. The scale factor of an universe domi- too might be anthropically constrained [17]. Taking the point of ∼ 2 2 nated by V φ expands by N MPl/g e-folds while transiting the view that a multiverse remains needed, we explore the role that classical slow-roll region. the above ingredients a) and b), assumed to be generic enough, Eternal inﬂation occurs for ﬁeld values such that V φ V class: might play in a multiverse context. Is an anthropically accept- starting from any given point φ<φ the ﬁeld eventually ﬂuctu- able vacuum more easily found by random chance or through the class ates down to φ after N ∼|φ|M2 /g3 e-folds. The Fokker-Planck mechanism of [11]? class Pl equation for the probability density P(φ, N) in comoving coordi- In section 2 we consider in isolation the ingredient a), ﬁnding that all observers eventually end up in an anti-de-Sitter crunch, nates of ﬁnding the scalar ﬁeld at the value φ has the form of a that can be late enough to be compatible with cosmological data. leaky box [18] In section 3 we consider in isolation the ingredient b), ﬁnding that 2 ∂ P ∂ M ∂ H H3/2 ∂ it modiﬁes the multiverse structure, in particular leading to multi- = Pl P + (H3/2 P ) . (4) t 4 8 2 ple cycles of a “temporal multiverse”. ∂ ∂φ π ∂φ π ∂φ Adding both ingredients a) and b), in section 4 we show that This equation admits stationary solutions where P decreases going the mechanism of [11]can have a dominant multiverse probability deeper into the quantum region (while being non-normalizable), of forming universes with an anthropically acceptable vacuum en- and leaks into the classical region. ergy. In such a case, the small discrepancy left by usual anthropic A large density ρ of radiation and/or matter is present during selection (the measured vacuum energy V 0 is 100 times below its the early big-bang phase. The scalar φ, similarly to a cosmological most likely value) can be alleviated or avoided. Conclusions are constant, is irrelevant during this phase. The variation in the scalar given in section 5. potential energy due to its slow-roll is negligible as long as 2. Rolling: a bottom-less scalar in cosmology | | 2 V φ H MPl. (5) Indeed A scalar potential with a small slope but no bottom is one of the ingredients of [11]. We here study its cosmology irrespectively V 2 dVφ dφ φ 2 2 of the other ingredients. We consider a scalar ﬁeld φ with La- = V = ρ ∼ H M . (6) dN φ dN 3H2 Pl grangian 3 Thereby the evolution of a scalar ﬁeld with a very small slope g 2 (∂μφ) becomes relevant only at late times when the energy density ρ Lφ = − V φ(φ), (1) 2 becomes small enough, ρ V φ . Fig. 1 shows the cosmological evolution of our universe, assum- where the quasi-ﬂat potential can be approximated as V φ(φ) 3 ing different initial values of the vacuum energy density V φ (φin). −g φ with small g. We consider a ﬂat homogeneous universe with If such vacuum energy is negative, a crunch happens roughly as in scale-factor a(t) (with present value a0) in the presence of φ and 3 3 standard cosmology, after a time of non-relativistic matter with density ρm(a) = ρm(a0)a /a , as in 0 our universe at late times. Its cosmological evolution is described amax da π MPl by the following equations tcrunch = 2 = aH 6 −V φ(φin) a¨ 4π G 0 =− (ρ + 3p) (2a) a 3 V 0 ˙ ≈ 3.6 × 1010 yr. (7) a − φ¨ =−3 φ˙ − V (2b) V φ(φin) a φ Unlike in standard cosmology the Universe ﬁnally undergoes a 2 where G = 1/M is the Newton constant; ρ = ρφ +ρm and p = pφ Pl crunch even if V φ(φin) ≥ 0, because φ starts dominating the en- are the total energy density and pressure with ergy density (like a cosmological constant) and rolls down (unlike a cosmological constant).