Letters B 797 (2019) 134901

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Physics Letters B


Cosmological constant: Relaxation vs ∗ 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

Article history: We consider a scalar field with a bottom-less potential, such as g3φ, finding that unavoidably 2/3 1/3 Received 18 June 2019 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 energy by a small enough amount, this dynamics Available online 30 August 2019 Editor: G.F. Giudice selects a small and becomes the most likely source of with anthropically small 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 Relaxation (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 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 scale tom (at least in the field space probed cosmologically), such as (e.g. ) 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 is dimensionless 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 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 . 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 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 , in particular light 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 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) 2 ple cycles of a “temporal multiverse”. ∂t ∂φ 4π ∂φ 8π ∂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 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)

A scalar potential with a small slope but no bottom is one of Indeed  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 Thereby the evolution of a scalar field with a very small slope g3 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)  ing different initial values of the vacuum energy density V (φ ). −g3φ with small g. We consider a flat homogeneous universe with φ in 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)a0/a , as in 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 with ergy density (like a cosmological constant) and rolls down (unlike a cosmological constant). The crunch happens in the future for the φ˙2 φ˙2 ρφ = + V φ, pφ = − V φ. (3) observed value of the cosmological constant and for the value of 2 2 3 = 2 g H0 MPl assumed in Fig. 1. Here H0 is the present Hubble con- In an inflationary phase with negligible radiation and matter den- stant. For larger g the crunch happens earlier. We do not need N sity ρm the scale factor grows as a ∝ e and φ undergoes classical to show plots with different values of g because, up to a rescal- ˙ −  −  2 ing of the time-scale, the cosmological evolution only depends on slow-roll φ V φ/3H i.e. dφ/dN V φ/3H as well as quan- 3 2 ∼ 2 = 2 g /H∗ MPl where H∗ is the Hubble constant when matter stops tum fluctuations δφ H/2π per e-fold, where H 8π V /3MPl. We assume that all other scalars eventually settle to their mini- dominating (in our universe, H∗ is the present Hubble constant H up to order one factors). This means that large enough val- mum, such that we can assume V = V φ , up to a constant that can 0 be reabsorbed in a shift in φ. ues of the vacuum energy behave as a cosmological constant for a while. Fig. 2 shows the time evolution of the dark-energy parameter = 1 Notice that the mechanism of [12]can relax vacuum energies up to ∼ TeV4, w pφ/ρφ for cosmologies that reproduce the present value of while having a cutoff in the MeV range or lower. the matter and densities and for different values of A. Strumia, D. Teresi / Physics Letters B 797 (2019) 134901 3

Fig. 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.

Fig. 2. Left: We consider cosmologies that reproduce, at early times, the measured vacuum energy density V 0, for different values of the slope parameter g. We plot the time evolution of the dark-energy parameter w = pφ /ρφ . We consider both models with thermal friction (dashed curves) and without thermal friction (continuous curves,  = 0). Right: iso-contours of w today. The shaded regions are disfavoured at 1 and 2 standard deviations by current data. the slope parameter g. The observed present value w0 =−1.01 ± where ρ and p are the total energy density and pressure  2/3 1/3 0.04 [19,20]implies the experimental bound g 1.2H0 MPl . This means that the anthropic restriction on the vacuum energy ρ = ρφ + ρm + ρbath, p = pφ + pbath. (10) remains essentially the same as in standard cosmology (where vacuum energy is a cosmological constant), despite that all cos- Equation (9c), dictated by energy conservation, tells the evolution mologies (even for large positive cosmological constant) eventually of the energy density of the bath ρbath in view of the expansion end with a crunch. of the universe and of the energy injection from φ. The pressure pbath equals ρbath/3(0) for a relativistic (non-relativistic) bath. 2.1. Cosmology with a falling scalar and thermal friction The presence of a bath can modify the expansion phase, even adding a qualitatively new intermediate period during which φ The mechanism of [11]employs some interaction that, during ˙ ∼ ˙2 rolls down the potential acquiring an asymptotic velocity φ the crunch, converts the kinetic energy φ /2into a thermal bath  V /(T ) while the bath, populated by the φ kinetic energy, by particle production due to the evolution of φ, thus reheating φ bath acquires a corresponding quasi-stationary temperature T ∼ the Universe up to potentially high temperatures. The scalar φ can bath (g6/H)1/4. The final crunch gets delayed but it eventually hap- have interactions compatible with its lightness. Indeed, φ might be pens as illustrated by the dashed curves in Fig. 2, and as we now a Goldstone boson with derivative interactions e.g. to extra vectors show analytically. Conservation of ‘energy’ gives a first integral of Fμν or eqs. (9)well known as Friedmann’s equation, H2 = 8π Gρ/3. By φ ˜ ¯ differentiating it and using (9b) and (9c)one obtains Lbath = Fμν Fμν + yφ γ5 . (8) fφ ˙ In the presence of a cosmological thermal bath of these parti- H =−4π G (ρ + p) ≤ 0 (11) cles φ acquires a thermal friction  without acquiring a thermal ˙ mass [11]. For example  ∼ 4g6 T 3 /π f 2 in the bath of vectors in which the contribution of  cancels. In general, H is non- φ bath φ + ≥ with gauge coupling g . We study its effects during the expansion positive because the null- ρ p 0is satisfied. φ = = phase. The cosmological equations of eq. (9) generalise to At the turning point H 0one has ρ 0 thanks to a cancella- tion between a positive ρbath and negative ρφ : for our system this a¨ 4π G implies p > 0such that H˙ is strictly negative and the Universe =− (ρ + 3p) (9a) a 3 starts collapsing. This avoids the Boltzmann-brain paradox that af- ˙ fects cosmologies with positive cosmological constant [21–23]. The ¨ =− a + ˙ −  φ (3 )φ V φ (9b) right panel of Fig. 2 shows that interactions relax the observa- a −1/3 a˙ tional bound on g by an amount proportional to fφ , for g large ˙ ˙2 ρbath =−3 (ρbath + pbath) + φ (9c) enough. a 4 A. Strumia, D. Teresi / Physics Letters B 797 (2019) 134901

3. Rebouncing: a temporal multiverse Due to b), contracting regions with small negative vacuum en- ergy density ∼ V end eventually rebounce, becoming expanding uni- It is usually assumed that anti-de Sitter regions with negative verses with vacuum energy varied by ∼ V rebounce. As in the previ- vacuum density collapse to a big-crunch singularity. The resolution ous section, a temporal multiverse is created: different values of of this singularity is not known (for example in perturbative string the vacuum energy are sampled in different cycles. theory [24]), so it makes sense to consider the opposite possibility A new feature arises due to the presence of both ingredients: b): that collapsing anti-de-Sitter regions rebounce into an expand- all cycles now last a finite time. A single patch samples different ing space. The mechanism of [11] assumes that the vacuum energy values of physical parameters (for example vacuum energies) with density changes by a small amount V rebounce in the process. a given probability distribution. We refer to this as temporal mul- Following the usual assumptions that anti-de Sitter vacua are tiverse. ‘terminal’, various authors tried to compute the statistical distri- A disordered temporal multiverse arises if the extra vacuum en- bution of vacua in a multiverse populated by eternal inflation, in ergy generated during the rebounce, V rebounce, is typically larger terms of vacuum decay rates κIJ from vacuum I to vacuum J [25]. than V class, such that the small vacuum energy selected by the These rates are defined up to unknown multiverse factors, because rolling mechanism at the previous cycle is lost. Small vacuum en- eternal inflation gives an infinite multiverse, so probabilities are ergy is not a special point of this dynamics, and the usual an- affected by divergences. Some measures lead to paradoxes (see for thropic selection argument discussed by Weinberg [2] applies: the instance [21–23,26]). Furthermore, even if the multiverse statistics most likely value of the cosmological constant is 2 orders of mag- were known, its use would be limited by our ability of observ- nitude above its observed value (for the observed value of the − ing only one event (our universe). Despite these drawbacks and amount of primordial inhomogeneities, δρ/ρ ∼ 10 5). This dis- difficulties many authors tried addressing the issue (see e.g. [25, crepancy by 2 orders of magnitude (or worse if δρ/ρ can vary) 27–31]). possibly signals that anthropic selection is not enough to fully ex- If vacua with negative cosmological constant are not terminal, plain the observed small value of the cosmological constant. multiverse dynamics would change as follows (see also [32,33]). An ordered temporal multiverse arises if, instead, the con- For simplicity we consider a toy multiverse with 3 vacua: S (de traction/bounce phase changes the vacuum energy density by an Sitter), M (Minkowski) and A (anti de Sitter). The evolution of the amount V rebounce smaller enough than V class such that, when one fraction of ‘time’ spent by an ‘observer’ in the vacua is described cycle starts, it proceeds forever giving rise at some point to an an- thropically acceptable vacuum energy. We refer to this possibility by an equation of the form (see e.g. [25]) as ‘hiccupping’. A small vacuum energy ∼ V is a special point ⎛ ⎞ ⎛ ⎞ end f S −κSM − κSA 00 of this dynamics, such that, depending on details of hiccupping, d ⎝ ⎠ ⎝ ⎠ f M = κSM −κMA 0 the probability distribution of the vacuum energy can peak below dt the maximal value allowed by anthropic selection. In the absence f A κSA κMA 0 ⎛ ⎞ ⎛ ⎞ of knowledge of a clear rebouncing mechanism (while some recent 00 κAS f S attempts have been done, see e.g. [34]), having one rather than the + ⎝ ⎠ · ⎝ ⎠ 00 κAM f M (12) other possibility shoud be considered as an assumption. − − 00 κAS κAM f A Going beyond the two limiting cases discussed above, an inter- If anti-de-Sitter vacua are terminal only the first term is present: mediate situation can be broadly characterised by different scales: V rebounce (the amount of randomness in vacuum energy at each then, in a generic context, the frequencies f I are dominated by rebounce); V ∼ g2 M2 (the critical value of V above which φ decays from the most long-lived de Sitter vacuum [25]. If anti-de- class Pl φ 4 starts fluctuating); V max ∼ M (the maximal energy scale in the Sitter vacua are not terminal, the second term containing the κAJ max recycling coefficients is present, allowing for a steady state solution theory). The observed value V 0 of the vacuum energy can either be ⎛ ⎞ ⎛ ⎞ reached trough the rolling dynamics of [11]or by the usual ran- f S κASκMA dom sampling the multiverse. The relative probability of these two ⎝ ⎠ ⎝ ⎠ f M ∝ κAMκS + κASκSM (13) histories is f κ κ A MA S ℘ [ V rebounce/V 0 ] rolling ∼ min 1,(V class/V rebounce) = + . (14) where κS κSA κSM. ℘random V 0/V max If anti-de-Sitter crunches rebounce due to some generic mecha- We ignored possible fine structures within each scale. If the mech- nism when they reach Planckian-like energies, the κ coefficients AJ anism of [11] provides the dominant source of anthropically- might be universal and populate all lower-energy vacua. The mech- acceptable vacua (those with V  103 V ), the observed value V anism of [11]needs a milder hiccupping mechanism, that only 0 0 of the vacuum energy density can have a probability larger than in populates vacua with vacuum energy slightly higher than the spe- the usual multiverse scenario. cific AdS vacuum that crunched. 4.1. Possible hiccuping dynamics 4. Rolling and rebouncing: the hiccupping multiverse Let us now discuss the value of V from a theoreti- Finally we consider the combined action of the ingredients a) rebounce cal point of view. If the rebounce occurs when the contracting and b) of [11]. region heats up to temperatures T (or, more in general, Due to a), observers in de Sitter regions unavoidably end up rebounce energy density ∼ T 4 ), one can expect that scalars lighter sliding down the V φ potential until the vacuum energy becomes rebounce 3 than T can jump to different minima (assuming that po- small and negative, of order V end ∼−g MPl. This happens even if rebounce the vacuum energy is so big that quantum fluctuations of φ ini- tential barriers are characterised by the mass). If the rebounce tially dominate over its classical slow-roll. This process can end de happens when contraction reaches Planckian densities, one expects ∼ 4 ∼ Sitter faster than quantum tunnelling to vacua with lower energy V rebounce MPl. A very small value of V rebounce V 0 could arise densities, as the vacuum decay rates are exponentially suppressed assuming that the contraction/rebounce/expansion phase triggers  by possibly large factors. movement of some lighter fields φ with potentials such that A. Strumia, D. Teresi / Physics Letters B 797 (2019) 134901 5 vacua close by in field space have similar energies. ‘Ordered’ land- 4.2. Probability distribution of the cosmological constant scapes of this kind have been considered, for instance, in [35–41].2 An example of this hiccupping structure is provided by Abbott’s Finally, we discuss the probability distribution of the cosmolog-  model [35], i.e. a light scalar φ with potential that can be (at least ical constant Pobs(V ) measured by random observers taking their locally) approximated as anthropic selection into account. In our case Pobs(V ) is simply  given by the product of P(V ) times an astrophysical factor Pant(V ) 3  4 φ V φ =−g  φ − cos (15) that estimates how many observers form as function of the vac- φ  fφ uum energy V . As the volume of a flat universe is infinite, some V 3 3  3 regularising volume reg is needed [4,48,49]: with, again, a very small slope gφ , such that gφ fφ V 0. Dur-  ing a given cycle the field φ is trapped in a local minimum  Pobs(V ) ∝ P(V )Pant(V ), (that we may take at φ = 0) provided that is large enough 2 to quench tunnelling. At the end of the cycle, during the contrac- d nobs Pant(V ) ∝ dt Vreg (V ). (18) tion/rebounce/expansion phase, the barriers become irrelevant for dt dV   some time and the field φ is free to diffuse from φ = 0by ther- The temporal integral is over the finite lifetime of a single cycle. mal or de Sitter fluctuations. We focus on de Sitter fluctuations, 2 The quantity d nobs/dt dV is the observer production rate per unit given that a phase of the usual inflation with Hubble constant time and comoving volume.4 The anthropic factor depends on the Hinfl is probably needed to explain the observed primordial in- prescription adopted to regularise the number of observers. Fol- homogeneities. For gφ Hinfl the quantum evolution dominates  lowing Weinberg [5]we consider the number of observers per unit (the classical rolling of φ gives a negligible variation in V , of or- of mass, which corresponds to Vreg = 1in eq. (18). This measure ∼ 6 2  der Ninfl gφ /Hinfl) and the field φ acquires a probability density prefers vacuum-energy densities 2 orders of magnitude larger than P ∼ − 2 2 2 the observed V . This unsatisfactory aspect of the standard spatial 1 exp( 2π φ /Hinfl Ninfl), where Ninfl is the number of e-folds 0 of inflation. Hence, when barriers become relevant again, the vac- multiverse can be limited by choosing appropriate regularisation uum energy has probability density volumes, such as the causal-diamond measure (see, for instance, [7–10]). 1 V 2 Without needing such choices, a temporal multiverse can give P1(V ) ∼ exp − 2 2 a probability distribution of V peaked around its observed value. V rebounce This needs an ordered landscape with small V rebounce . Fig. 3 shows H 3 infl numerical result for Pobs(V ) assuming V end V 0: with V rebounce = g  Ninfl. (16) φ 2π • Quantum fluctuations happen differently in different Hubble The red curve considers a disordered hiccup, or a ordered hic-  3 patches: after inflation regions in different vacua progressively cup with V rebounce 10 V 0: they both give a flat P(V ) around return in causal contact, and the region with lower vacuum en- V 0, reproducing the usual CDM anthropic selection [2,5]: vacuum energy densities 2 orders of magnitude larger than V 0 ergy density expands into the other regions. If fφ Hinfl there is a order unity probability that this is happening now on horizon are preferred. • scales, giving rise to gravitational waves [46] (and to other signals The yellow and blue curves assume an ordered asymmetric  as in [47]if φ couples to photons). The field φ (for = 0) can hiccup, that cuts large positive values of V , but not negative 2 large values. be identified with φ provided that Ninfl  (2π MPl/Hinfl) is large • The green curve assumes an ordered symmetric hiccup, that enough that V rebounce ≥|V end|. The above hiccup mechanism can be part of the scenario of [11], that tries avoiding the multiverse. cuts large (positive and negative) values of V . Assuming a small V ∼ V gives a P (V ) peaked around the ob- This hiccup mechanism preserves, on average, the value of V . rebounce 0 obs served V , while the measure-dependent anthropic factor Since the field φ classically rolls down a bit whenever a cycle starts 0 P (V ) becomes irrelevant, being approximatively constant with V > V , V gradually decreases and after a large number of and end in such a small V interval. cycles the probability distribution of V becomes, for V rebounce  |V | end 5. Conclusions 1 V < −V rebounce P(V ) ∝ − + 2 2 (17) The authors of [11]proposed a dynamical mechanism that e (V V rebounce) /2σ V ≥−V rebounce makes the small vacuum energy density observed in cosmology with σ  1.3V rebounce according to numerical simulations. We refer less fine-tuned from the point of view of . This pos- to this as asymmetric hiccup. sibility was put forward as an alternative to anthropic selection An alternative speculative possibility is that the downward av- in a multiverse. However, given that multiple vacua are anyhow erage drift of V is avoided by some symmetric hiccup mecha- needed by the mechanism of [11], and that a multiverse of many nism that gives a distribution of V peaked around the special point of the dynamics V = V end also for cycles with negative vac- uum energy (e.g. thermalisation might cause loss of memory). In 4 As the literature is not univocal, we adopt the following choice. For positive such a case P(V ) = P (V ) may be peaked around some small cosmological constants we take the observer production rate from the numerical 1 simulations in [50], which qualitatively agree with the semi-analytical approach scale. of [51]. For the observer model, we choose the “stellar-formation-rate plus fixed- delay” model [52], where the rate of formation of observers is taken as proportional to the formation rate of stars, with a 5 Gyr fixed time delay inspired by the for- 2 In our context, this property must only be obeyed by some lighter fields, not by mation of complex-enough life on Earth. For negative cosmological constants, we the full landscape. approximate the star formation rate as the zero cosmological constant rate supple- 3 Such small slopes are typical, for instance, of relaxion models [36], where they mented by a hard cut-off at the crunch time of eq. (7). In doing so, we neglect a can be generated dynamically by the clockwork mechanism, either in its multi- possible new phase of star formation during contraction since we assume a fixed field [42,43]or extra-dimensional [44,45] versions. time delay ≈ 5 Gyr for the formation of observers. 6 A. Strumia, D. Teresi / Physics Letters B 797 (2019) 134901

Fig. 3. Possible probability distribution of eq. (18)of the vacuum energy V in units of the observed value V 0. The red curve shows the case of the usual spatial multiverse [5], that in our context can arise from a disordered hiccupping. The yellow and blue curves assume an ordered landscape with asymmetric hiccupping as in eq. (17)for different values of V rebounce , whereas the green curve assumes a symmetric hiccup. vacua is anyhow suggested by independent considerations, we ex- [2] S. Weinberg, Anthropic bound on the cosmological constant, Phys. Rev. Lett. 59 plored how the ingredients proposed in [11]behave in a multi- (1987) 2607, https://doi .org /10 .1103 /PhysRevLett .59 .2607. verse context. [3] J.D. Barrow, J.F. Tipler, The Anthropic Cosmological Principle, Oxford U. 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