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.
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. ), 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 . 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  (see also ) 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  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  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  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 . The weak scale too might be anthropically constrained . 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 ? 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  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 . 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 -44 14 now 10 t = 10 t = -34 2.24 1 a=a0 1 a=a0 -24 34 0.10 -14 0.10 Scale factor a / Scale factor a / 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 |Vφ| 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 ∼ |φ|MPl/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 
∂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 -0.8 now -0.97 t = w
1 1 in GeV -0.99
3 ϕ -0.93
-0.9 / g 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. 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, Γ = 0). Right: iso-contours of w today. The shaded regions are disfavoured at 1 and 2 standard deviations by current data. in the scalar potential energy due to its slow-roll is negligible as long as
0 2 |Vφ| H MPl. (5)
Indeed dV dφ V 02 φ = V 0 = φ ρ ∼ H2M 2 . (6) dN φ dN 3H2 Pl Thereby the evolution of a scalar ﬁeld with a very small slope g3 becomes relevant only at late < times when the energy density ρ becomes small enough, ρ ∼ Vφ. Fig.1 shows the cosmological evolution of our universe, assuming diﬀerent initial values of the vacuum energy density Vφ(φin). If such vacuum energy is negative, a crunch happens roughly as in standard cosmology, after a time s Z amax r da π MPl V0 10 tcrunch = 2 = p ≈ 3.6 × 10 yr. (7) 0 aH 6 −Vφ(φin) −Vφ(φin)
Unlike in standard cosmology the Universe ﬁnally undergoes a crunch even if Vφ(φin) ≥ 0, because φ starts dominating the energy density (like a cosmological constant) and rolls down (unlike a cosmological constant). The crunch happens in the future for the observed value of 3 2 the cosmological constant and for the value of g = H0 MPl assumed in ﬁg.1. Here H0 is the present Hubble constant. For larger g the crunch happens earlier. We do not need to show plots with diﬀerent values of g because, up to a rescaling of the time-scale, the cosmological evolution 3 2 only depends on g /H∗ MPl where H∗ is the Hubble constant when matter stops dominating
5 (in our universe, H∗ is the present Hubble constant H0 up to order one factors). This means that large enough values of the vacuum energy behave as a cosmological constant for a while.
Fig.2 shows the time evolution of the dark-energy parameter w = pφ/ρφ for cosmologies that reproduce the present value of the matter and dark energy densities and for diﬀerent
values of the slope parameter g. The observed present value w0 = −1.01 ± 0.04 [15,16] implies < 2/3 1/3 the experimental bound g ∼ 1.2H0 MPl . This means that the anthropic restriction on the vacuum energy remains essentially the same as in standard cosmology (where vacuum energy is a cosmological constant), despite that all cosmologies (even for large positive cosmological constant) eventually end with a crunch.
2.1 Cosmology with a falling scalar and thermal friction The mechanism of  employs some interaction that, during the crunch, converts the kinetic energy φ˙2/2 into a thermal bath by particle production due to the evolution of φ, thus reheating the Universe up to potentially high temperatures. The scalar φ can have interactions compatible with its lightness. Indeed, φ might be a Goldstone boson with derivative interactions e.g. to extra vectors Fµν or fermions Ψ
φ ˜ ¯ Lbath = FµνFµν + yφΨγ5Ψ. (8) fφ In the presence of a cosmological thermal bath of these particles φ acquires a thermal friction 6 3 2 Γ without acquiring a thermal mass . For example Γ ∼ 4gφTbath/πfφ in the bath of vectors
with gauge coupling gφ. We study its eﬀects during the expansion phase. The cosmological equations of eq. (9) generalise to a¨ 4πG = − (ρ + 3p) (9a) a 3 a˙ φ¨ = −(3 + Γ)φ˙ − V 0 (9b) a φ a˙ ρ˙ = −3 (ρ + p ) + Γφ˙2 (9c) bath a bath bath where ρ and p are the total energy density and pressure
ρ = ρφ + ρm + ρbath, p = pφ + pbath. (10)
Equation (9c), dictated by energy conservation, tells the evolution of the energy density of the
bath ρbath in view of the expansion of the universe and of the energy injection from φ. The
pressure pbath equals ρbath/3 (0) for a relativistic (non-relativistic) bath. The presence of a bath can modify the expansion phase, even adding a qualitatively new intermediate period during which φ rolls down the potential acquiring an asymptotic velocity ˙ 0 φ ∼ Vφ/Γ(Tbath) while the bath, populated by the φ kinetic energy, acquires a corresponding
6 6 1/4 quasi-stationary temperature Tbath ∼ (g /HΓ) . The ﬁnal crunch gets delayed but it even- tually happens as illustrated by the dashed curves in ﬁg.2, and as we now show analytically. Conservation of ‘energy’ gives a ﬁrst integral of eq.s (9) well known as Friedmann’s equation, H2 = 8πGρ/3. By diﬀerentiating it and using (9b) and (9c) one obtains
H˙ = −4πG (ρ + p) ≤ 0 (11)
in which the contribution of Γ cancels. In general, H˙ is non-positive because the null-energy condition ρ + p ≥ 0 is satisﬁed. At the turning point H = 0 one has ρ = 0 thanks to a
cancellation between a positive ρbath and negative ρφ: for our system this implies p > 0 such that H˙ is strictly negative and the Universe starts collapsing. This avoids the Boltzmann-brain paradox that aﬀects cosmologies with positive cosmological constant [17–19]. The right panel of ﬁg.2 shows that interactions relax the observational bound on g by an amount proportional −1/3 to fφ , for g large enough.
3 Rebouncing: a temporal multiverse
It is usually assumed that anti-de Sitter regions with negative vacuum density collapse to a big- crunch singularity. The resolution of this singularity is not known (for example in perturbative string theory ), so it makes sense to consider the opposite possibility b): that collapsing anti-de-Sitter regions rebounce into an expanding space. The mechanism of  assumes that
the vacuum energy density changes by a small amount Vrebounce in the process. Following the usual assumptions that anti-de Sitter vacua are ‘terminal’, various authors tried to compute the statistical distribution of vacua in a multiverse populated by eternal in-
ﬂation, in terms of vacuum decay rates κIJ from vacuum I to vacuum J . These rates are deﬁned up to unknown multiverse factors, because eternal inﬂation gives an inﬁnite mul- tiverse, so probabilities are aﬀected by divergences. Some measures lead to paradoxes (see for instance [17–19, 22]). Furthermore, even if the multiverse statistics were known, its use would be limited by our ability of observing only one event (our universe). Despite these drawbacks and diﬃculties many authors tried addressing the issue (see e.g. [21,23–27]). If vacua with negative cosmological constant are not terminal, multiverse dynamics would change as follows (see also [28, 29]). For simplicity we consider a toy multiverse with 3 vacua: S (de Sitter), M (Minkowski) and A (anti de Sitter). The evolution of the fraction of ‘time’ spent by an ‘observer’ in the vacua is described by an equation of the form (see e.g. ) f −κ − κ 0 0 0 0 κ f d S SM SA AS S f = κ −κ 0 + 0 0 κ · f (12) dt M SM MA AM M fA κSA κMA 0 0 0 −κAS − κAM fA If anti-de-Sitter vacua are terminal only the ﬁrst term is present: then, in a generic context, the frequencies fI are dominated by decays from the most long-lived de Sitter vacuum . If
7 anti-de-Sitter vacua are not terminal, the second term containing the κAJ recycling coeﬃcients is present, allowing for a steady state solution fS κASκMA fM ∝ κAM κS + κASκSM (13) fA κMAκS where κS = κSA + κSM . If anti-de-Sitter crunches rebounce due to some generic mechanism when they reach Planckian- like energies, the κAJ coeﬃcients might be universal and populate all lower-energy vacua. The mechanism of  needs a milder hiccupping mechanism, that only populates vacua with vac- uum energy slightly higher than the speciﬁc AdS vacuum that crunched.
4 Rolling and rebouncing: the hiccupping multiverse
Finally we consider the combined action of the ingredients a) and b) of .
Due to a), observers in de Sitter regions unavoidably end up sliding down the Vφ potential 3 until the vacuum energy becomes small and negative, of order Vend ∼ −g MPl. This happens even if the vacuum energy is so big that quantum ﬂuctuations of φ initially dominate over its classical slow-roll. This process can end de Sitter faster than quantum tunnelling to vacua with lower energy densities, as the vacuum decay rates are exponentially suppressed by possibly large factors.
Due to b), contracting regions with small negative vacuum energy density ∼ Vend eventually rebounce, becoming expanding universes with vacuum energy varied by ∼ Vrebounce. As in the previous section, a temporal multiverse is created: diﬀerent values of the vacuum energy are sampled in diﬀerent cycles. A new feature arises due to the presence of both ingredients: all cycles now last a ﬁnite time. A single patch samples diﬀerent values of physical parameters (for example vacuum energies) with a given probability distribution. We refer to this as temporal multiverse. A disordered temporal multiverse arises if the extra vacuum energy generated during the rebounce, Vrebounce, is typically larger than Vclass, such that the small vacuum energy selected by the rolling mechanism at the previous cycle is lost. Small vacuum energy is not a special point of this dynamics, and the usual anthropic selection argument discussed by Weinberg  applies: the most likely value of the cosmological constant is 2 orders of magnitude above its observed value (for the observed value of the amount of primordial inhomogeneities, δρ/ρ ∼ 10−5). This discrepancy by 2 orders of magnitude (or worse if δρ/ρ can vary) possibly signals that anthropic selection is not enough to fully explain the observed small value of the cosmological constant. An ordered temporal multiverse arises if, instead, the contraction/bounce phase changes the vacuum energy density by an amount Vrebounce smaller enough than Vclass such that, when
8 one cycle starts, it proceeds forever giving rise at some point to an anthropically acceptable vacuum energy. We refer to this possibility as ‘hiccupping’. A small vacuum energy ∼ Vend is a special point of this dynamics, such that, depending on details of hiccupping, the probability distribution of the vacuum energy can peak below the maximal value allowed by anthropic selection. In the absence of knowledge of a clear rebouncing mechanism (while some recent attempts have been done, see e.g. ), having one rather than the other possibility shoud be considered as an assumption. Going beyond the two limiting cases discussed above, an intermediate situation can be broadly characterised by diﬀerent scales: Vrebounce (the amount of randomness in vacuum energy 2 2 at each rebounce); Vclass ∼ g MPl (the critical value of Vφ above which φ starts ﬂuctuating); 4 Vmax ∼ Mmax (the maximal energy scale in the theory). The observed value V0 of the vacuum energy can either be reached trough the rolling dynamics of  or by the usual random sampling the multiverse. The relative probability of these two histories is
℘ min[1, (V /V )Vrebounce/V0 ] rolling ∼ class rebounce . (14) ℘random V0/Vmax We ignored possible ﬁne structures within each scale. If the mechanism of  provides the < 3 dominant source of anthropically-acceptable vacua (those with V ∼ 10 V0), the observed value V0 of the vacuum energy density can have a probability larger than in the usual multiverse scenario.
4.1 Possible hiccuping dynamics
Let us now discuss the value of Vrebounce from a theoretical point of view. If the rebounce occurs when the contracting region heats up to temperatures Trebounce (or, more in general, energy 4 density ∼ Trebounce), one can expect that scalars lighter than Trebounce can jump to diﬀerent minima (assuming that potential barriers are characterised by the mass). If the rebounce 4 happens when contraction reaches Planckian densities, one expects Vrebounce ∼ MPl. A very small value of Vrebounce ∼ V0 could arise assuming that the contraction/rebounce/expansion phase triggers movement of some lighter ﬁelds φ0 with potentials such that vacua close by in ﬁeld space have similar energies. ‘Ordered’ landscapes of this kind have been considered, for instance, in [31–37].2 An example of this hiccupping structure is provided by Abbott’s model , i.e. a light scalar φ0 with potential that can be (at least locally) approximated as
0 3 0 4 φ 0 Vφ = −gφ0 φ − Λ cos (15) fφ0
2In our context, this property must only be obeyed by some lighter ﬁelds, not by the full landscape.
9 3 3 3 0 0 with, again, a very small slope gφ0 , such that gφ0 fφ V0. During a given cycle the ﬁeld φ is trapped in a local minimum (that we may take at φ0 = 0) provided that Λ is large enough to quench tunnelling. At the end of the cycle, during the contraction/rebounce/expansion phase, the barriers become irrelevant for some time and the ﬁeld φ0 is free to diﬀuse from φ0 = 0 by thermal or de Sitter ﬂuctuations. We focus on de Sitter ﬂuctuations, given that a phase of the
usual inﬂation with Hubble constant Hinﬂ is probably needed to explain the observed primordial 0 inhomogeneities. For gφ0 Hinﬂ the quantum evolution dominates (the classical rolling of φ 6 2 0 gives a negligible variation in V , of order ∼ Ninﬂgφ0 /Hinﬂ) and the ﬁeld φ acquires a probability 2 02 2 density P1 ∼ exp(−2π φ /HinﬂNinﬂ), where Ninﬂ is the number of e-folds of inﬂation. Hence, when barriers become relevant again, the vacuum energy has probability density 2 1 V 3 Hinﬂ p P1(V ) ∼ exp − 2 with Vrebounce = gφ0 Ninﬂ. (16) 2 Vrebounce 2π Quantum ﬂuctuations happen diﬀerently in diﬀerent Hubble patches: after inﬂation regions in diﬀerent vacua progressively return in causal contact, and the region with lower vacuum energy density expands into the other regions. If fφ0 Hinﬂ there is a order unity probability that this is happening now on horizon scales, giving rise to gravitational waves  (and to other signals as in  if φ couples to photons). The ﬁeld φ0 (for Λ = 0) can be identiﬁed with φ > 2 provided that Ninﬂ ∼ (2πMPl/Hinﬂ) is large enough that Vrebounce ≥ |Vend|. The above hiccup mechanism can be part of the scenario of , that tries avoiding the multiverse. This hiccup mechanism preserves, on average, the value of V . Since the ﬁeld φ classically rolls down a bit whenever a cycle starts with V > Vend, V gradually decreases and after a large number of cycles the probability distribution of V becomes, for Vrebounce |Vend| ( 1 V < −Vrebounce P(V ) ∝ 2 2 (17) −(V +Vrebounce) /2σ e V ≥ −Vrebounce with σ ' 1.3Vrebounce according to numerical simulations. We refer to this as asymmetric hiccup. An alternative speculative possibility is that the downward average drift of V is avoided by some symmetric hiccup mechanism that gives a distribution of V peaked around the special point of the dynamics V = Vend also for cycles with negative vacuum energy (e.g. thermalisation might cause loss of memory). In such a case P (V ) = P1(V ) may be peaked around some small scale.
4.2 Probability distribution of the cosmological constant
Finally, we discuss the probability distribution of the cosmological constant Pobs(V ) measured
by random observers taking their anthropic selection into account. In our case Pobs(V ) is simply
3Such small slopes are typical, for instance, of relaxion models , where they can be generated dynamically by the clockwork mechanism, either in its multi-ﬁeld [38, 39] or extra-dimensional [40, 41] versions.
10 1.0 Hiccup mechanism
0.8 disordered / d V
obs 0.6 asymmetric,V rebounce=10V0
0.4 asymmetric,V rebounce=V0 | V d V V 0.2 symmetric, rebounce= 0
0.0 102 101 100 10-1 100 101 102
Figure 3: Possible probability distribution of eq. (18) of the vacuum energy V in units of the
observed value V0. The red curve shows the case of the usual spatial multiverse , 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 Vrebounce, whereas the green curve assumes a symmetric hiccup.
given by the product of P(V ) times an astrophysical factor Pant(V ) that estimates how many observers form as function of the vacuum energy V . As the volume of a ﬂat universe is inﬁnite,
some regularising volume Vreg is needed [44,3,45]:
Z d2n P (V ) ∝ P(V )P (V ), P (V ) ∝ dt V obs (V ). (18) obs ant ant reg dt dV
2 The temporal integral is over the ﬁnite lifetime of a single cycle. The quantity d nobs/dt dV is the observer production rate per unit time and comoving volume.4 The anthropic factor depends on the prescription adopted to regularise the number of observers. Following Weinberg  we consider the number of observers per unit of mass, which corresponds to Vreg = 1 in eq. (18). This measure prefers vacuum-energy densities 2 orders of magnitude larger than the observed
V0. This unsatisfactory aspect of the standard spatial multiverse can be limited by choosing appropriate regularisation volumes, such as the causal-diamond measure (see, for instance, [6–9]).
4As the literature is not univocal, we adopt the following choice. For positive cosmological constants we take the observer production rate from the numerical simulations in , which qualitatively agree with the semi-analytical approach of . For the observer model, we choose the “stellar-formation-rate plus ﬁxed-delay” model , 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 formation of complex-enough life on Earth. For negative cosmological constants, we approximate the star formation rate as the zero cosmological constant rate supplemented by a hard cut-oﬀ at the crunch time of eq. (7). In doing so, we neglect a possible new phase of star formation during contraction since we assume a ﬁxed time delay ≈ 5 Gyr for the formation of observers.
11 Without needing such choices, a temporal multiverse can give a probability distribution of
V peaked around its observed value. This needs an ordered landscape with small Vrebounce.
Figure3 shows numerical result for Pobs(V ) assuming Vend V0:
3 • The red curve considers a disordered hiccup, or a ordered hiccup with Vrebounce 10 V0:
they both give a ﬂat P (V ) around V0, reproducing the usual ΛCDM anthropic selection [2,
49]: vacuum energy densities 2 orders of magnitude larger than V0 are preferred.
• The yellow and blue curves assume an ordered asymmetric hiccup, that cuts large positive values of V , but not negative large values.
• The green curve assumes an ordered symmetric hiccup, that cuts large (positive and
negative) values of V . Assuming a small Vrebounce ∼ V0 gives a Pobs(V ) peaked around the
observed V0, while the measure-dependent anthropic factor Pand(V ) becomes irrelevant, being approximatively constant in such a small V interval.
The authors of  proposed a dynamical mechanism that makes the small vacuum energy density observed in cosmology less ﬁne-tuned from the point of view of particle physics. This possibility was put forward as an alternative to anthropic selection in a multiverse. However, given that multiple vacua are anyhow needed by the mechanism of , and that a multi- verse of many vacua is anyhow suggested by independent considerations, we explored how the ingredients proposed in  behave in a multiverse context. A ﬁrst ingredient of  is a scalar with a bottom-less potential and small slope that re- laxes the cosmological constant down to small negative values. In section2 we computed the resulting cosmology. In particular, we found that any universe eventually undergoes a phase of contraction, leading to a crunch, even starting from a positive cosmological constant. This avoids the possible Boltzmann-brain paradox generated e.g. by the observed positive cosmolog- ical constant. We calculated the parameter space compatible with present observations, with the novel behaviour starting in the future. A second ingredient of  is a mechanism that rebounces a contracting universe into an expanding one and mildly changes its cosmological constant. In section3 we explored how rebounces would aﬀects attempts of computing probabilities in the multiverse. In particular a steady-state temporal multiverse becomes possible, as anti-de Sitter vacua are no longer terminal and rebounce into expanding regions. In section4 we combined both ingredients above. Any region now undergoes cycles of ex- pansion, contraction and rebounce in a ﬁnite time5. This temporal universe is not aﬀected by issues that often plague the spatial multiverse. For instance, the Boltzmann-brain paradox and
5This turns out to be similar, in spirit, to the proposal of Ref.  based on the Ekpyrotic Universe, where
12 the youngness paradox  are avoided because there are no exponentially inﬂating regions nucleating habitable universes. In a part of its parameter space, the mechanism of  can provide the most likely source of universes with vacuum energy density below anthropic bound- aries. One can devise speciﬁc models where the probability distribution of the vacuum energy improves on the situation present in the usual anthropic selection, where the most likely value of the cosmological constant seems 2 orders of magnitude above its observed value.
Acknowledgements This work was supported by the ERC grant NEO-NAT. We thank Dario Buttazzo, Juan Garcia-Bellido, Luca Di Luzio, Antonio Riotto and Enrico Trincherini for discussions.
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