Testable anthropic predictions for dark energy

In the context of models where the dark energy density $\rD$ is a random variable, anthropic selection effects may explain both the"old"cosmological constant problem and the"time coincidence". We argue that this type of solution to both cosmological constant problems entails a number of definite predictions, which can be checked against upcoming observations. In particular, in models where the dark energy density is a discrete variable, or where it is a continuous variable due to the potential energy of a single scalar field, the anthropic approach predicts that the dark energy equation of state is $p_D=-\rho_D$ with a very high accuracy. It is also predicted that the dark energy density is greater than the currently favored value $\Omega_D\approx 0.7$. Another prediction, which may be testable with an improved understanding of galactic properties, is that the conditions for civilizations to emerge arise mostly in galaxies completing their formation at low redshift, $z\approx 1$. Finally, there is a prediction which may not be easy to test observationally: our part of the universe is going to recollapse eventually. However, the simplest models predict that it will take more than a trillion years of accelerated expansion before this happens.


I. INTRODUCTION
The "old" cosmological constant problem -why don't we see the large vacuum energy density ρ Λ which is expected from particle physics? -and the "time coincidence" problem -why do we live at the epoch when the dark energy component ρ D starts dominating?may find a natural explanation in models where ρ D is a random variable. The idea is to introduce a dynamical dark energy component X whose contribution ρ X varies from place to place, due to processes which occurred in the early universe. Then ρ D = ρ Λ + ρ X will also vary from place to place, and the old cosmological constant problem takes a different form. The question is not why ρ Λ is much smaller than η 4 , where η is some high energy physics mass scale, such as the supersymmetry breaking scale η ∼ T eV , but why do we happen to live in a place where ρ Λ is almost exactly cancelled by ρ X . This line of enquiry is rather quantitative, since we can ask what is the probability for us to observe certain values of ρ D ∼ 10 −11 (eV ) 4 , or what is the probability for the time coincidence.
Explicit particle physics models for a variable ρ X have been reviewed in [1]. Two examples which have been thoroughly discussed in the literature are a four-form field strength, which can vary through nucleation of membranes [2,3], and a scalar field with a very low mass [3,4]. Assuming one such mechanism, and using a theory of initial conditions such as inflation, one can calculate the "a priori" probability distribution P * (ρ D )dρ D . This is defined as the fraction of co-moving volume which at some fiducial initial time (which we conventionally take to be the time of recombination) had the value of the dark energy density in the interval dρ D . Inflation is also responsible for smoothing out the value of ρ D over comoving distances much larger than the size of our presently observable universe.
By itself, P * is not sufficient to calculate probabilities for our observations. Selection effects which bias the measurement of ρ D must be included, and the most important one in this case is anthropic [5][6][7][8][9]. 1 While |ρ D | may be very large in most places, there is nobody there to observe such extreme values. If ρ D > 0, galaxy formation stops once the dark energy becomes dominant over the matter density. Some galaxies are seen at redshifts of order z ∼ 5, but not much higher, indicating [9] that galaxies will not form in regions where ρ D (1 + z EG ) 3 ρ 0 . Here, ρ 0 is the matter density at the present time t 0 , and z EG ≈ 5 is the redshift at the time t EG ∼ (1 + z EG ) −3/2 t 0 when the earliest galaxies formed. Also, for a negative ρ D the universe recollapses on a time scale t D ∼ |Gρ D | −1/2 , where G is Newton's constant. This time should be larger than the earliest time t EI which is required for intelligence to develop [7,13]. Thus, observers will only exist within a tiny "anthropic range" It should be noted that, aside from the above minimal requirements, anthropic selection includes all other ways in which ρ D disfavours the existence of observers. For instance, in regions where ρ D < 0, the matter density is larger than |ρ D | throughout the cosmic evolution. If |ρ D | is too large, all galaxies formed in that region will be very dense, and as a result, very inhospitable. This occurs also for a large ρ D > 0, since galaxies must form before ρ D starts dominating. We shall come back to this issue in Sections III and V.
The selection effect can be implemented quantitatively by assuming the mediocrity principle, according to which our civilization is typical in the ensemble of all civilizations in the universe. The probability to find ourselves in a region with given values of ρ D is thus given by [14] dP(ρ D ) ∝ P * (ρ D )n civ (ρ D )dρ D .
Here, n civ (ρ D ) refers to the number of civilizations which will ever form per unit co-moving volume in regions where the dark energy density was equal to ρ D at the time of recombination. 2 Needless to say, the determination of both factors in the r.h.s. of Eq. (2) leaves room for some uncertainties. However, we shall argue that there are reasons to be optimistic. If the distribution (2) is to explain both cosmological constant problems, then a number of rather generic predictions can be made, rendering these ideas very testable.
In the next section, we review the calculation of the prior probability distribution P * (ρ D ). The anthropic factor n civ (ρ D ) is discussed in section III. In the same section, we argue that the anthropic approach can succeed only if the conditions for civilizations to evolve arise mostly in galaxies formed at low redshifts, z ∼ 1. Anthropic predictions for the dark energy 1 Anthropic selection effects associated with the possible variation of the amplitude of density fluctuations [10,11] and of the baryon to photon ratio [10,12] have also been discussed in the literature. 2 As we shall argue, in models where both cosmological constant problems can be solved anthropically, ρ D has not varied appreciably since the time of recombination, and therefore it can be treated as constant in time. equation of state, for the energy density ρ D , and for the Hubble parameter h are discussed in sections IV, V. The prediction for the future of the universe in unveiled in section VI. Finally, our conclusions are briefly summarized and discussed in section VII.

II. THE PRIOR DISTRIBUTION
The first task in determining (2) is to estimate P * . The vacuum energy density is of order ρ Λ ∼ η 4 (T eV ) 4 , and therefore ρ D must have a natural range of variation of order η 4 or larger. Weinberg noted [9] that a function P * (ρ D ) that varies smoothly on scales ρ D ∼ η 4 , should behave as a constant in the utterly narrower interval (1) -unless of course, the function would happen to have a zero or a pole in that interval (which would be an utter coincidence). This led him to conjecture that for values of ρ D in the anthropic range the prior probability would be constant, Outside of this range the form of P * is irrelevant, because the factor n civ vanishes. Weinberg's conjecture is subject to verification. As mentioned in the Introduction, P * is calculable, provided that the dynamics of ρ X is known, and assuming an inflationary model which would determine its spatial distribution at the time of recombination. Analysis of explicit models shows that (3) is not automatically guaranteed [4], but it does seem to be satisfied in generic models.
There are basically two reasons [1,4] why a non-flat P * may result from the process of randomization of ρ D which occurs during inflation (this randomization is due to quantum diffusion in the case where X is a scalar field, or to nucleation of membranes in the case when X is a four-form). The first reason is the differential expansion induced by the dark energy component. During inflation, the expansion rate is determined by Although ρ D is very small compared with the inflationary potential V inf , its effect may build up over time, in such a way that more thermalized volume is generated with high values of ρ D . In this way, P * (ρ D ) could be biased towards large values. Let us denote by τ (X, H) the characteristic time needed for the dynamics of X to sample (at a fixed point in space) all values of ρ D within the anthropic range (∆ρ D ) anth . The differential expansion is characterized by the parameter If q ≫ 1, then P * is exponentially steep in the range of interest. This case is ruled out by observations, because it predicts a very large ρ D , even after selection effects have been factored in. If q ∼ 1, the distribution P * may have a moderate dependence on ρ D within the anthropic range. This dependence affects the position of the peak of the distribution for the observed values of ρ D , Eq. (2), and hence it affects our predictions. While models of this sort are not ruled out, they require a very unnatural adjustment of parameters, since q is determined by a combination of rather different pieces of dynamics. Hence, we shall disregard this marginal possibility as non-generic. Finally, there is a wide class of models where q ≪ 1 is satisfied without any fine tuning [1,4], and hence we shall take this to be the generic case. Numerical simulations confirm that in this case the bias effect due to differential expansion is insignificant [18].
The second reason why P * may be non-flat is the following. Even if the differential expansion is negligible, and the prior distribution for X is flat, this does not automatically guarantee that the prior for ρ D will be flat, unless the relation between X and ρ D is linear in the range of interest. Through this effect, it is possible to have a moderate variation of P * (ρ D ) within the anthropic range. But again, this would require a contrived adjustment of parameters and we shall dismiss this case as non-generic (see also [19] for a discussion of this issue).
As an example, let us consider the case where ρ X = V (φ) is the potential energy density of a scalar field φ, The field must change very slowly on a cosmological time-scale, so that its potential energy behaves as an effective cosmological constant. This requires the slow-roll conditions [4] |V to be satisfied up to the present time (when ρ D ∼ ρ 0 , with ρ 0 the present matter density).
During inflation, the scalar field is randomized by quantum fluctuations, and at recombination it is distributed according to the "length" in field space, Therefore 3 , Thus, the flatness of the prior depends on how much V ′ changes in the anthropic range. As we shall see, variations in this range may occur, but they do not bias the probability distribution for ρ D in any significant way, unless we adjust some parameters specifically for this purpose. Consider a potential of the form where µ 2 ρ Λ < 0, so that it is possible to have |ρ D | very small even if |ρ Λ | is large. Eqs. (6) lead to the condition [4] |µ| ≪ 10 −120 m 3 Such a small mass parameter may seem unrealistic, but it can naturally arise, for instance, in a low energy effective theory with a suitable discrete symmetry [3] (for other proposals, see [1,19,20] and references therein). Note that (11) does not correspond to a fine tuning, but just to a strong suppression. The condition (7) translates into where H 0 is the present Hubble rate, and in the last step we have used H ∼ 10 −7 m p , corresponding to a GUT scale of inflation. The conditions (11) and (12) leave very many orders of magnitude available for the parameter µ, and so fine tuning is not necessary. From (5), where φ 2 0 = −2ρ Λ /µ 2 and κ = µ 2 φ 0 . We are interested in the vicinity of ρ D = 0, where it is easy to show from (9) that [4] Since ρ D ≪ ρ Λ in the anthropic range, the distribution is indeed flat to a very good accuracy. For contrast, we may consider the "washboard" potential where κ was given above and M and η are different mass scales. Let us assume that 4 Then the field will typically be found away from the local minima, with a probability distribution Both κ and M 4 /η should be much smaller than H 2 0 m p in order to satisfy the slow roll condition. In the case κ ≫ M 4 /η, the distribution (17) is still flat, as in (3). In the opposite case, where M 4 /η ≫ κ, the a priori distribution can have a sizable variation within the anthropically allowed range. If η ≪ m p , this range is very wide in the field space, δφ ρ 0 /κ ≫ m p . This means that the oscillations in P * will average out on scales much smaller than the anthropic range, and effectively we recover (3). Clearly, the only way to avoid this averaging effect is if η m p , and The last equation is to ensure that a significant range of values of φ/η is sampled in the anthropic range (∆ρ D ) anth 10 3 ρ 0 , so that changes in the slope of the potential are appreciable. Otherwise the distribution for P * will be almost flat. Thus, aside from the fact that the washboard potential is already a somewhat contrived example [19], Eq. (18) implies an otherwise unnecessary adjustment of the parameter M.
In what follows, we shall only consider models where there is no such ad-hoc adjustment. In this sense, our predictions may not be completely inescapable, but they can be considered generic. The situation can be compared with the predictions of inflation that the density parameter is Ω = 1 and the spectrum of density perturbations is nearly flat. It is certainly possible, in the context of inflation, to have an open universe with Ω < 1, or to have a markedly non-flat spectrum of density perturbations. But to achieve this, additional parameters must be introduced and adjusted to the desired outcome. 4 If M 4 ≫ H 2 0 m p η, the slow roll condition is not satisfied today and the field φ will be in any one of the local minima of the washboard. With some generic requirements on the inflationary parameters, the minima will have equal a priori probability within the anthropic range [1].

III. THE ANTHROPIC FACTOR
We now consider the effect of the anthropic factor n civ in Eq. (2). The physical situation is rather different for positive and negative ρ D , so we consider these two cases separately.
For positive ρ D , the main change introduced by n civ is that the time of earliest galaxy formation t EG in the anthropic range (1) is effectively replaced by the time at which the bulk of galaxy formation occurs. This is because a few early birds will not make a difference once we apply the principle of mediocrity. More precisely, we should take into consideration that the morphology of some galaxies could make them less suitable for the development of civilizations, and therefore Here, α denotes the set of parameters characterizing the type of galaxy (e.g. its size, density, etc.), n(α, ρ D ) is the number density of such galaxies that form per co-moving volume in regions characterized by ρ D , and N civ (α) is the number of civilizations per galaxy of type α. Suppose that the above integral receives a dominant contribution from galaxies of type α G . Then and the relevant time for anthropic considerations is the time at which this type of galaxies form, which we shall denote by t G . With the assumption of a flat prior P * , it was shown in [11,39] that the most probable value for a positive ρ D is the one characterized by This fact was used in order to explain the observed time coincidence The last relation follows from (21), assuming that stars and civilizations develop on a timescale not much greater than t G , and therefore t G is comparable to t 0 , defined as the time when most civilizations make their first determination of ρ D . Connected with the above discussion, there is a prediction of the anthropic approach, which can be checked by a combination of observations and theoretical analysis. In a not so distant future, our understanding of galactic evolution and morphology may improve to the point where we can tell with some confidence which galaxies are suitable for sustaining planetary systems similar to our own, where civilizations can develop. The anthropic approach to the cosmological constant problems (CCPs) predicts that the conditions for civilizations to emerge will be found mostly in galaxies that formed (or completed their formation) at a low redshift, z ∼ 1.
In the standard cold dark matter cosmology, galaxy formation is a hierarchical process, with smaller objects merging to form more and more massive ones. We know from observations that some galaxies existed already at z = 5, and the theory predicts that some dwarf galaxies and dense central parts of giant galaxies could form as early as z = 10 or even 20. The fraction of matter bound in giant galaxies (M ∼ 10 12 M ⊙ ) at z = 1 (∼ 20%) is somewhat less than that in objects of mass ∼ 10 9 M ⊙ at z = 3, or in objects of mass ∼ 10 7 M ⊙ at z = 5 [30]. If civilizations were as likely to form in early galaxies as in late ones, then Eq. (21) would indicate that, for a typical observer, the cosmological constant should start dominating at a redshift z G 5. The corresponding dark energy density, would be far greater than observed. Clearly, the agreement becomes much better if we assume that the conditions for civilizations to emerge arise mainly in the types of galaxies which form at lower redshifts, z G ∼ 1.
We now point to some directions along which the choice of z G ∼ 1 may be justified. One problem with dwarf galaxies is that if the mass of a galaxy is too small, then it cannot retain the heavy elements dispersed in supernova explosions. Numerical simulations suggest that the fraction of heavy elements retained is ∼ 30% for a 10 9 M ⊙ galaxy and is negligible for much smaller galaxies [37]. The heavy elements are necessary for the formation of planets and of observers, and thus one has to require that the structure formation hierarchy should evolve up to mass scales ∼ 10 9 M ⊙ or higher prior to the dark energy domination. This gives the condition z G 3, but falls short of explaining z G ∼ 1.
Another point to note is that smaller galaxies, formed at earlier times, have a higher density of matter. This may increase the danger of nearby supernova explosions and the rate of near encounters with stars, large molecular clouds, or dark matter clumps. Gravitational perturbations of planetary systems in such encounters could send a rain of comets from the Oort-type cloud towards the inner planets, causing mass extinctions 5 .
Our own Galaxy has definitely passed the test for the evolution of intelligence, and the principle of mediocrity suggests that most observers may live in galaxies of this type. Our Milky Way is a giant spiral galaxy. The dense central parts of such galaxies were formed at a high redshift z 5, but their discs were assembled at z ∼ 1 or later [32]. Our Sun is located in the disc, at a distance ∼ 8.5 kpc from the Galactic center 6 . If this situation is typical, then the relevant epoch to use in Eq. (23) is the epoch z G ∼ 1 associated with the formation of discs of giant galaxies.
The above remarks may or may not be on the right track, but we emphasize once again that if CCPs have an anthropic resolution, then, for one reason or another, the evolution of intelligent life should require conditions which are found mainly in giant galaxies, which completed their formation at z G ∼ 1.
In order to estimate n(α G , ρ D ) in Eq. (20), we shall need a simple quantitative criterion to specify the relevant type of galaxies. The most important parameter characterizing a 5 The cross-section for disruption of planetary orbits is much smaller, and it would take a rather substantial increase of the density for this process to become statistically important. A.V. is grateful to David Spergel for a discussion of this issue. 6 It has been noted [31] that this distance is close to the corotation radius, where the orbital velocity of the stars coincides with the rotational velocity of the spiral pattern. In other words, the motion of the Sun relative to the spiral arms is rather slow, and as a result, the periods between spiral arm crossings are rather long (∼ 10 8 yrs). Spiral arms are the primary sites of supernova explosions. They are also rich in giant molecular clouds, and are therefore very hazardous to life. It has been argued in [31] that spiral arm crossings are responsible for the major mass extinctions observed in the fossil record. Then one expects that habitable planetary systems are to be found mainly in the vicinity of the corotation radius, since mass extinctions at a rate much greater than once in 10 8 yrs may be too frequent for intelligent life to evolve. (Note that it took us 6.5 × 10 7 yrs to evolve since the last great extinction.) galaxy is its mass M. For the Milky Way it is M M W ∼ 10 12 M ⊙ [33], and the above discussion suggests that we identify the relevant galaxies with gravitationally bound halos of this mass. (Note that this is also the typical mass of L * galaxies, which contain most of the luminous stars in the Universe.) It should be recognized, however, that the choice of this characteristic mass scale is somewhat uncertain, so we shall illustrate how our results are affected by choosing a larger or a smaller mass.
Our Galaxy is a member of the Local Group cluster, whose mass has been estimated as [34] M LG ∼ 4 × 10 12 M ⊙ . It is conceivable that the gas captured in this cluster is later accreted onto the member galaxies and thus affects the properties of their discs. There seems to be no justification to consider larger mass objects, and we shall regard M LG as an upper bound on the potentially relevant mass scales. On the lower mass end, we shall use M ∼ 10 11 M ⊙ , which is roughly the mass of the bright part of our Galaxy, up to ∼ 10 kpc from the center. (We note that M M W is probably a more reasonable choice, because the properties of the disc depend on the total mass of the halo [35]. ) We now consider negative ρ D . The scale factor of a universe filled with nonrelativistic matter and dark energy with ρ D < 0 is given by where t D ≡ (1/6πG|ρ D |) 1/2 . The matter density ρ M initially decreases while the universe expands, but at t = πt D /2, when it reaches the value ρ M = −ρ D , the universe stops its expansion and starts recontraction. The matter density grows in the contracting phase, and thus ρ M ≥ |ρ D | throughout the evolution. The structure formation in a universe with a negative ρ D proceeds as usual until t ∼ t D , but then the growth of density perturbations accelerates during the contraction, so that all overdensities collapse to form bound objects prior to the big crunch. For t D t 0 , giant galaxies will form at about the same time as they did in our part of the universe and will have similar properties (with a possible caveat indicated below). However, for t D ≪ t 0 halos of the galactic size will be forced to collapse at a much earlier time t ∼ t D , and their density will therefore be much higher than that of our Galaxy. This would probably make such halos unsuitable for life.
These considerations suggest that the anthropic factor effectively constrains t D to be in the range for both positive and negative ρ D . There is, however, an additional factor that could make negative ρ D less probable. For ρ D > 0, structure formation effectively stops at t > t D , and the existing structures evolve more or less in isolation. This may account for the fact that discs of giant galaxies take their grand-design spiral form only relatively late, at z ∼ 0.3. The discs are already in place at z ∼ 1, but they have a very unsettled, irregular appearance [32]. On the other hand, for ρ D < 0 the clustering hierarchy only speeds up at t > t D , and quiescent discs which may be necessary for the evolution of fragile creatures like ourselves may never be formed. Another factor to consider is the characteristic time t I needed for intelligence to develop. For positive ρ D , this factor is unimportant, since the time after the dark energy domination is practically unlimited, but for negative ρ D the available time is bounded by t < πt D , and the effect of t I requires a closer examination.
We first note that t I ≪ t 0 is unlikely, since then it is not clear why it took so long for intelligence to develop on Earth. (The total time of biological evolution, from the origin of life on Earth till present, is estimated at ∼ 3.5 ×10 9 yrs.) For t I ≫ t 0 , we note that the main sequence lifetime of stars believed to be suitable to harbor life is t ⋆ ∼ (5 − 20) × 10 9 yrs ∼ t 0 (see [11] for a discussion of this point). If t I ≫ t 0 ∼ t ⋆ , most of these stars will explode as red giants before intelligence has a chance to develop. Carter [6] has argued that this is the most likely scenario 7 . In this case, the number N civ is suppressed by a factor ∼ min{t ⋆ , t D }/t I ∼ t 0 /t I , where we have used (25) in the last step. For positive ρ D , the suppression is by a factor ∼ t ⋆ /t I , which is of the same order of magnitude.
We conclude that the precise value of t I has little effect on the relative probability of positive and negative ρ D . If the accelerated clustering hierarchy is detrimental for life, then the probability for negative ρ D is suppressed; otherwise the two signs of ρ D are equally likely.
In either case, we should not be surprised that ρ D is positive in our part of the universe. In the following sections we shall focus on the positive values of ρ D .

IV. PREDICTION FOR THE EQUATION OF STATE
A rather generic prediction of models where both CCP's are solved anthropically is that the equation of state of dark energy is given by p D = wρ D , with The error bars correspond to the precision to which the observable universe can be aproximated by a homogeneous and isotropic model. In models where ρ X is the energy density of a four-form field, this equation of state is guaranteed by the fact that the four-form energy density is a constant and can only change by the nucleation of branes (other than that, it behaves exactly like an additional cosmological constant). If ρ X is a generic scalar field potential, the slow roll conditions (6) are likely to be satisfied by excess, by many orders of magnitude, rather than marginally. For instance, for the quadratic potential (10), these conditions imply the constraint (11). It would be contrived to arrange for the condition to be satisfied marginally, since the whole point of the present approach is to have ρ Λ cancelled regardless of its precise value (which is not known to us even by order of magnitude). If the slow roll conditions are satisfied by excess by just more than three orders of magnitude, then the kinetic energy of the scalar field will be less than its potential energy by more than six orders of magnitude, and Eq. (26) follows. There are certainly models for dark energy, some of them with anthropic input, were (26) is not satisfied. For instance, Kallosh and Linde [13] recently considered a supergravity model where the time coincidence problem is solved anthropically, and where (26) does not hold. However, their model does not solve the old CCP, since it is assumed that the cosmological constant vanishes in the observable matter sector due to some unspecified mechanism. Likewise, (26) does not hold in the usual quintessence models [21], which have no anthropic input at all, but which do not address the CCP's [1,19], or in models of k-essence [22], where only the time coincidence is partially addressed.
A possibility worth discussing is the case of models where the slow roll parameters are themselves random fields. Consider, for instance, the following model: If the probability distribution for the new scalar field ψ were such that all values of µ 2 are equiprobable, then one might imagine that the order of magnitude of µ 2 would be such that the slow roll conditions would be marginally satisfied. 8

V. PREDICTIONS FOR Ω D AND h.
Currently favoured values for the dark energy density and for the Hubble parameter are Ω D ≈ .7 and h ≈ .7 [24,25], both with error bars of the order of 10%. While observations are not very accurate, we would like to challenge the status quo and boldly use the anthropic approach to the CCP's to make predictions for these two parameters. As we shall see, this approach predicts that Ω D is likely to be somewhat higher, and that h is likely to be smaller than those currently favoured values.
The basic reason why we expect Ω D to be larger is the following [14,15]. The growth of density fluctuations in a universe with a positive cosmological constant effectively stops at the redshift z D when the cosmological constant starts dominating. This is given by (1 + z D ) ∼ (Ω D /Ω M ) 1/3 , where Ω M = 1 − Ω D is the matter density parameter. According to (21), we expect z D ∼ z G , where z G is the epoch when the relevant galaxies were formed. With z G ∼ 1, this corresponds to (Ω D /Ω M ) ∼ 8, which in turn implies Ω D ∼ .9. (For z G > 1, we would obtain an even higher value for ρ D .) This prediction can be made more 8 The new field ψ must also be a light field and hence its prior distribution is in principle calculable. We can actually consider a more general form of the potential for ψ and φ, Around any point (ψ 0 , φ 0 ) on the curve γ defined by V (ψ, φ) = −ρ Λ , the potential can be approximated by a linear function of the fields. Moreover, we can always rotate coordinates in field space so that ψ is directed along γ, and φ is orthogonal to it, Here V φ is the gradient of the potential at that point. During inflation, both fields ψ and φ are randomized by quantum fluctuations. Hence, the prior probability distribution is given by the area in field space P * dψdφ ∝ dψdφ, which leads to Along the curve γ, the values of V φ that will carry more weight per unit distance along the curve are those for which the slope is smaller. In the published version of this paper, it was (incorrectly) concluded from this observation that "given a model where the slope of the potential is variable, smaller values of the slope are preferred a priori, and there is no reason to expect that the slow roll conditions should be satisfied only marginally." However, this is not necessarily correct in general, since the slope could be large for a very large range of ψ, and large slopes may end up dominating the integral in Eq. (29). A detailed analysis of this point has been given in [42], where it is shown that there is a wide class of two-field models for which a small slope is favoured a priori (and for which the conclusions of Section IV do indeed apply). However, there is an equally broad class of models for which a large slope is favoured a priori. In that case, the posterior probability distribution for the slope is such that the slow roll condition is satisfied only marginally, and some departure from the vacuum equation of state may be expected (see [43,44]).
quantitative [17] by using the distribution (2). As we shall see, the precise predictions depend not only on Ω D but also on h. Throughout this section, we shall assume that ρ D > 0 as part of our prior. In a universe filled with pressureless matter and with a dark energy component ρ D > 0, the scale factor behaves as where t D ≡ (1/6πGρ D ) 1/2 . A primordial overdensity will eventually collapse, provided that its value at the time of recombination is larger than a certain value δ rec c . In the spherical collapse model, this is estimated as δ rec c (ρ D ) = 1.13x [27]. Here, we have introduced the variable The number of galaxies n(M, ρ D ) of mass M that will form per unit comoving volume in a region characterized by the value ρ D of the dark energy density, is proportional to the fraction of matter that eventually clusters into this type of galaxies. In the Press-Schechter approximation [26,27], this is given by Here, erfc is the complementary error function, and σ rec (M) is the dispersion in the density contrast at the time of recombination t rec . As argued in the preceeding section, we shall assume that most civilizations are formed in galaxies characterized by a mass M ∼ M M W ≈ 10 12 M ⊙ (although we shall also consider slightly larger and smaller masses). The factor n civ depends on the parameter σ rec , which in turn depends on the amplitude of density perturbations generated during inflation. The value of σ rec can be inferred from the normalization of CMB anisotropies, but for this task, both the present value of Ω D and the value of the Hubble parameter h would be needed. Since these are the parameters we wish to make predictions about, it would be somewhat contrived to use them at this point to make an inference about σ rec .
Another factor to consider is that σ rec may be different in distant regions of the universe (where, as a consequence, galaxies would form earlier or later). In models where the inflaton field has only one component, the value of σ rec is the same in all regions of the universe. However, if the inflaton field has more than one component, the amplitude of density perturbations depends on the path followed by the inflaton on its way to the minimum of the potential. In such models, it is possible for σ rec to vary over distances much larger than the presently observable universe.
To make our discussion sufficiently general, we shall consider that σ rec is itself a random variable with unspecified prior. This prior may be determined by processes occurring during inflation, or it may just reflect our ignorance of the actual value of the fixed parameter σ rec . Then, Eq. (2) is generalized to dP(ρ D , σ rec ) ≈ n civ P * (ρ D , σ rec )dρ D dσ rec .
In this context, the generic expectation that the prior does not depend on ρ D in the anthropic range [see Eq. Substituting (31) into (32), we have dP(ρ D , σ rec ) ∝ erfc .80x where we have used that Ω M (t rec ) ≈ 1 in all regions of interest, so that dρ D ∝ dx rec . Introducing y = x rec σ −3 rec , the change of variables (x rec , σ rec ) → (y, σ rec ) produces a Jacobian proportional to σ 3 rec , and we have 9 dP(y, σ rec ) ≈ f (y)σ 3 rec P * (σ rec )dydσ rec , where f (y) does not depend on σ rec . Integrating over σ rec leads to the normalized distribution dP(y) = (.80) 3 π −1/2 erfc(.80y 1/3 ) y d ln y, which is uncorrelated with σ rec . The variable y can be expressed in terms of observable quantities, as we shall see below, and from (34) we should expect y ∼ 1 by order of magnitude (See Fig. 1). More precisely, 9 The appearance of the factor σ 3 rec in the posterior distribution was noted in Ref. [11]. This factor implies that the prior P * (σ rec ) should decay faster than σ −3 rec at large values of the density contrast. Otherwise, the posterior distribution is not normalizable (and we should expect both a large value of the effective vacuum energy and of the linear density contrast, in contradiction with observations). In the published version of the present paper this factor was omitted, due to an incorrect normalization of Eq. (33). We thank Takahiro Tanaka for drawing our attention to this point.
We shall denote these two equations as the 1σ and 2σ confidence level predictions for y. Let us now show how these translate into confidence level curves for the expected values of the parameters Ω D and h. Here, and in what follows, Ω D will denote the present value of the dark energy density parameter in our observable universe. Let us first express the "observed" value of y, which we shall denote as y 0 , in terms of Ω D and h. The density contrast at present is given by σ 0 = G(x 0 , x rec )σ rec , where, assuming z rec ≫ 1, the growth factor is given by [17] Therefore, The linearized density contrast at present σ 0 can be inferred from measurements of CMB temperature anisotropies, as described e.g. in [17,28]. Since the spectrum is expressed as a function of wavelength, the mass scale has to be converted into a length-scale. A halo of mass M corresponds to a co-moving radius R(M) = (3M/4πρ 0 ) 1/3 . The mean matter density of the universe is given by ρ 0 = 1.88 × 10 −29 Ω M h 2 g/cm 3 , which leads to

Mpc.
Assuming an adiabatic primordial spectrum of scalar density perturbations, characterized by a spectral index n, we have Here, c 100 = 2.9979 is the speed of light in units of 100km s −1 and ] is the so-called shape parameter, with Ω b the density parameter in baryons. For numerical estimates, we shall take Ω b h 2 ≈ .02. The dimensionless amplitude of cosmological perturbations inferred from the COBE DMR experiment is given by [28,29]  The parameter r denotes the ratio of tensor to scalar amplitudes. Note that the effect of tensors is to make δ H a bit smaller (although not very significantly). Finally, where the transfer function is given by T (q) = (2.34q) −1 ln(1 + 2.34q)[1 + 3.89q + (16.1q) 2 + (5.46q) 3 + (6.71q) 4 ] −1/4 and the window function is given by W (u) = 3u −3 (sin u − u cos u).
Substituting (39) in (38), and using Ω D + Ω M = 1, we obtain the function y 0 = y 0 (Ω D , h). Contour lines of this function, corresponding to the 1σ and 2σ predictions represented by Eqs. (35)(36), are plotted in Fig. 2, assuming that the dominant contribution to n civ is in galaxies of mass M = M M W = 10 12 M ⊙ (thick solid lines). We also consider the predictions for different choices of the mass, as discussed in Section III. The short dashed curves correspond to the mass of the local group M LG = 4 × 10 12 M ⊙ , and the long dashed curves correspond to the mass of the bright inner part of our galaxy M = 10 11 M ⊙ . The effect of a tilt in the spectral index is plotted in Fig. 3. Both of these figures ignore the effect of tensor modes in the normalization (40). Tensor modes tend to lower the value of δ H , and hence they tend to make the bounds somewhat less stringent. The effect, however, is not dramatic. Even for r as large as .5, the effect on the curves is comparable to the effect of lowering the spectral index by .05.
Expressions similar to Eqs. (34)- (38) were already contained in the exhaustive analysis of the problem given by Martel, Shapiro and Weinberg (MSW) in [17], where σ rec was treated as a fixed parameter. However, our use of these expressions is somewhat different. MSW noted that the existing observations indicate a value of Ω D ∼ 0.6 − 0.7 and used Eq. (33), with h = 0.7 to show that this range corresponds to probabilities from 2% to 12%, depending on the values chosen for the galactic scale M and the spectral index of perturbations n. They concluded that "anthropic considerations do fairly well as an explanation of a cosmological constant with [Ω D ] in the range 0.6 − 0.7". However, one cannot help but feel disappointed by the somewhat low values of the probabilities.
Our approach here is that anthropic models should be used as any other models -to make testable predictions. Thus, the goal is not so much to explain the value of Ω D after it is determined by observations, but to predict that value at a specified confidence level. The contour lines in Figs These predictions can be turned around. If the values Ω D ≤ 0.7, h ≥ 0.7 are confirmed by future measurements, then our model will be ruled out at a 95% confidence level, again assuming M ∼ M M W and a scale invariant spectrum. For a tilted spectrum, slightly lower values of Ω D are allowed at the same confidence level. The observational situation at the time of this writing is far from being clear. CMB and supernovae measurements yield [23,24] Ω D ≈ 0.7 , while the observations of galaxy clustering give [38] Ω M = 0.18 ± 0.8, and thus Ω D ≈ 0.8.

VI. THE FUTURE OF THE UNIVERSE
We finally discuss the anthropic prediction which is not likely to be tested any time soon. In all anthropic models, ρ D can take both positive and negative values, so the observed positive dark energy will eventually start decreasing and will turn negative, and our part of the universe will recollapse to a big crunch.
To be specific, we shall consider a scalar field model with a very flat potential. In the anthropic range (1), the potential can be approximated as a linear function, where V ′ 0 is a constant and we have set φ = 0 at V = 0. Once the dark energy dominates, the evolution is described by the usual slow roll equations where H =ȧ/a and a(t) is the scale factor. The solution of (42), (43) is where −φ * is the present value of φ and is the time from the present to the beginning of recollapse. The slow roll condition (6) implies that φ * m p . As we discussed in Sec. IV, we do not expect this condition to be only marginally satisfied, and thus φ * ≫ m p . Then it follows from Eqs. (46) and (45) that t * ≫ 8πt D and therefore we should expect our region of the universe to undergo accelerated expansion for at least another trillion years before recollapse. 10 The slow roll approximation breaks down at φ ∼ −m p , so the above equations cannot be used to describe the evolution at φ > 0, where the potential becomes negative. A general analysis of models with negative potentials has been given in [36], where it is shown that at φ ≫ m p the dynamics becomes dominated by the kinetic energy of the field,φ 2 ≫ |V (φ)|. The corresponding evolution is described by where t c is the time of the big crunch. The linear approximation (41) for the potential breaks down at sufficiently large φ, but in this regime the form of the potential is unimportant and Eqs. (47), (48) still apply.
During the dark energy dominated expansion, the ordinary nonrelativistic matter is diluted by the exponential factor (45). When the contraction starts, the density of matter begins to grow as ρ M ∝ (t c − t) −1 . However, the kinetic energy of the field φ grows much faster,φ 2 ∝ (t c − t) −2 , and thus ordinary matter forever remains a subdominant component of the universe.

VII. CONCLUSIONS
We now summarize the predictions that follow from the anthropic approach to the CCP's. (1) In the simplest models where the dark energy density takes discrete values, or where the dark energy density is due to the potential energy of a single scalar field, the dark energy equation of state is predicted to be that of the vacuum, where w = −1 with a very high accuracy. This distinguishes the anthropic models we discussed here from other approaches, such as quintessence [21] or k-essence [22]. 11 (2) The anthropic predictions for the dark energy density Ω D and for the Hubble parameter h are given in Figs (3) Conditions for intelligent life to evolve are expected to arise mainly in giant galaxies that form (or complete their formation) at low redshifts, z G 1.
(4) The accelerated expansion will eventually stop and our part of the universe will recollapse, but, at least in the framework of the simplest models, it will take more than a trillion years for this to happen. Of course, this prediction is not likely to be tested anytime soon. 13 .
The above predictions apply to models where both CCP's are solved anthropically. For comparison, we may consider other models. For instance, it is conceivable that a small value of the cosmological constant will eventually be explained within the fundamental theory. (We note the interesting recent proposal by Dvali, Gabadadze and Shifman [40] in this regard.) Even then, the coincidence problem will still have to be addressed. One possibility 11 For a discussion of multifield models, see [42]. There is a class of two field models where the above prediction does not hold. These are the models where the prior distribution favours a large slope of the potential. In general, the equation of state p D = w(z)ρ D depends on the redshift z, and increases with time as the dark energy fields pick up kinetic energy, on their way to negative values of the potential. The prediction for w(z) in the general case has been discussed in [43,44]. 12 These predictions are implicit in the earlier analysis by Martel, Shapiro and Weinberg [17] 13 Again, this prediction is somewhat different in two field models where the prior distribution favours a large slope of the potential [42] is that ρ D is truly a constant, while the amplitude of the density fluctuations σ rec is a stochastic variable. With some mild assumptions about the prior probability distribution P * (σ rec ), it can be shown [1] that most galaxies are then formed at about the time of vacuum domination. In this class of models, predictions (1) and (3) still hold, while the other two predictions no longer apply.
Another possibility has been recently discussed by Kallosh and Linde [13]. They assumed an M-theory inspired potential with a stochastic variable Λ. An interesting property of this potential is that its curvature is correlated with its height (at φ = 0). As a result, the universe tends to recollapse within a few Hubble times after the dark energy comes to dominate. Assuming that other contributions to the vacuum energy are somehow cancelled (that is, that the old CCP is solved by some unspecified mechanism), Kallosh and Linde argue that the coincidence t D ∼ t I is to be expected, where t I is the time it takes intelligent life to evolve (they assume it to be ∼ 10 10 yrs). Predictions (1)- (3) are not applicable to this model. The model does predict recollapse of the universe, but the corresponding timescale (∼ 10 10 yrs) is much shorter than the anthropic prediction (4). Anthropic arguments are sometimes perceived as handwaving, unpredictive and unfalsifiable lore, of questionable scientific validity. In our view, the results presented in this paper should dispel this notion. Here, we have used the anthropic approach to make several quantitative predictions, some of which may soon be checked against observations. It should also be emphasized that, for the particular case of dark energy, there are at present no alternative theories explaining both CCP's, or making generic predictions of comparable accuracy.
The present bound on the equation of state parameter w from the CMB and supernovae measurements is [41] w < −0.7, which is consistent with the anthropic prediction of w = −1. The value of w = −1 is usually associated with a plain cosmological constant. However, if in addition to this equation of state, observations confirm some of the other predictions presented above, this may be taken as an indication that the dark energy is dynamical. Thus, a better understanding of structure formation and galactic evolution may in fact reveal a crucial property of dark energy, with important implications for particle physics.