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Physics Letters B 600 (2004) 15–21 www.elsevier.com/locate/physletb
Anthropic distribution for cosmological constant and primordial density perturbations
Michael L. Graesser a,StephenD.H.Hsub, Alejandro Jenkins a,MarkB.Wisea
a California Institute of Technology, Pasadena, CA 91125, USA b Institute of Theoretical Science and Department of Physics, University of Oregon, Eugene, OR 97403, USA Received 2 August 2004; accepted 26 August 2004
Editor: H. Georgi
Abstract The Anthropic Principle has been proposed as an explanation for the observed value of the cosmological constant. Here we revisit this proposal by allowing for variation between universes in the amplitude of the scale-invariant primordial cosmological density perturbations. We derive a priori probability distributions for this amplitude from toy inflationary models in which the parameter of the inflaton potential is smoothly distributed over possible universes. We find that for such probability distributions, the likelihood that we live in a typical, anthropically-allowed universe is generally quite small. 2004 Elsevier B.V. Open access under CC BY license.
The Anthropic Principle has been proposed as a on the observation that the existence of life capable possible solution to the two cosmological constant of measuring Λ requires a universe with cosmological problems: why the cosmological constant Λ is orders structures such as galaxies or clusters of stars. A uni- of magnitude smaller than any theoretical expectation, verse with too large a cosmological constant either and why it is non-zero and comparable today to the does not develop any structure, since perturbations that energy density in other forms of matter [1–3].This could lead to clustering have not gone non-linear be- anthropic argument, which predates direct cosmologi- fore the universe becomes dominated by Λ,orelsehas cal evidence of the dark energy, is the only theoretical a very low probability of exhibiting structure-forming prediction for a small, non-zero Λ [3,4]. It is based perturbations, because such perturbations would have to be so large that they would lie in the far tail-end of the cosmic variance. The existence of the string theory landscape, in which causally disconnected regions can E-mail addresses: [email protected] have different cosmological and particle physics prop- (M.L. Graesser), [email protected] (S.D.H. Hsu), [email protected] (A. Jenkins), [email protected] erties, adds support to the notion of an anthropic rule (M.B. Wise). for selecting a vacuum.
0370-2693 2004 Elsevier B.V. Open access under CC BY license. doi:10.1016/j.physletb.2004.08.061 16 M.L. Graesser et al. / Physics Letters B 600 (2004) 15–21
How well does this principle explain the observed a priori probability distribution for Q, making only value of Λ in our universe? Careful analysis by [4] modest assumptions about the behavior of the a priori finds that 5% to 12% of universes would have a cos- distribution for the parameter of the inflaton poten- mological constant smaller than our own. In everyday tial in the anthropically-allowed range. Cosmological experience we encounter events at this level of confi- and particle physics parameters other than Λ and Q dence,1 so as an explanation this is not unreasonable. are held fixed as initial conditions at recombination. If the value of Λ is not fixed a priori, then one We provisionally adopt Tegmark and Rees’s anthropic might expect other fundamental parameters to vary be- bound on Q: a factor of 10 above and below the value tween universes as well. This is the case if one sums measured in our universe. Even though this interval is over wormhole configurations in the path integral for small, we find that the likelihood that our universe has quantum gravity [5], as well as in the string theory a typical cosmological constant is drastically reduced. landscape [6–9].In[9] it was emphasized that all the The likelihood tends to decrease further if larger inter- parameters of the low energy theory would vary over vals are considered. the space of vacua (“the landscape”). Douglas [7] has Weinberg determined in [3] that, in order for an initiated a program to quantify the statistical proper- overdense region to go non-linear before the energy ties of these vacua, with additional contributions by density of the universe becomes dominated by Λ,the others [8]. value of the overdensity δ ≡ δρ/ρ must satisfy In [10], Aguirre stressed that life might be possible in universes for which some of the cosmological pa- 729Λ 1/3 δ> . (1) rameters are orders of magnitude different from those 500ρ¯ of our own universe. The point is that large changes in one parameter can be compensated by changes in In a matter-dominated universe this relation has no ex- another in such a way that life remains possible. An- plicit time dependence. Here ρ¯ is the energy density thropic predictions for a particular parameter value in non-relativistic matter. Perturbations not satisfying will therefore be weakened if other parameters are al- the bound cease to grow once the universe becomes lowed to vary between universes. One cosmological dominated by the cosmological constant. For a fixed parameter that may significantly affect the anthropic amplitude of perturbations, this observation provides argument is Q, the standard deviation of the ampli- an upper bound on the cosmological constant compat- tude of primordial cosmological density perturbations. ible with the formation of structure. Throughout our Rees [11] and Tegmark and Rees [12] have pointed analysis we assume that at recombination Λ ¯ρ. out that if the anthropic argument is applied to uni- To quantify whether our universe is a typical, verses where Q is not fixed but randomly distributed, anthropically-allowed universe, additional assump- then our own universe becomes less likely because tions about the distribution of cosmological parame- universes with both Λ and Q larger than our own ters and the spectrum of density perturbations across are anthropically allowed. The purpose of this Letter the ensemble of universes are needed. is to quantify this expectation within a broad class of A given slow-roll inflationary model with reheating inflationary models. Restrictions on the a priori proba- leads to a Friedman–Roberston–Walker universe with bility distribution for Q necessary for obtaining a suc- a (late-time) cosmological constant Λ and a spectrum cessful anthropic prediction for Λ, were considered in of perturbations that is approximately scale-invariant [13,14]. and Gaussian with a variance In our analysis we let both Λ and Q vary between universes and then quantify the anthropic likelihood Q2 ≡ δ˜2 . (2) of a positive cosmological constant less than or equal HC to that observed in our own universe. We offer a class The expectation value is computed using the ground of toy inflationary models that allow us to restrict the state in the inflationary era and perturbations are eval- uated at horizon-crossing. The variance is fixed by the parameters of the inflationary model together with 1 For instance, drawing two pairs in a poker hand. some initial conditions. Typically, for single-field φ M.L. Graesser et al. / Physics Letters B 600 (2004) 15–21 17 slow-roll inflationary models, Additional model-dependence occurs in the introduc- tion of the parameter s given by the ratio of the vol- H 4 Q2 ∼ . (3) ume of overdense sphere to the volume of the under- ˙2 φ HC dense shell surrounding the sphere. We will set s = 1 This leads to spatially separated over- or under-dense throughout. regions with an amplitude δ that for a scale-invariant Since the anthropically allowed values for Λ are spectrum are distributed (at recombination) according so much smaller than any other mass scale in particle to physics, and since we assume that Λ = 0 is not a spe- cial point in the landscape, we follow [4,17] in using 2 1 − 2 2 N (σ, δ) = e δ /2σ . (4) the approximation P(Λ) P(Λ= 0) for Λ within the π σ anthropically allowed window.2 The requirement that (The linear relation between Q and the filtered σ in the universe not recollapse before intelligent life has Eq. (4) is discussed below.) had time to evolve anthropically rules out large nega- By Bayes’s theorem, the probability for an anthro- tive Λ [2,19]. We will assume that the anthropic cutoff pically-allowed universe (i.e., the probability that the for negative Λ is close enough to Λ = 0thatallΛ<0 cosmological parameters should take certain values, may be ignored in our calculations. given that life has evolved to measure them) is pro- As an example of a concrete model for the variation portional to the product of the a priori probability dis- in Q between universes, we consider inflaton poten- tribution P for the cosmological parameters, times the tials of the form (see, for example, [20]) probability that intelligent life would evolve given that 2p choice of parameter values. Following [4], we estimate V = Λ + λφ , (8) that second factor as being proportional to the mean where p is a positive integer.3 We assume there are F fraction (σ, Λ) of matter that collapses into galaxies. additional couplings that provide an efficient reheating The latter is obtained in a universe with cosmologi- mechanism, but are unimportant for the evolution of φ cal parameters Λ and σ by spatially averaging over all during the inflationary epoch. The standard deviation over- or under-dense regions, so that [4] of the amplitude of perturbations gives ∞ + √ φp 1 F(σ, Λ) = dδN(σ,δ)F(δ, Λ). (5) Q = A HC λ 3 , (9) MPl δmin where A is a constant, and φ is the value of the The lower limit of integration is provided by the HC field when the mode of wave number k leaves the hori- anthropic bound of Eq. (1), which gives δ ≡ min zon. This φ has logarithmic dependence on λ and k, (729Λ/500ρ)¯ 1/3. The anthropic probability distribu- HC which we neglect. Randomness in the initial value for tion is φ affects only those modes that are (exponentially) P(σ, Λ) = P(Λ,σ)F(σ, Λ) dΛ dσ. (6) well outside our horizon. Throughout this Letter, we will set the spectral index to one and ignore its run- Computing the mean fraction of matter collapsed ning. Eq. (9) then gives λ ∝ Q2. into structures requires a model for the growth and Next, suppose that the fundamental parameters of collapse of inhomogeneities. The Gunn–Gott model the Lagrangian are not fixed, but vary between uni- [15,16] describes the growth and collapse of an over- verses, as might be expected if one sums over worm- dense spherical region surrounded by a compensat- hole configurations in the path integral for quantum ing underdense shell. The weighting function F(δ, Λ) gives the fraction of mass in the inhomogeneous re- gion of density contrast δ that eventually collapses 2 Garriga and Vilenkin point to examples of quintessence models (and then forms galaxies). To a good approximation in which the approximation P(Λ) P(Λ= 0) in the anthropically- it is given by [4] allowed range is not valid [18]. 3 Recent analysis of astronomical data disfavors the λφ4 infla- F = 1 tionary model [21], but for generality we will consider an arbitrary p (δ, Λ) δ . (7) in Eq. (8). δ + δmin 18 M.L. Graesser et al. / Physics Letters B 600 (2004) 15–21 gravity [5] or in the string theory landscape [6–9].To In terms of these variables and following [4],the obtain the correct normalization for the density per- probability distribution of Eq. (6) is found to be turbations observed in our universe, the self-coupling ∞ must be extremely small. As the standard deviation e−x P = NdσdyP(ˆ σ)ˆ dx , (12) Q will be allowed to vary by an order of magnitude 1/2 + 1/2 − β x around 10 5, for this model the self-coupling in alter- β nate universes will be very small as well. where We may then perform an expansion about λ = 0for / the a priori probability distribution of λ. The small- 1 729y 2 3 β ≡ , (13) ness of λ suggests that we may keep only the leading 2σˆ 2 500 term in that expansion. If the a priori probability dis- and N is the normalization constant. tribution extends to negative values of λ (which are Notice that, since x β,largeβ implies that P ∼ anthropically excluded due to the instability of the re- − e β 1. For a fixed σˆ ,largey implies large β. sulting action for φ), we expect it to be smooth near Thus, for fixed σˆ , large cosmological constants are λ = 0, and the leading term in the power series expan- anthropically disfavored. But if σˆ is allowed to in- sion to be zeroth order in λ (i.e., a constant). Therefore crease, then β ∼ O(1) may be maintained at larger y. we expect a flat a priori probability distribution for λ. Garriga and Vilenkin have pointed out that the distrib- The a priori probability distribution for Q is then ution in Eq. (12) may be rewritten using the change of dλ variables (σ,y)ˆ → (σ,β)ˆ [14]. The Jacobian for that P(Q)∝ ∼ Q, (10) dQ transformation is a function only of σˆ .Eq.(12) then factorizes into two parts: one depending only on σˆ , where the normalization constant is determined by the other only on β. Integration over σˆ produces an the range of integration in Q. Note that this distrib- overall multiplicative factor that cancels out after nor- ution favors large Q. On the other hand, if the a priori malization, so that any choice of P(σ)ˆ will give the probability distribution for the coupling λ only has same distribution for the dimensionless parameter β. support for λ>0thenλ = 0 is a special point and In that sense, even in a scenario where σˆ is randomly we cannot argue that P(Q)∝ Q. However, since the distributed, the computation in [4] may be seen as an anthropically-allowed values of λ are very small, the anthropic prediction for β.4 The measured value of β a priori distribution for λ should be dominated, in the is, indeed, typical of anthropically-allowed universes, anthropically-allowed window, by a leading term such but an anthropic explanation for β alone does not ad- as P(λ) ∼ λq . Normalizability requires q>−1. Us- + dress the problem of why both Λ and Q should be so ing λ ∝ Q2,thisgivesP(Q)∼ Q2q 1. small in our universe. Before proceeding, it is convenient to transform to Implementing the Anthropic Principle requires the new variables: making an assumption about the minimum mass of Λ ρ¯ 1/3 “stuff”, collapsed into stars, galaxies, or clusters of y ≡ , σˆ ≡ σ . (11) ρ∗ ρ∗ galaxies, that is needed for the formation of life. It is more convenient to express the minimum mass M Here ρ¯ is the energy-density in non-relativistic mat- min in terms of a comoving scale R: M = 4πρa¯ 3 R3/3 ter at recombination, which we take to be fixed in min eq (by convention a = 1 today, so R is a physical scale). all universes, and ρ∗ is the value for the present-day We do not know the precise value of R. A better un- energy density of non-relativistic matter in our own derstanding of biology would in principle determine universe. For a matter-dominated universe σˆ is time- its value, which should only depend on chemistry, the independent, whereas y is constant for any era. Here fraction of matter in the form of baryons, and New- and throughout this Letter, a subscript ∗ denotes the ton’s constant. In our analysis these are all fixed initial value that is observed in our universe for the corre- conditions at recombination. In particular, we would sponding quantity. The only quantities whose varia- tion from universe to universe we will consider are y and σˆ . 4 We thank Garriga and Vilenkin for explaining this point to us. M.L. Graesser et al. / Physics Letters B 600 (2004) 15–21 19
5 not expect Mmin to depend on Λ or Q. Therefore, COBE normalized amplitude of fluctuations at horizon −5 −0.785–0.05∗lnΩ∗ even though the relation between Mmin and R de- crossing, Q∗ = 1.94 × 10 Ω∗ . pends on present-day cosmological parameters, the As we have argued, the dependence of C∗ on the value of this threshold will be constant between uni- cosmological constant is not relevant for our pur- −2 verses because it depends only on parameters that we poses. For our calculations we use Ω∗ = 0.134h∗ , are treating as fixed initial conditions. Thus, in com- and h∗ = 0.73 (consistent with their observed best-fit puting the probability distribution over universes, we values [22]). The smoothing scale R will be taken to will fix R. Since we do not know what is the correct be either 1 Mpc or 2 Mpc, and the corresponding val- anthropic value for R, we will present our results for ues for C∗ are 5.2 × 104 and 3.8 × 104. both R = 1and2Mpc.(R on the order of a few Mpc The values chosen for the range of Q are moti- corresponds to requiring that structures as large as our vated by the discussion in [12] about anthropic limits galaxy be necessary for life.) on the amplitude of the primordial density perturba- We then proceed to filter out perturbations with tions. The authors of [12] argue that Q between 10−3 wavelength smaller than R, leading to a variance σ 2 and 10−1 leads to the formation of numerous super- that depends on the filtering scale. Expressed in terms massive black holes which might obstruct the emer- of the power spectrum evaluated at recombination, gence of life.6 They then claim that universes with −6 ∞ Q less than 10 are less likely to form stars, or if 1 star clusters do form, that they would not be bound σ 2 = dkk2P(k)W2(kR) (14) 2π2 strongly enough to retain supernova remnants. Since 0 there is considerable uncertainty in these limits, we where W is the filter function, which we take to be a carry out calculations using both the range indicated 2 7 Gaussian W(x)= e−x /2. P(k)is the power spectrum, by [12] as well as a range that is somewhat broader. which we assume to be scale-invariant. (For P(k) we Previous work on applying the Anthropic Principle use Eq. (39) of [4], setting n = 1.) to variable Λ and Q has assumed a priori distributions k Evaluating (14) at recombination gives, for our uni- P(Q) that fall off as 1/Q for large Q, with k 3 verse, [13,14]. Such distributions were chosen in order to keep the anthropic probability P(y, Q) normalizable, σˆ∗ = C∗Q∗. (15) and they usually yield anthropic predictions for the cosmological constant similar to those that were ob- The number C∗ contains the growth factor and transfer tained in [4] by fixing Q to its observed value, because function evaluated from horizon crossing to recombi- they naturally favor a Q as small as its observed value nation and only depends on physics from that era. We in our universe. For instance, for P(Q) ∝ 1/Q3 in assume Λ is small enough so that at recombination the range Q∗/10
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