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Anthropic Distribution for Cosmological Constant and Primordial Density Perturbations 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. All rights reserved. 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/$ – see front matter 2004 Elsevier B.V. All rights reserved. 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 2 ≡ ˜2 Q δ HC. (2) of a positive cosmological constant less than or equal 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) = HC Q A 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.
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