dimension less physical constants

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Dimensionless constants, cosmology and other dark matters

Max Tegmark1,2, Anthony Aguirre3, Martin J. Rees4 & Frank Wilczek2,11 MIT Kavli Institute for Astrophysics and Space Research, Cambridge, MA 02139 2 Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139

3Department of Physics, UC Santa Cruz, Santa Cruz, CA 95064 and 4Institute of Astronomy, University of Cambridge, Cambridge CB3 OHA, UK

(Dated: Submitted December 1 2005; accepted December 7; Phys. Rev. D, 73, 023505)

We identify 31 dimensionless physical constants required by particle physics and cosmology, andemphasize that both microphysical constraints and selection effects might help elucidate their origin.Axion cosmology provides an instructive example, in which these two kinds of arguments must bothbe taken into account, and work well together. If a Peccei-Quinn phase transition occurred before orduring inflation, then the axion dark matter density will vary from place to place with a probabilitydistribution. By calculating the net dark matter halo formation rate as a function of all four relevantcosmological parameters and assessing other constraints, we find that this probability distribution,computed at stable solar systems, is arguably peaked near the observed dark matter density. If cosmologically relevant WIMP dark matter is discovered, then one naturally expects comparabledensities of WIMPs and axions, making it important to follow up with precision measurements todetermine whether WIMPs account for all of the dark matter or merely part of it.

I. INTRODUCTION

Although the standard models of particle physics andcosmology have proven spectacularly successful, they to-gether require 31 free parameters (Table 1). Why weobserve them to have these particular values is an out-standing question in physics.

A. Dimensionless numbers in physics

This parameter problem can be viewed as the logicalcontinuation of the age-old reductionist quest for simplic-ity. Realization that the material world of chemistry andbiology is built up from a modest number of elementsentailed a dramatic simplification. But the observationof nearly one hundred chemical elements, more isotopes,and countless excited states eroded this simplicity.

The modern SU (3) × SU (2) × U (1) standard modelof particle physics provides a much more sophisticatedreduction. Key properties (spin, electroweak and colorcharges) of quarks, leptons and gauge bosons appear aslabels describing representations of space-time and in-ternal symmetry groups. The remaining complexity isencoded in 26 dimensionless numbers in the Lagrangian(Table 1)1. All current cosmological observations can befit with 5 additional parameters, though it is widely antic-ipated that up to 6 more may be needed to accommodate

more refined observations (Table 1).Table 2 expresses some common quantities in terms of

these 31 fundamental ones2, with ∼ denoting cruder ap-proximations than ≈. Many other quantities commonly

1 Here µ2 and λ are defined so that the Higgs potential is V (Φ) =µ2|Φ|2 + λ|Φ|4.2 The last six entries are mere order-of-magnitude estimates [1, 2].

In the renormalization group approximation for α in Table 2,

referred to as parameters or constants (see Table 3 for asample) are not stable characterizations of properties of the physical world, since they vary markedly with time[3]. For instance, the baryon density parameter Ωb, thebaryon density ρb, the Hubble parameter h and the cos-mic microwave background temperature T all decreasetoward zero as the Universe expands and are, de facto,alternative time variables.

Our particular choice of parameters in Table 1 is acompromise balancing simplicity of expressing the funda-mental laws (i.e., the Lagrangian of the standard modeland the equations for cosmological evolution) and ease of measurement. All parameters except µ2, ρΛ, ξb, ξc andξν are intrinsically dimensionless, and we make these finalfive dimensionless by using Planck units (for alternatives,see [7, 8]). Throughout this paper, we use “extended”Planck units defined by c = G = h = |qe| = kB = 1. Weuse h = 1 rather than h = 1 to minimize the number of (2π)-factors elsewhere.

B. The origin of the dimensionless numbers

So why do we observe these 31 parameters to have theparticular values listed in Table 1? Interest in that ques-tion has grown with the gradual realization that someof these parameters appear fine-tuned for life, in thesense that small relative changes to their values wouldresult in dramatic qualitative changes that could pre-clude intelligent life, and hence the very possibility of reflective observation. As discussed extensively elsewhere[9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], thereare four common responses to this realization:

only fermions with mass below mZ should be included.

http://arxiv.org/abs/astro-ph/0511774v3




































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Table 2: Derived physical parameters, in extended Planck units c = h = G = kb = |qe| = 1.

Parameter Meaning Definition Measured value

e Electromagnetic coupling constant at mZ g sin θW ≈ 0.313429 ± 0.000022

α(mz) Electromagnetic interaction strength at mZ e2/4π = g2 sin2 θW /4π ≈ 1/(127.918 ± 0.018)

αw Weak interaction strength at mZ g2/4π ≈ 0.03383 ± 0.00001

αs Strong interaction strength at mZ g2s/4π ≈ 0.1186 ± 0.



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Fundamental laws

Effective laws(“Bylaws”, “initial conditions”)

* Planets are spherical

* Orbits are circular

Initial matter distribution

(Aµ

, Jµ,

ψ, ...)

Kepler,

Newton

Classical

physics

Landscape

+ inflation

* SU(3)xSU(2)xU(1) symmetry

* Dimensionality of space

* Constants (α, ρΛ, ...)

TOE equationsLevel IV

multiverse?

FIG. 1: The shifting boundary (horizontal lines) between fun-damental laws and environmental laws/effective laws/initialconditions. Whereas Ptolemy and others sought to explainroughly spherical planets and circular orbits as fundamentallaws of nature, Kepler and Newton reclassified such propertiesas initial conditions which we now understand as a combina-tion of dynamical mechanisms and selection effects. Classicalphysics removed from the fundamental law category also theinitial conditions for the electromagnetic field and all otherforms of matter and energy (responsible for almost all thecomplexity we observe), leaving the fundamental laws quitesimple. A prospective theory of everything (TOE) incorporat-ing a landscape of solutions populated by inflation reclassifiesimportant aspects of the remaining “laws” as initial condi-tions. Indeed, those laws can differ from one post-inflationary

region to another, and since inflation generically makes eachsuch region enormous, its inhabitants might be fooled intomisinterpreting regularities holding within their particular re-gion as Universal (that is, multiversal) laws. Finally, if theLevel IV multiverse of all mathematical structures [23] exists,then even the “theory of everything” equations that physi-cists are seeking are merely local bylaws in Rees’ terminology[11], that vary across a wider ensemble. Despite such retreatsfrom ab initio explanations of certain phenomena, physics hasprogressed enormously in explanatory power.

illustrated in Figure 1.

There is quite a lot that physicists might once havehoped to derive from fundamental principles, for whichthat hope now seems naive and misguided [24]. Yet itis important to bear in mind that these philosophicalretreats have gone hand in hand with massive progressin predictive power. While Kepler and Newton discred-ited ab initio attempts to explain planetary orbits andshapes with circles and spheres being “perfect shapes”,Kepler enabled precise predictions of planetary positions,and Newton provided a dynamical explanation of the ap-proximate sphericity of planets and stars. While classi-

cal physics removed all initial conditions from its predic-tive purview, its explanatory power inspired awe. Whilequantum mechanics dashed hopes of predicting when aradioactive atom would decay, it provided the founda-tions of chemistry, and it predicts a wealth of surprisingnew phenomena, as we continue to discover.

C. Testing fundamental theories observationally

Let us group the 31 parameters of Table 1 into a 31-dimensional vector p. In a fundamental theory whereinflation populates a landscape of possibilities, some orall of these parameters will vary from place to place asdescribed by a 31-dimensional probability distributionf (p). Testing this theory observationally correspondsto confronting that theoretically predicted distributionwith the values we observe. Selection effects make thischallenging [9, 12]: if any of the parameters that canvary affect the formation of (say) protons, galaxies or ob-servers, then the parameter probability distribution dif-

fers depending on whether it is computed at a randompoint, a random proton, a random galaxy or a randomobserver [12, 14]. A standard application of conditionalprobabilities predicts the observed distribution

f (p) ∝ f prior(p)f selec(p), (1)

where f prior(p) is the theoretically predicted distributionat a random point at the end of inflation and f selec(p) isthe probability of our observation being made at thatpoint. This second factor f selec(p), incorporating theselection effect, is simply proportional to the expectednumber density of reference objects formed (say, protons,galaxies or observers).

Including selection effects when comparing theoryagainst observation is no more optional than the correctuse of logic. Ignoring the second term in equation (1) caneven reverse the verdict as to whether a theory is ruledout or consistent with observation. On the other hand,anthropic arguments that ignore the first term in equa-tion (1) are likewise spurious; it is crucial to know whichof the parameters can vary, how these variations are cor-related, and whether the typical variations are larger orsmaller than constraints arising from the selection effectsin the second term.

D. A case study: cosmology and dark matter

Examples where we can compute both terms in equa-tion (1) are hard to come by. Predictions of fundamen-tal theory for the first term, insofar as they are plau-sibly formulated at present, tend to take the form of functional constraints among the parameters. Famil-iar examples are the constraints among couplings aris-ing from gauge symmetry unification and the constraintθQCD ≈ 0 arising from Peccei-Quinn symmetry. At-tempts to predict the distribution of inflation-related



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cosmological parameters are marred at present by reg-ularization issues related to comparing infinite volumes[31, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Addi-tional difficulties arise from our limited understanding asto what to count as an observer, when we consider varia-tion in parameters that affect the evolution of life, such as(m p, α , β ), which approximately determine all propertiesof chemistry.

In this paper, we will focus on a rare example wherewhere there are no problems of principle in computingboth terms: that of cosmology and dark matter, involv-ing variation in the parameters (ξc, ρΛ, Q) from Table 1,i.e., the dark matter density parameter, the dark energydensity and the seed fluctuation amplitude. Since none of these three parameters affect the evolution of life at thelevel of biochemistry, the only selection effects we need toconsider are astrophysical ones related to the formationof dark matter halos, galaxies and stable solar systems.Moreover, as discussed in the next section, we have spe-cific well-motivated prior distributions for ρΛ and (forthe case of axion dark matter) ξc. Making detailed dark

matter predictions is interesting and timely given the major efforts underway to detect dark matter both directly[53] and indirectly [54] and the prospects of discoveringsupersymmetry and a WIMP dark matter candidate inLarge Hadron Collider operations from 2008.

For simplicity, we do not include any of the cur-rently optional cosmological parameters, i.e., we takens = Ωtot = 1, αn = r = nt = 0, w = −1. Theremaining two non-optional cosmological parameters inTable 1 are the density parameters ξν for neutrinos andξb for baryons. It would be fairly straightforward togeneralize our treatment below to include ξν along thelines of [83, 84], since it too affects f selec only through

astrophysics and not through subtleties related to bio-chemistry. Here, for simplicity, we will ignore it; in anycase, it has been observed to be rather unimportant cos-mologically (ξν ≪ ξc). When computing cosmologicalfluctuation growth, we will also make the simplifying ap-proximation that ξb ≪ ξc (so that ξ ∼ ξc), althoughwe will include ξb as a free parameter when discussinggalaxy formation and solar system constraints. (For verylarge ξb, structure formation can change qualitatively;see [105].) We will see below that ξc/ξb ≫ 1 is not onlyobservationally indicated (Table 1 gives ξc/ξb ∼ 6), butalso emerges as the theoretically most interesting regimeif ξb is considered fixed.

The rest of this paper is organized as follows. In Sec-tion II, we discuss theoretical predictions for the firstterm of equation (1), the prior distribution f prior. In Sec-tion III, we discuss the second term f selec, computing theselection effects corresponding to halo formation, galaxyformation and solar system stability. We combine theseresults and make predictions for dark-matter-related pa-rameters in Section IV, summarizing our conclusions inSection V. A number of technical details are relegatedto Appendix A.

II. PRIORS

In this section, we will discuss the first term in equa-tion (1), specifically how the function f prior(p) dependson the parameters ξc, ρΛ and Q. In the case of ξc, we willconsider two dark matter candidates, axions and WIMPs.

A. Axions

The axion dark matter model offers an elegant examplewhere the prior probability distribution of a parameter(in this case ξc) can be computed analytically.

The strong CP problem is the fact that the dimension-less parameter θqcd in Table 1, which parameterizes apotential CP-violating term in quantum chromodynam-ics (QCD), the theory of the strong interaction, is sosmall. Within the standard model, θqcd is a periodicvariable whose possible values run from 0 to 2π, so itsnatural scale is of order unity. Selection effects are of little help here, since values of θqcd far larger than the

observed bound |θqcd| ∼< 10−9 would seem to have noserious impact on life.

Peccei and Quinn [55] introduced microphysical modelsthat address the strong CP problem. Their models ex-tend the standard model so as to support an appropriate(anomalous, spontaneously broken) symmetry. The sym-metry is called Peccei-Quinn (PQ) symmetry, and the en-ergy scale at which it breaks is called the Peccei-Quinnscale. Weinberg [56] and Wilczek [57] independently re-alized that Peccei-Quinn symmetry implies the existenceof a field whose quanta are extremely light, extremelyfeebly interacting particles, known as axions. Later itwas shown that axions provide an interesting dark mat-

ter candidate [58, 59, 60].Major aspects of axion physics can be understood byreference to a truncated toy model where θqcd is the com-plex phase angle of a complex scalar field Φ that developsa potential of the type

V (Φ) = (|Φ|2 − f 2a)2 + Λ4qcd(1 − f −1a ReΦ), (2)

where Λqcd ∼ 200 MeV is ultimately determined by theparameters in Table 1; roughly speaking, it is the energyscale where the strong coupling constant αs(Λqcd) = 1.

At the Peccei-Quinn (PQ) symmetry breaking scalef a, assumed to be much larger than Λqcd, this complexscalar field Φ feels a Mexican hat potential and seeks tosettle toward a minimum Φ = f aeiθqcd . In the contextof cosmology, this will occur at temperatures not muchbelow f a. Initially the angle θqcd = θ0 is of negligibleenergetic significance, and so it is effectively a randomfield on superhorizon scales. The angular part of thisfield is called the axion field a = f aθqcd. As the cosmicexpansion cools our universe to much lower temperaturesapproaching the QCD scale, the approximate azimuthalsymmetry of the Mexican hat is broken by the emergenceof the second term, a periodic potential 1 − f −1a Re Φ =



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1 − cos θqcd = 2sin2 θqcd (induced by QCD instantons)whose minimum corresponds to no strong CP-violation,i.e., to θqcd = 0. The axion field oscillates around thisminimum like an underdamped harmonic oscillator witha frequency ma corresponding to the second derivative of the potential at the minimum, gradually settling towardsthis minimum as the oscillation amplitude is damped byHubble friction. That oscillating field can be interpreted

as a Bose condensate of axions. It obeys the equation of state of a low-pressure gas, which is to say it provides aform of cold dark matter.

By today, θqcd is expected to have settled to an an-gle within about 10−18 of its minimum [60], comfortablybelow the observational limit |θqcd| ∼< 10−9, and thus dy-namically solving the strong CP problem. (The exactlocation of the minimum is model-dependent, and notquite at zero, but comfortably small in realistic models[61].)

The axion dark matter density per photon in the cur-rent epoch is estimated to be [58, 59, 60]

ξc = ξ∗

sin2 θ0

2, ξ

∗ ∼f 4

a. (3)

If axions constitute the cold dark matter and the Peccei-Quinn phase transition occurred well before the end of inflation, then the measurement ξc ∼ 3 × 10−28 thus im-plies that

f a ∼ 10−7

sinθ02

−1/2∼ 1012 GeV ×

sin

θ02

−1/2,

(4)where θ0 is the the initial misalignment angle of the axionfield in our particular Hubble volume.

Frequently it has been argued that this implies f a ∼1012 GeV, ruling out GUT scale axions with f a ∼10

16

GeV. Indeed, in a conventional cosmology the hori-zon size at the Peccei-Quinn transition corresponds toa small volume of the universe today, and the observeduniverse on cosmological scales would fully sample therandom distribution θ0. However, the alternative possi-bility that |θ0| ≪ 1 over our entire observable universewas pointed out already in [58]. It can occur if an epochof inflation intervened between Peccei-Quinn symmetrybreaking and the present; in that case the observed uni-verse arises from within a single horizon volume at thePeccei-Quinn scale, and thus plausibly lies within a cor-relation volume. Linde [32] argued that if there werean anthropic selection effect against very dense galaxies,then models with f a

≫1012 GeV and

|θ0

| ≪1 might

indeed be perfectly reasonable. Several additional as-pects of this scenario were discussed in [33, 34]. Muchof the remainder of this paper arose as an attempt tobetter ground its astrophysical foundations, but most of our considerations are of much broader application.

We now compute the axion prior f prior(ξc). Since thesymmetry breaking is uncorrelated between causally dis-connected regions, θ0 is for all practical purposes a ran-dom variable that varies with a uniform distribution be-tween widely separated Hubble volumes. Without loss of

generality, we can take the interval over which θ0 variesto be 0 ≤ θ0 ≤ π. This means that the probability of ξbeing lower than some given value ξ0 ≤ ξ∗ is

P (ξ < ξ0) = P (ξ∗ sin2 θ02

< ξ0) = P

θ0 < 2sin−1

ξ1/20

ξ1/2∗

=2

π

sin−1ξ1/20

ξ1/2∗

. (5)

Differentiating this expression with respect to ξ0 gives theprior probability distribution for the dark matter densityξ∗:

f ξ(ξ) =1

π (ξ∗ξ)1/2

1 − ξ

ξ∗

(6)

For the case at hand, we only care about the tail of theprior corresponding to unusually small θ0, i.e., the caseξ ≪ ξ∗, for which the probability distribution reduces tosimply

f ξ(ξ) ∝1

√ξ . (7)

Although this may appear to favor low ξ, the probabilityper logarithmic interval ∝ √

ξ, and it is obvious fromequation (5) that the bulk of the probability lies near thevery high value ξ ∼ ξ∗.

A striking and useful property of equation (7) is that itcontains no free parameters whatsoever. In other words,this axion dark matter model makes an unambiguous pre-diction for the prior distribution of one of our 31 pa-rameters, ξc. Since the axion density is negligible atthe time of inflation, this prior is immune to the infla-tionary measure-related problems discussed in [6], and

no inflation-related effects should correlate ξc with otherobservable parameters. Moreover, this conclusion appliesfor quite general axion scenarios, not merely for our toymodel — the only property of the potential used to derive

the ξ−1/2c -scaling is its parabolic shape near any mini-

mum. Although many theoretical subtleties arise regard-ing the axion dark matter scenario in the contexts of in-flation, supersymmetry and string theory [62, 63, 64, 65],

the ξ−1/2c prior appears as a robust consequence of the

hypothesis f a ≫ 1012 GeV.We conclude this section with a brief discussion of

bounds from axion fluctuations. Like any other masslessfield, the axion field a = f aθ acquires fluctuations of or-

der H during inflation, where H ∼ E 2

inf /mpl = E 2

inf andE inf is the inflationary energy scale, so δθ0 ∼ E 2inf /f a.

For our |θ0| ≪ 1 case, equation (3) gives θ0 ∼ ξ1/2c /f 2a ,so we obtain the axion density fluctuations amplitude

Qa ≡ δξcξc

≈ 2δθ0θ0

∼ E 2inf f a

ξ1/2c

. (8)

Such axion isocurvature fluctuations (see, e.g., [66] fora review) would contribute acoustic peaks in the cos-mic microwave background (CMB) out of phase with



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the those from standard adiabatic fluctuations, allow-ing an observational upper bound Qa ∼< 0.3Q ∼ 10−5

to be placed [68, 69]. Combining this with the ob-served ξc-value from Table 1 gives the bound E 2inf f a ∼Qaξ

1/2c ∼< 10−19, bounding the inflation scale. The tra-

ditional value f a ∼ 1012 GeV ∼ 10−7 gives the famil-iar bound E inf ∼< 10−6 ∼ 1013 GeV [66, 67]. A higherf a gives a tighter limit on the inflation scale: increas-

ing f a to the Planck scale (f a ∼ 1) lowers the bound toE inf ∼< 10−9 ∼ 1010GeV — the constraint grows strongerbecause the denominator in δθ0/θ0 must be smaller toavoid an excessive axion density.

For comparison, inflationary gravitational waves haveamplitude rQ ∼ H ∼ E 2inf , so they are unobservablysmall unless E inf ∼> 1016 GeV. Although various loop-holes to the axion fluctuation bounds have been proposed(see e.g., [66, 67, 70, 71]), it is interesting to note thatthe simplest axion dark matter models therefore makethe falsifiable prediction that future CMB experimentswill detect no gravitational wave signal [67].

B. WIMPs

Another popular dark matter candidate is a stableweakly interacting massive particle (WIMP), thermallyproduced in the early universe and with its relic abun-dance set by a standard freezeout calculation. See, e.g.,[72, 73] for reviews. Stability could, for instance, be en-sured by the WIMP being the lightest supersymmetricparticle.

At relevant (not too high) temperatures the thermallyaveraged WIMP annihilation cross section takes the form[72]

σv = γ α2w

m2wimp

, (9)

where γ is a dimensionless constant of order unity. Inour notation, the WIMP number density is nwimp =ρwimp/mwimp = nγξwimp/mwimp. The WIMP freezout isdetermined by equating the WIMP annihilation rate Γ ≡σvnwimp = σvnγξwimp/mwimp with the radiation-

dominated Hubble expansion rate H = (8πργ/3)1/2.Solving this equation for ξwimp and substituting the ex-pressions for αw, nγ and ργ from Tables 2 and 3 gives

ξwimp =

29

π11

45m

3

wimpζ (3)γg4T

≈ 1522 m

3

wimpγg4T

(10)

The WIMP freezeout temperature is typically found tobe of order T ∼ mwimp/20 [72]. If we further assumethat the WIMP mass is of order the electroweak scale(mwimp ∼ v) and that the annihilation cross section pref-actor γ ∼ 1, then equation (10) gives

ξwimp ∼ 105m2

wimp

g4∼ 105

v2

g4∼ 10−28 (11)

for the measured values of v and g from Table 1. Thiswell-known fact that the predicted WIMP abundanceagrees qualitatively with the measured dark matter den-sity ξc is a key reason for the popularity of WIMP darkmatter.

In contrast to the above-mentioned axion scenario, wehave no compelling prior for the WIMP dark matter den-sity parameter ξwimp. Let us, however, briefly explore the

interesting scenario advocated by, e.g., [74], where thetheory prior determines all relevant standard model pa-rameters except the Higgs vacuum expectation value v,which has a broad prior distribution.3 (It should be said,however, that the connection mwimp ∼ v is somewhat ar-tificial in this context.) It has recently been shown that vis subject to quite strong microphysical selection effectsthat have nothing to do with dark matter, as nicely re-viewed in [75]. As pointed out by [76, 77], changing vup or down from its observed value v0 by a large factorwould correspond to a dramatically less complex universebecause the slight neutron-proton mass difference has aquark mass contribution (md − mu) ∝ v that slightlyexceeds the extra Coulomb repulsion contribution to theproton mass:

1. For v/v0 ∼< 0.5, protons (uud) decay into neutrons(udd) giving a universe with no atoms.

2. For v/v0 ∼> 5, neutrons decay into protons even in-side nuclei, giving a universe with no atoms excepthydrogen.

3. For v/v0 ∼> 103, protons decay into ∆++ (uuu) par-ticles, giving a universe with only Helium-like ∆++

atoms.

Even smaller shifts would qualitatively alter the synthe-sis of heavy elements: For v/v0

∼< 0.8, diprotons and

dineutrons are bound, producing a universe devoid of,e.g., hydrogen. For v/v0 ∼> 2, deuterium is unstable,drastically altering standard stellar nucleosynthesis.

Much stronger selection effects appear to result fromcarbon and oxygen production in stars. Revisiting theissue first identified by Hoyle [78] with numerical nu-clear physics and stellar nucleosynthesis calculations, [79]quantified how changing the strength of the nucleon-nucleon interaction altered the yield of carbon and oxy-gen in various types of stars. Combining their resultswith those of [80] that relate the relevant nuclear physicsparameters to v gives the following striking results:

1. For v/v0∼< 0.99, orders of magnitude less carbon is

produced.

3 If only two microphysical parameters from Table 1 vary by manyorders of magnitude across an ensemble and are anthropicallyselected, one might be tempted to guess that they are ρΛ ∼10−123 and µ2 ∼ −10−33, since they differ most dramaticallyfrom unity. A broad prior for −µ2 would translate into a broadprior for v that could potentially provide an anthropic solutionto the so-called “hierarchy problem” that v ∼ 10−17 ≪ 1.



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2. For v/v0 ∼> 1.01, orders of magnitude less oxygenis produced.

Combining this with equation (11), we see that this couldpotentially translate into a percent level selection effecton ξwimp ∝ v2.

In the above-mentioned scenario where v has a broadprior whereas the other particle physics parameters (inparticular g) do not [74], the fact that microphysical se-lection effects on v are so sharp translates into a narrowprobability distribution for ξwimp via equation (11). Aswe will see, the astrophysical selection effects on the darkmatter density parameter are much less stringent. Weshould emphasize again, however, that this constraintrelies on the assumption of a tight connection betweenξwimp and v, which could be called into question.

C. ρΛ and Q

As discussed in detail in the literature (e.g., [6, 10,44, 88, 107, 108, 109, 110]), there are plausible reasonsto adopt a prior on ρΛ that is essentially constant andindependent of other parameters across the narrow rangewhere |ρΛ| ∼< Q3ξ4 ∼ 10−123 where f selec is non-negligible(see Section III). The conventional wisdom is that sinceρΛ is the difference between two much larger quantities,and ρΛ = 0 has no evident microphysical significance, noultrasharp features appear in the probability distributionfor ρΛ within 10−123 of zero. That argument holds evenif ρΛ varies discretely rather than continuously, so longas it takes ≫ 10123 different values across the ensemble.

In contrast, calculations of the prior distribution for Qfrom inflation are fraught with considerable uncertainty[6, 81, 82]. We therefore avoid making assumptions about

this function in our calculations.



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FIG. 2: Many selection effects that we discuss are convenientlysummarized in the plane tracking the virial temperatures and den-sities in dark matter halos. The gas cooling requirement preventshalos below the heavy black curve from forming galaxies. Close en-counters make stable solar systems unlikely above the downward-sloping line. Other dangers include collapse into black holes anddisruption of galaxies by supernova explosions. The location andshape of the small remaining region (unshaded) is independent of all cosmological parameters except the baryon fraction.

III. SELECTION EFFECTS

We now consider selection effects, by choosing our “se-lection object” to be a stable solar system, and focusingon requirements for creating these. In line with the pre-ceding discussion, our main interest will be to exploreconstraints in the 4-dimensional cosmological parameterspace (ξb, ξc, ρΛ, Q). Since we can only plot one or twodimensions at a time, our discussion will be summarizedby a table (Table 4) and a series of figures showing var-ious 1- and 2-dimensional projections: (ξb, ξc), (ρΛ, Qξ),Q3ξ4, ξ, ρΛ/Q3ξ4, (ξ, Q).

Many of the the physical effects that lead to these con-straints are summarized in Figure 2 and Figure 3, show-ing temperatures and densities of galactic halos. Theconstraints in this plane from galaxy formation and solarsystem stability depend only on the microphysical pa-rameters (mp, α , β ) and sometimes on the baryon frac-tion ξb/ξc, whereas the banana-shaped constraints fromdark matter halo formation depend on the cosmologi-cal parameters (ξb, ξc, ξν , ρΛ, Q), so combining them con-strains certain parameter combinations. Crudely speak-ing, f selec(p) will be non-negligible only if the cosmolog-ical parameters are such that part of the “banana” falls

FIG. 3: Same as the previous figure, but including the banana-shaped contours showing the the halo formation/destruction rate.Solid/greenish contours correspond to positive net rates (haloproduction) at 0.5, 0.3, 0,2, 0.1, 0.03 and 0.01 of maximum,whereas dashed/reddish contours correspond to negative net rates(halo destruction from merging) at -0.3, -0,2, -0.1, -0.03 and -0.01 of maximum, respectively. The heavy black contour corre-sponds to zero net formation rate. The cosmological parameters(ξb, ξc, ξν , , Q , ρΛ) have a strong effect on this banana: ξb and ξcshift this banana-shape vertically, Q shifts if along the parallel di-agonal lines and ρΛ cuts it off below 16ρΛ (see the next figure).Roughly speaking, there are stable habitable planetary systemsonly for cosmological parameters where a greenish part of the ba-

nana falls within the allowed white region from Figure 2.

within the “observer-friendly” (unshaded) region sand-wiched between the galaxy formation and solar systemstability constraints

In the following three subsections, we will now dis-cuss the three above-mentioned levels of structure for-mation in turn: halo formation, galaxy formation andsolar-system stability.

A. Halo formation and the distribution of halo

properties

Previous studies (e.g., [6, 10, 44, 88, 107, 108, 109,110]) have computed the total mass fraction collapsedinto halos, as a function of cosmological parameters.Here, however, we wish to apply selection effects based onhalo properties such as density and temperature, and willcompute the formation rate of halos (the banana-shapedfunction illustrated in Figure 3) in terms of dimensionlessparameters alone.



11

Q

f b

ξ

ρΛ

FIG. 4: Same as Figure 3, but showing effect of changing thecosmological parameters. Increasing the matter density parameterξ shifts the banana up by a factor ξ4 (top left panel), increasingthe fluctuation level Q shifts the banana up by a factor Q3 andto the right by a factor Q (top right panel), increasing the baryonfraction f b = ξb/ξ shifts the banana up by a factor f b and affectsthe cooling and 2nd generation constraints (bottom left panel), andincreasing the dark energy density ρΛ bites off the banana belowρvir = 16ρΛ (bottom right panel). In these these four panels, theparameters have been scaled relative to their observed values asfollows: up by 1/4, 2/4,...,9/4 orders of magnitude (ξ), up by 1/3,2/3,...,9/3 orders of magnitude (Q), down by 1 and 2 orders of magnitude (f b), and down by 0, 7, 12 and ∞ orders of magnitude

(ρΛ; other panels have ρΛ = 0).

1. The dependence on mass and time

As previously explained, we will focus on the dark-matter-dominated case ξc ≫ ξb, ξc ≫ ξν , so we ignoremassive neutrinos and have ξ = ξb + ξc ∼ ξc. For our cal-culations, it is convenient to define a new dimensionlesstime variable

x ≡ ρΛρm

=ΩΛ

Ωm= Ω−1

m − 1 (12)

and a new dimensionless mass variable

µ ≡ ξ2M. (13)

In terms of the usual cosmological scale factor a, our newtime variable therefore scales as x ∝ a3. It equals unityat the vacuum domination epoch when linear fluctuationgrowth grinds to a halt. The horizon mass at matter-radiation equality is of order ξ−2 [85], so µ can be in-terpreted as the mass relative to this scale. It is a keyphysical scale in our problem. It marks the well-known

break in the matter power spectrum; fluctuation modeson smaller scales entered the horizon during radiationdomination, when they could not grow.

We estimate the fraction of matter collapsed into darkmatter halos of mass M ≥ ξ−2µ by time x using thestandard Press-Schechter formalism [86], which gives

F (µ, x) = erfc δc(x)

√2σ(µ, x) . (14)

Here σ(µ, x) is the r.m.s. fluctuation amplitude at timex in a sphere containing mass M = ξ−2µ, so F is theprobability that a fluctuation lies δc standard deviationsout in the tail of a Gaussian distribution. As shown inAppendix A 2, σ is well approximated as4

σ(µ, x) ≈ σ∗Qξ4/3

ρ1/3Λ

s(µ)GΛ(x), (15)

where

σ∗ ≡ 45 · 21/3ζ (3)4/3

π14/3

1 +

21

8 4

114/3 ≈ 0.206271 (16)

and the known dimensionless functions s(µ) and GΛ(x)do not depend on any physical parameters. s(µ) andGΛ(x) give the dependence on scale and time, respec-tively, and appear in equations (A13) and (A1) in Ap-pendix A. The scale dependence is s(µ) ∼ µ−1/3 on largescales µ ≫ 1, saturating to only logarithmic growthtowards small scales for µ ≪ 1. Fluctuations growas GΛ(x) ≈ x1/3 ∝ a for x ≪ 1 and then asymp-tote to a constant amplitude corresponding to G∞ ≡5Γ

23

Γ56

/3

√π ≈ 1.43728 as a → ∞ and dark energy

dominates. (This is all for ρΛ > 0; we will treat ρΛ < 0in Section IIIA3 and find that our results are roughly

independent of the sign of ρΛ, so that we can sensiblyreplace ρΛ by |ρΛ|.)

Returning to equation (14), the collapse densitythreshold δc(x) is defined as the linear perturbationtheory overdensity that a top-hat-averaged fluctuationwould have had at the time x when it collapses. It wascomputed numerically in [83], and found to vary onlyvery weakly (by about 3%) with time, dropping from thefamiliar cold dark matter value δc(0) = (3/20)(12π)2/3 ≈1.68647 early on to the limit δc(∞) = (9/5)2−2/3G∞ ≈1.62978 [88] in the infinite future. Here we simply ap-proximate it by the latter value:

δc(x) ≈ δc(∞) = 9Γ23

Γ56

3π1/222/3 ≈ 1.62978. (17)

4 Although the baryon density affects σ mainly via the sum ξ =ξb + ξc, there is a slight correction because fluctuations in thebaryon component do not grow between matter-radiation equal-ity and the drag epoch shortly after recombination [ 87]. Sincethe resulting correction to the fluctuation growth factor (∼ 15%for the observed baryon fraction ξb/ξc ∼ 1/6) is negligible forour purposes, we ignore it here.



12

Substituting equations (15) and (17) into equation (14),we thus obtain the collapsed fraction

F (µ, x) = erfc

Aρ

1/3Λ

ξ4/3QGΛ(x)s(µ)

, (18)

where

A ≡ δc(∞)√2σ∗

=

1 + 42

224/3

π 25

6 Γ23

Γ56

30

√2ζ (3)4/3

≈ 5.58694.

(19)Below we will occasionally find it useful to rewrite equa-tion (18) as

F (µ, x) = erfc

A(ρΛ/ρ∗)1/3

GΛ(x)s(µ)

, ρ∗ ≡ ξ4Q3. (20)

Let us build some intuition for equation (18). It tells usthat early on when x ∼ 0, no halos have formed (F ≈ 0),and that as time passes and x increases, small halos form

before any large ones since s(µ) is a decreasing function.Moreover, we see that since GΛ(x) and s(µ) are at mostof order unity, no halos will ever form if ρΛ ≫ ξ4Q3. Werecognize the combination ρ∗ ≡ ξ4Q3 as the character-istic density of the universe when halos would form inthe absence of dark energy [85]. If ρΛ ≫ ρ∗, then darkenergy dominates long before this epoch and fluctuationsnever go nonlinear. Figure 5 illustrates that the densitydistributions corresponding to equation (20) are broadlypeaked around ρvir ∼ 102ρ∗ when ρ∗ ≫ ρΛ, are exponen-tially suppressed for ρ∗ ≪ ρΛ, and in all cases give nohalos with ρvir < 16ρΛ.

2. The dependence on temperature and density

We now discuss how our halo mass µ and formationtime x transform into the astrophysically relevant pa-rameters (T vir, nvir) appearing in Figure 3.

As shown in Appendix A 3, halos that virialize at timex have a characteristic density of order

ρvir ∼ 16ρΛ

9π2

8x

107200

+ 1

200107

, (21)

i.e., essentially the larger of the two terms 16ρΛ and18π2ρm(x). For a halo of total (baryonic and darkmatter) mass M in Planck units, this corresponds toa characteristic size R ∼ (M/ρvir)

1/3, velocity vvir ∼(M/R)1/2 ∼ (M 2ρvir)1/6 and virial temperature T vir ∼m pv2vir ∼ mpM 2/3ρ1/3vir , so

T vir ∼ m p

16ρΛµ2

ξ4

1/3

9π2

8x

107200

+ 1

200321

. (22)

FIG. 5: Halo density distribution for various values of ρ∗/ρΛ,where ρ∗ ≡ [s(µ)/A]3ρ∗ ∼ Q3ξ4. The dashed curves show the cu-mulative distribution F = erfc

(ρΛ/ρ∗)1/3/GΛ(x[ρvir])

and the

solid curves show the probability distribution −∂F/∂ lg ρvir for(peaking from right to left) ρ∗/ρΛ = 105, 104, 103, 102, 10, 1(heavy curves), 0.1 and 0.05. Note that no halos with ρvir < 16ρΛare formed.

Inverting equations (21) and (22) gives

µ ∼ ξ4T 3virm3 pρvir , (23)

x ∼ 9π2

8

ρvir

16ρΛ

107200

− 1

− 200107

. (24)

For our applications, the initial gas temperature will benegligible and the gas density will trace the dark mat-ter density, so until cooling becomes important (Sec-tion IIIB1), the proton number density is simply

nvir =ξb

ξmpρvir. (25)

According to the Press-Schechter approximation, thederivative − ∂F

∂ lgµ (µ, x) can be interpreted as the so-called

mass function , i.e., as the distribution of halo masses attime x. We can therefore interpret the second derivative

f µx(µ, x) ≡ − ∂ 2F

∂ (lg µ)∂ (lg x)(26)

as the net formation rate of halos as a function of massand time. Transforming this rate from (lg µ, lg x)-space



13

to (lg T vir, lg ρvir)-space, we obtain the function whosebanana-shaped contours are shown in Figure 3:

f (T vir, nvir) ≡ |J |f µx(µ, x) = |J | ∂ 2F

∂ (lg µ)∂ (lg x), (27)

where J is a Jacobian determinant that can be ignoredsafely.5

Let us now build some intuition for this importantfunction f (T vir, nvir). First of all, since equation (21)shows that ρvir ∝ nvir decreases with time and is µ-independent, we can reinterpret the vertical nvir-axis inFigure 3 as simply the time axis: as our Universe ex-pands, halos can form at lower densities further and fur-ther down in the plot. Since nothing ever forms withρvir < 16ρΛ, the horizontal line nvir = 16ρΛξb/ξmp cor-responds to t → ∞. Second, f is the net formation rate,which means that it is negative if the rate of formationof new halos of this mass is smaller than the rate of de-struction from merging into larger halos. The destructionstems from the fact that, to avoid double-counting, thePress-Schechter approximation counts a given proton asbelonging to at most one halo at a given time, definedas the largest nonlinear structure that it is part of. 6

Figure 3 shows that halos of any given mass (defined bythe lines of slope 3) are typically destroyed in this fash-ion some time after its formation unless ρΛ-dominationterminates the process of fluctuation growth.

We will work out the detailed dependence of this ba-nana on physical parameters in the next section. Fornow, we merely note that Figure 3 shows that the firsthalos form with characteristic density ρvir ∼ ρ∗ ∼ ξ4Q3,nvir ∼ ξbξ3Q3/mp, and (assuming ρΛ ≪ ρ∗) the largest

halos approach virial velocities vvir ∼ Q1/2c and temper-atures T vir

∼Q

×mpc2 [85].

5 It makes sense to treat f µx as a distribution, since f µxd(ln µ) d(ln x) equals the total collapsed fraction. When

transforming it, we therefore factor in the Jacobian of the trans-formation from (lg µ, lg x) to (lg T vir, lg ρvir),

J ≡

∂ lgµ∂ lgT

∂ lgµ∂ lg ρ

∂ lgx∂ lgT

∂ lg x∂ lg ρ

,

=

32

− 12

0 −

1 −

16ρΛρvir

159200

−1 , (28)

with determinant

J ≡ |J| = −3

21 −

16ρΛ

ρvir

159200

−1

= −3

2

ρvirx

18π2

ρΛ

159200

. (29)

So the Jacobian is an irrelevant constant J = −3/2 for ρvir ≫16ρΛ. Since the entire function f vanishes for ρvir < 16ρΛ,the Jacobian only matters near the ρvir = 16ρΛ boundary,where it has a rather unimportant effect (the divergence is inte-grable). Equation (28) shows that away from that boundary, the(lg T vir, lg ρvir)-banana is simply a linear transformation of the(lg µ, lg x)-banana ∂ 2F/∂ lg µ∂ lg x, with slanting parallel lines of slope 3 in Figure 3 corresponding to constant µ-values.6 Our treatment could be improved by modeling halo substructure

survival, since subhalos that harbor stable solar systems may becounted as part of an (apparently inhospitable) larger halo.

3. How mp, ξb, ξc, Q and ρΛ affect “the banana”

Substituting the preceding equations into equa-tion (27) gives the explicit expression

f (T vir, nvir; mp, ξb, ξc, Q , ρΛ) =

=3a[GΛ(x)

2s(µ)2−2a2]xGΛ′(x)µs′(µ)

√πGΛ(x)4s(µ)4 exp a

GΛ(x)s(µ)2

ρvirx18π2ρΛ

159200

,

a ≡ Aρ1/3Λ

Qξ4/3, (30)

where µ and x are determined by T vir and nvir via equa-tions (23), (24) and (25). This is not very illuminating.We will now see how this complicated-looking function f of seven variables can be well approximated and under-stood as a fixed banana-shaped function of merely twovariables, which gets translated around by variation of mp, ξb, ξc and Q, and truncated from below at a valuedetermined by ρΛ.

Let us first consider the limit ρΛ → 0 where dark en-ergy is negligible. Then x

≪1 so that GΛ(x) = x1/3 =

(ρΛ/ρm)1/3, causing equation (18) to simplify to

F (µ, x) = erfc

Aρ

1/3m

ξ4/3Qs(µ)

. (31)

In the same limit, equation (21) reduces to ρvir =18π2ρm, so using equation (25) gives ρm = ρvir/18π2 =ξmpnvir/18π2ξb. Using this and equation (23), we canrewrite equation (31) in the form

F = erfc

Ampnvirξbξ3Q3

1/3

(18π2

)

1/3

s T vir

mpQ3/2 mpnvir

ξbξ3Q3−1/2

= B

mpnvir

ξbξ3Q3,

T virmpQ

, (32)

where the “standard banana” is characterized by a func-tion of only two variables,

B(X, Y ) = erfc

AX 1/3

(18π2)1/3 s

Y 3/2X −1/2

. (33)

Thus B(X, Y ) determines the banana shape, and the pa-rameters mp, ξb, ξ and Q merely shift it on a log-logplot: increasing mp shifts it down and to the right alonglines of slope −1, increasing ξb shifts it upward, increas-ing ξ shifts it upward three times faster, and increasingQ shifts it upward and to the right along lines of slope 3.The halo formation rate defined by its derivatives (equa-tion 27) and plotted in Figure 3 clearly scales in exactlythe same way with these four parameters, since in theρΛ → 0 limit that we are considering, the Jacobian J issimply a constant matrix.

Let us now turn to the general case of arbitrary ρΛ. Asas discussed above, time runs downward in Figure 3 since



14

the cosmic expansion gradually dilutes the matter den-sity ρm. The matter density completely dwarfs the darkenergy density at very early times. They key point is thatsince ρΛ has no effect until our Universe has expandedenough for the matter density to drop near the dark en-ergy density, the part of the banana in Figure 3 thatlies well above the vacuum density ρΛ will be completelyindependent of ρΛ. As the matter density ρm drops be-

low ρΛ (i.e., as x grows past unity), fluctuation growthgradually stops. This translates into a firm cutoff belowρvir = 16ρΛ in Figure 3 (i.e., below nvir = 16ξbρΛ/mpξ),since equation (21) shows that no halos ever form withdensities below that value.

Viewed at sensible resolution on our logarithmic plot,spanning many orders of magnitude in density, the transi-tion from weakly perturbing the banana to biting it off isquite abrupt, occurring as the density changes by a factorof a few. For the purposes of this paper, it is appropriateto approximate the effect of ρΛ as simply truncating thebanana below the cutoff density.

Putting it together, we can approximate the non-intuitive equation (30) by the much simpler

f (T vir, nvir; mp, ξb, ξc, Q , ρΛ) ≈≈ b

T virmpQ

, mpnvirξbξ3Q3

θ

nvir − 16ξbρΛmpξ

, (34)

where θ is the Heaviside step function (θ(x) = 0 for x < 0,θ(x) = 1 for x ≥ 0) and b is the differentiated “standardbanana” function

b(X, Y ) ≡ ∂ 2

lg X lg Y B(X, Y ). (35)

This is useful for understanding constraints on the cos-mological parameters: the seemingly complicated depen-dence of the halo distribution on seven variables fromequation (30) can be intuitively understood as the stan-dard banana shape from Figure 3 being rigidly translatedby the four parameters mp, ξb, ξ and Q and truncatedfrom below with a cutoff nvir > 16ξbρΛ/mpξ. All this isillustrated in Figure 4, which also confirms numericallythat the approximation of equation (35) is quite accurate.

4. The case ρΛ < 0

Above we assumed that ρΛ ≥ 0, but for the purposesof this paper, we can obtain a useful approximate gen-eralization of the results by simply replacing ρΛ by |ρΛ|in equation (34). This is because ρΛ has a negligible ef-fect early on when |ρΛ| ≪ ρm and a strongly detrimentalselection effect when |ρΛ| ∼ ρm, either by suppressinggalaxy formation (for ρΛ > 0) or by recollapsing our uni-verse (for ρΛ < 0), in either case causing a rather sharplower cutoff of the banana. As pointed out by Wein-berg [88], one preferentially expects ρΛ > 0 as observedbecause the constraints tend to be slightly stronger fornegative ρΛ. If ρΛ > 0, observers have time to evolve

long after |ρΛ| = ρm as long as galaxies had time toform before while ρΛ was still subdominant. If ρΛ < 0,however, both galaxy formation and observer evolutionmust be completed before ρΛ dominates and recollapsesthe universe; thus, for example, increasing Q so as tomake structure form earlier will not significantly improveprospects for observers.

B. Galaxy formation

1. Cooling and disk formation

Above we derived the time-dependent (or equivalently,density-dependent) fraction of matter collapsed into ha-los above a given mass (or temperature), derived fromthis the formation rate of halos of a given temperatureand density, and discussed how both functions dependedon cosmological parameters. For the gas in such a halo tobe able to contract and form a galaxy, it must be able to

dissipate energy by cooling [96, 97, 98, 99] — see [100] fora recent review. In the shaded region marked “No cool-ing” in Figure 2, the cooling timescale T /T exceeds theHubble timescale H −1.7 We have computed this familiarcooling curve as in [85] with updated molecular coolingfrom Tom Abel’s code based on [101], which is availableat http://www.tomabel.com/PGas/. From left to right,the processes dominating the cooling curve are molec-ular hydrogen cooling (for T ∼< 104K), hydrogen linecooling (1st trough), helium line cooling (2nd trough),Bremsstrahlung from free electrons (for T ∼> 105K) andCompton cooling against cosmic microwave backgroundphotons (horizontal line for T ∼> 109K). The curve cor-responds to zero metallicity, since we are interested in

whether the first galaxies can form.The cooling physics of course depends only on atomic

7 This criterion is closely linked to the question of whether ob-servers can form if one is prepared to wait an arbitrarily longtime. First of all, if the halo fails to contract substantially in aHubble time, it is likely to lose its identity by being merged into alarger halo on that timescale unless x ≫ 1 so that ρΛ has frozenclustering growth. Second, if T vir ∼

< 104K so that the gas islargely neutral, then the cooling timescale will typically be muchlonger than the timescale on which baryons evaporate from thehalo, and once less than 0.08M ⊙ of baryons remain, star forma-tion is impossible. Specifically, for a typical halo profile, about1% of the baryons in the high tail of the Bolzmann velocity dis-tribution exceed the halo escape velocity, and the halo thereforeloses this fraction of its mass each relaxation time (when thishigh tail is repopulated by collisions between baryons). In con-trast, the cooling timescale is linked to how often such collisionslead to photon emission. This question deserves more work toclarify whether all T vir≫ 104K halos would cool and form starseventually (providing their protons do not have time to decay).However, as we will see below in Section IV, this question isunimportant for the present paper’s prime focus on axion darkmatter, since once we marginalize over ρΛ, it is rather the upper

limit on density in Figure 3 that affects our result.

http://www.tomabel.com/PGas/

http://www.tomabel.com/PGas/



15

processes, i.e., on the three parameters (α, β, mp). Re-quiring the cooling timescale to not exceed some fixedtimescale would therefore give a curve independent of allcosmological parameters. Since we are instead requir-ing the cooling timescale (∝ n−1 for all processes involv-ing particle-particle collisions) not to exceed the Hubble

timescale (∝ ρ−1/2m ∝ (n/f b)−1/2), our cooling curve will

depend also on the baryon fraction f b, with all parts ex-

cept the Compton piece to the right scaling vertically asf −1b .

2. Disk fragmentation and star formation

We have now discussed how fundamental parametersdetermine whether dark matter halos form and whethergas in such halos can cool efficiently. If both of theseconditions are met, the gas will radiate away kinetic en-ergy and settle into a rotationally supported disk, andthe next question becomes whether this disk is stable orwill fragment and form stars. As described below, this

depends strongly on the baryon fraction f b ≡ ξb/ξ, ob-served to be f b ∼ 1/6 in our universe. The constraintsfrom this requirement propagate directly into Figure 12in Section IV D rather than Figure 3.

First of all, stars need to have a mass of at least M min ≈0.08M ⊙ for their core to be hot enough to allow fusion.Requiring the formation of at least one star in a halo of mass M therefore gives the constraint

f b >M min

M . (36)

An interesting point [111] is that this constraint placesan upper bound on the dark matter density parame-

ter. Since the horizon mass at equality is M eq ∼ ξ−2[85], the baryon mass within the horizon at equality isM eqξb/ξ ∼ ξb/ξ3, which drops with increasing ξ. Thisscale corresponds to the bend in the banana of Figure 3,so unless [111]

f b ∼> M minξ2, (37)

one needs to wait until long after the first wave of haloformation to form the first halo containing enough gasto make a star, which in turn requires a correspondinglysmall ρΛ-value. The ultimate conservative limit is re-quiring that M hb ∼ ρbH −3, the baryon mass within thehorizon, exceeds M min. Since M hb increases during mat-

ter domination and decreases during vacuum domination,taking its maximum value M hb ∼ f bρ

−1/2Λ when x ∼ 1,

the M hb > M min requirement gives

f b ∼> M minρ1/2Λ . (38)

However, these upper limits on the dark matter param-eter density parameter ξc are very weak: for the ob-served values of ξb and ρΛ from Table 1, equations (37)and (38) give ξc < ξ ∼< (ξb/M min)1/3 ∼ 10−22 and

ξc < ξ ∼< ξb/M minρ1/2Λ ∼ 10−4, respectively, limits many

orders of magnitude above the observed value ξc ∼ 10−28.It seems likely, however, that M min is a gross underes-

timate of the baryon requirement for star formation. Thefragmentation instability condition for a baryonic disc isessentially that it should be self-gravitating in the “ver-tical” direction. This is equivalent to requiring the bary-onic density in the disc exceed that of the dark matter

background it is immersed in. The first unstable mode isthen the one that induces breakup into spheres of radiusof order the disc thickness. This conservative criterion isweaker than the classic Toomre instability criterion [102],which requires the disc to be self-gravitating in the radial

direction and leads to spiral arm formation. If the halowere a singular isothermal sphere, then [103] instabilitywould require

f b ∼> λ

T min

T vir

1/2

, (39)

Here T min is the minimum temperature that the gascan cool to and λ is the dimensionless specific angu-

lar momentum parameter, which has an approximatelylognormal distribution centered around 0.08 [103]. Foran upper limit on the dark matter parameter ξc, whatmatters is thus the upper limit on λ, the upper limiton T vir and the lower limit on T min, all three of whichare quite firm. Ignoring probability distributions, takingT min ∼ α2βmp/6 ln α−1 ∼ 104K ∼ 1 eV (atomic Hydro-gen line cooling) and T vir ∼ 500eV (Milky Way) givesT vir/T min ∼ 500, ξc ∼< 300ξb from equation (39). Tak-ing the extreme values T min ∼ 500K (H 2-cooling freeze-out) and T vir ∼ mpQ ∼ 20 keV (largest clusters) gives

T vir/T min ∼ 1010Q and ξ ∼< 106Q1/2ξb ∼ 104ξb. On theother extreme, if we argue that T vir

∼T min for the very

first galaxies to form, then equation (39) gives ξ/ξb ∼< 12,which is interestingly close to what we observe.

For NFW potentials [104], λ-dependence is more com-plicated and the local velocity dispersion of the dark mat-ter near the center is lower than the mean T vir.

Even if the baryon fraction were below the threshold of equation (39), stars could eventually form because viscos-ity (even just “molecular” viscosity) would redistributemass and angular momentum so that the gas becomesmore centrally condensed. The condition then becomesthat the mass of spherical blob of radius R ∼ M d/T min

exceed the dark mass M d within that radius. For theisothermal sphere, this would automatically happen, but

for a realistic flat-bottomed or NFW potential, then thisgives a non-trivial inequality. For instance, in a parabolicpotential well, the requirement would be that

f b ∼>

T min

T vir

3/2

. (40)

Although a weaker limit limit than equation (39), givingξc ∼< 104ξb for the above example with T vir/T min ∼ 500,it is still stronger than those of equation (37) and equa-tion (38). A detailed treatment of these issues is beyond



16

the scope of the present paper; it should include mod-eling of the t → ∞ limit as well as merger-induced starformation and the effect of dark matter substructure dis-turbing and thickening the disk.

C. Second generation star formation

Suppose that all the above conditions have been metso that a halo has formed where gas has cooled andproduced at least one star. The next question becomeswhether the heavy elements produced by the death of thefirst star(s) can be recycled into a solar system around asecond generation star, thereby allowing planets and per-haps observers made of elements other than hydrogen,helium and the trace amounts of deuterium and lithiumleft over from big bang nucleosynthesis.

The first supernova explosion in the halo will releasenot only heavy metals, but also heat energy of order

E = 10−3m−2p ∼ 1046J = 1051erg. (41)

Here m−2p is the approximate binding energy of a Chan-

drashekar mass at its Schwarzschild radius, with the pref-actor incorporating the fact that neutron stars are usu-ally somewhat larger and heavier and, most importantly,that about 99% of the binding energy is lost in the formof neutrinos.

By the virial theorem, the gravitational binding energyE of the halo equals twice its total kinetic energy, i.e.,

E ≈ M v2vir ∼

T 5virm5

pρvir, (42)

where we have used equation (23) in the last step. If

E sn ≫ E , the very first supernova explosion will thereforeexpel essentially all the gas from the halo, precluding theformation of second generation stars. Combining equa-tions (41) and (42), we therefore obtain the constraint

ρvir ∼< 106T 5virmp

. (43)

Figure 2 illustrates this fact that lines of constant bindingenergy have slope 5, and shows that the second genera-tion constraint rules out an interesting part of the the(nvir, T vir) plane that is allowed by both cooling and dis-ruption constraints.

While we have ignored the important effect of coolingby gas in the supernova’s immediate environment, thisconstraint is probably nonetheless rather conservative,and a more detailed calculation may well move it furtherto the right. First of all, many supernovae tend to go off in close succession in a star formation site, thereby jointlyreleasing more energy than indicated by equation (42).Second, it is likely that many supernovae are required toproduce sufficiently high metallicity. Since ∼ 1 supernovaforms per 100M ⊙ of star formation, releasing ∼ 1M ⊙

of metals, raising the mean metallicity in the halo tosolar levels (∼ 10−2) would require an energy input of order E/(100m⋆/mp) ∼ 10−5mp ∼ 10 keV per proton.A careful calculation of the corresponding temperaturewould need to model the gas cooling occurring betweenthe successive supernova explosions.

D. Encounters and extinctions

The effect of halo density on solar system destructionwas discussed in [85] making the crude assumption thatall halos of a given density had the same characteristicvelocity dispersion. Let us now review this issue, whichwill play a key role in determining predictions for theaxion density, from the slightly more refined perspectiveof Figure 2. Consider a habitable planet orbiting a star of mass M suffering a close encounter with another star of mass M †, approaching with a relative velocity v† and animpact parameter b. There is some “kill” cross sectionσ†(M, M †, v†) = πb2 corresponding to encounters close

enough to make this planet uninhabitable. There areseveral mechanisms through which this could happen:

1. It could become gravitationally unbound from itsparent star, thereby losing its key heat source.

2. It could be kicked into a lethally eccentric orbit.

3. The passing star could cause disastrous heating.

4. The passing star could perturb an Oort cloud in theouter parts of the solar system, triggering a lethalcomet impact.

The probability of the planet remaining unscathed for atime t is then e−γ†t, where the destruction rate is γ † is

n⋆σ†v† appropriately averaged over incident velocities v†and stellar masses M and M †. Assuming that n⋆ ∝ nvir,

v† ∝ vvir ∝ T 1/2vir and σ† is independent of nvir and T vir,contours of constant destruction rate in Figure 2 are thuslines of slope −1/2. The question of which such contour isappropriate for our present discussion is highly uncertain,and deserving of future work that would lie beyond thescope of the present paper. Below we explore only acouple of crude estimates, based on direct and indirectimpacts, respectively.

1. Direct encounters

Lightman [112] has shown that if the planetary surfacetemperature is to be compatible with life as we know it,the orbit around the central star should be fairly circularand have a radius of order

rau ∼ α−5m−3/2p β −2 ∼ 1011m, (44)

roughly our terrestrial “astronomical unit”, precessingone radian in its orbit on a timescale

torb ∼ α−15/2m−5/4p β −3 ∼ 0.1 year. (45)



17

An encounter with another star with impact parameterr ∼< rau has the potential to throw the planet into a highlyeccentric orbit or unbind it from its parent star.8 For n⋆,what matters here is not the typical stellar density in ahalo, but the stellar density near other stars, includingthe baryon density enhancement due to disk formationand subsequent fragmentation. Let us write

n⋆ = f ⋆nvirN ⋆

, (47)

where N ⋆ ∼ m−3p ∼ 1057 is the number of protons

in a star and the dimensionless factor f ⋆ parametrizesour uncertainty about the extent to which stars concen-trate near other stars. Substituting characteristic valuesnvir ∼ 103/m3 for the Milky Way and n⋆ ∼ (1pc)−3

for the solar neighborhood gives a concentration factorf ⋆ ∼ 105. Using this value, v† ∼ vvir and σ† = πr2auexcludes the dark shaded region to the upper right inFigure 2 if we require γ −1† to exceed tmin ∼ 109 years

∼ α2α−3/2g β −2, the lifetime of a bright star [2]. It is of

course far from clear what is an appropriate evolutionaryor geological timescale to use here, and there are manyother uncertainties as well. It is probably an overesti-mate to take v† ∼ vvir since we only care about relativevelocities. On the other hand, our value of σ† is an under-estimate since we have neglected gravitational focusing.The value we used for f ⋆ is arguably an underestimate aswell, stellar densities being substantially higher in giantmolecular clouds at the star formation epoch.

2. Indirect encounters

In the above-mentioned encounter scenarios 1-3, theincident directly damages the habitability of the planet.In scenario 4, the effect is only indirect, sending a hailof comets towards the inner solar system which may ata later time impact the planet. This has been argued toplace potentially stronger upper limits on Q than directencounters [81].

Although violent impacts are commonplace in our par-ticular solar system, large uncertainties remain in thestatistical details thereof and in the effect that chang-ing n⋆ and v† would have. It is widely believed that

8 Encounters have a negligible effect on our orbit if they are adi-abatic, i.e., if the impact duration r/v ≫ torb so that the solarsystem returned to its unperturbed state once the encounter wasover. Encounters are adiabatic for

v ∼< r

ctorb∼

α5/2β

m1/4p

∼ 0.0001 ∼ 30km/s, (46)

the typical orbital speed of a terrestrial planet. As long as theimpact parameter ∼

< rau, however, the encounter is guaranteedto be non-adiabatic and hence dangerous, since the infalling starwill be gravitationally accelerated to at least this speed.

solar systems are surrounded by a rather spherical cloudof comets composed of ejected leftovers. Our own par-ticular Oort cloud is estimated to contain of order 1012

comets, extending out to about a lightyear (∼ 105AU)from the Sun. Recent estimates suggest that the impactrate of Oort cloud comets exceeding 1 km in diameteris between 5 and 700 per million years [113]. These im-pacts are triggered by gravitational perturbations to the

Oort cloud sending a small fraction of the comets intothe inner solar system. About 90% of these perturba-tions are estimated to be caused by Galactic tidal forces(mainly related to the motion of the Sun with respect tothe Galactic midplane), with random passing stars be-ing responsible for most of the remainder and randompassing molecular clouds playing a relatively minor role[113, 114, 115].

It is well-known that Earth has suffered numerous vi-olent impacts with celestial bodies in the the past, andthe 1994 impact of comet Shoemaker-Levy on Jupiterillustrated the effect of comet impacts. Although thenearly 10 km wide asteroid that hit the Yucatan 65 mil-lion years ago [116] may actually have helped our ownevolution by eliminating dinosaurs, a larger impact of a30 km object 3.47 billion years ago [117] may have causeda global tsunami and massive heating, killing essentiallyall life on Earth (For comparison, the Shoemaker-Levyfragments were less than 2 km in size.) We thereforecannot dismiss out of hand the possibility that we arein fact close to the edge in parameter space, with only amodest increase in comet impact rates on planets causinga significant drop in the fraction of planets evolving ob-servers. This possibility is indicated by the light-shadedexcluded region in the upper right of Figure 2.

However, there are large uncertainties here of twotypes. First, we lack accurate risk statistics for our ownparticular solar system. Although the lunar crater radiusdistribution is roughly power law of slope −2 at high end,we still have very limited knowledge of the size distribu-tion of comet nuclei [118] and hence cannot accurately es-timate the frequency of extremely massive impacts. Sec-ond, we lack accurate estimates of what would happen indenser galaxies. There, the more frequent close encoun-ters with other stars could rapidly strip stars of muchof their dangerous Oort cloud, so it is far from obviousthat the risk rises as n⋆v†. One interesting possibility isthat the inner Oort cloud at radii ∼< 1000AU contributesa substantial fraction of our impact risk, in which casesuch tidal stripping of most of the cloud by volume willdo little to reduce risk.

An indirect hint that comet impacts are anthropicallyimportant may be the observation [119] that the orbit of our Sun through the galaxy appears fine-tuned to min-imize Oort cloud perturbations and resulting comet im-pacts: compared to similar stars, its orbit has an un-usually low eccentricity and small amplitude of verticalmotion relative to the Galactic disk.



18

3. Nearby explosions

A final category of risk that deserves further explo-ration is that from nearby supernova explosions andgamma-ray bursts. Since these are independent of stel-lar motion and thus depend only on n⋆, not on T vir, theywould correspond to a horizontal upper cutoff in Figure 2.For example, the Ordovician extinction 440 million years

ago has been blamed on a nearby gamma-ray burst. Thiswould have depleted the ozone layer causing a massive in-crease in ultraviolet solar radiation exposure and couldalso have triggered an ice age [120].

E. Black hole formation

There are two potentially rather extreme selection ef-fects involving black holes.

First, Figure 2 illustrates a vertical constraint on theright side corresponding to large T vir-values. Since theright edge of the halo banana is at vvir

∼Q1/2c, typical

halos will form black holes for Q-values of order unity asdiscussed in [85]. Specifically, typical fluctuations wouldbe of black-hole magnitude already by the time they en-tered the horizon, converting some substantial fractionof the radiation energy into black holes shortly after theend of inflation and continually increasing this fractionas longer wavelength fluctuations entered and collapsed.For a scale-invariant spectrum, extremely rare fluctua-tions that are Q−1 standard deviations out in the Gaus-sian tail can cause black hole domination if [85]

Q ∼> 10−1. (48)

This constraint is illustrated in Figure 12 below ratherthan in Figure 3.

A second potential hazard occurs if halos form withhigh enough density that collapsing gas can trap photons.This makes the effective γ -factor close to 4/3 so that theJeans mass does not fall as collapse proceeds, and col-lapse proceeds in a qualitatively different way becausethere will be no tendency to fragment. This might leadto production of single black hole (instead of a myriadof stars) or other pathological objects, but what actu-ally happens would depend on unknown details, such aswhether angular momentum can be transported outwardso as to prevent the formation of a disk.

F. Constraints related to the baryon density

To conclude our discussion of selection effects, Figure 6illustrates a number of constraints that are independentof Q and ρΛ, depending on the density parameters ξband ξc for baryonic and dark matter either through theirratio or their sum ξ ≡ ξb + ξc.

As we saw in Section IIIB2, a very low baryon fractionf b ≡ ξb/ξ ∼< 300 may preclude disk fragmentation and

FIG. 6: Qualitatively different parts of the (ξb, ξc)-plane. Ourobserved universe corresponds to the star, and altering the param-eters to leave the white region corresponds to qualitative changes.The hatched regions delimit baryon fractions f b = ξb/ξc > 300and f b < 1, which may suppress galaxy formation by impedingdisk fragmentation and by Silk damping out Galaxy-scale mat-ter clustering, respectively, The shaded regions delimit interestingranges of constant total matter density parameter ξ = ξb+ ξc — towhat extent these qualitative transitions are detrimental to galaxyformation deserves further work.

star formation. On the other hand, a baryon fractionof order unity corresponds to dramatically suppressedmatter clustering on Galactic scales, since Silk dampingaround the recombination epoch suppresses fluctuationsin the baryon component and the fluctuations that cre-ated the Milky Way were preserved through this epochmainly by dark matter [90].

Recombination occurs at T ∼Ry/50 ∼ mpα2β/100 ∼3000K, whereas matter-domination occurs at T eq ∼0.2ξ. If ξ ∼< 0.05mpα2β , recombination would thereforeprecede matter-radiation equality, occurring during theradiation-dominated epoch. It is not obvious whetherthis would have detrimental effects on galaxy formation,

but it is interesting to note that, as illustrated in Figure 6,our universe is quite close to this boundary in parameterspace.

If we instead increase ξ, a there are two qualitativetransitions.

In the limit x = ρΛ/ρm ≪ 1, using equation (15),equation (A4) and ρm = ξnγ gives

σ =

π2

2ζ (3)

1/3

σ∗s(µ)ξQ

T ≈ 0.33028 s(µ)

ξQ

T . (49)



19

This means that the first Galactic mass structures(s(µ) ≈ 28) go nonlinear at T ∼ 9ξQ, i.e., before recom-bination if ξ ∼> 10−3mpα2βmp/Q. A baryonic cloud ableto collapse before or shortly after recombination whilethe ionization fraction remained substantial would trapradiation and, as mentioned above in Section IIIE, po-tentially produce a single black hole instead of stars.

Big Bang Nucleosynthesis (BBN) occurs at T ∼mpβ n ∼ 1 MeV, so if we further increase ξ so thatξ ∼> mpβ n, then the universe would be matter-dominatedbefore BBN, producing dramatic (but not necessarily fa-tal [105]) changes in primordial element abundances.

IV. PUTTING IT ALL TOGETHER

As discussed in the introduction, theoretical predic-tions for physical parameters are confronted with obser-vation using equation (1), where neither of the two factorsf prior(p) and f selec(p) are optional. Above we discussedthe two factors for the particular case study of cosmol-

ogy and dark matter, covering the prior term in Section IIand the selection effect term in Section III. Let us nowcombine the two and investigate the implications for theparameters ξ, Q and ρΛ.

A. An instructive approximation

For this exercise, we would ideally want to compute,say, the function f selec(p) defined as the fraction of pro-tons ending up in stable habitable planets as functionof (ρΛ, ξ , ξb, Q), leaving the particle physics parametersfixed at their observed values. Figure 3 shows that this

is quite complicated, for two reasons. First, as discussedabove we have computed only certain integrated versionsof f selec. Second, the other selection effects discussedinvolve substantial uncertainties. To provide useful qual-itative intuition, let us therefore start by working outthe implications of the simple approximation that thereare sharp upper and lower halo density cutoffs, i.e., thatobservers only form in halos within some density rangeρmin ≤ ρvir ≤ ρmax, where these two density limits maydepend on Q, ξ and ξb. Based on the empirical obser-vation that typical galaxies lie near the right side of thecooling curve in Figure 2, one would expect ρmin to be de-termined by the bottom of the cooling curve and ρmax bythe intersection of the cooling curve with the encounterconstraint.

Equation (20) showed that the fraction of all protons collapsing into a halo of mass µ iserfc[A(ρΛ/ρ∗)1/3/s(µ)G∞], where ρ∗ ≡ Q3ξ4 canbe crudely interpreted as the characteristic densityof the first halos to form. Roughly, A/s(µ)G∞ ∼ 1,ξ4 is the matter-radiation equality density and Q3 isthe factor by which our Universe gets diluted betweenequality (when fluctuations start to grow) and galaxyformation. More generally, of these protons, the fraction

FIG. 7: The fraction of all protons that end up in halos withdensity in the range ρmin < ρvir < ρmax is shown as a functionof the cosmological parameters ρΛ and ρ∗ ≡ Q3ξ4. The con-tours show fractions 0.3, 0.05 and 0.003 for the example ρmin =10−128, ρmax = 10−120 (the two dashed lines) and µ = 10−5

(corresponding roughly to 1012M ⊙ halos and giving s(µ) ≈ 28).When ρΛ ≫ Q3ξ4, essentially no halos at all are formed. WhenQ3ξ4 ∼

< 10−3ρmin, essentially all halos have ρvir < ρmin. When

Q3ξ4 ∼> 102ρmax, essentially all halos have ρvir > ρmax. The star

shows our observed parameter values.

f h in halos within a density range ρmin ≤ ρvir ≤ ρmax is

f h(ρΛ, ξ , ξb, Q) = erfc

(ρΛ/ρ∗)1/3

GΛ(x(ρmin))

−erfc

(ρΛ/ρ∗)1/3

GΛ(x(ρmax))

(50)

where the function x(ρ) is given by equation (24) ex-tended so that x(ρ) = ∞ for ρ ≤ 16ρΛ. For convenience,we have here defined the rescaled density

ρ∗ ≡

s(µ)

A

3ρ∗. (51)

Figure 7 is a contour plot of this function, and illustratesthat in addition to classical constraint ρΛ

∼< Q3ξ4 dat-

ing back to Weinberg and others [10, 44, 88, 107, 108,109, 110], we now have two new constraints as well: ρΛ-independent upper and lower bounds on ρ∗ ≡ Q3ξ4.

B. Marginalizing over ρΛ

As mentioned in Section II, there is good reason toexpect the prior f prior(ρΛ, ξ , Q) to be independent of ρΛ across the tiny relevant range |ρΛ| ∼< Q3ξ4 where



20

FIG. 8: The probability factor f h(ρ∗) after marginalizing over ρΛ.The black line of slope 1 is for ρmin = 0, ρmax = ∞. Increas-ing ρmin shifts the exponential cutoff on the left side progressivelyfurther the right: ρmin = 10−5, 10−4, 10−3 and 10−2 times thedensity at the dotted horizontal line gives the four curves on theleft. Decreasing ρmax shifts left the break to a −1/3 slope furtherto the right: ρmax = 100, 10, 1 and 0.1 times the density at thedotted horizontal line gives the four downward-sloping curves onthe right, respectively.

f selec(ρΛ, ξ , Q) is non-negligible. This implies that we

can write the predicted parameter probability distribu-tion from equation (1) as

f (ρΛ, ξ , ξb, Q) ∝ f prior(ξ, ξb, Q)f o(ξ, ξb, Q)f h(ρΛ, ξ , ξb, Q),(52)

where f o(ξ, ξb, Q) is the product of all other selectioneffects that we have discussed — we saw that these wereall independent of ρΛ.

The key point here is that the only ρΛ-dependence inequation (52) comes from the f h-term, i.e., from selec-tion effects involving the halo banana. It is therefore in-teresting to marginalize over ρΛ and study the resultingpredictions for Q and ξ. Thus integrating equation (52)over ρΛ, we obtain the theoretically predicted probabilitydistribution

f (ξ, Q) ∝ f prior(ξ, ξb, Q)f o(ξ, ξb, Q)f h(ξ, ξb, Q), where

f h(ξ, ξb, Q) ≡ ∞0

f h(ρΛ, ξ , ξb, Q)dρΛ. (53)

Note that if ρmin and ρmax are constants, then f h(ξ, Q) isreally a function of only one variable: since equation (50)shows that ξ and Q enter in f h(ρΛ, ξ , Q) only in the com-bination ρ∗ ≡ Q3ξ4, the marginalized result f h(ξ, Q)

will be a function of ρ∗ alone. Figure 8 shows thisfunction evaluated numerically for various values of ρmin

and ρmax, as well as the following approximation (dottedcurves) which becomes quite accurate for ρ ≪ ρmin andρ ≫ ρmax:

f h≈

724.9

ρ1/3∗

ρ4/3max

+

√π

G∞3ρ∗erfc

ρmin

18π2ρ∗

1/3

−1

(54)This approximation is valid also when ρmin and ρmax arefunctions of the cosmological parameters, as long as theyare independent of ρΛ. To understand this result qual-itatively, consider first the simple case where we countbaryons in all halos regardless of density, i.e., ρmin = 0,ρmax = ∞. Then a straightforward change of variablesin equation (52) gives f h(ρ∗) ∝ ρ∗ = Q3ξ4, as was pre-viously derived in [6]. This result troubled the authorsof both [6] and [81], since any weak prior on Q frominflation would be readily overpowered by the Q3-factor,

leading to the incorrect prediction of a much larger CMBfluctuation amplitude than observered. The same resultwould also spell doom for the axion dark matter modeldiscussed in Section IIA, since the ξ−1/2-prior would beoverpowered by the ξ4-factor from the selection effect anddramatically overpredict the dark matter abundance.

Figure 8 shows that the presence of selection effects onhalo density has the potential to solve this problem. Boththe problem and its potential resolution can be intuitivelyunderstood from Figure 7. Note that f h(ρ∗) is simply thehorizontal integral of this two-dimensional distribution.For ρmin ≪ ρ∗ ≪ ρmax, the integrand ≈ 1 out to theρ∗ ∼ ρΛ-line marked “NO HALOS”, giving f h(ρ∗) ∝ ρ∗.In the limit ρ

∗ ≫ρmax

, however, we are integratingacross the region marked ‘TOO DENSE” in Figure 8, and

the result drops as f h(ρ∗) ∝ ρ4/3max/ρ

1/3∗ = ρ

4/3max/Qξ4/3.

This is why Figure 8 shows a break in slope from +1to −1/3 at a location ρ∗ ∼ ρmax/200, as illustrated bythe four curves with different ρmax-values. Conversely, inthe limit ρ∗ ≪ ρmin, we are integrating across the expo-nentially suppressed region marked ‘TOO DIFFUSE” inFigure 7, causing a corresponding exponential suppres-sion in Figure 7 for ρ∗ ≪ ρmin.

Figures 9 and 10 illustrate this for the simple exampleof using the Section II A axion prior f prior(ξ) ∝ ξ−1/2,treating Q as fixed at its observed value, ignoring addi-tional selection effects (f o(ξ) = 1), and imposing a halodensity cutoff ρmax of 5000 times the current cosmic mat-ter density.

The key features of this plot follow directly from ouranalytic approximation of equation (54). First, for ρ∗ ≪ρmax, we saw that f h ∝ ρ∗ ∝ ξ4, so equation (53) givesthe probability growing as f ∝ ξ−1/2ξ4ξ = ξ9/2 (thelast ξ-factor comes from the fact that we are plottingthe probability distribution for log ξ rather than ξ). Thismeans that imposing a lower density cutoff ρmin wouldhave essentially no effect. This also justifies our approxi-



21

FIG. 9: Probability distribution for the axion dark matter den-sity parameter ξ measured from a random 1012M ⊙ halo withvirial density below 5000 times the present cosmic matter den-sity. Green/light shading indicates the 95% confidence interval.The dotted vertical line shows our observed value ξ ≈ 4 eV, ingood agreement with the prediction. The dashed curves show theanalytic asymptotics ξ9/2 and ξ−5/6, respectively.

mation of using the total matter density parameter ξ as aproxy for the dark matter density parameter ξc: we have

a baryon fraction ξb/ξc ≪ 1 in the interesting regime,with a negligible probability for ξc ∼< ξb ∼ 0.6eV (theprobability curve continues to drop still further to theleft even without the f h-factor). Second, for ρ∗ ≫ ρmax,

we saw that f h ∝ ρ−1/3∗ ∝ ξ−4/3, so equation (53) gives

the probability falling as f (ln ξ) ∝ ξ−1/2ξ−4/3ξ = ξ−5/6.

Although this simple example was helpful for build-ing intuition, a more accurate treatment is required be-fore definitive conclusions about the viability of the axiondark matter model can be drawn. One important issue,to which we return below in Section IV, is the effect of the unknown Q-prior, since the preceding equations onlyconstrained a combination of Q and ξ. A second impor-

tant issue, to which we devote the remainder of this sub-section, has to do with properly incorporating the con-straints from Figure 3. If we only consider the encounterconstraint and make the unphysical simplification of ig-noring the effect of halo velocity dispersion, then our up-per limit should be not on dark matter density ( ρmax

constant) but on baryon density (ρmaxξb/ξ constant). In-

serting this into equation (54) makes the term ρ1/3∗ /ρ

4/3max

independent of ξ as ξ → ∞, replacing the f h ∝ Q−1ξ−4/3

cutoff by f h ∝ Q−1ξ−4/3b , which is constant if ξb is. The

FIG. 10: Same as previous figure, but showing the probabilitydistribution for ξ1/2, the quantity which has a uniform prior in ouraxion model. The shape of this curve therefore reflects the selectioneffect alone.

axion prior would then predict a curve f (ln ξ) ∝ ξ1/2 ris-ing without bound. This failure, however, results fromignoring some of the physics from Figure 3. Consider-ing a series of nominally more likely domains with pro-

gressively larger ξ, they typically have more dark energy(ρΛ ∝ ξ) and higher characteristic halo baryon density(nvir ∝ ξ3ξb), so stable solar systems are found only inrare galaxies that formed exceptionally late, just beforeρΛ-domination, with nvir ∝ ρvirξb/ξ ∼> 16ρΛξb/ξ ∝ ξbroughly ξ-independent and below the the maximum al-lowed value. Since the increased dark matter densityboosts virial velocities, this failure mode correspondsto moving from the star in Figure 3 straight to theright, running right into the constraints from cooling andvelocity-dependent encounters. In other words, a morecareful calculation would be expected to give a predictionqualitatively similar to that of Figure 9. The lower cutoff would remain visually identical to that of Figure 9 as longas ρmin can be neglected in the calculation, whereas theupper cutoff would become either steeper or shallowerdepending on the details of the encounter and coolingconstraints. This interesting issue merits further workgoing well beyond the scope of the present paper, ex-tending our treatment of halo formation, halo mergersand galaxy formation into a quantitative probability dis-tribution in the plane of Figure 3, i.e., into a functionf (T vir, nvir; Q, ρΛ, ξ , ξb) that could be multiplied by theplotted constraints and marginalized over (T vir, nvir) to



22

FIG. 11: Probability distribution for the quantity R ≡ ρΛ/ξ4Q3

measured from a random 1012M ⊙ halo, using a uniform prior for Rand ignoring other selection effects. This is equivalent to treating ξand Q as fixed. Green/light shading indicates the 95% confidenceinterval, the dotted line indicates the observed value R ≈ 15.

give f selec(Q, ρΛ, ξ , ξb).

C. Predictions for ρΛ

This above results also have important implicationsfor the dark energy density ρΛ. The most commonway to predict its probability distribution in the liter-ature has been to treat all other parameters as fixedand assume both that the number of observers is propor-tional to the matter fraction in halos (which in our no-tation means f selec( p) = erfc[A(ρΛ/Q3ξ4)1/3/s(µ)G∞])and that f prior(ρΛ) is constant across the narrow range|ρΛ| ∼< Q3ξ4 where f selec is non-negligible. This gives thefamiliar result shown in Figure 11: a probability distribu-tion consistent with the observed ρΛ-value but favoringslightly larger values. The numerical origin of the pre-

dicted magnitude ρΛ ∼ 10−123 is thus the measured valueof ρ∗ ≡ Q3ξ4.

Generalizing this to the case where Q and/or ξ can varyacross an inflationary multiverse, the predictions will de-pend on the precise question asked. Figure 11 then showsthe successful prediction for ρΛ given (conditionalized on)our measured values for Q and ξ. However, when test-ing a theory, we wish to use all opportunities that wehave to potentially falsify it, and each predicted param-eter offers one such opportunity. For our axion example,

the theory predicts a 2-dimensional distribution in the(ρΛ, ξ)-plane of which Figure 9 is the marginal distribu-tion for ξ. The corresponding marginal distribution forρΛ (marginalized over ξ) will generally differ from Fig-ure 11 in both its shape and in the location of its peak.It will differ in shape because the (ρΛ, ξ)-distribution isnot generally separable: the selection effects (as in Fig-ure 11) will not be a function of ρΛ times a function of ξ;

in other words, a uniform prior on ρΛ will not correspondto a uniform prior on the quantity R ≡ ρΛ/Q3ξ4 plottedin Figure 7 once non-separable selection effects such asones on the halo density parameter ρ∗ ≡ Q3ξ4 are in-cluded. Moreover, the distribution will in general notpeak where that in Figure 11 does because the predictedmagnitude ρΛ ∼ 10−123 no longer comes from condition-ing on astrophysical measurements of Q and ξ, but fromother parameters like α, β and mp that determine the se-lection effects in Figure 3 and the maximum halo densityρmax. In this case, whether the observed ρΛ-value agreeswith predictions or not thus depends sensitively on howstrong the selection effects against dense halos are.

In summary, the prediction of ρΛ

given Q and ξ is anunequivocal success, whereas predictions for Q, ξ and theentire joint distribution for (ρΛ, ξ , Q) are fraught with theabove-mentioned uncertainties. Since a uniform ρΛ-priordoes not imply a uniform R ≡ ρΛ/Q3ξ4-prior, we cannotconclude that anthropic arguments succeed in predictingR ≡ ρΛ/Q3ξ4 [81] without additional hypotheses.

D. Constraint summary

Table 4 summarizes the constraints that we have dis-cussed above. Those not superseded by other strongerones are also illustrated in Figure 12, with ξb fixed at itsobserved value. Let us briefly comment on how they allfit together.

The first key point to note in Figure 12 is that thevarious constraints involve many different combinationsof Q and ξ, thereby breaking each other’s degeneraciesand providing interesting constraints on both ξ and Qseparately. The fragmentation constraint (requiring diskinstability) constrains ξ alone when ξb is given whereasthere are constraints on Q alone from both line cool-ing freezeout and primordial black hole trouble. The en-counter and cooling constraints both have different slopes(−1 and −2/3, respectively) because they involve differ-ent powers of the baryon fraction. Whereas the non-

linear halo constraint depends on the total halo density∝ Q3ξ4, the encounter constraint depends on the halobaryon density ∝ Q3ξ3ξb. The cooling constraint (whichis equation (11) in [85] after changing variables to reflectthe notation in this paper) comes from equating the cool-ing timescale (given by the inverse of the baryon density∝ Q3ξ3ξb) and the dynamical timescale (given by the to-

tal density as ∝ ρ−1/2vir ∝ (Q3ξ4)−1/2, thus constraining

the combination Q3ξ2ξ2b.The second key point in Figure 12 is that this combi-



23

Table 4: These constraints (see text) are summarized in Figure 12

Constraint Generally Fixing (α,β,mp) Fixing all but (Q, ξ)

Need nonlinear halos |ρΛ| ∼< ρ∗ |ρΛ|/ξ4Q3 ∼

< 1 Q ∼> 10−5(ξ/ξ0)−4/3

Avoid line cooling freezeout Q ∼> α2β Q ∼

> 10−8 Q ∼> 10−8

Primordial black hole excess Q ∼< 10−1 Q ∼

< 10−1 Q ∼< 10−1

Need cooling in Hubble time Q3ξ2b

ξ2 ∼> α−3 ln[α−2]−16/3β 4m6p/125 Q3ξ2

bξ2 ∼

> 10−129 Q ∼> 10−6(ξ/ξ0)−2/3

Avoid close encounters Q3ξbξ3

∼< 10−123 Q ∼

< 10−4(ξ/ξ0)−1

Go nonlinear after decoupling ξQ ∼< 10−3α2βmp ξQ ∼

< 10−30 Q ∼< 10−2(ξ/ξ0)−1

Need equality before decoupling? ξ ∼>

0.05α2

βmp ξ ∼>

10−28

(ξ/ξ0) ∼>

1/3Avoid severe Silk damping f b ∼

< 1/2 ξ/ξb ∼> 2 (ξ/ξ0) ∼

> 1/3

Need disk instability f b ∼< 102 ξc/ξb ∼

< 102 (ξ/ξ0) ∼< 20

FIG. 12: Crude summary of constraints (ξ, Q)-plane constraintsfrom Table 4. The star shows our observation (ξ, Q) ≈ (4eV, 2 ×10−5). The parallel dotted lines are lines of constant characteristichalo density, so the constraint that dark matter halos form (thatthey go nonlinear before vacuum domination freezes fluctuationgrowth) rules out everything to the lower left of these lines, whichfrom left to right correspond to ρΛ/ρobsΛ = 10−3, 1, 103. Thismeans that if ξ is fixed, marginalizing over ρΛ pushes things to thetriangle at larger Q (the “smoothness problem” [6]), but if ξ toocan vary, marginalizing over ρΛ instead pushes things to the squareat smaller Q.

nation of constraints reverses previously published con-clusions about Q. Where we should expect to find our-selves in Figure 12 depends on the priors for both Q andξ. Recall that halos form only above whichever dottedline corresponds to the ρΛ-value. If the prior for ρΛ isindeed uniform, then it is a priori (before selection ef-fects are taken into account) 1000 times more likely forthe halo constraint to be the upper dotted line than themiddle one, and this is in turn 1000 more likely thanthe bottom dotted line. In other words, marginalizing

over ρΛ relentlessly pushes us towards the upper rightin Figure 12, towards larger value of ρ∗ = Q3ξ4. Thismeans that if ξ is fixed and only Q can vary across theensemble, marginalizing over ρΛ pushes things towardsthe triangle at larger Q. This “smoothness problem”was pointed out in [85] and elaborated in [6, 121, 122],suggesting that unless the f prior(Q) falls off as Q−4 orfaster, it is overpowered by the selection effect and onepredicts a clumpier universe than observed (the star in

Figure 12 should be shifted vertically up against the edgeof the encounter constraint, to the triangle). Figure 12shows that if ξ too can vary, marginalizing over ρΛ in-stead pushes things towards smaller Q, to the square,since this gives the greatest allowed halo density ∝ Q3ξ4.In [6], it was found that certain low-energy inflation sce-narios naturally predicted a rather flat priors for lg Q —to determine whether they are ruled out or not thus re-quires more detailed modeling of the effect of the darkmatter density on galaxy formation and encounters.

Table 4 illustrates that (except for one rather unim-portant logarithm), all constraints we have consideredcorrespond to hyperplanes in the 7-dimensional space

parametrized by

(lg α, lg β, lg mp, lg ρΛ, lg Q, lg ξ, lg ξb), (55)

providing a simple geometric interpretation of the selec-tion effects. This means that if all priors are power laws,the preferred parameter values are determined by solvinga simple linear programming problem, and will generi-cally correspond to one of the corners of the convex 7-dimensional allowed region. Above we have frequentlyused such sharp inequalities to help build intuition for theunderlying physics. For future extensions of this work,however, it should be borne in mind that such inequalitiesare not necessarily sufficient to capture the key featuresof the problem, and that they can either overstate orunderstate the importance of selection effects. The ulti-mate goal is to test theories by computing the predictedprobability distribution from equation (1), and almost noselection effects f selec(p) correspond to sharp cutoffs re-sembling a Heaviside step function. Some (like the haloconstraint) are exponential and thus able to overpowerany power law prior. Others, however, may be softer, andit is therefore crucial to check whether their functionalform falls of faster than than the relevant prior grows —



24

if not, the “constraint” will be penetrated, and the mostlikely parameter values may lie in the allegedly “disal-lowed region”. Conversely, if f (p) = f prior(p)f selec(p)is rather flat in a large “allowed region”, then the mostlikely parameter values can lie well inside it, far from anyedges, since it corresponds to the peak of the probabilitydistribution rather than its edge.

V. CONCLUSIONS

We have discussed predictions for the 31-dimensionalvector p of dimensionless physical constants in the con-text of “ensemble theories” producing multiple Hubblevolumes (“universes”) between which one or more of theparameters can differ. The prediction is the probabil-ity distribution of equation (1), where neither of the twofactors is optional. Although it is generally difficult toevaluate either one of the terms, it is nonetheless cru-cial: if candidate theories involve such ensembles, theyare neither testable nor falsifiable unless their probabilis-

tic predictions can be computed.Although many scientists hope that f prior in equa-tion (1) will be a multidimensional Dirac δ-function, ren-dering selection effects irrelevant, to elevate this hopeinto an assumption would, ironically, be to push theanthropic principle to a hedonistic extreme, suggestingthat nature must be devised so as to make mathematicalphysicists happy. We must therefore face some difficultquestions both about what, in principle, to select on,and how to quantitatively calculate that selection effect.For both statistical mechanics and quantum mechanics,it proved challenging to correctly predict probability dis-tributions. If the analogous challenge can be overcomefor theories predicting dimensionless constants, they toowill become testable in the same fashion. The fact thatwe can only observe once is not a show-stopper: a sin-gle observation of a vertically polarized photon passingthrough a polarizer θ = 89.9 from vertical would rule outquantum mechanics at 1 − cos2 θ ≈ 99.9997% confidence.The outstanding difficulty is therefore not a “philosoph-ical” problem that we cannot observe the full ensemble,but rather that we generally do not know how to computethe theoretical prediction f (p).

A. Axion dark matter

We have tackled this problem quantitatively for a par-ticular example where both factors in equation (1) are

computable: that of axion dark matter with its phasetransition well before the end of inflation. We foundthat f prior(ξ) ∝ ξ−1/2, i.e., that the prior dark mat-ter distribution has no free parameters even though theaxion model itself does. We saw that this useful pre-dictability gain has the same basic origin as that of the zero-parameter (flat) ρΛ-prior of [88]: the anthrop-ically relevant range over which f selec is non-negligible

is much smaller than the natural range of f prior, soonly a particular property of f prior matters. For the fa-mous ρΛ-case, the natural f prior-range involves either thesupersymmetry-breaking scale or the Planck scale |ρΛ| ∼<ρplanck whereas the relevant range is |ρΛ| ∼< 10−123ρplanck,so the property that f prior(ρΛ) is smooth on that tinyscale implies that it is for all practical purposes constant.For our ξ-case, the natural f prior-range is 0 ≤ ξ ≤ ξ∗

where ξ1/4∗ lies somewhere between the inflation scaleand the Planck scale, whereas the relevant range is muchsmaller, so the only property of f prior(ξ) that matters isits asymptotic scaling as ξ → 0.

In computing the selection effect factor f selec(ρΛ, ξ , Q),we took advantage of the fact that none of these pa-rameters affect chemistry or biology directly. We couldtherefore avoid poorly understood biochemical issues re-lated to life and consciousness, and could limit our cal-culations to astrophysical selection effects involving haloformation, galaxy, solar system stability, etc. The samesimplifying argument has been previously applied to ρΛand the neutrino density (e.g., [84]). We discovered that

combining various astrophysical selection effects with allthree parameters (ρΛ, Q , ξ) as variables reversed previ-ous conclusions (Figure 12), pushing towards lower ratherthan higher Q-values. In particular, we found that onecan reach misleading conclusions by simply taking f selecto be the matter fraction collapsing into halos, neglectingthe fact that very dense halos limit planetary stability. Inparticular, we found that if we impose a stiff upper limiton the density of habitable halos, the predicted proba-bility distribution for the dark matter density parameter(Figure 9) can be brought into agreement with the mea-sured ξ-value for reasonable assumptions, i.e., that thepre-inflationary axion model is a viable dark matter the-ory.

B. Multiple dark matter components

Suppose that supersymmetry were to be discovered atthe Large Hadron Collider, indicating the existence of aweakly interacting massive particle (WIMP) with a relicdensity approximately equal to the observed dark matterdensity, i.e., with ξwimp ∼ ξc. There would then be astrong temptation to declare the dark matter problemsolved. Based on our results, however, that would bequite premature!

As detailed in Section IIB, it is not implausible thatthe function f prior(ξwimp) to be taken as input to the as-trophysics calculation is a sharply peaked function whoserelative width is much narrower than that of f selec(ξc),perhaps only a few percent. This situation could arise if the fundamental theory predicted a broad prior only forthe µ2-parameter in Table 1 (which controls the Higgsvacuum expectation value), since this may arguably bedetermined to within a few percent from selection effectsin the nuclear physics sector, notably stellar carbon andoxygen production. If the pre-inflationary axion model



25

is correct, we would then have

ξc = ξwimp + ξaxion, (56)

where f prior(ξaxion) ∝ ξ−1/2axion and ξwimp would be for all

practical purposes a constant. This gives f prior(ξc) ∝(ξc − ξwimp)−1/2 for ξc ≥ ξwimp, zero otherwise, and thetestable prediction becomes this times the astrophysical

factor selection effects factor f selec(ξc) that we have dis-cussed. Repeating the calculation of [14], this impliesthat we should expect to find roughly comparable densi-ties of WIMP and axion dark matter.

As shown in [14], removing the assumption thatf prior(ξwimp) is sharply peaked does not alter this qual-itative conclusion. This conclusion would also hold forany other axion-like fields that were energetically irrele-vant during inflation, even unrelated to the strong CP-problem. More generally, string-inspired model buildingsuggests the possibility that multiple species of “GIMPS”(gravitationally interacting massive particles that do notcouple through the electromagnetic, weak or strong in-

teractions) may be present. According to recent under-standing, their presence may be expected for modularinflation, though not for brane inflation [123]. If suchparticles exist and have generic not-too-extreme priors(falling no faster than ξ−1 and rising no faster than theselection effect cutoff), then our anthropic constraints onthe sum of their densities again predict a comparable den-sity for all of species of dark matter that have a risingprior [14].

Even before the LHC turns on, more careful calcu-lation of the astrophysical selection effects determiningf selec could give interesting hints. If the constraints ondense halos turn out to be rather weak so that the im-proved version of Figure 9 predicts substantially moreaxion dark matter than observed, this will favor an alter-native model such as WIMPs where f prior(ξc) is so nar-row that ξc is effectively determined non-anthropically.Alternatively, if Figure 9 remains consistent with obser-vation, then it will be crucial to follow up an initial LHCSUSY or other WIMP detection, if it occurs, with de-tailed measurements of the relevant parameters to estab-lish whether the WIMP density accounted for (which alinear collider might, in favorable cases, pinpoint withina few percent [73]) is exactly equal to the cosmologicallymeasured dark matter density, or whether a substantialfraction of the dark matter still remains to be accountedfor.

The authors wish to thank Ed Bertschinger, DouglasFinkbeiner, Alan Guth, Craig Hogan, Andrei Linde andMatias Steinmetz for helpful comments, and Sam Rib-nick for some earlier, exploratory calculations. Thanks toTom Abel for molecular cooling software. This work wassupported by NASA grant NAG5-11099, NSF CAREERgrant AST-0134999, and fellowships from the David andLucile Packard Foundation and the Research Corpora-tion. The work of AA was supported in part by NSF

grant AST-0507117. The work of FW is supported inpart by funds provided by the U.S. Department of En-ergy under cooperative research agreement DE-FC02-94ER40818.

APPENDIX A: FITTING FUNCTIONS USED

In this Appendix, we derive a variety of fitting func-tions used in the paper, explicitly highlighting the de-pendence of standard cosmological structure formationon the dimensionless parameters ξ, ρΛ, Q and µ. Sincesome of our approximations are quite accurate, we retainthe exact numerical coefficients where appropriate.

1. The fluctuation time dependence G(x)

As shown in [83], the function

GΛ(x) ≈ x1/3

1 +

xG∞

3

α−1/3α , (A1)

where α = 159/200 = 0.795 and

G∞ ≡ 5Γ23

Γ56

3√

π≈ 1.43728, (A2)

describes how, in the absence of massive neutrinos, fluc-tuations in a ρΛ > 0 ΛCDM universe grow as the cos-mic scale factor a as long as dark energy is negligible(GΛ(x) ≈ x1/3 ∝ a ∝ (1 + z)−1 for x ≪ 1) and thenasymptote to a constant value as t

→ ∞and dark en-

ergy dominates (GΛ(x) → G∞ as x → ∞).Including the effect of radiation domination early on

but ignoring baryon-photon coupling, the total growthfactor is well approximated by [83]

G(x) ≈ 1 +3

2x−1/3eq GΛ(x), (A3)

which is accurate to better than 1.5% for all x [83]. Inessence, fluctuations grow as δ ∝ a ∝ x1/3 between mat-ter domination (x = xeq) and dark energy domination

(x = 1), giving a net growth of x−1/3eq . Equation (A3)

shows that they grow by an extra factor of 1.5 by start-

ing slightly before matter domination and by an extrafactor GΛ(∞) ≈ 1.44 by continuing to grow slightly afterdark energy domination.

Let us now simplify this result further. From the defi-nition of ξ, the matter density can be written ρm = ξnγ ,where nγ is the number density of photons. The latteris given (in Planck units) by the standard black bodyformula

nγ =2ζ (3)

π2T 3, ζ (3) ≈ 1.20206. (A4)



26

The corresponding standard formula for the energy den-sity is

ργ =π2

15T 4 (A5)

for photons and

ργν =21

8 4

114/3

ργ ≈ 0.68132ργ (A6)

for three standard species of relativistic massless neutri-nos. The matter-radiation equality temperature whereρm = ργ + ργν is thus

T eq =30ζ (3)

π4

1 +

21

8

4

11

4/3−1

ξ ≈ 0.220189ξ, (A7)

measured to be about 9400K in our Hubble volume.Since xeq = ρΛ/ρeqm = ρΛ/ξnγ(T eq), we thus obtain

xeq = π14

2 · 303ζ (3)4

1 + 21

8

4

11

4/3

3ρΛξ4

≈ 384.554 ρΛξ4

(A8)The above relations are summarized in Tables 2 and 3.In all cases of relevance to the present paper, the growthfactor G(x) ≫ 1, allowing initial fluctuations Q ≪ 1 togrow enough to go nonlinear. We can therefore drop thefirst term in equation (A3), obtaining the useful result

G(x) ≈ 45 · 21/3ζ (3)4/3

π14/3

1 +

21

8

4

11

4/3−1

ξ4/3

ρ1/3Λ

GΛ(x)

≈ 0.206271

ξ4/3

ρ1/3Λ

GΛ(x). (A9)

2. The fluctuation scale dependence s(µ)

The r.m.s. fluctuation amplitude σ in a sphere of radiusR are given by [106]

σ2 = 4π

∞0

sin x − x cos x

x3/3

2P (k)

k2dk

(2π)3, (A10)

where x ≡ kR. We are interested in the linear regime

long after the relevant fluctuation modes have entered thehorizon, when all modes grow at the same rate (assum-ing a cosmologically negligible contribution from massiveneutrinos as per [89]. This means that σ can be factoredas a product of a function of time x and a function of comoving scale:

σ = QG(x)s(µ). (A11)

Here Q is the amplitude of primordial fluctuations cre-ated by, e.g., inflation, defined as the scalar fluctuation

amplitude δH on the horizon. As our measure of comov-ing scale, we use the dimensionless quantity

µ ≡ ξ2M, (A12)

where M is the comoving mass enclosed by the above-mentioned sphere.

The horizon mass at matter-radiation equality is of

order ξ−2

, so µ can be interpreted as the mass rela-tive to this scale. This equality horizon scale is thekey physical scale in the problem, and the one on whichthe matter power spectrum has the well-known break inits slope. (fluctuation modes on smaller scales enteredthe horizon before equality when they could not grow).For reference, the measured vales from WMAP+SDSSgive ξ ≈ 3 × 10−28, ξ−2 ≈ 1017 solar masses andµ ≈ R/(60h−1Mpc)3 for a sphere of comoving radius R.Numerically, using equation (A10) and employing [90] tocompute P (k), we find that the scale-dependence s(µ) isapproximately given by

s(µ) ≈ (9.1µ−

2/3

)β

+ (50.5 lg(834 + µ−1/3

) − 92)β1/β

,(A13)

where β = −0.27. This assumes that the primordialfluctuations are approximately scale-invariant and thatmassive neutrinos have no major effect, as indicated byrecent measurements [89, 91, 92]. For our observed ξ-value a galactic mass scale M = 1012M ⊙ thus corre-sponds to µ = ξ2M ≈ 10−5 and s(µ) ≈ 28. Again usingmeasured parameter values, the common reference scaleR = 8h−1Mpc corresponds to µ ≈ 0.002 and s(µ) ≈ 11.

3. The virial density ρvir(x)

We will now derive the virial density of a halo formedat time x in a ΛCDM universe, generalizing the standardresult ρvir ∼ 18π2ρm(x). Whereas past work on thissubject [94, 95] has focused on accurate numerical fitsto the observationally relevant present epoch and recentpast (x ∼< 1), we need a formula accurate for all times x.

Consider first the evolution of a top hat overdensityin a ΛCDM universe. By Birkhoff’s theorem, it is givenby the Friedman equation corresponding to a closed uni-verse:

3H 2

8πG

= ρΛ + ρm

−k

a2

= (1

−κ

A2

+1

A3

)ρΛ, (A14)

where we have defined a dimensionless scale factor A ≡(ρΛ/ρm)1/3 and a dimensionless curvature parameter κ ≡(A/a)2k. One readily finds that this will give H = 0 andrecollapse for any curvature κ > κmin ≡ 3/22/3 ∼ 1.8899,so the scale factor Amax(κ) at turnaround is given bysolving equation (A14) for H = 0, i.e., by numericallyfinding the smallest positive root A of the cubic polyno-mial A3−κA+1 = 0. Amax(κmin) = 2−1/3 ≈ 0.79, fallingoff as Amax(κ) ≈ 1/κ for κ ≫ 1.



27

FIG. 13: The top panel shows the turnaround (minimum) matterdensity ρturn of a top hat halo as a function of the time x whenits collapses. The virial density of the resulting dark matter halois ρvir ∼ 8ρturn. The solid curve is the exact result, with theasymptotic behavior ρturn → 9π2ρm/4x as x → 0 and ρturn → 2ρΛas x → ∞ (dotted lines). The three analytic fits correspond toequation (A18) (short-dashed), equation (A19) (dotted) and thatof [94, 95], which in our notation is (18π2/x + 18π2− 82 − 39x/(1 +x))/8. The first two are seen to be accurate to better than 18%and 4% respectively (bottom panel).

The age of the universe tturn when the top hat turnsaround is given by integrating dt = d ln A/(d ln A/dt) =dA/(AH ) using equation (A14), i.e.,

H Λtturn(κ) =

Amax(κ)

0

dA A2 − κ + 1

A

, (A15)

where H Λ ≡ (8πGρΛ/3)1/2. For the unperturbed back-ground ΛCDM universe where we used x as a more con-

venient time variable, we have κ = 0 and A = x1/3turn

when the overtensity turns around, giving tturn in termsof xturn:

H Λtturn(xturn) = x1/3turn

0dA

A2 + 1A

. (A16)

Eliminating tturn between equations (A15) and (A16) al-lows us to numerically compute the function κ(xturn).

Putting everything together now allows us to numeri-cally compute the density of our top hat overdensity atturnaround: Since ρm = ρΛ/A3, we have

ρturn(xturn) =ρΛ

Amax(κ(xturn))3. (A17)

The result is shown in Figure 13 together with three an-alytic fitting functions. The simple fit

ρturn(x) ≈ 2

9π2

8x

107200

+ 1

200107

ρΛ (A18)

is accurate to better than 18% for all x and becomes

exact in the two limits x → 0 and x → ∞. The fit

ρturn(x) ≈

9π2

4x+

6.6

x0.3+ 2

ρΛ (A19)

is accurate to better than 4% for all x and also becomesexact in both limits, but we will nontheless use the lessaccurate equation (A18) because it is analytically invert-ible and still accurate enough for our purposes. Apply-ing the virial theorem gives a characteristic virial densityρvir ∼ 8ρturn, with a weak additional dependence on ρΛ[93]9. This means that a halo that virialized at time xhas

ρvir ∼ (18π2ρm(x))

107200 + (16ρΛ)

107200 200107

, (A20)

i.e., essentially the larger of the two terms 18π2ρm(x)and 16ρΛ. Inverting this, we obtain

x ∼ 9π2

8

ρvir

16ρΛ

107200

− 1

− 200107

. (A21)

A familiar quantity in the literature is the collapseoverdensity ∆c ≡ ρvir/ρm = ρvirx/ρΛ, equaling 18π2 ≈178 for a flat ρΛ = 0 universe. Using equation (A18),equation (A19) and the fit in [95] to the calculations of

[94] gives

∆c(x) ≈ 18π2

1 +

8x

9π2

107200

200107

, (A22)

∆c(x) ≈ 18π2 + 52.8x0.7 + 16x, (A23)

∆c(x) ≈ 18π2 + (18π2 − 82)x − 39x2

1 + x, (A24)

respectively. Figure 13 shows that our approximationssubstantially improve the accuracy, equation (A23) beinggood to 4%. This improvement not surprising, since theapproximation (A24) was not designed to work well for

x ≫ 1.

9 The detailed calculation of [93] shows that the standard collapsefactor of 2 varies at thr 20% level with ρΛ at the 20% level.Qualitatively, positive (repulsive) ρΛ increases the collapse factorbecause shells have to fall further to acquire a velocity which willbring the system into equilibrium. We will not explicitly modelthis correction here given the larger uncertainties pertaining toother aspects of our treatment.



28

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