Multiverse Predictions for Habitability: Number of Potentially Habitable Planets
uuniverse
Article
Multiverse Predictions for Habitability:Number of Potentially Habitable Planets
McCullen Sandora Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA;[email protected] Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA 19104, USAReceived: 14 May 2019; Accepted: 21 June 2019; Published: 25 June 2019 (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:1) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Abstract:
How good is our universe at making habitable planets? The answer to this depends onwhich factors are important for life: Does a planet need to be Earth mass? Does it need to be insidethe temperate zone? are systems with hot Jupiters habitable? Here, we adopt different stances onthe importance of each of these criteria to determine their effects on the probabilities of measuringthe observed values of several physical constants. We find that the presence of planets is a genericfeature throughout the multiverse, and for the most part conditioning on their particular propertiesdoes not alter our conclusions much. We find conflict with multiverse expectations if planetary sizeis important and it is found to be uncorrelated with stellar mass, or the mass distribution is toosteep. The existence of a temperate circumstellar zone places tight lower bounds on the fine structureconstant and electron to proton mass ratio.
Keywords: multiverse; habitability; planets
1. Introduction
This paper is a continuation of [1], which aims to use our current understanding from a variety ofdisciplines to estimate the number of observers in a universe N obs , and track how this depends on themost important microphysical quantities such as the fine structure constant α “ e {p π q , the ratio of theelectron to proton mass β “ m e { m p , and the ratio of the proton mass to the Planck mass γ “ m p { M pl .Determining these dependences as accurately as possible allows us to compare the measured values ofthese constants with the multiverse expectation that we are typical observers within the ensemble ofallowable universes [2]. In doing so, there remain key uncertainties that reflect our ignorance of whatprecise conditions must be met in order for intelligent life to arise. Rather than treating this obstacle asa reason to delay this endeavor until we have reached a more mature understanding of all the complexprocesses involved, we instead view this as a golden opportunity: since the assumptions we makealter how habitability depends on parameters, sometimes drastically, several of the leading schoolsof thought for what is required for life are incompatible with the multiverse hypothesis. Generically,if we find that our universe is no good at a particular thing, then it should not be necessary forlife because, if it were, we would most likely have been born in a universe which is better at thatthing. Conversely, if our universe is preternaturally good at something, we expect it to play a rolein the development of complex intelligent life, otherwise there would be no reason we would be inthis universe. The requirements for habitability are in the process of being determined with muchgreater rigor through advances in astronomy, exoplanet research, climate modeling, and solar systemexploration, so we expect that in the not too distant future our understanding of the requirementsfor intelligent life will be much more complete. At this stage of affairs, then, we are able to use themultiverse hypothesis to generate predictions for which of these habitability criteria will end up Universe , niverse , , 157 2 of 34 being true. These will either be vindicated, lending credence to the multiverse hypothesis, or not,thereby falsifying it.In this work, we define the habitability of a universe H as the total number of observers it produces.In estimating the number of observers the universe contains, a great many factors must be taken intoconsideration. Thankfully, there has long been a useful way of organizing these factors: the Drakeequation, which in a slightly modified form along the lines of [3] reads H “ ż λ min d λ p IMF p λ q ˆ N ‹ ˆ f p p λ q ˆ n e p λ q ˆ f bio p λ q ˆ f int p λ q ˆ N obs p λ q . (1)Here, N ‹ is the number of stars in the universe, f p is the fraction of stars containing planets, n e is theaverage number of habitable planets around planet-bearing stars, f bio is the fraction of planets thatdevelop life, f int is the fraction of life bearing worlds that develop intelligence, and N obs is the numberof intelligent observers per civilization. We have included dependence of the size of the host star λ “ M ‹ {pp π q { M pl { m p q , in order to more accurately reflect the fact that these quantities may dependon this. We then integrate over the stellar initial mass function given in [4], which approximates abroken power law with turnover at 0.2 M d . This can be related to the probability of being in ouruniverse by incorporating the relative occurrence rates of different universes as P p prior H . It wasargued in [1] that a reasonable choice of prior is given by p prior {p βγ q . However, this is ultimatelyset by physics at high energies, and so may in principle be something else.Previously, the fact that the strength of gravity γ can be two orders of magnitude higher causedthe biggest problems with finding a successful criterion, since most stars are in universes with strongergravity. Though we had set out to focus solely on the properties of stars, it was only when we weightedthe habitability of a system by the total entropy processed by its planets over its entire lifetime that wehit upon a fully satisfactory criterion. This was also reliant on the condition that starlight be in thephotosynthetic range, colloquially referred to as ‘yellow’ light here: conservatively, this correspondsto the 600–750 nm range. Relaxing this range to be from 400–1100 nm will not qualitatively affectour results. The other factors we considered may be freely included at will without hindering thisconclusion. For the majority of this paper, we take the entropy and yellow light conditions as ourbaseline minimal working model, and incorporate factors that influence the availability and propertiesof planets to determine how these alter our estimates for habitability. While our previous analysis wasnot heavily reliant on cutting edge results from the field of astronomy, our understanding of planets,from their population statistics to their formation pathways, has undergone rapid expansion in thepast decade, and a state-of-the-art analysis needs to reflect that.To this end, we begin by estimating the fraction of stars with planets in Section 2. Most notable isthe recent determination of a threshold metallicity below which rocky planets are not found [5], as wellas the understanding of the origin of this threshold. We also find the conditions for the lifetime ofmassive stars to be shorter than the star formation time and the fraction of galaxies able to retain metalsto be sizable, but find that these only impose mild constraints. Using these allows us to determine thedependence of f p on the underlying physical parameters. Additionally, we incorporate the fraction ofsystems that host hot Jupiters into our analysis, and find conditions for this process to not wreck allplanetary systems.In Section 3, we turn to the average number of habitable planets n e . Two commonly usedrequirements for a planet to be habitable are that it needs to be both terrestrial and temperate, and sowe optionally include both of these when estimating habitability. Recent results indicate that thedistribution of rocky planets peaks at 1.3 R C [6–9], which is rather close to the terrestrial radiuscapable of supporting an Earthlike atmosphere. We track how this quantity changes in alternateuniverses, and what implications this effect has on the number of habitable planets in these universes.Additionally, we track the location and width of the temperate zone and compare this to the typicalinter-planet spacing that results from the dynamical evolution of stellar systems, which provides a niverse , , 157 3 of 34 rough estimate for the probability that a planet will end up in the temperate zone. Finally, we discussthe importance of planet migration and how this changes in other universes.An appendix is provided to collect the relevant formulas for the dependence on the physicalconstants on the variety of processes and quantities that are needed, in order to avoid distraction fromthe main text.The overarching message we derive from this analysis is that the presence of planets is notthat important in determining our location in this universe, a direct consequence of the fact that thepresence of planets is a nearly universal phenomenon throughout the multiverse. Including theseeffects barely alters the probabilities we derived before. We find that most effects act as thresholds,serving to limit the allowed parameter range rather than alter the probability distribution of observingany particular value. We find several new anthropic bounds, including the most stringent lowerbound on the electron to proton mass ratio in the literature. We find that these results are relativelyinsensitive to the exact models of planet formation and occurrence rate used, and so are robust to thesecurrent uncertainties.This is not the first work to address the question of whether planets are still present for alternativevalues of the physical parameters. Limitations on the strength of gravity and electromagnetismimposed by the existence of habitable planets was investigated in [10], where it was found thatlong lived, temperate, terrestrial planets can exist over a wide range of parameter space. This wascontinued in [11] with the investigation of the influence of the density of galaxies on planetary stability,again finding a broad allowable parameter region. Our current work is novel in not just examiningthe possibility of the existence of planets with desirable properties, but also taking care to incorporatemodern theories of planet formation into determining whether planets with these properties areindeed produced.Taken together, we analyze 12 distinct possible criteria for habitability in this paper (not countingmigration or the different views in the planetary size distribution we consider). Coupled to the 40 weconsidered in [1], this represents a total of 480 different hypotheses to compare. The quantities used tocompute these are displayed in Table 1 for convenience. Table 1.
The quantities computed in this work. Here, Q is the amplitude of perturbations, κ parameterizes the density of galaxies, λ parameterizes stellar size, GI stands for giant impact formationmechanism, and iso stands for isolation production mechanism. Quantity Description Expression f gal fraction of stars in galaxies that retain supernova ejecta erfc p Q ´ α β { q f fraction of stars born after supernova enrichment exp p´ κ { α ´ { β ´ { γ ´ q f Z fraction of stars with high enough Z for planets θ p ´ λ { α ´ β ´ { γ { q f hj fraction of stars without hot Jupiters 1 ´ ˆ κ λ α ´ { β ´ n p average number of planets around a star GI: Equation (25), iso: Equation (31) f terr fraction of terrestrial planets GI: Equation (27), iso: Equation (30) f temp fraction of temperate planets 0.0053 κ ´ { λ ´ α { β { γ ´ {
2. Fraction of Stars with Planets f p Two of the factors in the Drake equation regard the existence of planets, which will be consideredin turn in this paper. The first quantity to determine is f p , the fraction of stars that form planets. Here,we represent this as a product of factors: f p “ f gal ˆ f ˆ f Z ˆ p f hj q p hj . (2) niverse , , 157 4 of 34 In succession, we have: f gal , the fraction of stars in galaxies large enough to retain supernova ejecta, f , the fraction of stars born after supernova enrichment, f Z , the fraction of stars born with highenough metallicity for planets to form, and an optional f hj , the fraction of stars that do not producehot Jupiters. Here, the exponent p hj P t
0, 1 u is introduced as a choice of whether to include this lastcriterion or not. The other two are not treated as optional. The requirement for a galaxy to be habitable is that it must retain its supernova ejecta in orderto reprocess it into another round of metal-rich stars [12]. This sets a minimum galactic mass by thecondition that the velocity of supernova ejecta is less than the escape velocity, v „ G M ret R ret . (3)This is the asymptotic speed the supernova ejecta attains, and, to find this, a bit of the ejecta dynamicsmust be used. The initial speed can be found from energy balance [13]: this can be written in the form v „ d T SN A m p , (4)where the temperature of the supernova is set by the Gamow energy, which is the amount required toovercome the repulsive nuclear barrier and force fusion, T SN „ α m p . The mass of a typical particle inthe ejecta is related to the atomic number A „
50, which cannot conceivably vary by much. We findthat v „ T H „ α m e { „ K,below which Hydrogen becomes predominantly neutral and no more cooling takes place [14].How far does the ejecta of a supernova spread before it completely merges with the interstellarmedium, and, more importantly for our purposes, why? The observed value is around 100 pc [15],and this is after the blast has gone through several successive phases. The first is known as the blastwave phase, where the ejecta spread out at their initial velocity for roughly 100 yr, traveling a totalof a few pc. After the amount of interstellar material encountered rivals the initial mass of the ejecta,which occurs at d ST „ p M ej { ρ gal q { , the blast enters the self-similar Sedov–Taylor phase, where theblast slows and expands considerably. According to standard theory [15], the self-similarity of thedynamics dictates that the temperature of the blast falls off as T d ´ . During this phase, the velocitywill decrease with distance as v p d q “ { p d ST { d q { until the snowplow phase begins, at which pointthe speed essentially does not decrease any further. The snowplow phase occurs after the temperaturereaches the molecular cooling threshold, where it expands by a factor of a few until the density of thematerial falls to that of the surrounding medium, at which point it is completely merged. Since thebulk of the expansion takes place in the Sedov–Taylor phase, the size of the blast will be dictated bythe dynamics that take place there, and so the ultimate speed is given by v SN “ v ˆ T SN T H ˙ { . (5)Using the above relations, the asymptotic speed of supernova ejecta is found to be v SN “ α β { . (6) niverse , , 157 5 of 34 Now, we can find an expression for the minimum mass by using M min „ ρ gal R . Using thedensity of galaxies given in the appendix, this gives M ret „ M pl v ρ { “ α m { e M pl κ { m { p , (7)where the coefficient in the last expression has been chosen to reproduce the observed minimal mass of M ret “ M d [16]. As explained in the appendix, the quantity κ determines the density of galaxiesin terms of cosmological parameters.While this critical mass is important conceptually, it is not as relevant to the retention of ejecta asthe gravitational potential itself, which directly sets the escape velocity of the overdensity. The fractionof initial overdensities that exceed a potential of a given strength Φ is given by the Press–Schechterformalism as f “ erfc p Φ {p? Q qq [17], where Q is the primordial amplitude of perturbations. Usually,this expression is immediately expressed in terms of mass and density, but this more primitive formwill suffice for our purposes. With this, we can derive the fraction of matter that resides in potentialwells deep enough to produce a second generation of stars as f gal “ erfc ˜ α β { Q ¸ . (8)Perhaps somewhat interestingly, this does not depend on the strength of gravity γ at all. This effectivelyacts as a step function, severely diminishing the habitability of universes where M ret ą M typ . Note thatthis is a pessimistic estimate, as it ignores the potential for subsequent evolution that causes potentialwells to deepen with time. Nevertheless, it is a very mild bound, and so a more thorough treatment isnot called for. Since the formation of metal-rich systems is reliant on the evolution of the first stars throughto their completion, if the lifetime of massive stars exceeds the duration of star formation, then nosystems will form with any substantial metallicity. The second generation stars are not necessarilyenriched enough to produce planets, but this serves as a sufficient condition for planets to be formed atall. Then, the fraction of stars with planets can be estimated as those that are born after a few massivestellar lifetimes have elapsed. It is worth considering how these two timescales compare for generalvalues of the physical constants.The star formation rate, averaged throughout the universe, is found to decline exponentially,as gas is depleted from the initial reservoir [18]. The most naive treatment one can perform is torelate the timescale of this depletion to the free fall time of a galaxy, t dep “ {p (cid:101) SFR a G ρ q , where forsimplicity the efficiency coefficient is taken to not vary with parameters. Then, the fraction of starsborn after a time t is f ‹ p t q “ e ´ t { t dep .This needs to be compared to the lifetime of massive stars, where massive here is taken to meanlarge enough to become a type II supernova. This threshold is eight solar masses in our universe,and is set by the inner core having a high enough temperature to undergo carbon fusion [19]. As such,this threshold is parametrically similar to the minimum stellar mass, which was shown in [20] to scaleas λ α { β ´ { , which equated the Gamow energy of fusion with the internal temperature of the star.The resultant mass is about two orders of magnitude larger than the minimal mass, stemming fromthe larger repulsive barrier for large nuclei. Then, using the stellar lifetime from [1], we find t SN t dep “ κ { α { β { γ . (9) niverse , , 157 6 of 34 The normalization has been set to the value of 0.01. This has been a bit loose on several counts,namely the assumption that the second stars are always metallic enough to form planets, and theneglect of even higher mass stars, which have correspondingly shorter lifetimes. However, in practice,these worries are of no consequence, precisely because the scales are separated by such a large amount.We find that, for all practical values of the parameters, the star formation timescale exceeds the lifetimeof massive stars, so that f „ f ‹ p t SN q is always close to 1. Terrestrial planets must be formed out of heavy elements, and though even a minusculeamount would suffice in terms of actual planetary mass, the planet formation processes withinthe protoplanetary accretion disk can only occur after a high enough metallicity is reached. A recentanalysis [5] has done a nice job characterizing the metallicity needed (as a function of stellar mass,and distance to the host star) in order for planets to form out of the initial protoplanetary disk.The underlying physical picture is that there are two timescales, the lifetime of the disk t disk andthe dust settling time t dust . If the second exceeds the first, the disk will dissipate before sufficientdust may settle into the midplane, and will disperse without forming planets. Both of these dependon metallicity monotonically, and so only above some critical value will the conditions for planetaryformation hold. We detail the scaling of each in turn.The physics of accretion disks was first laid out in [21]. An initially uncollapsed cloud firstcondenses, and then through angular momentum transfer begins to form a disk. The disk remainsaround the star until either UV light from the star photoevaporates the gas or it escapes thermally,and the disk evaporates. This phase of evolution occurs on the viscous timescale of the disk, which ismuch smaller than the total disk lifetime [22].Though the precise time at which this crossover occurs depends on the exact mechanism ofphotoevaporation (X-ray versus extreme ultraviolet, other stellar sources in clusters, presence of highactivity tau phase) the drop-off of accretion is set by the disk’s secular evolution time [23], withcrossover occurring on the order of this time. Therefore, the only relevant physics that needs to bekept track of is the free fall time. Since this is given by a cloud that has approximately virialized afterJeans collapse, it is simply set by the condition that the gravitational energy is equal to the thermalenergy. Then, the accretion rate is given by M “ c s { G „ ´ M d { yr, and the timescale is givenby t disk “ M ‹ { M „ Myr. The sound speed and the temperature are given by bremsstrahlung frommolecular line cooling, as found in the appendix. Then, t disk “ ˆ λ m { p M pl α m { e ˆ ZZ d ˙ p . (10)In [24], the metallicity dependence was studied, where it was found that cooling quickens withmetallicity Z . The exact process was open to interpretation, with a handful of viable candidatemechanisms, but the overarching explanation is that the less shielding there is, the faster the disk willdissipate. Observationally and theoretically, it was found that the scaling with metallicity is consistentwith p “ {
2, and so we will adopt this for our analysis.The dust settling timescale is dictated by the rate at which dust grains sink into the midplanefrom their initial positions in the protoplanetary disk. It is given in terms of the Keplerian timescaleof the disk, t dust “ t Kepler { Z , as derived in [5]. There it was related to the growth rate of grains,and found to depend inversely on the metallicity, since only dust can participate in the accretionprocess. This is a function of orbital distance, but specifying to planets that form within what willbecome a temperate orbit for simplicity, we have t dust “ ˆ λ { m { p M { pl α { m e Z d Z . (11) niverse , , 157 7 of 34 Here, we are explicitly assuming that significant migration does not occur, which would alter themetallicity needed for planet formation, given its dependence on orbit.Equating these two timescales yields the critical metallicity Z min “ ˆ ´ λ { γ { α β { . (12)The critical value is found to be Z min “ ˆ ´ . As compared to the solar value Z d “ Z min “ Z d [5].This may be used to determine the fraction of stars that host planets by considering the amountof stars that are formed above this metallicity. This requires a model for how metallicity builds upinside a galaxy (of a given mass, as well as the distribution of galaxy masses). A full calculation ofthis sort would take us too far afield here so we will return to it in a later publication. However,a very reasonable approximation is to compare this threshold metallicity to the asymptotic metallicitya galaxy attains—this usually affords percent-level accuracy. In our universe, this is found to be Z “ α β { γ ´ { ą α ą { α ą {
685 based on galactic cooling found in [26,27]. A more forgiving bound is found using thesmallest stellar mass, which is α { β { γ ´ { ą α , β and γ . Figure 1.
The distribution of observers when taking metallicity effects into account, in the α – γ , α – β ,and β – γ subplanes. What is plotted is the logarithm of the probability of measuring any value of thecoupling constants, with red being more probable than blue, and the black dot corresponding to ourobserved values. Included thresholds are an upper bound on α for hydrogen stability, as well as alower bound for galactic cooling, and an upper bound on β for proton–proton fusion. The teal andbrown curves correspond to the metallicity and supernova lifetime thresholds, respectively, and thesupernova retention threshold is not relevant in this range. Incorporating the above effects determining the fraction of stars that host planets, and againusing the entropy and yellow light conditions, we can find that the probability of observing ourparticular values of each constant. These are defined as P p x obs q “ min t P p x ą x obs q , P p x ă x obs qu for niverse , , 157 8 of 34 any observable quantity x , the others being integrated over, and the probability of measuring anyvalue given by Equation (1). With this, we have P p α obs q “ P p β obs q “ P p γ obs q “ . The constraints above were all quite mild, indicating that the presence of planets is a fairly genericfeature of the multiverse. However, we can make a further refinement by incorporating one of the mostfamous statistical correlations in the field of exoplanets, the hot Jupiter–metallicity correlation [28].This finds that the fraction of stars that possess hot Jupiters, that is, Jupiter sized planets on orbitsextremely close to the star increase with metallicity as Z . The general (though not universal— [29])consensus is that these planets must have formed in the outer system before moving inward, to avoidthe necessity of a disk that would be so massive as to be unstable [30]. In this scenario, the migrationof the planet through the inner solar system would have certainly ejected any preexisting planets fromtheir orbits (or worse), precluding them from sustaining life. However, the authors in [31] hypothesizethat this entire process could happen early enough that the main stage of planet formation couldoccur after this migration had already taken place. If this turns out to be the case, then there may infact be no correlation between hot Jupiters and habitability (barring other factors that may impacthabitability [32]). If one wishes to include this effect, however, then the fraction of stars that host Earthsand not Hot Jupiters is given by f hj “ ´ ¯ Z Z . (14)Here, we have used the mean metallicity rather than averaging this fraction over the metallicitydistribution, but this approximation is sufficient for a first analysis. The normalization Z max is thethreshold above which the stellar system is almost assured to possess a hot Jupiter. It is set toreproduce the observed abundance of hot Jupiter systems of 3% as Z max “ Z d . In a multiversesetting, we would expect to inhabit a universe where Z max is safely above the average metallicity,beyond which any further increase would result in little increase in habitability. Somewhat in line withthis expectation, then, is the fact that in our universe hot Jupiters exist in only a few percent of systems,and mainly in those that are highly metal-rich.To investigate whether this is the result of some selection effect, we must know what determinesthis metallicity. The functional dependence is a clue: as the effect becomes more pronounced with thesquare of the nongaseous material present, this is indicative of an interaction process. What remains,however, is the question of whether this migration is a result of planet–planet or planet–diskinteractions. In fact, both explanations have been considered in the literature: references may befound in [33] . The planet–planet hypothesis is supported by the fact that the eccentricities of observedhot Jupiters are correlated with metallicity as well, indicating a more chaotic, violent origin, ratherthan the steady, deterministic process indicative of planet–disk interaction. Additionally, [35] notea substantial misalignment between the orbits of known hot Jupiters and the spins of their hoststars, which is most easily explained through a violent migration scenario. However, systems likeWASP-47 [36], which possess both a hot Jupiter and smaller companions, demonstrate that more The code to compute all probabilities discussed in the text is made available at https://github.com/mccsandora/Multiverse-Habitability-Handler. A third explanation was additionally given in [34] that the disk dispersal timescale increases with metallicity, allowing alonger period of accretion onto seed cores. niverse , , 157 9 of 34 dynamically quiet migration pathways are possible, if not necessarily the norm. Here, we onlyexpound upon the planet–planet scenario, though the others could just as readily be incorporated intoour analysis.We start by determining the value of Z max due to planet–planet interactions, as proposed in [37].As noted in [38], the coexistence of hot Jupiters and low mass planets is impossible in this paradigm,as migration occurs after the disk has dissipated. From [33], this is set by the expected number ofJupiter mass planets initially formed in the outer system. They provide a framework for estimatingthis by determining the probability that a core will attain Jupiter mass as a result of planetesimalaccretion as p ∆ M { M crit , where M crit „ M C is the mass above which runaway gas accretion ispossible and ∆ M is the typical total accreted mass. For this, we use the analytic expression for theaccretion rate from [39], M « p π q { G M crit M { ‹ Σ planetesimals ρ { a . (15)This employs the strong gravitational focusing limit, and treats the typical relative velocities ofplanetesimals as roughly given by the Hill velocity. This can be used to determine the total massaccreted by simply multiplying by the disk lifetime given in Equation (10) (making use of thesimplifications that the nonlinear oligarchic regime is not quite reached, and the initial isolationtimescale is small compared to the disk lifetime). The probability that there will be at least two gasgiants to trigger the instability will scale as p pě q „ N p , where N jup is the typical number ofplanets in the outer system, and we have assumed that p is small. The quantity N can be found bydividing the total mass of the planetary disk by the typical mass of a planet at the typical location offormation. Here, we use the initial seed being set by the isolation mass, have fixed the orbital radius tobe given by the snow line, and have taken the disk temperature to be given by viscous accretion, all ofwhich are discussed in the appendix. This gives N jup „ M disk M crit „ ˆ ´ λα { β { . (16)This scales linearly with stellar mass, in agreement with the observations in [40]. The maximal value ofmetallicity is found to be Z max „ N jup M crit M t disk “ ˆ ´ α { β { κ λ . (17)This quantity is somewhat sensitive to both α and β , but not at all to γ . This also defines a stellarmass λ hj “ ˆ ´ α { β { { κ : stars above this mass, equal to 11 M d for our values, will alwayshost hot Jupiters. This will be below the smallest stellar mass if 789 α { β ă
1, which will occur whenthe electron to proton mass ratio is about 10 times smaller. However, this criteria does not alter theprobabilities much: P p α obs q “ P p β obs q “ P p γ obs q “
3. Number of Habitable Planets per Star n e Next, we focus on the number of habitable planets per star, n e . The determination of habitabilitymay depend on many factors, such as amount of water, eccentricity, presence of any moons, magnetic niverse , , 157 10 of 34 field, distance from its star, atmosphere, composition, etc. Here, we focus on two: temperature andsize, and determine the fraction of stars that have planets with each of these characteristics.As usual, it is possible that habitability is completely independent from these properties;which viewpoint one adopts depends on how habitable one expects environments without liquidsurface water and thin atmospheres can be. In this work, we remain agnostic to either expectation andreport the number of observers for all combinations of choices, where a planet will only be habitable ifit is approximately Earthlike, and where the size and/or temperature of the planet have no effect onits habitability.It should be noted that we are restricting our attention here to surface dwelling life on planetsorbiting their star. Thus, life in subsurface oceans and/or on icy moons like Enceladus [41], or evenmore exotic types of life (e.g., [42,43]), are not considered. It is our plan to consider these alternativeenvironments in future work.The number of habitable planets can be broken down as n e “ n p ˆ p f terr q p terr ˆ p f temp q p temp . (19)Here, the average total number of planets around a star is n p . The fraction of terrestrial massplanets is denoted f terr , and f temp is the fraction of planets that reside within the temperate zone.The exponents p i P t
0, 1 u parameterize the choice of whether to include these conditions in thedefinition of habitability or not. These quantities all depend on stellar mass, giving preference to largestars because they make larger planets, and small stars in that their temperate zone is wider comparedto the interplanetary spacing.Estimating these quantities is somewhat muddled by the current uncertainties in planet formationtheory. Not only is the distribution of planet masses contested in the literature, but the exact formationpathways, as well as the physics that dictates the results, is not completely settled. Where we comeacross disagreement, we separately try each proposal, in order to understand the sensitivity of ouranalysis to present uncertainties. While the results for the overall probabilities can vary by a factor of2, the upshot is that our estimates are relatively robustx. The size of a planet is of crucial importance because it dictates what kind of atmosphere it canretain. If it is too small, all atmospheric gases will eventually escape, whereas if it is too large, it willretain a thick hydrogen and helium envelope, leading to a runaway growth process. Terrestrialplanets must have a very specific size in order that the escape velocity exceeds the thermal velocityfor heavy gases such as H O, CO and N , but not that of the lightest gas H and He. In our universe,and for temperatures within the range where liquid surface water is possible, this restricts the rangeof planetary radii to be within 0.7 and 1.6 that of Earth’s [44]. This is a narrow sliver compared to theeight orders of magnitude mass range of spherical, non-fusing bodies, ranging from the potato radiusof 200 km to 10 Jupiter masses. In terms of fundamental parameters, this requirement gives the massto be M terr “ α { M pl m { e m { p . (20)The coefficient has been set to reproduce Earth’s mass, but the allowed spread in masses is taken to bebetween 0.3–4 M C .There are compelling arguments that complex may only be possible on terrestrial planets,with atmospheres composed of only heavy gases [45]. Any smaller, and the planet would beMarslike, an apparently barren wasteland incapable of sustaining any appreciable liquid ar atmosphere.The other extreme would be Neptunelike, with its hellish surface temperatures, pressures and windspeeds. Of course, these arguments may be misguided, but here we explore the consequences ofadopting them for the multiverse computations we perform. niverse , , 157 11 of 34 It is important to note that the conditions that set the presence of atmospheres are completelyseparate from the physics that dictates the size of planets, which is set by the clumping of the initialcircumstellar disk . Nonetheless, the observed population of rocky planets is thought to peak at onlyslightly super-Earth mass, making the production of terrestrial planets the norm for stars throughoutthe universe. To be fair, the current exoplanet samples are biased towards large mass planets andbecome very incomplete below Earth mass [48], but a number of different groups have concludedthat a detectable turnover is present near Earth masses: the authors of Ref. [9] find a good fit to alog-normal distribution that peaks at 1.3 R C . In Ref. [7] a Rayleigh distribution with width 3 M C is used.The authors of Ref. [8] advocate for a broken power law with turnover at 5 M C . It was noted in Ref. [6]that the distribution appears to be flat below 2.8 R C . Use of these differing proposed distributionsmake very little difference to our final outcome.However, not everyone is convinced that the mass distribution exhibits a peak, and even ifthere is one, it is just as reasonable to assume that there are many more smaller mass planets forevery planet of Earth size, as the plethora of small asteroids and comets in our system indicates.Because of the incompleteness of current exoplanet surveys for small mass planets, there is roomfor such disagreement at the current moment. Additionally, even if the mass peak is real, it is onlyobserved for close in exoplanets, and so requires an extrapolation to Earthlike orbits, where differentdynamics may be at play. One possibility is that the peak at super-Earth mass may be due to theirenhanced migration capability [49]. We will consider each scenario in turn.3.1.1. What Sets the Size of Planets?Why is the turnover so nearly equal to Earth mass planets, out of the potentially eight ordersof magnitude that could have been selected instead? Simulations provide a means to address thisquestion: it was found in Ref. [50] that the mass of planets is directly proportional to the amount ofinitial material present in the disk, so that increasing disk mass makes larger, rather than more, planets.In this scenario, nearly all the material initially present in the disk eventually gets constituted intoplanets, with negligible (perhaps a factor of two, but not an order of magnitude) losses throughout theevolution of the system. Determining the final planet mass in this setup requires knowledge of theinitial disk mass, as well as the fraction of material within the inner solar system. This boundary is setby the snow line, the difference in composition interior and exterior to which dictates the formationof rocky versus icy planets. In the following, we adopt the conventional view that little migrationtakes place during planet formation, and comment further on alternative pathways when migrationis discussed.Current observations indicate that the initial mass of heavy elements in protoplanetary disksis roughly proportional to disk mass, M disk “ M ‹ [51]. In Ref. [52], a stronger dependence wasfound, but the scatter of 0.5 dex is larger than the trend, and the observed trend was suggested topossibly be due to a selection effect arising from processing into undetectably large grains, so weomit this stronger scaling for now. For solar mass stars, this works out to be roughly 10 M Jupiter , or3300 M C . This mass is distributed out to a radius of „
100 AU, which is set by the conservation ofangular momentum, and the initial size of the collapsing cloud. From the appendix, we arrive at thefollowing expression for disk size: r disk “ ˆ ´ λ { κ M pl m p . (21) This is not entirely true: there is known to be some feedback between the irradiation of initial atmospheres from the hoststar that favors both small and large atmospheres [46]. While this leads to the interesting bimodal distribution of observedradii [47], this is driven by atmospheric size, and is certainly not enough to affect the terrestrial core. niverse , , 157 12 of 34 This may be significantly altered if the young star is in a dense environment [53], but this scaling willsuffice for our purposes.To determine the amount of material present within the snow line, one must take the surfacedensity profile into account. For this, we use the expression for Σ p a q found in the appendix, which isinversely proportional to a . Using these, the typical mass of an interior planet becomes M inner „ η ă a snow R disk M disk , (22)where we have included the quantity η ă „ a snow “ λ M pl α { m { e m { p . (23)Note that, perhaps unsurprisingly, the snow line is situated outside the temperate zone for all relevantparameter values. The dependence on stellar mass was found taking M λ , but is generically foundto scale as a snow λ { ´ [56]. This is now enough to determine the parameter dependence of M inner ,which will ultimately be used to derive the expected number of planets per star as a function ofthese parameters.Though we tend to favor accretion dominated disks throughout this work, irradiation from thecentral star can actually play a significant role as well [22]. If this is the dominant mode of heating,then the snow line will instead be given by the expression a snow “ λ { α ´ m ´ { e m ´ { p M { pl .Though this is functionally quite similar to the accretion dominated case from above, in Table 2,we investigate the effect of assuming this form instead. Actually, both are almost equally relevant indetermining the position of the snow line, which helps to greatly complicate the disk structure (as wellas enhance the variability between different star systems [57]). Additionally, irradiation from otherneighboring stars may be important as well, especially in clusters [58], but we do not consider thiscontribution in this work. Table 2.
Display of the insensitivity of the probabilities of our observables to the choices made regardingplanet formation. In addition to the options displayed, they may have been combined, but this wouldbelabor the point.
Choices P p α obs q P p β obs q P p γ obs q standard 0.229 0.260 0.409Rayleigh distribution 0.229 0.251 0.411irradiation 0.255 0.320 0.426with no λ dependence 0.380 0.254 0.007shot noise 0.232 0.260 0.306 Determining the average planet mass is somewhat involved, given the many distinct stages ofgrowth that occur as microscopic dust grains agglomerate to the size of planets. For reviews of thismulti-stage process, see [59–61]. In brief, planetesimals form characteristic masses as the final outcomeof pebble (or dust) accretion. This arrangement is unstable, and ultimately leads to a phase of giantimpacts, wherein planetesimals collide together to form full planets.An estimate for the maximum mass a planet can attain after a phase of growth through chaoticgiant impacts was found in [62]. There, they assumed no migration, small eccentricity, and determined niverse , , 157 13 of 34 the width of the ‘feeding zone’ to be ∆ a “ v esc { Ω by noting that within this region planet–planetinteractions result in collisions rather than velocity exchange. This yields M planet „ ˜ π Σ a { ρ { M { ‹ ¸ { , (24)where ρ is the average density of the planet and Σ is the disk density. With this, we can use theappendix to reinstate parameter dependence into the expressions for the average number of planets(normalized to 3 for the solar system) as well as the typical planet mass: n p “ α { β { κ { λ { , (25) M planet M terr “ ˆ κ { λ { α { β { . (26)The latter has quite a steep dependence on stellar size. This is expected from the simulations [50],with a dependence closer to linear, but is not particularly observed in exoplanet catalogs due tolarge scatter [63]. In our calculations, we explore the effect of ignoring this dependence altogether,displayed in Table 2. A full treatment would take the dependence on semimajor axis into account,rather than simply evaluating at the snow line: in fact, this would be somewhat unnecessary, as thescatter observed in exoplanet surveys, simulations, and indeed within the solar system masks anydependence that may exist.For the fraction of planets that are terrestrial, we use the log-normal distribution of [9] since thisis what is expected of the core accretion process, though we check that the actual distribution useddoes not alter the outcome much. Under these assumptions, we can find the probability that a planetwill be terrestrial as f terr “
12 erf ¨˝ log ´ M terr M planet ¯ ` σ ? σ M ˛‚ ´
12 erf ¨˝ log ´ .3 M terr M planet ¯ ` σ ? σ M ˛‚ . (27)This function peaks at M planet „ M terr , and approaches are when M planet is very different from M terr .Being a two-parameter distribution, this requires not just the mean, but also the variance. As it is notcurrently known what sets this quantity, here we explore two options: the first uses maximizationof entropy production to set σ M “ {?
6, which is fairly widely observed in natural processes [64].This estimate should occur for large systems, but for small systems one would expect the variance tobe set by shot noise instead, σ „ a M iso { M inner , making use of the isolation mass defined in the nextsection. The dependence on the various parameters is displayed in Figure 2.The probabilities of observing the observed values of our constants are computed for the variouschoices we made in Table 2. These can be compared with Equation (13) that only considered thefraction of stars with planets. The most significant change for our most favored prescription is theprobability of observing our electron to proton mass ratio, which is decreased by less than a factor of 2.Such insensitivity hardly constitutes any evidence for whether life should only appear on terrestrialplanets. More interestingly, if the dependence on stellar mass is neglected, our strength of gravitybecomes quite uncommon to observe. This gives us strong reason to suppose that, if life requiresterrestrial planets, we will begin to see a correlation between the two soon, and, if we don’t, then liferequiring Earth mass planets is incompatible with the multiverse at the 2.7 σ level. niverse , , 157 14 of 34 Figure 2.
The distribution of observers if life can only arise on a terrestrial planet in the α – γ , α – β ,and β – γ subplanes, size being dictated by a giant impact phase. Here, we have used a log-normaldistribution with σ “ {?
6, excluded λ dependence on the average planet mass, and assumed accretiondominated disks. Aside from this, though, the choices we made to come up with these estimates do not affect theoutcome very much at all. On the one hand, this is disappointing, as the stronger the dependencethese probabilities have on the assumptions of planet formation, the stronger our predictions canbe about which to expect to be dominant. However, this is also heartening: because the currentuncertainties about planet formation do not affect the outcome all that much, we are able to trust thebroad conclusions we have reached a bit better.3.1.2. Is Life Possible on Planetesimals?The mass discussed above really refers to the maximal planet mass of a system, which formas a result of the secondary stage of collisions after the isolated planetesimals form. However,this agglomeration will likely not completely deplete the system of its primordial planetesimals,and so there are also expected to be numerous smaller planets accompanying each large one, as is thecase in and around the solar system’s asteroid belt. If these smaller bodies are considered as potentialabodes for life as well, the distribution continues past Earth masses, rather than having a peak there.As a planet is condensing out of the protoplanetary disk, it eventually reaches what is termed asthe pebble isolation mass, which from the appendix is given by M iso “ ˆ κ { λ M pl α { m { e m { p . (28)The isolation mass is a function of the semi-major axis, but if we evaluate it at the edge of theinner system given by Equation (23), we find, using the value for the disk surface density from theAppendix A, M iso M terr “ ˆ κ { λ α β { . (29)Here, the density of the galaxy comes into play in setting the outer edge of the disk. The dependenceon stellar mass in this expression is quite close to that found in [55], M iso λ { .It remains to specify the distribution of planetary masses in order to find the fraction that areterrestrial in this picture. It is generically expected to take a power law form that continues to the smallmass cutoff, so that N p M q “ p M iso { M q q . However, different authors prefer different values for theslope: Ref. [65] find q “ ˘ niverse , , 157 15 of 34 which translates into q “ ˘ M r . Refs. [39,67] find q “ q “
1. Ref. [68]favors a value of q “ q to investigate its influence on the probabilities. The fraction of terrestrial planets is then f terr “ min " ˆ M iso M terr ˙ q * ´ min " ˆ M iso M terr ˙ q * . (30)This is an increasing function of stellar mass until M iso ą M terr , reflecting the expectation [69] thatearthlike planets should be rare among low mass stars. With this prescription, stars above a certainmass will not produce any earthlike planets because the isolation mass will exceed the largest terrestrialplanet size. This defines the largest viable stellar mass λ iso “ κ ´ { α β { , which correspondsto 6.3 M d in our universe. This will only exceed the minimal stellar mass if α { β { κ ´ { ą q ă
1, the average mass for this distribution is formally infinite, which presents a problem forusing our expression for the expected number of planets in a system as given by n p “ M inner {x M p y .However, the total mass in the inner disk introduces a large mass cutoff, for which we have n p “ q ´ q ´ M inner M iso ¯ ´ ´ M inner M iso ¯ ´ q ´ ´ M inner M iso ¯ ´ q . (31)This interpolates between p q ´ q{ qM inner { M iso for q ą p q ´ q{ q for q ă
0. The ratio of masses is M inner M iso “ α { β { κ { λ { , (32)thus that this equates to 30 with our constants. With this viewpoint, the distribution of observers isplotted in Figure 3. Figure 3.
The distribution of terrestrial planetesimals that result from isolated accretion in the α – γ , α – β ,and β – γ subplanes, without the subsequent phase of giant impacts. The slope here is q “ {
3. The tealline represents the region where the largest star capable of hosting terrestrial planets is smaller than thesmallest possible star.
The overall probabilities are calculated for different representative values of the power law inTable 3. It can be seen that, for increasing q , the probability for α increases, while the other two decrease. niverse , , 157 16 of 34 In particular, for q “
2, the probability of observing our value of the electron to proton mass ratio isdisfavored by 2 σ . Table 3.
Display of the probabilities of our observables for different values of the power law slope.
Exponent P p α obs q P p β obs q P p γ obs q q “ { q “ { q “ q “ Perhaps the most commonly employed habitability criteria is that a planet must be positioned asuitable distance away from its host star to maintain liquid water on its surface. This assumption isso pervasive that this region is usually referred to as the circumstellar habitable zone. In the spirit ofremaining agnostic toward the conditions required for life, we will adhere to the recently proposedrenaming as the ‘temperate zone’ [70]. If this is indeed essential for life, then the expected number ofhabitable planets orbiting a star will depend both on the interplanetary spacing, as well as the widthand location of the temperate zone. A rather clement feature of our universe is that the width of thetemperate zone is comparable to the interplanetary spacing (for sunlike stars). Because of this, it isrelatively common that one of the planets in any stellar system is situated inside the temperate zone,no matter its particular arrangement. This could be contrasted to the hypothetical case where thetemperate zone were much narrower than the interplanetary spacing, in which case the odds of aplanet being situated inside it would be quite low. However, this coincidence of distance scales isnot automatic: both these quantities are dependent on the underlying physical parameters, and so inuniverses with different parameter values the expected number of potentially habitable planets perstar will be altered. We go through these length scales in turn, and then fold them into our estimate forthe overall habitability of the universe.The boundaries of the temperate zone depend on the characteristics of the planet in question,such as the atmospheric mass and composition [71], its orbital period [72], etc. However, these detailsonly alter the location of the temperate zone to subleading order, and do not affect the scaling withfundamental parameters we are interested in. If the planet is assumed to be a simple blackbody,then the temperature is set solely by the amount of incident flux. In this case, the location of thetemperate zone will be a temp “ T ‹ E H O R ‹ “ λ { m { p M { pl α m e . (33)Albedo and greenhouse effects will change the coefficient, but not the overall scaling. Determiningthe width of the temperate zone entails finding the temperatures at which runaway climate processesoccur, but we will now argue that both the inner and outer edge are dictated by (broadly speaking) thesame underlying physical process of phase change, and so the width of the temperate zone will scalein the same way as its location.The inner edge of the temperate zone is set by the runaway greenhouse effect: this occurs when atemperature threshold is crossed that allows an appreciable amount of water vapor to be sustainedin the atmosphere. Since this serves to trap infrared light from escaping to space, this will increasethe temperature further, in turn driving a further increase in atmospheric water vapor. Once theatmosphere is comprised primarily of water, it will be photodissociated and/or escape to space,leaving the Earth in a dry and Venus-like state [73]. Since there is always some level of outward flux,the ocean will escape into space eventually if a long enough time has elapsed. Therefore, the exactthreshold for this process is defined as when the timescale for this process is of the order of one billion niverse , , 157 17 of 34 years, which in turn will depend on the mass of the planet’s ocean. However, key to our discussion isthat the change in the atmospheric water fraction occurs very abruptly when the temperature crossesthe latent heat of vaporization, going from equilibrium values of 10 ´ ´ increases the albedo of aplanet, leading to a runaway icehouse effect [73]. Because the amount of atmospheric carbon dioxideis also set by reactions due to intermolecular forces, it scales the same as above. Therefore, both theinner and outer edges of the temperate zone are set by molecular binding energies, and so the widthwill always be of the same order as the mean .The exact delineations are subject to the uncertainties of the atmospheric model used, but,for definiteness, we take ∆ R HZ “ ˘ R Hill .As mentioned in [62], the typical multiple of mutual Hill radii is not universal, but can be shownto depend on the width of a planet’s ‘feeding zone’ to be given by p ρ a { M ‹ q { , where ρ is the densityof matter and a is the semimajor axis. Though above we were interested in the typical size of a planetduring this process, here we may use this condition to find the typical spacing as a spacing „ ˜ π Σ a { ρ { M { ‹ ¸ { “ κ { λ { γ { α { β { (34)This was evaluated at the center of the temperate zone to give the ratio of these two length scalesin terms of fundamental constants. Since this ratio loosely sets the fraction of planets within thetemperate zone, we arrive at f temp „ ∆ a temp a spacing “ α { β { κ { λ { γ { . (35) This neglects other potential thresholds, such as the inner boundary set by the photosynthetic threshold of carbon dioxideabundance, which is remarkably close to the inner boundary discussed here [76]. niverse , , 157 18 of 34 Set this to the value of 1.6 in our solar system. Notice that, for α or β much smaller or γ much larger,the fraction of temperate planets will be diminished. Additionally, this quantity is larger for smallermass stars, reflecting the comparatively broad temperate zones there.This distribution is illustrated in Figure 4. The corresponding probabilities for observing ourvalues of the constants are P p α obs q “ P p β obs q “ P p γ obs q “ Figure 4.
Distribution of observers if life may only exist within the temperate zone, in the α – γ , α – β ,and β – γ subplanes. The teal line is the boundary for which planetary disks are smaller than thetemperate zone. Let us also take this opportunity to determine what values of parameters would render diskssmaller than the habitable orbit, thereby precluding temperate planets from ever forming. Based offthe expressions above and in the appendix, we have a temp R disk “ ˆ κ λ { γ { α β . (37)Aside from the obvious condition that, if the mean free path of the galaxy were 100 times smaller,the protoplanetary disk would be shrunk by the same amount, this also places restrictions on thefundamental parameters. Though these introduce lower bounds for α and β , this is the region with veryfew temperate observers anyway, and so this does not change the habitability estimate appreciably,especially since these scales would have to shift by two orders of magnitude before the threshold isreached. A more stringent boundary would require that disks be larger than the orbits of the gas giants.If not, they would be precluded from forming in the first place, and, if these were essential for life onEarth, universes like this would be altogether barren. Traditional models of planet formation assume that little orbital migration takes place duringplanet formation. However, the large number of gas and ice giants found situated extremely close totheir host stars [82], the presence of orbital resonances found in some exoplanet systems [83], and icycomposition of planets within the snow line [84] all point to the presence of migration. Migration may niverse , , 157 19 of 34 strongly affect the habitability of planetary systems, as if it proceeds for too long, all inner planetswill drift toward the inner edge of the disk at around 0.014 AU [85], and giant planets migratingacross the temperate zone would destabilize the orbits of any existing planets. The vast diversity ofsystems found, as well as analytic and numeric simulations of protoplanetary disks [86], point to anexquisitely complex array of migration scenarios, which will be selectively operational dependingon the characteristics of the initial system such as disk mass, viscosity, and the size and locationsof the planets. Nevertheless, it is possible to perform rough estimates for when migration will bepresent in a given system. In this section, we do not attempt to characterize all the complex features ofplanet migration in universes with alternate physical constants, but rather wish to provide a usefuldiagnostic for when migration will be important. We will find conditions that the physical constantsmust satisfy so that all planets do not migrate into their host stars at an early stage of evolution, ourobserved values being intermediate such that this scenario only afflicts some percentage of planetarysystems with unlucky characteristics. It should be noted that some amount of migration appears tohave occurred even in the outer solar system [87]. However, we evidently ended up with at least onehabitable location. In our particular instance, the particular positions of the giant planets ultimatelyhalted migration, preventing the obliteration of the inner solar system [88].Migration occurs when a planet’s influence on the surrounding disk produces a net torque on theplanet itself, usually driving it inward. As such, migration halts after the time t disk when the disk hascleared out. This may be compared to the migration timescale t mig „ a a „ L Γ „ Ω a M Γ , (38)where M is the planet’s mass, L its angular momentum and Γ the torque it experiences. A heuristiccondition for when migration will be significant was found in [89] as t mig À t disk . There are variouscontributions to the torque, and which is dominant organizes migration into several different types,which depend on the circumstances of the case at hand. These are classified into type I, which ariseswhen a trailing overdensity of dust behind a planet (and preceding in front) exerts a torque, and type II,in which the planet is capable of opening up a gap in the disk, resulting in a torque imbalance from theabsence of material (for a recent review see [86]). The latter is more relevant for larger planets capableof significantly altering the disk structure, and the former more relevant for smaller planets, which arenot. They will each be considered in turn, resulting in conditions on the fundamental constants thatmust be satisfied in order for a system with given characteristics to retain its initially habitable planets.For type I migration, the timescale is derived in [90]: t I “ h M ‹ Σ Ω a M , (39)where h is the disk height, set by the sound speed. Then, when using the expressions from theAppendix A and Equation (10), and specifying to terrestrial planets situated in the temperate zone,we find t I t disk λ { β { γ { κ α { . (40)If this quantity becomes too large, then Earthlike planets in all systems will migrate into their stars,and the universe would be uninhabitable in the traditional sense. Note the dependence λ , indicatingthat this type of migration is more important for low mass stars.This estimate would also be relevant for the production mechanism for Trappist-1 type planetsproposed in [91], whereby earthlike planets form behind the snow line before migrating inwards.Such an unconventional pathway is needed to explain this system [92], which has multiple Earthsized planets orbiting the temperate zone of a 0.08 solar mass star with planets near mean motionresonances [93] and possessing icy compositions [84]. This will not be undertaken here, however. niverse , , 157 20 of 34 Even if we restrict our attention to parameter values where Earthlike planets do not undergosubstantial migration, we may still run into trouble if Jupiter-like planets routinely barrel through theirplanetary systems. As this is governed by the physics of type II migration, the conditions that mustbe satisfied for (the majority of) these planets to stay put are somewhat different. Here, we restrictour attention to the case where a full gap is opened in the disk, and where the mass of the planet issubstantially smaller than the mass contained in the disk. In this case, the timescale of this migrationprocess is given by [86]: t II “ a ν , (41)where ν is the viscosity. Strictly speaking, these quantities are supposed to be evaluated not at theposition of the planet, but rather the place of maximum angular momentum deposition, which can beapproximated as a ` R Hill . For our purposes, however, this correction is negligible for the scalingarguments. Then, the ratio of this timescale to the disk lifetime is t II t disk λ { γ { α disk α β { . (42)Here, α disk is the standard parameterization of disk viscosity, discussed further in the appendix.Since this is inversely proportional to disk mass, type II migration is more important for smallerdisks, opposite to the type I case. Additionally, since only the scaling with β is flipped from before,the conjunction of these two migration scenarios is capable of putting an upper bound on α anda lower bound on γ , if the other quantity is fixed. The thresholds for both types of migration aredisplayed in Figure 5. However, the absolute normalization of each of these timescales is uncertain,so we do not derive how the probabilities are altered due to these effects. Figure 5.
Display of when runaway migration takes hold, in the α – γ , α – β , and β – γ subplanes. Theteal line is for type I migration, the orange line is for type II, and the purple line is when the massof the fastest migrating body is equal to the terrestrial mass. All curves have been normalized so thetimescales in our universe are five times larger than the critical value.
4. Discussion: Comparing 480 Hypotheses
Having spent the last two sections detailing the physics behind a multitude of processes that mayinfluence the habitability of a system, we now synthesize these into an estimate of the probability ofobserving our measured parameter values, for each combination of individual habitability hypotheses.To summarize our results so far, we have firstly included several conditions necessary for planetformation, namely that the majority of galaxies should be larger than the minimal retentive mass thatmassive stars should have shorter lifetimes than the star formation timescale, and that the minimum niverse , , 157 21 of 34 metallicity needed to form planets should be smaller than the asymptotic value. Of these three,the third was most constraining. These are also presumably not optional, as opposed to the rest of thecriteria that were considered: these were the absence of hot Jupiters, the production of terrestrial planetsboth through giant impact and isolation formation pathways, and the fraction of planets that end upin the temperate zone. Whether these are necessary for life are still intensely debated, and so eachprovides a prime opportunity to determine its compatibility with the multiverse hypothesis. Becausethey are all independent criteria, taken in conjunction they lead to a total of 2 ˆ ˆ “
12 possibilities.This does not include the 16 different choices we made in terms of planet formation, as well as thepotentially continuous parameter signifying the slope of the power law for smaller planets. We displaythe probabilities of observing our measured values in Table 4 for the criteria mentioned in this paper.Note that, though the spread in probabilities is around 2–3 for each, all choices are well within anacceptable range to explain our observations within the multiverse context.
Table 4.
Probabilities of different combinations of the habitability hypotheses discussed in the text.Here, yellow stands for the photosynthesis criterion, S the entropy criterion (both explained belowEquation (1)), temp for the temperate zone, GI and iso for the terrestrial planet criterion with the giantimpact and isolation production mechanisms, and HJ the hot Jupiter condition.
Criteria P p α obs q P p β obs q P p γ obs q L yellow S 0.189 0.437 0.318 2.57yellow temp S 0.241 0.363 0.151 1.54yellow GI S 0.229 0.26 0.409 6.72yellow GI temp S 0.377 0.196 0.42 6.65yellow iso S 0.234 0.419 0.436 5.02yellow iso temp S 0.313 0.48 0.245 3.54yellow HJ S 0.181 0.428 0.308 2.59yellow HJ temp S 0.237 0.359 0.147 1.54yellow HJ GI S 0.23 0.261 0.41 6.72yellow HJ GI temp S 0.378 0.197 0.419 6.64yellow HJ iso S 0.231 0.424 0.431 5.05yellow HJ iso temp S 0.311 0.483 0.241 3.54 When combined with the 40 additional combinations from [1] (including combinations of the tidallocking criterion, which posits that only planets that are not tidally locked are habitable, the biologicaltimescale criterion, positing that only stars that last several billion years are habitable, the convectivecriterion, which states that only stars which are not purely convective are habitable, the photosynthesisand yellow criteria, which state that photosynthesis is necessary for complex life, defined with anoptimistic and pessimistic wavelength range, respectively, and the entropy criterion, where habitabilityis proportional to the total amount of entropy processed by a system), there are a total of 480 separatehabitability criteria that may be considered. Of these, only 43% of them give rise to probabilities ofobserving all three constants we consider of greater than 1%. The full suite of criteria is displayed inTable 5 at the end of this manuscript. For brevity, we omit the convective criteria of [1] because it onlyever marginally changes the numerical values, and in all instances its inclusion does not affect theviability of the combination of other hypotheses one way or the other. Of the 190 habitability criteriawhich give probabilities of over 10%, all make use of the entropy condition. A further 16 which do notinclude the entropy condition have probabilities greater than 1%—all of them benefit from an interplaybetween the yellow, tidal locking and biological timescale criteria, which place both upper and lowerbounds on the types of allowed stars. The rest of the habitability criteria can safely be regarded asincompatible with the multiverse hypothesis.The inclusion of multiple hypotheses leads to nonlinear effects, as the interplay between thedistribution of purported observers and the anthropic boundaries alter the overall probabilities in niverse , , 157 22 of 34 sometimes surprising ways. That being said, none of the criteria have that drastic of an effect on theprobabilities, especially when including the entropy condition.Some criteria, namely the terrestrial and temperate conditions, introduce lower bounds to somecombination of α and β . In fact, lower bounds on these quantities are somewhat hard to come by in theanthropics literature, though it has always been clear that they should exist, as a world with masslesselectrons or no electromagnetism would certainly be very different from our own. The bounds we findare stronger than those that exist in the literature.We also introduce a new measure of our universe’s fitness, which we term the luxuriance. Thisis defined as the expected number of observers in our universe divided by the average number ofobservers per universe, restricting to universes that do have observers: L “ P p α obs , β obs , γ obs q ş d (cid:126) α θ p P p α , β , γ qq ş d (cid:126) α P p α , β , γ q . (43)Here, the integration is over all three constants, and θ p x q is the Heaviside step function: θ p q “ θ p x q “ x ą
0. The rationale for including this is that, for most habitability criteria, the vastmajority of universes will be sterile, obfuscating comparisons between different criteria. Restrictingto universes that only contain life gives a better feeling for how good our universe is at satisfyingthe chosen criteria. If our universe is better than typical at making life, then this quantity will begreater than 1. While this is not actually the guiding principle for evaluating whether our observationsare consistent with the multiverse, it is a somewhat interesting quantity to consider. It gives someindication of how strongly observers may cluster within the multiverse- and the strong dependence ofthe properties we discuss on physical constants leads us to expect that they will, so that the majority ofobservers do find themselves in overly productive universes. The luxuriance ranges by two orders ofmagnitude for the different possibilities we consider, but the maximum is L “ niverse , , 157 23 of 34 environments thus does not seem to be the most important factor in determining which of the potentialuniverses we find ourselves situated in. The other factors of the Drake equation which we will explorein subsequent works [94,95] will uncover many additional predictions for the requirements of life.
5. Conclusions
We have demonstrated that there are plenty of habitability conditions that are completelyincompatible with the multiverse: what this illustrates is that, if any of the ones we have uncoveredso far are shown to be the correct condition for the emergence of intelligent life, then we will beable to conclude to a very high degree of confidence (up to 5.2 σ ) that the multiverse must be wrong.It should be stressed that there is a great deal more that these conditions omit: nothing at all is saidabout how habitability is affected by things like planetary eccentricity, elemental composition, waterabundance, or a host of other potentially paramount aspects of a planetary system [96,97]. The de factostance on all omissions is that they have no bearing on habitability, and it will only be through futurework, including all possibly relevant aspects that a fully coherent list of predictions may be assembled.Further still, placing a priority on the relative availability of each type of universe based on reasonablygeneric arguments, the precise probabilities, and the conclusions that follow will tremendously benefitfrom a way of being able to derive this prior with absolute surety. Since only a single one of themyriad habitability criteria is ultimately true, and since we will eventually be able to determine whichone that is once we have a large enough sample of life-bearing planets, this demonstrates that themultiverse is capable of generating strong experimentally testable predictions that are capable of beingverified or falsified on a reasonable timescale, the hallmark of a sensible scientific theory. Table 5.
Probabilities of various hypotheses, including those from [1] (continued on following pages).In addition to the abbreviations from Table 4, the shorthand is: photo: photosynthesis (optimistic),yellow: photosynthesis (conservative), TL: tidal locking, bio: biological timescale, S: entropy, temp:temperate zone, GI and iso: terrestrial planet with giant impact and isolation production mechanisms,resp. and HJ: hot Jupiter.
Criteria P p α obs q P p β obs q P p γ obs q L number of stars 0.381 0.355 8.06 ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ TL temp 0.455 0.257 1.93 ˆ ´ ˆ ´ niverse , , 157 24 of 34 Table 5.
Cont . Criteria P p α obs q P p β obs q P p γ obs q L TL GI 0.421 0.283 6.42 ˆ ´ ˆ ´ TL GI temp 0.237 0.29 8.16 ˆ ´ ˆ ´ ˆ ´ TL iso temp 0.454 0.356 3.63 ˆ ´ ˆ ´ ˆ ´ TL HJ temp 0.454 0.256 1.8 ˆ ´ ˆ ´ TL HJ GI 0.422 0.284 6.38 ˆ ´ ˆ ´ TL HJ GI temp 0.236 0.291 8.12 ˆ ´ ˆ ´ ˆ ´ TL HJ iso temp 0.455 0.355 3.55 ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ photo temp S 0.338 0.292 0.207 7.07photo GI 0.016 0.296 8.09 ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ photo HJ temp S 0.337 0.291 0.205 7.08photo HJ GI 0.016 0.297 8.04 ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ niverse , , 157 25 of 34 Table 5.
Cont . Criteria P p α obs q P p β obs q P p γ obs q L photo bio HJ 0.0631 0.103 1.71 ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ photo TL temp S 0.453 0.288 0.283 10.8photo TL GI 0.016 0.312 4.34 ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ photo TL HJ S 0.36 0.394 0.355 11.2photo TL HJ temp 0.412 0.197 2.51 ˆ ´ ˆ ´ photo TL HJ temp S 0.458 0.276 0.272 10.9photo TL HJ GI 0.016 0.312 4.31 ˆ ´ ˆ ´ ˆ ´ ˆ ´ niverse , , 157 26 of 34 Table 5.
Cont . Criteria P p α obs q P p β obs q P p γ obs q L photo TL bio HJ iso S 0.243 0.49 0.397 20.7photo TL bio HJ iso temp 0.0966 0.43 0.000255 0.0693photo TL bio HJ iso temp S 0.348 0.375 0.367 25.3yellow 0.486 0.162 8.78 ˆ ´ ˆ ´ yellow temp 0.492 0.161 2.31 ˆ ´ ˆ ´ yellow GI 0.00555 0.292 6.88 ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ yellow HJ 0.486 0.162 8.64 ˆ ´ ˆ ´ yellow HJ temp 0.492 0.161 2.27 ˆ ´ ˆ ´ yellow HJ GI 0.00554 0.292 6.83 ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ yellow bio 0.0351 0.102 1.72 ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ ˆ ´ niverse , , 157 27 of 34 Table 5.
Cont . Criteria P p α obs q P p β obs q P p γ obs q L yellow TL HJ GI 0.00345 0.38 4.82 ˆ ´ ˆ ´ ˆ ´ ˆ ´ Funding:
This research received no external funding.
Acknowledgments:
I would like to thank Cullen Blake, Diana Dragomir, Scott Kenyon, Jabran Zahid, and LiZeng for useful discussions.
Conflicts of Interest:
The author declares no conflict of interest.
Appendix A. Planetary Parameters
In this appendix, we collect results on how various quantities relevant to our estimates in thiswork depend on orbital parameters of the stellar system, as well as fundamental quantities. To begin,we display the typical molecular binding energy: T mol “ d α m mol r “ α m { e m { p . (A1)This also defines the temperature required for liquid water. Planets:
The mass of a terrestrial planet, based on the criteria that carbon dioxide but not heliumis gravitationally bound to the surface at these temperatures, is M terr “ ´ α { m { e M pl m { p . (A2) niverse , , 157 28 of 34 Next, we display the range of habitable orbits from the host star, using estimates from [1,98]: a temp “ ´ λ { m { p M { pl α m e , (A3)which is based off of the temperature of a perfect blackbody located at that position, T p a q “ T ‹ c R ‹ a . (A4)The speed of a circularly orbiting planet at a given location is v “ c GM ‹ a “ Ω a (A5)and Ω is the angular frequency. This also defines the orbital period as t Kepler “ π { Ω .The region of influence of a planet of mass M and orbiting a star at semimajor axis a is known asthe Hill sphere, and has the radius R Hill “ ˆ M M ‹ ˙ { a . (A6) Galaxies: the average galactic density is set by the galaxy at time of virialization, with asubsequent era of contraction due to equilibration [11,99]: ρ gal “ ˆ κ m p (A7)corresponding to 10 ´ g/cm . Here, κ “ Q p ηω q { “ ´ is a composite of the primordial amplitudeof perturbations Q “ ˆ ´ , the baryon to photon ratio η “ ˆ ´ , and the total matter tobaryon ratio ω “ t ff “ b G ρ gal “ ˆ ´ M pl κ { m p , (A8)which equates to 3 ˆ yr. In addition, the typical temperature of the interstellar gas is relevant,which is set by the threshold for H cooling, at roughly 10 K: T H “ α m e . (A9)This also sets the velocity dispersion in the galaxy as v „ a T H { m p „
20 km/s.
Disks:
We take the disk mass to be set by M disk “ M ‹ . (A10)The size of the disk is set by the conservation of angular momentum of the initial collapsing cloud. If ithas a typical angular momentum of L „ Mvr , with v given by the dispersion velocity of molecularclouds, we find r disk „ ˜ M ‹ ρ gal ¸ { “ ˆ ´ λ { M pl κ m p (A11)set to be 100 AU for sunlike stars. niverse , , 157 29 of 34 For infalling dust, the accretion rate M is just given in terms of the typical free fall timescaleas [100] M “ c s G N “ ˆ ´ λ α m { e M pl m { p . (A12)In evaluating this quantity, the molecular energy has been used, reflecting that star formationexclusively forms within molecular clouds, where increased levels of cooling facilitate collapse. A diskwhose dominant form of heating is given by accretion will have a temperature set by [55] T “ π G M ‹ Ma . (A13)The speed of sound in the disk is given by c s „ a T p a q{ m p . This sets the height of the disk as thetypical sound dispersion during one oscillation period. Using the accretion temperature, h „ c s Ω “ .34 α { m { e a { λ { m { p M { pl . (A14)This is normalized to 0.08 AU for earthlike orbits around sunlike stars. The viscosity of the disk isusually parameterized as ν “ α disk c s Ω . (A15)Typical values for the coefficient of proportionality are α disk „ ´ ´ ´ . In equilibrium, the surfacedensity of the disk can be found to be given by Σ “ M π ν . (A16)The mass loss is usually taken to be independent of position [60], so that the surface density profileis dictated by the temperature. Early models took Σ a ´ { , but this is now considered unlikely forequilibrium disks. The now standard dependence is Σ { a [51], which is both observed in simulationsand understood from a theoretical perspective [54]. We will take our profile to be Σ p a q “ M disk r disk a θ p r disk ´ a q . (A17)The snow line is the distance beyond which the disk is cool enough for water to condense intoa solid phase, which is found by setting the temperature equal to the vibrational energy of watermolecules, a snow “ λ M pl α { m { e m { p . (A18)The size a planetesimal attains is given by the isolation mass, M iso „ ˆ π Σ a R Hill , [60].The rationale behind this quantity is that, once a planet is above this mass, it will have alreadydepleted all the material within its Hill radius, and so it will no longer have a supply to continue itsgrowth. Note that the Hill radius is itself a function of the planet mass, so that solving this equationfor the mass yields the expression: M iso „ p π Σ a q { M { ‹ “ ˆ κ { λ { m p M { pl a { . (A19)The coefficient is set by noting that the isolation mass is about Mars sized for the solar system. niverse , , 157 30 of 34 References
1. Sandora, M. Multiverse Predictions for Habitability I: The Number of Stars and Their Properties. arXiv , arXiv:1901.04614.2. Vilenkin, A. Predictions from Quantum Cosmology.
Phys. Rev. Lett. , , 846–849. [CrossRef] [PubMed]3. Frank, A.; Sullivan, W.T., III. A New Empirical Constraint on the Prevalence of Technological Species in theUniverse. Astrobiology , , 359–362. [CrossRef] [PubMed]4. Maschberger, T. On the function describing the stellar initial mass function. Mon. Not. R. Astron. Soc. , , 1725–1733. [CrossRef]5. Johnson, J.L.; Li, H. The first planets: The critical metallicity for planet formation. Astrophys. J. , , 81.[CrossRef]6. Petigura, E.A.; Marcy, G.W.; Howard, A.W. A plateau in the planet population below twice the size of Earth. Astrophys. J. , , 69. [CrossRef]7. Owen, J.E.; Wu, Y. The evaporation valley in the Kepler planets. Astrophys. J. , , 29. [CrossRef]8. Ginzburg, S.; Schlichting, H.E.; Sari, R. Core-powered mass loss sculpts the radius distribution of smallexoplanets. arXiv , arXiv:1708.01621.9. Zeng, L.; Jacobsen, S.B.; Sasselov, D.D.; Vanderburg, A. Survival function analysis of planet size distributionwith Gaia Data Release 2 updates. Mon. Not. R. Astron. Soc. , , 5567–5576. [CrossRef]10. Adams, F.C. Constraints on Alternate Universes: Stars and habitable planets with different fundamentalconstants. J. Cosmol. Astropart. Phys. , , 042. [CrossRef]11. Adams, F.C.; Coppess, K.R.; Bloch, A.M. Planets in other universes: Habitability constraints on densityfluctuations and galactic structure. J. Cosmol. Astropart. Phys. , , 030. [CrossRef]12. Weinberg, S. Anthropic bound on the cosmological constant. Phys. Rev. Lett. , , 2607–2610. [CrossRef][PubMed]13. Thielemann, F.K.; Nomoto, K.; Hashimoto, M.A. Core-collapse supernovae and their ejecta. Astrophys. J. , , 408. [CrossRef]14. Rees, M.J.; Ostriker, J. Cooling, dynamics and fragmentation of massive gas clouds: Clues to the masses andradii of galaxies and clusters. Mon. Not. R. Astron. Soc. , , 541–559. [CrossRef]15. Padmanabhan, T. Theoretical Astrophysics: Volume 2, Stars and Stellar Systems ; Cambridge University Press:Cambridge, UK, 2001.16. Tremonti, C.A.; Heckman, T.M.; Kauffmann, G.; Brinchmann, J.; Charlot, S.; White, S.D.; Seibert, M.;Peng, E.W.; Schlegel, D.J.; Uomoto, A.; et al. The origin of the mass-metallicity relation: Insights from 53,000star-forming galaxies in the sloan digital sky survey.
Astrophys. J. , , 898. [CrossRef]17. Press, W.H.; Schechter, P. Formation of galaxies and clusters of galaxies by self-similar gravitationalcondensation. Astrophys. J. , , 425–438. [CrossRef]18. Dayal, P.; Ward, M.; Cockell, C. The habitability of the Universe through 13 billion years of cosmic time. arXiv , arXiv:1606.09224.19. Woosley, S.E.; Heger, A.; Weaver, T.A. The evolution and explosion of massive stars. Rev. Mod. Phys. , , 1015–1071. [CrossRef]20. Burrows, A.S.; Ostriker, J.P. Astronomical reach of fundamental physics. Proc. Natl. Acad. Sci. USA , , 2409–2416. [CrossRef]21. Shakura, N.I.; Sunyaev, R.A. Black holes in binary systems. Observational appearance. Astron. Astrophys. , , 337–355.22. Alexander, R.; Pascucci, I.; Andrews, S.; Armitage, P.; Cieza, L. The dispersal of protoplanetary disks. arXiv , arXiv:1311.1819.23. Apai, D.; Lauretta, D.S. Protoplanetary Dust: Astrophysical and Cosmochemical Perspectives ; CambridgeUniversity Press: Cambridge, UK, 2010; Volume 12.24. Ercolano, B.; Clarke, C. Metallicity, planet formation and disc lifetimes.
Mon. Not. R. Astron. Soc. , , 2735–2743. [CrossRef]25. Zahid, H.J.; Dima, G.I.; Kudritzki, R.P.; Kewley, L.J.; Geller, M.J.; Hwang, H.S.; Silverman, J.D.; Kashino, D.The universal relation of galactic chemical evolution: The origin of the mass-metallicity relation. Astrophys. J. , , 130. [CrossRef] niverse , , 157 31 of 34
26. Schellekens, A.N. Life at the Interface of Particle Physics and String Theory.
Rev. Mod. Phys. , , 1491–1540. [CrossRef]27. Tegmark, M.; Rees, M.J. Why Is the Cosmic Microwave Background Fluctuation Level 10 ´ ? Astrophys. J. , , 526–532. [CrossRef]28. Fischer, D.A.; Valenti, J. The Planet-Metallicity Correlation. Astrophys. J. , , 1102–1117. [CrossRef]29. Batygin, K.; Bodenheimer, P.H.; Laughlin, G.P. In situ formation and dynamical evolution of hot Jupitersystems. Astrophys. J. , , 114. [CrossRef]30. Dawson, R.I.; Johnson, J.A. Origins of Hot Jupiters. arXiv , arXiv:1801.06117.31. Raymond, S.N.; Mandell, A.M.; Sigurdsson, S. Exotic Earths: Forming Habitable Worlds with Giant PlanetMigration. Science , , 1413–1416. [CrossRef]32. Smallwood, J.L.; Martin, R.G.; Lepp, S.; Livio, M. Asteroid impacts on terrestrial planets: The effects ofsuper-Earths and the role of the ν Mon. Not. R. Astron. Soc. , , 295–305. [CrossRef]33. Buchhave, L.A.; Bitsch, B.; Johansen, A.; Latham, D.W.; Bizzarro, M.; Bieryla, A.; Kipping, D.M. JupiterAnalogues Orbit Stars with an Average Metallicity Close to that of the Sun. arXiv , arXiv:1802.06794.34. Ndugu, N.; Bitsch, B.; Jurua, E. Planet population synthesis driven by pebble accretion in clusterenvironments. Mon. Not. R. Astron. Soc. , , 886–897. [CrossRef]35. Fabrycky, D.; Tremaine, S. Shrinking binary and planetary orbits by Kozai cycles with tidal friction. Astrophys. J. , , 1298. [CrossRef]36. Becker, J.C.; Vanderburg, A.; Adams, F.C.; Rappaport, S.A.; Schwengeler, H.M. WASP-47: A hot Jupitersystem with two additional planets discovered by K2. Astrophys. Lett. , , L18. [CrossRef]37. Chatterjee, S.; Ford, E.B.; Matsumura, S.; Rasio, F.A. Dynamical outcomes of planet-planet scattering. Astrophys. J. , , 580. [CrossRef]38. Spalding, C.; Batygin, K. A Secular Resonant Origin for the Loneliness of Hot Jupiters. Astron. J. , , 93. [CrossRef]39. Johansen, A.; Lambrechts, M. Forming Planets via Pebble Accretion. Annu. Rev. Earth Planet. Sci. , ,359–387. [CrossRef]40. Johnson, J.A.; Aller, K.M.; Howard, A.W.; Crepp, J.R. Giant planet occurrence in the stellar mass-metallicityplane. Publ. Astron. Soc. Pac. , , 905. [CrossRef]41. Taubner, R.S.; Pappenreiter, P.; Zwicker, J.; Smrzka, D.; Pruckner, C.; Kolar, P.; Bernacchi, S.; Seifert, A.H.;Krajete, A.; Bach, W.; et al. Biological methane production under putative Enceladus-like conditions. Nat. Commun. , , 748. [CrossRef]42. Bains, W. Many chemistries could be used to build living systems. Astrobiology , , 137–167. [CrossRef][PubMed]43. Schulze-Makuch, D.; Irwin, L.N. The prospect of alien life in exotic forms on other worlds. Naturwissenschaften , , 155–172. [CrossRef] [PubMed]44. Rogers, L.A. Most 1.6 Earth-radius planets are not rocky. Astrophys. J. , , 41. [CrossRef]45. Ward, P.D.; Brownlee, D. Rare Earth: Why Complex Life Is Uncommon in the Universe ; Copernicus Books:New York, NY, USA, 2003.46. Owen, J.E.; Lai, D. Photoevaporation and high-eccentricity migration created the sub-Jovian desert.
Mon. Not.R. Astron. Soc. , , 5012–5021. [CrossRef]47. Fulton, B.J.; Petigura, E.A.; Howard, A.W.; Isaacson, H.; Marcy, G.W.; Cargile, P.A.; Hebb, L.; Weiss, L.M.;Johnson, J.A.; Morton, T.D.; et al. The California-Kepler survey. III. A gap in the radius distribution of smallplanets. Astron. J. , , 109. [CrossRef]48. Fressin, F.; Torres, G.; Charbonneau, D.; Bryson, S.T.; Christiansen, J.; Dressing, C.D.; Jenkins, J.M.; Walkowicz,L.M.; Batalha, N.M. The false positive rate of Kepler and the occurrence of planets. Astrophys. J. , , 81.[CrossRef]49. Raymond, S.N.; Boulet, T.; Izidoro, A.; Esteves, L.; Bitsch, B. Migration-driven diversity of super-Earthcompositions. Mon. Not. R. Astron. Soc. Lett. , , L81–L85. [CrossRef]50. Kokubo, E.; Kominami, J.; Ida, S. Formation of terrestrial planets from protoplanets. I. Statistics of basicdynamical properties. Astrophys. J. , , 1131. [CrossRef] niverse , , 157 32 of 34
51. Williams, J.P.; Cieza, L.A. Protoplanetary disks and their evolution.
Annu. Rev. Astron. Astrophys. , , 67–117. [CrossRef]52. Pascucci, I.; Testi, L.; Herczeg, G.; Long, F.; Manara, C.; Hendler, N.; Mulders, G.; Krijt, S.; Ciesla, F.;Henning, T.; et al. A steeper than linear disk mass–stellar mass scaling relation. Astrophys. J. , , 125.[CrossRef]53. Bate, M.R. On the diversity and statistical properties of protostellar discs. Mon. Not. R. Astron. Soc. , , 5618–5658. [CrossRef]54. Morbidelli, A.; Lambrechts, M.; Jacobson, S.; Bitsch, B. The great dichotomy of the Solar System: Smallterrestrial embryos and massive giant planet cores. Icarus , , 418–429. [CrossRef]55. Kennedy, G.M.; Kenyon, S.J. Planet formation around stars of various masses: The snow line and thefrequency of giant planets. Astrophys. J. , , 502. [CrossRef]56. Kennedy, G.M.; Kenyon, S.J. Planet formation around stars of various masses: Hot Super-Earths. Astrophys. J. , , 1264. [CrossRef]57. Ida, S.; Guillot, T.; Morbidelli, A. The radial dependence of pebble accretion rates: A source of diversity inplanetary systems-I. Analytical formulation. Astron. Astrophys. , , A72. [CrossRef]58. Adams, F.C.; Hollenbach, D.; Laughlin, G.; Gorti, U. Photoevaporation of circumstellar disks due to externalfar-ultraviolet radiation in stellar aggregates. Astrophys. J. , , 360. [CrossRef]59. Morbidelli, A.; Lunine, J.I.; O’Brien, D.P.; Raymond, S.N.; Walsh, K.J. Building terrestrial planets. Annu. Rev.Earth Planet. Sci. , , 251–275. [CrossRef]60. Youdin, A.N.; Kenyon, S.J. From disks to planets. In Planets, Stars and Stellar Systems ; Springer: New York,NY, USA, 2013; pp. 1–62.61. Izidoro, A.; Raymond, S.N. Formation of Terrestrial Planets. In
Handbook of Exoplanets ; Springer: New York,NY, USA, 2018; pp. 1–59.62. Schlichting, H.E. Formation of close in super-Earths and mini-Neptunes: Required disk masses and theirimplications.
Astrophys. J. Lett. , , L15. [CrossRef]63. Sinukoff, E.; Fulton, B.; Scuderi, L.; Gaidos, E. Below One Earth: The Detection, Formation, and Properties ofSubterrestrial Worlds. Space Sci. Rev. , , 71–99. [CrossRef]64. Wu, Z.N.; Li, J.; Bai, C.Y. Scaling relations of lognormal type growth process with an extremal principle ofentropy. Entropy , , 56. [CrossRef]65. Cumming, A.; Butler, R.P.; Marcy, G.W.; Vogt, S.S.; Wright, J.T.; Fischer, D.A. The Keck planet search:Detectability and the minimum mass and orbital period distribution of extrasolar planets. Publ. Astron.Soc. Pac. , , 531. [CrossRef]66. Zeng, L.; Jacobsen, S.B.; Sasselov, D.D.; Vanderburg, A. Survival Function Analysis of Planet OrbitDistribution and Occurrence Rate Estimate. arXiv , arXiv:1801.03994.67. Simon, J.B.; Armitage, P.J.; Li, R.; Youdin, A.N. The mass and size distribution of planetesimals formed bythe streaming instability. I. The role of self-gravity. Astrophys. J. , , 55. [CrossRef]68. Mordasini, C. Planetary population synthesis. Handbook of Exoplanets ; Springer: New York, NY, USA, 2018;pp. 1–50.69. Raymond, S.N.; Scalo, J.; Meadows, V.S. A decreased probability of habitable planet formation aroundlow-mass stars.
Astrophys. J. , , 606. [CrossRef]70. Tasker, E.; Tan, J.; Heng, K.; Kane, S.; Spiegel, D.; Brasser, R.; Casey, A.; Desch, S.; Dorn, C.; Hernlund, J.; et al.The language of exoplanet ranking metrics needs to change. Nat. Astron. , , 0042. [CrossRef]71. Kopparapu, R.K.; Ramirez, R.; Kasting, J.F.; Eymet, V.; Robinson, T.D.; Mahadevan, S.; Terrien, R.C.;Domagal-Goldman, S.; Meadows, V.; Deshpande, R. Habitable zones around main-sequence stars: Newestimates. Astrophys. J. , , 131. [CrossRef]72. Yang, J.; Boué, G.; Fabrycky, D.C.; Abbot, D.S. Strong dependence of the inner edge of the habitable zone onplanetary rotation rate. Astrophys. J. Lett. , , L2. [CrossRef]73. Kasting, J.F.; Whitmire, D.P.; Reynolds, R.T. Habitable zones around main sequence stars. Icarus , , 108–128. [CrossRef] niverse , , 157 33 of 34
74. Leconte, J.; Forget, F.; Charnay, B.; Wordsworth, R.; Pottier, A. Increased insolation threshold for runawaygreenhouse processes on Earth-like planets.
Nature , , 268. [CrossRef]75. Walker, J.C.; Hays, P.; Kasting, J.F. A negative feedback mechanism for the long-term stabilization of Earth’ssurface temperature. J. Geophys. Res. Ocean. , , 9776–9782. [CrossRef]76. Rushby, A.J.; Johnson, M.; Mills, B.J.; Watson, A.J.; Claire, M.W. Long-Term Planetary Habitability and theCarbonate-Silicate Cycle. Astrobiology , , 469–480. [CrossRef]77. Barnes, R.; Quinn, T. The (in) stability of planetary systems. Astrophys. J. , , 494. [CrossRef]78. Dawson, R.I. Tightly Packed Planetary Systems. In Handbook of Exoplanets ; Springer: New York, NY, USA,2017; pp. 1–18.79. Holman, M.J.; Wisdom, J. Dynamical stability in the outer solar system and the delivery of short periodcomets.
Astron. J. , , 1987–1999. [CrossRef]80. Fang, J.; Margot, J.L. Are planetary systems filled to capacity? A study based on Kepler results. Astrophys. J. , , 115. [CrossRef]81. Raymond, S.N.; Barnes, R.; Veras, D.; Armitage, P.J.; Gorelick, N.; Greenberg, R. Planet-planet scatteringleads to tightly packed planetary systems. Astrophys. J. Lett. , , L98. [CrossRef]82. Borucki, W.J.; Koch, D.; Basri, G.; Batalha, N.; Brown, T.; Caldwell, D.; Caldwell, J.; Christensen-Dalsgaard, J.;Cochran, W.D.; DeVore, E.; et al. Kepler planet-detection mission: Introduction and first results. Science , 327, 977–980. [CrossRef] [PubMed]83. Snellgrove, M.; Papaloizou, J.; Nelson, R. On disc driven inward migration of resonantly coupled planetswith application to the system around GJ876.
Astron. Astrophys. , , 1092–1099. [CrossRef]84. Unterborn, C.T.; Desch, S.J.; Hinkel, N.R.; Lorenzo, A. Inward migration of the TRAPPIST-1 planets asinferred from their water-rich compositions. Nat. Astron. , , 297–302 . [CrossRef]85. Trilling, D.E.; Lunine, J.I.; Benz, W. Orbital migration and the frequency of giant planet formation. Astron. Astrophys. , , 241–251. [CrossRef]86. Baruteau, C.; Masset, F. Recent developments in planet migration theory. In Tides in Astronomy andAstrophysics ; Springer: New York, NY, USA, 2013; pp. 201–253.87. Morbidelli, A.; Crida, A. The dynamics of Jupiter and Saturn in the gaseous protoplanetary disk.
Icarus , , 158–171. [CrossRef]88. Masset, F.; Snellgrove, M. Reversing type II migration: Resonance trapping of a lighter giant protoplanet. Mon. Not. R. Astron. Soc. , , L55–L59. [CrossRef]89. Ida, S.; Lin, D.N. Toward a deterministic model of planetary formation. I. A desert in the mass and semimajoraxis distributions of extrasolar planets. Astrophys. J. , , 388. [CrossRef]90. Tanaka, H.; Takeuchi, T.; Ward, W.R. Three-dimensional interaction between a planet and an isothermalgaseous disk. I. Corotation and Lindblad torques and planet migration. Astrophys. J. , , 1257.[CrossRef]91. Ormel, C.W.; Liu, B.; Schoonenberg, D. Formation of TRAPPIST-1 and other compact systems. Astron.Astrophys. , , A1. [CrossRef]92. Gillon, M.; Triaud, A.H.; Demory, B.O.; Jehin, E.; Agol, E.; Deck, K.M.; Lederer, S.M.; De Wit, J.; Burdanov, A.;Ingalls, J.G.; et al. Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1. Nature , , 456. [CrossRef] [PubMed]93. Tamayo, D.; Rein, H.; Petrovich, C.; Murray, N. Convergent Migration Renders TRAPPIST-1 Long-lived. Astrophys. J. Lett. , , L19. [CrossRef]94. Sandora, M. Multiverse Predictions for Habitability III: Fraction of Planets That Develop Life. arXiv ,arXiv:1903.06283.95. Sandora, M. Multiverse Predictions for Habitability IV: Fraction of Life that Develops Intelligence. arXiv , arXiv:1904.11796.96. Schulze-Makuch, D.; Méndez, A.; Fairén, A.G.; Von Paris, P.; Turse, C.; Boyer, G.; Davila, A.F.;António, M.R.d.S.; Catling, D.; Irwin, L.N. A two-tiered approach to assessing the habitability of exoplanets. Astrobiology , , 1041–1052. [CrossRef]97. Cockell, C.S.; Bush, T.; Bryce, C.; Direito, S.; Fox-Powell, M.; Harrison, J.; Lammer, H.; Landenmark, H.;Martin-Torres, J.; Nicholson, N.; et al. Habitability: A review. Astrobiology , , 89–117. [CrossRef] niverse , , 157 34 of 34
98. Press, W.H.; Lightman, A.P. Dependence of macrophysical phenomena on the values of the fundamentalconstants.
Philos. Trans. R. Soc. Lond. Ser. A , , 323–334. [CrossRef]99. Tegmark, M.; Aguirre, A.; Rees, M.J.; Wilczek, F. Dimensionless constants, cosmology, and other darkmatters. Phys. Rev. D , , 023505. [CrossRef]100. Shu, F.H. Self-similar collapse of isothermal spheres and star formation. Astrophys. J. ,214