Multiverse Predictions for Habitability: Number of Habitable Planets

How good is our universe at making habitable planets? The answer to this depends on which factors are important for life: Does a planet need to be Earth mass? Does it need to be inside the temperate zone? Are systems with hot Jupiters habitable? Adopting different stances on the importance of each of these criteria, as well as the underlying physical processes involved, can affect the probability of being in our universe; this can help to determine whether the multiverse framework is correct or not.


Introduction
This paper is a continuation of [Sandora, 2019a], which aims to use our current understanding from a variety of disciplines to estimate the number of observers in a universe N obs , and track how this depends on the most important microphysical quantities such as the fine structure constant α " e 2 {p4πq, the ratio of the electron 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 of these constants with the multiverse expectation that we are typical observers within the ensemble of allowable universes [Vilenkin, 1995]. In doing so, there remain key uncertainties that reflect our ignorance of what precise conditions must be met in order for intelligent life to arise. Rather than treating this obstacle as a reason to delay this endeavor until we have reached a more mature understanding of all the complex processes involved, we instead view this as a golden opportunity: since the assumptions we make alter how habitability depends on parameters, sometimes drastically, several of the leading schools of 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 for life, because if it were, we would most likely have been born in a universe which is better at that thing. Conversely, if our universe is preternaturally good at something, we expect it to play a role in the development of complex intelligent life, otherwise there would be no reason we would be in this universe. The requirements for habitability are in the process of being determined with much greater rigor through advances in astronomy, exoplanet research, climate modeling, and solar system exploration, so we expect that in the not too distant future our understanding of the requirements for intelligent life will be much more complete. At this stage of affairs, then, we are able to use the multiverse hypothesis to generate predictions for which of these habitability criteria will end up being true. These will either be vindicated, lending credence to the multiverse hypothesis, or not, thereby falsifying it.
In estimating the number of observers the universe contains, a great many factors must be taken into consideration. Thankfully, there has long been a useful way of organizing these factors: the Drake equation, which in a slightly modified form along the lines of [Frank and Sullivan, 2016] Here, N ‹ is the number of stars in the universe, f p is the fraction of stars containing planets, n e is the average number of habitable planets around planet-bearing stars, f bio is the fraction of planets that develop life, f int is the fraction of life bearing worlds that develop intelligence, and N obs is the number of intelligent observers per civilization. We have included dependence of the size of the host star λ " M ‹ {pp8πq 3{2 M 3 pl {m 2 p q, in order to more accurately reflect the fact that these quantities may depend on this. We then integrate over the stellar initial mass function given in [Maschberger, 2012], which approximates a broken power law with turnover at .2M d . This can be related to the probability of being in our universe by incorporating the relative occurrence rates of different universes as P 9p prior H, where p prior 91{pβγq.
Previously, the fact that the strength of gravity γ can be two orders of magnitude higher caused the biggest problems with finding a successful criterion, since most stars are in universes with stronger gravity. Though we had set out to focus solely on the properties of stars, it was only when we weighted the habitability of a system by the total entropy processed by its planets over its entire lifetime that we hit upon a fully satisfactory criterion. This was also reliant on the condition that starlight be in the photosynthetic range, though the other factors we considered may be freely included at will without hindering this conclusion. For the majority of this paper we take the entropy and yellow light conditions as our baseline minimal working model, and incorporate factors that influence the availability and properties of planets to determine how these alter our estimates for habitability. While our previous analysis was not 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 the past 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 is the recent determination of a threshold metallicity below which rocky planets are not found [Johnson and Li, 2012], as well as the understanding of the origin of this threshold. We also find the conditions for the lifetime of massive stars to be shorter than the star formation time and the fraction of galaxies able to retain metals to be sizable, but find that these only impose mild constraints. Using these allows us determine the dependence of f p on the underlying physical parameters. Additionally, we incorporate the fraction of systems that host hot Jupiters into our analysis, and find conditions for this process to not wreck all planetary systems.
In section (3) we turn to the average number of habitable planets n e . Two commonly used requirements for a planet to be habitable are that it needs to be both terrestrial and temperate, and so we optionally include both of these when estimating habitability. Recent results indicate that the distribution of rocky planets peaks at 1.3R C [Petigura et al., 2013, Owen and Wu, 2017, Ginzburg et al., 2017, Zeng et al., 2018b, which is rather close to the terrestrial radius capable of supporting an Earthlike atmosphere. We track how this quantity changes in alternate universes, 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 typical inter-planet spacing that results from the dynamical evolution of stellar systems, which provides a rough estimate for the probability that a planet will end up in the temperate zone. Finally, we discuss the 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 physical constants on the variety of processes and quantities that are needed, to avoid distraction from the main text.
The overarching message we derive from this analysis is that the presence of planets is not that important in determining our location in this universe, a direct consequence of the fact that the presence of planets is a nearly universal phenomenon throughout the multiverse. Including these effects 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 observing any particular value. We find several new anthropic bounds, including the most stringent lower bound on the electron to proton mass ratio in the literature. We find that these results are relatively insensitive to the exact models of planet formation and occurrence rate used, and so are robust to these current uncertainties. This is not the first work to address the question of whether planets are still present for alternative values of the physical parameters. Limitations on the strength of gravity and electromagnetism imposed by the existence of habitable planets was investigated in [Adams, 2016], where it was found that long lived, temperate, terrestrial planets can exist over a wide range of parameter space. This was continued in [Adams et al., 2015] 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 examining the possibility of the existence of planets with desirable properties, but also taking care to incorporate modern theories of planet formation into determining whether planets with these properties are indeed produced.
Taken together, we analyze 12 distinct possible criteria for habitability in this paper (not counting migration or the different views in the planetary size distribution we consider). Coupled to the 40 we considered in [Sandora, 2019a], this represents a total of 480 different hypotheses to compare.

Fraction of Stars with Planets f p
Two of the factors in the Drake equation regard the existence of planets, which will be considered in 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: In succession, we have: f gal , the fraction of stars in galaxies large enough to retain supernova ejecta, f 2nd gen , the fraction of stars born after supernova enrichment, f Z , the fraction of stars born with high enough metallicity for planets form, and an optional f hj , the fraction of stars that do not produce hot Jupiters. Here the exponent p hj P t0, 1u is introduced as a choice of whether to include this last criterion or not. The other two are not treated as optional.
What sets the size of the smallest metal-retaining galaxy?
The requirement for a galaxy to be habitable is that it must retain its supernova ejecta in order to reprocess it into another round of metal-rich stars [Weinberg, 1987]. This sets a minimum galactic mass by the condition that the velocity of supernova ejecta is less than the escape velocity, This is the asymptotic speed the supernova ejecta attains, and to find this a bit of the ejecta dynamics must be used. The initial speed can be found from energy balance [Thielemann et al., 1996]: this can be written in the form where the temperature of the supernova is set by the Gamow energy, which is the amount required to overcome the repulsive nuclear barrier and force fusion, T SN " α 2 m p . The mass of a typical particle in the ejecta is related to the atomic number A " 50, which cannot conceivably vary by much. We find that v 0 SN " .03. However, as the ejecta moves through the intergalactic medium, it cools and slows until it merges completely. The asymptotic speed is that at which the temperature becomes equal to (about an order of magnitude less than) the hydrogen binding energy T H 2 " α 2 m e {32 " 10 4 K, below which Hydrogen becomes predominantly neutral and no more cooling takes place [Rees and Ostriker, 1977].
How far does the ejecta of a supernova spread before it completely merges with the interstellar medium, and, more importantly for our purposes, why? The observed value is around 100 pc [Padmanabhan, 2001], and this is after the blast has gone through several successive phases. The first is known as the blast wave phase, where the ejecta spread out at their initial velocity for roughly 100 yr, traveling a total of a few pc. After the amount of interstellar material encountered rivals the initial mass of the ejecta, which occurs at d ST " pM ej {ρ gal q 1{3 , the blast enters the self-similar Sedov-Taylor phase, where the blast slows and expands considerably. According to standard theory [Padmanabhan, 2001], the self-similarity of the dynamics dictates that the temperature of the blast falls off as T 9d´3. During this phase the velocity will decrease with distance as vpdq " 2{5pd ST {dq 5{2 until the snowplow phase begins, at which point the speed essentially does not decrease any further. The snowplow phase occurs after the temperature reaches the molecular cooling threshold, where it expands by a factor of a few until the density of the material falls to that of the surrounding medium, at which point it is completely merged. Since the bulk of the expansion takes place in the Sedov-Taylor phase, the size of the blast will be dictated by the dynamics that take place there, and so the ultimate speed is given by Using the above relations, the asymptotic speed of supernova ejecta is found to be Now, we can find an expression for the minimum mass by using M min " ρ gal R 3 . Using the density of galaxies given in the appendix, this gives Where the coefficient in the last expression has been chosen to reproduce the observed minimal mass of M ret " 10 9.5 M d [Tremonti et al., 2004]. As explained in the appendix, the quantity κ determines the density of galaxies in terms of cosmological parameters.
While this critical mass is important conceptually, it is not as relevant to the retention of ejecta as the gravitational potential itself, which directly sets the escape velocity of the overdensity. The fraction of initial overdensities that exceed a potential of a given strength Φ is given by the Press-Schechter formalism as f " erfcpΦ{p ? 2Qqq [Press and Schechter, 1974], where Q is the primordial amplitude of perturbations. Usually, this expression is immediately expressed in terms of mass and density, but this more primitive form will suffice for our purposes. With this, we can derive the fraction of matter that resides in potential wells deep enough to produce a second generation of stars as Perhaps somewhat interestingly, this does not depend on the strength of gravity γ at all. This effectively acts as a step function, severely diminishing the habitability of universes where M ret ą M typ . Note that this is a pessimistic estimate, as it ignores the potential for subsequent evolution that causes potential wells to deepen with time. Nevertheless, it is a very mild bound, and so a more thorough treatment is not called for.
Is massive star lifetime always shorter than star formation time?
Since the formation of metal-rich systems is reliant on the evolution of the first stars through to their completion, if the lifetime of massive stars exceeds the duration of star formation, then no systems will form with any substantial metallicity. The second generation stars are not necessarily enriched enough to produce planets, but this serves as a sufficient condition for planets to be formed at all. Then, the fraction of stars with planets can be estimated as those that are born after a few massive stellar lifetimes have elapsed. It is worth considering how these two timescales compare for general values 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 [Dayal et al., 2016]. The timescale of this depletion is simply related to the free fall time of a galaxy, t dep " 1{p SFR ? Gρq, where for simplicity the efficiency coefficient is taken to not vary with parameters. Then the fraction of stars born after a time t is f ‹ ptq " e´t {t dep . This needs to be compared to the lifetime of massive stars, where massive here is taken to mean large enough to become a type II supernova. This threshold is 8 solar masses in our universe, and is set by the inner core having a high enough temperature to undergo carbon fusion [Woosley et al., 2002]. As such, this threshold is parametrically similar to the minimum stellar mass, which was shown in [Burrows and Ostriker, 2014] to scale as λ9α 3{2 β´3 {4 , the two orders of magnitude difference being only the energy barrier to overcome. Then, using the stellar lifetime from [Sandora, 2019a], we find The normalization has been set to the value of .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 the neglect 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 lifetime of massive stars, so that f 2nd gen " f ‹ pt SN q is always close to 1.
What is the metallicity needed to form planets?
Terrestrial planets must be formed out of heavy elements, and though even a minuscule amount would suffice in terms of actual planetary mass, the planet formation processes within the protoplanetary accretion disk can only occur after a high enough metallicity is reached. A recent analysis [Johnson and Li, 2012] 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 and the dust settling time t dust . If the second exceeds the first, the disk will dissipate before sufficient dust may settle into the midplane, and will disperse without forming planets. Both of these depend on metallicity monotonically, and so only above some critical value will the conditions for planetary formation hold. We detail the scaling of each in turn.
The physics of accretion disks was first laid out in [Shakura and Sunyaev, 1973]. An initially uncollapsed cloud first condenses, and then through angular momentum transfer begins to form a disk. Matter continually falls in at a rate that decreases proportional to t´3 {2 [Lynden-Bell and Pringle, 1974], so that accretion continually dwindles. Once it falls below the rate at which UV light from the star photoevaporates the gas, accretion effectively stops, and the disk is blown away. This phase of evolution occurs on the viscous timescale of the disk, which is much smaller than the accretion timescale [Alexander et al., 2013].
Though the precise time at which this crossover occurs depends on the exact mechanism of photoevaporation (x-ray versus EUV, other stellar sources in clusters, presence of high activity tau phase) the drop-off of accretion is set by the disk's secular evolution time [Apai and Lauretta, 2010], with crossover occurring on the order of this time. Therefore, the only relevant physics that needs to be kept track of is the free fall time. Since this is given by a cloud that has approximately virialized after Jeans collapse, it is simply set by the condition that the gravitational energy is equal to the thermal energy. Then the accretion rate is given by 9 M " c 3 s {G " 10´6M d {yr, and the timescale is given by t disk " M ‹ { 9 M " Myr. The sound speed and the temperature are given by bremsstrahlung from molecular line cooling, as found in the appendix. Then In [Ercolano and Clarke, 2010] the metallicity dependence was studied, where it was found that cooling quickens with metallicity Z. The exact process was open to interpretation, with a handful of viable candidate mechanisms, but the overarching explanation is that the less shielding there is, the faster the disk will dissipate. Observationally and theoretically it was found that the scaling with metallicity is consistent with p " 1{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 midplane from their initial positions in the protoplanetary disk. It is given in terms of the Keplerian timescale of the disk, t dust " .72t Kepler {Z, as derived in [Johnson and Li, 2012]. 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 accretion process. This is a function of orbital distance, but specifying to planets that form within what will become a temperate orbit for simplicity, we have Here, we are explicitly assuming that significant migration does not occur, which would alter the metallicity needed for planet formation, given its dependence on orbit. Equating these two timescales yields the critical metallicity The critical value is found to be Z min " 6.3ˆ10´4. As compared to the solar value Z d " .02, we find Z min " .03Z d [Johnson and Li, 2012]. This may be used to determine the fraction of stars that host planets by considering the amount of stars that are formed above this metallicity. This requires a model for how metallicity builds up inside a galaxy (of a given mass, as well as the distribution of galaxy masses). A full calculation of this 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 metallicity a galaxy attains-this usually affords percent-level accuracy. In our universe this is found to be Z 8 " .011 [Tremonti et al., 2004, Zahid et al., 2014, which is actually a factor of two below solar due to the intrinsic scatter in metallicities. This asymptotic value is set by the fraction of stellar mass that is transformed into heavy elements during stellar fusion, and will not depend sensitively on the fundamental constants.
If attention is restricted to solar mass stars, the requirement that the threshold metallicity be lower than the asymptotic value equates to α 3 β 1{2 γ´1 {2 ą .80, which is most sensitive to the fine structure constant. Leaving the other parameters fixed, this gives α ą 1{361, which is stronger than the bound α ą 1{685 based on galactic cooling found in [Schellekens, 2013, Tegmark andRees, 1998]. A more forgiving bound is found using the smallest stellar mass, which is α 5{2 β 17{12 γ´2 {3 ą .32. This latter boundary is displayed in Fig. 1, where the distribution of observers is plotted as a function of the three variables α, β and γ.
Incorporating the above effects determining the fraction of stars that host planets, and again using the entropy and yellow light conditions, we find that the probability of observing our particular values of each constant are Ppα obs q " .19, Ppβ obs q " .45, Ppγ obs q " .32 These values are indistinguishable from those that were found in [Sandora, 2019a] without including these effects. We conclude that the presence of planets is a generic feature throughout most of the multiverse, and it does not alter where we expect to be situated in the slightest. Figure 1: The distribution of observers when taking metallicity effects into account. What is plotted is the logarithm of the probability of measuring any value of the coupling constants, with red being more probable than blue, and the black dot corresponding to our observed values. Included thresholds are an upper bound on α for hydrogen stability, as well as a lower bound for galactic cooling, and an upper bound on β for proton-proton fusion. The teal and brown curves correspond to the metallicity and supernova lifetime thresholds, respectively, and the supernova retention threshold is not relevant in this range.
Are hot Jupiter systems habitable?
The constraints above were all quite mild, indicating that the presence of planets is a fairly generic feature of the multiverse. However, we can make a further refinement by incorporating one of the most famous statistical correlations in the field of exoplanets, the hot Jupiter-metallicity correlation [Fischer and Valenti, 2005]. This finds that the fraction of stars that possess hot Jupiters, that is, Jupiter sized planets on orbits extremely close to the star, increases with metallicity as Z 2 . The general (though not universal- [Batygin et al., 2016]) consensus is that these planets must have formed in the outer system before moving inward, to avoid the necessity of a disk that would be so massive as to be unstable [Dawson and Johnson, 2018]. In this scenario, the migration of the planet through the inner solar system would have certainly ejected any preexisting planets from their orbits (or worse), precluding them from sustaining life. However, [Raymond et al., 2006] hypothesize that this entire process could happen early enough that the main stage of planet formation could occur after this migration had already taken place. If this turns out to be the case, then there may in fact be no correlation between hot Jupiters and habitability (barring other factors that may impact habitability [Smallwood et al., 2017]). If one wishes to include this effect, however, then the fraction of stars that host Earths and not Hot Jupiters is given by Here we have used the mean metallicity rather than averaging this fraction over the metallicity distribution, but this approximation is sufficient for a first analysis. The normalization Z max is the threshold above which the stellar system is almost assured to possess a hot Jupiter. It is set to reproduce the observed abundance of hot Jupiter systems of 3% as Z max " 5.77Z d . In a multiverse setting, 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 with this 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 determines this metallicity. The functional dependence is a clue: as the effect becomes more pronounced with the square 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-disk interactions. In fact, both explanations have been considered in the literature: references may be found in [Buchhave et al., 2018] 1 . The planetplanet hypothesis is supported by the fact that the eccentricities of observed hot Jupiters are correlated with metallicity as well, indicating a more chaotic, violent origin, rather than the steady, deterministic process indicative of planet-disk interaction. Additionally, [Fabrycky and Tremaine, 2007] note a substantial misalignment between the orbits of known hot Jupiters and the spins of their host stars, which is most easily explained through a violent migration scenario. Here we only expound upon the planet-planet scenario, though the others could just as readily be incorporated into our analysis.
We start by determining the value of Z max due to planet-planet interactions, as proposed in [Chatterjee et al., 2008]. As noted in [Spalding and Batygin, 2017], the coexistence of hot Jupiters and low mass planets is impossible in this paradigm, as migration occurs after the disk has dissipated. From [Buchhave et al., 2018], this is set by the expected number of Jupiter mass planets initially formed in the outer system. They provide a framework for estimating this by determining the probability that a core will attain Jupiter mass as a result of planetesimal accretion as p9∆M {M crit , where M crit " 10M C is the mass above which runaway gas accretion is possible and ∆M is the typical total accreted mass. For this we use the analytic expression for the accretion rate from [Johansen and Lambrechts, 2017], This employs the strong gravitational focusing limit, and treats the typical relative velocities of planetesimals as roughly given by the Hill velocity. This can be used to determine the total mass accreted by simply multiplying by the disk lifetime given in eqn (10) (making use of the simplifications that the nonlinear oligarchic regime is not quite reached, and the initial isolation timescale is small compared to the disk lifetime). The probability that there will be at least two gas giants to trigger the instability will scale as ppě 2q " N 2 jup p 2 , where N jup is the typical number of planets in the outer system, and we have assumed that p is small. The quantity N can be found by dividing the total mass of the planetary disk by the typical mass of a planet at the typical location of formation. Here, we use that the initial seed is set by the isolation mass, have fixed the orbital radius to be given by the snow line, and have taken the disk temperature to be given by viscous accretion, all of which are discussed in the appendix. This gives This scales linearly with stellar mass, in agreement with the observations in [Johnson et al., 2010]. The maximal value of metallicity is found to be This quantity is somewhat sensitive to both α and β, but not at all to γ. This also defines a stellar mass λ hj " 2.3ˆ10´6α 13{6 β 3{2 {κ: stars above this mass, equal to 11 M d for our values, will always host hot Jupiters. This will be below the smallest stellar mass if 789α 8{27 β ă 1, which will occur when the electron to proton mass ratio is about 10 times smaller. However, this criteria does not alter the probabilities much: Ppα obs q " .18, Ppβ obs q " .44, Ppγ obs q " .31 With this, notice that the probabilities are only changed from eqn (13) by a few percent. So, even when including the demand that hot Jupiter systems are uninhabitable, the fraction of systems with planets seems to be relatively insensitive to the physical constants. We now turn to the next factor in the Drake equation, which deals with the characteristics of planets, rather than just their presence.

Number of Habitable Planets per Star n e
Next, we focus on the number of habitable planets per star, n e . The determination of habitability may depend on many factors, such as amount of water, eccentricity, presence of any moons, magnetic field, distance from its star, atmosphere, composition, etc. Here, we focus on two: temperature and size, and determine the fraction of stars that have planets with each of these characteristics.
As usual, it is possible that habitability is completely independent of these properties; which viewpoint one adopts depends on how habitable one expects environments without liquid surface water and thin atmospheres can be. In this work we remain agnostic to either expectation and report the number of observers for all combinations of choices, where a planet will only be habitable if it is approximately Earthlike, and where the size and/or temperature of the planet have no effect on its habitability.
The number of habitable planets can be broken down as Here, the average total number of planets around a star is n p . The fraction of terrestrial mass planets is denoted f terr , and f temp is the fraction of planets that reside within the temperate zone. The exponents p i P t0, 1u parameterize the choice of whether to include these conditions in the definition of habitability or not. These quantities all depend on stellar mass, giving preference to large stars because they make larger planets, and small stars in that their temperate zone is wider compared to the interplanetary spacing. Estimating these quantities is somewhat muddled by the current uncertainties in planet formation theory. Not only is the distribution of planet masses contested in the literature, but the exact formation pathways, as well as the physics that dictates the results, is not completely settled. Where we come across disagreement, we separately try each proposal, in order to understand the sensitivity of our analysis to present uncertainties. While the results for the overall probabilities can vary by a factor of 2, the upshot is that our estimates are relatively robust.
Why does our universe naturally make terrestrial planets?
The size of a planet is of crucial importance, because it dictates what kind of atmosphere it can retain. If it is too small, all atmospheric gases will eventually escape, whereas if it is too large, it will retain a thick hydrogen and helium envelope, leading to a runaway growth process. Terrestrial planets must have a very specific size in order that the escape velocity exceeds the thermal velocity for heavy gases such as H 2 O, CO 2 and N 2 , 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 range of planetary radii to be within .7 and 1.6 that of Earth's [Rogers, 2015], a narrow sliver compared to the eight orders of magnitude range of planetary masses. In terms of fundamental parameters, this requirement gives the mass to be The coefficient has been set to reproduce Earth's mass, but the allowed spread in masses is taken to be between .3´4M C . The conditions that set the presence of atmospheres, however, is completely separate from the physics that dictates the size of planets in the first place, which is set by the clumping of the initial circumstellar disk 2 . Nonetheless, the observed population of rocky planets is thought to peak at only slightly super-Earth mass, making the production of terrestrial planets the norm for stars throughout the universe. To be fair, the current exoplanet samples are biased towards large mass planets and become very incomplete below Earth mass [Fressin et al., 2013], but a number of different groups have concluded that a detectable turnover is present near Earth masses: [Zeng et al., 2018b] find a good fit to a log-normal distribution that peaks at 1.3R C . [Owen and Wu, 2017] use a Rayleigh distribution with width 3M C . [Ginzburg et al., 2017] advocate for a broken power law with turnover at 5M C . [Petigura et al., 2013] note that the distribution appears to be flat below 2.8R C . Use of these differing proposed distributions make very little difference to our final outcome.
However, not everyone is convinced that the mass distribution exhibits a peak, and even if there one, it is just as reasonable to assume that there are many more smaller mass planets for every 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 room for such disagreement at the current moment. Additionally, even if the mass peak is real, it is only observed for close in exoplanets, and so requires an extrapolation to Earthlike orbits, where different dynamics may be at play. One possibility is that the peak at super-Earth mass may be due to their enhanced migration capability [Raymond et al., 2018]. We will consider each scenario in turn.

What sets the size of planets?
Why is the turnover so nearly equal to Earth mass planets, out of the potentially eight orders of magnitude that could have been selected instead? Simulations provide a means to address this question: it was found in [Kokubo et al., 2006] that the mass of planets is directly proportional to the amount of initial 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 into planets, with negligible (perhaps a factor of two, but not an order of magnitude) losses throughout the evolution of the system. Determining the final planet mass in this setup requires knowledge of the initial disk mass, as well as the fraction of material within the inner solar system. This boundary is set by the snow line, the difference in composition interior and exterior to which dictates the formation of rocky versus icy planets.
Current observations indicate that the initial mass of protoplanetary disks are roughly proportional to disk mass, M disk " .01M ‹ [Williams and Cieza, 2011]. In [Pascucci et al., 2016] a stronger dependence was found, but the scatter of .5 dex is larger than the trend, and the observed trend was suggested to possibly be due to a selection effect arising from processing into undetectably large grains, so we omit this stronger scaling for now. For solar mass stars, this works out to be roughly 5´10M Jupiter , or 1500M C . This mass is distributed out to a radius of " 100 AU, which is set by the conservation of angular momentum, and the initial size of the collapsing cloud. From the appendix, we arrive at the following expression mospheres from the host star that favors both small and large atmospheres [Owen and Lai, 2018]. While this leads to the interesting bimodal distribution of observed radii [Fulton et al., 2017], this is driven by atmospheric size, and is certainly not enough to affect the terrestrial core.
for disk size: This may be significantly altered if the young star is in a dense environment [Bate, 2018], but this scaling will suffice for our purposes.
To determine the amount of material present within the snow line, one must take the surface density profile into account. For this, we use the expression for Σpaq found in the appendix, which is inversely proportional to a. Using these, the typical mass of an interior planet becomes where we have included the quantity η ă " .25 to account for the difference in composition interior to the snow line [Morbidelli et al., 2015]. The location of the snow line is the point in the disk beyond which water condenses, equal to 2.7 AU in our solar system [Kennedy and Kenyon, 2008b]. In order to determine its location, the temperature as a function of radius must be used, which will depend on the dominant source of heating. For most of the present paper, we use the temperature given by viscous accretion, and find Note that, perhaps unsurprisingly, the snow line is situated outside the temperate zone for all relevant parameter values. The dependence on stellar mass was found taking 9 M 9λ 2 , but is generically found to scale as a snow 9λ 2{3´2 [Kennedy and Kenyon, 2008a]. 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 of these parameters.
Though we tend to favor accretion dominated disks throughout this work, irradiation from the central star can actually play a significant role as well [Alexander et al., 2013]. If this is the dominant mode of heating, then the snow line will instead be given by the expression a snow " 297λ 43{30 α´4m´4 pl . Though this is functionally quite similar to the accretion dominated case from above, in Table 1 we investigate the effect of assuming this form instead. Actually, both are almost equally relevant in determining the position of the snow line, which helps to greatly complicate the disk structure (as well as enhance the variability between different star systems [Ida et al., 2016]). Additionally, irradiation from other neighboring stars may be important as well, especially in clusters [Adams et al., 2004], but we do not consider this contribution in this work.
Determining the average planet mass is somewhat involved, given the many distinct stages of growth that occur as microscopic dust grains agglomerate to the size of planets. For reviews of this multi-stage process, see [Morbidelli et al., 2012, Youdin and Kenyon, 2013. In brief, planetesimals form characteristic masses as the final outcome of pebble (or dust) accretion. This arrangement is unstable, and ultimately leads to a phase of giant impacts, wherein planetesimals collide together to form full planets.
An estimate for the maximum mass a planet can attain after a phase of growth through chaotic giant impacts was found in [Schlichting, 2014]. There, they assumed no migration, small eccentricity, and determined the width of the 'feeding zone' to be ∆a " 2v esc {Ω by noting that within this region planet-planet interactions result in collisions rather than velocity exchange. This yields Where ρ is the average density of the planet and Σ is the disk density. With this, we can use the appendix 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: The latter has quite a steep dependence on stellar size. This is expected from the simulations [Kokubo et al., 2006], with a dependence closer to linear, but is not particularly observed in exoplanet catalogs due to large scatter [Sinukoff et al., 2013]. In our calculations we explore the effect of ignoring this dependence altogether, displayed in Table 1. 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 the scatter observed in exoplanet surveys, simulations, and indeed within the solar system masks any dependence that may exist.
For the fraction of planets that are terrestrial we use the log-normal distribution of [Zeng et al., 2018b] since this is what is expected of the core accretion process, though we check that the actual distribution used does not alter the outcome much. Under these assumptions, we can find the probability that a planet will be terrestrial as This function peaks at M planet " M terr , and approaches 0 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 not currently known what sets this quantity, here we explore two options: the first uses maximization of entropy production to set σ M " 1{ ? 6, which is fairly widely observed in natural processes [Wu et al., 2017]. This estimate should occur for large systems, but for small systems one would expect the variance to be set by shot noise instead, σ " a M iso {M inner , making use of the isolation mass defined in the next section. The dependence on the various parameters is displayed in Fig. 2 The probabilities of observing the observed values of our constants are computed for the various choices we made in Table 1. These can be compared with eqn (13) that only ? 6, excluded λ dependence on the average planet mass, and assumed accretion dominated disks.
considered the fraction of stars with planets. The most significant change for our most favored prescription is the probability 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 terrestrial planets. More interestingly, if the dependence on stellar mass is neglected, our strength of gravity becomes quite uncommon to observe. This gives us strong reason to suppose that if life requires terrestrial planets, we will begin to see a correlation between the two soon, and if we don't, then life requiring Earth mass planets is incompatible with the multiverse at the 2.7σ level.
Aside from this, though, the choices we made to come up with these estimates do not affect the outcome very much at all. On the one hand this is disappointing, as the stronger the dependence these probabilities have on the assumptions of planet formation, the stronger our predictions can be about which to expect to be dominant. However, this is also heartening: because the current uncertainties about planet formation do not affect the outcome all that much, we are able to trust the broad conclusions we have reached a bit better.
Is life possible on planetesimals?
The mass discussed above really refers to the maximal planet mass of a system, which form as 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 the case in and around the solar system's asteroid belt. If these smaller bodies are considered as potential abodes 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 as the pebble isolation mass, which from the appendix is given by The isolation mass is a function of the semi-major axis, but if we evaluate it at the edge of the inner system given by eqn (23) we find, using the value for the disk surface density from the appendix, Here, the density of the galaxy comes into play in setting the outer edge of the disk. The dependence on stellar mass in this expression is quite close to that found in [Kennedy and Kenyon, 2008b], M iso 9λ 7{4 . It remains to specify the distribution of planetary masses in order to find the fraction that are terrestrial in this picture. It is generically expected to take a power law form that continues to the small mass cutoff, so that N pM q " pM iso {M q q . However, different authors prefer different values for the slope: [Cumming et al., 2008] find q " .31˘.2. [Zeng et al., 2018a] find a nearly scale invariant distribution for the radius, which translates into q " .30˘.03 if we use M 9r 3 . [Johansen andLambrechts, 2017, Simon et al., 2016] find q " .6 for simulations of planetesimals, and [Morbidelli et al., 2015] extrapolate from the known asteroid population to find q " 1. [Mordasini, 2018] favors a value of q " 2 from population synthesis methods. Here, we report with various values of q to investigate its influence on the probabilities. The fraction of terrestrial planets is then This is an increasing function of stellar mass until M iso ą .3M terr , reflecting the expectation [Raymond et al., 2007] that earthlike planets should be rare among low mass stars. With this prescription, stars above a certain mass will not produce any earthlike planets, because the isolation mass will exceed the largest terrestrial planet size. This defines the largest viable stellar mass λ iso " .0013κ´3 {4 α 2 β 21{16 , which corresponds to 6.3 M d in our universe. This will only exceed the minimal stellar mass if α 1{2 β 33{16 κ´3 {4 ą 169. This condition is most sensitive to the electron to proton mass ratio and, holding the other constants fixed, will be violated if it drops to about 11% of its observed value. For q ă 1, the average mass for this distribution is formally infinite, which presents a problem for using our expression for the expected number of planets in a system as given by n p " M inner {xM p y. However, the total mass in the inner disk introduces a large mass cutoff, for which we have This interpolates between pq´1q{qM inner {M iso for q ą 1 and pq´1q{q for q ă 0. The ratio of masses is So that this equates to 30 with our constants. With this viewpoint, the distribution of observers is plotted in Fig. 3. Figure 3: The distribution of terrestrial planetesimals that result from isolated accretion, without the subsequent phase of giant impacts. The slope here is q " 1{3. The teal line represents the region where the largest star capable of hosting terrestrial planets is smaller than the smallest possible star.
The overall probabilities are calculated for different representative values of the power law in Table 2. It can be seen that for increasing q, the probability for α increases, while the other two decrease. In particular, for q " 2, the probability of observing our value of the electron to proton mass ratio is disfavored by 2σ. Why is the interplanet spacing equal to the width of the temperate zone?
Perhaps the most commonly employed habitability criteria is that a planet must be positioned a suitable distance away from its host star to maintain liquid water on its surface. This assumption is so pervasive that this region is usually referred to as the circumstellar habitable zone. In the spirit of remaining agnostic toward the conditions required for life, we will adhere to the recently proposed renaming as the 'temperate zone' [Tasker et al., 2017]. If this is indeed essential for life, then the expected number of habitable planets orbiting a star will depend both on the interplanetary spacing, as well as the width and location of the temperate zone. A rather clement feature of our universe is that the width of the temperate zone is comparable to the interplanetary spacing (for sunlike stars). Because of this, it is relatively 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 the temperate zone were much narrower than the interplanetary spacing, in which case the odds of a planet being situated inside it would be quite low. However, this coincidence of distance scales is not automatic: both these quantities are dependent on the underlying physical parameters, and so in universes with different parameter values the expected number of potentially habitable planets per star will be altered. We go through these length scales in turn, and then fold them into our estimate for the 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 [Kopparapu et al., 2013], its orbital period [Yang et al., 2014], etc. However, these details only alter the location of the temperate zone to subleading order, and do not affect the scaling with fundamental 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 the temperate zone will be a temp " 1 2 Albedo and greenhouse effects will change the coefficient, but not the overall scaling. Determining the width of the temperate zone entails finding the temperatures at which runaway climate processes occur, but we will now argue that both the inner and outer edge are dictated by (broadly speaking) the same underlying physical process of phase change, and so the width of the temperate zone will scale in the same way as its location. The inner edge of the temperate zone is set by the runaway greenhouse effect: this occurs when a temperature threshold is crossed that allows an appreciable amount of water vapor to be sustained in the atmosphere. Since this serves to trap infrared light from escaping to space, this will increase the temperature further, in turn driving a further increase in atmospheric water vapor. Once the atmosphere is comprised primarily of water, it will be photodissociated and/or escape to space, leaving the Earth in a dry and Venus-like state [Kasting et al., 1993]. Since strictly speaking the ocean will escape into space eventually anyway if one is prepared to wait for an eternity, the exact threshold for this is defined as when the timescale for this process is of the order of one billion years, which in turn will depend on the mass of the planet's ocean. However, key to our discussion is that the change in the atmospheric water fraction occurs very abruptly when the temperature crosses the latent heat of vaporization, going from equilibrium values of 10´5´1 in the span of 250-420 K. Because of this, the exact mass of the planet, ocean or atmosphere will only play a subleading role in determining this threshold, which instead is dictated solely by molecular processes. Since this transition is set by what is essentially the intermolecular binding energy, this scales identically with the condition for liquid water. For Earth, this occurs at a temperature of 330 K, which corresponds to .95 AU [Leconte et al., 2013].
Similarly, the outer edge of the temperate zone is set by the runaway icehouse process. The temperature of a planet is set not only by the incident flux, but also by the carbon dioxide content content of the atmosphere and albedo. Carbonate weathering is an important regulatory feedback mechanism that serves to stabilize the temperature of a planet to remain within the temperate range by adjusting atmospheric carbon dioxide content to compensate for a change in stellar flux [Walker et al., 1981] (it does this because liquid water is necessary for efficient weathering to occur). However, this only works to an extent, since beyond a critical concentration atmospheric CO 2 increases the albedo of a planet, leading to a runaway icehouse effect [Kasting et al., 1993]. Because the amount of atmospheric carbon dioxide is also set by reactions due to intermolecular forces, it scales the same as above. Therefore, both the inner and outer edges of the temperate zone are set by molecular binding energies, and so the width will always be of the same order as the mean 3 .
The exact delineations are subject to the uncertainties of the atmospheric model used, but for definiteness we take ∆R HZ " .73 AU, from [Kopparapu et al., 2013].
The next task is to determine how far apart planets typically reside from each other. It was hypothesized in [Barnes and Quinn, 2004] that stellar systems are dynamically packed, in that they are filled to capacity, and the insertion of any additional planet would render the system unstable. Though this may not strictly hold all the time [Dawson, 2017], this spacing is roughly observed in our solar system [Holman and Wisdom, 1993], in Kepler data on multiplanetary systems [Fang and Margot, 2013], and found to occur naturally in simulations in [Raymond et al., 2009]. The essential idea is that planetary scattering either ejects or collides excess planets from an initially overpacked system until the remainder are far enough apart to be stable on the timescale of the system. The stability condition is that planets are further than a certain multiple of Hill radii away, nominally around 10, and it was found in [Fang and Margot, 2013] that the distribution of separations was a shifted Rayleigh distribution with usual separation 21.7˘9.5R Hill .
As mentioned in [Schlichting, 2014], the typical multiple of mutual Hill radii is not universal, but can be shown to depend on the width of a planet's 'feeding zone' to be given by pρa 3 {M ‹ q 1{4 , where ρ is the density of matter and a is the semimajor axis. Though above we were interested in the typical size of a planet during this process, here we may use this condition to find the typical spacing as This was evaluated at the center of the temperate zone to give the ratio of these two length scales in terms of fundamental constants. Since this ratio loosely sets the fraction of planets within the temperate zone, we arrive at Set 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 smaller mass stars, reflecting the comparatively broad temperate zones there. This distribution is illustrated in Fig. 4. The corresponding probabilities for observing our values of the constants are Ppα obs q " .24, Ppβ obs q " .37, Ppγ obs q " .15 Though the probabilities shift around by a factor of a few, the inclusion of this criterion does not actually affect the results very much. This is a generic conclusion, even when this is included with other criteria. Whether or not this condition is essential for life, the multiverse hypothesis is relatively insensitive to it. Let us also take this opportunity to determine what values of parameters would render disks smaller than the habitable orbit, thereby precluding temperate planets from ever forming. Based off the expressions above and in the appendix, we have 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 the fundamental parameters. Though these introduce lower bounds for α and β, this is the region with very few 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 is reached. 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 on Earth, universes like this would be altogether barren.

Planet Migration
Traditional models of planet formation assume that little orbital migration takes place during planet formation. However, the large number of gas and ice giants found situated extremely close to their host stars [Borucki et al., 2010], the presence of orbital resonances found in some exoplanet systems [Snellgrove et al., 2001], and icy composition of planets within the snow line [Unterborn et al., 2018], all point to the presence of migration. Migration may strongly affect the habitability of planetary systems, as if it proceeds for too long, all inner planets will drift toward the inner edge of the disk at around .014 AU [Trilling et al., 2002], and giant planets migrating across the temperate zone would destabilize the orbits of any existing planets. The vast diversity of systems found, as well as analytic and numeric simulations of protoplanetary disks [Baruteau and Masset, 2013] point to an exquisitely complex array of migration scenarios, which will be selectively operational depending on the characteristics of the initial system such as disk mass, viscosity, and the size and locations of the planets. Nevertheless, it is possible to perform rough estimates for when migration will be present in a given system. In this section we do not attempt to characterize all the complex features of planet migration in universes with alternate physical constants, but rather wish to provide a useful diagnostic for when migration will be important. We will find conditions the physical constants must satisfy in order that all planets do not migrate into their host stars at an early stage of evolution, our observed values being intermediate such that this scenario only afflicts some percentage of planetary systems with unlucky characteristics. Migration occurs when a planet's influence on the surrounding disk produces a net torque on the planet itself, usually driving it inward. As such, migration halts after the time t disk when the disk has cleared out. This may be compared to the migration timescale where M is the planet's mass, L its angular momentum and Γ the torque it experiences. A heuristic condition for when migration will be significant was found in [Ida and Lin, 2004] as t mig À 10 t disk . There are various contributions 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 arises when 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 the absence of material (for a recent review see [Baruteau and Masset, 2013]). The latter is more relevant for larger planets capable of significantly altering the disk structure, and the former more relevant for smaller planets, which are not. They will each be considered in turn, resulting in conditions on the fundamental constants that must 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 [Tanaka et al., 2002]: Where h is the disk height, set by the sound speed. Then, when using the expressions from the appendix and eqn (10), and specifying to terrestrial planets situated in the temperate zone, we find 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 .9 , indicating that 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 planets proposed in [Ormel et al., 2017], whereby earthlike planets form behind the snow line before migrating inwards. Such an unconventional pathway is needed to explain this system [Gillon et al., 2017], which has multiple Earth sized planets orbiting the temperate zone of a .08 solar mass star with planets near mean motion resonances [Tamayo et al., 2017] and possessing icy compositions [Unterborn et al., 2018]. This will not be undertaken here, however.
Even if we restrict our attention to parameter values where Earthlike planets do not undergo substantial migration, we may still run into trouble if Jupiter-like planets routinely barrel through their planetary systems. As this is governed by the physics of type II migration, the conditions that must be satisfied for (the majority of) these planets to stay put are somewhat different. Here, we restrict our attention to the case where a full gap is opened in the disk, and where the mass of the planet is substantially smaller than the mass contained in the disk. In this case, the timescale of this migration process is given by [Baruteau and Masset, 2013]: where ν is the viscosity. Strictly speaking, these quantities are supposed to be evaluated not at the position of the planet, but rather the place of maximum angular momentum deposition, which can be approximated as a`2.5R Hill . For our purposes, however, this correction is negligible for the scaling arguments. Then the ratio of this timescale to the disk lifetime is 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 smaller disks, 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 and a lower bound on γ, if the other quantity is fixed. The thresholds for both types of migration are displayed in Fig. 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. The teal line is for type I migration, the orange line is for type II, and the purple line is when the mass of the fastest migrating body is equal to the terrestrial mass. All curves have been normalized so the timescales in our universe are 5 times larger than the critical value.

Discussion: Comparing 480 Hypotheses
Having spent the last two sections detailing the physics behind a multitude of processes that may influence the habitability of a system, we now synthesize these into an estimate of the  Table 3: Probabilities of different combinations of the habitability hypotheses discussed in the text. Here yellow stands for the photosynthesis criterion, S the entropy criterion, temp for the temperate zone, GI and iso for the terrestrial planet criterion with the giant impact and isolation production mechanisms, and HJ the hot Jupiter condition.
probability of observing 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 planet formation, namely that the majority of galaxies should be larger than the minimal retentive mass, that massive stars should have shorter lifetimes than the star formation timescale, and that the minimum 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 the criteria that were considered: these were the absence of hot Jupiters, the production of terrestrial planets both through giant impact and isolation formation pathways, and the fraction of planets that end up in the temperate zone. Whether these are necessary for life are still intensely debated, and so each provides a prime opportunity to determine its compatibility with the multiverse hypothesis. Because they are all independent criteria, taken in conjunction they lead to a total of 2ˆ3ˆ2 " 12 possibilities. This does not include the 16 different choices we made in terms of planet formation, as well as the potentially continuous parameter signifying the slope of the power law for smaller planets. We display the probabilities of observing our measured values in Table 3 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 an acceptable range to explain our observations within the multiverse context. When combined with the 40 additional combinations from [Sandora, 2019a], there are a total of 480 separate habitability criteria that may be considered. Of these, only 43% of them give rise to probabilities of observing all three constants we consider of greater than 1%. The full suite of criteria is displayed in Table 4 at the end of this manuscript. For brevity we omit the convective criteria of [Sandora, 2019a] because it only ever marginally changes the numerical values, and in all instances its inclusion does not affect the viability of the combination of other hypotheses one way or the other. Of the 190 habitability criteria which give probabilities of over 10%, all make use of the entropy condition. A further 16 which do not include the entropy condition have probabilities greater than 1%-all of them benefit from an interplay between the yellow, tidal locking and biological timescale criteria, which place both upper and lower bounds on the types of allowed stars. The rest of the habitability criteria can safely be regarded as incompatible with the multiverse hypothesis.
The inclusion of multiple hypotheses leads to nonlinear effects, as the interplay between the distribution of purported observers and the anthropic boundaries alter the overall probabilities in sometimes surprising ways. That being said, none of the criteria have that drastic an effect on the probabilities, especially when including the entropy condition.
Some criteria, namely the terrestrial and temperate conditions, introduce lower bounds to some combination of α and β. In fact, lower bounds on these quantities are somewhat hard to come by in the anthropics literature, though it has always been clear that they should exist, as a world with massless electrons or no electromagnetism would certainly be very different from our own. The bounds we find are stronger than those that exist in the literature.
We also introduce a new measure of our universe's fitness, which we term the luxuriance. This is defined as the expected number of observers in our universe divided by the average number of observers per universe, restricting to universes that do have observers.

L "
Ppα obs , β obs , γ obs q ş d α θpPpα, β, γqq ş d α Ppα, β, γq Here the integration is over all three constants, and θpxq is the Heaviside step function: θp0q " 0, θpxq " 1 for x ą 0. The rationale for including this is that for most habitability criteria the vast majority of universes will be sterile, obfuscating comparisons between different criteria. Restricting to universes that only contain life gives a better feeling for how good our universe is at satisfying the chosen criteria. If our universe is better than typical at making life, then this quantity will be greater than 1. While this is not actually the guiding principle for evaluating whether our observations are consistent with the multiverse, it is a somewhat interesting quantity to consider. It gives some indication for how strongly observers may cluster within the multiverse-and the strong dependence of the properties we discuss on physical constants leads us to expect that they will, so that the majority of observers do find themselves in overly productive universes. The luxuriance ranges by two orders of magnitude for the different possibilities we consider, but the maximum is L " 78.4 for the condition which includes the yellow, tidal locking, biological timescale, hot Jupiter, isolation planet production mechanism, temperate, and entropy criteria. Also recall that certain choices for the physical processes involved in determining the structure of planets greatly affected some of the probabilities: the absence of a dependence on the mass of a planet with stellar mass, and too large a slope for the isolation mass, are both disfavored from the multiverse perspective. These scenarios are not otherwise excluded, but it would count as strong evidence against the multiverse if either of these were verified to be the case.
The one conclusion that should be drawn from this study is that planets are a rather generic feature, and not especially atypical for our particular values of the fundamental constants. Metal buildup is generic, massive stars burn out quickly, and disks tend to clump faster than they dissipate, so the inclusion of these criteria barely influenced the numerical values of the probabilities at all, apart from adding some mild anthropic boundaries. Further, even specifying the characteristics of the planets that are formed, or considering different scenarios of their formation, did not alter the probabilities by more than a factor of few for the most part. This points to a reassuring robustness of our predictions, that they are not as highly sensitive to the vagaries of incompletely known planet formation processes as one may have feared.
While it is simple to imagine a universe where planets are almost never made (and indeed before the current plethora of exoplanets was discovered many wondered if our universe was of this character), the parameter values needed to realize this possibility are quite extreme. Additionally, there are several features of planets in our universe that are tantalizingly coincidental, seeming to beckon for an anthropic explanation: the tendency to produce Earth mass planets, the similarity of the interplanetary spacing around sunlike stars and the width of the habitable zone, and the small but nonzero fraction of stars with hot Jupiters. Nevertheless, this reasoning was not borne out when incorporating these criteria into our analysis. The multiverse hypothesis is consistent with the expectation that life may only arise on temperate, terrestrial planets in systems without hot Jupiters, but it is essentially as compatible with the complete converse. The existence of suitable planetary environments thus does not seem to be the most important factor in determining which of the potential universes we find ourselves situated in. The other factors of the Drake equation which we will explore in subsequent works [Sandora, 2019b, Sandora, 2019c will uncover many additional predictions for the requirements of life.
However, we have demonstrated that there are plenty of habitability conditions that are completely incompatible with the multiverse: what this illustrates is that if any of the ones we have uncovered so far is shown to the correct condition for the emergence of intelligent life, then we will be able to conclude to a very high degree of confidence (up to 5.2σ) that the multiverse must be wrong. Since there is a single true habitability criterion which we will eventually determine, this manifestly demonstrates that the multiverse is capable of generating strong experimentally testable predictions that are capable of being verified or falsified on a reasonable timescale, the hallmark of a sensible scientific theory.

Appendix: Planetary Parameters
In this appendix we collect results on how various quantities relevant to our estimates in this work depend on orbital parameters of the stellar system, as well as fundamental quantities.
To begin we display the typical molecular binding energy: This also defines the temperature required for liquid water. Planets: The mass of a terrestrial planet, based off the criteria that carbon dioxide but not helium is gravitationally bound to the surface at these temperatures, is Next, we display the range of habitable orbits from the host star, using estimates from [Sandora, 2019a, Press andLightman, 1983]: Which is based off the temperature of a perfect blackbody located at that position, The speed of a circularly orbiting planet at a given location is and Ω is the angular frequency. This also defines the orbital period as t Kepler " 2π{Ω. The region of influence of a planet of mass M and orbiting a star at semimajor axis a is known as the Hill sphere, and has the radius Galaxies: the average galactic density is set by the galaxy at time of virialization, with a subsequent era of contraction due to equilibration [Tegmark et al., 2006, Adams et al., 2015: Corresponding to 10´2 3 g/cm 3 . Here, κ " Qpηωq 4{3 " 10´1 6 is a composite of the primordial amplitude of perturbations Q " 1.8ˆ10´5, the baryon to photon ratio η " 6ˆ10´1 0 , and the total matter to baryon ratio ω " 6.4. This defines the typical freefall timescale as Which equates to 3ˆ10 7 yr. Also relevant is the typical temperature of the interstellar gas, which is set by the threshold for H 2 cooling, at roughly 10 4 K: This also sets the velocity dispersion in the galaxy as v " a 5T H 2 {m p " 20 km/s. Disks: We take the disk mass to be set by The size of the disk is set by the conservation of angular momentum of the initial collapsing cloud. If it has a typical angular momentum of L " .01M vr, with v given by the dispersion velocity of molecular clouds, we find The mass loss is usually taken to be independent of position [Youdin and Kenyon, 2013], so that the surface density profile is dictated by the temperature. Early models took Σ9a´3 {2 , but this is now considered unlikely for equilibrium disks. The now standard dependence is Σ91{a [Williams and Cieza, 2011], which is both observed in simulations and understood from a theoretical perspective [Morbidelli et al., 2015]. We will take our profile to be The snow line is the distance beyond which the disk is cool enough for water to condense into a solid phase, which is found by setting the temperature equal to the vibrational energy of water molecules, The size a planetesimal attains is given by the isolation mass, M iso " 10ˆ2πΣ a R Hill , [Youdin and Kenyon, 2013]. The rationale behind this quantity is that once a planet is above this mass it will have already depleted all the material within its Hill radius, and so it will no longer have a supply to continue its growth. Note that the Hill radius is itself a function of the planet mass, so that solving this equation for the mass yields the expression: The coefficient is set by noting that the isolation mass is about Mars sized for the solar system.