###### Abstract

We investigate the dependence of elemental abundances on physical constants,
and the implications this has for the distribution of complex life for various
proposed habitability criteria. We consider three main sources of abundance
variation: differing supernova rates, alpha burning in massive stars, and
isotopic stability, and how each affects the metal-to-rock ratio and the
abundances of carbon, oxygen, nitrogen, phosphorus, sulfur, silicon,
magnesium, and iron. Our analysis leads to several predictions for which
habitability criteria are correct by determining which ones make our
observations of the physical constants, as well as a few other observed
features of our universe, most likely. Our results indicate that carbon-rich
or carbon-poor planets are uninhabitable, slightly magnesium-rich planets are
habitable, and life does not depend on nitrogen abundance too sensitively. We
also find suggestive but inconclusive evidence that metal-rich planets and
phosphorus-poor planets are habitable. These predictions can then be checked
by probing regions of our universe that closely resemble normal environments
in other universes. If any of these predictions are found to be wrong, the
multiverse scenario would predict that the majority of observers are born in
universes differing substantially from ours, and so can be ruled out, to
varying degrees of statistical significance.

###### keywords:

multiverse; habitability; element abundances

1 1 0 Academic Editor: Dirk Schulze-Makuch 03 November 2022 02 December 2022
https://doi.org/ Multiverse Predictions for Habitability: Element Abundances
Multiverse Predictions for Habitability: Element Abundances McCullen Sandora
1,*, Vladimir Airapetian 2,3, Luke Barnes 4, Geraint F. Lewis 5 and Ileana
Pérez-Rodríguez 6 McCullen Sandora, Vladimir Airapetian, Luke Barnes, Geraint
F. Lewis and Ileana Pérez-Rodríguez Sandora, M.; Airapetian, V.; Barnes, L.;
Lewis, G.F.; Pérez-Rodríguez, I. Correspondence<EMAIL_ADDRESS>
##  1 Introduction

There is seemingly an element of arbitrariness to the laws of physics. This
has prompted some to speculate that the laws we observe may not be unique, but
instead may vary from place to place (Carter, 1974); this has become known as
the multiverse hypothesis (Carr and Ellis, 2008). If this is correct, we will
never be able to explain our physical laws in the same way we explain other
features of the universe, such as the size of an atom, through relating them
to more fundamental quantities (Weinberg, 2007). In this scenario, many of the
universe’s peculiar features and behaviors are instead understood by an
alternative type of explanation known as the anthropic principle (Carr and
Rees, 1979; Barrow and Tipler, 1986). By invoking this, we start with the
tautology that we can only exist in universes capable of supporting complex
life, and use ensuing selection effects to explain otherwise puzzling aspects
of our universe. For example, it was put forward in (Weinberg, 1987) that our
universe is as immense as it is, even though the theoretical expectation for
typical universe size is minuscule, because a universe must be practically as
large as ours to host galaxies large enough to retain the heavy elements
necessary for planet formation. A major criticism of these anthropic arguments
is that we do not know the conditions complex life requires to arise and
thrive, since our current understanding of life is based solely on the single
example of Earth life, which shares a common ancestor. Consequently, invoking
anthropics to explain any fact about our whereabouts, its critics would have,
makes strong assumptions about the nature of habitability. This leaves us
vulnerable to overlooking a more fundamental, reductionistic explanation
(Wilczek, 2007), akin to how early anthropic explanations for the length of
the year overlooked the fact that this can be derived through Kepler’s third
law. Because it is unlikely that we will ever have the opportunity to directly
verify the existence of other universes, anthropic explanations run the risk
of not meeting the basic criteria that define scientific practices, and invite
a potentially unending debate (Kragh, 2009).

This situation can be remedied by inverting this anthropic logic; instead of
committing to a set of assumptions about habitability and inventing post-hoc
explanations for our observations, we may consider many potential habitability
criteria and determine which are compatible with the fact that we are in this
universe. This reasoning uncovers preferences for particular habitability
criteria over others, generating predictions for which conditions are expected
to be necessary for complex life. Here, compatibility with observations is
judged by characterizing how the features of universes (such as, for example,
the lifetime of stars and the size of planets) vary, allowing us to determine
which properties of our universe are generic throughout the multiverse, and
which are exceptional. By calculating the probability distributions of
observing these features within the multiverse, we can compare different
habitability conditions on the basis of how likely each makes our location
here. If some particular assumptions about habitability indicate that the
majority of complex life is in other universes with different properties,
these assumptions would imply that our presence in this universe is
exceedingly atypical. By invoking the principle of mediocrity (Vilenkin,
1995), which states that we expect to be typical among all observers who make
a particular observation, we can declare that conditions making our
observations unlikely are incompatible with multiverse expectations. In turn,
this generates predictions for which conditions are necessary for
habitability. In the coming decades, we will have the opportunity to learn
much more about the nature of habitability, and will be able to probe regions
of our universe that closely resemble normal environments in universes that
are predicted to be uninhabitable. If our findings run counter to these
predictions, this will serve to falsify the multiverse hypothesis, often to a
very high degree of confidence.

This task requires us to be able to quantitatively assess the probability of
being in a particular universe. Among other things to be discussed below, the
probability of being in a particular universe is proportional to the number of
observers that universe contains. The term ‘observer’ usually means something
like ‘a conscious entity’ but is left ambiguous here, as to date it has no
consensus definition. We may brush aside philosophical discussions about who
exactly qualifies because we make the simplifying assumption that observers
roughly correlate with complex life, here defined as multicellular organisms
(this definition may in fact reflect untoward bias; see (Booth and Doolittle,
2015)). In addition, we make an even greater assumption for the present paper,
which is that the conditions necessary for complex life do not differ
substantially for those necessary for simple life (unicellular organisms). To
be sure, unicellular organisms can survive over a broader range of conditions
than multicellular life, but, when compared to the full range of possible
environmental variability, the differences no longer appear so significant.
Both of these assumptions are meant to be approximations aimed at facilitating
our calculations, and have been partially examined in previous papers of this
series (Sandora, 2019a, b). However, further work is needed to determine what
interesting conclusions arise from considering the differences in
distributions between simple life, complex life, and observers. These
assumptions allow us to refer to the habitability of a universe as the number
of observers it contains. Via our assumptions, the habitability of a universe
is then also equated with the amount of simple life it can support.

Our strategy for evaluating different habitability conditions can be employed
in a rather formulaic manner toward any that one deems important. Life’s
occurrence depends on a great many different conditions, each of which inhibit
or promote habitability to different degrees when parameters are changed. The
present paper is part of a series that sets to task programmatically
incorporating as many habitability conditions into this framework as possible.
To begin, we focused on how the number and properties of stars vary with
physical constants (taken to mean masses of particles and strengths of forces
throughout) in (Sandora, 2019c), followed by the occurrence and properties of
planets (Sandora, 2019d), the fraction of planets that develop life (Sandora,
2019a), and the rates of mass extinctions (Sandora, 2019b). This exercise has
so far yielded around a dozen predictions for various habitability criteria,
any of which would be capable of ruling out the multiverse in the range of
$2-6\,\sigma$ (with probability ranging from 0.05 to $10^{-10}$) if found
false. However, our analysis is by no means complete, and sustained effort is
required to thoroughly fulfill the potential of this enterprise.

Any reliance of life on the particular chemistry present in an environment was
neglected in our previous works, enforcing a de facto stance that habitability
of an environment should be completely insensitive to its chemical
composition. Here, we investigate the effects of adopting various expectations
for the role of chemical abundances and life. For this, we consider three
different effects that can drastically alter elemental abundances, and their
consequences on purportedly essential elements: variations in the metal-to-
rock ratio, alpha burning in massive stars, and isotope stability.

The first is the ratio of metals to rocks, which is set by the relative yields
of the supernovae which produce each type of element. In Section 2, we find
that an upper bound on this ratio, proposed to maintain the stability of
liquid surface water, is disfavored by, though only mildly incompatible with,
the multiverse, and that any lower bound bears no consequence on the
probabilities of our observations.

In Section 3, we investigate the dependence of alpha burning in massive stars
on the Hoyle resonance, and the implications this has on the carbon-to-oxygen
ratio, the magnesium-to-silicon ratio, and the organic-to-rock ratio (defined
as (C+O)/(Mg+Si), with no intent to imply that all carbon and oxygen are
contained in biomolecules, or discount the importance of hydrogen). We find,
in line with some of the original anthropic arguments, that the organic-to-
rock ratio must be important for life to be compatible with the multiverse.
Further, a planetary threshold for the magnesium-to-silicon ratio which is
quite close to Earth’s value must be unimportant.

Lastly, in Section 4 we consider the variations in isotope stability with the
physical constants. We find that this induces large and drastic changes to the
abundances of every element throughout parameter space, and consider the
effects this has on nitrogen, phosphorus, sulfur, and iron. The expectations
that nitrogen and phosphorus are crucial for habitability are disfavored,
though only mildly incompatible with the multiverse, and no predictions can be
made for sulfur and iron. Including the carbon-to-oxygen criterion even
alleviates the nitrogen and phosphorus incompatibilities with the multiverse.
Other elements essential for Earth biochemistry, such as hydrogen, boron,
sodium, and chlorine, do not exert appreciable influence on our calculations,
and so are not considered in depth.

In our analysis, we make use of previously described criteria that were
determined to give the best probabilities for observing our values of the
physical constants. These are: the yellow condition, which enforces that
stellar radiation is within a narrow window of atomic energies (taken to be
600–750 nm), so that photosynthesis is possible (Sandora, 2019c), and the
entropy condition, which treats the probability of the emergence of life on a
planet as directly proportional to the incident entropy, in the form of
stellar radiation (Sandora, 2019a). Note that the yellow condition is
equivalent to restricting attention to stars with long lived, large habitable
zones (Underwood et al., 2003) and rarefied, cooler winds (Kudritzki and Puls,
2000), which are almost necessarily corollaries of stellar surface temperature
being similar to molecular binding energies. These are not the unique criteria
so far uncovered that would make all observations that we consider viable, and
we discuss alternative possibilities more thoroughly in Section 5.

We also assume that a planet must be temperate (so that surface liquid water
is stable) and terrestrial (to retain heavy but not light gases) in order to
qualify as habitable. These restrictions are based on the expectation that
liquid water is necessary for life, in accordance with (Ball, 2008, 2017).
Indeed, the presence of water is observed to place rather strict limits for
life, with very few microbes capable of being biologically active under
extreme desiccation (Stevenson et al., 2017; Hallsworth, 2019). Our discussion
presupposes both that water retains its properties which are deemed essential
for habitability as the physical constants are varied, that water will be
present on planets where it has the potential to be liquid, and, for some
planets at least, in moderate enough abundance to maintain some land. These
may not necessarily be true, but are not investigated here. Ignoring these
effects is tantamount to assuming that these properties are not essential,
assumptions which will be investigated in more detail in future work. The list
of habitability assumptions we employ, along with the conditions we
investigate in this work, is displayed in Table 1.

Table 1: List of habitability conditions assumed/examined in this work. Each condition is assumed necessary for habitability, with justification/analysis provided in the specified locations. Conditions not explicitly mentioned, such as the importance of tidal locking or the properties of water, are by default assumed not to be relevant for habitability in this work, but many have been/will be examined in other papers of this series. Assumption |  Discussion
---|---
photosynthesis |  (Sandora, 2019c)
temperate planets |  (Sandora, 2019d)
terrestrial planets |  (Sandora, 2019d)
life $\propto$ entropy |  (Sandora, 2019a)
observers $\approx$ complex life |  (Sandora, 2019b)
complex life $\approx$ simple life |  (Sandora, 2019a)
water |  future work
metal/rock |  Section 2
C, O, Mg, Si |  Section 3
N, P, S, Fe (Na, Cl) |  Section 4

Our previous analyses were restricted in scope to variations of only three of
the physical constants of nature: the fine structure constant
$\alpha=e^{2}/(4\pi)$, the electron-to-proton mass ratio, $\beta=m_{e}/m_{p}$,
and the strength of gravity $\gamma=m_{p}/M_{pl}$, where $e$ is electric
charge, $m_{e}$ is the electron mass, $m_{p}$ the proton mass, and $M_{pl}$
the Planck mass. These sufficed for the previous macroscopic properties we
considered. However, since the elemental abundances also depend sensitively on
the quark masses, we take this opportunity to enlarge the parameters which we
vary to include the up and down quark masses, $m_{u}$ and $m_{d}$. The
additional quantities we use to parameterize these are denoted by
$\delta_{u}=m_{u}/m_{p}$ and $\delta_{d}=m_{d}/m_{p}$. Discussion of how these
are implemented into our existing computations are given in Section 6.

Evaluating a given habitability hypothesis involves determining the
probability of observing our universe’s values of each of these constants,
computed according to the formula $p(\vec{x})\propto
N_{\text{obs}}(\vec{x})p_{\text{prior}}(\vec{x})$, where $\vec{x}$ are the
constants, $N_{\text{obs}}$ is the number of observers (conscious life forms)
in a given universe, and $p_{\text{prior}}$ is the underlying distribution of
universesThough our calculus explicitly counts the number of observers in the
universe, throughout this paper we consider hypotheses for which conditions
are clement for microbial life forms. Though these are certainly hardier than
complex, macroscopic organisms, the presumption is that the abundance of the
latter will track the former. We investigated various ways in which this
assumption may be violated in a manner that impacts our calculations in
(Sandora, 2019b), but so far have found few indications that the distinction
makes a meaningful difference.. As discussed in (Sandora, 2019c), we take as a
reasonable measure that $p_{\text{prior}}\propto
1/(\beta\gamma\delta_{u}\delta_{d})$, though most of our results do not depend
on this precise choice too muchA related issue is the measure problem, whereby
if universes are taken to be spatially infinite, the expected number of
observers per universe will also be infinite (see e.g., Freivogel (2011)).
This introduces a problem because the comparison of number of observers per
universe is then ill-defined, and highly sensitive to the method used to
regulate these infinite values. Thankfully for our purposes, however, these
troubles mainly affect the cosmological parameters, and not the microphysical
parameters we are concerned with here.. Far more important is the quantity
$N_{\text{obs}}$, which depends sensitively on the assumed requirements for
complex life. The probability of observing our measured value of any constant
is defined as
$\mathbb{P}(x_{\text{obs}})=\text{min}(P(x<x_{\text{obs}}),P(x>x_{\text{obs}}))$.
For reference, the probabilities of observing the five values of our constants
with the yellow and entropy baseline assumptions are

$\displaystyle\mathbb{P}(\alpha_{obs})=0.423,\,\,\mathbb{P}(\beta_{obs})=0.273,\,\,\mathbb{P}(\gamma_{obs})=0.234,\,\,\mathbb{P}(\delta_{u\phantom{!}obs})=0.167,\,\,\mathbb{P}(\delta_{d\phantom{!}obs})=0.494$
(1)

These will serve as the basis of comparison for all other habitability
criteria we consider in this paper.

##  2 What Metal-to-Rock Ratio Is Required for Life?

Though life is assembled primarily out of dissolved rocks, metals play a
crucial role as well. In many environments, they can serve as the limiting
nutrient, setting population sizes (Sigman and Hain, 2012). Metals are also
implicated in leading origin of life scenarios (Wächtershäuser, 1990; Ritson
and Sutherland, 2012), and have been argued to be essential for the origin of
many primordial anabolic processes (Muchowska et al., 2019) given that they
serve as cofactors catalysing many important biochemical processes.
Conversely, the high reactivity of metals makes them toxic when present in
large quantities (Stohs and Bagchi, 1995). Indeed, regions of high metal
pollution are typically restricted to microorganisms adapted to such
environments (Gadd, 2010). Therefore, from the biological context, it is not
unreasonable to think that there are tolerability limits to the amount of
metals, outside of which life would be extremely indigent, if not altogether
absent. However, the optimist may disregard such hesitations, pointing out
that several other essential molecules can be just as deleterious, including
both free oxygen (Bilinski, 1991) and water (Benner, 2014). The precise
habitability limits on metals are therefore open to debate at the moment,
especially when considering the diversity of single-celled and multicellular
lifeforms- as well as what it means to be alive, dead or dormant. Since the
metal-to-rock ratio varies throughout the multiverse, we can determine which
limits are compatible with human existence in this universe, and use this to
make predictions for what to expect these limits to be.

Aside from the biochemical aspects, metals have a significant effect on the
planetary system as a whole. These can be examined by considering the metal-
to-rock ratio $R$, where we take the word ‘metal’ here to mean the first row
of transition metals in the periodic table, and the word ‘rock’ to be all
other light elements above atomic number 5. The primary effect the metal-to-
rock ratio has on a planet is in the determination of core size. Several
studies have addressed the effects an altered core size would have on an
Earth-like planet: (Noack et al., 2014) investigate the influence of core size
on atmospheric outgassing, and found insufficient outgassing for maintaining
surface liquid water in planetary bodies where $R$ is greater than $1.9$ times
Earth’s value. They also stress that if the Earth’s metal-to-rock ratio were
substantially different, plate tectonics would be disrupted, leading to a
potentially uninhabitable planet (Walker et al., 1981); if $R$ is increased to
eight times its value, cooling of the much thinner convective region becomes
more efficient, and plate tectonics quickly shuts down. If $R$ is decreased to
0.08, heat dissipation is very slow and too much atmospheric CO2 builds up.
These conclusions are corroborated in the recent analysis done in (O’Neill et
al., 2020)Even within the Solar System this ratio varies from body to body,
ranging from $0.7\%$ for our Moon to $55\%$ for Mercury. However, both of
these are a result of giant impacts: typically, planetary composition will
closely follow that of the host star in many respects (McDonough and Sun,
1995).

In principle, extreme metal paucity would adversely affect planetary
habitability as well, most notably since the liquid core is responsible for
maintaining Earth’s magnetic field, buffering the atmosphere from cosmic rays.
However, this threshold is incredibly low: since a planet’s magnetosphere is
linearly proportional to core radius (Stevenson, 2003), the Earth’s would be
just as effective at shielding the atmosphere against solar wind above
$R=0.001$. Lastly, a lack of metals will have an adverse effect on stellar
dynamos, which act as engines for stellar wind while preventing the erosion of
thick planetary atmospheres and non-equilibrium prebiotic chemistry (see
(Airapetian et al., 2019) for a recent review). However, this threshold is
similarly quite weak. Though at this time we cannot be certain which of these
thresholds guarantee the absence of complex life, we are in a position to see
which are compatible with the multiverse.

Heavy elements are created in stellar fusion, and distributed throughout the
surroundings during supernovae events. Of note is the diversity of stellar
fates, as the properties of a supernova depend on stellar mass, spin,
metallicity, and whether a nearby companion is present. Several types of
supernovae are particularly relevant for our purposes: rock-like elements
(typified by oxygen and carbon) are produced in type II supernovae, which
result when a massive star exhausts its nuclear fuel. A related source of
rock-like elements is in the stellar winds and explosions of intermediate
stars, which have reached the asymptotic giant branch (or Wolf-Rayet) stage of
their evolution. Type Ia supernova arise when a white dwarf siphons enough gas
from a nearby companion to exceed the critical Chandrasekhar mass, triggering
a collapse and producing most of the metals present in our universe. Because
these two processes have such different origins, their yields will depend on
physical constants differently, and the ratio between these two processes will
vary in other universes. Additionally, type Ia supernovae are typically much
more delayed, causing the ratio to change with time. Using a simple star
formation model, we compute the distribution of values for a given set of
constants. We then integrate this criterion into our existing framework for
multiverse probability calculations for several proposed threshold values. To
do this, we first consider the two production rates in turn.

The type II production rate is comparatively simple: the rock production rate
is given by:

$\dot{M}_{\text{rock}}(t)=\psi_{\text{SFR}}(t-t_{\text{II}})\,f_{\text{II}}\,f_{\text{ej}}\,\langle
M\rangle_{\text{II}}$ (2)

Here, $\psi_{\text{SFR}}$ is the star formation rate, $f_{\text{II}}$ is the
fraction of stars that are large enough to undergo complete nucleosynthesis up
to nickel and iron, $f_{\text{ej}}$ is the fraction of the star’s mass which
is ejected as heavy elements during the supernova, and $\langle
M\rangle_{\text{II}}$ is the average mass of a massive star. The star
formation rate is evaluated at a slightly delayed time $t-t_{\text{II}}$ to
reflect the typical lifetime of these massive stars, but in practice this
matters little because the lifetime of massive stars is orders of magnitude
smaller than the star formation timescale for practically all parameter
values.

The fraction of stars which undergo type II supernovae explosions is about
$0.01$ and is, to first approximation, independent of physical constants.
Briefly, the reason is that this mass scale is dictated by whether the stellar
interior will have enough pressure to overcome a nuclear burning threshold
(Sandora, 2019d). Since both the minimum stellar mass and the knee of the
stellar initial mass function also only depend on a similar criterion (albeit
for hydrogen burning rather than oxygen), the fraction of massive stars is
independent of the details which set these thresholds, and thus does not
depend on physical constants.

Similarly, the fraction of mass ejected as heavy elements $f_{\text{ej}}$ is
roughly independent of the values of the physical constants that we are
considering. As described originally in (Carr and Rees, 1979), the ejection
mechanism relies on the fact that the neutrino outflow during the final stages
of the burning process interacts with the outer material, triggering a
blowout. This, together with changes in the weak scale, would make the
neutrino interaction too weak or too strong, and the material ejected would be
significantly diminished. Enforcing that type II supernovae are operational
effectively imposes a relatively narrow range of values the weak scale (Higgs
vacuum expectation value) may take, though fully incorporating a varying weak
scale into our analysis is left for future work. The ejected mass fraction is
independent of the constants, essentially because a majority of nuclei within
the star must participate in order to trigger a supernova. This scaling was
verified by computations performed in (D’Amico et al., 2019). Equally
important to note is that the type of material ejected is also independent of
constants: the fact that the ejecta is primarily oxygen, even though heavier
elements are produced in the star’s center, hinges on the fact that the
nuclear burning timescale during the final stages of the supernova are much
shorter than the Kelvin-Helmholtz turnover timescale, so that the outer
material does not have a chance to completely fuse. The fact that the inner
material is not ejected is similarly robust, as the radius at which material
escapes is always larger than the core (Marassi et al., 2019). The rock
production rate, after all these considerations, is then
$\dot{M}_{\text{rock}}\sim\psi_{\text{SFR}}\alpha^{3/2}\beta^{-3/4}M_{pl}^{3}/m_{p}^{2}$.

We briefly consider elemental enrichment due to intermediate mass stars: these
yield comparable amounts as type II supernovae, though which process dominates
depends on environmental conditions and element (Dray and Tout, 2003).
Depending on the element, the majority of production in intermediate mass
stars can take place from stellar winds (Dray et al., 2003). However, like in
type II supernovae, the total yield is an O(1) fraction of the star’s initial
mass, since each element is either ejected through wind until depleted or
released in the final supernova. Because of this, the yield from this source
scales in the same way as our estimates for rock production above, and so can
be treated as a rescaling of our previous estimate. It is also worth
mentioning that intermediate mass stars contribute subdominantly to the
production of many metals. Therefore, as long as this process is operational,
it may prevent universes from being completely metal free. However, as we will
see, the lower bound on $R$ is not constraining, and so neglecting this source
will not alter our conclusions.

Modeling the type Ia rate is more involved, as the physics dictating this
process is more complex. We can express it as

$\displaystyle\dot{M}_{\text{metal}}(t)=\int_{0}^{t}dt_{\text{I}}\,\psi_{\text{SFR}}(t-t_{\text{I}})\,f_{\text{WD}}\,f_{\text{binary}}(a_{\text{Roche}})\,f_{\text{IMF}}(t_{\text{I}})\,\langle
M\rangle_{\text{WD}}$ (3)

There are several important factors here: $f_{\text{WD}}$ is the fraction of
stars which are large enough to undergo a supernova explosion within the
lifetime of the system, but small enough that the remnant is a white dwarf
rather than a black hole or neutron starIn principle, one could worry that for
some parameter values the minimal mass black hole is smaller than the typical
supernova remnant, precluding Ia supernovae. If we take $18M_{\odot}$ as the
smallest black hole creating progenitor (Brown and Bethe, 1994), we find that
white dwarfs will exist and comprise the majority of supernova remnants as
long as $\alpha^{2}<31.3\beta$. This consideration only practically affects
regions for parameter space where $\beta$ is 300 times smaller, and so can be
safely disregarded.. Secondly, $f_{\text{binary}}(a_{\text{Roche}})$ is the
fraction of stars which occur in binaries, which are sufficiently closely
spaced for Roche lobe overflow to trigger catastrophic mass transfer,
resulting in explosion. Most importantly, $f_{\text{IMF}}(t_{\text{I}})$ is
the fraction of companions which deplete their hydrogen within time
$t_{\text{I}}$. This results a tenfold expansion in stellar radius
(independent of constants), such that nearly all type Ia supernovae that occur
are for companions which have entered the helium burning phase (Podsiadlowski,
2014). Lastly, $\langle M\rangle_{\text{WD}}$ is the average white dwarf mass,
which, like the average large star mass, is dimensionally set by the
Chandrasekhar mass, $M_{\text{Ch}}\sim M_{Pl}^{3}/m_{p}^{2}$. Here we must
integrate over the lifetime of these systems, as the spectrum of stellar
lifetimes is quite broad, and ranges over the galactic depletion timescales.

Our treatment of the fraction of binary stars
$f_{\text{binary}}(a_{\text{Roche}})$ will be cursory and superficial here,
with the intent to convince the reader that this does not depend much on the
constants we vary. A more detailed account is given in the Appendix A.
However, it suffices to note that the distribution of initial separations is
log-uniform (Korntreff et al., 2012). As such, the fraction closer than any
given threshold is only logarithmically sensitive to the parameters in
question. We find that, over the entire range of parameters we vary, this
quantity only changes by several percent.

Turning now to $f_{\text{IMF}}(t_{\text{I}})$, we use the simple Salpeter
power law form for the initial mass function,
$f_{\text{IMF}}(\lambda)=(\beta_{\text{IMF}}-1)\lambda_{\text{min}}^{\beta_{\text{IMF}}-1}/\lambda^{\beta_{\text{IMF}}}$,
where $\lambda$ is stellar mass, in units of the Chandrashekar mass,
$\lambda_{\text{min}}$ is the smallest stellar mass, and
$\beta_{\text{IMF}}=2.35$ (Salpeter, 1955). This neglects two things: firstly,
the power law form used here is not valid for small stellar masses. This is
unimportant for the regimes we are interested in, as those stars will not
start helium burning early enough to be relevant, and so will only affect a
shift in the normalization of this distribution. Secondly, it treats binary
companion masses as uncorrelated. This assumption does not hold for close
binaries (Kratter, 2011), where it is found that the mass ratio tends to 1.
However, use of this distribution is justified, as the companion star can be
treated as drawn randomly from the initial mass function, and so marginalizing
over this reproduces the original distribution, regardless of any correlation.

To turn this into a distribution for stellar lifetimes, we use the following
expression for dependence of lifetime on mass:
$t_{\star}(\lambda)=110\alpha^{2}M_{pl}^{2}/(\lambda^{5/2}m_{e}^{2}m_{p})$
(Sandora, 2019c). Lastly, we use the following expression for the star
formation rate:
$\psi_{\text{SFR}}(t)=\psi_{0}\,\text{exp}(-t/t_{\text{dep}})$, where the
depletion time is simply related to the galactic freefall time, so that
$t_{\text{dep}}\sim(G\rho)^{-1/2}\sim 0.070M_{pl}/(\kappa^{3/2}m_{p}^{2})$
(Sandora, 2019d), normalized to 3.6 Gyr in our universe from (Kennicutt Jr et
al., 1994). The density parameter $\kappa=Q(\eta\omega)^{4/3}$, with $Q$ the
amplitude of primordial fluctuations, $\eta$ the baryon-to-photon ratio, and
$\omega$ the matter-to-dark matter ratio. This star formation prescription is
perhaps somewhat simplistic, but it suffices to indicate how strongly the
metal-to-rock ratio varies. We plan on returning to more realistic models of
star formation, taking galactic substructure and superstructure into account,
in future work.

The production rate then becomes

$\displaystyle\dot{M}_{\text{metal}}(t)=f_{\text{binary}}\,\langle
M\rangle_{\text{WD}}\,\psi_{0}\,e^{-t/t_{\text{dep}}}\times\int_{0}^{t/t_{\text{dep}}}dz\,e^{z}\,\frac{d}{dz}\left(\frac{z}{z_{\text{max}}}\right)^{\zeta_{\text{IMF}}}$
(4)

Here
$z_{\text{max}}=t_{\star}(\lambda_{\text{min}})/t_{\text{dep}}=62100\,\kappa^{3/2}/(\alpha^{7/4}\beta^{1/8}\gamma)$
and $\zeta_{\text{IMF}}=2/5(\beta_{\text{IMF}}-1)=0.54$. Then the metal-to-
rock ratio of stars produced at time $t$ (normalized to equal 1 for solar
composition for ease of exposition) becomes

$R(t)=7311\,\alpha^{-3/2}\,\beta^{3/4}\,z_{\text{max}}^{-\zeta_{\text{IMF}}}\,\Gamma_{\zeta_{\text{IMF}}}\left(-\frac{t}{t_{\text{dep}}}\right)$
(5)

Here, $\Gamma_{k}(x)$ is the lower incomplete gamma function (subsuming a
factor of $(-1)^{\zeta_{\text{IMF}}}$ that enforces reality), which tends to
$x^{k}/k$ for small $x$ and $x^{k-1}e^{x}$ for large $x$. From this, it will
be seen that the ratio is nearly 0 at early times, when only type II
supernovae are operational, and tends toward infinity at late times, when star
formation has effectively ended but type Ia supernovae are still exploding.

We then invert this expression to find the fraction of stars below a given
metal-to-rock ratio:

$f(R)=1-\text{exp}\left(-\Gamma_{\zeta_{\text{IMF}}}^{-1}\left(0.053\,R\,\frac{\kappa^{0.81}\,\alpha^{0.56}}{\beta^{0.82}\,\gamma^{0.54}}\right)\,\right)$
(6)

This fraction ranges from 0 for small $R$ to 1 for large $R$. The coefficient
has been normalized so that the fraction of stars with $R$ greater than solar
is 0.88 (Madau and Dickinson, 2014) (also in agreement with (Kennicutt Jr et
al., 1994)), we find that the various candidate thresholds alter the
probabilities of measuring our values of the constants as displayed in Table
2.

Table 2: The probabilities of observing the values of our physical constants for various metal-to-rock thresholds. All utilize the entropy and yellow habitability criteria. For a habitability criterion to be considered consistent with the multiverse hypothesis, all probabilities must be reasonably close to 1. Restriction | Reasoning | $\,\mathbb{P}(\alpha_{\text{obs}})$ |  $\mathbb{P}(\beta_{\text{obs}})$ |  $\mathbb{P}(\gamma_{\text{obs}})$ |  $\mathbb{P}(\delta_{u\phantom{!}obs})$ |  $\mathbb{P}(\delta_{d\phantom{!}obs})$
---|---|---|---|---|---|---
None | baseline | 0.423 | 0.273 | 0.234 | 0.167 | 0.494
$R<1.9$ | outgassing | 0.262 | 0.042 | 0.461 | 0.183 | 0.327
$R<8$ | plate tectonics | 0.343 | 0.120 | 0.339 | 0.175 | 0.407
$R>0.08$ |  plate tectonics | 0.428 | 0.276 | 0.225 | 0.168 | 0.498

As mentioned above, we take what we designated as the entropy and yellow
habitability criteria in (Sandora, 2019a) as a baseline. We evaluate the
compatibility of our additional habitability criteria by comparing their
probabilities to those of the baseline. Generically, we say that a
habitability criterion is disfavored by the multiverse hypothesis if the
probabilities are significantly lower, and favored if the probabilities are
significantly higher. If we take $R<1.9$, as suggested by the outgassing
criterion of (Noack et al., 2014), the probability of observing our value of
the electron-to-proton mass ratio dips to $4.2\%$. If all probabilities are
treated as independent, this criterion has a Bayes factor (ratio of product of
probabilities) 7.3 times smaller than the baseline scenario. Though in the
standard terminology of (Kass and Raftery, 1995) this gives ‘substantial’
reason to expect the outgassing condition to not be important for
habitability, the evidence gained if it is would not be enough to exclude the
multiverse to any significance, since the chance of one of the probabilities
being this small is only about one in five. The other two considerations have
even less of a bearing on the probabilities. Even if the lower bound on $R$ is
taken to be arbitrarily close to our value, the probabilities are affected by
at most $50\%$, because this only excludes regions of parameter space that are
not particularly fecund anyway. While one could say that the multiverse gives
better odds for high metal planets being habitable, we conclude that the
selection for a prime metal-to-rock ratio does not seem to be a determining
factor for why we live in this universe. Likewise, the multiverse does not
give us very strong expectations for which of these habitability criteria are
correct.

##  3 Organics and Rocks

###  3.1 What Carbon-to-Oxygen Ratio Is Required for Life?

The life-giving properties and ubiquity of carbon have long been a fixture of
modern anthropic arguments. A large part of (Henderson, 1913) is devoted to
extolling carbon’s natural ability to form large biomolecules, dissolve in
water, partake in multiple phases, and act as a mild acid. Today it is
recognized as the key element comprising molecules important to living
organisms, and defines organic chemistry. There is a long history of debate
among astrobiologists over whether carbon is essential for life (Pace, 2001),
or whether life may be based on some other substrate elsewhere in the universe
(Bains, 2004; Schulze-Makuch and Irwin, 2006) that continues to this day (see
e.g., (Petkowski et al., 2020) for a recent review). Since the relative
amounts of these elements can vary quite strongly depending on the constants,
it is worthwhile to consider whether taking a stance on the necessity of
carbon affects the probabilities of observing our universe’s physical
constants, and thus whether multiverse expectations make a prediction about
its importance.

A major spur in the anthropic discussion of carbon’s importance was Hoyle’s
1954 observation that stellar carbon production is several orders of magnitude
greater than naive expectations due to an improbable coincidence in the energy
levels of nuclear states (Hoyle, 1954). This coincidence causes a near
equivalence between the energy of three helium nuclei and an excited level of
the carbon-12 nucleus, which enhances the rate of this unusual fusion pathway,
dubbed the triple-alpha reaction. Modern measurements put this energy
difference at 380 keV (Epelbaum et al., 2013), an impressively small number,
given that it results from the sum of several terms all of the order 10 MeV.
Due to this, the carbon abundance is seven orders of magnitude larger than the
beryllium abundance, despite the trend for heavy elements to be less abundant
than lighter ones. Equally important is the fact that the reaction to fuse
oxygen from a carbon and a helium is slightly endothermic, so that the carbon
initially created is not immediately destroyed (Livio et al., 1989). Since
this is controlled by nuclear binding energies as well, there is only a tiny
region of parameter space where both carbon and oxygen exist simultaneously
(Oberhummer et al., 2000), and we happen to be situated in it.

This amount of fine tuning has been taken as evidence for the necessity of a
highly specific C/O ratio, as otherwise it would be a strong coincidence that
we find ourselves in such an apparently anomalous universe (Barnes, 2017).
However, this has not yet been incorporated into our framework, where one
tries to tally up the number of observers per universe for different
habitability hypotheses regarding the importance of this ratio to determine
which are compatible with our measurements of the physical constants. We
undertake this analysis here, and also show how the traditional fine tuning
arguments can be subsumed into our analysis.

For the dependence of the C/O ratio on the physical constants, we use the
dependence of the various abundances calculated in (Huang et al., 2019) on
$E_{R}=E_{12}^{*}-3E_{4}$, the Hoyle resonance energy. There they modify this
value in stellar evolution codes and keep track of nuclear interactions out to
silicon, which allows us to study several relevant abundance ratios. These are
displayed in Figure 1, which is reconstructed from Figure 9 of (Huang et al.,
2019). Note that this represents the yield averaged over solar metallicity
supernovae, and does not take other processes, such as yields from medium mass
stars, into account. As such, it represents the current best estimate for the
dependence of these abundances on the constants, but it will not reflect the
dependence exactly. The overall conclusions we draw from using this
approximation are expected to hold, barring some unforeseen reason why the
contributions from the neglected processes act to perversely undo these
changes.

Figure 1: Abundance ratios of various elements as a function of the Hoyle
resonance energy. The quantity $\Delta E_{R}=0$ for our values of the
constants.

To relate the change in the Hoyle resonance to physical constants, we use the
relations found in (Epelbaum et al., 2013) to arrive at:

$E_{R}=\left(0.626\,\left(\delta_{u}+\delta_{d}\right)+0.58\,\alpha-0.004204\right)\,m_{p}$
(7)

This is a lowest order Taylor expansion, but will be sufficient for our
purposes, since the relevant thresholds are so close to our observed value. By
combining this expression with the curves in Figure 1, we can determine
thresholds on these parameters for any given ratio threshold.

The C/O ratio strongly affects planetary systems, so there is good reason to
suspect it may have an effect on habitability. Its effects on planetary
composition are very nearly stepwise around the value C/O=1. This is a
consequence of the fact that essentially all available C and O in the
protoplanetary disk will combine to form carbon monoxide (CO); this leaves
behind only the more abundant of the two to interact with other compounds
(Larimer, 1975). For the solar value, C/O=0.55 (Asplund et al., 2009), which
is very near the cosmic value of 0.6, the surplus oxygen forms silicates and
oxides. If carbon were the more abundant element, carbide rocks like SiC and
Fe3C would condense instead (Larimer, 1975). These are much sturdier rocks
than silicates, are more resistant to weathering, and may not support plate
tectonics as easily. These planets would have crusts made of graphite rather
than quartz and feldspar (Lodders and Fegley Jr, 1997), and may have oceans of
hydrocarbons or tar rather than water (Kuchner and Seager, 2005).

Likewise, we may consider that universes with much lower C/O ratio would have
correspondingly less material available for the construction of organic
compounds, resulting in reduced biomass. A particular subject of study has
been hydrogen cyanide (HCN), a key constituent of nucleic acids, and the
elementary constituent of most, if not all, proposed prebiotic RNA synthesis
pathways (Benner et al., 2019). Primordial production of this compound was
discussed in (Rimmer and Rugheimer, 2019), where it was found that HCN is not
produced in any appreciable abundance below C/O=0.5, where CO2 becomes the
overwhelmingly dominant atmospheric constituent, and CH4 is essentially
absent.

It should be noted that stellar values of C/O need not necessarily correspond
to the cosmic mean, and planetary values need not correspond to their host
stellar values. The C/O ratio increases toward the galactic center (Esteban et
al., 2004), and also with time (Kuchner and Seager, 2005), though the
enrichment timescale is several times the current age of the universe, and
robustly longer than the star formation epoch. As a consequence, some stars in
our universe have C/O $>1$ (Petigura and Marcy, 2011). The condensation of CO
and CO2 beyond their respective ice lines in the disk significantly affects
the abundances of the bodies that form at that particular location; between
the H2O and CO snow lines the C/O ratio can be significantly depleted from the
stellar value, (Öberg et al., 2011). Chemical and disk evolution can further
complicate the relation between planetary and disk C/O (Hussein et al., 2020).
Atmospheric C/O is highly dependent on the planet’s stochastic accretion
history and subsequent evolution (Thiabaud et al., 2015). Lastly, local
planetary C/O ratios can be substantially altered due to volcanic sources
(Etiope and Milkov, 2004). Here, we restrict our attention to planets inside
the water snow line, and do not take variation about the cosmic mean into
account. Though potentially important, the amount of scatter is dictated by
galactic more than microphysical processes, and so a proper treatment of this
will necessarily be delayed until we fold these galactic effects into our
analysis, and vary cosmic parameters as well.

We can now test how the habitability hypothesis that carbon-rich systems are
uninhabitable fares with multiverse reasoning. Using the ratio C/O=1 to
restrict our attention to silicate planets, which corresponds to $\Delta
E_{R}=-8.9$ keV, we have the following probabilities of measuring our observed
values of the constants:

$\displaystyle\mathbb{P}(\alpha_{obs})=0.159,\,\,\mathbb{P}(\beta_{obs})=0.192,\,\,\mathbb{P}(\gamma_{obs})=0.251,\,\,\mathbb{P}(\delta_{u\phantom{!}obs})=0.211,\,\,\mathbb{P}(\delta_{d\phantom{!}obs})=0.371$
(8)

As can be seen, these are all extremely reasonable values, and do not differ
significantly from the results we get when this criterion is not assumed. We
may also include a lower bound on C/O: if we take the threshold to be 0.5 to
ensure the production of HCN, corresponding to $\Delta E_{R}=3.7$ keV, the
probabilities change by at most $40\%$:

$\displaystyle\mathbb{P}(\alpha_{obs})=0.292,\,\,\mathbb{P}(\beta_{obs})=0.213,\,\,\mathbb{P}(\gamma_{obs})=0.280,\,\,\mathbb{P}(\delta_{u\phantom{!}obs})=0.228,\,\,\mathbb{P}(\delta_{d\phantom{!}obs})=0.300$
(9)

Therefore, considering _only these observables_ does not give us strong reason
to expect this ratio to be important one way or the other.

However, we may augment these standard probabilities with additional measures.
One observable corresponds to the probability of measuring our value of
$E_{R}$. Note that this is equivalent to considering the probability of
measuring C/Si to be at least as large as our value. The probability without
restricting to any C/O range is $\mathbb{P}(E_{R\phantom{!}obs})=0.456$, while
including an upper bound of 1 leads to
$\mathbb{P}(E_{R\phantom{!}obs})=0.063$, including a lower bound of 0.5 leads
to $\mathbb{P}(E_{R\phantom{!}obs})=0.006$, and including both yields
$\mathbb{P}(E_{R\phantom{!}obs})=0.301$. The lower bound fares poorest,
indicating that if HCN production is important, then silicate planets have to
be as well.

Another quantity we can consider is the ‘organic-to-rock ratio’,
(C+O)/(Si+Mg). As can be seen in Figure 1, we are situated very close to the
peak of this quantity, so that our universe is exceptionally rich in organic
material. If we consider the probability of measuring such a high value of
this quantity without assuming that a particular composition is necessary for
habitability, we find $\mathbb{P}(\text{(C+O)/(Si+Mg)}_{obs})=1.2\times
10^{-4}$, a highly unlikely value. However, if we restrict to universes with
$0.5<$ C/O $<1$, we instead find
$\mathbb{P}(\text{(C+O)/(Si+Mg)}_{obs})=0.287$. This is one way of formalizing
the fine tuning argument that is often discussed regarding the triple-alpha
process: if habitability does not depend much on the chemical content of the
universe, then it is very coincidental that we happen to reside in a universe
with such high organic content. However, unlike the standard anthropic
argument, which only takes note of the fine tuning, our account of the
relative probabilities of measuring this quantity ascribes a definite
statistical significance to this coincidence. Correspondingly, this sets the
significance by which the multiverse would be disfavored if this prediction is
found to be wrong, and life is insensitive to organic-to-rock ratio. These
results are summarized in Table 3. Lastly, note that hydrogen, the other key
organic element, is the most abundant element in the universe by many orders
of magnitude, and this fact does not change with the parameters we vary
(though see (Hall et al., 2014) for a discussion on how the presence of
hydrogen places constraints on the weak scale).

Table 3: The probabilities of our observations for various habitability criteria described in the main text. Each row indicates a purported habitability criterion (or combination), the associated restrictions on elemental abundance ratios, and the range of Hoyle resonance energies compatible with those restrictions. $\mathbb{P}(E_{R\phantom{!}obs})$ is the probability of observing the Hoyle resonance energy to be at least as small as our measured value, and $\mathbb{P}(\text{(C+O)/(Si+Mg)}_{obs})$ is the probability of measuring the ‘organic-to-rock’ ratio to be at least as large as our measured value. Probabilities greater than 0.1 are displayed in bold. All probabilities are computed utilizing the entropy and yellow habitability criteria. Reasoning | Thresholds | Range (keV) | $\,\mathbb{P}(E_{R\phantom{!}obs})$ |  $\mathbb{P}\left(\frac{C+O}{Si+Mg}\big{|}_{\text{obs}}\right)\,$
---|---|---|---|---
baseline | none | - | 0.456 | 0.00012
silicates | C/O $<1$ | $-8.9<\Delta E_{R}$ | 0.063 | 0.026
HCN |  $0.5<$ C/O | $\Delta E_{R}<3.7$ | 0.006 | 0.006
silicates + HCN |  $0.5<$ C/O $<1$ | $-8.9<\Delta E_{R}<3.7$ | 0.301 | 0.287
olivine |  $0.7<$ Mg/Si | $\Delta E_{R}<6.2$ | 0.010 | 0.006
plate tectonics |  $1<$ Mg/Si | $\Delta E_{R}<0.73$ | 0.001 | 0.001
mantle viscosity | Mg/Si $<1.5$ | $-83.9<\Delta E_{R}$ | 0.404 | 0.017
plate tectonics + mantle viscosity |  $1<$ Mg/Si $<1.5$ |  $-83.9<\Delta E_{R}<0.73$ | 0.007 | 0.007
silicates + olivine |  C/O $<$ 1, & 0.7 $<$ Mg/Si | $-8.9<\Delta E_{R}<6.2$ | 0.410 | 0.243
silicates + plate tectonics | C/O $<1<$ Mg/Si | $-8.9<\Delta E_{R}<0.73$ | 0.063 | 0.063

###  3.2 What Magnesium-to-Silicon Ratio Is Required for Life?

As with the C/O ratio, the initial Mg/Si of a planetary system has a large
effect on the mineral content of its ensuing planets. As can be seen in Figure
1, this ratio is even more sensitive to the Hoyle resonance energy, and we are
situated even more closely to an apparent boundary. Though the effects of the
various purported thresholds on habitability are still under active
investigation, each can be incorporated into our analysis to generate
predictions for which ones we expect to be important.

The cosmic Mg/Si ratio is 1.04, the spread is between 0.8 and 2 (Shim et al.,
2015), and Earth’s is 1.02 (Thiabaud et al., 2015). Major thresholds for this
quantity are the values of 1 and 2; because the minerals that form from these
either contain Mg and Si in 1:1 or 2:1 ratios, the relative abundance
determines which rocks are formed in the protoplanetary disk, which species is
left over to form other minerals, and the large scale properties of planetary
interiors. For Mg/Si $<$ 1, almost all magnesium is incorporated into pyroxene
(given loosely by MgSiO3), and the remaining Si forms silicates. Between 1 and
2, a mixture of pyroxene and olivine (loosely Mg2SiO4) form, as is the case
for Earth. Above 2, and all Si is contained in olivine, with the remaining Mg
in oxides such as periclase (MgO) (Thiabaud et al., 2015). Thus the Earth, and
our universe, represent an intermediate phase between planets with excess
magnesium and planets with excess silicon.

Planetary mantles which are dominated by these different minerals can have
extremely different properties and behaviors. Mg acts very similarly to Fe in
many mineral systems, and of the abundant elements, interacts most strongly
with O in the primordial mantle (Unterborn et al., 2014). The different
magnesium compounds have different stereochemical structures, and so will have
varying bulk properties. Pyroxene can be viewed as packs of long chains,
making for a highly viscous mantle. Olivine, on the other hand, is comprised
of packed silicate tetrahedra, leading to a less viscous mantle. Periclase is
comprised of still smaller units, resulting in the most inviscid mantle of all
three. Because mantle viscosity is exponentially sensitive to chemical binding
energies, mantles dominated by periclase can be 100 times less viscous than
our own (Ammann et al., 2011). This will have a profound impact on a planet’s
tectonics, but whether for the better or worse is unclear; Ref. (Pagano et
al., 2015) point out that such extreme volcanism would likely lead to an
excess of methane, which could lead to a runaway greenhouse effect.
Conversely, a low Mg/Si ratio can lead to a thermally stratified mantle that
can arrest convection and just as significantly affect tectonic regime and
outgassing rate (Spaargaren et al., 2020). A lower limit of Mg/Si $<$ 0.7 was
argued to lead to the absence of plate tectonics altogether in (Shim et al.,
2015).

Thus the thresholds we have to consider are Mg/Si$=$ 0.7, 1, and 2. The
abundance ratio never exceeds 1.86, which occurs at $\Delta E_{R}=-59$ keV.
The other two occur at 0.73 keV and 6.2 keV, respectively. We display these,
along with the C/O thresholds discussed in the previous section, in Table 3.

Most rows in this table contain at least one anomalously small probability.
Habitability criteria can only be said to be consistent with multiverse
expectations if all probabilities are close to 1. Only by assuming that the
C/O ratio lies within a narrow range do all probabilities exceed $10\%$.
Taking the extreme stance that systems with Mg/Si $<$ 1 are uninhabitable
leads to exceedingly low probabilities, though this is alleviated to some
extent if one also adopts the view that the C/O ratio is important. The less
stringent bound of Mg/Si $=$ 0.7, however, is perfectly compatible with these
observations. This leads us to the following predictions: systems with
marginally lower Mg/Si should be just as habitable as Earth, whereas systems
with grossly different C/O should not be. If this is found not to be the case,
this has the potential to cast severe doubt on the multiverse.

##  4 Additional Elements

On Earth, a wide range of elements are essential for biochemistry. These
include the main constituents carbon, hydrogen, nitrogen, oxygen, phosphorus
and sulfur, as well as trace metals such as iron. The ubiquity of these is
controlled not only by which nuclides are produced in stars, but also which
isotope is most stable for a given atomic number. This latter facet is just as
sensitive to the physical constants as the production mechanisms, and because
of the stepwise nature of nuclear stability, small changes to the values of
the constants can lead to $\mathcal{O}(10-100)$ changes to the elemental
abundances. In this section, we explore in detail how these stability
thresholds affect several of the most important elements for life, namely
nitrogen, phosphorus, sulfur, and the iron peak elements.

To investigate the conditions for nuclear stability, we use the semi-empirical
mass formula, which is a phenomenological model that makes extrapolating
nuclear properties to different values of the physical constants possible
(Krane, 1987). With this, the binding energy of a nucleus with $A$ nucleons
and charge $Z$ is

$\displaystyle
E_{b}(A,Z)=a_{v}A-a_{s}A^{2/3}-a_{c}\frac{Z(Z-1)}{A^{1/3}}-a_{sym}\frac{(A-2Z)^{2}}{A}+\delta$
(10)

The first terms are the volume and surface energies, which result from nearest
neighbor strong interactions. The third term is due to Coulomb repulsion, the
fourth is due to Fermi repulsion, and gives preference to nuclei with equal
numbers of protons and neutrons, and the fifth encapsulates spin coupling,
giving preference to nuclei with even numbers of protons and neutrons.

The dependence on $\alpha$ and $m_{p}$ of these coefficients are taken to be:

$\displaystyle a_{c}$ $\displaystyle=$ $\displaystyle
0.714\,\frac{\alpha}{\alpha_{0}}\,\frac{m_{p}}{m_{p0}}\text{ MeV},$
$\displaystyle a_{sym}$ $\displaystyle=$ $\displaystyle
23\,\frac{m_{p}}{m_{p0}}\text{ MeV},$ $\displaystyle\delta$ $\displaystyle=$
$\displaystyle
34\,\frac{(-1)^{Z}}{A^{1/2}}\,\frac{m_{p}}{m_{p0}}\,\chi_{evens}(A)\text{
MeV}$ (11)

The function $\chi$ is the indicator function, which equals 1 when $A$ is even
and vanishes otherwise. We also take the difference between the proton and
neutron masses to be given by (Hall and Nomura, 2008):

$m_{n}-m_{p}=m_{d}-m_{u}-1.21\frac{\alpha}{\alpha_{0}}\,\frac{m_{p}}{m_{p0}}\text{
MeV}$ (12)

This is used to determine which element $Z$ is the most energetically
favorable for a given nuclear weight $A$ by considering the process of beta
decay. This process can only occur if the quantity

$\Delta E=E_{b}(A,Z)-E_{b}(A,Z-1)+m_{n}-m_{p}-m_{e}>0.$ (13)

The total abundance of a given element will the the sum of all the isotopes
for which that element is stable, multiplied by the abundance of each of those
isotopes. In Figure 2, we display the abundances of the second and third row
elements according to this criterion.

In this figure, we have kept track of isotopes up to atomic number 86, and
have used the abundance values given in (Lodders, 2003), who record abundances
for each nuclear weight $A$. It is these values which are most appropriate for
our purposes, as they are insensitive to the physical constants and can be
treated as fixed. The elemental abundances are then calculated by grouping
these abundances according to which element $Z$ is energetically preferred for
every $A$. In general, these thresholds will occur along hyperplanes in
parameter space of the form $\beta+\delta_{d}-\delta_{u}=k\alpha+c$, where $k$
ranges from 0 to 2, with an average over the first 40 isotopes of 1.005. To
display the extent of this effect, then, we plot it in the maximally sensitive
direction, where the change in $-\beta-\delta_{d}+\delta_{u}$ is proportional
to the change in $\alpha$. This is given by the dotted line in Figure 3. Bear
in mind, however, that restricting to this subspace is only a first order
approximation, and the details of Figure 2 can be stretched out in various
directions of parameter space.

Figure 2: The abundances of second and third row elements, restricted to the
maximally sensitive direction in parameter space described in the text, and
parameterized by $\alpha$. The black line denotes our observed values. The
elements are ordered from the upper right downward. Figure 3: Nuclear
stability thresholds for elements from the first three rows of the periodic
table. The black dot represents our values. The black dashed line is the
maximally sensitive direction, averaged over these isotopes. Each domain in
the figure has fixed abundance ratios, and the thresholds correspond to the
abrupt changes displayed in Figure 2.

Several features of these plots should be remarked upon. First is the fact
that many transitions are present, beyond which nearly the entirety of a
particular element is swapped out for its neighbor. These transitions are
abrupt, so that there are stretches of parameter space where a particular
element may be nearly absent, but beyond which conditions are changed to the
point where that element is produced as a completely different isotope. Notice
that the abundances are organized into several horizontal stripes in Figure 2
representing the different nuclear weights $A$, and as $\alpha$ increases, the
preferred charge $Z$ decreases stepwise. The overall trend for larger $\alpha$
is to produce universes with less heavy and more light elements. An exception
to this can be seen at the very right edge of the plot, where the argon
abundance spikes up. This occurs because the stability of the iron peak,
$A=56$, has traversed to the point where it has entered the third row of the
periodic table, which we will discuss more thoroughly below. Even charges are
generically stable for larger regions of parameter space than odd charges due
to the pairing term.

Because there are so many relevant isotopes, and there are up to a dozen such
transitions for each isotope as parameters are varied, hundreds of such
transitions occur within the range of several times the observed value of
$\alpha$. As such, we expect some transition to occur whenever the constants
are varied by even a fraction of a percent. The result is a highly
heterogeneous checkerboard of elemental abundances throughout parameter space,
producing universes which can drastically differ with regards to certain,
possibly key properties.

Before we go on to consider several specific elements, let us state a few
caveats in this analysis. As stated before, we normalize the baseline
abundances to those in our universe. As explored in previous sections, these
abundances may change if the yields of massive stars, or the ratio of yields
between differing supernovae, are altered. Our analysis in this section can be
seen as complementary to these discussions; this is in part to disentangle the
origins of these different effects, and in part because a full account,
incorporating all three separate effects, is a much more ambitious
undertaking, that will require calculating the parameter dependence of the
supernova yield of every element. Additionally, we do not consider instability
due to electron capture, and do not treat cases where unstable elements have
half life longer than the age of the universe as effectively stable. These
borderline cases will shift the thresholds somewhat, but not appreciably.

With these preliminaries, we can now specialize our discussion to several
nearby thresholds, and the consequences each has for the abundances of some of
our most cherished elements.

###  4.1 Is Nitrogen Essential for Life?

Nitrogen is essential for life as we know it, being a constituent of proteins,
DNA, and RNA. It is the linchpin of the peptide bond, which forms the
structural backbone of proteins. The reason this has been selected by
biochemistry over other potential scaffolds seems to be nitrogen’s ability to
form three bonds, which allows it to bind two molecules together while
maintaining a positive charge (Blanco et al., 2018). This increases its
stability in aqueous solution and its affinity for nucleic acids. Nitrogen’s
triple bond is not wholly beneficial for life, though, as it results in the
majority of nitrogen atoms pairing off, rendering it a mostly inert gas. As
such, nitrogen is often a limiting nutrient in ecosystems (Elser et al.,
2007). Still, one may be able to convince themselves that microbial life, and
the complex life it supports, would do just as fine in nitrogen poor universes
by prioritizing fixation (Graf et al., 2014) and/or tightening cycling of
bioavailable N chemical species (Morris et al., 2013). However, without a
noncondensible gas in the atmosphere, liquid surface water would be much less
stable on terrestrial planets (Pierrehumbert, 2010).

Despite its ubiquity, nitrogen is actually one of the most precariously stable
element in our universe. The vast majority of nitrogen-$99.6\%$\- is
nitrogen-14, produced during the CNO cycle of massive stars, with some
contribution from smaller stars as well (Woosley and Weaver, 1995). Its
binding energy is extremely small; from the semi-empirical mass formula,
Equation (13):

$\displaystyle\Delta
E\left({}^{14}\text{N}\right)=-0.0017\,m_{p}+0.696\,\alpha\,m_{p}-m_{e}-m_{d}+m_{u}$
(14)

This equates to just 156 keV for our valuesApplication of the SEMF in this
case actually predicts this reaction to be marginally allowed in our universe,
a consequence of the actual value being below the SEMF’s typical accuracy
threshold. To correct for this, we set the additive multiple of the proton
mass to correspond to the value we infer from the reverse reaction..

The flip side of such marginal stability is the fact that in our universe,
carbon-14 decays with extremely tiny reaction energy. The primary consequence
of this is a drastic increase of its half life by a factor of $10^{6}$, making
it by far the lightest element that decays on kyr timescales. This can be
attributed to an accidental cancellation between density-dependent nuclear
forces (Holt et al., 2010). The tuning is quite significant- if $\alpha$ were
increased by just 3.4$\%$ nitrogen-14 would decay into carbon-14, leading to a
cosmic nitrogen abundance reduced to a factor of 0.0037 the observed value.

In light of the importance of nitrogen, we may consider the hypothesis that
universes with such significant reduction in nitrogen are uninhabitable.
Adopting this stance leads to the following probabilities:

$\displaystyle\mathbb{P}(\alpha_{obs})=0.049,\,\,\mathbb{P}(\beta_{obs})=0.071,\,\,\mathbb{P}(\gamma_{obs})=0.163,\,\,\mathbb{P}(\delta_{u\phantom{!}obs})=0.080,\,\,\mathbb{P}(\delta_{d\phantom{!}obs})=0.182$
(15)

A comparison to the baseline in Equation (1) (and again treating each
probability as independent) leads to a Bayes factor of 268 lower for this
criterion, meaning that, if nitrogen is taken to be essential for life, the
probability of our observations would be lowered by this amount compared to
the baseline case. Evidently, this expectation is comparatively quite highly
disfavored from multiverse expectations. This may be somewhat surprising,
given the many important roles nitrogen plays. We hesitate to conclude that
this is incompatible with the multiverse, however, since all probability
values are not unreasonably small.

To compare these two hypotheses, the recommendation is to search for life in
nitrogen poor regions of our universe. If atmospheric nitrogen is the result
of tectonic forces, we may look at super-Earths which are too large to support
plate tectonics (Van Den Berg et al., 2019), or, if it is the result of influx
from the outer system, we may look at planets that have been subjected to less
late stage delivery (Grewal et al., 2019). If we indeed find that these
environments are unfit for life (controlling for other concomitant factors),
this would run counter to our predictions above, and so count as a strike
against the multiverse hypothesis.

We note that if the C/O habitability criterion is included in conjunction with
the N criterion, the probabilities are significantly ameliorated. This will be
discussed further in Section 5.

###  4.2 Are Phosphorus and Sulfur Essential for Life?

The most utilized third row elements in biochemistry are phosphorus and
sulfur. The main utility of these third row elements is that they are capable
of forming additional chemical bonds. Phosphorus is particularly important for
life, phosphate (PO4) being a key component of DNA, RNA, ATP, and
phospholipids. These essential roles derive from several properties of
phosphorus: first, it has the ability to link multiple atoms while maintaining
a (single) negative charge, which inhibits reactions and promotes its
stability. Second, phosphate is kinetically unstable and so can act as an
energy source, but this process is thermodynamically unlikely without an
enzyme, allowing its controlled release (Westheimer, 1987). Phosphorus is
often a limiting nutrient (Tyrrell, 1999), and an increase in the global
phosphorus cycle has been argued to be the cause of the Cambrian explosion
(Reinhard et al., 2017). In (Hinkel et al., 2020), phosphorus abundance was
argued to have a strong effect on planetary habitability.

Some hypothetical biochemistries have been put forward that would replace
phosphorus with arsenic, though whether this would ultimately be suitable
remains to be seen. As has been pointed out for example in (Tawfik and Viola,
2011), arsenic is much less stable in water, and maintaining the molecular
configuration of an arsenic-containing biomolecule represents a significant
challenge. Additionally, being a fourth row element, the abundance of arsenic
is 1000 times lower than phosphorus.

Essentially all phosphorus and sulfur are produced during oxygen burning in
massive stars (Woosley et al., 2002), the resultant isotopes being 31P and
32S. Since sulfur is an alpha element, it is a factor of 53 times more
abundant than phosphorus in our universe.

However, this situation will be reversed if sulfur-32 is unstable. It will
decay to phosphorus-32 if the energy difference from Equation (13) is
negative:

$\displaystyle\Delta
E\left({}^{32}\text{S}\right)=-0.005\,m_{p}+1.16\,\alpha\,m_{p}-m_{e}-m_{d}+m_{u}$

This is actually the closest nuclear stability threshold (of the cases we
consider here). This threshold is crossed if $\alpha$ is increased by $0.7\%$,
at which point the phosphorus abundance is increased by a factor of 51.5, and
the sulfur abundance is decreased by a factor of 0.049.

This region of enhanced phosphorus and decreased sulfur does not go on
indefinitely, however. As can be seen from Figure 2, after a certain point
chlorine-35 becomes unstable and will decay into sulfur, and beyond that
phosphorus-32 will decay to silicon. These thresholds are given by

$\displaystyle\Delta
E\left({}^{35}\text{Cl}\right)=0.00565\,m_{p}+1.197\,\alpha\,m_{p}-m_{e}-m_{d}+m_{u}$

and

$\displaystyle\Delta
E\left({}^{32}\text{P}\right)=-0.0070\,m_{p}+1.096\,\alpha\,m_{p}-m_{e}-m_{d}+m_{u}$
(16)

The first of these thresholds is crossed at a $5.0\%$ increase of $\alpha$ and
results in a universe with 0.25 the amount of sulfur as ours, and 0.01 the
amount of chlorine. The second is crossed with an increase in $\alpha$ of
$14.6\%$, and results in a universe with 0.40 times the amount of phosphorus
as ours.

There are a number of different stances one may take on the habitability of
these various regions. If we take the habitability of a location to be
proportional to the amount of phosphorus it contains, then the neighboring
region of parameter space would be 50 times more habitable than our universe.
With this ansatz, the probabilities of observing our constants become

$\displaystyle\mathbb{P}(\alpha_{obs})=0.076,\,\,\mathbb{P}(\beta_{obs})=0.482,\,\,\mathbb{P}(\gamma_{obs})=0.338,\,\,\mathbb{P}(\delta_{u\phantom{!}obs})=0.299,\,\,\mathbb{P}(\delta_{d\phantom{!}obs})=0.088$
(17)

These values are disfavored compared to the baseline by a Bayes factor of 6.8,
suggesting that habitability should not be proportional to phosphorus content.
This expectation is not unreasonable, given that on Earth phosphorus, while
being a limiting nutrient, is recycled on average some 500 times before
exiting an ecosystem (Ward et al., 2018). In principle this could be even
higher on planets where phosphorus is even more limited. This could be checked
by searching for signs of life around stars that have anomalously low
phosphorus content, as described for instance in (Maas et al., 2017).

Alternatively, one could worry that universes with decreased sulfur abundance
would be detrimental to life. In this case, the probabilities become

$\displaystyle\mathbb{P}(\alpha_{obs})=0.436,\,\,\mathbb{P}(\beta_{obs})=0.285,\,\,\mathbb{P}(\gamma_{obs})=0.250,\,\,\mathbb{P}(\delta_{u\phantom{!}obs})=0.163,\,\,\mathbb{P}(\delta_{d\phantom{!}obs})=0.494$
(18)

These are all within $4\%$ of the baseline values, making the probability of
our observations almost independent of whether a dearth of sulfur is
detrimental to life or not. As such, the multiverse does not provide any
strong predictions for this habitability criterion.

###  4.3 Is Iron Essential for Life?

Differing values of the constants will lead to different end products of the
iron peak. While this does not lead to any strong predictions, we include it
here for completeness, and because it may be of some passing interest to the
reader.

Stellar fusion will always stop at nickel-56; this isotope is ‘doubly magic’,
leading to an enhanced binding energy. In our universe, this nickel
subsequently undergoes two beta decays to ultimately yield iron. The
energetically preferred isotope varies with the constants, however. This
dependence was initially explored in (Sandora, 2016), though attention was
restricted to $\alpha$, and no attempt at inferring predictions for
habitability was made. Generically, the endpoint element $Z$ can be computed
from Equation (13) to be

$\displaystyle Z_{56}=\text{floor}\big{[}\big{(}\\{\begin{subarray}{c}26.54\\\
25.94\end{subarray}\\}-0.23\left(\hat{\alpha}-1\right)+0.14\left(\hat{\beta}-1\right)$
$\displaystyle+1.27\left(\hat{\delta}_{d}-1\right)-0.59\left(\hat{\delta}_{u}-1\right)\big{)}/\left(1+0.10\left(\hat{\alpha}-1\right)\right)\big{]}$
(19)

where the top coefficient is relevant for even $Z$, and the bottom is relevant
for odd $Z$. We use the notation $\hat{x}=x/x_{obs}$. Figure 4 displays the
abundances of the fourth row elements, which are primarily dictated by the
location of this peak. One can see that for different values of the constants,
essentially all the iron on Earth would be replaced by another element. For
nearby values these would be another transition metal, and so may not produce
such a drastic change. We will leave it to the reader to imagine how different
the world would be if iron were replaced with potassium, or beyond that the
noble gas argon. The general trend is that the higher $\alpha$ is, the smaller
the elemental constituents of the universe are. For $\alpha$ beyond 3 times
its observed value, essentially all fourth row elements will be absent.

Figure 4: The abundances of the fourth row elements, restricted to the
maximally sensitive direction in parameter space described in the text, and
parameterized by $\alpha$. The black line denotes our observed values. The
elements are ordered from the upper right downward.

The closest thresholds are as follows: if $\alpha$ is increased by $11.4\%$
(again, restricting to the line in parameter space which is maximally
sensitive to the change in constants), the iron abundance is decreased by a
factor of 0.067, replaced instead by manganese. An increase of $19.1\%$ would
result in chromium, which would be enhanced by a factor of 68. A decrease in
$\alpha$ by $23.3\%$ would lead to a cobalt universe, and a decrease by
$29.8\%$ would make nickel stable.

The most conservative hypothesis would be to take universes for which iron is
not the end product of this decay chain as uninhabitable. This stance is
bolstered by the recent finding that iron may have been an essential component
for many of the most essential reactions in biochemistry (Muchowska et al.,
2019). Taking iron to be essential, the values of the probabilities become

$\displaystyle\mathbb{P}(\alpha_{obs})=0.303,\,\,\mathbb{P}(\beta_{obs})=0.312,\,\,\mathbb{P}(\gamma_{obs})=0.289,\,\,\mathbb{P}(\delta_{u\phantom{!}obs})=0.318,\,\,\mathbb{P}(\delta_{d\phantom{!}obs})=0.247$
(20)

These are not substantially different from the probabilities that are
completely agnostic to the exact metal present. Similar probabilities result
for any metal-specific habitability criterion one may adopt (any metal above
chromium, for instance). Therefore, no predictions regarding the habitability
of different metals can be made.

As a final note, let us comment on the absence of other key elements in our
discussion, namely sodium and chlorine. The sodium abundance is relatively
stable across parameter space compared to other elements, and the closest
thresholds actually increase its abundance rather than decrease it.
Chlorine-36 does become unstable if $\alpha$ changes by $+9.1\%$ ($-12.6\%$),
resulting in a decrease in abundance by a factor of 72 (23), but these
thresholds are far enough away that they do not induce a significant change in
any of the probabilities.

##  5 Discussion

In a multiverse setting, many of the properties of our universe are not
unique, but instead could conceivably have been different. We expect some
range of conditions to be habitable, and that our experiences are not too
atypical. This leads us to the conclusion that we should probably inhabit a
universe that makes a relatively large number of observers. Since the number
of observers that can exist within a universe is highly dependent on the
assumptions we make about the requirements for complex life, only some of
these habitability assumptions will be compatible with these generic
multiverse expectations.

It is tempting to use this reasoning to explain every unique facet of our
universe. Since our universe has unusually high abundances of many of the
elements which are principle components of living systems, for instance, we
might like to conclude that universes with different elemental palettes are
sterile. Such reasoning is preemptive, however, and dangerously circular; are
we made out of CHNOPS because these are the only elements capable of
comprising biochemical compounds, or simply because they are the most
abundant? Surely not every feature of our universe would be tuned to be
maximally conducive to life, and typical observers ought to experience several
facets of their surroundings to be suboptimal. Especially since we have found
that very abrupt changes to the cosmic abundances are common, a more thorough
investigation of the impact of each habitability assumption is necessary
before any conclusions can be drawn.

Throughout, we investigated three sources of abundance variations: the
relative outputs of the two primary supernova sources, the final yield
dependence on nuclear resonances, and the changes to the stable isotopes. All
three effects were found to produce changes that can profoundly alter the
basic structure of the macroscopic world, sometimes when our physical
constants are varied by only a fraction of a percent. All this begs the
question as to whether we really expect our minute region of parameter space
to contain the only recipe capable of producing life, or whether universes are
just starkly diverse, and each of them contain observers marveling at how
perfectly theirs is suited to their own brand of life.

The answer we have found lies somewhere in between these extremes. Some
thresholds truly should be significant, or else our presence in this universe
would be highly atypical. However, the majority of thresholds we consider,
when treated as significant, were either shown to have very little effect on
the probability of our observations, or else decrement them. This conclusion
was only reached by considering a wide range of potential features: the metal-
to-rock ratio (Table 2), the carbon-to-oxygen ratio, the magnesium-to-silicon
ratio, the organic-to-rock ratio (all Table 3), the nitrogen abundance
(Equation (15)), the phosphorus abundance (Equation (17)), the sulfur
abundance (Equation (18)), and the iron abundance (Equation (20)). Of these
eight potentially important habitability criteria, only the carbon-to-oxygen
ratio was determined to be important to very high significance. Even though
the others are affected by nearby thresholds within parameter space, ascribing
special significance to these was found to be erroneous.

As was the explicit intent of this exercise, these findings have implications
for our expectations for which regions in our universe we expect to be
habitable: planets with more metals should be habitable. Carbon poor (or rich)
planets should not. Highly magnesium rich planets may or may not be habitable,
but mildly magnesium rich planets should be habitable. Nitrogen and phosphorus
poor planets should be habitable. If any of these predictions turns out to be
wrong, then the majority of observers would have arisen in universes different
from ours, and the multiverse would be ruled out to potentially very high
statistical significance, depending on the case.

We should stress at this point that all these conclusions take the base model
that habitability is proportional to the entropy processed by a planet, and
that oxygenic photosynthesis is necessary for (or at the very least greatly
facilitates) complex life. These are not the only base assumptions which are
compatible with our observations, and we chose them for sake of
narrativization only. This highlights a major shortcoming in our analysis thus
far; since we have chosen a habitability criteria that ‘works’ as our
baseline, the addition of any other criteria is bound to be either
insignificant or detrimental. There may equally well be habitability
hypotheses from our initial list, dismissed at first because they are
untenable on their own, which become viable when combined with one of these
additional considerations. Additionally, we did not consider the 256
possibilities that result from including multiple abundance assumptions
simultaneously. A fuller exploration is needed, and we remedy these oversights
here.

When taken in conjunction with the criteria we considered in previous papers,
it is becoming impracticable to compute all possible combinations. An
exhaustive computation would thus far include 509,607,936 unique combinations,
from 22 different habitability considerations, corresponding to over 100 CPU-
years. This number grows exponentially with the number of factors; since we
will include even more habitability criteria in the future, clearly some way
of mapping this space of possibilities without going through every combination
is needed. To do this, we neglect the habitability criteria that had no
bearing on previous calculations (such as mass extinction rate and hot Jupiter
rate), and furthermore we restrict our consideration to the presence of three
or less criteria at a time. This gives fairly good coverage of effects that
may occur due to the interaction of different boundaries, but admittedly does
not fully sample the space of possibilities. However, this restricted
computation scales cubically with the number of habitability criteria, and the
number of combinations remains under 10,000 until 40 criteria are considered.

As a base, we consider the following habitability hypotheses for conditions
required for complex life: (i): photosynthesis (optimistic and pessimistic
bounds, denoted photo/yellow), (ii): tidally unlocked planets (TL), (iii):
stellar lifetimes of several billion years (bio), (iv): plate tectonics
(plates), (v): terrestrial mass planets (terr), (vi): temperate planets
(temp), and that the probability of the emergence of life is proportional to:
(vii): stellar lifetime (time), (viii): planet area (area), and (ix): entropy
(entropy). To these, we add the additional criteria from this paper: that
complex life requires: (x): a certain metal-to-rock ratio (metal), (xi): a
certain C/O or Mg/Si ratio (C/O, Mg/Si), (xii): nitrogen, phosphorus, or
sulfur (N,P,S), and (xiii): iron (Fe). These 13 conditions result in 756
combinations to check.

For each of these, we compute the probabilities of the five physical constants
we have been considering, $\alpha$, $\beta$, $\gamma$, $\delta_{u}$, and
$\delta_{d}$, as well as the probability of orbiting a star as massive as our
sun, and of the observed organic-to-rock ratio. To give the reader a sense of
which combinations are viable, we report all those for which all probabilities
are greater than 0.01:

C/O + entropy + ( photo or yellow or TL)

C/O + area $\pm$ (photo or yellow or TL or bio or plates or terr or metal or N
or P or S or Fe)

Mg/Si + area + N

The full list, with numerical values, is included in the code repository
mentioned in the methods section below Section 6.

Several important lessons should be taken from this: in addition to the
entropy + yellow branch we have been focusing on (which gives the overall
highest probabilities), there is an additional branch centered around the area
condition. In all but one viable criteria, a specific C/O ratio is required.
In contrast to the cases when considered in isolation, the nitrogen,
phosphorus, metal-to-rock ratio, and plate tectonics criteria are compatible
habitability criteria when the area condition is important. This last point
partially holds on the C/O + entropy branch as well, though these were not
covered by this analysis which terminated at combinations of three criteria;
the addition of the nitrogen and phosphorus criteria become compatible with
our observations, but the plate tectonics and metal-to-rock ratio criteria
remain disfavored.

A histogram of the smallest probability for the 756 combinations considered is
displayed in Figure 5. Of the combinations considered, only in $2\%$ do all
values exceed 0.01, only $6\%$ exceed 0.001, and $32\%$ exceed 0.0001.

Figure 5: The smallest probability of the combinations of the 756 habitability
criteria considered. All combinations of 3 or less criteria discussed in this
paper and the previous four in this series are included. The probabilities
considered are of observing the five physical constants being varied, as well
as the probability of orbiting a star of solar mass and witnessing our
organic-to-rock ratio.

This should be compared to the expected smallest value of seven random
numbers, which has a cumulative distribution function of $1-(1-p)^{7}$. Thus,
the fact that there are seven chances for one number to be unnaturally small
decreases the significance of any value by that same factor, but this has
little bearing on the majority of cases.

The majority of potential habitability criteria are incompatible with the
multiverse, and so, if found to be true, would rule out the multiverse to a
high statistical significance. The multiverse predicts relatively specific
conditions that should be important to life, and equally predicts the
unimportance of others. This goes a long way toward establishing the
multiverse as a predictive, testable scientific framework. To the list of
about a dozen predictions that have been made in previous parts of this
series, we now add a few more, as well as recommendations for how to test each
with upcoming (or more distant) future missions. Further work will yield even
more predictions, maximizing both the chances and certainty of testing this
framework.

##  6 Methods

In the previous papers of this series, we focused our attention on only three
physical constants: $\alpha$, $\beta$, and $\gamma$, and held the others
fixed. The chemical abundances, however, depend sensitively on the light quark
masses, which necessitates the inclusion of these quantities in this paper. We
briefly describe the changes this required to our numerical computations, and
the updates to our previous conclusions this brings about. The full python
code is available at https://github.com/mccsandora/Multiverse-Habitability-
Handler accessed on 02 December 2022.

Our code is built to numerically compute the probabilities quoted throughout
the paper, of observing our universe’s values of the 5 physical constants we
vary, the probability of being around a sunlike star, the probability of
observing such a small Hoyle resonance energy, and the probability of
observing such a high organic-to-rock ratio. These are computed using the
Monte Carlo integration technique, which generates random points throughout
the range and equates the desired integral with the expectation value of the
integrand. To generate the pseudo-random points, we use the Sobol sequence
method, which avoids the clustering inherent in truly random processes,
increasing accuracy (Sobol, 1967). The sample size is set to achieve our
target accuracy of $1\%$, which is determined by randomly splitting the
generated sequence and comparing the values of the two resultant
probabilities.

While Monte Carlo methods are well suited for high dimensional integrals, the
inclusion of the anthropic boundaries reduces the number of points utilized
from a hypercube covering the relevant parameter space to $2\%$. Including the
plate tectonics and/or C/O boundaries reduces this by a further 2 orders of
magnitude each, making a brute force Monte Carlo approach untenable. For each
of the four cases, we rescaled the initially generated points to the relevant
parallelotopes, enhancing efficiency and accuracy. The combination of the
plate tectonics and C/O boundaries is particularly economical with this
rescaling approach, using $30\%$ of the generated points, because the
conjunction of the two boundary regions restricts the constants to lie within
a narrow range of parameter space.

The anthropic boundaries we include are those found in (Barr and Khan, 2007).
Stated briefly, they are as follows: (i) the proton and neutron should be the
most stable nuclei, rather than the $\Delta^{++}$ or $\Delta^{-}$, (ii) heavy
elements are stable, (iii) the proton is stable in nuclei, (iv) hydrogen is
stable, both to positron emission and electron capture, (v) proton-proton
fusion is exothermic, (vi) the deuteron is stable, both to strong and weak
decays, and (vii) the diproton is unstable. Each of these boundaries have been
contested (Hogan, 2006; Jaffe et al., 2009; MacDonald and Mullan, 2009;
Bradford, 2009; Barnes, 2015; Adams and Grohs, 2017; Barnes and Lewis, 2017),
some by the present authors, but the exclusion of most individual bounds does
not have a large effect on the probabilities we compute (this holds especially
true for lower bounds). Because the intent of this paper is to generate
potential tests of the multiverse based off probing regions of our universe
that closely resemble normal environments within other universes, and because
there are no regions of our universe that resemble a universe where any of
these bounds do not hold, we do not gain anything by relaxing any of these
bounds.

Conceptualization, all authors; Methodology, M.S.; Formal Analysis, M.S.;
Validation, V.A., L.B., G.F.L. and I.P.-R.; Writing—Original Draft
Preparation, M.S.; Writing—Review and Editing, V.A., L.B., G.F.L. and I.P.-R.
All authors have read and agreed to the published version of the manuscript.
This research received no external funding.

All code to generate data and analysis is located at
https://github.com/mccsandora/Multiverse-Habitability-Handler accessed on 02
December 2022. The authors declare no conflict on interest.

yes

##  Appendix A Binary Evolution

Our calculation of the type Ia supernova rate required estimating
$f_{\text{binary}}(a_{\text{Roche}})$, the fraction of binary stars whose
separation is close enough for significant mass transfer. This can be
estimated by considering the distribution of initial separations, the critical
separation, here taken to be the Roche radius $a_{\text{Roche}}$, and a model
for orbital evolution of the system, which accounts for effects that can
significantly tighten the pair’s orbit over the course of its evolution.
Ultimately, the dependence of this quantity on physical constants will be
shown to be very weak, but we include the calculation here for completeness.

The initial separations are found to be given by a log-uniform distribution,
$f_{\text{binary-0}}(a)=1/a/\log(a_{\text{max}}/a_{\text{min}})$, from (Poveda
et al., 2006). Here, the maximum separation is given by the typical
intracluster separation, $a_{\text{max}}\sim n_{\text{cluster}}^{-1/3}$, and
the minimum separation is given by the separation at which the system becomes
unstable to merging. This latter scale is given by the Roche limit, which
scales as the stellar radius $a_{\text{min}}\sim R_{\star}$ (Podsiadlowski,
2014). A milder version of this criterion sets the separation needed for
significant mass transfer in terms of the stellar radius as well, but
multiplied by a factor of a few.

There are four sources of orbital decay that could potentially play an
important role in determining the fraction of binary systems that become
supernovae within a given amount of time. We will detail each of these in
turn. The first comes from gas drag within the initial cluster (Korntreff et
al., 2012). This leads to an evolution given by

$\dot{a}=-\,\frac{\xi_{\text{d}}\,G^{3}\,M_{\star}^{2}\,\rho}{c_{s}^{5}},\quad
a(t_{\text{cluster}})=a_{i}-\frac{\xi_{\text{d}}\,G^{3}\,M_{\star}^{2}\,\rho}{c_{s}^{5}}\,t_{\text{cluster}}$
(21)

Here $q$ is the mass ratio of the system, $\rho$ is the cluster density, and
$c_{s}$ is the gas speed of sound. Throughout this section, the $\xi$s
represent $\mathcal{O}(1)$ quantities, many of which must be fit to
simulations. As can be seen, this leads to a constant shift inward,
independent of initial separation. Using the cluster lifetime
$t_{\text{cluster}}\sim 1/\sqrt{G\rho}$ and the virial speed
$c_{s}\sim\sqrt{G\rho^{1/3}M_{\text{cluster}}^{2/3}}$, we find this shift to
be parametrically of the order $\Delta a\sim
a_{\text{max}}/N_{\text{cluster}}^{5/3}$, where $N_{\text{cluster}}$ is the
number of stars in the cluster. This is always much smaller than the maximal
separation, but significantly sculpts short period systems, causing the
distribution to resemble a log-normal (Korntreff et al., 2012).

Another source of intracluster evolution comes from close encounters with
other stellar systems. For systems whose binding energy exceeds the typical
cluster energy, an encounter usually results in the system becoming tighter
(Heggie, 1975). The hardening rate for this process is given by

$\dot{a}=-\frac{\xi_{\text{e}}\,G\,\rho}{c_{s}}\,a^{2},\quad
a(t_{\text{cluster}})=\left(a_{i}^{-1}+\frac{\xi_{\text{e}}\,G\,\rho}{c_{s}}\,t_{\text{cluster}}\right)^{-1}$
(22)

This contribution to orbital evolution becomes less effective for small
separations, so, while it can lead to a skewing of the distribution, it is
usually not directly responsible for coalescence.

A third component to orbital evolution comes from gravitational waves. This
evolution is given by (Carroll, 2019)

$\dot{a}=-\frac{\xi_{\text{g}}\,G^{3}\,M_{\star}^{3}}{4\,a^{3}},\quad
a(t)=\left(a_{i}^{4}-\xi_{\text{g}}\,G^{3}\,M_{\star}^{3}\,t\right)^{1/4}$
(23)

This effect is very weak throughout all of parameter space, however, and will
only affect those systems with very small separations.

Finally, the most relevant contribution to orbital evolution is given by
magnetic braking. For systems whose orbits are close enough for the stellar
rotation to become orbitally locked, the magnetic drag from the star’s
differential rotation, which ordinarily causes rotational spindown, can
backreact to cause orbital spindown as well (Verbunt and Zwaan, 1981). Taking
the rotational magnetic braking to be given by the usual Skumanich form,
angular momentum loss is related to angular velocity through
$\dot{J}=-\tau\,\Omega^{3}$. The resultant orbital evolution is given by

$\dot{a}=-\frac{\xi_{\text{s}}\,\tau\,G}{5\,a^{4}},\quad
a(t)=\left(a_{i}^{5}-\xi_{\text{s}}\,\tau\,G\,t\right)^{1/5}$ (24)

What remains is to determine the origin of the parameter $\tau$ in the
Skumanich formula: from (Lau et al., 2011; Talon and Zahn, 1997), we can
express the rotational spindown as $\dot{J}\sim
UR_{\star}^{2}\text{max}(R_{\star}^{2},R_{\text{Alfven}}^{2})\Omega^{3}/G$,
where $R_{\text{Alfven}}$ is the Alfven radius, and
$U\sim\sqrt{T_{\star}/m_{p}}$ is the convective speed. Specializing to the
regime where $R_{\star}>R_{\text{Alfven}}$, this gives $\tau\sim
UR_{\star}^{4}/G$.

A full time evolution equation can be obtained by summing the four individual
components, and taking into account that the first two are only operational
while the pair is within its birth cluster. This can then be used to infer the
lifetime of a system with given initial separation. The fraction of binary
stars within a critical separation $a_{\text{crit}}$ which will coalesce
before a time $t$ is found to be
$f_{\text{binary}}(a_{\text{crit}})=\log(a_{\text{crit}}/a_{\text{min}})/\log(a_{\text{max}}/a_{\text{min}})$.
Finally, the spectrum of system lifetimes can be integrated over in Equation
(3), yielding the type Ia supernova rate. However, this exercise is
unnecessary for an initial estimate of the supernova rate; because the
dependence is logarithmic, this fraction only varies by a few percent over
most of the interesting parameter range.

[custom]

References

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