1. Introduction
Mass Extinctions
2. Rates
2.1. Catastrophes
2.2. What Sets the Recovery Time?
3. Comets
3.1. Comet Dynamics
3.2. What Sets the Size of Deadly Comets?
3.2.1. Sulfate
3.2.2. Dust
4. Volcanism, Glaciations, and Sea Level Change
4.1. Glaciations
4.2. Volcanoes
5. Gamma Ray Bursts
5.1. Gamma Ray Bursts and Extinctions
5.2. GRB Rate
5.3. What Sets ${r}_{\u2020}$?
6. Discussion
6.1. Multiple Causes
6.2. Why Are We in This Universe?
7. Conclusions
Funding
Acknowledgments
Conflicts of Interest
References
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1.  
2.  This hierarchy does not actually hold for all parameters, but the full implications of this will be explored in future work. 
3.  The inner edge will exceed the outer for stars above the mass
$${\lambda}_{\mathrm{none}}=5.7\times {10}^{12}\frac{{\alpha}^{5/2}\phantom{\rule{0.166667em}{0ex}}{\beta}^{15/8}}{{\kappa}^{3/2}}$$

4.  This is the reason comets can be deadlier than asteroids, even though they are less dense: since they can come from any direction rather than being roughly coorbital with the Earth, the average speed will be $\sqrt{3}{v}_{\oplus}=52\phantom{\rule{4.pt}{0ex}}\mathrm{km}/\mathrm{s}$, rather than $(\sqrt{3}2\sqrt{2}){v}_{\oplus}=12.4\phantom{\rule{4.pt}{0ex}}\mathrm{km}/\mathrm{s}$, a factor of 17.5 more energy. Therefore, comets can be smaller and still impart more energy than asteroids, and so their rate of deadly impacts will be more numerous (which depends on their relative population sizes as well, of course). 
5.  The code to compute all probabilities discussed in the text is made available at https://github.com/mccsandora/MultiverseHabitabilityHandler. 
6.  This Gaussian approximation can be improved upon by treating the sum as an integral and using the saddle point approximation: this yields ${f}_{\mathrm{rad}}=2.7e^\left(1+\ell (1x){e}^{l(1x)}\right)$, with $x=1/\sqrt{144\alpha}$ and $\ell =52.72+log\left(\widehat{{t}_{\mathrm{rec}}/{t}_{\mathrm{decay}}}\right)$, where the hatted quantity inside the log evaluates to 1 for our values. Using this more accurate expression does not affect our results at all. 
Setback Model  

${\mathit{t}}_{\mathbf{rec}}$  Molecular  Year  
$\mathbb{P}\left({\alpha}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\beta}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\gamma}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\alpha}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\beta}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\gamma}_{\mathbf{obs}}\right)$  
comets  0.125  0.0809  0.287  0.169  0.217  0.117 
grbs  0.323  0.233  0.042  0.241  0.277  0.019 
volcanoes  0.193  0.426  0.303  0.192  0.425  0.293 
glaciations  0.193  0.423  0.304  0.193  0.426  0.305 
comets+grbs  0.208  0.0761  0.0757  0.207  0.242  0.0284 
comets+glaciations  0.123  0.116  0.267  0.169  0.25  0.154 
comets+volcanoes  0.124  0.115  0.265  0.169  0.25  0.148 
grbs+glaciations  0.299  0.253  0.0527  0.233  0.289  0.0309 
grbs+volcanoes  0.299  0.253  0.052  0.232  0.289  0.0301 
glaciations+volcanoes  0.193  0.423  0.303  0.193  0.426  0.299 
comets+grbs+glac  0.198  0.1  0.0905  0.207  0.255  0.037 
comets+grbs+vol  0.198  0.0998  0.0895  0.206  0.256  0.036 
comets+glac+vol  0.125  0.146  0.262  0.169  0.274  0.171 
grbs+glac+vol  0.283  0.262  0.0697  0.228  0.297  0.0387 
all  0.193  0.123  0.0995  0.205  0.266  0.043 
Reset Model  

${\mathit{t}}_{\mathbf{noo}}$  Molecular  Year  
$\mathbb{P}\left({\alpha}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\beta}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\gamma}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\alpha}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\beta}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\gamma}_{\mathbf{obs}}\right)$  
comets  0.105  0.0717  0.251  0.168  0.26  0.136 
grbs  0.294  0.243  0.0332  0.226  0.293  0.0194 
volcanoes  0.19  0.436  0.317  0.188  0.432  0.31 
glaciations  0.192  0.43  0.312  0.19  0.43  0.31 
comets+grbs  0.179  0.0804  0.0617  0.202  0.273  0.0351 
comets+glaciations  0.114  0.132  0.248  0.167  0.288  0.174 
comets+volcanoes  0.114  0.131  0.247  0.168  0.289  0.168 
grbs+glaciations  0.265  0.262  0.0506  0.22  0.308  0.038 
grbs+volcanoes  0.265  0.262  0.0497  0.22  0.309  0.0368 
glaciations+volcanoes  0.191  0.431  0.313  0.189  0.431  0.31 
comets+grbs+glac  0.175  0.119  0.07  0.201  0.288  0.0428 
comets+grbs+vol  0.175  0.119  0.0692  0.201  0.288  0.042 
comets+glac+vol  0.12  0.189  0.257  0.167  0.303  0.19 
grbs+glac+vol  0.25  0.271  0.0616  0.217  0.317  0.0467 
all  0.173  0.145  0.0874  0.2  0.297  0.0504 
Intermediate Disturbance Hypothesis Model  

${\mathit{t}}_{\mathbf{dist}}$  Molecular  Year  
$\mathbb{P}\left({\alpha}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\beta}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\gamma}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\alpha}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\beta}_{\mathbf{obs}}\right)$  $\mathbb{P}\left({\gamma}_{\mathbf{obs}}\right)$  
comets  0.106  0.119  0.251  0.151  0.462  0.4 
grbs  0.2  0.329  0.0984  0.129  0.384  0.0963 
volcanoes  0.128  0.356  0.368  0.156  0.404  0.262 
glaciations  0.131  0.323  0.337  0.148  0.466  0.455 
comets+grbs  0.165  0.135  0.0818  0.18  0.453  0.124 
comets+glaciations  0.113  0.205  0.256  0.144  0.498  0.454 
comets+volcanoes  0.113  0.202  0.261  0.147  0.496  0.489 
grbs+glaciations  0.185  0.353  0.113  0.142  0.414  0.146 
grbs+volcanoes  0.185  0.351  0.113  0.141  0.423  0.162 
glaciations+volcanoes  0.129  0.345  0.359  0.153  0.435  0.4 
comets+grbs+glac  0.16  0.18  0.132  0.175  0.46  0.158 
comets+grbs+vol  0.16  0.179  0.131  0.179  0.472  0.166 
comets+glac+vol  0.123  0.281  0.274  0.147  0.438  0.489 
grbs+glac+vol  0.167  0.349  0.178  0.138  0.443  0.186 
all  0.159  0.227  0.154  0.172  0.482  0.187 
Setback  Reset  IDH  

${\mathit{t}}_{\mathbf{rec}}$  mol  Year  mol  Year  mol  Year 
comets  140  71  132  55  70  12 
grbs  424  1061  422  813  96  95 
volcanoes  10  10  10  10  10  10 
glaciations  10  10  10  10  10  10 
comets+grbs  160  640  201  473  102  68 
comets+glaciations  74  40  69  32  39  10 
comets+volcanoes  75  45  70  36  40  13 
grbs+glaciations  216  537  215  412  46  58 
grbs+volcanoes  218  540  217  415  50  52 
glaciations+volcanoes  10  10  10  10  10  10 
comets+grbs+glac  114  430  138  319  69  52 
comets+grbs+vol  117  433  139  321  69  47 
comets+glac+vol  53  33  49  27  29  10 
grbs+glac+vol  148  364  148  280  33  41 
all  91  328  107  243  53  41 
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