# Gravitational Pulse Wave Scattering and the Geroch group

Robert F. Penna<EMAIL_ADDRESS>Institute for Advanced Study
Einstein Drive, Princeton, NJ 08540 USA

###### Abstract

Cylindrical gravitational pulse waves are cylindrical gravitational waves with
a pulse profile in the radial direction. The dynamics of cylindrical
gravitational pulse waves is governed by a two dimensional integrable sigma
model with an infinite dimensional hidden symmetry called the Geroch group.
Piran, Safier, and Katz obtained an exact solution for a single pulse wave by
making a double analytic continuation of the Kerr metric. Their solution
describes a pulse wave that arrives from infinity, bounces off the symmetry
axis, and returns to infinity. In this note, we obtain an exact solution for a
double pulse wave by making a double analytic continuation of the double Kerr
metric. The new solution describes a pair of pulse waves that arrive from
infinity, scatter through each other, and return to infinity. The pulses pass
through each other and emerge with their shapes intact, as in ordinary soliton
scattering. Unlike ordinary soliton scattering, there is no time delay. The
metric is free of the conical singularity that appears in the double Kerr
metric. In the future, it would be interesting to understand how the Geroch
group manifests in the quantum version of gravitational pulse wave scattering.

##  I Introduction

The dimensional reduction of general relativity to two spacetime dimensions is
an integrable sigma model with an infinite dimensional hidden symmetry called
the Geroch group Geroch (1971, 1972); Breitenlohner and Maison (1987); Nicolai
(1991). Four dimensional vacuum metrics with two commuting Killing vectors can
be realized as solutions of this two dimensional sigma model.

One class of solutions that can be realized this way are stationary
axisymmetric black holes. In this case, the integrable sigma model lives on a
two dimensional Euclidean half-plane with coordinates $\rho\sim r\sin\theta>0$
and $z\sim r\cos\theta$. The Kerr black hole corresponds to a single soliton
of the sigma model. The double Kerr black hole corresponds to a double soliton
Kramer and Neugebauer (1980); Tomimatsu and Sato (1981); Dietz and Hoenselaers
(1985).

Another class of solutions that can be realized this way are cylindrical
gravitational waves. The sigma model lives on a two dimensional Lorentzian
half plane with coordinates $\rho>0$ and time, $t$. Long ago, Piran, Safier,
and Katz Piran  _et al._ (1986) made the remarkable observation that an exact
solution for a cylindrical pulse wave can be obtained by a double analytic
continuation of the Kerr metric that interchanges the $t$ and $z$ directions.
The new solution describes a cylindrical gravitational wave with a pulse
profile in the radial direction. The pulse arrives from infinity, bounces off
$\rho=0$, and returns to infinity. The pulse maintains its shape as it travels
111Actually, the shape of the pulse steepens as it approaches $\rho=0$ because
it travels on a half-line., as expected for a soliton.

In this note, we obtain a double pulse wave solution by making a double
analytic continuation of the double Kerr metric. The new metric describes a
pair of cylindrical pulse waves that arrive from infinity, scatter, and return
to infinity. The pulses pass through each other and emerge with their shapes
intact, so their soliton nature is confirmed. Unlike ordinary solitons, they
suffer no time delay after scattering. The conical singularity of the double
Kerr metric is not present in the new metric. So the double pulse wave is, in
some sense, a more physical solution. One can contemplate the generalization
to $N$ pulse waves but we will not do so here.

Earlier works Carr and Verdaguer (1983); Ibanez and Verdaguer (1983); Ibaez
and Verdaguer (1985); Boyd  _et al._ (1991) have studied gravitational soliton
scattering in cosmological spacetimes. In those solutions, the solitons emerge
from a “big bang” singularity at $t=0$. An interesting feature of the solution
described in the present work is that the pulses emerge from (and return to)
asymptotic infinity. Related works Piran  _et al._ (1985); Tomimatsu (1989);
Tomizawa and Mishima (2015) on gravitational soliton scattering have studied
the gravitational analogue of Faraday rotation.

An interesting challenge for the future is to understand the quantum version
of gravitational pulse wave scattering. The Poisson brackets of Geroch group
charges generate a Yangian symmetry Korotkin and Samtleben (1998). In other
quantum integrable sigma models, Yangian symmetry manifests through the
absence of particle production and the factorization of the $S$ matrix
Zamolodchikov and Zamolodchikov (1978, 1990); Lüscher (1978). It would be
interesting to understand how the Yangian symmetry of dimensionally reduced
general relativity manifests in the quantum version of gravitational pulse
wave scattering.

General relativity with a negative cosmological constant admits cylindrical
black hole solutions Lemos (1995). So it might be possible to find integrable
models of cylindrical pulse waves interacting with a cylindrical black hole.
Unfortunately, it is not known how to include a cosmological constant in the
theory of the Geroch group (but see Leigh  _et al._ (2014); Petkou  _et al._
(2015) for work in that direction). Perhaps the recently introduced twistor
Chern-Simons theory Woodhouse and Mason (1988); Costello ; Bittleston and
Skinner (2020); Penna (2021) could offer a useful new perspective on the
problem of including a cosmological constant in the theory of the Geroch
group.

It would be worthwhile to perform an asymptotic symmetry analysis of the pulse
wave metrics described herein along the lines of Ashtekar  _et al._ (1997a,
b); Barnich and Compere (2007). It would also be interesting to study the
effect of small deviations from cylindrical symmetry on the dynamics. In that
case the Geroch symmetry will be weakly broken, but even weakly broken Geroch
symmetry might be useful for organizing the dynamics, in the spirit of
effective field theory.

##  II Result

The double Kerr metric has the form Kramer and Neugebauer (1980); Tomimatsu
and Sato (1981); Dietz and Hoenselaers (1985)

$ds^{2}=-f(d{\hat{t}}-\Omega
d\varphi)^{2}+f^{-1}\left(e^{2\gamma}(d\rho^{2}+d{\hat{z}}^{2})+\rho^{2}d\varphi^{2}\right).$
(1)

There are three functions in the metric: $f$, $\Omega$, and $\gamma$. They are
functions of $\rho$ and ${\hat{z}}$ only.

$\rho$ is a radial coordinate ($\rho>0$) and ${\hat{z}}$ is the vertical
coordinate ($-\infty<{\hat{z}}<\infty$). The metric has two commuting Killing
vectors, $\partial_{\hat{t}}$ and $\partial_{\varphi}$. ${\hat{t}}$ is the
time coordinate.

It is convenient to replace $\Omega$ with a new function, $\psi$, defined by

$\partial_{\rho}\psi=\frac{f^{2}}{\rho}\partial_{\hat{z}}\Omega\,,\quad\partial_{\hat{z}}\psi=-\frac{f^{2}}{\rho}\partial_{\rho}\Omega\,.$
(2)

The vacuum Einstein equations give

$\displaystyle\partial_{\rho}\gamma$
$\displaystyle=\frac{\rho}{4f^{2}}\left[(\partial_{\rho}f)^{2}-(\partial_{\hat{z}}f)^{2}+(\partial_{\rho}\psi)^{2}-(\partial_{\hat{z}}\psi)^{2}\right],$
(3) $\displaystyle\partial_{\hat{z}}\gamma$
$\displaystyle=\frac{\rho}{2f^{2}}\left[(\partial_{\rho}f)(\partial_{\hat{z}}f)+(\partial_{\rho}\psi)(\partial_{\hat{z}}\psi)\right].$
(4)

We can solve these equations for $\gamma$ once we know $f$ and $\psi$.
Actually, these equations only fix $\gamma$ up to a constant. Fix this
constant by setting $\gamma=0$ at infinity.

Combine $f$ and $\psi$ into a single complex scalar field, the Ernest
potential,

$\epsilon=f+i\psi\,.$ (5)

The Ernest potential for the double Kerr metric is

$\epsilon=\frac{1-\xi}{1+\xi}\,,$ (6)

where

$\xi=\frac{\begin{vmatrix}1&1&1&1\\\ s_{1}&s_{2}&s_{3}&s_{4}\\\
k_{1}&k_{2}&k_{3}&k_{4}\\\
k_{1}^{2}&k_{2}^{2}&k_{3}^{2}&k_{4}^{2}\end{vmatrix}}{\begin{vmatrix}1&1&1&1\\\
s_{1}&s_{2}&s_{3}&s_{4}\\\ k_{1}&k_{2}&k_{3}&k_{4}\\\
k_{1}s_{1}&k_{2}s_{2}&k_{3}s_{3}&k_{4}s_{4}\end{vmatrix}}\,.$ (7)

We recover $f$ and $\psi$ as the real and imaginary parts of the Ernst
potential.

Figure 1:  Gravitational pulse wave scattering with $m_{1}=4$, $m_{2}=2$,
$q_{1}=q_{2}=2$ and $t_{1}=-t_{2}=50$. The function plotted is
$\partial_{\rho}\gamma$, which is related to the energy Thorne (1965). The
$\varphi$ and $z$ directions are not shown.

The numerator and denominator of $\xi$ are determinants of $4\times 4$
matrices. The $k_{m}$ ($m=1,\dots,4$) in equation (7) are four real constants.
The spacetime dependence of the Ernest potential enters through

$s_{m}=r_{m}e^{i\omega_{m}}\,,$ (8)

where the $\omega_{m}$ ($m=1,\dots,4$) are an additional four real constants
and

$r_{m}=\left(\rho^{2}+({\hat{z}}-k_{m})^{2}\right)^{1/2}\,.$ (9)

So the metric depends on $\rho$ and ${\hat{z}}$ only through $r_{m}$. The
metric has a total of eight real parameters, $k_{m}$ and $\omega_{m}$
($m=1,\dots,4$).

To get a more familiar set of parameters, let

$\displaystyle k_{1}$ $\displaystyle={\hat{z}}_{1}+m_{1}p_{1}\,,$ (10)
$\displaystyle k_{2}$ $\displaystyle={\hat{z}}_{1}-m_{1}p_{1}\,,$ (11)
$\displaystyle k_{3}$ $\displaystyle={\hat{z}}_{2}+m_{2}p_{2}\,,$ (12)
$\displaystyle k_{4}$ $\displaystyle={\hat{z}}_{2}-m_{2}p_{2}\,,$ (13)

and

$\displaystyle\omega_{1}$ $\displaystyle=\alpha_{1}+\lambda_{1}\,,$ (14)
$\displaystyle\omega_{2}$ $\displaystyle=\alpha_{1}-\lambda_{1}+\pi\,,$ (15)
$\displaystyle\omega_{3}$ $\displaystyle=\alpha_{2}+\lambda_{2}\,,$ (16)
$\displaystyle\omega_{4}$ $\displaystyle=\alpha_{2}-\lambda_{2}+\pi\,,$ (17)

where

$\displaystyle\lambda_{i}$
$\displaystyle=\frac{1}{i}\log(p_{i}+i{\hat{q}}_{i})\,,$ (18) $\displaystyle
p_{i}$ $\displaystyle=\sqrt{1-{\hat{q}}_{i}^{2}}\,.$ (19)

The new set of eight parameters is $m_{i}$, ${\hat{q}}_{i}$, $\alpha_{i}$, and
${\hat{z}}_{i}$ ($i=1,2$). When the black holes are widely separated, $m_{i}$,
${\hat{q}}_{i}$, $\alpha_{i}$ and ${\hat{z}}_{i}$ are, respectively, the
masses, dimensionless spin parameters, NUT parameters, and vertical
displacements of the holes. We set the NUT parameters to zero from now on
($\alpha_{1}=\alpha_{2}=0$).

To get the double pulse wave metric, we make the double analytic continuation

${\hat{t}}=iz\,,\quad{\hat{z}}=it\,,$ (20)

where $t$ and $z$ are real coordinates. To get a real metric, we need to
choose

${\hat{q}}_{i}=iq_{i}\,,\quad{\hat{z}}_{i}=it_{i}\quad(i=1,2),$ (21)

where $q_{i}$ and $t_{i}$ are real parameters.

The new metric describes a pair of pulse waves that arrive from infinity,
scatter through each other, and return to infinity. Figure 1 depicts a
solution with $m_{1}=4$, $m_{2}=2$, $q_{1}=q_{2}=2$, and $t_{1}=-t_{2}=50$.
The most interesting feature is the interaction region where the pulse waves
momentarily merge. The pulses emerge from the interaction region with their
original shapes intact. This is a characteristic feature of soliton
scattering.

It is often the case in soliton scattering that the solitons emerge from an
interaction with a time delay. However, there is no time delay in the present
example. This is illustrated in Figure 2. The second pulse reaches $\rho=0$ at
$t=50$, just as it would with no scattering. The absence of time delays for
cosmological solitons was observed by Boyd  _et al._ (1991).

Figure 2:  Same as Figure 1 but viewed from above. It is clear from this
figure that there is no time delay. The second pulse reaches $\rho=0$ at
$t=50$, just as it would if there were no interaction.

The black holes in the double Kerr metric are connected by a conical
singularity that runs along the axis at $\rho=0$. The conical singularity acts
as a strut and holds the black holes apart.

There is no such conical singularity in the double pulse wave metric. To see
why, consider equation (4) at $\rho=0$:

$(g_{{\hat{t}}{\hat{t}}})^{2}\partial_{\hat{z}}\gamma=0\,.$ (22)

It follows from this equation that $\gamma$ is a piecewise constant function
of ${\hat{z}}$ at $\rho=0$. In the double Kerr metric, there are jumps in
$\gamma$ at the “ergosphere,” where $(g_{{\hat{t}}{\hat{t}}})^{2}=0$. The
conical singularity appears at the jumps in $\gamma$.

In the case of the double pulse metric, we have instead

$(g_{zz})^{2}\partial_{t}\gamma=0\,.$ (23)

Now $(g_{zz})^{2}$ is strictly positive, so there are no jumps in $\gamma$ and
no conical singularity.

The single pulse wave solution of Piran, Safier, and Katz Piran  _et al._
(1986) requires periodic identifications in the $\varphi$ and $z$ directions
which are related to the thermodynamics of the Kerr black hole. We expect the
periodicity of the double pulse wave metric in the $\varphi$ and $z$
directions to be related to the rather more complicated thermodynamics of the
double Kerr metric Herdeiro  _et al._ (2010).

Acknowledgments

I am grateful to Ibrahima Bah for helpful comments. This material is based
upon work supported by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics under Award Number DE-SC0009988.

## References

  * Geroch (1971) R. Geroch, Journal of Mathematical Physics 12, 918 (1971).
  * Geroch (1972) R. Geroch, Journal of Mathematical Physics 13, 394 (1972).
  * Breitenlohner and Maison (1987) P. Breitenlohner and D. Maison, in  _Annales de l’IHP Physique Théorique_ , Vol. 46 (1987) pp. 215–246.
  * Nicolai (1991) H. Nicolai, in  _Recent Aspects of Quantum Fields_ (Springer, 1991) pp. 231–273.
  * Kramer and Neugebauer (1980) D. Kramer and G. Neugebauer, Physics Letters A 75, 259 (1980).
  * Tomimatsu and Sato (1981) A. Tomimatsu and H. Sato, Progress of Theoretical Physics Supplement 70, 215 (1981).
  * Dietz and Hoenselaers (1985) W. Dietz and C. Hoenselaers, Annals of Physics 165, 319 (1985).
  * Piran  _et al._ (1986) T. Piran, P. Safier, and J. Katz, Physical Review D 34, 331 (1986).
  * Note (1) Actually, the shape of the pulse steepens as it approaches $\rho=0$ because it travels on a half-line.
  * Carr and Verdaguer (1983) B. Carr and E. Verdaguer, Physical Review D 28, 2995 (1983).
  * Ibanez and Verdaguer (1983) J. Ibanez and E. Verdaguer, Physical review letters 51, 1313 (1983).
  * Ibaez and Verdaguer (1985) J. Ibaez and E. Verdaguer, Physical Review D 31, 251 (1985).
  * Boyd  _et al._ (1991) P. T. Boyd, J. M. Centrella, and S. A. Klasky, Physical Review D 43, 379 (1991).
  * Piran  _et al._ (1985) T. Piran, P. N. Safier, and R. F. Stark, Physical Review D 32, 3101 (1985).
  * Tomimatsu (1989) A. Tomimatsu, General Relativity and Gravitation 21, 613 (1989).
  * Tomizawa and Mishima (2015) S. Tomizawa and T. Mishima, Physical Review D 91, 124058 (2015).
  * Korotkin and Samtleben (1998) D. Korotkin and H. Samtleben, Nuclear Physics B 527, 657 (1998).
  * Zamolodchikov and Zamolodchikov (1978) A. B. Zamolodchikov and A. B. Zamolodchikov, Nuclear Physics B 133, 525 (1978).
  * Zamolodchikov and Zamolodchikov (1990) A. B. Zamolodchikov and A. B. Zamolodchikov, in  _Yang-Baxter Equation In Integrable Systems_ (World Scientific, 1990) pp. 82–120.
  * Lüscher (1978) M. Lüscher, Nuclear physics B 135, 1 (1978).
  * Lemos (1995) J. S. Lemos, Physics Letters B 353, 46 (1995).
  * Leigh  _et al._ (2014) R. G. Leigh, A. C. Petkou, P. M. Petropoulos, and P. K. Tripathy, Classical and Quantum Gravity 31, 225006 (2014).
  * Petkou  _et al._ (2015) A. C. Petkou, P. M. Petropoulos, and K. Siampos, PoS PLANCK2015, 104 (2015), arXiv:1512.04970 [hep-th] .
  * Woodhouse and Mason (1988) N. M. J. Woodhouse and L. J. Mason, Nonlinearity 1, 73 (1988).
  * (25) K. Costello, “Topological strings, twistors, and skyrmions,” Available at http://web.math.ucsb.edu/~drm/WHCGP/ (4/27/2020).
  * Bittleston and Skinner (2020) R. Bittleston and D. Skinner, (2020), arXiv:2011.04638 [hep-th] .
  * Penna (2021) R. F. Penna, Journal of High Energy Physics 2021, 1 (2021).
  * Ashtekar  _et al._ (1997a) A. Ashtekar, J. Bičák, and B. G. Schmidt, Physical Review D 55, 669 (1997a).
  * Ashtekar  _et al._ (1997b) A. Ashtekar, J. Bičák, and B. G. Schmidt, Physical Review D 55, 687 (1997b).
  * Barnich and Compere (2007) G. Barnich and G. Compere, Classical and Quantum Gravity 24, F15 (2007).
  * Thorne (1965) K. S. Thorne, Physical Review 138, B251 (1965).
  * Herdeiro  _et al._ (2010) C. Herdeiro, E. Radu, and C. Rebelo, Physical Review D 81, 104031 (2010).


# The Fine-Tuning of the Universe for Life

Luke A. Barnes Electronic address<EMAIL_ADDRESS>
School of Science
Western Sydney University
Penrith, NSW 2751, Australia

##  1 Introduction: Fine-Tuning in Physics

When a physicist says that a theory is _fine-tuned_ , they mean that it must
make a suspiciously precise assumption in order to explain a certain
observation. This is evidence that the theory is deficient or incomplete. As a
simple example, consider a geocentric model of the Solar System. Naively, at
any particular time, the Sun and planets could be anywhere in their orbits
around the Earth. However, in our night sky, Mercury is never observed to be
more than 28∘ from the Sun, and Venus is never seen more than 47∘ from the
Sun.

Can a geocentric model explain this observation? Yes, but only by adding a
postulate. In Ptolemy’s geocentric model, Mercury and Venus travel on
epicycles, and those epicycles are centred on a line joining the Earth to the
Sun (Figure 1). This explains the data, so the model does not fail. However,
in the context of the model, this assumption is unmotivated and suspiciously
precise. Given only that the planets and Sun orbit the Earth, there is no
reason to expect such an arrangement.

Figure 1: The Ptolemaic model of the Solar System. All the planets travel on
epicycles (smaller circles) in orbit around the Earth, but in the case of
Mercury and Venus it must be assumed that the centres of their epicycles are
fixed to a line that connects in the Earth and the Sun. In this way, Mercury
and Venus are never seen too far from the Sun. But in the context of the
model, this assumption is unmotivated and suspiciously precise. This presents
an opportunity for another model to explain this data more naturally: Mercury
and Venus are never seen too far from the Sun because the planets orbit the
Sun, not the Earth.

This fine-tuning of the geocentric model doesn’t necessarily mean that it is
wrong, but it should make us wary. We should search for a model in which the
data is explained more naturally: Mercury and Venus are never seen too far
from the Sun because the planets orbit the Sun, not the Earth.111This isn’t
how it happened historically, but it does illustrate the principle.

Similar arguments play an important role in modern cosmology and particle
physics. A standard cosmology textbook case for cosmic inflation goes as
follows (e.g.. Peacock, 1998). In the standard model of cosmology, the
geometry of the universe can be negatively curved, flat, or positively curved,
depending on whether the density universe is less than, equal to, or greater
than the _critical density_. In this model, two facts seem to be in tension
with each other. Firstly, the matter in the universe causes the density of the
universe to evolve away from critical. Secondly, observations tell us that the
density of the universe is very close to critical.

What about in the past? If we extend the model back to nucleosynthesis, about
1 second after the beginning, then the density of the universe must be within
one part in $10^{15}$ of the critical density in order to still be close to
critical today. The further we push back, the closer the constraint: at the
Planck time, it is one part in $10^{55}$. As with Ptolemy’s model, the
standard model of cosmology can explain the data, but only with an unmotivated
and suspiciously precise assumption. We must simply assume that the density of
the universe was extremely close to critical in its earliest moments. This
motivates inflationary models, in which a early period of accelerating
expansion drives _towards_ critical density (see Ijjas in this volume).

A second example comes from particle physics (Dine, 2015). The observed mass
of the Higgs particle can be written in terms of a “bare” value and quantum
corrections. These quantities are independent in the model. However, the size
of the quantum corrections diverges quadratically with the scale up to which
the effective theory can be trusted. Dine says, “if the cutoff is the Planck
scale, this correction is enormous …about thirty four orders of magnitude
larger than [the observed value], corresponding to a fine tuning of the bare
parameters against the radiative correction at the part in $10^{34}$ level.”

Again, the model can explain the observed value, but only by making the
unmotivated and suspiciously precise assumption that the bare value almost
perfectly cancels out the quantum corrections (see also Donoghue, 2007).
Particle physicists tend to call this situation “unnatural” rather than fine-
tuned, but it’s a similar idea. As Dine notes, “naturalness has for many years
been a guiding principle in the search for physics beyond the Standard Model”.

The assumptions underlying these arguments have been the subject of much
theoretical attention, but the logic is quite widely accepted. The
cosmological constant problem, the flatness problem, the big- and little-
hierarchy problems of particle physics (see Jacquart in this volume) and the
strong CP problem (see Ijjas in this volume) can be framed as fine-tuning
problems.

One particular case of fine-tuning is particularly striking. The data in
question are not the precise measurements of cosmology or particle physics,
but a more general feature of our universe: it supports the existence of life.
Before we look at this in more detail, it will be helpful to place fine-tuning
in the context of Bayesian approaches to testing physical theories.

##  2 Bayesian Accounts of Fine-Tuning

The Bayesian approach to probability theory views probabilities as quantifying
the _degree of plausibility_ of some proposition, given other propositions.
Bayesians have argued that the familiar probability axioms of Kolmogorov
(1933) (or similar) also apply to degrees of plausibility. This can be shown
via Dutch book arguments, representational theorems that trace back to Ramsey
(1926), or (more common among physicists) the theorem of Cox (1946), which
proposes that degrees of plausibility obey some intuitive desiderata (see also
Jaynes, 2003; Caticha, 2009; Knuth & Skilling, 2012).

In the Bayesian approach, physical theories are tested as follows. Let,

  * •

$T$ = the proposed theory to be tested. As a concrete example, $T$ may
represent a set of symmetry principles, from which we can derive the
mathematical form of a Lagrangian (or, equivalently, the dynamical equations),
but not the values of its free parameters.

  * •

$U$ = our observations of this Universe.

  * •

$B$ = everything else we know. For example, we treat the findings of
mathematics and theoretical physics as given, so these are included in $B$. As
I have defined it for our purposes here, the information in $B$ does not give
us any information about which possible world is actual. The theoretical
physicist can explore models of the universe mathematically, without concern
for whether they describe reality.

We then would like to know how plausible $T$ is, in light of everything that
we know $UB$. If the _posterior_ probability $p(T|UB)$ — read “the probability
of $T$ given $U$ and $B$” — were to descend to us on a cloud from the heavens,
then our job would be done. Alternatively, we may need some help in
calculating the posterior, and so we turn to Bayes Theorem,

$p(T|UB)=\frac{p(U|TB)~{}p(T|B)}{p(U|B)}~{}.$ (1)

If the theory in question has free parameters, which we generically denote
$\boldsymbol{\alpha}_{T}$, then we must take into account our lack of
knowledge of these parameters in evaluating the _likelihood_ of the data given
the theory $p(U|TB)$. We can think of this as dividing the theory into a large
number of sub-theories, each with a different value of the free parameters. To
calculate the likelihood, we need to average over these sub-theories — this is
known as marginalizing over nuisance parameters. Sub-theories that can account
for the data bring the average up, and sub-theories that can’t bring the
average down.

As a simplistic model, suppose a free parameter varies uniformly over a range
$R$, but only a small range $\Delta\boldsymbol{\alpha}_{T}$ is consistent with
the data. Then the theory’s likelihood is penalized by a factor
$\Delta\boldsymbol{\alpha}_{T}/R$. The smaller the range of free parameters
that accounts for the data, relative to the range of the parameters dictated
by the theory, the more the likelihood is penalized. Fine-tuning can be
translated directly into improbability within a Bayesian approach (see also
Aguirre, 2007; Fowlie, 2014; Barnes, 2017, and references therein).

##  3 The Fine-Tuning of the Universe for Life

Part of exploring any physical model is calculating the effect of varying its
free parameters. As we have seen, this is necessary for calculating the
likelihood of the data given the theory (via marginalizing), and so this can
tell us whether the theory is fine-tuned or not. Beginning in the 1970’s,
physicists noted that seemingly small changes to the fundamental constants of
nature and the initial conditions of the cosmos not only brought our models in
conflict with precise measurements; they described universes in which no life
form could exist. The complexity and stability required by any known or thus-
far conceived form of life could be rather easily erased.

This _fine-tuning of the universe for life_ was first investigated by Carter
(1974), Silk (1977), Carr & Rees (1979), Davies (1983), and Barrow & Tipler
(1986), and has been reviewed recently by Hogan (2000), Barnes (2012),
Schellekens (2013) and Lewis & Barnes (2016). We will consider a few examples.

###  3.1 The Cosmological Constant

The cosmological constant problem is described in the textbook of Burgess &
Moore (2006) as “arguably the most severe theoretical problem in high-energy
physics today, as measured by both the difference between observations and
theoretical predictions, and by the lack of convincing theoretical ideas which
address it.” The problem is as follows. Quantum field theory describes
particles as configurations of a field. There is a particular configuration of
the field that corresponds to a state with zero particles; this is known as
the vacuum state. Because the field is still there, we can ask: how much
energy is contained in the vacuum?

The absolute energy of the field doesn’t effect the interactions of the
standard model of particle physics, which depend only on energy differences.
But gravity, on Einstein’s theory, responds to the absolute amount of energy.
In a homogeneous and isotropic universe, vacuum energy has the same effect as
Einstein’s cosmological constant. When cosmologists speak of the cosmological
constant, they usually mean the sum of the “bare” cosmological constant in
Einstein’s equation and all the forms of energy in the universe that behave in
the same way. This is the quantity that is constrained by cosmological data.
In Planck units ($\hbar=G=c=1$) and expressed as a density, the observed
cosmological constant has the value $\rho_{\Lambda}\approx 1.2\times
10^{-123}$.

We can estimate the contribution to the energy in the vacuum from a given
quantum field. Loosely speaking, even in the vacuum state, virtual particles
will be created and annihilate, forming loops in a Feynman diagram. The vacuum
energy depends on the energy scale up to which we trust the theory to describe
this process. Even if we only consider well-understood fields (e.g. the
electron field) up to energy scales that have been thoroughly investigated by
experiment (say, $\sim 100$ GeV), the contribution to the vacuum energy is
$\sim 10^{-68}$, or 55 orders of magnitude larger than the observed value. If
we extend the range of our theory up to a popular energy scale where new
physics is expected, the supersymmetry scale, then the contribution to the
vacuum energy is $\sim 10^{-64}$. If we extend all the way to the Planck
scale, where cannot trust our theories because they do not account for quantum
gravity effects, the contribution to the vacuum energy is $\sim 1$, 123 orders
of magnitude larger than the observed value.

This is a fine-tuning problem. Quantum field theory and general relativity
_can_ explain the small observed value of the cosmological constant, but only
by supposing that the different (positive and negative) contributions to the
vacuum energy from each quantum field happen to cancel each other to 123
decimal places. This requires an unmotivated but suspiciously precise
coincidence between a number of independent factors.

As an example of fine-tuning _for life_ , the cosmological constant problem is
a near-perfect storm.

  * •

It’s actually several problems. Each quantum field –– electron, quark, photon,
neutrino, etc. –– adds a very large (positive or negative) contribution to the
vacuum energy of the universe.

  * •

General Relativity won’t help. Einstein’s theory links energy and momentum to
spacetime geometry. It does not dictate what energy and momentum exists in the
universe. Universes that are no good for life are perfectly fine by the
principles of General Relativity.

  * •

Particle physics probably won’t help. All particle physics processes, being
described by quantum field theory, depend only on energy differences; only
gravity responds to absolute energies. Thus, particle physics is largely blind
to its effect on cosmology, and thereby life.

  * •

It isn’t just a problem at the Planck scale, so quantum gravity won’t
necessarily help. As noted above, we don’t need to trust quantum field theory
all the way up to the Planck energy in order to see the cosmological constant
problem. It is entrenched firmly within well-understood, well-tested physics.

  * •

Alternative forms of dark energy have very similar problems. They usually
posit some other kind of field, and so the problem of the vacuum energy of the
field remains, unchanged and unsolved. See Jacquart (this volume) for more
discussion of dark energy.

  * •

We can’t aim for zero. Before the accelerated expansion of the universe was
discovered 1998, it was thought that some principle or symmetry would set the
cosmological constant to zero. Even this was a speculative hope, and it has
since evaporated.

  * •

The quantum vacuum has observable consequences, and so cannot be dismissed as
mere fiction. In particular, an electron in an atom feels the influence of the
quantum vacuum (the _Lamb shift_). Our theory works beautifully for electrons
and atoms. Why doesn’t cosmic expansion feel the influence of the quantum
vacuum?

  * •

The cosmological constant has a very obvious and definitive effect on the
necessary conditions for life. A positive cosmological constant causes the
expansion of the universe to accelerate, freezing structure formation. Make
the cosmological constant a few orders of magnitude larger and structure
formation freezes before _anything_ has formed. The universe will be a thin,
uniform hydrogen and helium soup, a diffuse gas where the occasional particle
collision is all that ever happens. A very simple way to make a universe
lifeless is to make it devoid of any structure whatsoever. Alternatively, a
negative cosmological constant causes the universe to recollapse. If the
cosmological constant were $-10^{-68}$, then the universe would recollapse
$\sim\sqrt{10^{68}}~{}t_{\textrm{Planck}}\sim 10^{-10}$ seconds after the big
bang.

###  3.2 The Parameters of the Standard Model

The standard model of particle physics has 25 free parameters which are
constrained by experiment. Many of these play a crucial role in providing the
complexity required by life.

The Higgs field “gives mass” to the fundamental particles of the standard
model. We can write their masses in terms of the vacuum expectation value
(vev) of the field ($v$) as $m_{f}=\Gamma_{f}v/\sqrt{2}$, where $\Gamma_{f}$
is the particle’s dimensionless Yukawa parameter. As with vacuum energy,
quantum corrections to the bare Higgs vev are predicted to be of the same
order as the scale up to which we trust the theory. The observed value of
$v=1.0\times 10^{-17}$ is unnaturally small.

Similarly small changes to $v$ significantly affect how particles interact and
bind. Damour & Donoghue (2008) refine the approach of Agrawal et al. (1998) by
considering nuclear binding, and conclude that unless $0.78\times
10^{-17}\lesssim v/m_{\textrm{Planck}}\lesssim 3.3\times 10^{-17}$ hydrogen is
unstable to the reaction $p+e\rightarrow n+\nu_{e}$ (if $v$ is too small) or
else there is no nuclear binding at all (if $v$ is too large).

Similarly, the strengths of the fundamental forces are subject to anthropic
constraints. For example, unless $\alpha\lesssim(m_{d}-m_{u})/$141 MeV, the
electromagnetic contribution to the mass of the proton causes it to be heavier
than the neutron, making the proton unstable (Hogan, 2000; Hall & Nomura,
2008). If the strong force were a few percent weaker, the deuteron would be
unbound (Pochet et al., 1991). The first step in stellar burning would require
a three-body reaction to form helium-3. This requires such extreme
temperatures and densities that stable stars cannot form: anything big enough
to burn is too big to be stable (Barnes & Lewis, 2017)222The fine-tuning
required for stable, life-powering stars has been clarified by recent work by
Adams (2008); Barnes (2015); Adams (2016); Adams & Grohs (2016, 2017).. Weaken
the strong force by a few more percent, or increase the strength of
electromagnetism, and carbon and all larger elements are unstable (Barrow &
Tipler, 1986). The parameters of the standard model must walk a tight-rope in
order to form stable nuclei and support stable stars.

###  3.3 The Dimensionality of Spacetime

Spacetime is the arena in which physics takes place. At the length scales
relevant to nuclei, atoms, stars, and the observable universe, spacetime is
described by three dimensions of space and one of time. It is often
straightforward to write down our familiar laws of nature in any number of
dimensions. For example, in $m$ time dimensions ($t_{i}$) and $n$ space
dimensions ($x_{j}$), the wave equation is,

$\sum\limits_{i=1}^{m}\frac{\partial^{2}\rho}{\partial
t_{i}^{2}}=c_{s}^{2}\sum\limits_{j=1}^{n}\frac{\partial^{2}\rho}{\partial
x_{j}^{2}}~{},$ (2)

for the scalar wave variable $\rho$, and wave speed $c_{s}$.

Given that we can theoretically explore such universes, what would they be
like? This question has been addressed by Ehrenfest (1917), Whitrow (1955),
Barrow & Tipler (1986), and Tegmark (1997). It has been known for some time
that Newtonian gravity only predicts stable planetary orbits in three space
dimensions (Bertrand’s theorem). With four space dimensions, for example,
slightly non-circular orbits are spiralled, not elliptical — they would send
the planet into the star or off into empty space. The same applies to atomic
orbits described by the Schrodinger equation — there is no stable ground
state.

We can also vary the number of time dimensions. In such a universe, an
observer will have their own clock that measures time along their worldline;
but what would they experience? Tegmark (1997) notes that linear partial
differential equations, of which the wave equation is one example and by which
many known laws can be approximated locally, have interesting properties when
there is more than one time dimension. In our universe, we can approximately
predict the behavior of a physical system into the future on the basis of
knowledge of our immediate environment. (I don’t necessarily mean _predict_ in
a mathematical sense. A bird “predicts” the path of a flying insect to catch
it.) But if there were more (or less) than one time dimension, then the
problem would be mathematically ill-posed, being infinitely sensitive to the
initial conditions. The behaviour of one’s environment could not be predicted
using only local, finite accuracy data, making storing and processing
information impossible.

##  4 The Multiverse

Fine-tuning in physics serves as impetus to search for a better theory, one
which can account for the facts in a more natural way, without unmotivated
assumptions. But what could naturally explain a _life-permitting_ universe?

Perhaps we won the cosmic lottery: a life-permitting universe exists, despite
the seemingly overwhelming odds, because the universe as a whole consists of a
vast, variegated ensemble of sub-universes — a _multiverse_.

A viable multiverse model needs a few ingredients. The first is a physical
theory that goes beyond the standard models by promoting the constants of
nature and initial conditions to dynamic variables. We have some hints about
how to do this. The strengths of the fundamental forces of particle physics
are a function of energy, and seem to converge at an energy far above our
current experiments. This has led to the development of Grand Unified Theories
(GUT), in which the strong nuclear force, weak nuclear force, and
electromagnetism are manifestations of a single, unified force (see Raby,
2010). At low energy, the greater symmetry of the unified field is
_spontaneously broken_ : the strengths of the forces are not written in stone
in the fundamental equations, but rather are a frozen accident.

There are other ways to promote the constants to variables. In string theory,
there is a _landscape_ of solutions to the fundamental equations, with the
familiar “constants” of physics written into the various folds and holes of
the extra, _compactified_ spatial dimensions (Schellekens, 2013). They become
free parameters of the _solution_ to the equations, rather than appearing in
the equations themselves.

The second ingredient of a multiverse theory is a cosmological mechanism to
create domains of the universe with different values of the “constants”. The
leading contender today is cosmic inflation: in its earliest moments, the
universe expanded at an accelerating rate, driving it towards critical density
and laying down the seeds of cosmic structure.

The successful predictions of inflation require only that our observable
universe inflated, but it has been argued that inflation will naturally
produce a multiverse (see Linde, 2015). Most inflationary models posit a form
of energy called an _inflaton field_ that drives the expansion of the
universe. The physics of a quantum field is codified in its _potential_ : the
dynamics is analogous to a ball rolling on a hill, and the shape of the hill
tells us how the motion of the ball depends on the value of the field. For an
inflaton field to cause accelerating expansion, it must be rolling slowly on a
very flat section of the potential. Inflation ends when the field rolls off
the flat section, usually into a valley. As the field oscillates around the
bottom of the valley, reheating begins: the energy in the inflaton field is
transferred into ordinary matter and radiation, beginning the hot big bang
phase.

But the field is a quantum field, and so will not evolve deterministically
(depending on your interpretation of quantum mechanics). Somewhat
simplistically, consider an inflating region of the universe, in which the
inflaton field dominates the energy of the universe and is slowly rolling.
While in most of the region the field will roll into the valley and inflation
will end, there is a finite probability that the field in some sub-region will
evolve to a state further up the slope. This part of the universe will inflate
for longer. Because this sub-region keeps growing in size, it will soon be
larger than the original region, and so inflation will always continue
somewhere. Given a sufficiently large initial inflating region, post-big-bang
pockets form in an inflating background.

If the energy scale of reheating is above the symmetry breaking scale of the
fields in the universe, then the symmetry will break differently in different
sub-universes. This creates a population of sub-universes with different
‘constants’ and big bang ‘initial’ conditions.

The final ingredient is a selection effect (Wall & Jenkins, 2003; Bostrom,
2002; Neal, 2006). Consider the prediction of cosmic microwave background
(CMB) anisotropies in the standard big bang model, from which cosmologists
infer the values of various cosmic parameters. Like any thermodynamic system,
there are fluctuations in the recombining plasma. So, in a sufficiently large
universe, the probability of _someone_ observing the CMB that we see
approaches one regardless of the values of the cosmic parameters. If we tested
physical models by calculating the probability that _some_ observation in the
universe matches our actual observations, then any values of the cosmic
parameters would do in an infinite universe. We couldn’t infer their values
from observations. A multiverse would make this problem even worse.

To resolve this problem, remember that we don’t just know that _some_
observation has taken place, but that a _particular_ observer has made an
observation. Even if _some_ observer sees a misleading CMB, the vast majority
won’t, justifying our inference. We apply this to the multiverse: that _some_
region of the universe permits life is a good start but not sufficient. What
will a _typical_ observer see?

The anthropic prediction by Weinberg (1987) of the cosmological constant
provides an excellent test case. Given a large enough variety of sub-universes
with different values of the cosmological constant, somewhere will have a
value that permits structure to form. In such an ensemble, asks Weinberg, what
cosmological constant would a typical observer see? There is nothing in
fundamental physics as we know it that singles out $\rho_{\Lambda}=0$ as a
privileged value, so we assume for the moment that for values of
$\rho_{\Lambda}$ much smaller than the ‘natural’ Planck scale, the multiverse
produces a roughly uniform distribution of values. Then (considering positive
values of $\rho_{\Lambda}$ for now) what is the largest value of
$\rho_{\Lambda}$ that permits the formation of structure? Weinberg’s analytic
calculation gives an upper limit of $\rho_{\Lambda,\textrm{max}}\approx
550\rho_{0}\approx 3\times 10^{-121}$, where $\rho_{0}$ is the present cosmic
mass density. Weinberg made this prediction before observation showed that
$\rho_{\Lambda}\approx 1.2\times 10^{-123}$.

A typical observer would expect to observe a vacuum energy roughly comparable
with the anthropic upper bound. It can’t be larger, of course — there are no
observers in those sub-universes to make an observation. Weinberg’s
calculation gives the upper bound as being two orders of magnitude above the
actual value, which is close enough to take the calculation seriously. My
colleagues and I are currently repeating Weinberg’s calculation with more
sophisticated supercomputer models of galaxy formation.

If we had observed a value that was ten orders of magnitude smaller than the
upper bound, then we would conclude that one of the assumptions in our model
is probably wrong. We would look for a dynamical or symmetry-based
explanation, rather than an anthropic one.

This kind of case for the multiverse has been criticized as speculative and
untestable. But it should be remembered that such considerations are almost
unavoidable in cosmology. Just as the astronomer must understand their
telescope before they can understand what they see through it, when a
cosmologist models the universe, they are inevitably modeling a system that
contains themselves. We cannot pretend to stand outside the universe.
Selection effects cannot be ignored. We are not Dr Frankenstein; we are the
monster. We have woken up in a laboratory and are trying to understand how it
made us.

We can test the multiverse using Bayesian probability theory. In this case,
the “data” to be explained is the constants of nature. If the fine-tuning for
life implies that almost all observers in the multiverse would observe similar
constants to what we observe, then this could provide a major advantage for a
multiverse hypothesis over theories in which the constants are free parameters
(Aguirre, 2007; Barnes, 2017).

One way in which a multiverse theory can fail spectacularly is known as the
_Boltzmann Brain problem_. Physical theories predict observations, and so a
multiverse model should — in principle — be able to predict what _kind_ of
observer we would expect to be. One striking feature of our status as
observers is that we formed through a long, consistently entropy-increasing
process: gravitational collapse into galaxies and stars, stellar burning and
supernovae, planet formation, and biological evolution. In some multiverses,
including Boltzmann’s original multiverse (Boltzmann, 1895), most observers
form via a chance statistical fluctuation. Without a consistent thermodynamic
arrow of time, they will not observe records of the processes that formed them
(Hartle, 2004). They will observe as much free energy around them as is
required for their existence as observers, and almost certainly no more.

To be clear, this is not the philosophical “brain-in-a-vat” problem: how can I
know whether I’m a Boltzmann brain with false memories? This is a more
straightforward “theoretical prediction meets observation” scenario: a
cosmological theory predicts that a typical observer will be a Boltzmann
brain, _and_ will observe that they are a Boltzmann brain. And that prediction
is wrong. Whether multiverse models can naturally avoid this problem is an
open question; see, among many others, Page (2006); Linde (2007); Banks
(2007); de Simone et al. (2010); Aguirre, Carroll & Johnson (2011); Nomura
(2011); Boddy & Carroll (2013); Albrecht (2015); Boddy, Carroll & Pollack
(2015).

Misgivings about the whole multiverse project are hardly surprising. Are the
tests of multiverse theories enough to make it scientific? Unobservable sub-
universes are very different to unobservable quarks: we can constrain the
properties of quarks via experiment, but every other sub-universe in the
multiverse could disappear tomorrow and we would never know. The meagre tests
of the multiverse “prove nothing”, say Ellis & Silk (2014), ”Fundamentally,
the multiverse explanation relies on string theory, which is as yet
unverified, and on speculative mechanisms for realizing different physics in
different sister universes. It is not, in our opinion, robust, let alone
testable.”

A potentially tricky hurdle is the _measure problem_ , about which there is an
extensive literature. Many multiverse theories imply or assume that there are
an infinite number of other sub-universes. Given a finite population, deriving
probabilities is straightforward: what fraction of observers see a value of
$\rho_{\Lambda}$ as small as the one we observe? But in an infinite
multiverse, we cannot simply count sub-universes.

In particular, once we have a useful definition of an observer, it seems that
we should treat them all on equal footing. Think of this as permutation
symmetry — having arbitrarily numbered all the observers (or observer
moments), we should be able to shuffle the labels without changing the
prediction of the model. But there is no assignment of probabilities to an
infinite number of possibilities that respects this symmetry. This is often
taken as incentive to assign different probabilities. But it could be argued,
and with considerable force, that this means that an infinite multiverse
theory cannot justify probabilities and so cannot make predictions. “In an
infinite universe,” says Olum (2012), “everything which can happen will happen
an infinite number of times, so what does it mean to say that one thing is
more likely than another?”. These are open questions; see, among many others,
Vilenkin (1995); Garriga et al. (2006); Aguirre, Gratton, & Johnson (2007);
Vilenkin (2007a, b); Gibbons & Turok (2008); Page (2008); Bousso, Freivogel, &
Yang (2009); de Simone et al. (2010); Freivogel (2011); Bousso & Susskind
(2012); Garriga & Vilenkin (2013); Carroll (2017); Page (2017).

##  5 After Physics

In physics, fine-tuning problems afflict theories that seem to be successful,
that is, that can account for the data. The problem is not a falsified
prediction, as one might expect from a discounted or discarded theory. Recall
the lesson of Ptolemy’s model. Within the set of possible geocentric planetary
systems, an uncomfortably large proportion look very different to our Solar
System. A fine-tuning problem is raised by a large set of alternate
possibilities. This suggests an interesting thought experiment.

Suppose there is an ultimate theory of physics. At a future International
Meeting of Really Important Physicists, Alberta Einstein walks to the
chalkboard, scribbles a few equations, and fundamental physics comes to an
end. Like chess pieces who had discovered the laws of chess, no deeper rules
exist.

By hypothesis, this theory would be consistent with all scientific data. But
we may still glimpse a large set of alternate possibilities, and so a kind of
fine-tuning problem remains. Even if it contains no free parameters, Alberta’s
chalkboard will show one particular mathematical equation or structure. We
will be faced by a very old question: why _this_ universe? Of all the ways the
world could have been, why this way? Of all the mathematically consistent
chalkboards of equations, why Alberta’s?

Obviously, the answer is not yet-another chalkboard of equations. Neither is
it more observations of this universe. This is not the kind of question that
physics can answer, because we can’t prove from any set of equations that they
describe reality. Theories don’t predict their own success. But if not
physics, then what? What do we do when fundamental physics is over?

Perhaps we stop asking questions. Maybe reality doesn’t have any ultimate
reason for why it is the way it is. Explanations of the physical world reach
the ultimate laws, and stop. This is the supposition of _naturalism_ : the
natural world is all there is. For a modern defence, see Carroll (2016).

Alternatively, Tegmark (1998) has defended the “the ultimate ensemble theory”,
that “physical existence is equivalent to mathematical existence”. The actual
world is not chosen from a set of mathematical possibilities; rather, all
mathematical possibilities are equally real, and we are self-aware
substructures (SASs) within a particular mathematical structure. A
metaphysician might worry about the dissolution of the line between _abstract_
and _concrete_. The physicist who tries to test Tegmark’s idea via its
prediction that “the mathematical structure describing our world is the most
generic one that is consistent with our observations” faces a problem: we need
a probability distribution over the set of mathematical structures, but a
probability distribution is itself a mathematical structure. Tegmark says that
probabilities are “merely subjective”, but our subjective states of mind are
mathematical substructures, too.

By contrast, axiarchism (Leslie, 1989) and theism (e.g. Swinburne, 2004;
Collins, 2009) argue that beneath the mathematical structure of our universe
is a _reason_ : our universe is morally valuable, particularly its embodied,
free, conscious agents. Just as Tegmark promotes possibilities to reality on
mathematical grounds, axiarchism does so on moral grounds: the world exists
because it is good. Theism proposes that God exists necessarily in some sense,
and the physical world is the result of God’s free choice to create a morally
valuable world.

For each of these alternatives, the fine-tuning of the universe for life plays
an important role. For Tegmark, the complexity required by any SAS explains
why we see this universe/mathematical structure, rather than a simpler one.
For axiarchism and theism, fine-tuning for life shows how these ideas could
have explanatory power. Given the seemingly extraordinarily small proportion
of possibilities that permit the existence of embodied moral agents, the
axiarchist and theist can understand something of why Alberta’s blackboard is
the one has gone to all the bother of existing. Further examination of these
alternatives takes us beyond the philosophy of physics.

## Acknowledgments

Supported by a grant from the John Templeton Foundation. This publication was
made possible through the support of a grant from the John Templeton
Foundation. The opinions expressed in this publication are those of the author
and do not necessarily reflect the views of the John Templeton Foundation.

## References

  * Adams (2008) Adams, F. C. [2008]: ‘Stars in other universes: stellar structure with different fundamental constants’, _Journal of Cosmology and Astroparticle Physics_ , 8, 010.
  * Adams (2016) Adams, F. C. [2016]: ‘Constraints on alternate universes: stars and habitable planets with different fundamental constants’, _Journal of Cosmology and Astroparticle Physics_ , 2, 042.
  * Adams & Grohs (2016) Adams, F. C. and Grohs, E. [2016], ‘On the Habitability of Universes without Stable Deuterium’, _arXiv_ :1612.04741.
  * Adams & Grohs (2017) Adams, F. C. and Grohs, E. [2017]: ‘Stellar helium burning in other universes: A solution to the triple alpha fine-tuning problem’, _Astroparticle Physics_ , 87, 40.
  * Agrawal et al. (1998) Agrawal, V, S. M. Barr, John F. Donoghue, and D Seckel. [1998]: ‘Anthropic Considerations in Multiple-Domain Theories and the Scale of Electroweak Symmetry Breaking’, _Physical Review Letters_ , 80, pp. 1822–25.
  * Aguirre (2007) Aguirre, A. [2007]: ‘The inflationary multiverse’, in Carr, B. J. (ed.), _Universe or Multiverse?_ , Cambridge: Cambridge University Press.
  * Aguirre, Gratton, & Johnson (2007) Aguirre A., Gratton S., Johnson M. C. [2007]: ‘Hurdles for recent measures in eternal inflation’, _Physical Review D_ , 75, 123501
  * Aguirre, Carroll & Johnson (2011) Aguirre, Anthony, Sean M. Carroll, and Matthew C. Johnson [2011]: ‘Out of Equilibrium: Understanding Cosmological Evolution to Lower-Entropy States’, _arXiv preprint:_ 1108.0417.
  * Albrecht (2015) Albrecht, A. [2015]: ‘Tuning, Ergodicity, Equilibrium, and Cosmology’, _Physical Review D_ , 91, pp. 1–11.
  * Banks (2007) Banks, T. A. [2007]: ‘Entropy and Initial Conditions in Cosmology’, _arXiv preprint:_ hep-th/0701146.
  * Barnes (2012) Barnes, L. A. [2012]: ‘The Fine-Tuning of the Universe for Intelligent Life’, _Publications of the Astronomical Society of Australia_ , 29, pp. 529–564.
  * Barnes (2015) Barnes, L. A. [2015]: ‘Binding the diproton in stars: anthropic limits on the strength of gravity’, _Journal of Cosmology and Astroparticle Physics_ , 12, 050.
  * Barnes (2017) Barnes, L. A. [2017]: ‘Testing the multiverse: Bayes, fine-tuning and typicality’, in Chamcham et al. (ed.), _The Philosophy of Cosmology_ , forthcoming with Cambridge University Press.
  * Barnes & Lewis (2017) Barnes, L. A. and Lewis, G. F. [2017]: ‘Producing the deuteron in stars: anthropic limits on fundamental constants’, _Journal of Cosmology and Astroparticle Physics_ , 07, 036.
  * Barrow & Tipler (1986) Barrow, J. D. and F. J. Tipler [1986]: _The Anthropic Cosmological Principle_ , Oxford: Clarendon Press.
  * Boltzmann (1895) Boltzmann, Ludwig [1895]: ‘On Certain Questions of the Theory of Gases’, Nature, 51, 413–15
  * Boddy & Carroll (2013) Boddy, Kimberly K, and Sean M. Carroll [2013]: ‘Can the Higgs Boson Save Us From the Menace of the Boltzmann Brains?’, _arXiv preprint:_ 1308.4686.
  * Boddy, Carroll & Pollack (2015) Boddy, Kimberly K, Sean M. Carroll, and Jason Pollack [2015]: ‘Why Boltzmann Brains Don’t Fluctuate Into Existence From the De Sitter Vacuum’, _arXiv preprint:_ 1505.02780.
  * Bostrom (2002) Bostrom, N. [2002]: _Anthropic Bias: Observation Selection Effects in Science and Philosophy_ , New York: Routledge.
  * Bousso, Freivogel, & Yang (2009) Bousso R., Freivogel B. and Yang I.-S. [2009]: ‘Properties of the scale factor measure’, _Physical Review D_ , 79, 063513
  * Bousso & Susskind (2012) Bousso, R., & Susskind, L. [2012]: ‘Multiverse interpretation of quantum mechanics’, _Physical Review D_ , 85, 045007
  * Burgess & Moore (2006) Burgess C. and G. Moore [2006]: _The Standard Model: A Primer_ , Cambridge: Cambridge University Press.
  * Carr & Rees (1979) Carr, B. J. and M. J. Rees [1979]: ‘The Anthropic Principle and the Structure of the Physical World’, _Nature_ , 278, pp. 605–612.
  * Carroll (2016) Carroll, S. M. [2016]: _The Big Picture: On the Origins of Life, Meaning, and the Universe Itself_ , New York: Dutton.
  * Carroll (2017) Carroll, S. M. [2017]: ‘Why Boltzmann Brains Are Bad’, _arXiv preprint:_ 1702.00850
  * Carter (1974) Carter, B. [1974]: ‘Large Number Coincidences and the Anthropic Principle in Cosmology’, in M. S. Longair (ed.), _Confrontation of Cosmological Theories with Observational Data_ , Dordrecht: D. Reidel, pp. 291–298.
  * Caticha (2009) Caticha, A. [2009]: ‘Quantifying Rational Belief’, _AIP Conference Proceedings_ , 1193, pp. 60-8.
  * Collins (2009) Collins, R. [2009]: ‘The teleological argument: an exploration of the fine-tuning of the universe’, in W. L. Craig and J. P. Moreland (ed.), _The Blackwell Companion to Natural Theology_ , Oxford: Blackwell Publishing.
  * Cox (1946) Cox, R. T. [1946]: ‘Probability, frequency and reasonable expectation’, _American Journal of Physics_ , 17, pp. 1–13.
  * Damour & Donoghue (2008) Damour, Thibault, and John F. Donoghue [2008]: ‘Constraints on the Variability of Quark Masses from Nuclear Binding’, _Physical Review D_ , 78, 014014.
  * Davies (1983) Davies, P. C. W. [1983]: ‘The Anthropic Principle’, _Progress in Particle and Nuclear Physics_ , 10, pp. 1–38.
  * de Simone et al. (2010) de Simone, A., Guth, A. H., Linde, A., Noorbala, M., Salem, M. P., & Vilenkin, A. [2010]: ‘Boltzmann brains and the scale-factor cutoff measure of the multiverse’, _Physical Review D_ , 82, 063520
  * Dine (2015) Dine, M. [2015]: ‘Naturalness Under Stress’, _arXiv_ : 1501.01035.
  * Donoghue (2007) Donoghue, J. F. [2007]: ‘The fine-tuning problems of particle physics and anthropic mechanisms’, in Carr, B. J. (ed.), _Universe or Multiverse?_ , Cambridge: Cambridge University Press.
  * Ehrenfest (1917) Ehrenfest P. [1917]: ‘Can atoms or planets exist in higher dimensions?’, _Proceedings of the Amsterdam Academy_ , 20, 200.
  * Ellis & Silk (2014) Ellis, G.F.R. and Silk, J. [2014]: ‘Scientific method: Defend the integrity of physics’, _Nature_ , 516, pp. 321–323.
  * Fowlie (2014) Fowlie A. [2014]: ‘CMSSM, naturalness and the ”fine-tuning price” of the Very Large Hadron Collider’, _Physical Review D_ , 90, 015010.
  * Freivogel (2011) Freivogel B. [2011]: ‘Making predictions in the multiverse’, _Classical and Quantum Gravity_ , 28, 204007
  * Garriga et al. (2006) Garriga, J., Schwartz-Perlov, D., Vilenkin, A., & Winitzki, S. [2006]: ‘Probabilities in the inflationary multiverse’, _Journal of Cosmology and Astroparticle Physics_ , 1, 017
  * Garriga & Vilenkin (2013) Garriga J., Vilenkin A. [2013]: ‘Watchers of the multiverse’, _Journal of Cosmology and Astroparticle Physics_ , 5, 037
  * Gibbons & Turok (2008) Gibbons G. W., Turok N. [2008]: ‘Measure problem in cosmology’, _Physical Review D_ , 77, 063516
  * Hall & Nomura (2008) Hall L. and Nomura Y. [2008]: ‘ Evidence for the multiverse in the standard model and beyond’, _Physical Review D_ , 78, 035001.
  * Hartle (2004) Hartle, J. B. [2008]: ‘The Physics of Now’, _arXiv preprint:_ gr-qc/0403001
  * Hogan (2000) Hogan, C. J. [2000]: ‘Why the Universe Is Just so’, _Reviews of Modern Physics_ , 72, pp. 1149–1161.
  * Jaynes (2003) Jaynes, E. T. [2003]: _Probability Theory: the Logic of Science_ , Cambridge: Cambridge University Press.
  * Knuth & Skilling (2012) Knuth, K. H. and J. Skilling [2012]: ‘Foundations of Inference’, _Axioms_ , 1 (1), pp. 38-73.
  * Kolmogorov (1933) Kolmogorov, A. [1933]: _Foundations of the Theory of Probability_ , Berlin: Julius Springer.
  * Leslie (1989) Leslie J. [1989]: _Universes_. London: Routledge.
  * Lewis & Barnes (2016) Lewis, G. F. and L. A. Barnes [2016]: _A Fortunate Universe: Life in a Finely Tuned Cosmos_ , Cambridge: Cambridge University Press.
  * Linde (2007) Linde, A. [2007]: ‘Sinks in the Landscape, Boltzmann Brains and the Cosmological Constant Problem’, _Journal of Cosmology and Astroparticle Physics_ , 01, 22.
  * Linde (2015) Linde, A. [2015]: ‘A Brief History of the Multiverse’, _arXiv:_ 1512.01203.
  * Neal (2006) Neal R. M. [2006]: ‘Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning’, _arXiv:_ math/0608592
  * Nomura (2011) Nomura, Y. [2011]: ‘Physical Theories, Eternal Inflation, and the Quantum Universe’, _Journal of High Energy Physics_ , 11, 63.
  * Olum (2012) Olum, Ken D. [2012]: ‘Is There Any Coherent Measure for Eternal Inflation?’, _Physical Review D_ , 86, 1–6.
  * Page (2006) Page, D. N. [2006]: ‘Susskind’s Challenge to the Hartle-Hawking No-Boundary Proposal and Possible Resolutions’, _arXiv preprint:_ hep-th/0610199.
  * Page (2008) Page D. N. [2008]: ‘Cosmological measures without volume weighting’, _Journal of Cosmology and Astroparticle Physics_ , 10, 025
  * Page (2017) Page D. N. [2017]: ‘Bayes Keeps Boltzmann Brains at Bay’, _arXiv preprint:_ 1708.00449
  * Peacock (1998) Peacock, J. A. [1998]: _Cosmological Physics_ , Cambridge: Cambridge University Press.
  * Pochet et al. (1991) Pochet, T., J. M. Pearson, G. Beaudet, and H. Reeves [1991]: ‘The Binding of Light Nuclei, and the Anthropic Principle’, _Astronomy and Astrophysics_ , 243, pp. 1–4.
  * Raby (2010) Raby, S. [2010]: ‘Grand Unified Theories’, in Review of Particle Physics, Nakamura, K. et al. _Physical Review DJ. Phys._ G37: 075021.
  * Ramsey (1926) Ramsey, Frank P., [1926]:‘Truth and Probability’, in _The Foundations of Mathematics and other Logical Essays_ , R.B. Braithwaite (ed.), London: Kegan, Paul, Trench, Trubner & Co., pp. 156–-198.
  * Schellekens (2013) Schellekens, A. N. [2013]: ‘Life at the Interface of Particle Physics and String Theory’, _Reviews of Modern Physics_ , 85, pp. 1491–1540.
  * Silk (1977) Silk, J. [1977]: ‘Cosmogony and the Magnitude of the Dimensionless Gravitational Coupling Constant’, _Nature_ , 265, pp. 710–711.
  * Skilling (2014) Skilling, J. [2014]: ‘Foundations and Algorithms’, in Hobson, M. et al (eds.), _Bayesian Methods in Cosmology_ , Cambridge: Cambridge University Press.
  * Swinburne (2004) Swinburne, R. [2004]: _The Existence of God_ , Oxford: Clarendon Press.
  * Tegmark (1997) Tegmark M. [1997]: ‘On the dimensionality of spacetime’, _Classical and Quantum Gravity_ , 14, L69.
  * Tegmark (1998) Tegmark M. [1998]: ‘Is “the Theory of Everything” Merely the Ultimate Ensemble Theory?’, _Annals of Physics_ , 270, 1, pp. 1–51.
  * Vilenkin (1995) Vilenkin, A. [1995]: ‘Making predictions in an eternally inflating universe’, _Physical Review D_ , 52, 3365
  * Vilenkin (2007a) Vilenkin A. [2007]: ‘Freak observers and the measure of the multiverse’, Journal of High Energy Physics, 1, 092
  * Vilenkin (2007b) Vilenkin A. [2007]: ‘A measure of the multiverse’, _Journal of Physics A_ , 40, 6777.
  * Wall & Jenkins (2003) Wall, J. V. and Jenkins, C. R. [2003]: _Practical Statistics for Astronomers_ , Cambridge: Cambridge University Press.
  * Weinberg (1987) Weinberg, S. [1987]: ‘Anthropic Bound on the Cosmological Constant’, _Physical Review Letters_ , 59, pp. 2607–10.
  * Whitrow (1955) Whitrow G. J. [1955]: ‘Why physical space has three dimensions’, _The British Journal for the Philosophy of Science_ , VI, 13.