## 1. Cosmology's Standard Model



Physical cosmology has achieved a consensus Standard Model (SM), based
on extending the local physics governing gravity and the other forces
to describe the overall structure of the universe and its evolution.
According to the SM, the universe has evolved from an extremely high
temperature early state, by expanding, cooling, and developing
structures at various scales, such as galaxies and stars. This model
is based on bold extrapolations of existing theories--applying
general relativity, for example, at length scales 14 orders of
magnitude larger than the those at which it has been tested--and
requires several novel ingredients, such as dark matter and dark
energy. The last few decades have been a golden age of physical
cosmology, as the SM has been developed in rich detail and
substantiated by compatibility with a growing body of observations.
Here we will briefly introduce some of the central concepts of the SM
to provide the minimal background needed for the ensuing
 discussion.[2]



### 1.1 Spacetime Geometry



Gravity is the dominant interaction at large length scales. General
relativity introduced a new way of representing gravity: rather than
describing gravity as a force deflecting bodies from inertial motion,
bodies free from non-gravitational forces move along the analog of
straight lines, called geodesics, through a curved spacetime
 geometry.[3]
 The spacetime curvature is related to the distribution of energy and
matter through GR's fundamental equations (Einstein's
field equations, EFE). The dynamics of the theory are non-linear:
matter curves spacetime, and the curvature of spacetime determines how
matter moves; and gravitational waves interact with each other
gravitationally, and act as gravitational sources. The theory also
replaces the single gravitational potential, and associated field
equation, of Newton's theory, with a set of 10 coupled,
non-linear equations for ten independent
 potentials.[4]
 This complexity is an obstacle to understanding the general features
of solutions to EFE, and to finding exact solutions to describe
specific physical situations. Most exact solutions have been found
based on strong idealizations, introduced to simplify the
mathematics.



Remarkably, much of cosmology is based on an extremely simple set of
solutions found within a decade of Einstein's discovery of GR.
These Friedman-Lemaitre-Robertson-Walker (FLRW) solutions have,
in a precise sense, the most symmetry possible. The spacetime geometry
is constrained to be uniform, so that there are no preferred locations
or
 directions.[5]
 They have a simple geometric structure, consisting of a
"stack" of three-dimensional spatial surfaces
\(\Sigma(t)\) labeled by values of the cosmic time \(t\)
(topologically, \(\Sigma \times \mathbb{R}\)). The surfaces
\(\Sigma(t)\) are three-dimensional spaces (Riemannian manifolds) of
constant curvature, with three possibilities: (1) spherical space, for
the case of positive curvature; (2) Euclidean space, for zero
curvature; and (3) hyperbolic space, for negative
 curvature.[6]




These models describe an expanding universe, characterized fully by
the behavior of the scale factor \(R(t)\). The worldlines of
"fundamental observers", defined as at rest with respect
to matter, are orthogonal to these surfaces, and the cosmic time
corresponds to the proper time measured by the fundamental observers.
The scale factor \(R(t)\) represents the spatial distance in
\(\Sigma\) between nearby fundamental observers as a function of
cosmic time. The evolution of these models is described by a simple
set of equations governing \(R(t)\), implied by Einstein's field
equations (EFE): the Friedmann
 equation,[7]


  

\[\label{eq:Fried}
 \left(\frac{\dot{R}}{R}\right)^2 = \frac{8 \pi G \rho}{3} - \frac{k}{R^2} + \frac{\Lambda}{3},\] 


and the isotropic form of the Raychaudhuri equation: 

 

\[\label{eq:Ray}
 3 \frac{\ddot{R}}{R} = {- 4}\pi G \left(\rho + 3 p \right) + \Lambda.
\] 


The curvature of surfaces \(\Sigma(t)\) of constant cosmic time is
given by \(\frac{k}{R^2(t)}\), where \(k = \{-1,0,1\}\) for negative,
flat, and positive curvature (respectively). The assumed symmetries
force the matter to be described as a perfect
 fluid[8]
 with energy density \(\rho\) and pressure \(p\), which obey the
energy conservation equation 

 

\[\label{eq:cons}
 \dot{\rho} + (\rho + p) 3 \frac{\dot{R}}{R} = 0.\] 


The unrelenting symmetry of the FLRW models makes them quite simple
geometrically and dynamically. Rather than a set of coupled partial
differential equations, which generically follow from EFE, in the FLRW
models one only has to deal with 2 ordinary differential equations
(only two of \((\ref{eq:Fried})\)-\((\ref{eq:cons})\) are
independent) which are determinate once an equation of state \(p =
p(\rho)\) is given.



These equations reveal three basic features of these models. First,
these are dynamical models: it is hard to arrange an unchanging
universe, with \(\dot{R}(t) =0\). "Ordinary" matter has
positive total stress-energy density, in the sense that
\(\rho\_{\textit{grav}}\coloneqq \rho + 3p > 0\). From
(\(\ref{eq:Ray}\)), the effect of such ordinary matter is to
decelerate cosmic expansion, \(\ddot{R} < 0\)--gravity is a
force of attraction. This is only so for ordinary matter: a positive
cosmological constant, or matter with negative gravitational-energy
density \(\rho\_{\textit{grav}}\) leads, conversely, to accelerating
expansion, \(\ddot{R} > 0\). Einstein was only able to construct a
static model by delicately balancing the attraction of ordinary matter
with a precisely chosen value of \(\Lambda\); he unfortunately failed
to notice that the solution was unstable, and overlooked the dynamical
implications of his own theory.



Second, the expansion rate varies as different types of matter come to
dominate the dynamics. As shown by (\(\ref{eq:cons}\)), the energy
density for different types of matter and radiation dilutes at
different rates: for example, pressureless dust (\(p=0\)) dilutes as
\(\propto R^{-3}\), radiation (\(p=\rho/3\)) as \(\propto R^{-4}\),
and the cosmological constant (\(p=-\rho\)) remains (as the name
suggests) constant. The SM describes the early universe as having a
much higher energy density in radiation than matter. This
radiation-dominated phase eventually transitions to a matter-dominated
phase as radiation dilutes more rapidly, followed eventually, if
\(\Lambda > 0\), by a transition to a \(\Lambda\)-dominated phase;
if \(k \neq 0\) there may also be a curvature dominated phase.



Third, FLRW models with ordinary matter have a singularity at a finite
time in the past. Extrapolating back in time, given that the universe
is currently expanding, eqn. (\(\ref{eq:Ray}\)) implies that the
expansion began at some finite time in the past. The current rate of
expansion is given by the Hubble parameter, \(H\_0 =
(\frac{\dot{R}}{R})\_0\). Simply extrapolating this expansion rate
backward, from eqn. (\(\ref{eq:Ray}\)) the expansion rate must
increase at earlier times, so \(R(t) \rightarrow 0\) at a time less
than the Hubble time Hubble time \(H\_0^{-1}\) before now, if
\(\rho\_{\textit{grav}}\geq 0\). As this "big bang" is
approached, the energy density and curvature increase without bound
provided \(\rho\_{\textit{inert}}\coloneqq (\rho+p)>0\) (which
condition guarantees that \(\rho \rightarrow\infty\) as
\(R\rightarrow0\)). This reflects gravitational instability: as
\(R(t)\) decreases, the energy density and pressure both increase, and
they both appear with the same sign on the right hand side of eqn.
(\(\ref{eq:Ray}\)), hence pressure \(p>0\) does not help avoid
the singularity. Work in the 1960s, discussed below in
 SS4.1,
 established that the existence of a singularity holds in more
realistic models, and is not an artifact of the symmetries of the FLRW
models.



The SM adds small departures from strict uniformity in order to
account for the formation and evolution of structure. Due to
gravitational instability, such perturbations are enhanced
dynamically--the density contrast of an initial region that
differs from the average density grows with time. Sufficiently small
fluctuations can be treated as linear perturbations to a background
cosmological model, governed by an evolution equation that follows
from EFE. Yet as the fluctuations grow larger, linearized perturbation
theory no longer applies. According to the SM, structure grows
hierarchically with smaller length scales going non-linear first, and
larger structures forming via later mergers. Models of evolution of
structures at smaller length scales (e.g., the length scales of
galaxies) include physics other than gravity, such as gas dynamics, to
describe the collapsing clumps of matter. Cold dark matter (CDM) also
plays a crucial role in the SM's account of structure formation:
it clumps first, providing scaffolding for clumping of baryonic
matter.



A full account of structure formation requires integrating physics
over an enormous range of dynamical scales and including a
cosmological constant as well as baryonic matter, radiation, and dark
matter. This is an active area of research, primarily pursued using
sophisticated \(N\)-body computer simulations to study features of the
galaxy distribution produced by the SM, given various
 assumptions.[9]



### 1.2 Observations



There are two main ways in which cosmological observations support
perturbed FLRW models. First, cosmologists use matter and radiation in
the universe to probe the background spacetime geometry and its
evolution. The universe appears to be isotropic at sufficiently large
scales, as indicated by background radiation (most notably the cosmic
microwave background radiation (CMB), discussed below) and discrete
sources (e.g., galaxies). Isotropy observed along a single worldline
is, however, not sufficient to establish the universe is well
described by an FLRW geometry. A further assumption that our worldline
is not the *only* vantage point from which the universe appears
isotropic, often called the Copernican principle, is needed. Granting
this principle, there are theorems establishing that observations of
almost isotropic background radiation implies that the spacetime
geometry is almost
 FLRW.[10]
 The principle itself cannot be established directly via observations
(see
 SS2).
 Given that we live in an almost FLRW models, we need to determine its
parameters such as the Hubble constant \(H\_0\) and the deceleration
parameter \(q\_0 \coloneqq {-}\ddot{R}/(RH\_0^2)\), which measures how
the rate of expansion is changing, and the normalized density
parameters \(\Omega\_m\coloneqq \rho\_m/(3H\_0^2)\) for each matter or
energy density component \(m\). There are a variety of ways to
determine the accuracy of the background evolution described by the
FLRW models, which depends on these parameters. For this purpose,
cosmologists seek effective standard candles and standard
rulers--objects with a known intrinsic luminosity and length,
respectively, which can then be used to measure the expansion history
of the universe.



The second main avenue of testing focuses on the SM's account of
structure formation, which describes the evolution of small
perturbations away from the background FLRW geometry in terms of a
small number of parameters such as the tilt \(n\_s\) and the scalar to
tensor ratio \(r\). Observations from different epochs, such as
temperature anisotropies in the CMB and the matter power spectrum
based on galaxy surveys, can be used as independent constraints on
these parameters as well as on the background parameters (indeed such
observations turn out to give the best constraints on the background
model parameters). These two routes to testing almost FLRW spacetime
geometry are closely linked because the background model provides the
context for the evolution of perturbations under the dynamics
described by general relativity.



The remarkable success of perturbed FLRW models in describing the
observed universe has led many cosmologists to focus almost
exclusively on them, yet there are drawbacks to such a myopic
approach. For example, the observations at best establish that the
observed universe can be well-approximated by an almost FLRW model
within some (large) domain. But they are not the only models that fit
the data: there are other cosmological models that mimic FLRW models
in the relevant domain, yet differ dramatically elsewhere (and
elsewhen). Specifically, on the one hand there are a class of
spatially homogeneous and anisotropic models (Bianchi models) that
exhibit "intermediate isotropization": namely, they have
physical properties that are arbitrarily close to (isotropic) FLRW
models over some time scale
 \(T\).[11]
 Agreement over the time interval \(T\) does not imply global
agreement, however, as these models have large anisotropies at other
times. Relying on the FLRW models in making extrapolations to the
early or late universe requires some justification for ignoring
models, such as these Bianchi models, that mimic their behavior for a
finite time interval. On the other hand there are inhomogeneous
spherically symmetric models that can reproduce exactly the background
model observations (number counts versus redshifts and angular
diameter distance versus redshift, for example) with or without a
cosmological constant (Mustapha et al. 1997). These can be excluded by
direct observations with good enough standard candles (Clarkson et al.
2008) or by observations of structure formation features in such
universes (Clarkson & Maartens 2010); but that exclusion cannot
take place unless one indeed examines such models and their
observational consequences.



Lack of knowledge of the full space of solutions to EFE makes it
difficult to assess the fragility of various inferences cosmologists
make based on perturbed FLRW models. A fragile inference depends on
the properties of the model holding exactly, contrasted with robust
inferences that hold even if the models are good approximations (up to
some tolerable error) that will hold even if the model is perturbed.
The singularity theorems (Hawking & Ellis 1973), for example,
establish that the existence of an initial singularity is robust:
rather than being features specific to the FLRW models, or other
highly symmetric models, singularities are generic in models
satisfying physically plausible assumptions. The status of various
other inferences cosmologists make is less clear. For example, how
sensitively does the observational case in favor of dark energy, which
contributes roughly 70% of the total energy density of the universe in
the SM, depend upon treating the universe as having almost-FLRW
spacetime geometry? As mentioned above, recent work has pursued the
possibility of accounting for the same observations based upon
large-scale inhomogeneities or local back-reaction, without recourse
to dark
 energy.[12]
 Studies along these lines are needed to evaluate the possibility that
subtle dynamical effects, absent in the FLRW models, provide
alternative explanations of observed phenomena. The deduction also
depends on the assumption that the EFE hold at cosmological scales -
which may not be true: maybe for example some form of scalar-tensor
theory should be used. More generally, an assessment of the
reliability of a variety of cosmological inferences requires detailed
study of a larger space of cosmological models.


### 1.3 Historical Epochs



The SM's account of the evolution of the matter and radiation in
the universe reflects the dynamical effect of expansion. Consider a
cube of spacetime in the early universe, filled with matter and
radiation. The dynamical effects of the universe's expansion are
locally the same as slowly stretching the cube. For some stages of
evolution the contents of the cube interact sufficiently quickly that
they reach and stay in local thermal equilibrium as the cube changes
volume. (Because of isotropy, equal amounts of matter and radiation
enter and leave the cube from neighboring cubes.) But when the
interactions are too slow compared to the rate of expansion, the cube
changes volume too rapidly for equilibrium to be maintained. As a
result, particle species "freeze out" and decouple, and
entropy increases. Without a series of departures from equilibrium,
cosmology would be boring--the system would remain in equilibrium
with a state determined solely by the temperature, without a trace of
things past. The rate of expansion of the cube varies with cosmic
time. Because radiation, matter, and a cosmological constant term (or
dark energy) dilute with expansion at different rates, an expanding
universe naturally falls into separate epochs, characterized by
different expansion rates.



There are several distinctive epochs in the history of the universe,
according to the SM, including the following:


* *Quantum gravity*: Classical general relativity is
expected to fail at early times, when quantum effects will be crucial
in describing the gravitational degrees of freedom. There is
considerable uncertainty regarding physics at this scale.
* *Inflation*: A period of exponential, quasi-De Sitter
expansion driven by an "inflaton" field (or fields),
leading to a uniform, almost flat universe with Gaussian linear nearly
scale invariant density perturbations. During inflation pre-existing
matter and radiation are rapidly diluted; the universe is repopulated
with matter and energy by the decay of the inflaton field into other
fields at the end of inflation ("re-heating").
* *Big Bang Nucleosynthesis*: At \(t \approx 1\) second, the
constituents of the universe include neutrons, protons, electrons,
photons, and neutrinos, tightly coupled and in local thermal
equilibrium. Synthesis of light elements occurs during a burst of
nuclear interactions that transpire as the universe falls from a
temperature of roughly \(10^9\) K to \(10^8\) K after neutrinos fall
out of equilibrium and consequent onset of neutron decay. The
predicted light-element abundances depend on physical features of the
universe at this time, such as the total density of baryonic matter
and the baryon to photon ratio. Agreement between theory and
observation for a specific baryon to photon ratio (Steigman 2007) is a
great success of the SM.
* *Decoupling*: As the temperature drops below \(\approx
4,000 K\), electrons become bound in stable atoms, and photons
decouple from the matter with a black-body spectrum. With the
expansion of the universe, the photons cool adiabatically but retain a
black-body spectrum with a temperature \(T \propto 1/R\). This
"cosmic background radiation" (CBR) has been aptly called
the cosmic Rosetta stone because it carries so much information about
the state of the universe at decoupling (Ade et al. 2016).
* *Dark Ages*: After decoupling, baryonic matter consists
almost entirely of neutral hydrogen and helium. Once the first
generation of stars form, the dark ages come to an end with light from
the stars, which re-ionizes the universe.
* *Structure Formation*: Cold dark matter dominates the
early stages of the formation of structure. Dark matter halos provide
the scaffolding for hierarchical structure formation. The first
generation of stars aggregate into galaxies, and galaxies into
clusters. Massive stars end their lives in supernova explosions and
spread through space heavy elements that have been created in their
interiors, enabling formation of second generation stars surrounded by
planets.
* *Dark Energy Domination*: Dark energy (or a non-zero
cosmological constant) eventually comes to dominate the expansion of
the universe, leading to accelerated
 expansion.[13]
 This expansion will be never-ending if the dark energy is in fact a
cosmological constant.


### 1.4 Status of the Standard Model



The development of a precise cosmological model compatible with the
rich set of cosmological data currently available is an impressive
achievement. Cosmology clearly relies very heavily on theory; the
cosmological parameters that have been the target of observational
campaigns are only defined given a background model. The strongest
case for accepting the SM rests on the evidence in favor of the
underlying physics, in concert with the overdetermination of
cosmological parameters. The SM includes several free parameters, such
as the density parameters characterizing the abundance of different
types of matter, each of which can be measured several
 ways.[14]
 These methods have distinctive theoretical assumptions and sources of
error. For example, the abundance of deuterium produced during big
bang nucleosynthesis depends sensitively on the baryon density.
Nucleosynthesis is described using well-tested nuclear physics, and
the light element abundances are frozen in within the "first
three minutes". The amplitudes of the acoustic peaks in the CMB
angular power spectrum depend on the baryon density at the time of
decoupling. Current measurements fix the baryon density to an accuracy
of one percent, and the values determined by these two methods agree
within observational error. This agreement is one of many consistency
checks for the
 SM.[15]
 There are important discrepancies, such as that between local versus
global measurements of the Hubble parameter \(H\_0\) (Lukovic et
al. 2016; Bernal et al. 2016). The significance and further
implications of these discrepancies is not clear.



The SM from nucleosynthesis on can be regarded as well supported by
many lines of evidence. The independence and diversity of the
measurements provides some assurance that the SM will not be
undermined by isolated theoretical mistakes or undetected sources of
systematic error. But the SM is far from complete, and there are three
different types of significant open issues.



First, we do not understand three crucial components of the SM that
require new physics. We do not have a full account of the nature, or
underlying dynamics, of dark matter (Bertone et al. 2005), dark energy
(Peebles & Ratra 2003), or the inflaton field (Lyth & Riotto
1999; Martin et al. 2014). These are well-recognized problems that
have inspired active theoretical and observational work, although as
we note below in
 SS2.4
 they will be challenging to resolve due to inaccessibility of physics
at the appropriate scale.



The second set of open questions regards structure formation. While
the account of structure formation matches several significant
observed features, such as the correlations among galaxies in large
scale surveys, there are a number of open questions about how galaxies
form (Silk 2017). Many of these, such as the cusp-core problem
(Weinberg et al. 2015), and the dark halos problem (a great many more
small dark halos are predicted around galaxies than observed) regard
features of galaxies on relatively small scales, which require
detailed modeling of a variety of astrophysical processes over an
enormous dynamical range. This is also a very active area of research,
driven in particular by a variety of new lines of observational
research and large-scale numerical simulations.



The third and final set of open issues regards possible observations
that would show that the SM is substantially wrong. Any scientific
theory should be incompatible with at least some observations, and
that is the case for the SM. In the early days of relativistic
cosmology, the universe was judged to be younger than some stars or
globular clusters. This conflict arose due to a mistaken value of the
Hubble constant. There is currently no such age problem for the SM,
but obviously discovering an object older than 13.7 Gyr would force a
major re-evaluation of current cosmological models. Another example
would be if there was not a dipole in matter number counts that agrees
with the CMB dipole (Ellis & Baldwin 1984).


### 1.5 Local vs. Global Interplay in Cosmology



Although cosmology is generally seen as fitting into the general
physics paradigm of everything being determined in a bottom up manner,
as in the discussion above, there is another tradition that sees the
effect of the global on the local in cosmology.



The traditional issues of this kind (Bondi 1960; Ellis & Sciama
1972; Ellis 2002) are


* *Mach's Principle*: the idea that the origin of
inertia is due to the very distant matter in the universe (Barbour
& Pfister 1995), nowadays understood as being due to the fact that
the vorticity \(\omega\) of the universe is very low at present (it
could have been otherwise);
* *Olber's Paradox*: the issue of why the sky is dark
at night (Harrison 1984), resolved by evolution of the universe
together with the redshift factor of about 1000 since the surface of
last scattering (which determines that the temperature of the night
sky is the 2.73K of the CMB everywhere except for the small fraction
of the sky covered by stars and galaxies)
* *The Arrow of Time*: where does the arrow of time come
from, if the underlying physics is time symmetric? This has to be due
to special initial conditions at the start of the universe (Ellis
2007). This is related to the Sommerfeld outgoing radiation condition
and Penrose's Weyl curvature hypothesis (Penrose 2016).



In each case, global boundary conditions have an important effect on
local physics. More recent ones relate to


* *Nucleosynthesis*, where the course of nuclear reactions is
determined by the \(T(t)\) relation that is controlled by cosmological
evolution (Steigman 2007) (the temperature \(T\) being a coarse
grained variable with evolution determined by the average density
\(\rho\) of matter in the universe through the Friedmann
equation)
* *Structure formation* due to gravitational instability
(Mukhanov et al. 1992), which is affected crucially by the expansion
of the universe, which turns what would have been an exponential
growth of inhomogeneity(in a static universe) to a power law growth.
It is because of this effect that studies of structure such as the BAO
and CMB anisotropies give us strong limits on the parameters of the
background model (Ade et al. 2016).
* *The Anthropic Principle*, discussed below
 (SS4.1),
 whereby large-scale conditions in the universe (such as the value of
the cosmological constant and the initial amplitude of inhomogeneities
in the early universe) provide local conditions suitable for life to
come into being.



Relevant to all this is the idea of an "effective
horizon": the domain that has direct impact on structures
existing on the Earth, roughly 1 Mpc co-moving sphere, see Ellis &
Stoeger 2009. This is the part of the universe that actually has a
significant effect on our history.


## 2. Underdetermination



Many philosophers hold that evidence is not sufficient to determine
which scientific theory we should choose. Scientific theories make
claims about the natural world that extend far beyond what can be
directly established through observations or experiments. Rival
theories may fare equally well with regard to some body of data, yet
give quite different accounts of the world. Philosophers often treat
the existence of such rivals as inevitable: for a given theory, it is
always possible to construct rival theories that have "equally
good fit" with available data. Duhem (1914 [1954]) gave an
influential characterization of the difficulty in establishing
physical theories conclusively, followed a half century later by
Quine's arguments for a strikingly general version of
underdetermination (e.g., Quine 1970). The nature of this proposed
underdetermination of theory by evidence, and appropriate responses to
it, have been central topics in philosophy of science (Stanford 2009
[2016]). Although philosophers have identified a variety of distinct
senses of underdetermination, they have generally agreed that
underdetermination poses a challenge to justifying scientific
theories.



There is a striking contrast with discussions of underdetermination
among scientists, who often emphasize instead the enormous difficulty
in constructing compelling rival
 theories.[16]
 This contrast reflects a disagreement regarding how to characterize
the empirical content of theories. Suppose that the empirical content
of theory consists of a set of observational claims implied by the
theory. Philosophers then take the existence of rival theories to be
straightforward. Van Fraassen (1980), for example, defines a theory as
"empirically adequate" if what it says about observable
phenomena is true, and argues that for any successful theory there are
rival theories that disagree about theoretical claims. If we demand
more of theories than empirical adequacy in this sense, it is possible
to draw distinctions among theories that philosophers would regard as
underdetermined. Furthermore, even when scientists do face a choice
among competing theories, they are almost never rivals in the
philosopher's sense. Instead, they differ in various ways:
intended domain of applicability, explanatory scope, importance
attributed to particular problems, and so on.



The scientists' relatively dismissive attitude towards alleged
underdetermination threats may be based on a more demanding conception
of empirical
 success.[17]
 Scientists demand much more of their theories than mere compatibility
with some set of observational claims: they must fit into a larger
explanatory scheme, and be compatible with other successful theories.
Given a more stringent account of empirical success it is much more
challenging to find rival theories. (We return to this issue in
 SS5
 below.)



One aspect of underdetermination (emphasized by Stanford 2006) is of
more direct relevance to scientific debates: current theories may be
indistinguishable, within a restricted domain, from a successor
theory, even though the successor theory makes different predictions
for other domains. This raises the question of how far we can rely on
extrapolating a theory to a new domain. For example, despite its
success in describing objects moving with low relative velocities in a
weak gravitational field, where it is nearly indistinguishable from
general relativity, Newtonian gravity does not apply to other regimes.
How far, then, can we rely on a theory to extend our reach? The
obstacles to making such reliable inferences reflect the specific
details of particular domains of inquiry. Below we will focus on the
obstacles to answering theoretical questions in cosmology due to the
structure of the universe and our limited access to phenomena.


### 2.1 Underdetermination in Cosmology



Given the grand scope of cosmology, one might expect that many
questions must remain unresolved. Basic features of the SM impose two
fundamental limits to the ambitions of cosmological theorizing. First,
the finitude of the speed of light ensures that we have a limited
observational window on the universe due to existence of the visual
horizon, representing the most distant matter from which we can
receive and information by electromagnetic radiation, and the particle
horizon, representing the most distant matter with which we can have
had any causal interaction (matter up to that distance can influence
what we see at the visual horizon). Recent work has precisely
characterized what can be established via idealized astronomical
observations, regarding spacetime geometry within, or outside, our
past light cone (the observationally accessible region). Second, in
addition to enormous extrapolations of well-tested physics in the SM,
cosmologists have explored speculative ideas in physics that can only
be tested through their implications for cosmology; the energies
involved are too high to be tested by any accelerator on Earth. Ellis
(2007) has characterized these speculative aspects of cosmology as
falling on the far side of a "physics horizon". We will
briefly discuss how this second type of horizon poses limits for
cosmological theorizing. In both cases, the type of underdetermination
that arises differs from that discussed in the philosophical
literature.


### 2.2 Global Structure



To what extent can observations determine the spacetime geometry of
the universe directly? The question can be posed more precisely in
terms of the region that is, in principle, accessible to an observer
at a location in spacetime \(p\)--the *causal past*,
\(J^-(p)\), of that point. This set includes all regions of spacetime
from which signals traveling at or below the speed of light can reach
\(p\). What can observations confined to \(J^-(p)\), assuming that GR
is valid, reveal about the spacetime geometry of \(J^-(p)\) itself,
and the rest of spacetime?



The observational cosmology program (Kristian & Sachs 1966; Ellis
et al. 1985) clarifies the extent to which a set of ideal observations
can determine the spacetime geometry directly with minimal
cosmological assumptions. (By contrast, the standard approach starts
by assuming a background cosmological model and then finding an
optimal parameter fit.) Roughly put, the ideal data set consists of a
set of astrophysical objects that can be used as standard candles and
standard rulers. If the intrinsic properties and evolution of a
variety of sources are given, observations can directly determine the
area (or luminosity) distance of the sources, and the distortion of
distant images determines lensing effects. These observations thus
directly constrain the spacetime geometry of the past light cone
\(C^-(p)\). Number counts of discrete sources (such as galaxies or
clusters) can be used to infer the total amount of baryonic matter,
again granting various assumptions. Ellis et al. (1985) proved the
remarkable result that an appropriate idealized data set of this kind
is sufficient, if we grant that EFE hold, to fully fix the spacetime
geometry and distribution of matter on the past light cone \(C^-(p)\),
and from that, in the causal past \(J^-(p)\) of the observation point
\(p\).[18]
Observers do not have access to anything like the ideal data set,
obviously, and in practice cosmologists face challenges in
understanding the nature of sources and their evolution with
sufficient clarity that they can be used to determine spacetime
geometry, so this is the ideal situation.



What does \(J^-(p)\) reveal about the rest of spacetime? In classical
GR, we would not expect the physical state on \(J^-(p)\) to determine
that of other regions of spacetime--even the causal past of a
point just to the future of 
\(p\).[19] 
There are some models in which \(J^-(p)\)
does reveal more: "small universe" models are closed
models with a finite maximum length in all directions that is smaller
than the visual horizon (Ellis & Schreiber 1986). Observers in
such a model would be able to "see around the universe" in
all directions, and establish some global properties via direct
observation because they would be able to see all matter that
exists.[20]




Unless this is the case, the causal past for a single observer, and
even a collection of causal pasts, place very weak constraints on the
global properties of spacetime. The global properties of a spacetime
characterize its causal structure, such as the presence or absence of
 singularities.[21]
 General relativity tolerates a wide variety of global properties,
since EFE impose only a local constraint on the spacetime geometry.
One way to make this question precise is to consider whether there are
any global properties shared by spacetimes that are constructed as
follows. For a given spacetime, construct an indistinguishable
counterpart that includes the collection of causal pasts
\(\{J^-(p)\}\) for all points in the original spacetime. The
constructed spacetime is indistinguishable from the first, because for
any observer in the first spacetime there is a "copy" of
their causal past in the counterpart. It is possible, however, to
construct counterparts that do not have the same global properties as
the original spacetime. The property of having a Cauchy surface, for
example, need not be shared by an indistinguishable
 counterpart.[22]
 More generally, the only properties that are guaranteed to hold for
an indistinguishable counterpart are those that can be established
based on the causal past of a single point. This line of work
establishes that (some) global properties cannot be established
observationally, and raises the question of whether there are
alternative justifications.


### 2.3 Establishing FLRW Geometry?



The case of global spacetime geometry is not a typical instance of
underdetermination of theory by evidence, as discussed by
philosophers, for two reasons (see Manchak 2009, Norton 2011,
Butterfield 2014). First, this whole discussion assumes that classical
GR holds; the question regards discriminating among models of a given
theory, rather than a choice among competing theories. Second, these
results establish that all observations available to us that are
compatible with a given spacetime, with some appealing global
property, are equally compatible with its indistinguishable
counterparts. But as is familiar from more prosaic examples of the
problem of induction, evidence of past events is compatible, in a
similar sense, with many possible futures. Standard accounts of
inductive inference aim to justify some expectations about the future
as more reasonable, e.g., those based on extending past uniformities.
The challenge in this case is to articulate an account of inductive
inferences that justifies accepting one spacetime over its
indistinguishable counterparts.



As a specific instance of this challenge, consider the status of the
cosmological principle, the global symmetry assumed in the derivation
of the FLRW models. The results above show that all evidence available
to us is equally compatible with models in which the cosmological
principle does or does not hold. One might take the principle as
holding *a priori*, or as a pre-condition for cosmological
theorizing (Beisbart 2009). A recent line of work aims to justify the
FLRW models by appealing to a weaker general principle in conjunction
with theorems relating homogeneity and isotropy. Global isotropy
around every point implies global homogeneity, and it is natural to
seek a similar theorem with a weaker antecedent formulated in terms of
observable quantities. The Ehlers-Geren-Sachs theorem (Ehlers et al.
1968) shows that if all geodesic fundamental observers in an expanding
model find that freely propagating background radiation is exactly
isotropic, then their spacetime is an FLRW model. If our causal past
is "typical", observations along our worldline will
constrain what other observers should see. This is often called the
Copernican principle--namely, no point \(p\) is distinguished
from other points \(q\) by any spacetime symmetries or lack thereof
(there are no "special locations"). There are indirect
ways of testing this principle empirically: the
Sunyaev-Zel'dovich effect can be used to indirectly measure the
isotropy of the CBR as observed from distant points. Other tests are
direct tests with a good enough set of standard candles, and an
indirect test based on the time drift of cosmological redshift. This
line of work provides an empirical argument that the observed universe
is well-approximated by an FLRW model, thus changing that assumption
from a philosophically based starting point to an observationally
tested foundation.


### 2.4 Physics Horizon



The Standard Model of particle physics and classical GR provide the
structure and framework for the SM. But cosmologists have pursued a
variety of questions that extend beyond these core theories. In these
domains, cosmologists face a form of underdetermination: should a
phenomena be accounted for by extending the core theories, or by
changing physical or astrophysical assumptions?



The Soviet physicist Yakov Zel'dovich memorably called the early
universe the "poor man's accelerator", because
relatively inexpensive observations of the early universe may reveal
features of high-energy physics well beyond the reach of even the most
lavishly funded earth-bound accelerators. For many aspects of
fundamental physics, including quantum gravity in particular,
cosmology provides the only feasible way to assess competing ideas.
This ambitious conception of cosmology as the sole testing ground for
new physics extends beyond the standard model of particle physics
(which is generally thought to be incomplete, even though there are no
observations that contradict it). Big bang nucleosynthesis, for
example, is an application of well-tested nuclear physics to the early
universe, with scattering cross-sections and other relevant features
of the physics fixed by terrestrial experiments. While working out how
nuclear physics applied in detail required substantial effort, there
was little uncertainty regarding the underlying physics. By contrast,
in some domains cosmologists now aim to explain the universe's
history while at the same time evaluating new physics used in
constructing it.



This contrast can be clarified in terms of the "physics
horizon" (Ellis 2007), which delimits the physical regime
accessible to terrestrial experiments and observations, roughly in
terms of energy scales associated with different interactions. The
horizon can be characterized more precisely for a chosen theory, by
specifying the regions of parameter space that can be directly tested
by experiments and
 observations.[23]
 Aspects of cosmological theories that extend past the physics horizon
cannot be independently tested through non-cosmological experiments or
observations; the only empirical route to evaluating these ideas is
through their implications for cosmology. (This is not to deny that
there may be strong theoretical grounds to favor particular proposals,
as extensions of the core theories.)



Cosmological physics extending beyond the physics horizon faces an
underdetermination threat due to the lack of independent lines of
relevant evidence. The case of dark matter illustrates the value of
such independent evidence. Dark matter was first proposed to account
for the dynamical behavior of galaxy clusters and galaxies, which
could not be explained using Newtonian gravitational theory with only
the luminous matter observed. Dark matter also plays a crucial role in
accounts of structure formation, as it provides the scaffolding
necessary for baryonic matter to clump, without conflicting with the
uniformity of the
 CMB.[24]
 Both inferences to the existence of dark matter rely on gravitational
physics, raising the question of whether we should take these
phenomena as evidence that our gravitational theory fails, rather than
as evidence for a new type of matter. There is an active research
program (MOND, for *Mo*dified *N*ewtonian
*D*ynamics) devoted to accounting for the relevant phenomena by
modifying gravity. Regardless of one's stance on the relative
merits of MOND vs. dark matter (obviously MOND needs to be extended to
a relativistic theory), direct evidence of existence of dark matter,
or indirect evidence via decay products, would certainly reshape the
debate. Efforts have been underway for some time to find dark matter
particles through direct interactions with a detector, mediated by the
weak force. A positive outcome of these experiments would provide
evidence of the existence of dark matter that does not depend upon
gravitational
 theory.[25]




Such independent evidence is not available for two prominent examples
of new physics motivated by discoveries in cosmology. "Dark
energy" was introduced in studies of structure formation, which
employed a non-zero cosmological constant to fit observational
constraints (the \(\Lambda\)CDM models). Subsequent observations of
the redshift-distance relation, using supernovae (type Ia) as a
standard candle, led to the discovery that the expansion of the
universe is
 accelerating.[26]
 (For \(\ddot{R}>0\) in an FLRW model, there must be a contribution
that appears in eqn. (\(\ref{eq:Ray}\)) like a positive \(\Lambda\)
term.) Rather than treating these observations as simply determining
the value of a parameter in the SM, many cosmologists have developed
phenomenological models of "dark energy" that leads to an
effective \(\Lambda\). Unlike dark matter, however, the properties of
dark energy insure that any attempt at non-cosmological detection
would be futile: the energy density is so small, and uniform, that any
local experimental study of its properties is practically impossible.
Furthermore these models are not based in well-motivated physics: they
have the nature of 'saving the phenomena' in that they are
tailored to fitting the cosmological observations by curve
 fitting.[27]




Inflationary cosmology originally promised a powerful unification of
particle physics and cosmology. The earliest inflationary models
explored the consequences of specific scalar fields introduced in
particle physics (the then supposed Higgs field for the strong
interactions). Yet theory soon shifted to treating the scalar field
responsible for inflation as the "inflaton" field, leaving
its relationship to particle physics unresolved, and the promise of
unification unfulfilled. If the properties of the inflaton field are
unconstrained, inflationary cosmology is extremely flexible; it is
possible to construct an inflationary model that matches any chosen
evolutionary history of the early
 universe.[28]
 Specific models of inflation, insofar as they specify the features of
the field or fields driving inflation and its initial state, do have
predictive content. In principle, cosmological observations could
determine some of the properties of the inflaton field and so select
among them (Martin et al. 2014). This could in principle then have
implications for a variety of other experiments or observations; yet
in practice the features of the inflaton field in most viable models
of inflation guarantee that it cannot be detected in other regimes.
The one exception to this is if the inflaton were the electroweak
Higgs particle detected at the LHC (Ellis & Uzan 2014). This
remains a viable inflaton candidate, so testing if it is indeed the
inflaton is an important task (Bezrukov & Gorbunov 2012).



The physics horizon poses a challenge because one particularly
powerful type of evidence--direct experimental detection or
observation, with no dependence on cosmological assumptions--is
unavailable for the physics relevant at earliest times (before
inflation, and indeed even for baryosynthesis after inflation). Yet
this does not imply that competing theories, such as dark matter vs.
modified gravity, should be given equal credence. The case in favor of
dark matter draws on diverse phenomena, and it has been difficult to
produce a compelling modified theory of gravity, consistent with GR,
that captures the full range of phenomena as an alternative to dark
matter. Cosmology typically demands a more intricate assessment of
background assumptions, and the degree of independence of different
tests, in evaluating proposed extensions of the core theories. Yet
this evidence may still be sufficiently strong, in the sense discussed
more fully in
 SS5
 below, to justify new physics.


### 2.5 Cosmic Variance



There is a distinctive form of underdetermination regarding the use of
statistics in cosmology, due to the uniqueness of the universe. To
compare the universe with the statistical predictions of the SM, we
conceptualize it as one realization of a family of possible universes,
and compare what we actually measure with what is predicted to occur
in the ensemble of hypothetical models. When they are significantly
different, the key issue is: Are these just statistical fluctuations
we can ignore? Or are they serious anomalies that need an
explanation?



This question arises in several concrete cases:


* Existence of low CMB anisotropy power at high and angular scales
relative to that predicted by the SM (Schwarz et al. 2016; Knight
& Knox 2017)
* Existence of a CMB cold spot of substantial size (Zhang &
Huterer 2010; Schwarz et al. 2016).
* Disagreement about the value of the Hubble parameter as measured
directly in the local region on the one hand, and as deduced from CMB
anisotropies on the other (Lukovic et al. 2016; Bernal et al.
2016).



How do we decide? This will depend on the particular measurement (see
e.g., Kamionkowski & Loeb 1997; Marra et al. 2013), but in general
because of the uniqueness of the universe, we don't know if
these potential anomalies are real, pointing to serious problems with
the models, or not real--just statistical flukes in the way the
family of models differs from the one instance that we have at hand,
the unique universe that actually exists. In all the physical
sciences, this is a unique problem of
 cosmology.[29]



## 3. Origins of the Universe



Cosmology confronts a distinctive challenge in accounting for the
origin of the universe. In most other branches of physics the initial
or boundary conditions of a system do not call out for theoretical
explanation. They may reflect, for example, the impact of the
environment, or an arbitrary choice regarding when to cut off the
description of a subsystem of interest. But in cosmology there are
heated debates regarding what form a "theory of the initial
state" should take, and what it should contribute to our
understanding of the universe. This basic question regarding the
nature of aims of a theory of origins has significant ramifications
for various lines of research in cosmology.


### 3.1 The Initial State



Contemporary cosmology at least has a clear target for a theory of
origins: the SM describes the universe as having expanded and evolved
over 13.7 billion years from an initial state where many physical
quantities diverged. In the FLRW models, the cosmic time \(t\) can be
measured by the total proper time elapsed along the worldline of a
fundamental observer, from the "origin" of the universe
until the present epoch. Extrapolating backwards from the present,
various quantities diverge as the cosmic time \(t \rightarrow
0\)--for example, \(R(t) \rightarrow 0\) and the matter density
goes to
 infinity.[30]
 The worldlines of observers cannot be extended arbitrarily far into
the past. Although there is no "first moment" of time,
because the very concept of time breaks down as \(t\rightarrow 0\),
the age of the universe is the maximum length of these worldlines.


### 3.2 Singularity Theorems



The singularity theorems proved in the 60s (see, in particular,
Hawking & Ellis 1973) show that the universe is finite to the past
in a broad class of cosmological models. Past singularities, signaled
by the existence of inextendible geodesics with bounded length, must
be present in models with a number of plausible features. (Geodesics
are the curves of extreme length through curved spacetime, and freely
falling bodies follow timelike geodesics.) Intuitively, extrapolating
backwards from the present, an inextendible geodesic reaches, within
finite distance, an "edge" beyond which it cannot be
extended. There is not a uniquely defined "cosmic time",
in general, but the maximum length of these curves reflects the finite
age of the universe. The singularity theorems plausibly apply to the
observed universe, within the domain of applicability of general
relativity. There are various related theorems differing in detail,
but one common ingredient is an assumption that there is sufficient
matter and energy present to guarantee that our past light cone
 refocuses.[31]
 The energy density of the CMB alone is sufficient to justify this
assumption. The theorems also require an energy condition: a
restriction on the types of matter present in the model, guaranteeing
that gravity leads to focusing of nearby geodesics. (In eqn.
(\(\ref{eq:Ray}\)) above, this is the case if \(\rho\_{\textit{grav}}
> 0\) and \(\Lambda=0\); it is possible to avoid a singularity with
a non-zero cosmological constant, for example, since it appears with
the opposite sign as ordinary matter, counteracting this focusing
effect.)



The prediction of singularities is usually taken to be a deep flaw of
 GR.[32]
 One potential problem with singularities is that they may lead to
failures of determinism, because the laws "break down" in
some sense. This concern only applies to some kinds of singularities,
however. Relativistic spacetimes that are globally hyperbolic have
Cauchy surfaces, and appropriate initial data posed on such surfaces
fix a unique solution throughout the spacetime. Global hyperbolicity
does not rule out the existence of singularities, and in particular
the FLRW models are globally hyperbolic in spite of the existence of
an initial singularity. The threat to determinism is thus more
qualified: the laws do not apply "at the singularity
itself" even though the subsequent evolution is fully
deterministic, and there are some types of singularities that pose
more serious threats to determinism.



Another common claim is that the presence of singularities establish
that GR is incomplete, since it fails to describe physics "at
the
 singularity".[33]
 This is difficult to spell out fully without a local analysis of
singularities, which would give precise meaning to talk of
"approaching" or being "near" the singularity.
In any case, it is clear that the presence of a singularity in a
cosmological model indicates that spacetime, as described by GR, comes
to an end: there is no way of extending the spacetime through the
singularity, without violating mathematical conditions needed to
insure that the field equations are well-defined. Any description of
physical conditions "before the big bang" must be based on
a theory that supersedes GR, and allows for an extension through the
singularity.



There are two limitations regarding what we can learn about the
origins of the universe based on the singularity theorems. First,
although these results establish the existence of an initial
singularity, they do not provide much guidance regarding its
structure. The spacetime structure near a "generic"
initial singularity has not yet been fully characterized. Partial
results have been established for restricted classes of solutions; for
example, numerical simulations and a number of theorems support the
BKL conjecture, which holds that isotropic, inhomogeneous models
exhibit a complicated form of chaotic, oscillatory behavior. The
resulting picture of the approach to the initial singularity contrasts
sharply with that in FLRW
 models.[34]
 It is also possible to have non-scalar singularities (Ellis &
King 1974).



Second, classical general relativity does not include quantum effects,
which are expected to be relevant as the singularity is approached.
Crucial assumptions of the singularity theorems may not hold once
quantum effects are taken into account. The standard energy conditions
do not hold for quantum fields, which can have negative energy
densities. This opens up the possibility that a model including
quantum fields may exhibit a "bounce" rather than collapse
to a singularity. More fundamentally, GR's classical spacetime
description may fail to approximate the description provided by a full
theory of quantum gravity. According to recent work applying loop
quantum gravity to cosmology, spacetime collapses to a minimum finite
size rather than reaching a true singularity (Ashtekar & Singh
2011; Bojowald 2011). On this account, GR fails to provide a good
approximation in the region of the bounce, and the apparent
singularity is an artifact. Classical spacetime "emerges"
from a state to which familiar spacetime concepts do not apply. There
are several accounts of the early universe, motivated by string theory
and other approaches, that similarly avoid the initial singularity due
to quantum gravity effects.


### 3.3 Puzzling Features of the Initial State



In practice, cosmologists often take the physical state at the
expected boundary of the domain of applicability of GR as the
"initial state". (For example, this might be taken as the
state specified on a spatial hypersurface at a very early cosmic time.
However, the domain of applicability of GR is not well understood,
given uncertainty about quantum gravity.) Projecting observed features
of the universe backwards leads to an initial state with three
puzzling
 features:[35]



* *Uniformity*: The FLRW models have a finite particle
horizon distance, much smaller than the scales at which we observe the
 CMB.[36]
 Yet the isotropy of the CMB, among other observations, indicate that
distant regions of the universe have uniform physical properties.
* *Flatness*: An FLRW model close to the "flat"
model, with nearly critical density at some specified early time is
driven rapidly away from critical density under FLRW dynamics if
\(\Lambda = 0\) and \(\rho+3p>0\). Given later observations, the
initial state has to be *very* close to the flat model (or,
equivalently, *very* close to critical density, \(\Omega=1\))
at very early
 times.[37]
* *Perturbations*: The SM includes density perturbations that
are coherent on large scales and have a specific amplitude,
constrained by observations. It is challenging to explain both
properties dynamically. In the standard FLRW models, the perturbations
have to be coherent on scales much larger than the Hubble radius at
early
 times.[38]



On a more phenomenological approach, the gravitational degrees of
freedom of the initial state could simply be chosen to fit with later
observations, but many proposed "theories of initial
conditions" aim to account for these features based on new
physical principles. The theory of inflation discussed below aims to
explain these issues.


### 3.4 Theories of the Initial State



There are three main approaches to theories of the initial state, all
of which have been pursued by cosmologists since the late 60s in
different forms. Expectations for what a theory of initial conditions
should achieve have been shaped, in particular, by inflationary
cosmology. Inflation provided a natural account of the three otherwise
puzzling features of the initial state emphasized in the previous
section. Prior to inflation, these features were regarded as
"enigmas" (Dicke & Peebles 1979), but after inflation,
accounting for these features has served as an eligibility requirement
for any proposed theory of the early universe.



The first approach aims to reduce dependence on special initial
conditions by introducing a phase of attractor dynamics. This phase of
dynamical evolution "washes away" the traces of earlier
states, in the sense that a probability distribution assigned over
initial states converges towards an equilibrium distribution. Misner
(1968) introduced a version of this approach (his "chaotic
cosmology program"), proposing that free-streaming neutrinos
could isotropize an initially anisotropic state. Inflationary
cosmology was initially motivated by a similar idea: a
"generic" or "random" initial state at the
Planck time would be expected to be "chaotic", far from a
flat FLRW model. During an inflationary stage, arbitrary initial
states are claimed to converge towards a state with the three features
described above.



The second approach regards the initial state as extremely special
rather than generic. Penrose, in particular, has argued that the
initial state must be very special to explain time's arrow; the
usual approaches fail to take seriously the fact that gravitational
degrees of freedom are not excited in the early universe like the
others (Penrose 2016). Penrose (1979) treats the second law as arising
from a law-like constraint on the initial state of the universe,
requiring that it has low entropy. Rather than introducing a
subsequent stage of dynamical evolution that erases the imprint of the
initial state, we should aim to formulate a "theory of initial
conditions" that accounts for its special features.
Penrose's conjecture is that the Weyl curvature tensor
approaches zero as the initial singularity is approached; his
hypothesis is explicitly time asymmetric, and implies that the early
universe approaches an FLRW solution. (It does not account for the
observed perturbations, however.) Later he proposed the idea of
Conformal Cyclic Cosmology, where such a special initial state at the
start of one expansion epoch is the result of expansion in a previous
epoch that wiped out almost all earlier traces of matter and radiation
(Penrose 2016).



A third approach rejects the framework accepted by the other two
proposals, and regards the "initial state" as a misnomer:
it should instead by regarded as a "branch point" where
our pocket universe separated off from a larger multiverse. (There are
still, of course, questions regarding the initial state of the
multiverse ensemble, if one exists.) We will return to this approach
in
 SS5
 below.



A dynamical approach, even if it is successful in describing a phase
of the universe's evolution, arguably does not offer a complete
solution to the problem of initial conditions: it collapses into one
of the other two approaches. For example, an inflationary stage can
only begin in a region of spacetime if the inflaton field and the
geometry are uniform over a sufficiently large region, such that the
stress-energy tensor is dominated by the potential term (implying that
the derivative terms are small) and the gravitational entropy is
small. There are other model-dependent constraints on the initial
state of the inflaton field. One way to respond is to adopt
Penrose's point of view, namely that this reflects the need to
choose a special initial state, or to derive one from a previous
expansion phase. The majority of those working in inflationary
cosmology instead appeal to the third approach: rather than treating
inflation as an addition to standard big-bang evolution in a single
universe, we should treat the observed universe as part of a
multiverse, discussed below. But even this must have a theory of
initial conditions.


### 3.5 The Limits of Science



Cosmology provokes questions about the limits of scientific
explanation because it lacks many of the features that are present in
other areas of physics. Physical laws are usually regarded as
capturing the features of a type of system that remain invariant under
some changes, and explanations often work by placing a particular
event in larger context. Theories of the initial state cannot appeal
to either idea: we have access to only one universe, and there is no
larger context to appeal to in explaining its properties. This
contrast between the types of explanation available in cosmology and
other areas of physics has often led to dissatisfaction (see, e.g.,
Unger & Smolin 2014). At the very least, cosmology forces us to
reconsider basic questions about modalities, and what constitutes
scientific explanation.



One challenge to establishing theories of the initial state is
entirely epistemic. As emphasized in
 SS2.4,
 we lack independent experimental probes of physics at the relevant
scales, so the extensions of core theories described above are only
tested indirectly through their implications for cosmology. This
limitation reflects contingent facts about the universe, namely the
contrast between the energy scales of the early universe and those
accessible to us, and does not follow from the uniqueness of the
universe per se. Yet this limitation does not imply that it would be
impossible to establish laws. There are cases in the history of
physics, such as celestial mechanics, where confidence in a
theory's laws is based primarily on successful application under
continually improving standards of precision.



A further conceptual challenge regards whether it even makes sense to
seek "laws" in cosmology (Munitz 1962; Ellis 2007). Laws
are usually taken to cover multiple instances of some type of
phenomena, or family of objects. What can we mean by
"laws" for a unique object (the universe as a whole) or a
unique event (its origin)?



Competing philosophical analyses of laws of nature render different
verdicts on the possibility of cosmological laws. Cosmological laws,
if possible, differ from local physical laws in a variety of
ways--they do not apply to subsystems of the universe, they lack
multiple instances, and etc. Philosophical accounts of laws take
different features to be essential to law-hood. For example, the
influential Mill-Ramsey-Lewis account takes the laws to be axioms of
the deductive system capturing some body of physical knowledge that
optimally balances strength (the scope of derived claims) and
simplicity (the number of axioms) (see, e.g., Loewer 1996). It is
quite plausible that a constraint on the initial state, such as
Penrose's Weyl curvature hypothesis, would count as a law on
this account. By contrast, accounts that take other features, such as
governing evolution, as essential, reach the opposite verdict.



Finally, there are a number of conceptual pitfalls regarding what
would count as an adequate "explanation" of the origins of
the universe. What is the target of such explanations, and what can be
used in providing an explanation? The target might be the state
defined at the earliest time when extrapolations based on the SM can
be trusted. The challenge is that this state then needs to be
explained in terms of a physical theory, quantum gravity, whose basic
concepts are still obscure to us. This is a familiar challenge in
physics, where substantial work is often required to clarify how
central concepts (such as space and time) are modified by a new
theory. An explanation of origins in this first sense would explain
how it is that classical spacetime emerges from a quantum gravity
regime. While any such proposals remain quite speculative, the form of
the explanation is similar to other cases in physics: what is
explained is the applicability of an older, less fundamental theory
within some domain. Such an explanation does not address ultimate
questions regarding why the universe exists--instead, such
questions are pushed back one step, into the quantum gravity
regime.



Many discussions of origins pursue a more ambitious target: they aim
to explain the creation of the universe "from
 nothing".[39]
 The target is the true initial state, not just the boundary of
applicability of the SM. The origins are supposedly then explained
without positing an earlier phase of evolution; supposedly this can be
achieved, for example, by treating the origin of the universe as a
fluctuation away from a vacuum state. Yet obviously a vacuum state is
not nothing: it exists in a spacetime, and has a variety of
non-trivial properties. It is a mistake to take this explanation as
somehow directly addressing the metaphysical question of why there is
something rather than
 nothing.[40]



## 4. Anthropic Reasoning and Multiverse


### 4.1 Anthropic Reasoning



The physical conditions necessary for our existence impose a selection
effect on what we observe. The significance of this point for
cosmological theorizing is exemplified by Dicke's criticism of
Dirac's speculative "large number hypothesis". Dirac
(1937) noted the age of the universe expressed in terms of fundamental
constants in atomic physics is an extremely large number (roughly
\(10^{39}\)), which coincides with other large, dimensionless numbers
defined in terms of fundamental constants. Inspired by this
coincidence, he proposed that the large numbers vary to maintain this
order of magnitude agreement, implying (for example) that the
gravitational "constant" \(G\) is a function of cosmic
time. Dicke (1961) noted
that creatures like us, made of carbon produced in an earlier
generation of red giants and sustained by the light and heat of a main
sequence star, can only exist within a restricted interval of cosmic
times, and that Dirac's coincidence holds for observations made
within this interval. Establishing that the coincidence holds at a
randomly chosen \(t\) would support Dirac's hypothesis, however
slightly, but Dicke's argument shows that *our* evidence
does not do so.



Dicke's reasoning illustrates how taking selection effects into
account can mitigate surprise, and undermine the apparent implications
of facts like those noted by Dirac (see Roush 2003). These facts
reflect biases in the evidence available to us, rather than supporting
his hypothesis. It is also clear that Dicke's argument is
"anthropic" in only a very limited sense: his argument
does not depend on a detailed characterization of human observers. All
that matters is that we can exist at a cosmic time constrained by the
time scales of stellar evolution.



How to account for selection effects, within a particular approach to
confirmation theory, is one central issue in discussions of anthropic
reasoning. This question is intertwined with other issues that are
more muddled and contentious. Debates among cosmologists regarding
"anthropic principles" ignited in the 70s, prompted by the
suggestion that finely-tuned features of the universe--such as
the universe's isotropy (Collins & Hawking 1973)--can
be explained as necessary conditions for the existence of
observers.[41]
More recently, a number of cosmologists have argued that cosmological
theories should be evaluated based on predictions for what a
"typical" observer should expect to see. These ideas have
dovetailed with work in formal epistemology. A number of philosophers
have developed extensions of Bayesianism to account for
"self-locating" evidence, for
example.[42]
This kind of evidence includes indexical information characterizing an
agent's beliefs about their identity and location. At present
work in this area has not reached a consensus, and we will present a
brief overview of some of the considerations that have motivated
different positions in these debates.



In cosmology the most famous example of an "anthropic
prediction" is Weinberg (1987)'s prediction for
 \(\Lambda\).[43]
 One part of Weinberg's argument is similar to Dicke's: he
argued that there are anthropic bounds on \(\Lambda\), due to its
impact on structure formation. The existence of large, gravitationally
bound structures such as galaxies is only possible if \(\Lambda\)
falls within certain bounds. Weinberg went a step further than Dicke,
and considered what value of \(\Lambda\) a "typical
observer" should see. He assumed that observers occupy different
locations within a multiverse, and that the value of \(\Lambda\)
varies across different regions. Weinberg further argues that the
prior probability assigned to different values of \(\Lambda\) should
be uniform within the anthropic bounds. Typical observers should
expect to see a value close to the mean of the anthropic bounds,
leading to Weinberg's prediction for \(\Lambda\).



Essential to Weinberg's argument is an appeal to the principle
of indifference, applied to a class of
 observers.[44]
 We should calculate what we expect to observe, that is, as if we are
a "random choice" among all possible
 observers.[45]
 Bostrom (2002) argues that indifference-style reasoning is necessary
to respond to the problem of "freak observers". As Bostrom
formulates it, the problem is that in an infinite universe,
*any* observation \(O\) is true for *some* observer
(even if only for an observer who has fluctuated into existence from
the vacuum). His response is that we should evaluate theories based
not on the claim that *some observer* sees \(O\), but on an
indexical claim: that is, *we* make the observation \(O\). He
assumes that we are a "random" choice among the class of
possible observers. (How to justify such a strong claim is a major
challenge for this line of thought.) If we grant the assumption, then
we can assign low probability to the observations of the
"freak" observers, and recover the evidential value of
\(O\).



There are three immediate questions regarding this proposal. The first
is called the "reference class" problem. The assignments
of probabilities to events requires specifying how they are grouped
 together.[46]
 Obviously, what is typical with respect to one reference class will
not be typical with respect to another (compare, for example,
"conscious observers" with "carbon-based
life"). Second, the principle of indifference has been
thoroughly criticized as a justification for probability in other
contexts; what justifies the use of indifference in this case? Why
should we take ourselves as "randomly chosen" among an
appropriate reference class? The third problem reflects the intended
application of these ideas: Bostrom and other authors in this line of
work are particularly concerned with observes that may occupy an
infinite universe. There is no proof that the universe is in fact
infinite. These are all pressing problems for those who hold that the
principle of indifference is essential to making cosmological
predictions.



Furthermore, one way of implementing this approach leads to absurd
consequences. The Doomsday Argument, for example, claims to reach a
striking conclusion about the future of the human species without any
empirical input (see, e.g., Leslie 1992; Gott 1993; Bostrom 2002).
Suppose that we are "typical" humans, in the sense of
having a birth rank that is randomly selected among the collection of
all humans that have ever lived. We should then expect that there are
nearly as many humans before and after us in overall birth rank. For
this to be true, given current rates of population growth, there must
be a catastrophic drop in the human population
("Doomsday") in the near future. The challenge to
advocates of indifference applied to observers is to articulate
principles that avoid such consequences, while still solving (alleged)
problems such as that of freak observers.



In sum, one approach to anthropic reasoning aims to clarify the rules
of reasoning applicable to predictions made by observers in a large or
infinite universe. This line of work is motivated by the idea that
without such principles we face a severe skeptical predicament, as
observations would not have any bearing on the theory. Yet there is
still not general agreement on the new principles required to handle
these cases, which are of course not scientifically testable
principles: they are philosophically based proposals. According to an
alternative approach, selection effects can and should be treated
within the context of a Bayesian approach to inductive inference (see
Neal 2006; Trotta 2008). On this line of thought,
"predictions" like those that Bostrom and others hope to
analyze play no direct role in the evaluation of cosmological
theories, so further principles governing anthropic reasoning are
simply not necessary. There is much further work to be done in
clarifying and assessing these (and other) approaches to anthropic
 reasoning.[47]



### 4.2 Fine-Tuning



Fine-tuning arguments start from a conflict between two different
perspectives on certain features of cosmology (or other physical
theories). On the first perspective, the existence of creatures like
us seems to be sensitive to a wide variety of aspects of cosmology and
physics. To be more specific, the prospects for life depend
sensitively on the values of the various fundamental constants that
appear in these theories. The SM includes about 10 constants, and the
particle physics standard model includes about 20
 more.[48]
 Tweaking the SM, or the standard model of particle physics, by
changing the values of these constants seems to lead to a barren
 cosmos.[49]
 Focusing on the existence of "life" runs the risk of
being too provincial; we don't have a good general account of
what physical systems can support intelligent life. Yet it does seem
plausible that intelligence requires an organism with complex
structural features, living in a sufficiently stable environment.



At a bare minimum, the existence of life seems to require the
existence of complex structures at a variety of scales, ranging from
galaxies to planetary systems to macro-molecules. Such complexity is
extremely sensitive to the values of the fundamental constants of
nature. From this perspective, the existence of life in the universe
is fragile in the sense that it depends sensitively on these aspects
of the underlying theory.



This view contrasts sharply with the status of the constants from the
perspective of fundamental physics. Particle physicists typically
regard their theories as effective field theories, which suffice for
describing interactions at some specified energy scale. These theories
include various constants, characterizing the relative strength of the
interactions they describe, that cannot be further explained by the
effective field theory. The constants can be fixed by experimental
results, but are not derivable from fundamental physical principles.
(If the effective field theory can be derived from a more fundamental
theory, the value of the constants can in principle be determined by
integrating out higher-energy degrees of freedom. But this merely
pushes the question back one step: the constants appearing in the more
fundamental theory are determined experimentally.) Similarly, the
constants appearing in the SM are treated as contingent features of
the universe. There is no underlying physical principle that sets, for
example, the cosmological densities of different kinds of matter, or
the value of the Hubble constant.



So features of our theories that appear entirely contingent, from the
point of view of physics, are necessary to account for the complexity
of the observed universe and the very possibility of life. The
fine-tuning argument starts from a sense of unease about this
situation: shouldn't something as fundamental as the complexity
of the universe be explained by the *laws* or *basic
principles* of the theory, and not left to brute facts regarding
the values of various constants? The unease develops into serious
discomfort if the specific values of the constants are taken to be
extremely unlikely: how could the values of all these constants be
*just right*, by sheer coincidence?



In many familiar cases, our past experience is a good guide to when an
apparent coincidence calls for further explanation. As Hume
emphasized, however, intuitive assessments from everyday life of
whether a given event is likely, or requires a further explanation, do
not extend to cosmology. Recent formulations of fine-tuning arguments
often introduce probabilistic considerations. The constants are
"fine-tuned", meaning that the observed values are
"improbable" in some sense. Introducing a well-defined
probability over the constants would provide a response to Hume:
rather than extrapolating our intuitions, we would be drawing on the
formal machinery of our physical theories to identify fine-tuning.
Promising though this line of argument may be, there is not an obvious
way to define physical probabilities over the values of different
constants, or over other features of the laws. There is nothing like
the structure used to justify physical probabilities in other
contexts, such as equilibrium statistical
 mechanics.[50]




There are four main responses to fine-tuning:


* *Empiricist Denial*: This response follows Hume in denying
that a clear problem has even been identified. One form of this
response challenges appeals to probability, undermining the claim that
there are unexplained coincidences. Alternatively, fine-tuning is
taken to reveal that the laws alone are not sufficient to account for
some features of nature; these features are properly explained by the
laws in conjunction with various contingent facts.
* *Designer*: Newton famously argued, for example, that the
stability of the solar system provides evidence of providential
design. For the hypothesized Designer to be supported by fine-tuning
evidence, we require some way of specifying what kind of universe the
Designer is likely to create; only such a specific Design hypothesis,
based in some theory of the nature of the Designer, can offer an
explanation of fine-tuning.
* *New Physics*: The fine-tuning can be eliminated by
modifying physical theory in a variety of ways: altering the dynamical
laws, introducing new constraints on the space of physical
possibilities (or possible values of the constants of nature),
etc.
* *Multiverse*: Fine-tuning is explained as a result of
selection, from among a large space of possible universes (or
multiverse).



In the next section we discuss the last response in more detail; see
 SS3
 for further discussion of the third response.


### 4.3 Multiverse



The multiverse response replaces a single, apparently finely-tuned
universe within an ensemble of universes, combined with an appeal to
anthropic selection. Suppose that all possible values of the
fundamental constants are realized in individual elements of the
ensemble. Many of these universes will be inhospitable to life. In
calculating the probabilities that we observe specific values of the
fundamental constants, we need only consider the subset of universe
compatible with the existence of complexity (or some more specific
feature associated with life). If we have some way of assigning
probabilities over the ensemble, we could then calculate the
probability associated with our measured values. These calculations
will resolve the fine-tuning puzzles if they show that we observe
typical values for a complex (or life-permitting) universe.



Many cosmologists have argued in favor of a specific version of the
multiverse called eternal inflation
 (EI).[51]
 On this view, the rapid expansion hypothesized by inflationary
cosmology continues until arbitrarily late times in some regions, and
comes to an end (with a transition to slower expansion) in others.
This leads to a global structure of "pocket" universes
embedded within a larger multiverse.



On this line of thought, the multiverse should be accepted for the
same reason we accept many claims about what we cannot directly
observe--namely, as an inevitable consequence of an established
physical theory. It is not clear, however, that EI is inevitable, as
not all inflationary models, arguably including those favored by CMB
observations, have the kind of potential that leads to
 EI.[52]
 Accounts of how inflation leads to EI rely on speculative
 physics.[53]
 Furthermore, if inflation does lead to EI, that threatens to
undermine the original reasons for accepting inflation (Smeenk 2014):
rather than the predictions regarding the state produced at the end of
inflation taken to provide evidence for inflation, EI seems to imply
that, as Guth (2007) put it, in EI "anything that can happen
will happen; in fact, it will happen an infinite number of
times".



There have been two distinct approaches to recovering some empirical
content in this
 situation.[54]
 First, there may be traces of the early formation of the pocket
universes, the remnants of collisions between neighboring
"bubbles", left on the CMB sky (Aguirre & Johnson
2011). Detection of a distinctive signature that cannot be explained
by other means would provide evidence for the multiverse. However,
there is no expectation that a multiverse theory would generically
predict such traces; for example, if the collision occurs too early
the imprint is erased by subsequent inflationary expansion.



The other approach regards predictions for the fundamental constants,
such as Weinberg's prediction of \(\Lambda\) discussed above.
The process of forming the pocket universes is assumed to yield
variation in the local, low-energy physics in each pocket. Predictions
for the values of the fundamental constants follow from two things:
(1) a specification of the probabilities for different values of the
constant over the ensemble, and (2) a treatment of the selection
effect imposed by restricting consideration to pocket universes with
observers and then choosing a "typical" observer.



The aim is to obtain probabilistic predictions for what a typical
observer should see in the EI multiverse. Yet there are several
challenges to overcome, alongside those mentioned above related to
anthropics. The assumption that the formation of pocket universes
leads to variation in constants is just an assumption, which is not
yet justified by a plausible, well-tested dynamical theory. The most
widely discussed challenge in the physics literature is the
"measure problem": roughly, how to assign
"size" to different regions of the multiverse, as a first
step towards assigning probabilities. It is difficult to define a
measure because the EI multiverse is usually taken to be an infinite
ensemble, lacking in the kinds of structure used in constructing a
measure. On our view, these unmet challenges undercut the hope that
the EI multiverse yields probabilistic predictions. And without such
an account, the multiverse proposal does not have any testable
consequences. If everything happens somewhere in the ensemble, then
any potential observation is compatible with the theory.



Supposing that we grant a successful resolution of all these
challenges, the merits of a multiverse solution of fine-tuning
problems could then be evaluated by comparison with competing ideas.
The most widely cited evidence in favor of a multiverse is
Weinberg's prediction for the value of \(\Lambda\), discussed
above. There are other proposals to explain the observed value of
\(\Lambda\); Wang, Zhu, and Unruh (2017), for example, treat the
quantum vacuum as extremely inhomogeneous, and argue that resonance
among the vacuum fluctuations leads to a small \(\Lambda\).



The unease many have about multiverse proposals are only reinforced by
the liberal appeals to "infinities" in discussion of the
 idea.[55]
 Many have argued, for example, that we must formulate an account of
anthropic reasoning that applies to a truly infinite, rather than
merely very large, universe. Claims that we occupy one of infinitely
many possible pocket universes, filled with an infinity of other
observers, rest on an enormous and speculative extrapolation. Such
claims fail to take seriously the concept of infinity, which is not
merely a large number. Hilbert (1925 [1983]) emphasized that while
infinity is required to complete mathematics, it does not occur
anywhere in the *accessible* physical universe. One response is
to require that infinities in cosmology should have a restricted use.
It may be useful to introduce infinity as part of an explanatory
account of some aspect of cosmology, as is common practice in
mathematical models that introduce various idealizations. Yet this
infinity should be eliminable, such that the explanation of the
phenomena remains valid when the idealization is
 removed.[56]
 Even for those who regard this demand as too stringent, there
certainly needs to be more care in clarifying and justifying claims
regarding infinities.



In sum, interest in the multiverse stems primarily from speculations
about the consequences of inflation for the global structure of the
universe. The main points of debate regard whether EI is a disaster
for inflation, undermining the possibility of testing inflation at
all, and how much predictions such as that for \(\Lambda\) lend
credence to these
 speculations.[57]
 Resolution of these questions is needed to decide whether the
multiverse can be tested in a stronger sense, going beyond the special
cases (such as bubble collisions) that may provide more direct
evidence.


## 5. Testing models



As mentioned at the start, the uniqueness of the universe raises
specific problems as regards cosmology as a science. First we consider
issues to do with verification of cosmological models, and then make a
comment as regards interpreting the human implications of
cosmology


### 5.1 Criteria



The basic challenge in cosmology regards how to test and evaluate
cosmological models, given our limited access to the unique universe.
As discussed above, current cosmological models rely in part on
extrapolations of well-tested local physics along with novel
proposals, such as the inflaton field. The challenge is particularly
pressing in evaluating novel claims that only have cosmological
implications, due to the physics horizon
 (SS2.4).
 Distinctions that are routinely employed in other areas of physics,
such as that between laws and initial conditions, or chance and
necessity, are not directly applicable, due to the uniqueness of the
universe.



Recent debates regarding the legitimacy of different lines of research
in cosmology reflect different responses to this challenge. One
response is to retreat to hypothetico-deductivism (HD): a hypothesis
receives an incremental boost in confidence when one of its
consequences is verified (and a decrease if it is
 falsified).[58]
 Proponents of inflation argue, for example, that inflation should be
accepted based on its successful prediction of a flat universe with a
specific spectrum of density perturbations. Some advocates of the
multiverse take its successful prediction of the value of \(\Lambda\)
as the most compelling evidence in its favor.



Despite its appeal, there are well-known problems with taking HD as a
sufficient account of how evidence supports theories (this is often
called "naive HD"). In particular, the naive
view lacks the resources to draw distinctions among underdetermined
rival theories that make the same predictions (see Crupi 2013 [2016]).
We take it as given that scientists do draw distinctions among
theories that naive HD would treat as on par, as is reflected in
judgments regarding how much a given body of evidence supports a
particular theory. Scientists routinely distinguish among, for
example, theories that may merely "fit the data" as
opposed to those that accurately capture laws governing a particular
domain, and evaluate some successful predictions as being far more
revealing than others.



A second response is that the challenge requires a more sophisticated
methodology. This may take the form of acknowledging explicitly the
criteria that scientists use to assess desirability of scientific
theories (Ellis 2007), which include considerations of explanatory
power, consistency with other theories, and other factors, in addition
to compatibility with the evidence. These come into conflict in
unexpected ways in cosmology, and these different factors should be
clearly articulated and weighed against one another. Alternatively,
one might try to show that some of these desirable features, such as
the ability to unify diverse phenomena, should be taken as part of
what constitutes empirical
 success.[59]
 This leads to a more demanding conception of empirical success,
exemplified by historical cases such as Perrin's argument in
favor of the atomic constitution of matter.


### 5.2 Scope of Cosmological Theories and Data



Finally, a key issue is what scope do we expect our theories to have.
Ellis (2017) makes a distinction between *Cosmology*, which is
the physically based subject dealt with in the textbooks listed in
this article, dealing with the expansion of the universe, galaxies,
number counts, background radiation, and so on, and
*Cosmologia*, where one takes all that as given but adds in
consideration about the meaning this all has for life. Clearly the
anthropic discussions mentioned above are a middle
 ground.[60]
 However a number of popular science books by major scientists are
appearing that make major claims about Cosmologia, based purely in
arguments from fundamental physics together with astronomical
observations. We will make just one remark about this here. If one is
going to consider Cosmologia seriously, it is incumbent on one to take
seriously the full range of data appropriate to that enterprise. That
is, the data needed for the attempted scope of such a theory must
include data to do with the meaning of life as well as data derived
from telescopes, laboratory experiments, and particle colliders. It
must thus include data about good and evil, life and death, fear and
hope, love and pain, writings from the great philosophers and writers
and artists who have lived in human history and pondered the meaning
of life on the basis of their life experiences. This is all of great
meaning to those who live on Earth (and hence in the Universe). To
produce books saying that science proves there is no purpose in the
universe is pure myopia. It just means that one has shut ones eyes to
all the data that relates to purpose and meaning; and that one
supposes that the only science is physics (for psychology and biology
are full of purpose).