Cosmology (the study of the physical universe) is a science that, dueto both theoretical and observational developments, has made enormousstrides in the past 100 years. It began as a branch of theoreticalphysics through Einstein’s 1917 static model of the universe(Einstein 1917) and was developed in its early days particularlythrough the work of Lemaître (1927).^[1] As recently as 1960, cosmology was widely regarded as a branch ofphilosophy. It has transitioned to an extremely active area ofmainstream physics and astronomy, particularly due to the applicationto the early universe of atomic and nuclear physics, on the one hand,and to a flood of data coming in from telescopes operating across theentire electromagnetic spectrum on the other. However, there are twomain issues that make the philosophy of cosmology unlike that of anyother science. The first is,

The uniqueness of the Universe: there exists only one universe, sothere is nothing else similar to compare it with, and the idea of“Laws of the universe” hardly makes sense.

This means it is the historical sciencepar excellence: itdeals with only one unique object that is the only member of its classthat exists physically; indeed there is no non-trivial class of suchobjects (except in theoreticians’ minds) precisely for thisreason. This issue will recur throughout this discussion. The secondis

Cosmology deals with the physical situation that is the context in thelarge for human existence: the universe has such a nature that ourlife is possible.

This means that although it is a physical science, it is of particularimportance in terms of its implications for human life. This leads toimportant issues about the explanatory scope of cosmology, which wereturn to at the end.

• 1. Cosmology’s Standard Model
  • 1.1 Spacetime Geometry
  • 1.2 Observations
  • 1.3 Historical Epochs
  • 1.4 Status of the Standard Model
  • 1.5 Local vs. Global Interplay in Cosmology
• 2. Underdetermination
  • 2.1 Underdetermination in Cosmology
  • 2.2 Global Structure
  • 2.3 Establishing FLRW Geometry?
  • 2.4 Physics Horizon
  • 2.5 Cosmic Variance
• 3. Origins of the Universe
  • 3.1 The Initial State
  • 3.2 Singularity Theorems
  • 3.3 Puzzling Features of the Initial State
  • 3.4 Theories of the Initial State
  • 3.5 The Limits of Science
• 4. Anthropic Reasoning and Multiverse
  • 4.1 Anthropic Reasoning
  • 4.2 Fine-Tuning
  • 4.3 Multiverse
• 5. Testing models
  • 5.1 Criteria
  • 5.2 Scope of Cosmological Theories and Data
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Cosmology’s Standard Model

Physical cosmology has achieved a consensus Standard Model (SM), basedon extending the local physics governing gravity and the other forcesto describe the overall structure of the universe and its evolution.According to the SM, the universe has evolved from an extremely hightemperature early state, by expanding, cooling, and developingstructures at various scales, such as galaxies and stars. This modelis based on bold extrapolations of existing theories—applyinggeneral relativity, for example, at length scales 14 orders ofmagnitude larger than the those at which it has been tested—andrequires several novel ingredients, such as dark matter and darkenergy. The last few decades have been a golden age of physicalcosmology, as the SM has been developed in rich detail andsubstantiated by compatibility with a growing body of observations.Here we will briefly introduce some of the central concepts of the SMto provide the minimal background needed for the ensuing discussion.^[2]

1.1 Spacetime Geometry

Gravity is the dominant interaction at large length scales. Generalrelativity introduced a new way of representing gravity: rather thandescribing gravity as a force deflecting bodies from inertial motion,bodies free from non-gravitational forces move along the analog ofstraight lines, called geodesics, through a curved spacetime geometry.^[3] The spacetime curvature is related to the distribution of energy andmatter through GR’s fundamental equations (Einstein’sfield equations, EFE). The dynamics of the theory are non-linear:matter curves spacetime, and the curvature of spacetime determines howmatter moves; and gravitational waves interact with each othergravitationally, and act as gravitational sources. The theory alsoreplaces the single gravitational potential, and associated fieldequation, 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 featuresof solutions to EFE, and to finding exact solutions to describespecific physical situations. Most exact solutions have been foundbased on strong idealizations, introduced to simplify themathematics.

Remarkably, much of cosmology is based on an extremely simple set ofsolutions found within a decade of Einstein’s discovery of GR.These Friedman-Lemaître-Robertson-Walker (FLRW) solutions have,in a precise sense, the most symmetry possible. The spacetime geometryis constrained to be uniform, so that there are no preferred locationsor 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) ofconstant curvature, with three possibilities: (1) spherical space, forthe case of positive curvature; (2) Euclidean space, for zerocurvature; and (3) hyperbolic space, for negative curvature.^[6]

These models describe an expanding universe, characterized fully bythe behavior of the scale factor \(R(t)\). The worldlines of“fundamental observers”, defined as at rest with respectto matter, are orthogonal to these surfaces, and the cosmic timecorresponds 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 ofcosmic time. The evolution of these models is described by a simpleset of equations governing \(R(t)\), implied by Einstein’s fieldequations (EFE): the Friedmann equation,^[7]

and the isotropic form of the Raychaudhuri equation:

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

The unrelenting symmetry of the FLRW models makes them quite simplegeometrically and dynamically. Rather than a set of coupled partialdifferential equations, which generically follow from EFE, in the FLRWmodels one only has to deal with 2 ordinary differential equations(only two of \((\ref{eq:Fried})\)–\((\ref{eq:cons})\) areindependent) 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 unchanginguniverse, with \(\dot{R}(t) =0\). “Ordinary” matter haspositive 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 todecelerate cosmic expansion, \(\ddot{R} < 0\)—gravity is aforce of attraction. This is only so for ordinary matter: a positivecosmological constant, or matter with negative gravitational-energydensity \(\rho_{\textit{grav}}\) leads, conversely, to acceleratingexpansion, \(\ddot{R} > 0\). Einstein was only able to construct astatic model by delicately balancing the attraction of ordinary matterwith a precisely chosen value of \(\Lambda\); he unfortunately failedto notice that the solution was unstable, and overlooked the dynamicalimplications of his own theory.

Second, the expansion rate varies as different types of matter come todominate the dynamics. As shown by (\(\ref{eq:cons}\)), the energydensity for different types of matter and radiation dilutes atdifferent 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 namesuggests) constant. The SM describes the early universe as having amuch higher energy density in radiation than matter. Thisradiation-dominated phase eventually transitions to a matter-dominatedphase 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 finitetime in the past. Extrapolating back in time, given that the universeis currently expanding, eqn. (\(\ref{eq:Ray}\)) implies that theexpansion began at some finite time in the past. The current rate ofexpansion is given by the Hubble parameter, \(H_0 =(\frac{\dot{R}}{R})_0\). Simply extrapolating this expansion ratebackward, from eqn. (\(\ref{eq:Ray}\)) the expansion rate mustincrease at earlier times, so \(R(t) \rightarrow 0\) at a time lessthan the Hubble time Hubble time \(H_0^{-1}\) before now, if\(\rho_{\textit{grav}}\geq 0\). As this “big bang” isapproached, the energy density and curvature increase without boundprovided \(\rho_{\textit{inert}}\coloneqq (\rho+p)>0\) (whichcondition guarantees that \(\rho \rightarrow\infty\) as\(R\rightarrow0\)). This reflects gravitational instability: as\(R(t)\) decreases, the energy density and pressure both increase, andthey both appear with the same sign on the right hand side of eqn.(\(\ref{eq:Ray}\)), hence pressure \(p>0\) does not help avoidthe singularity. Work in the 1960s, discussed below in§4.1, established that the existence of a singularity holds in morerealistic models, and is not an artifact of the symmetries of the FLRWmodels.

The SM adds small departures from strict uniformity in order toaccount for the formation and evolution of structure. Due togravitational instability, such perturbations are enhanceddynamically—the density contrast of an initial region thatdiffers from the average density grows with time. Sufficiently smallfluctuations can be treated as linear perturbations to a backgroundcosmological model, governed by an evolution equation that followsfrom EFE. Yet as the fluctuations grow larger, linearized perturbationtheory no longer applies. According to the SM, structure growshierarchically with smaller length scales going non-linear first, andlarger structures forming via later mergers. Models of evolution ofstructures at smaller length scales (e.g., the length scales ofgalaxies) include physics other than gravity, such as gas dynamics, todescribe the collapsing clumps of matter. Cold dark matter (CDM) alsoplays a crucial role in the SM’s account of structure formation:it clumps first, providing scaffolding for clumping of baryonicmatter.

A full account of structure formation requires integrating physicsover an enormous range of dynamical scales and including acosmological constant as well as baryonic matter, radiation, and darkmatter. This is an active area of research, primarily pursued usingsophisticated \(N\)-body computer simulations to study features of thegalaxy distribution produced by the SM, given various assumptions.^[9]

1.2 Observations

There are two main ways in which cosmological observations supportperturbed FLRW models. First, cosmologists use matter and radiation inthe universe to probe the background spacetime geometry and itsevolution. The universe appears to be isotropic at sufficiently largescales, as indicated by background radiation (most notably the cosmicmicrowave background radiation (CMB), discussed below) and discretesources (e.g., galaxies). Isotropy observed along a single worldlineis, however, not sufficient to establish the universe is welldescribed by an FLRW geometry. A further assumption that our worldlineis not theonly vantage point from which the universe appearsisotropic, often called the Copernican principle, is needed. Grantingthis principle, there are theorems establishing that observations ofalmost isotropic background radiation implies that the spacetimegeometry is almost FLRW.^[10] The principle itself cannot be established directly via observations(see§2). Given that we live in an almost FLRW models, we need to determine itsparameters such as the Hubble constant \(H_0\) and the decelerationparameter \(q_0 \coloneqq {-}\ddot{R}/(RH_0^2)\), which measures howthe rate of expansion is changing, and the normalized densityparameters \(\Omega_m\coloneqq \rho_m/(3H_0^2)\) for each matter orenergy density component \(m\). There are a variety of ways todetermine the accuracy of the background evolution described by theFLRW models, which depends on these parameters. For this purpose,cosmologists seek effective standard candles and standardrulers—objects with a known intrinsic luminosity and length,respectively, which can then be used to measure the expansion historyof the universe.

The second main avenue of testing focuses on the SM’s account ofstructure formation, which describes the evolution of smallperturbations away from the background FLRW geometry in terms of asmall number of parameters such as the tilt \(n_s\) and the scalar totensor ratio \(r\). Observations from different epochs, such astemperature anisotropies in the CMB and the matter power spectrumbased on galaxy surveys, can be used as independent constraints onthese parameters as well as on the background parameters (indeed suchobservations turn out to give the best constraints on the backgroundmodel parameters). These two routes to testing almost FLRW spacetimegeometry are closely linked because the background model provides thecontext for the evolution of perturbations under the dynamicsdescribed by general relativity.

The remarkable success of perturbed FLRW models in describing theobserved universe has led many cosmologists to focus almostexclusively on them, yet there are drawbacks to such a myopicapproach. For example, the observations at best establish that theobserved universe can be well-approximated by an almost FLRW modelwithin some (large) domain. But they are not the only models that fitthe data: there are other cosmological models that mimic FLRW modelsin the relevant domain, yet differ dramatically elsewhere (andelsewhen). Specifically, on the one hand there are a class ofspatially homogeneous and anisotropic models (Bianchi models) thatexhibit “intermediate isotropization”: namely, they havephysical properties that are arbitrarily close to (isotropic) FLRWmodels over some time scale \(T\).^[11] Agreement over the time interval \(T\) does not imply globalagreement, however, as these models have large anisotropies at othertimes. Relying on the FLRW models in making extrapolations to theearly or late universe requires some justification for ignoringmodels, such as these Bianchi models, that mimic their behavior for afinite time interval. On the other hand there are inhomogeneousspherically symmetric models that can reproduce exactly the backgroundmodel observations (number counts versus redshifts and angulardiameter distance versus redshift, for example) with or without acosmological constant (Mustapha et al. 1997). These can be excluded bydirect observations with good enough standard candles (Clarkson et al.2008) or by observations of structure formation features in suchuniverses (Clarkson & Maartens 2010); but that exclusion cannottake place unless one indeed examines such models and theirobservational consequences.

Lack of knowledge of the full space of solutions to EFE makes itdifficult to assess the fragility of various inferences cosmologistsmake based on perturbed FLRW models. A fragile inference depends onthe properties of the model holding exactly, contrasted with robustinferences that hold even if the models are good approximations (up tosome 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 otherhighly symmetric models, singularities are generic in modelssatisfying physically plausible assumptions. The status of variousother inferences cosmologists make is less clear. For example, howsensitively does the observational case in favor of dark energy, whichcontributes roughly 70% of the total energy density of the universe inthe SM, depend upon treating the universe as having almost-FLRWspacetime geometry? As mentioned above, recent work has pursued thepossibility of accounting for the same observations based uponlarge-scale inhomogeneities or local back-reaction, without recourseto dark energy.^[12] Studies along these lines are needed to evaluate the possibility thatsubtle dynamical effects, absent in the FLRW models, providealternative explanations of observed phenomena. The deduction alsodepends on the assumption that the EFE hold at cosmological scales -which may not be true: maybe for example some form of scalar-tensortheory should be used. More generally, an assessment of thereliability of a variety of cosmological inferences requires detailedstudy of a larger space of cosmological models.

1.3 Historical Epochs

The SM’s account of the evolution of the matter and radiation inthe universe reflects the dynamical effect of expansion. Consider acube of spacetime in the early universe, filled with matter andradiation. The dynamical effects of the universe’s expansion arelocally the same as slowly stretching the cube. For some stages ofevolution the contents of the cube interact sufficiently quickly thatthey reach and stay in local thermal equilibrium as the cube changesvolume. (Because of isotropy, equal amounts of matter and radiationenter and leave the cube from neighboring cubes.) But when theinteractions are too slow compared to the rate of expansion, the cubechanges volume too rapidly for equilibrium to be maintained. As aresult, particle species “freeze out” and decouple, andentropy increases. Without a series of departures from equilibrium,cosmology would be boring—the system would remain in equilibriumwith a state determined solely by the temperature, without a trace ofthings past. The rate of expansion of the cube varies with cosmictime. Because radiation, matter, and a cosmological constant term (ordark energy) dilute with expansion at different rates, an expandinguniverse naturally falls into separate epochs, characterized bydifferent 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 isexpected to fail at early times, when quantum effects will be crucialin describing the gravitational degrees of freedom. There isconsiderable uncertainty regarding physics at this scale.
• Inflation: A period of exponential, quasi-De Sitterexpansion driven by an “inflaton” field (or fields),leading to a uniform, almost flat universe with Gaussian linear nearlyscale invariant density perturbations. During inflation pre-existingmatter and radiation are rapidly diluted; the universe is repopulatedwith matter and energy by the decay of the inflaton field into otherfields at the end of inflation (“re-heating”).
• Big Bang Nucleosynthesis: At \(t \approx 1\) second, theconstituents of the universe include neutrons, protons, electrons,photons, and neutrinos, tightly coupled and in local thermalequilibrium. Synthesis of light elements occurs during a burst ofnuclear interactions that transpire as the universe falls from atemperature of roughly \(10^9\) K to \(10^8\) K after neutrinos fallout of equilibrium and consequent onset of neutron decay. Thepredicted light-element abundances depend on physical features of theuniverse at this time, such as the total density of baryonic matterand the baryon to photon ratio. Agreement between theory andobservation for a specific baryon to photon ratio (Steigman 2007) is agreat success of the SM.
• Decoupling: As the temperature drops below \(\approx4,000 K\), electrons become bound in stable atoms, and photonsdecouple from the matter with a black-body spectrum. With theexpansion of the universe, the photons cool adiabatically but retain ablack-body spectrum with a temperature \(T \propto 1/R\). This“cosmic background radiation” (CBR) has been aptly calledthe cosmic Rosetta stone because it carries so much information aboutthe state of the universe at decoupling (Ade et al. 2016).
• Dark Ages: After decoupling, baryonic matter consistsalmost entirely of neutral hydrogen and helium. Once the firstgeneration of stars form, the dark ages come to an end with light fromthe stars, which re-ionizes the universe.
• Structure Formation: Cold dark matter dominates theearly stages of the formation of structure. Dark matter halos providethe scaffolding for hierarchical structure formation. The firstgeneration of stars aggregate into galaxies, and galaxies intoclusters. Massive stars end their lives in supernova explosions andspread through space heavy elements that have been created in theirinteriors, enabling formation of second generation stars surrounded byplanets.
• Dark Energy Domination: Dark energy (or a non-zerocosmological constant) eventually comes to dominate the expansion ofthe universe, leading to accelerated expansion.^[13] This expansion will be never-ending if the dark energy is in fact acosmological constant.

1.4 Status of the Standard Model

The development of a precise cosmological model compatible with therich set of cosmological data currently available is an impressiveachievement. Cosmology clearly relies very heavily on theory; thecosmological parameters that have been the target of observationalcampaigns are only defined given a background model. The strongestcase for accepting the SM rests on the evidence in favor of theunderlying physics, in concert with the overdetermination ofcosmological parameters. The SM includes several free parameters, suchas the density parameters characterizing the abundance of differenttypes of matter, each of which can be measured several ways.^[14] These methods have distinctive theoretical assumptions and sources oferror. For example, the abundance of deuterium produced during bigbang nucleosynthesis depends sensitively on the baryon density.Nucleosynthesis is described using well-tested nuclear physics, andthe light element abundances are frozen in within the “firstthree minutes”. The amplitudes of the acoustic peaks in the CMBangular power spectrum depend on the baryon density at the time ofdecoupling. Current measurements fix the baryon density to an accuracyof one percent, and the values determined by these two methods agreewithin observational error. This agreement is one of many consistencychecks for the SM.^[15] There are important discrepancies, such as that between local versusglobal measurements of the Hubble parameter \(H_0\) (Luković etal. 2016; Bernal et al. 2016). The significance and furtherimplications of these discrepancies is not clear.

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

First, we do not understand three crucial components of the SM thatrequire new physics. We do not have a full account of the nature, orunderlying dynamics, of dark matter (Bertone et al. 2005), dark energy(Peebles & Ratra 2003), or the inflaton field (Lyth & Riotto1999; Martin et al. 2014). These are well-recognized problems thathave inspired active theoretical and observational work, although aswe note below in§2.4 they will be challenging to resolve due to inaccessibility of physicsat the appropriate scale.

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

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

1.5 Local vs. Global Interplay in Cosmology

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

The traditional issues of this kind (Bondi 1960; Ellis & Sciama1972; Ellis 2002) are

• Mach’s Principle: the idea that the origin ofinertia is due to the very distant matter in the universe (Barbour& Pfister 1995), nowadays understood as being due to the fact thatthe vorticity \(\omega\) of the universe is very low at present (itcould have been otherwise);
• Olber’s Paradox: the issue of why the sky is darkat night (Harrison 1984), resolved by evolution of the universetogether with the redshift factor of about 1000 since the surface oflast scattering (which determines that the temperature of the nightsky is the 2.73K of the CMB everywhere except for the small fractionof the sky covered by stars and galaxies)
• The Arrow of Time: where does the arrow of time comefrom, if the underlying physics is time symmetric? This has to be dueto special initial conditions at the start of the universe (Ellis2007). This is related to the Sommerfeld outgoing radiation conditionand Penrose’s Weyl curvature hypothesis (Penrose 2016).

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

• Nucleosynthesis, where the course of nuclear reactions isdetermined by the \(T(t)\) relation that is controlled by cosmologicalevolution (Steigman 2007) (the temperature \(T\) being a coarsegrained variable with evolution determined by the average density\(\rho\) of matter in the universe through the Friedmannequation)
• Structure formation due to gravitational instability(Mukhanov et al. 1992), which is affected crucially by the expansionof the universe, which turns what would have been an exponentialgrowth of inhomogeneity(in a static universe) to a power law growth.It is because of this effect that studies of structure such as the BAOand CMB anisotropies give us strong limits on the parameters of thebackground model (Ade et al. 2016).
• The Anthropic Principle, discussed below (§4.1), whereby large-scale conditions in the universe (such as the value ofthe cosmological constant and the initial amplitude of inhomogeneitiesin the early universe) provide local conditions suitable for life tocome into being.

Relevant to all this is the idea of an “effectivehorizon”: the domain that has direct impact on structuresexisting on the Earth, roughly 1 Mpc co-moving sphere, see Ellis &Stoeger 2009. This is the part of the universe that actually has asignificant effect on our history.

2. Underdetermination

Many philosophers hold that evidence is not sufficient to determinewhich scientific theory we should choose. Scientific theories makeclaims about the natural world that extend far beyond what can bedirectly established through observations or experiments. Rivaltheories may fare equally well with regard to some body of data, yetgive quite different accounts of the world. Philosophers often treatthe existence of such rivals as inevitable: for a given theory, it isalways possible to construct rival theories that have “equallygood fit” with available data. Duhem (1914 [1954]) gave aninfluential characterization of the difficulty in establishingphysical theories conclusively, followed a half century later byQuine’s arguments for a strikingly general version ofunderdetermination (e.g., Quine 1970). The nature of this proposedunderdetermination of theory by evidence, and appropriate responses toit, have been central topics in philosophy of science (Stanford 2009[2016]). Although philosophers have identified a variety of distinctsenses of underdetermination, they have generally agreed thatunderdetermination poses a challenge to justifying scientifictheories.

There is a striking contrast with discussions of underdeterminationamong scientists, who often emphasize instead the enormous difficultyin constructing compelling rival theories.^[16] This contrast reflects a disagreement regarding how to characterizethe empirical content of theories. Suppose that the empirical contentof theory consists of a set of observational claims implied by thetheory. Philosophers then take the existence of rival theories to bestraightforward. Van Fraassen (1980), for example, defines a theory as“empirically adequate” if what it says about observablephenomena is true, and argues that for any successful theory there arerival theories that disagree about theoretical claims. If we demandmore of theories than empirical adequacy in this sense, it is possibleto draw distinctions among theories that philosophers would regard asunderdetermined. Furthermore, even when scientists do face a choiceamong competing theories, they are almost never rivals in thephilosopher’s sense. Instead, they differ in various ways:intended domain of applicability, explanatory scope, importanceattributed to particular problems, and so on.

The scientists’ relatively dismissive attitude towards allegedunderdetermination threats may be based on a more demanding conceptionof empirical success.^[17] Scientists demand much more of their theories than mere compatibilitywith some set of observational claims: they must fit into a largerexplanatory scheme, and be compatible with other successful theories.Given a more stringent account of empirical success it is much morechallenging to find rival theories. (We return to this issue in§5 below.)

One aspect of underdetermination (emphasized by Stanford 2006) is ofmore direct relevance to scientific debates: current theories may beindistinguishable, within a restricted domain, from a successortheory, even though the successor theory makes different predictionsfor other domains. This raises the question of how far we can rely onextrapolating a theory to a new domain. For example, despite itssuccess in describing objects moving with low relative velocities in aweak gravitational field, where it is nearly indistinguishable fromgeneral relativity, Newtonian gravity does not apply to other regimes.How far, then, can we rely on a theory to extend our reach? Theobstacles to making such reliable inferences reflect the specificdetails of particular domains of inquiry. Below we will focus on theobstacles to answering theoretical questions in cosmology due to thestructure of the universe and our limited access to phenomena.

2.1 Underdetermination in Cosmology

Given the grand scope of cosmology, one might expect that manyquestions must remain unresolved. Basic features of the SM impose twofundamental limits to the ambitions of cosmological theorizing. First,the finitude of the speed of light ensures that we have a limitedobservational window on the universe due to existence of the visualhorizon, representing the most distant matter from which we canreceive and information by electromagnetic radiation, and the particlehorizon, representing the most distant matter with which we can havehad any causal interaction (matter up to that distance can influencewhat we see at the visual horizon). Recent work has preciselycharacterized what can be established via idealized astronomicalobservations, regarding spacetime geometry within, or outside, ourpast light cone (the observationally accessible region). Second, inaddition to enormous extrapolations of well-tested physics in the SM,cosmologists have explored speculative ideas in physics that can onlybe tested through their implications for cosmology; the energiesinvolved are too high to be tested by any accelerator on Earth. Ellis(2007) has characterized these speculative aspects of cosmology asfalling on the far side of a “physics horizon”. We willbriefly discuss how this second type of horizon poses limits forcosmological theorizing. In both cases, the type of underdeterminationthat arises differs from that discussed in the philosophicalliterature.

2.2 Global Structure

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

The observational cosmology program (Kristian & Sachs 1966; Elliset al. 1985) clarifies the extent to which a set of ideal observationscan determine the spacetime geometry directly with minimalcosmological assumptions. (By contrast, the standard approach startsby assuming a background cosmological model and then finding anoptimal parameter fit.) Roughly put, the ideal data set consists of aset of astrophysical objects that can be used as standard candles andstandard rulers. If the intrinsic properties and evolution of avariety of sources are given, observations can directly determine thearea (or luminosity) distance of the sources, and the distortion ofdistant images determines lensing effects. These observations thusdirectly constrain the spacetime geometry of the past light cone\(C^-(p)\). Number counts of discrete sources (such as galaxies orclusters) can be used to infer the total amount of baryonic matter,again granting various assumptions. Ellis et al. (1985) proved theremarkable result that an appropriate idealized data set of this kindis sufficient, if we grant that EFE hold, to fully fix the spacetimegeometry 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 inunderstanding the nature of sources and their evolution withsufficient clarity that they can be used to determine spacetimegeometry, so this is the ideal situation.

What does \(J^-(p)\) reveal about the rest of spacetime? In classicalGR, we would not expect the physical state on \(J^-(p)\) to determinethat of other regions of spacetime—even the causal past of apoint just to the future of \(p\).^[19] There are some models in which \(J^-(p)\)does reveal more: “small universe” models are closedmodels with a finite maximum length in all directions that is smallerthan the visual horizon (Ellis & Schreiber 1986). Observers insuch a model would be able to “see around the universe” inall directions, and establish some global properties via directobservation because they would be able to see all matter thatexists.^[20]

Unless this is the case, the causal past for a single observer, andeven a collection of causal pasts, place very weak constraints on theglobal properties of spacetime. The global properties of a spacetimecharacterize 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 areany global properties shared by spacetimes that are constructed asfollows. For a given spacetime, construct an indistinguishablecounterpart that includes the collection of causal pasts\(\{J^-(p)\}\) for all points in the original spacetime. Theconstructed spacetime is indistinguishable from the first, because forany observer in the first spacetime there is a “copy” oftheir causal past in the counterpart. It is possible, however, toconstruct counterparts that do not have the same global properties asthe original spacetime. The property of having a Cauchy surface, forexample, need not be shared by an indistinguishable counterpart.^[22] More generally, the only properties that are guaranteed to hold foran indistinguishable counterpart are those that can be establishedbased on the causal past of a single point. This line of workestablishes that (some) global properties cannot be establishedobservationally, and raises the question of whether there arealternative justifications.

2.3 Establishing FLRW Geometry?

The case of global spacetime geometry is not a typical instance ofunderdetermination of theory by evidence, as discussed byphilosophers, for two reasons (see Manchak 2009, Norton 2011,Butterfield 2014). First, this whole discussion assumes that classicalGR holds; the question regards discriminating among models of a giventheory, rather than a choice among competing theories. Second, theseresults establish that all observations available to us that arecompatible with a given spacetime, with some appealing globalproperty, are equally compatible with its indistinguishablecounterparts. But as is familiar from more prosaic examples of theproblem of induction, evidence of past events is compatible, in asimilar sense, with many possible futures. Standard accounts ofinductive inference aim to justify some expectations about the futureas more reasonable, e.g., those based on extending past uniformities.The challenge in this case is to articulate an account of inductiveinferences that justifies accepting one spacetime over itsindistinguishable counterparts.

As a specific instance of this challenge, consider the status of thecosmological principle, the global symmetry assumed in the derivationof the FLRW models. The results above show that all evidence availableto us is equally compatible with models in which the cosmologicalprinciple does or does not hold. One might take the principle asholdinga priori, or as a pre-condition for cosmologicaltheorizing (Beisbart 2009). A recent line of work aims to justify theFLRW models by appealing to a weaker general principle in conjunctionwith theorems relating homogeneity and isotropy. Global isotropyaround every point implies global homogeneity, and it is natural toseek a similar theorem with a weaker antecedent formulated in terms ofobservable quantities. The Ehlers-Geren-Sachs theorem (Ehlers et al.1968) shows that if all geodesic fundamental observers in an expandingmodel find that freely propagating background radiation is exactlyisotropic, then their spacetime is an FLRW model. If our causal pastis “typical”, observations along our worldline willconstrain what other observers should see. This is often called theCopernican principle—namely, no point \(p\) is distinguishedfrom other points \(q\) by any spacetime symmetries or lack thereof(there are no “special locations”). There are indirectways of testing this principle empirically: theSunyaev-Zel’dovich effect can be used to indirectly measure theisotropy of the CBR as observed from distant points. Other tests aredirect tests with a good enough set of standard candles, and anindirect test based on the time drift of cosmological redshift. Thisline of work provides an empirical argument that the observed universeis well-approximated by an FLRW model, thus changing that assumptionfrom a philosophically based starting point to an observationallytested foundation.

2.4 Physics Horizon

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

The Soviet physicist Yakov Zel’dovich memorably called the earlyuniverse the “poor man’s accelerator”, becauserelatively inexpensive observations of the early universe may revealfeatures of high-energy physics well beyond the reach of even the mostlavishly funded earth-bound accelerators. For many aspects offundamental 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 fornew physics extends beyond the standard model of particle physics(which is generally thought to be incomplete, even though there are noobservations that contradict it). Big bang nucleosynthesis, forexample, is an application of well-tested nuclear physics to the earlyuniverse, with scattering cross-sections and other relevant featuresof the physics fixed by terrestrial experiments. While working out hownuclear physics applied in detail required substantial effort, therewas little uncertainty regarding the underlying physics. By contrast,in some domains cosmologists now aim to explain the universe’shistory while at the same time evaluating new physics used inconstructing it.

This contrast can be clarified in terms of the “physicshorizon” (Ellis 2007), which delimits the physical regimeaccessible to terrestrial experiments and observations, roughly interms of energy scales associated with different interactions. Thehorizon can be characterized more precisely for a chosen theory, byspecifying the regions of parameter space that can be directly testedby experiments and observations.^[23] Aspects of cosmological theories that extend past the physics horizoncannot be independently tested through non-cosmological experiments orobservations; the only empirical route to evaluating these ideas isthrough their implications for cosmology. (This is not to deny thatthere may be strong theoretical grounds to favor particular proposals,as extensions of the core theories.)

Cosmological physics extending beyond the physics horizon faces anunderdetermination threat due to the lack of independent lines ofrelevant evidence. The case of dark matter illustrates the value ofsuch independent evidence. Dark matter was first proposed to accountfor the dynamical behavior of galaxy clusters and galaxies, whichcould not be explained using Newtonian gravitational theory with onlythe luminous matter observed. Dark matter also plays a crucial role inaccounts of structure formation, as it provides the scaffoldingnecessary for baryonic matter to clump, without conflicting with theuniformity of the CMB.^[24] Both inferences to the existence of dark matter rely on gravitationalphysics, raising the question of whether we should take thesephenomena as evidence that our gravitational theory fails, rather thanas evidence for a new type of matter. There is an active researchprogram (MOND, forModifiedNewtonianDynamics) devoted to accounting for the relevant phenomena bymodifying gravity. Regardless of one’s stance on the relativemerits of MOND vs. dark matter (obviously MOND needs to be extended toa relativistic theory), direct evidence of existence of dark matter,or indirect evidence via decay products, would certainly reshape thedebate. Efforts have been underway for some time to find dark matterparticles through direct interactions with a detector, mediated by theweak force. A positive outcome of these experiments would provideevidence of the existence of dark matter that does not depend upongravitational theory.^[25]

Such independent evidence is not available for two prominent examplesof new physics motivated by discoveries in cosmology. “Darkenergy” was introduced in studies of structure formation, whichemployed a non-zero cosmological constant to fit observationalconstraints (the \(\Lambda\)CDM models). Subsequent observations ofthe redshift-distance relation, using supernovae (type Ia) as astandard candle, led to the discovery that the expansion of theuniverse is accelerating.^[26] (For \(\ddot{R}>0\) in an FLRW model, there must be a contributionthat appears in eqn. (\(\ref{eq:Ray}\)) like a positive \(\Lambda\)term.) Rather than treating these observations as simply determiningthe value of a parameter in the SM, many cosmologists have developedphenomenological models of “dark energy” that leads to aneffective \(\Lambda\). Unlike dark matter, however, the properties ofdark energy insure that any attempt at non-cosmological detectionwould be futile: the energy density is so small, and uniform, that anylocal experimental study of its properties is practically impossible.Furthermore these models are not based in well-motivated physics: theyhave the nature of ‘saving the phenomena’ in that they aretailored to fitting the cosmological observations by curve fitting.^[27]

Inflationary cosmology originally promised a powerful unification ofparticle physics and cosmology. The earliest inflationary modelsexplored the consequences of specific scalar fields introduced inparticle physics (the then supposed Higgs field for the stronginteractions). Yet theory soon shifted to treating the scalar fieldresponsible for inflation as the “inflaton” field, leavingits relationship to particle physics unresolved, and the promise ofunification unfulfilled. If the properties of the inflaton field areunconstrained, inflationary cosmology is extremely flexible; it ispossible to construct an inflationary model that matches any chosenevolutionary history of the early universe.^[28] Specific models of inflation, insofar as they specify the features ofthe field or fields driving inflation and its initial state, do havepredictive content. In principle, cosmological observations coulddetermine some of the properties of the inflaton field and so selectamong them (Martin et al. 2014). This could in principle then haveimplications for a variety of other experiments or observations; yetin practice the features of the inflaton field in most viable modelsof inflation guarantee that it cannot be detected in other regimes.The one exception to this is if the inflaton were the electroweakHiggs particle detected at the LHC (Ellis & Uzan 2014). Thisremains a viable inflaton candidate, so testing if it is indeed theinflaton is an important task (Bezrukov & Gorbunov 2012).

The physics horizon poses a challenge because one particularlypowerful type of evidence—direct experimental detection orobservation, with no dependence on cosmological assumptions—isunavailable for the physics relevant at earliest times (beforeinflation, and indeed even for baryosynthesis after inflation). Yetthis does not imply that competing theories, such as dark matter vs.modified gravity, should be given equal credence. The case in favor ofdark matter draws on diverse phenomena, and it has been difficult toproduce a compelling modified theory of gravity, consistent with GR,that captures the full range of phenomena as an alternative to darkmatter. Cosmology typically demands a more intricate assessment ofbackground assumptions, and the degree of independence of differenttests, in evaluating proposed extensions of the core theories. Yetthis evidence may still be sufficiently strong, in the sense discussedmore fully in§5 below, to justify new physics.

2.5 Cosmic Variance

There is a distinctive form of underdetermination regarding the use ofstatistics in cosmology, due to the uniqueness of the universe. Tocompare the universe with the statistical predictions of the SM, weconceptualize it as one realization of a family of possible universes,and compare what we actually measure with what is predicted to occurin the ensemble of hypothetical models. When they are significantlydifferent, the key issue is: Are these just statistical fluctuationswe can ignore? Or are they serious anomalies that need anexplanation?

This question arises in several concrete cases:

• Existence of low CMB anisotropy power at high and angular scalesrelative 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 measureddirectly in the local region on the one hand, and as deduced from CMBanisotropies on the other (Luković et al. 2016; Bernal et al.2016).

How do we decide? This will depend on the particular measurement (seee.g., Kamionkowski & Loeb 1997; Marra et al. 2013), but in generalbecause of the uniqueness of the universe, we don’t know ifthese potential anomalies are real, pointing to serious problems withthe models, or not real—just statistical flukes in the way thefamily of models differs from the one instance that we have at hand,the unique universe that actually exists. In all the physicalsciences, this is a unique problem of cosmology.^[29]

3. Origins of the Universe

Cosmology confronts a distinctive challenge in accounting for theorigin of the universe. In most other branches of physics the initialor boundary conditions of a system do not call out for theoreticalexplanation. They may reflect, for example, the impact of theenvironment, or an arbitrary choice regarding when to cut off thedescription of a subsystem of interest. But in cosmology there areheated debates regarding what form a “theory of the initialstate” should take, and what it should contribute to ourunderstanding of the universe. This basic question regarding thenature of aims of a theory of origins has significant ramificationsfor various lines of research in cosmology.

3.1 The Initial State

Contemporary cosmology at least has a clear target for a theory oforigins: the SM describes the universe as having expanded and evolvedover 13.7 billion years from an initial state where many physicalquantities diverged. In the FLRW models, the cosmic time \(t\) can bemeasured by the total proper time elapsed along the worldline of afundamental observer, from the “origin” of the universeuntil the present epoch. Extrapolating backwards from the present,various quantities diverge as the cosmic time \(t \rightarrow0\)—for example, \(R(t) \rightarrow 0\) and the matter densitygoes to infinity.^[30] The worldlines of observers cannot be extended arbitrarily far intothe 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 pastin a broad class of cosmological models. Past singularities, signaledby the existence of inextendible geodesics with bounded length, mustbe present in models with a number of plausible features. (Geodesicsare the curves of extreme length through curved spacetime, and freelyfalling bodies follow timelike geodesics.) Intuitively, extrapolatingbackwards from the present, an inextendible geodesic reaches, withinfinite distance, an “edge” beyond which it cannot beextended. There is not a uniquely defined “cosmic time”,in general, but the maximum length of these curves reflects the finiteage of the universe. The singularity theorems plausibly apply to theobserved universe, within the domain of applicability of generalrelativity. There are various related theorems differing in detail,but one common ingredient is an assumption that there is sufficientmatter and energy present to guarantee that our past light cone refocuses.^[31] The energy density of the CMB alone is sufficient to justify thisassumption. The theorems also require an energy condition: arestriction on the types of matter present in the model, guaranteeingthat 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 witha non-zero cosmological constant, for example, since it appears withthe opposite sign as ordinary matter, counteracting this focusingeffect.)

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 tofailures of determinism, because the laws “break down” insome sense. This concern only applies to some kinds of singularities,however. Relativistic spacetimes that are globally hyperbolic haveCauchy surfaces, and appropriate initial data posed on such surfacesfix a unique solution throughout the spacetime. Global hyperbolicitydoes not rule out the existence of singularities, and in particularthe FLRW models are globally hyperbolic in spite of the existence ofan initial singularity. The threat to determinism is thus morequalified: the laws do not apply “at the singularityitself” even though the subsequent evolution is fullydeterministic, and there are some types of singularities that posemore serious threats to determinism.

Another common claim is that the presence of singularities establishthat GR is incomplete, since it fails to describe physics “atthe singularity”.^[33] This is difficult to spell out fully without a local analysis ofsingularities, 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 acosmological model indicates that spacetime, as described by GR, comesto an end: there is no way of extending the spacetime through thesingularity, without violating mathematical conditions needed toinsure that the field equations are well-defined. Any description ofphysical conditions “before the big bang” must be based ona theory that supersedes GR, and allows for an extension through thesingularity.

There are two limitations regarding what we can learn about theorigins of the universe based on the singularity theorems. First,although these results establish the existence of an initialsingularity, they do not provide much guidance regarding itsstructure. The spacetime structure near a “generic”initial singularity has not yet been fully characterized. Partialresults have been established for restricted classes of solutions; forexample, numerical simulations and a number of theorems support theBKL conjecture, which holds that isotropic, inhomogeneous modelsexhibit a complicated form of chaotic, oscillatory behavior. Theresulting picture of the approach to the initial singularity contrastssharply 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 oncequantum effects are taken into account. The standard energy conditionsdo not hold for quantum fields, which can have negative energydensities. This opens up the possibility that a model includingquantum fields may exhibit a “bounce” rather than collapseto a singularity. More fundamentally, GR’s classical spacetimedescription may fail to approximate the description provided by a fulltheory of quantum gravity. According to recent work applying loopquantum gravity to cosmology, spacetime collapses to a minimum finitesize rather than reaching a true singularity (Ashtekar & Singh2011; Bojowald 2011). On this account, GR fails to provide a goodapproximation in the region of the bounce, and the apparentsingularity is an artifact. Classical spacetime “emerges”from a state to which familiar spacetime concepts do not apply. Thereare several accounts of the early universe, motivated by string theoryand other approaches, that similarly avoid the initial singularity dueto quantum gravity effects.

3.3 Puzzling Features of the Initial State

In practice, cosmologists often take the physical state at theexpected boundary of the domain of applicability of GR as the“initial state”. (For example, this might be taken as thestate 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 featuresof the universe backwards leads to an initial state with threepuzzling features:^[35]

• Uniformity: The FLRW models have a finite particlehorizon distance, much smaller than the scales at which we observe the CMB.^[36] Yet the isotropy of the CMB, among other observations, indicate thatdistant 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 isdriven rapidly away from critical density under FLRW dynamics if\(\Lambda = 0\) and \(\rho+3p>0\). Given later observations, theinitial state has to bevery 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 thatare coherent on large scales and have a specific amplitude,constrained by observations. It is challenging to explain bothproperties dynamically. In the standard FLRW models, the perturbationshave to be coherent on scales much larger than the Hubble radius atearly times.^[38]

On a more phenomenological approach, the gravitational degrees offreedom of the initial state could simply be chosen to fit with laterobservations, but many proposed “theories of initialconditions” aim to account for these features based on newphysical principles. The theory of inflation discussed below aims toexplain these issues.

3.4 Theories of the Initial State

There are three main approaches to theories of the initial state, allof which have been pursued by cosmologists since the late 60s indifferent forms. Expectations for what a theory of initial conditionsshould achieve have been shaped, in particular, by inflationarycosmology. Inflation provided a natural account of the three otherwisepuzzling features of the initial state emphasized in the previoussection. Prior to inflation, these features were regarded as“enigmas” (Dicke & Peebles 1979), but after inflation,accounting for these features has served as an eligibility requirementfor any proposed theory of the early universe.

The first approach aims to reduce dependence on special initialconditions by introducing a phase of attractor dynamics. This phase ofdynamical evolution “washes away” the traces of earlierstates, in the sense that a probability distribution assigned overinitial states converges towards an equilibrium distribution. Misner(1968) introduced a version of this approach (his “chaoticcosmology program”), proposing that free-streaming neutrinoscould isotropize an initially anisotropic state. Inflationarycosmology was initially motivated by a similar idea: a“generic” or “random” initial state at thePlanck time would be expected to be “chaotic”, far from aflat FLRW model. During an inflationary stage, arbitrary initialstates are claimed to converge towards a state with the three featuresdescribed above.

The second approach regards the initial state as extremely specialrather than generic. Penrose, in particular, has argued that theinitial state must be very special to explain time’s arrow; theusual approaches fail to take seriously the fact that gravitationaldegrees of freedom are not excited in the early universe like theothers (Penrose 2016). Penrose (1979) treats the second law as arisingfrom a law-like constraint on the initial state of the universe,requiring that it has low entropy. Rather than introducing asubsequent stage of dynamical evolution that erases the imprint of theinitial state, we should aim to formulate a “theory of initialconditions” that accounts for its special features.Penrose’s conjecture is that the Weyl curvature tensorapproaches zero as the initial singularity is approached; hishypothesis is explicitly time asymmetric, and implies that the earlyuniverse approaches an FLRW solution. (It does not account for theobserved perturbations, however.) Later he proposed the idea ofConformal Cyclic Cosmology, where such a special initial state at thestart of one expansion epoch is the result of expansion in a previousepoch that wiped out almost all earlier traces of matter and radiation(Penrose 2016).

A third approach rejects the framework accepted by the other twoproposals, and regards the “initial state” as a misnomer:it should instead by regarded as a “branch point” whereour pocket universe separated off from a larger multiverse. (There arestill, of course, questions regarding the initial state of themultiverse ensemble, if one exists.) We will return to this approachin§5 below.

A dynamical approach, even if it is successful in describing a phaseof the universe’s evolution, arguably does not offer a completesolution to the problem of initial conditions: it collapses into oneof the other two approaches. For example, an inflationary stage canonly begin in a region of spacetime if the inflaton field and thegeometry are uniform over a sufficiently large region, such that thestress-energy tensor is dominated by the potential term (implying thatthe derivative terms are small) and the gravitational entropy issmall. There are other model-dependent constraints on the initialstate of the inflaton field. One way to respond is to adoptPenrose’s point of view, namely that this reflects the need tochoose a special initial state, or to derive one from a previousexpansion phase. The majority of those working in inflationarycosmology instead appeal to the third approach: rather than treatinginflation as an addition to standard big-bang evolution in a singleuniverse, we should treat the observed universe as part of amultiverse, discussed below. But even this must have a theory ofinitial conditions.

3.5 The Limits of Science

Cosmology provokes questions about the limits of scientificexplanation because it lacks many of the features that are present inother areas of physics. Physical laws are usually regarded ascapturing the features of a type of system that remain invariant undersome changes, and explanations often work by placing a particularevent in larger context. Theories of the initial state cannot appealto either idea: we have access to only one universe, and there is nolarger context to appeal to in explaining its properties. Thiscontrast between the types of explanation available in cosmology andother areas of physics has often led to dissatisfaction (see, e.g.,Unger & Smolin 2014). At the very least, cosmology forces us toreconsider basic questions about modalities, and what constitutesscientific explanation.

One challenge to establishing theories of the initial state isentirely epistemic. As emphasized in§2.4, we lack independent experimental probes of physics at the relevantscales, so the extensions of core theories described above are onlytested indirectly through their implications for cosmology. Thislimitation reflects contingent facts about the universe, namely thecontrast between the energy scales of the early universe and thoseaccessible to us, and does not follow from the uniqueness of theuniverse per se. Yet this limitation does not imply that it would beimpossible to establish laws. There are cases in the history ofphysics, such as celestial mechanics, where confidence in atheory’s laws is based primarily on successful application undercontinually improving standards of precision.

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

Competing philosophical analyses of laws of nature render differentverdicts on the possibility of cosmological laws. Cosmological laws,if possible, differ from local physical laws in a variety ofways—they do not apply to subsystems of the universe, they lackmultiple instances, and etc. Philosophical accounts of laws takedifferent features to be essential to law-hood. For example, theinfluential Mill-Ramsey-Lewis account takes the laws to be axioms ofthe deductive system capturing some body of physical knowledge thatoptimally balances strength (the scope of derived claims) andsimplicity (the number of axioms) (see, e.g., Loewer 1996). It isquite plausible that a constraint on the initial state, such asPenrose’s Weyl curvature hypothesis, would count as a law onthis account. By contrast, accounts that take other features, such asgoverning evolution, as essential, reach the opposite verdict.

Finally, there are a number of conceptual pitfalls regarding whatwould count as an adequate “explanation” of the origins ofthe universe. What is the target of such explanations, and what can beused in providing an explanation? The target might be the statedefined at the earliest time when extrapolations based on the SM canbe trusted. The challenge is that this state then needs to beexplained in terms of a physical theory, quantum gravity, whose basicconcepts are still obscure to us. This is a familiar challenge inphysics, where substantial work is often required to clarify howcentral concepts (such as space and time) are modified by a newtheory. An explanation of origins in this first sense would explainhow it is that classical spacetime emerges from a quantum gravityregime. While any such proposals remain quite speculative, the form ofthe explanation is similar to other cases in physics: what isexplained is the applicability of an older, less fundamental theorywithin some domain. Such an explanation does not address ultimatequestions regarding why the universe exists—instead, suchquestions are pushed back one step, into the quantum gravityregime.

Many discussions of origins pursue a more ambitious target: they aimto explain the creation of the universe “from nothing”.^[39] The target is the true initial state, not just the boundary ofapplicability of the SM. The origins are supposedly then explainedwithout positing an earlier phase of evolution; supposedly this can beachieved, for example, by treating the origin of the universe as afluctuation away from a vacuum state. Yet obviously a vacuum state isnot nothing: it exists in a spacetime, and has a variety ofnon-trivial properties. It is a mistake to take this explanation assomehow directly addressing the metaphysical question of why there issomething rather than nothing.^[40]

4. Anthropic Reasoning and Multiverse

4.1 Anthropic Reasoning

The physical conditions necessary for our existence impose a selectioneffect on what we observe. The significance of this point forcosmological theorizing is exemplified by Dicke’s criticism ofDirac’s speculative “large number hypothesis”. Dirac(1937) noted the age of the universe expressed in terms of fundamentalconstants in atomic physics is an extremely large number (roughly\(10^{39}\)), which coincides with other large, dimensionless numbersdefined in terms of fundamental constants. Inspired by thiscoincidence, he proposed that the large numbers vary to maintain thisorder of magnitude agreement, implying (for example) that thegravitational “constant” \(G\) is a function of cosmictime. Dicke (1961) notedthat creatures like us, made of carbon produced in an earliergeneration of red giants and sustained by the light and heat of a mainsequence star, can only exist within a restricted interval of cosmictimes, and that Dirac’s coincidence holds for observations madewithin this interval. Establishing that the coincidence holds at arandomly chosen \(t\) would support Dirac’s hypothesis, howeverslightly, but Dicke’s argument shows thatour evidencedoes not do so.

Dicke’s reasoning illustrates how taking selection effects intoaccount can mitigate surprise, and undermine the apparent implicationsof facts like those noted by Dirac (see Roush 2003). These factsreflect biases in the evidence available to us, rather than supportinghis hypothesis. It is also clear that Dicke’s argument is“anthropic” in only a very limited sense: his argumentdoes not depend on a detailed characterization of human observers. Allthat matters is that we can exist at a cosmic time constrained by thetime scales of stellar evolution.

How to account for selection effects, within a particular approach toconfirmation theory, is one central issue in discussions of anthropicreasoning. This question is intertwined with other issues that aremore muddled and contentious. Debates among cosmologists regarding“anthropic principles” ignited in the 70s, prompted by thesuggestion that finely-tuned features of the universe—such asthe universe’s isotropy (Collins & Hawking 1973)—canbe explained as necessary conditions for the existence ofobservers.^[41]More recently, a number of cosmologists have argued that cosmologicaltheories should be evaluated based on predictions for what a“typical” observer should expect to see. These ideas havedovetailed with work in formal epistemology. A number of philosophershave developed extensions of Bayesianism to account for“self-locating” evidence, forexample.^[42]This kind of evidence includes indexical information characterizing anagent’s beliefs about their identity and location. At presentwork in this area has not reached a consensus, and we will present abrief overview of some of the considerations that have motivateddifferent positions in these debates.

In cosmology the most famous example of an “anthropicprediction” is Weinberg (1987)’s prediction for \(\Lambda\).^[43] One part of Weinberg’s argument is similar to Dicke’s: heargued that there are anthropic bounds on \(\Lambda\), due to itsimpact on structure formation. The existence of large, gravitationallybound 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 “typicalobserver” should see. He assumed that observers occupy differentlocations within a multiverse, and that the value of \(\Lambda\)varies across different regions. Weinberg further argues that theprior probability assigned to different values of \(\Lambda\) shouldbe uniform within the anthropic bounds. Typical observers shouldexpect 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 principleof indifference, applied to a class of observers.^[44] We should calculate what we expect to observe, that is, as if we area “random choice” among all possible observers.^[45] Bostrom (2002) argues that indifference-style reasoning is necessaryto respond to the problem of “freak observers”. As Bostromformulates it, the problem is that in an infinite universe,any observation \(O\) is true forsome observer(even if only for an observer who has fluctuated into existence fromthe vacuum). His response is that we should evaluate theories basednot on the claim thatsome observer sees \(O\), but on anindexical claim: that is,we make the observation \(O\). Heassumes that we are a “random” choice among the class ofpossible observers. (How to justify such a strong claim is a majorchallenge for this line of thought.) If we grant the assumption, thenwe 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 firstis called the “reference class” problem. The assignmentsof probabilities to events requires specifying how they are grouped together.^[46] Obviously, what is typical with respect to one reference class willnot be typical with respect to another (compare, for example,“conscious observers” with “carbon-basedlife”). Second, the principle of indifference has beenthoroughly criticized as a justification for probability in othercontexts; what justifies the use of indifference in this case? Whyshould we take ourselves as “randomly chosen” among anappropriate reference class? The third problem reflects the intendedapplication of these ideas: Bostrom and other authors in this line ofwork are particularly concerned with observes that may occupy aninfinite universe. There is no proof that the universe is in factinfinite. These are all pressing problems for those who hold that theprinciple of indifference is essential to making cosmologicalpredictions.

Furthermore, one way of implementing this approach leads to absurdconsequences. The Doomsday Argument, for example, claims to reach astriking conclusion about the future of the human species without anyempirical input (see, e.g., Leslie 1992; Gott 1993; Bostrom 2002).Suppose that we are “typical” humans, in the sense ofhaving a birth rank that is randomly selected among the collection ofall humans that have ever lived. We should then expect that there arenearly as many humans before and after us in overall birth rank. Forthis to be true, given current rates of population growth, there mustbe a catastrophic drop in the human population(“Doomsday”) in the near future. The challenge toadvocates of indifference applied to observers is to articulateprinciples 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 rulesof reasoning applicable to predictions made by observers in a large orinfinite universe. This line of work is motivated by the idea thatwithout such principles we face a severe skeptical predicament, asobservations would not have any bearing on the theory. Yet there isstill not general agreement on the new principles required to handlethese cases, which are of course not scientifically testableprinciples: they are philosophically based proposals. According to analternative approach, selection effects can and should be treatedwithin the context of a Bayesian approach to inductive inference (seeNeal 2006; Trotta 2008). On this line of thought,“predictions” like those that Bostrom and others hope toanalyze play no direct role in the evaluation of cosmologicaltheories, so further principles governing anthropic reasoning aresimply not necessary. There is much further work to be done inclarifying and assessing these (and other) approaches to anthropic reasoning.^[47]

4.2 Fine-Tuning

Fine-tuning arguments start from a conflict between two differentperspectives on certain features of cosmology (or other physicaltheories). On the first perspective, the existence of creatures likeus seems to be sensitive to a wide variety of aspects of cosmology andphysics. To be more specific, the prospects for life dependsensitively on the values of the various fundamental constants thatappear in these theories. The SM includes about 10 constants, and theparticle physics standard model includes about 20 more.^[48] Tweaking the SM, or the standard model of particle physics, bychanging the values of these constants seems to lead to a barren cosmos.^[49] Focusing on the existence of “life” runs the risk ofbeing too provincial; we don’t have a good general account ofwhat physical systems can support intelligent life. Yet it does seemplausible that intelligence requires an organism with complexstructural features, living in a sufficiently stable environment.

At a bare minimum, the existence of life seems to require theexistence of complex structures at a variety of scales, ranging fromgalaxies to planetary systems to macro-molecules. Such complexity isextremely sensitive to the values of the fundamental constants ofnature. From this perspective, the existence of life in the universeis fragile in the sense that it depends sensitively on these aspectsof the underlying theory.

This view contrasts sharply with the status of the constants from theperspective of fundamental physics. Particle physicists typicallyregard their theories as effective field theories, which suffice fordescribing interactions at some specified energy scale. These theoriesinclude various constants, characterizing the relative strength of theinteractions they describe, that cannot be further explained by theeffective field theory. The constants can be fixed by experimentalresults, but are not derivable from fundamental physical principles.(If the effective field theory can be derived from a more fundamentaltheory, the value of the constants can in principle be determined byintegrating out higher-energy degrees of freedom. But this merelypushes the question back one step: the constants appearing in the morefundamental theory are determined experimentally.) Similarly, theconstants appearing in the SM are treated as contingent features ofthe universe. There is no underlying physical principle that sets, forexample, the cosmological densities of different kinds of matter, orthe value of the Hubble constant.

So features of our theories that appear entirely contingent, from thepoint of view of physics, are necessary to account for the complexityof the observed universe and the very possibility of life. Thefine-tuning argument starts from a sense of unease about thissituation: shouldn’t something as fundamental as the complexityof the universe be explained by thelaws orbasicprinciples of the theory, and not left to brute facts regardingthe values of various constants? The unease develops into seriousdiscomfort if the specific values of the constants are taken to beextremely unlikely: how could the values of all these constants bejust right, by sheer coincidence?

In many familiar cases, our past experience is a good guide to when anapparent coincidence calls for further explanation. As Humeemphasized, however, intuitive assessments from everyday life ofwhether a given event is likely, or requires a further explanation, donot extend to cosmology. Recent formulations of fine-tuning argumentsoften introduce probabilistic considerations. The constants are“fine-tuned”, meaning that the observed values are“improbable” in some sense. Introducing a well-definedprobability over the constants would provide a response to Hume:rather than extrapolating our intuitions, we would be drawing on theformal machinery of our physical theories to identify fine-tuning.Promising though this line of argument may be, there is not an obviousway to define physical probabilities over the values of differentconstants, or over other features of the laws. There is nothing likethe structure used to justify physical probabilities in othercontexts, such as equilibrium statistical mechanics.^[50]

There are four main responses to fine-tuning:

• Empiricist Denial: This response follows Hume in denyingthat a clear problem has even been identified. One form of thisresponse challenges appeals to probability, undermining the claim thatthere are unexplained coincidences. Alternatively, fine-tuning istaken to reveal that the laws alone are not sufficient to account forsome features of nature; these features are properly explained by thelaws in conjunction with various contingent facts.
• Designer: Newton famously argued, for example, that thestability of the solar system provides evidence of providentialdesign. For the hypothesized Designer to be supported by fine-tuningevidence, we require some way of specifying what kind of universe theDesigner is likely to create; only such a specific Design hypothesis,based in some theory of the nature of the Designer, can offer anexplanation of fine-tuning.
• New Physics: The fine-tuning can be eliminated bymodifying physical theory in a variety of ways: altering the dynamicallaws, introducing new constraints on the space of physicalpossibilities (or possible values of the constants of nature),etc.
• Multiverse: Fine-tuning is explained as a result ofselection, from among a large space of possible universes (ormultiverse).

In the next section we discuss the last response in more detail; see§3 for further discussion of the third response.

4.3 Multiverse

The multiverse response replaces a single, apparently finely-tuneduniverse within an ensemble of universes, combined with an appeal toanthropic selection. Suppose that all possible values of thefundamental constants are realized in individual elements of theensemble. Many of these universes will be inhospitable to life. Incalculating the probabilities that we observe specific values of thefundamental constants, we need only consider the subset of universecompatible with the existence of complexity (or some more specificfeature associated with life). If we have some way of assigningprobabilities over the ensemble, we could then calculate theprobability associated with our measured values. These calculationswill resolve the fine-tuning puzzles if they show that we observetypical values for a complex (or life-permitting) universe.

Many cosmologists have argued in favor of a specific version of themultiverse called eternal inflation (EI).^[51] On this view, the rapid expansion hypothesized by inflationarycosmology continues until arbitrarily late times in some regions, andcomes to an end (with a transition to slower expansion) in others.This leads to a global structure of “pocket” universesembedded within a larger multiverse.

On this line of thought, the multiverse should be accepted for thesame reason we accept many claims about what we cannot directlyobserve—namely, as an inevitable consequence of an establishedphysical theory. It is not clear, however, that EI is inevitable, asnot all inflationary models, arguably including those favored by CMBobservations, 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 toundermine the original reasons for accepting inflation (Smeenk 2014):rather than the predictions regarding the state produced at the end ofinflation taken to provide evidence for inflation, EI seems to implythat, as Guth (2007) put it, in EI “anything that can happenwill happen; in fact, it will happen an infinite number oftimes”.

There have been two distinct approaches to recovering some empiricalcontent in this situation.^[54] First, there may be traces of the early formation of the pocketuniverses, the remnants of collisions between neighboring“bubbles”, left on the CMB sky (Aguirre & Johnson2011). Detection of a distinctive signature that cannot be explainedby other means would provide evidence for the multiverse. However,there is no expectation that a multiverse theory would genericallypredict such traces; for example, if the collision occurs too earlythe 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 yieldvariation in the local, low-energy physics in each pocket. Predictionsfor the values of the fundamental constants follow from two things:(1) a specification of the probabilities for different values of theconstant over the ensemble, and (2) a treatment of the selectioneffect imposed by restricting consideration to pocket universes withobservers and then choosing a “typical” observer.

The aim is to obtain probabilistic predictions for what a typicalobserver should see in the EI multiverse. Yet there are severalchallenges to overcome, alongside those mentioned above related toanthropics. The assumption that the formation of pocket universesleads to variation in constants is just an assumption, which is notyet justified by a plausible, well-tested dynamical theory. The mostwidely discussed challenge in the physics literature is the“measure problem”: roughly, how to assign“size” to different regions of the multiverse, as a firststep towards assigning probabilities. It is difficult to define ameasure because the EI multiverse is usually taken to be an infiniteensemble, lacking in the kinds of structure used in constructing ameasure. On our view, these unmet challenges undercut the hope thatthe EI multiverse yields probabilistic predictions. And without suchan account, the multiverse proposal does not have any testableconsequences. If everything happens somewhere in the ensemble, thenany potential observation is compatible with the theory.

Supposing that we grant a successful resolution of all thesechallenges, the merits of a multiverse solution of fine-tuningproblems could then be evaluated by comparison with competing ideas.The most widely cited evidence in favor of a multiverse isWeinberg’s prediction for the value of \(\Lambda\), discussedabove. There are other proposals to explain the observed value of\(\Lambda\); Wang, Zhu, and Unruh (2017), for example, treat thequantum vacuum as extremely inhomogeneous, and argue that resonanceamong the vacuum fluctuations leads to a small \(\Lambda\).

The unease many have about multiverse proposals are only reinforced bythe liberal appeals to “infinities” in discussion of the idea.^[55] Many have argued, for example, that we must formulate an account ofanthropic reasoning that applies to a truly infinite, rather thanmerely very large, universe. Claims that we occupy one of infinitelymany possible pocket universes, filled with an infinity of otherobservers, rest on an enormous and speculative extrapolation. Suchclaims fail to take seriously the concept of infinity, which is notmerely a large number. Hilbert (1925 [1983]) emphasized that whileinfinity is required to complete mathematics, it does not occuranywhere in theaccessible physical universe. One response isto require that infinities in cosmology should have a restricted use.It may be useful to introduce infinity as part of an explanatoryaccount of some aspect of cosmology, as is common practice inmathematical models that introduce various idealizations. Yet thisinfinity should be eliminable, such that the explanation of thephenomena remains valid when the idealization is removed.^[56] Even for those who regard this demand as too stringent, therecertainly needs to be more care in clarifying and justifying claimsregarding infinities.

In sum, interest in the multiverse stems primarily from speculationsabout the consequences of inflation for the global structure of theuniverse. The main points of debate regard whether EI is a disasterfor inflation, undermining the possibility of testing inflation atall, and how much predictions such as that for \(\Lambda\) lendcredence to these speculations.^[57] Resolution of these questions is needed to decide whether themultiverse can be tested in a stronger sense, going beyond the specialcases (such as bubble collisions) that may provide more directevidence.

5. Testing models

As mentioned at the start, the uniqueness of the universe raisesspecific problems as regards cosmology as a science. First we considerissues to do with verification of cosmological models, and then make acomment as regards interpreting the human implications ofcosmology

5.1 Criteria

The basic challenge in cosmology regards how to test and evaluatecosmological models, given our limited access to the unique universe.As discussed above, current cosmological models rely in part onextrapolations of well-tested local physics along with novelproposals, such as the inflaton field. The challenge is particularlypressing in evaluating novel claims that only have cosmologicalimplications, due to the physics horizon (§2.4). Distinctions that are routinely employed in other areas of physics,such as that between laws and initial conditions, or chance andnecessity, are not directly applicable, due to the uniqueness of theuniverse.

Recent debates regarding the legitimacy of different lines of researchin cosmology reflect different responses to this challenge. Oneresponse is to retreat to hypothetico-deductivism (HD): a hypothesisreceives an incremental boost in confidence when one of itsconsequences is verified (and a decrease if it is falsified).^[58] Proponents of inflation argue, for example, that inflation should beaccepted based on its successful prediction of a flat universe with aspecific spectrum of density perturbations. Some advocates of themultiverse 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 asufficient account of how evidence supports theories (this is oftencalled “naïve HD”). In particular, the naïveview lacks the resources to draw distinctions among underdeterminedrival theories that make the same predictions (see Crupi 2013 [2016]).We take it as given that scientists do draw distinctions amongtheories that naïve HD would treat as on par, as is reflected injudgments regarding how much a given body of evidence supports aparticular theory. Scientists routinely distinguish among, forexample, theories that may merely “fit the data” asopposed to those that accurately capture laws governing a particulardomain, and evaluate some successful predictions as being far morerevealing than others.

A second response is that the challenge requires a more sophisticatedmethodology. This may take the form of acknowledging explicitly thecriteria that scientists use to assess desirability of scientifictheories (Ellis 2007), which include considerations of explanatorypower, consistency with other theories, and other factors, in additionto compatibility with the evidence. These come into conflict inunexpected ways in cosmology, and these different factors should beclearly articulated and weighed against one another. Alternatively,one might try to show that some of these desirable features, such asthe ability to unify diverse phenomena, should be taken as part ofwhat constitutes empirical success.^[59] This leads to a more demanding conception of empirical success,exemplified by historical cases such as Perrin’s argument infavor 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 betweenCosmology, which isthe physically based subject dealt with in the textbooks listed inthis article, dealing with the expansion of the universe, galaxies,number counts, background radiation, and so on, andCosmologia, where one takes all that as given but adds inconsideration about the meaning this all has for life. Clearly theanthropic discussions mentioned above are a middle ground.^[60] However a number of popular science books by major scientists areappearing that make major claims about Cosmologia, based purely inarguments from fundamental physics together with astronomicalobservations. We will make just one remark about this here. If one isgoing to consider Cosmologia seriously, it is incumbent on one to takeseriously the full range of data appropriate to that enterprise. Thatis, the data needed for the attempted scope of such a theory mustinclude data to do with the meaning of life as well as data derivedfrom telescopes, laboratory experiments, and particle colliders. Itmust thus include data about good and evil, life and death, fear andhope, love and pain, writings from the great philosophers and writersand artists who have lived in human history and pondered the meaningof life on the basis of their life experiences. This is all of greatmeaning to those who live on Earth (and hence in the Universe). Toproduce books saying that science proves there is no purpose in theuniverse is pure myopia. It just means that one has shut ones eyes toall the data that relates to purpose and meaning; and that onesupposes that the only science is physics (for psychology and biologyare full of purpose).

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Acknowledgments

Work on this entry was supported by a grant from the John TempletonFoundation. The statements made here are those of the authors and arenot necessarily endorsed by the Foundation.