1 Introduction
Philosophers often take interpreting theories to be one of their professional obligations, to be discharged by providing an account of what the world would be like if a given theory were true. But exclusive focus on the theory–world relation, important though that may be, overlooks the insights to be gained by assessing how a theory fits into the space of possible alternative theories. Physicists routinely consider modifications, reformulations, and generalizations of a given theory in order to gain insight into its structure and viability. In this article, we will consider some of the reasons for exploring “theory space” and the potential benefits of doing so, as exemplified by recent work in cosmology.
We will argue that exploring theory space is not only common but also potentially fruitful. But the case we consider also illustrates some limitations of the approach. As we discuss, it is often challenging to sharply delineate apparently distinct directions or locations in theory space on empirical or even conceptual grounds. We will argue that this undermines some, but not all, of the interpretative benefits of shifting the focus of analysis up one level from theories to theory spaces. The upshot will be a nuanced view of what a theory-space approach can offer.
Our examples of theory-space exploration concern modifications to general relativity (GR). Alternatives to GR have been explored extensively over the last century, with at least three distinct motivations. First, a modified theory may provide a better path to a successor theory. The application of quantization techniques that have worked for other classical field theories to GR does not yield a perturbatively renormalizable theory. Among the many avenues of research that have been pursued in quantum gravity, one involves modifying classical GR to ease the application of quantization techniques or to yield a better outcome. Second, the exploration of alternatives has also been used to assess the “rigidity” of GR by showing, for example, that some modifications lead to pathological theories. It is appealing to seek physical theories that are overconstrained, in the sense that different structural features fit tightly together, with little scope for alterations. It is then more plausible that a few physical facts about the nature of gravity, along with general principles, are sufficient to pick out GR as the best (classical) theory of gravity.
We will focus primarily on a third motivation: the use of alternative theories to assess the robustness of empirical inferences. Cosmologists and astrophysicists routinely rely on GR to make inferences regarding the types of matter and energy in the universe and the nature of specific astrophysical systems. To what extent do the same conclusions follow if GR is replaced with an alternative theory?
The rationale for considering modifications of GR varies for different regimes of applicability. Physicists apply GR to an extremely wide range of physical environments, spanning several orders of magnitude in parameters measuring space-time curvature and the strength of the gravitational potential. GR is assumed to fail in the ultraviolet (high-energy, short-length-scale) regime, where quantum effects are expected to be relevant. In cosmology, the target systems are too big for GR to fail for this reason. There are no comparably persuasive theoretical arguments that suggest a “maximum length scale” for GR. Yet cosmology involves an enormous extrapolation from the length scales where GR has been most rigorously tested—the scale of the solar system or even smaller systems such as binary black holes.
Extrapolating over some
$14$
orders of magnitude is surely enough to make cautious empiricists wary. Systematic exploration of alternatives to GR at large length scales started in earnest in the early 2000s in response to the discovery that the universe’s rate of expansion is accelerating. In light of these observations, GR (along with standard cosmological assumptions) yields the striking conclusion that the vast majority of mass-energy in the universe comes in the form of an effective cosmological constant (
${\rm{\Lambda }}$
) or a type of matter that mimics its effects, “dark energy.” Although classical GR could accommodate an accelerating universe without modification, cosmologists have generally been reluctant to accept such a large
${\rm{\Lambda }}$
-like contribution. Instead, many have explored whether a cosmological constant can be avoided by modifying GR at cosmological scales.
Here, we focus on the benefits and challenges of exploring theory space as illustrated by this case of dark-energy research, broadly construed. We begin with a brief overview (in sec. 2) of contemporary dark-energy phenomenology, with an emphasis on the role of assumptions of different types (including GR but also cosmological assumptions) in supporting the inference to a large dark-energy contribution. In section 3, we discuss three different responses to these phenomenological results: treating
${\rm{\Lambda }}$
as a true constant, treating dark energy dynamically as a new form of matter, and modifying GR. We suggest that cosmologists’ exploration of these possibilities can be understood as a version of “closing the loop,” as described by George Smith (Reference Smith, Biener and Schliesser2014). This epistemic strategy, we suggest, has the potential to provide powerful evidence when it succeeds. But in this case, we argue, it has not succeeded. Instead, cosmologists are, to some degree, victims of their own success, with no opportunity to sharply distinguish between the theoretical options under consideration through further comparisons with observations.
We then turn in section 4 to question whether there is a clear contrast to be drawn at the foundational level between modified-gravity and dark-energy scenarios, and we critically assess one proposal for drawing that distinction. Finally, in the conclusion, we return to assess the implications of these difficulties for theory-space analysis of the sort considered here. It is feasible to assess the sensitivity of various empirical inferences to theoretical assumptions of different modifications without also classifying the modifications. The assessment of the plausibility of the modifications, however, or taking the phenomenology as a guide toward a successor theory, does require resolving the question in our title.
2 Measuring
${\bf{\Lambda }}$
The introduction of a cosmological constant
${\rm{\Lambda }}$
was, in effect, the first modification of GR: Einstein added the term 2 years after discovering his now-eponymous equation. Even for those wary of Einstein’s motivations, there are no compelling physical grounds to set its value to zero, as opposed to leaving it as a parameter to be determined by observations. Many midcentury treatments of relativistic cosmology set aside
${\rm{\Lambda }}$
, perhaps agreeing with Einstein’s regretful assessment that introducing it had been a mistake. Yet there have always been empirical hints that
${\rm{\Lambda }}$
should not be ignored. Early cosmological models faced an age crisis: some astrophysical objects (globular clusters, stars) appeared to be significantly older than the universe itself. A nonzero
${\rm{\Lambda }}$
severs the connection between the current expansion rate and the age of the universe responsible for this conflict. Several cosmologists preferred a model with flat spatial sections—such that the total matter-energy density sums to the so-called critical density. As observational estimates of matter density increased in accuracy, it became increasingly clear that reaching critical density would require a large contribution from
${\rm{\Lambda }}$
. These were two of the more prominent reasons cosmologists gave for taking
${\rm{\Lambda }}$
seriously, and there are several others of varying significance.
This situation changed dramatically in the late 1990s as cosmologists developed a compelling observational case that
${\rm{\Lambda }}$
accounts for roughly
$70{\rm{\% }}$
of the total mass-energy density of the universe—in other words,
${{\rm{\Omega }}_{\rm{\Lambda }}} \approx .7$
.
Footnote 1
During this period, cosmology transitioned from providing qualitative explanations of observed large-scale features of the universe to constraining parameters appearing in the “standard model” with increasing precision. Accepting the standard model allows cosmologists to bring a wide array of observations to bear to measure parameters, including the Hubble constant,
$H$
, and the density of different types of matter (
${{\rm{\Omega }}_m}$
, the total matter contribution, including baryonic and dark matter, and
${{\rm{\Omega }}_{\rm{\Lambda }}}$
). Cosmologists have clarified what types of observations provide the most precise constraints on different parameters and have also pursued sensitivity analysis, identifying the assumptions required to link observations to a parameter (or set of parameters) and then assessing to what extent this link is robust to variations.
Three types of observations provide the strongest constraints on the value of
${{\rm{\Omega }}_{\rm{\Lambda }}}$
. The first, going back to Hubble’s seminal observations of an approximately linear relationship between redshift and magnitude, takes galaxies as tracers of cosmic expansion. Granting that the universe at large scales can be approximately described by the simple homogeneous and isotropic models (the Friedmann–Lemaître–Robertson–Walker [FLRW] models), the dynamical equations support inferences from the redshift–magnitude relationship of “standard candles” to the densities and types of matter present. Ordinary matter leads to decelerating expansion—that is,
$\ddot R\left( t \right) \lt 0$
, where
$R\left( t \right)$
is the scale factor. The probative value of this kind of observation increases with distance, and starting in the 1990s, astrophysicists exploited supernovae (SNe) to extend to larger distances. Although they expected this to yield more precise measurements of deceleration, they instead discovered that the expansion rate appears to be accelerating. This implies that the dominant dynamical contribution (at late times) must come from a
${\rm{\Lambda }}$
-like source, needed to yield
$\ddot R\left( t \right) \gt 0$
. The two teams pursuing this line of observations both concluded that the “best-fit” FLRW model has
${{\rm{\Omega }}_m} \approx .3$
and
${{\rm{\Omega }}_{\rm{\Lambda }}} \approx .7$
.
This discovery came as a surprise to both teams, but two other lines of reasoning had already suggested a large value of
${\rm{\Lambda }}$
. Simulations of structure formation developed in the 1990s seemed to require a flat universe with nearly critical density. Given observational upper bounds on the total matter density, it was fairly routine to simply assume a large value of
${\rm{\Lambda }}$
. This was, at best, a qualitative constraint. Subsequent work has leveraged an understanding of structure formation to generate constraints on the underlying cosmological model through the study of baryon acoustic oscillations (BAOs). These oscillations provide cosmologists with a “standard ruler,” and observations of this length scale at different cosmological epochs constrain the expansion rate of the universe and the cosmological parameters.
Observations of the cosmic microwave background (CMB) also suggested the need for a large
${\rm{\Lambda }}$
. The angular position of the first acoustic peak in the power spectrum of temperature fluctuations in the CMB, for example, reveals that the universe at recombination had a flat spatial geometry. This implies that the total matter-energy density is close to the critical density (
${{\rm{\Omega }}_{{\rm{total}}}} = 1)$
, but it only weakly constrains individual contributions from
${{\rm{\Omega }}_m}$
and
${{\rm{\Omega }}_{\rm{\Lambda }}}$
. By the mid-1990s, however, several cosmologists made the case that other lines of evidence lead to an upper bound on
${{\rm{\Omega }}_m}$
, forcing the introduction of a large contribution from
${{\rm{\Omega }}_{\rm{\Lambda }}}$
.
Each of these three methods has limitations—the CMB and SNe measurements have significant degeneracy in the values of
${{\rm{\Omega }}_m}$
and
${{\rm{\Omega }}_{\rm{\Lambda }}}$
, whereas BAOs place tight constraints on
${{\rm{\Omega }}_m}$
but have little to say about
${{\rm{\Omega }}_{\rm{\Lambda }}}$
. Yet these degeneracies extend in different directions in parameter space, such that the three methods in conjunction do yield tight constraints on
$\left\{ {{{\rm{\Omega }}_m},{{\rm{\Omega }}_{\rm{\Lambda }}}} \right\}$
. Furthermore, each of these methods employs distinctive observational techniques and rests on different physical and cosmological assumptions. All three measurements rely on aspects of GR (to draw inferences from observed motions to the underlying sources), along with the approximate validity of FLRW models. They differ in several other respects, ranging from the kinds of objects studied to the physical and astrophysical details relevant to turning observations into parameter constraints.
Cosmologists have discovered something remarkable about the universe. Yet what is it? The phenomenology just summarized is compatible with three answers. First, treat
${\rm{\Lambda }}$
as a free parameter whose value is to be determined by observations and regarded as a contingent brute fact. In this view, the main question is then whether the different lines of evidence determine a consistent value of
${\rm{\Lambda }}$
. Second, attribute this phenomenology to a new type of mass-energy, “dark energy,” characterized by the fact that it mimics
${\rm{\Lambda }}$
in certain regimes. Hypothesized dynamics for dark energy could then lead to a richer phenomenology, differing from that of a true cosmological constant. Third, note that the previous inferences all rely on GR applied at large length scales, in conjunction with cosmological assumptions. Modifying GR in this regime may make it possible to account for the phenomenology without dark energy or a true cosmological constant.
3 Closing the cosmological loop?
Why, if well-established theory apparently accounts for all known observations, would anyone set aside the cosmological constant in favor of more exotic alternatives? Physicists are clearly not satisfied with merely measuring
${\rm{\Lambda }}$
and have introduced different possible ways of filling out the underlying physics.
One compelling answer that explains the appeal of doing so draws on George Smith’s account of the methodology of celestial mechanics (Smith Reference Smith, Biener and Schliesser2014). Over two centuries, astronomers iteratively developed more detailed accounts of the solar system by identifying robust physical sources for observational discrepancies and then adding these new sources back into an enriched theoretical model. There are three features of Smith’s analysis that are particularly salient. First, Smith emphasizes the need to have sufficient control over idealizations appearing in the description at any given stage of inquiry: the idealizations should identify regularities that would be exact in precisely specifiable (albeit counterfactual) circumstances. Second, measurements of fundamental quantities should remain stable even as the model undergoes further refinement. Third, the addition of robust sources has various impacts further downstream in the subsequent modeling that extend well beyond the resolution of the initial discrepancy. Smith argues that repeatedly closing the loop—by finding robust sources, adding them to the model, and pursuing the further implications of the enriched model—provides compelling evidence in favor of both the theory and the individual details incorporated at each successive stage.
It is clear why treating
${\rm{\Lambda }}$
as a true constant is unappealing to many cosmologists: it is sterile and fails to generate further consequences that can be pursued through theory or observational programs. It represents a dead end rather than a step toward further iterative refinements. It is much more appealing to assume that there is a robust physical source hiding under the facade of a true constant. Peebles and Ratra (Reference Peebles and Ratra2003), for example, note in the abstract of their review article: “Physics welcomes the idea that space contains energy whose gravitational effect approximates that of Einstein’s cosmological constant,
${\rm{\Lambda }}$
; today the concept is termed dark energy or quintessence” (559, our emphasis). It is crucial that the effects of dark energy only approximate that of a cosmological constant so that observational programs can isolate the differences and begin the process of iteratively learning more about the physics of dark energy.
One class of dark-energy models illustrates how the physics of dark energy could be revealed through cosmological observations. Quintessence models treat dark energy as a self-interacting scalar field, weakly coupled to other fields. There are two significant contrasts between these models and a true cosmological constant: the evolution of the scalar field with cosmic time generates a time-variation in the effective
${\rm{\Lambda }}$
, whereas a true
${\rm{\Lambda }}$
is fixed once and for all, and the field’s effective equation of state can vary (as a function of space and/or time). Cosmologists usually characterize these contrasts in terms of two parameters. The equation of state for a perfect fluid is given by
$w = {{p}\over{\rho }}$
, where
$p$
is the pressure, and
$\rho $
is the energy density. The first parameter
${w_0}$
reflects the current measured value of the equation of state; a second parameter
${w_a}$
characterizes its dependence on time (redshift). For a true cosmological constant,
${w_0} = - 1$
, and
${w_a} = 0$
; scalar field models are expected to lead to departures on large length and/or time scales.
After two decades of dedicated work, cosmologists have yet to find compelling evidence for such variations; the boring answer—a true cosmological constant, indicated by the dashed lines in figure 1—still fits the available data. As far as we are aware, there are, at present, no cosmological observations incompatible with taking
${\rm{\Lambda }}$
to be a true constant. To put it in slightly different terms, requiring that dark energy reproduces the phenomenology described earlier renders any new features inaccessible. The process of iterative development of dark-energy models cannot even get started.
Figure 1. The contours show parameter estimates at standard confidence levels for measurements of
$\left( {{w_0},{w_a}} \right)$
from cosmic microwave background (CMB), baryon acoustic oscillation (BAO), and supernova (SN) Ia. (From Huterer and Shafer Reference Huterer and Shafer2017, fig. 9.)
There are also foundational reasons for rejecting a true cosmological constant. Indeed, many cosmologists fail to consider the possibility that
${\rm{\Lambda }}$
could be regarded as a universal constant of nature, and they instead immediately identify it with vacuum energy density. In quantum field theory (QFT), the stress-energy tensor for vacuum energy density has the same form as a cosmological constant term. Conventional wisdom goes beyond this formal analogy to take the cosmological constant as nothing but vacuum energy; doing so leads to the “cosmological constant problem,” given the remarkable difference between the cosmological constraints on
${\rm{\Lambda }}$
discussed earlier and calculations of vacuum energy density in QFT.
Footnote 2
For several decades, physicists have sought new symmetries or other mechanisms that would provide a physical argument that vacuum energy density vanishes, which would imply (granting the presumed identity) a vanishing cosmological constant. The search continues, but even without a compelling argument along these lines, cosmologists have treated the first option as a nonstarter.
Finally, we turn to the possibility that modifications of GR account for the phenomenology described earlier. The measurements of
${\rm{\Lambda }}$
all depend on the dynamics of GR and the approximate validity of the FLRW models. Several cosmologists pursued the plausible suggestion that changing the long-range behavior of gravity could suffice to explain accelerating expansion (and other observations). One of the earliest proposals was based on
$f\left( R \right)$
theories, which replace the Einstein–Hilbert Lagrangian density,
${{\mathcal L}_{EH}} = kR$
, where
$k$
is a constant, and
$R$
is the scalar curvature, with some function of the scalar curvature,
${\mathcal L} = f\left( R \right)$
. Hu and Sawicki (Reference Hu and Sawicki2007), for example, constructed an
$f\left( R \right)$
theory that accounts for accelerated expansion without a cosmological constant or dark energy. The model can only reconcile this cosmological behavior with the constraints imposed by matching the success of GR on much smaller solar-system scales by introducing a mechanism to segregate the different scales. It is remarkable that the success of GR on solar-system scales places such tight constraints on GR at cosmological scales. But constructions like that of Hu and Sawicki (Reference Hu and Sawicki2007), and all other models we are aware of that match solar-system constraints, dramatically limit the possibility of further tests of the proposed novel physics. Modified-gravity scenarios are victims of GR’s success: the downstream consequences of the proposed new physics are simply not within empirical reach.
4 Foundations
Thus far, we have argued that there are in-principle challenges to empirically distinguishing modified-gravity from dark-energy scenarios or a true cosmological constant. In doing so, we have taken for granted that there is a clear conceptual difference. But as we will presently argue, it is not clear that a firm conceptual distinction between modified gravity and dark energy is possible, either. Space constraints forbid a complete discussion of this issue, but some brief remarks will point toward the difficulties.
To begin, we acknowledge that some theories appear to be unambiguous examples of modified gravity. Consider, for instance,
$f\left( R \right)$
theories (Sotiriou and Faraoni Reference Sotiriou and Faraoni2010). Such theories generally lead to higher-order generalizations of Einstein’s equation, but they do not lead to new degrees of freedom that might be taken for matter fields.
Footnote 3
But other theories are more ambiguous. Take Horndeski theories. These are theories described by the Horndeski Lagrangian density, which is the most general Lagrangian density, depending only on a Lorentzian metric and scalar field for which the Euler–Lagrange equations are second order (Horndeski Reference Horndeski1974). Horndeski theories have received considerable attention from theoretical cosmologists over the past decade because they encompass a very broad class of “reasonable” candidates for reproducing dark-energy phenomenology via a new dynamical field (Clifton et al. Reference Clifton, Ferreira, Padilla and Skordis2012). Studying the entire class of theories together, as special cases of a single very general Lagrangian density with several unknown functions, permits one to place constraints on many possible theories at once and rule out large swaths of theory space efficiently. This is a potentially powerful form of theory-space analysis.
But the generality of the Horndeski Lagrangian also introduces conceptual puzzles. Among the Horndeski theories are clear cases of dark-energy candidates, including quintessence (Caldwell et al. Reference Caldwell, Dave and Steinhardt1998). But also among the Horndeski theories are classic cases of modified-gravity theories, such as Brans–Dicke theory (Brans and Dicke Reference Brans and Dicke1961), which modifies GR by replacing the gravitational constant
${G^{ - 1}}$
with a dynamical field
$\varphi $
governed by its own dynamics. And one can also find hybrid theories that include elements of both. Indeed, even Brans–Dicke theory, viewed from this perspective, might be seen as an intermediate case because the dynamical gravitational constant in that theory also contributes to the stress-energy tensor and, in particular, can exchange energy and momentum with matter. This is so despite the fact that Brans and Dicke themselves intended it as a “modified relativistic theory of gravitation” (1961, abstract).
Cosmologists’ rare attempts to sharpen this distinction do not hold up under scrutiny. Footnote 4 Consider, for instance, the relatively influential proposal due to Joyce et al. (Reference Joyce, Lombriser and Schmidt2016), who argue that as a pragmatic matter, we should understand modified-gravity theories to be ones that lead to a “fifth force.” To make this precise, they invoke a version of the strong equivalence principle (SEP). The idea is that dark-energy theories will be ones that satisfy the SEP, whereas modified-gravity theories will not.
This is a compelling proposal, but the details do not work. Joyce et al. (Reference Joyce, Lombriser and Schmidt2016) propose their own statement of the SEP, which is the requirement that all massive bodies, including compact bodies such as black holes, follow geodesics (103). But as has been observed many times in the literature on geodesic motion in GR, there is no canonical or unambiguous way to associate an extended body with a curve in GR, so “geodesic motion” is not well defined (cf. Geroch and Weatherall Reference Geroch and Owen Weatherall2018). At best, this idea makes sense in a small-body limit. But even if one does consider the small-body limit, at first order in the size of the body, one expects deviations from geodesic motion due to finite-body effects, such as angular momentum (Gralla and Wald Reference Gralla and Wald2008). Thus, even in GR, one cannot expect compact bodies such as black holes to “follow geodesics,” except in some approximate or highly idealized sense. It seems geodesic motion for compact bodies cannot clearly distinguish theories that otherwise approximate GR.
One might hope that other statements of the SEP would do better. It is possible that they would. But several authors have recently argued that there are multiple such statements in the literature that do not generally agree, and in any case, many of them are not sufficiently clear to apply in cases where there is any actual dispute (Fletcher and Weatherall Reference Fletcher and Owen Weatherall2022). Moreover, other standard statements, which tend to concern the “locally special relativistic” character of matter field equations, do not appear to capture Joyce et al.’s (Reference Joyce, Lombriser and Schmidt2016) intuition that “fifth-force” effects are characteristic of modified gravity.
5 Conclusion
Physicists often devote serious time and attention to alternatives to well-established theories, even when empirical considerations do not force them to do so. Examples include the practice of phenomenological model building in particle theory, some work in the foundations of quantum theory, and as we have discussed here, the study of modified gravitational theories in dark-energy cosmology (among others). These activities can play many epistemic and other roles, and the practice deserves further scrutiny from philosophers of science. In this article, we have focused on the promise and perils of one role this sort of activity can play in modern physics, which is to identify potentially fruitful opportunities for theory-mediated measurements that, under the right circumstances, can provide a particularly strong form of evidence. In this endeavor, we took inspiration from the “closing the loop” methodology described by George Smith, who has argued that an iterative process of identifying robust physical sources for unexpected phenomena and then studying the downstream effects of those sources was key to developing the rich evidence for Newtonian gravitational theory that was accumulated between the 1680s and 1900.
But alas, although we maintain that this strategy is both powerful in principle and successfully used in other cases in cosmology, it has not yielded results in the present case, for reasons we suggest are informative about both cosmology and the method. In the case of dark-energy research, it is increasingly clear that both modifications of GR and exotic matter theories proposed to mimic a cosmological constant are highly constrained by established physics on other scales. Thus, they must be carefully tuned so that their only downstream effects concern precisely the phenomena that they were introduced to explain. For this reason, although some of these proposals would count as potentially robust physical sources for cosmic acceleration, that robustness cannot be established via their effects on other systems or in other regimes. The loop cannot close. To make matters worse, we argued, even conceptually, it turns out that it is difficult to sharply distinguish proposed sources of apparently different types. This means that even qualitative proposals, the details of which might be filled in in different ways, do not lead to clear observational signatures that might suggest that one path, rather than another, is the most fruitful way forward. The result is that every proposal that is viable mimics every other proposal that is viable, empirically and conceptually.
Of course, this does not mean that theory-space exploration is a useless enterprise or that it was a bad idea in this particular case. Instead, we take this discussion to show one reason that physicists might study theories with minimal empirical justification—as part of a broader research strategy—but also to highlight the failure modes for that kind of approach because the main example of a similar methodology in the literature is one of extraordinary success.
Acknowledgments
It is a pleasure to thank Helen Meskhidze for organizing the Philosophy of Science Association symposium where this work was presented, audience members for questions and discussion, and Patrick Duerr for comments on the manuscript.
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