2. The $\bf \Lambda$ CDM model

$\Lambda$ CDM takes perturbed Friedmann–Lemaître (FL) models to represent the large-scale structure of the universe. These are solutions of Einstein’s field equations (EFE) with maximal symmetry: they describe space-time as foliated by a collection of three-dimensional hypersurfaces of constant curvature over cosmic time, $\Sigma (t)$ (see, e.g., Weinberg Reference Weinberg2008). The world lines of “fundamental observers,” moving along geodesics and at rest with respect to matter, are orthogonal to these surfaces, and clocks carried by fundamental observers would measure cosmic time. The scale factor $R(t)$ represents spatial distance in $\Sigma$ between nearby fundamental observers, flowing along with the cosmic evolution. Spatial volume changes as a function of cosmic time, proportional to ${\dot{R}(t)}/{R(t)}$ . This quantity is called the (time-varying) Hubble “constant” $H(t)$ , with $H_0$ specifying the value at the present time. Owing to symmetries, EFE reduce to two ordinary differential equations for the scale factor, which can be solved given the equations of state and relative densities of the postulated material constituents. The pristine symmetry of these models has to be perturbed slightly to account for the existence of large-scale structures: structures like galaxies arise via gravitational enhancement of small “seed” perturbations in the early universe.

Free parameters in the $\Lambda$ CDM model characterize the space-time geometry and the constituents of the universe. One set of parameters characterizes the space-time geometry, given by a flat FL model (characterized in part by $H(t)$ ) and the seed perturbations (fully characterized by two parameters, if they are Gaussian). A second set specifies how much of each different type of matter is present, in terms of density parameters $\Omega _i$ . The list of different “types of matter” extends beyond familar types of matter (such as baryonic matter, $\Omega _b$ ) to include cold dark matter ( $\Omega _c$ ) and “dark energy.” The third and final set includes parameters that need to be specified to interpret observations, such as the ionization state of the early universe. There is some variation in the list of parameters cosmologists use in finding an optimal fit depending on the type of observations; cosmologists typically fit CMB data, for example, with six to nine parameters. The Hubble parameter is not included in this list for CMB observations but is treated instead as a derived quantity.

Cosmologists have developed a number of independent observational techniques for constraining these parameters, exemplified by the baryonic mass density $\Omega _b$ (for a detailed discussion, see Peebles Reference Peebles2020). Two of the most precise constraints on $\Omega _b$ derive from early-universe physics. According to big bang nucleosynthesis, primordial light element abundances are fixed by nuclear physics applied to the early universe. Primordial isotope abundances, in particular that of deuterium, depend sensitively on $\Omega _b$ (at the time of nucleosynthesis, $t \approx 10^2 s$ ). A second constraint follows from observations of the acoustic peaks in the CMB (from $t \approx 10^{11} s$ ). Prior to recombination, the speed of sound in the coupled plasma–radiation oscillations that produced these peaks depends on $\Omega _b$ , and the relative magnitude of the second peak measures its value. In both cases, cosmologists have derived equations relating an observable quantity—the primordial deuterium abundance or features of the acoustic peaks in the CMB—to $\Omega _b$ . The great appeal of these early-universe parameter constraints is the relative simplicity of the physics involved: the equations follow from well-understood nuclear and plasma physics, respectively. Astrophysical estimates of $\Omega _b$ (from the “late universe,” $t \approx 10^{17} s$ ) are subject to much greater uncertainty, as they are based on a census of known baryonic matter or employ assumptions about how luminous objects trace the mass distribution. For example, applying the virial theorem to the motion of galaxies in clusters yields constraints on $\Omega _b$ . Peebles (Reference Peebles2020, 337) reports a “crude estimate” of $\Omega _b h^2 \approx 0.015$ based on astrophysical constraints, compared to $\Omega _b h^2 =0.022 \pm 0.001$ from Planck observations of the CMB and $0.021 \pm 0.002$ from big bang nucleosynthesis.

All three techniques for estimating $\Omega _b$ are strongly “theory dependent”—but the theory and background knowledge in each case are quite different. There is also a clear contrast between two different ways that assumptions about the background cosmological model play a role: first, in establishing a link between a specific observable quantity and $\Omega _b$ , and then evaluating the stability of that link and how precision could be increased, and second, in making the case that different measurement techniques converge on the same physical quantity. Regarding the first, the connection between $\Omega _b$ and isotopic ratios, for example, does not depend on the more contentious aspects of the $\Lambda$ CDM model—dark matter and dark energy. The connection is also stable to variation of various other observable quantities, and physicists have clarified the impact of a wide variety of changes to background theory (including gravitational theory as well as nuclear physics). Cosmology enters into these calculations through the expansion timescale and its interplay with reaction rates, and little else.

The background cosmological model plays a more substantial part, by contrast, in making the case that the three measurements (and several others) target the same quantity. Dynamical evolution as described by the $\Lambda$ CDM model connects measurements of $\Omega _b$ at three different epochs—from $10^2 s$ all the way to $10^{17} s$ . Making the case in favor of convergence, so that constraints obtained in different epochs can be used as cross-checks, depends directly on the validity of the $\Lambda$ CDM model. Hence worries about circularity are more pressing in evaluating the case that we have multiple convergent measurements. To the point in slightly different terms, accepting the $\Lambda$ CDM model allows individual measurements of the fundamental parameters to be systematically interlinked. What the convergence of complementary measurement adds, over and above the fact that individual measurements are stable and well behaved, is evidence for these connections. In place of a general concern about circularity and theory dependence, cosmologists can then assess how robust the comparisons of different measurements of $\Omega _b$ are to specific modifications of the $\Lambda$ CDM model.Footnote ² As we will see next, we can also turn this inference around: assuming the $\Lambda$ CDM model makes it possible to assess systematic uncertainties in different measurement techniques.

3. Trouble with Hubble?

Hubble’s observations of a linear relationship between the redshift and distance of twenty-four nearby galaxies provided the first evidence that the universe is expanding. Astronomers have pursued more precise determinations of the Hubble constant ever since, with measured values decreasing substantially from Hubble’s value of $H_0 \approx 500$ (in units km s $^{-1}$ Mpc $^{-1}$ ). Astrophysical objects are not fundamental observers in an FL model, and their individual motions (also called “peculiar velocities”) result from gravitational interactions with nearby galaxies and clusters of galaxies, adding “noise” to the underlying “Hubble flow.” This is an obstacle to using local objects to measure $H_0$ , but the error decreases at larger distances because the peculiar velocities become smaller, in ratio, to the ever-increasing recessional velocities due to the Hubble expansion. Ideally, one could measure the Hubble flow based on the distance–redshift relation for a population of standard candles or rulers, objects with known luminosity or scale, at sufficiently large distances. Astronomers have a variety of techniques for determining distances to astrophysical objects that apply at a variety of (often overlapping) scales. The distance ladder extends rung by rung to cosmological scales, starting with parallax measurements for nearby stars, using these to calibrate different distance indicators, and iterating the process. Observations of large samples of type Ia supernovae have significantly extended the distance ladder out into the pure Hubble flow, leading to the discovery that the expansion rate is accelerating and enabling higher-precision determinations of $H_0$ .

As the precision of these “local” measurements of $H_0$ have increased, the contrast with “global” measurements based on observations of the CMB has become sharper. The Planck results led Bernal, etal. (Reference Bernal, Verde and Riess2016), for example, to prominently identify this as a serious tension in the $\Lambda$ CDM model. Local measurements favor a higher value, falling in the interval of 70–76, and are reasonably consistent given stated uncertainties, but they contrast with the lower value measured by the Planck Collaboration: $67.27 \pm 0.6$ (assuming a flat model as a prior, with the stated uncertainty at $1 \sigma$ ). Riess etal. (Reference Riess, Casertano, Yuan, Bowers, Macri, Zinn and Scolnic2021) find a value of $73.2 \pm 1.3$ , leading to a $4.2 \sigma$ discrepancy given the stated uncertainties.Footnote ³

CMB observations do not directly measure $H$ : they constrain the product of $H$ with other cosmological parameters, such as $\Omega _b$ and the total mass $\Omega _m$ . Observations of the acoustic peaks essentially determine the geometry of the FL model at the time of last scattering. The angular scale of the first peak depends on the scale factor at that time and our distance from the surface. But observations of the acoustic peaks provide further parameter constraints: the relative height of the second peak measures $\Omega _b h^2$ , and that of the third peak measures $\Omega _m h^2$ . The degeneracy can be broken by assuming a flat model or by using other data sets to fix parameter values, leading to the value just cited. Although the determination of $H$ is indirect and depends on the background $\Lambda$ CDM model, Planck can achieve such high precision in part because of the simplicity of the physics governing the CMB. Planck’s results agree well with those of other CMB observations, and it is striking that the conflict with local measurements seems to have had little impact in confidence on the Planck parameter values.

The Hubble tension has already generated a vast literature, with Di Valentino etal. (Reference Di Valentino, Mena, Pan, Visinelli, Yang, Melchiorri, Mota, Riess and Silk2021) counting more than 850 papers exploring different theoretical resolutions. The expansion history according to the $\Lambda$ CDM models connects the CMB and local measurements, and the proposed resolutions nearly all modify this linkage without modifying the understanding of the CMB measurements. There are some proposals based on modifying general relativity and adopting a quite different framework, but the majority of these responses reflect cosmologists’ confidence in the $\Lambda$ CDM model: rather than abandoning the model entirely, these are minor modifications to the model’s dynamics, such as introducing a new form of dark energy that has a targeted impact on the expansion history. Specific modifications have often been criticized as ad hoc, but to my mind, there is a more striking downside of such changes. In effect, these modifications break the connection between these two types of measurements that allowed them to function as cross-checks of each other. Although one of these modifications could be correct, in general, they make it much more challenging to constrain cosmological models.

A different line of response takes the conflict as a prompt to look for further possible sources of systematic error, a persistent challenge in measuring astrophysical distances since Hubble’s time. “Local” measurements extend into the Hubble flow using SNIa. There are different techniques to establish distances to the galaxies hosting the closest supernovae in order to calibrate the lowest rungs of the distance ladder. One approach relies primarily on Cepheids, a kind of variable star with a period–luminosity relationship discovered by Henrietta Leavitt. The advantages and disadvantages of using Cepheids as distance indicators have been scrutinized carefully, given the central role of this technique in earlier work, including the Hubble Space Telescope key project to measure $H_0$ . The SHoES collaboration (e.g., Riess etal. Reference Riess, Casertano, Yuan, Bowers, Macri, Zinn and Scolnic2021) uses Cepheids and claims to attain precision just below the 2 percent level, with a goal of reaching the 1 percent level based on new parallax measurements (compared to an approximately 5 percent level two decades ago).

The conflict with the Planck results suggests that unknown systematics may continue to plague local distance estimates. But a more compelling case has been made recently, relying on a new method using “tip of the red giant branch” (TRGB) stars. At the end of the red giant phase, stars undergo a core helium flash, leading to a sharp and readily observable discontinuity in the color-magnitude diagram, with a characteristic luminosity, for populations of low-mass stars. The CCHP project (e.g., Freedman etal. Reference Freedman, Madore, Hatt, Hoyt, Jang, Beaton and Burns2019) aims to avoid the systematics plaguing Cepheids by building an entirely new distance ladder using this feature as a standard candle. The results obtained so far provide compelling evidence that one or both of these projects has failed to identify some systematic errors (Freedman Reference Freedman2021). The two methods disagree even about the distance of nearby galaxies, with $2\sigma$ –3 $\sigma$ discrepancies. We can see the need for further scrutiny of the local distance ladder at this level of precision, even without comparison to early-universe parameter measurements. The resolution of this debate will directly impact the Hubble tension as well, if for no other reason than that the TRGB method leads to a lower value, $H_0 = 69.6 \pm 1.9$ . But in addition, on Freedman’s (2021) analysis, contrasting zero-point calibrations for SNIa is not the primary reason for the divergence between the estimates given by the two methods, and unraveling the full reasons for the discrepancy is essential for assessing the Hubble tension and its ramifications.

Although debates about the Hubble tension will certainly continue, to my mind, this episode illustrates the advantages of adopting the $\Lambda$ CDM model: it makes it possible to leverage high-precision measurements of the CMB to isolate systematic uncertainties as local distance ladder measurements are pushed to increasing levels of precision. This leverage is particularly valuable because the stability and agreement of closely related measurements often fail to expose systematic errors they all share. Smith and Seth (Reference Smith and Seth2020) pointedly note that this is true even of Perrin’s celebrated experimental measurements of Avogadro’s number: his preferred values fell in the range 65– $72 \times 10^{22}$ , and he substantially underestimated systematic errors shared by all these measurements. The introduction of novel techniques by Millikan and others led to a lower value and a reassessment of Perrin’s measurements.

4. Realism and the cosmological parameters

The strength of Peebles’s case that the $\Lambda$ CDM model is a permanent contribution to cosmology depends on the wide array of diverse data sets that yield stable, convergent measurements of its basic theoretical parameters. Obviously, I have the space here only to indicate the structure of the argument and to identify one area of active research and debate. Suppose, however, that we grant that the story is “close enough to consistency”: does that establish that we should regard the cosmological parameters as real properties of the universe? Here I will argue that even though we should, as van Fraassen (Reference Van Fraassen2009) suggested, resist a straightforwardly realist reading of the parameters, they can still anchor further work in cosmology.

A detailed description of the space-time geometry in the neighborhood of the Milky Way, according to general relativity, would resemble something like Swiss cheese—with the “holes” consisting of small-scale regions with large space-time curvature (say, near black holes), and sharp density contrasts, embedded within an expanding universe model. The parameters appearing in the $\Lambda$ CDM model characterize, by contrast, slight perturbations away from a completely uniform matter distribution at large scales. The relationship between the more realistic description and a “smoothed-out” or “averaged” model is far from straightforward in a nonlinear theory like general relativity. Ellis (Reference Ellis1984) showed that an “averaging operator” that smooths out nonuniformities below some length scale, when applied to a solution to EFE, does not generate a new solution (i.e., if the smoothing operator is applied to both the Einstein tensor and the stress-energy tensor for a given solution, the two new tensors generally do not satisfy EFE). More generally, there is not a straightforward procedure to generate a “best-fit” model from the bottom up, starting from a detailed description of space-time geometry at shorter length scales. It is a familiar lesson from other areas of physics that averaged parameters are not always physically meaningful—their utility depends on the relevant dynamics and justification for washing out details at different scales. These considerations block a straightforward reading of the parameters appearing in the $\Lambda$ CDM model as corresponding, in some direct sense, to actual properties of the universe at cosmological scales.

Setting aside a naive realist understanding of cosmological parameters does not, however, imply that they are merely artifacts, little more than a convenient way to summarize a large body of data. It is an old instrumentalist idea that the significance of a model should be characterized in terms of its role in guiding inquiry, not only in terms of fidelity to actual states of affairs. Miyake and Smith (Reference Miyake and Smith2021) articulate a version of this view in a discussion that has parallels to the cosmological case. As they describe in detail, molecular spectroscopy has led to the determination of (among other things) molecular constants that characterize the structural properties of diatomic molecules—represented as dumbbell-shaped bodies, with rotational and vibrational degrees of freedom. The constants specify structural features of this body, with the “rotation constant,” for example, interpreted as the inverse of its moment of inertia. Yet it would be a mistake to take this structural description as mapping onto the actual state of a molecule. For a molecule with an anharmonic potential, the rotational degrees of freedom couple to the vibrational modes; as a result, the atoms fluctuate around an equilibrium distance. Furthermore, taking electron–electron and electron–nuclei interactions into account requires time averaging over much shorter timescales. The full picture of a molecule includes fluctuations and interactions on several distinct timescales, and the simple interpretation of the rotation constant only applies if we, counterfactually, ignore all these details. Similarly, in the cosmological case, the parameters of the $\Lambda$ CDM model represent what the large-scale properties of the universe would be if we were to neglect complications due to structures at lower scales and the nonlinearities of general relativity. But, by contrast with spectroscopy, we lack a comparably clear understanding of the approximations involved in obtaining the large-scale model and the domain of validity of inferences based on it.

Miyake and Smith (Reference Miyake and Smith2021) propose that such counterfactual representations qualify as physically meaningful insofar as they can be used to guide inquiry, in the sense that systematic discrepancies between the model and observations have a physical source (as exemplified in the discovery of the isotopes of oxygen). On this view, a physically meaningful model is the first step in a series of successive approximations that are guided by ongoing comparisons with observations. Smith’s work has characteristically focused on long-term evaluation of the evidence developed in different areas of physics, to assess whether scientists have used physically meaningful models as a starting point and succeeded in identifying new features of nature through a process of refinement. Although we do not yet have a similarly long view in cosmology, the $\Lambda$ CDM model has clearly been the essential starting point for research in cosmology for several decades, and the model has (arguably) enabled the discovery of entirely new constituents of the universe (dark matter and dark energy).

Returning to the second methodological theme from the introduction, “taking the model seriously” is precisely to take it as physically meaningful in this sense and hence to treat any discrepancies as targets for further investigation. This should not be read as a straightforwardly realist commitment, because the model has an essentially counterfactual character. The contrast here is not between realism and instrumentalism but rather, following Stein (Reference Stein1989), between a commitment to seeing the model as a source of further physical insights, such as opening up possibilities for measurements of the fundamental parameters, rather than merely a convenient tool for organizing a body of knowledge.