Assessing observational constraints on dark energy

Abstract

Observational constraints on time-varying dark energy (e.g., quintessence) are commonly presented on a w0-wa plot that assumes the equation of state of dark energy strictly satisfies w(z) = w0 + wa z/(1 + z) as a function of the redshift z. Recent observations favor a sector of the w0-wa plane in which w0 is greater than minus one and w0 + wa is less than minus one, suggesting that the equation of state underwent a transition from violating the null energy condition (NEC) at large z to obeying it at small z. In this paper, we demonstrate that this impression is misleading by showing that simple quintessence models satisfying the NEC for all z predict an observational preference for the same sector. We also find that quintessence models that best fit observational data can predict a value for the dark energy equation of state at present that is significantly different from the best-fit value of w0 obtained assuming the parameterization above. In addition, the analysis reveals an approximate degeneracy of the w0-wa parameterization that explains the eccentricity and orientation of the likelihood contours presented in recent observational studies.

1. Introduction

Models of time-varying dark energy (e.g., quintessence) can be constrained by combining observations at different redshifts, such as the cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and Type IA supernovae (SNe Ia). A common convention is to display the observational constraints on a w0-wa plot assuming the variation of the dark energy equation of state w is well-described by a two-parameter function of the scale factor z that takes the Chevallier-Polarski-Linder form, w(z) = w0 + wa z/(1 + z). Recent observations appear to favor a sector of the w0-wa plane with w0 greater than minus one and w0 + wa less than minus one, in which case that form predicts that the null energy condition (NEC) and weak energy condition (WEC) are violated at large redshift, that is, w(z) less than minus one, but satisfied at small redshift, w(z) at or above minus one.

One must be careful when portraying observational results in this way, because cosmological observables depend directly on the Hubble parameter H(z) and its integrals but only indirectly on the dark energy equation of state w(z). In particular, the functional form of w(z) can differ significantly from the Chevallier-Polarski-Linder form in different models of quintessence, while still predicting very similar cosmological observables. In order to compare the predictions of different quintessence models to one another and to observations using a w0-wa plot, one can assign to each model the ordered pair of w0 and wa that predicts an H(z) most similar to that of the quintessence model. This is similar to the approach suggested by de Putter and Linder, who showed that even just by assigning wa in this way, it is possible to match H(z) to about 0.1 per cent or better for some models of quintessence.

In this paper, we use this mapping technique to predict where observational preferences should fall on the w0-wa plane for a range of quintessence models driven by a scalar field with canonical kinetic energy density and a potential. The models we consider, sometimes referred to as "thawing dark energy", have the property that w(z) approaches minus one at early times (large z) because Hubble friction during the matter- and radiation-dominated eras is large enough to freeze the scalar field. At late times (small z), the Hubble friction becomes negligible, the scalar field accelerates down the potential, and w(z) increases as z decreases. We show that, when matching Hubble parameters, these models are mapped onto the same sector of the w0-wa plot as observations currently prefer, even though they do not violate the NEC. As a result, observations favoring a region with w0 + wa less than minus one do not imply that dark energy must be NEC-violating at any redshift.

We detail our mapping protocol in Sec. 2 and discuss the uncertainties in this procedure (due partially to an approximate degeneracy in the w0-wa plane) in Sec. 3. The quintessence models we will test are presented in Sec. 4, and the results of their mapping onto the w0-wa plane are given in Sec. 5. Finally, in Sec. 6, we summarize our findings and discuss the implications for interpreting observational likelihood contours on a w0-wa plane. In particular, we point out how the observations bear on issues such as NEC violation, consistency with supergravity and string theory, and on whether accelerated expansion continues forever or terminates and transitions to contraction.

2. Methods

As noted above, our procedure relies on determining which combination of w0 and wa is the "best fit" to a given quintessence model of dark energy. Rather than choosing the fit that best mimics the quintessence field's equation of state, we choose the fit that best matches the evolution of the Hubble parameter in the quintessence model. This allows the fit to most nearly reproduce key late-time cosmological observables, such as the Hubble distance, the inverse of H(z); the co-moving angular diameter distance, the integral of the inverse Hubble parameter out to redshift z; and the physical luminosity distance, that same integral multiplied by one plus z — defined here for a spatially flat FRW universe. Notably, when optimizing to match H(z), the integrated angular diameter and luminosity distances are found to match with similar accuracy, as we will show for a sample model in Sec. 5.

Quantitatively, we define the "best fit" combination of w0 and wa as one that minimizes the error E, defined as the maximum, taken over redshifts less than four, of the absolute fractional difference between the fitted Hubble parameter and the quintessence model's Hubble parameter. The maximization is performed over that finite interval to roughly match the redshifts probed by BAO and SNe Ia measurements. These are also the redshifts during which the time-variation of dark energy is most relevant; at higher redshifts, the universe is strongly matter-dominated, and H(z) can be computed directly from the present-day matter density, independently of the nature or behavior of dark energy. Note that there is some flexibility to the above definition of error; for example, the approach of de Putter and Linder matches the angular diameter distance over all positive redshifts rather than the Hubble parameter over redshifts below four. In Sec. 3, we will discuss how this flexibility translates to an uncertainty in our best-fit results.

For some models of quintessence, the quintessence Hubble parameter can be easily calculated from analytically parameterized expressions for the equation of state. However, these expressions are approximate and only valid in the regime where one plus the equation of state is much less than one, which limits the range of models that can be studied. In this work, in order to find the Hubble parameter for a quintessence model driven by a canonical scalar field with a given potential, we switch variables from redshift z to the variable N, defined as minus the natural logarithm of one plus z, and numerically solve the full equations of motion: the scalar field equation with its Hubble-friction term and the derivative of the potential, the Friedmann equation written for the quintessence Hubble parameter in terms of the potential, the fractional dark energy density and the field velocity, the evolution equation for that fractional density, and the definition of the equation of state as the ratio of kinetic minus potential to kinetic plus potential energy. Here the fractional dark energy density is one minus the matter fraction, and primes denote derivatives with respect to N. Note that these equations assume a spatially flat universe. We set our initial conditions deep in the matter-dominated past, with the fractional dark energy density at one part in a million. The initial field value will depend on the particular model, but the field velocity can be initially set to zero without loss of generality, as it will evolve toward an attractor trajectory while the dark energy fraction remains very small. We end the simulation upon reaching a pre-selected fiducial value of the present-day dark energy fraction, which we take to be an input to the mapping procedure; in the examples we provide, we will assume that value is 0.7.

Next, we can generate a candidate fit by solving the corresponding system for the fitted Hubble parameter, the fitted dark energy fraction, and the fitted equation of state written in the Chevallier-Polarski-Linder form as w0 plus wa times one minus the exponential of N.

Here, we do not assume that the fitted fractional dark energy density today equals that of the quintessence model, nor do we assume that the fitted present-day Hubble parameter equals the model's. Instead, we treat these two initial values as free parameters, in addition to the combination of w0 and wa specifying the fitted equation of state. We do check, however, that the resulting best-fit matter density is within about one per cent of its value in the quintessence model. This ensures that the fit is compatible with CMB constraints in addition to measurements of the low-redshift universe.

The final step in our procedure is to scan over the four free parameters and identify the best-fit combination of w0, wa, the fitted present-day dark energy fraction and the fitted present-day Hubble parameter with the smallest error. We note that fitting all four parameters, rather than just wa as in the approach of de Putter and Linder, is necessary in order to properly identify the best fits for a wider range of quintessence models than those considered there. After projecting out the best-fit values of the present-day Hubble parameter and dark energy fraction, we are left with a single point on the w0-wa plane that best represents the quintessence model we started with. More generally, this protocol can be used to map a one-parameter family of quintessence models onto a best-fit curve on the w0-wa plane, or a multi-parameter family of models onto a best-fit region.

3. Degeneracy and uncertainty

In our fitting process, it is instructive to identify a region of "acceptable" combinations of w0 and wa in addition to the central or "best-fit" combination. There is an approximate degeneracy between w0 and wa when matching cosmological observables, causing this acceptable region to be highly eccentric. In particular, for any best-fit combination that matches a quintessence model's Hubble parameter with reasonable accuracy, there will be a large set of combinations satisfying a ratio of the change in wa to the change in w0 of about minus five that match the model with similar accuracy. The slope of this degeneracy can vary by about ten per cent depending on the model being fitted, but its qualitative appearance is clear and distinct.

To demonstrate that this effect is fundamental to the parameterization and not dependent on any particular choice of quintessence model, Figure 1 illustrates the degeneracy when fitting combinations of w0 and wa to a fiducial model strictly obeying the parameterized equation of state with w0 equal to minus 0.827 and wa equal to minus 0.75; these are the central values of the DESI, CMB and PantheonPlus constraints. The darkest regions in the figure correspond to the combinations that most accurately match the fiducial model's Hubble parameter, and the dashed line, whose slope in the change of wa against the change of w0 is minus 4.8, indicates the axis of degeneracy. One can see that many combinations aligned along this axis are accurate fits to the central model within an error of about 0.3 per cent or less.

This degeneracy is not only relevant for our theoretical fits to quintessence models, but it also makes a prediction about the orientation of constraint contours in the w0-wa plane produced by observational analyses. In particular, if the observed values of the Hubble parameter over a broad range of redshifts are well fit by one combination of w0 and wa, then we expect that it will also be reasonably fit by other combinations along the axis of degeneracy. The resulting eccentricity and orientation of the constraint contours are indeed apparent in the recent observational studies, with the slope most closely matching our prediction in the combined BAO, CMB and SNe Ia results from DESI.

Unlike the orientation of the degeneracy, the size of the degenerate region in theoretical fits is model-dependent. In the example shown in Figure 1, the best-fit error is zero by construction, and there is a correspondingly large region of acceptable fits, depending on the precision with which the Hubble parameter can be measured. In general, a larger error for the best-fit combination corresponds to a smaller region of acceptable fit.

When plotting a best-fit curve on the w0-wa plane corresponding to a one-parameter family of quintessence models, the acceptable-fit regions of each point merge together to form an acceptable-fit ribbon, shown in Figure 2 as the shaded region around the best-fit line. The ribbon is widest at the ΛCDM limit, where w0 is minus one and wa is zero, of any quintessence model, provided such a limit exists, but the rate at which the ribbon tapers off is model-dependent.

Finally, a second and entirely independent type of uncertainty reflects our confidence, or lack thereof, in the location of the best-fit curve itself on the w0-wa plane. This second uncertainty stems from a fundamental ambiguity in the notion of "best-fit", that is, in the definition of error. We have observed that choosing a different definition of error — say, a mean-square error instead of a maximum, or matching angular diameter distances instead of Hubble parameters — can affect the slope of best-fit lines by about ten per cent or less, depending on the model. The lines pivot about the ΛCDM limit, assuming such a limit exists in the models being fitted, where all definitions of error agree that the cosmological observables are best fit by w0 equal to minus one and wa equal to zero. This uncertainty in the slope is depicted by the dashed lines in Figure 2.

4. Models

In this work, we illustrate the methods outlined in the previous section using three classes of thawing quintessence models as examples: exponential potentials, hilltops, and plateaus. Each of these models is driven by a scalar field with canonical kinetic energy density rolling down a potential. The scalar field is initially held constant during the radiation- and matter-dominated epochs by Hubble friction, such that the equation of state approaches minus one at large redshift. Then, as the dark energy density becomes comparable to the matter density, the Hubble friction decreases, the field accelerates down the potential, and in turn the equation of state increases as the redshift decreases, or "thaws" away from minus one. This behavior corresponds to a negative value of wa in the parameterization, as appears to be preferred by recent observational constraints.

We parameterize the exponential potential as a constant times the exponential of lambda multiplied by the field, where the field is in units of the reduced Planck mass and the initial field value is set to zero, so that the constant is of order the present-day Hubble parameter squared times the reduced Planck mass squared. A theoretical motivation for studying this class of models is that scalar fields with exponential potentials are ubiquitous in supergravity, modified gravity, and superstring theories. Additionally, the exponential potential probes two interesting limits of quintessence models: one where the ratio of the potential's slope to the potential is constant and much smaller than the inverse reduced Planck mass, known as slow-roll thawing quintessence, and one where that ratio is constant but not small.

Hilltop potentials with a large, negative second derivative provide a complementary probe into the regime where the ratio of slope to potential is small compared to the inverse reduced Planck mass but not constant. Potentials of this type are good approximations to axion models, or pseudo-Nambu Goldstone bosons generally, provided the scalar field is initially frozen by Hubble friction near the top of its potential. We take a quadratic approximation and write the hilltop potential as a constant times the quantity one minus one half k squared times the field squared, where again the field is in units of the reduced Planck mass. In this work, we analyze a relatively flat hilltop, with k squared times the reduced Planck mass squared equal to one, and a more concave hilltop, with that quantity equal to one hundred. For each case, we will use a variety of initial field values to map out the set of possible best-fit combinations of w0 and wa for the given potential.

The plateau potential we consider is something of a hybrid between the previous two examples, pairing a region of smooth exponential decay with a sharp, cliff-like drop: a constant times the difference between the exponential of minus the field divided by a scale M and a coefficient kappa times the exponential of the field divided by a second scale m. This is an example of a case where the ratio of slope to potential cannot be assumed to be either small or constant. We choose an exponential cliff, rather than a power law, to match the model of Andrei, Ijjas and Steinhardt, where it was shown that such a potential can lead to a transition from accelerated expansion to a regime of slow contraction when the potential becomes negative, as can occur in a cyclic cosmology. For this model, we are free to set the initial field value to zero without loss of generality. As before, the field is in units of the reduced Planck mass, and kappa is a constant of order one or below. This model has three free parameters, kappa, M and m, which leads to a best-fit region on the w0-wa plane rather than a curve. Note that in any limit where the second term in the potential becomes negligible, we recover the exponential model with lambda equal to the inverse of M.

5. Results

The best-fit curves and regions for the three classes of models introduced in Sec. 4 are depicted in Figure 3, a w0-wa plot showing the predictions for the three types of canonical, NEC-satisfying quintessence potentials, overlaid with the best-fit contours from DESI BAO plus CMB plus either PantheonPlus or Union3, which prefer the sector below the dashed line where w0 plus wa is less than minus one. As mentioned in Sec. 2, this analysis assumes a fiducial present-day dark energy fraction of 0.7.

The orange curve corresponds to the exponential model. The limit of small lambda maps onto the ΛCDM parameters, w0 equal to minus one and wa equal to zero. Each fit along this curve has an error of about 0.1 per cent or less, which is the greatest level of accuracy among the models we tested.

The region between the exponential curve and the plateau boundary corresponds to the three-parameter family of plateau models. The upper boundary, the exponential curve, corresponds to the limit of small kappa in which the cliff is negligibly small. The lower boundary corresponds to large kappa and a substantial cliff that causes a sudden increase in the equation of state as the redshift decreases. This sudden increase causes the models to be mapped onto more negative values of wa. The errors near this boundary can be somewhat greater but still satisfy about 0.5 per cent or less for the cases we tested. We note that the lower boundary of this region is approximate, and other types of plateau models may not mimic the behavior of the specific potential examined in this work.

The two remaining curves correspond to the hilltop models, the flatter one and the more concave one. Their respective errors satisfy about 0.2 per cent and about 0.7 per cent within the region shown in the plot. The closer the initial field value is to the hilltop, the closer the best-fit model is to ΛCDM, and the smaller is the best-fit error.

These results are overlaid with the observational constraints obtained by the DESI collaboration, shown as two-sigma contours, when combining measurements of BAO, CMB, and SNe Ia. This allows the plot to be used for comparing quintessence models to each other and to the observations at the same time. For example, the likelihood contours in Figure 3 appear to favor quintessence potentials with sharp drops, like plateau models with the steepest cliffs or the more concave hilltop models. However, we make this point mainly for the purpose of illustrating the utility of w0-wa plots in general, and we would not suggest drawing any strong conclusions based on currently available data.

Notably, all the quintessence models we tested obey the NEC and have an equation of state above minus one at all redshifts, yet they are best fit by curves or regions on the w0-wa plane that satisfy w0 plus wa less than minus one. This boundary is marked by the dashed line in Figure 3. We therefore conclude that a preference for this sector of the plane, which would naively imply NEC violation at large redshift according to the parameterized equation of state, is actually compatible with NEC-satisfying models of quintessence. This result is a manifestation of the fact that even large differences in the equation of state over a wide range of redshifts much greater than one, where the dark energy fraction is negligibly small, have a negligible impact on cosmological observables such as the Hubble parameter.

At smaller redshifts, the Hubble parameter is more sensitive to differences in the equation of state, but only through an integral. As a result, two models can have very similar Hubble parameters even if their equations of state differ noticeably over a narrow range of redshifts near zero. In turn, this means that the best-fit values of w0 obtained when matching Hubble parameters need not be equal to — or even close to — the actual present-day equation of state for the quintessence model being considered. This is specifically the case for the hilltop and plateau models, in which the equation of state is increasing rapidly as the redshift approaches zero. We illustrate an example of this behavior for the more concave hilltop model in Figure 4, juxtaposing the fits to the Hubble parameter and the angular diameter distance, which have sub-percent level errors, with a comparison of the equations of state between the model and the fit, which differ by of order one hundred per cent. In fact, the best-fit curves drawn in Figure 3 include cases where the model's present-day equation of state is above minus one third, in which case the universe today is no longer accelerating, while the best-fit w0 is still at or below minus 0.65.

As discussed in Sec. 3, there is some uncertainty in our best-fit results due to the flexibility in how we define the error. This uncertainty is negligible for the exponential curve, but the plateau boundary and hilltop curves should be interpreted as having an error bar of about ten per cent on the value of wa at any given w0. Additionally, extending the plateau and hilltop curves beyond the region shown would produce increasingly large best-fit errors, ultimately reaching a point where there exists no good fit to the quintessence model within the w0-wa parameter space. In cases like this — including more general models of dark energy or modified gravity that are not well fit by any combination of w0 and wa — the safer approach would be to perform a separate analysis specific to the model in question, rather than trying to map it onto, and then constrain, the w0-wa parameter space.

Finally, we note that plots like Figure 3 carry no information about fine-tuning of either parameters or initial conditions for a given model. For example, in a hilltop model, only a small range of initial field values is simultaneously sufficiently close to the peak of the hilltop to produce a noticeable period of accelerated expansion and sufficiently far from the hilltop to be distinguishable from ΛCDM. This fine-tuning must be judged and weighed separately.

6. Discussion

In this work, we modified and illustrated a mapping protocol that assigns a best-fit combination of w0 and wa to any given model of quintessence based on matching the evolution of the Hubble parameter in a spatially flat universe. For the cases we tested, whose maps are shown in Figure 3, we confirmed that the parameterization was able to match the quintessence model's Hubble parameter with less than 0.7 per cent error at all redshifts. These results are not sensitive to the inclusion of spatial curvature given the observational constraints from, for example, DESI. We note that for each model considered in this work, the Swampland conjectures, thought to be required for a consistent theory of quantum gravity, are satisfied for a substantial range of parameters. The plateau and hilltop potentials, which are sharply decreasing and can become negative, also have a natural place in cyclic bouncing cosmology.

Though less precise than Bayesian model comparisons using Markov-Chain Monte Carlo simulations, this protocol provides a quick and useful way to visually compare different models of quintessence to each other and to observational data in a common two-dimensional parameter space. For example, if we take the constraints reported by DESI at face value, we see that plateau and hilltop models fare better than exponential models, with highly concave hilltop models being most compatible with the data. We reiterate, however, that this comparison test does not take into account any fine-tuning of the parameters or initial conditions of models that land within the observational constraint contours, and it is subject to the uncertainties discussed in Sec. 3.

This mapping protocol not only allows us to assess models of quintessence against observational data, but it also sheds new light on how to interpret the observational likelihood contours themselves. First, we have shown that the eccentricity and orientation of contours generated from measurements across a broad range of redshifts is a generic feature of the parameterization. Second, we have found that for some models of thawing quintessence, the best-fit value of w0 based on matching Hubble parameters can differ significantly from the true present-day value predicted by the model. Finally, we have pointed out that the thawing quintessence models analyzed in this work, all of which obey the NEC, are mapped onto combinations that satisfy w0 greater than minus one and w0 plus wa less than minus one. An observational preference for this sector, therefore, does not require the kinds of exotic field theories needed to enable a transition from NEC violation at large redshift to NEC compliance at small redshift.

This last finding has an important corollary: contrary to the suggestion of Cortês and Liddle, we have shown that it is not just reasonable but crucially important for observational analyses to include combinations of w0 and wa satisfying w0 plus wa less than minus one in their priors with high credence. Otherwise, these analyses would be inadvertently excluding families of simple, well-motivated models of thawing quintessence from consideration.

(Displayed equations, the four figures, the reference list and the acknowledgements are omitted for length; the complete text is at the source.)