When should we stop? Bayesian model comparison

In section 1.4.3.1, we explained that the measurement of the equation of statewcan exclude some classes of models, including the cosmological constant of ΛCDM. However, mostclasses of models allow the equation of state to be arbitrarily close to that of vacuum energy, w=−1, while still representing completely different physics. Since precision cannot be infinite, we need to propose an algorithm to determine how well this property should be measured. As we showed in section 1.4.3.2 above, inflation provides an example of a period that acceleration that, if it occurred at late times would have been judged as consistent withw=−1 given today’s constraints. We therefore should require a better measurement, but how much better?

We approach the answer to this question from the perspective of Bayesian evidence: at what precision does the non-detection of a deviation of the background expansion history signifies that we should prefer the simpler null hypothesis thatw=−1.

In our Bayesian framework, the first model, the null hypothesisM0, posits that the background

expansion is due to an extra component of energy density that has equation of state w = −1 at all times. The other models assume that the dark energy is dynamical in a way that is well parametrized either by an arbitrary constantw(modelM1) or by a linear fitw(a) =w0+ (1−a)wa

(modelM2).

Here we are using the constant and linear parametrization of wbecause on the one hand we can consider the constantwto be an effective quantity, averaged over redshift with the appropri- ate weighting factor for the observable, see [1100], and on the other hand because the precision targets for observations are conventionally phrased in terms of the figure of merit (FoM) given by 1/p|Cov(w0, wa)|. We will, therefore, find a direct link between the model probability and the

FoM. It would be an interesting exercise to repeat the calculations with a more general model, using e.g. PCA, although we would expect to reach a similar conclusion.

Bayesian model comparison aims to compute the relative model probability P(M0|d) P(M1|d) =P(d|M0) P(d|M1) P(M0) P(M1) (1.4.4) where we used Bayes formula and whereB01≡P(d|M0)/P(d|M1) is called the Bayes factor. The

Bayes factor is the amount by which our relative belief in the two models is modified by the data, with lnB01>0 (<0)indicating a preference for model 0 (model 1). Since modelM0is nested in

M1 at the pointw=−1 and in model M2 at (w0=−1, wa = 0), we can use the Savage–Dickey

(SD) density ratio [e.g. 1166]. Based on SD, the Bayes factor between the two models is just the ratio of posterior to prior at w = −1 or at (w0 = −1, wa = 0), marginalized over all other

parameters.

Let us start by following [1172] and consider the Bayes factor B01 between a cosmological

constant model w = −1 and a free but constant effective w. If we assume that the data are compatible withweff =−1 with an uncertaintyσ, then the Bayes factor in favor of a cosmological

constant is given by B = r 2 π ∆++ ∆− σ erfc −√∆+ 2σ −erfc _∆ − √ 2σ −1 , (1.4.5)

where for the evolving dark-energy model we have adopted a flat prior in the region −1−∆− ≤

weff ≤ −1 + ∆+and we have made use of the Savage–Dickey density ratio formula [see 1166]. The

prior, of total width ∆ = ∆++ ∆−, is best interpreted as a factor describing the predictivity of

the dark-energy model under consideration. In what follows we will consider example benchmark three models as alternatives tow=−1:

• Fluid-like: we assume that the acceleration is driven by a fluid the background configuration

of which satisfies both the strong energy condition and the null energy condition, i.e. we have that ∆+= 2/3,∆− = 0.

• Phantom: phantom models violate the null energy condition, i.e. are described by ∆+ =

0,∆−>0, with the latter being possibly rather large.

• Small departures: We assume that the equation of state is very close to that of vacuum

energy, as seems to have been the case during inflation: ∆+= ∆−= 0.01.

A model with a large ∆ will be more generic and less predictive, and therefore is disfavored by the Occam’s razor of Bayesian model selection, see Eq. (1.4.5). According to the Jeffreys’ scale for the strength of evidence, we have a moderate (strong) preference for the cosmological constant model for 2.5 < lnB01 < 5.0 (lnB01 > 5.0), corresponding to posterior odds of 12:1 to 150:1 (above

150:1).

Table 1: Strength of evidence disfavoring the three example benchmark models against a ΛCDM expansion history, using an indicative accuracy onw=−1 from present data,σ∼0.1.

Model (∆+,∆−) lnB today (σ= 0.1)

Phantom (0,10) 4.4 (strongly disfavored) Fluid-like (2/3,0) 1.7 (slightly disfavored) Small departures (0.01,0.01) 0.0 (inconclusive)

We plot in Figure 3 contours of constant observational accuracy σ in the model predictivity space (∆−,∆+) for lnB = 3.0 from Eq. (1.4.5), corresponding to odds of 20 to 1 in favor of a

Figure 3: Required accuracy on weff = −1 to obtain strong evidence against a model where

−1−∆−≤weff ≤ −1 + ∆+ as compared to a cosmological constant model,w=−1. For a given

σ, models to the right and above the contour are disfavored with odds of more than 20:1.

cosmological constant (slightly above the “moderate” threshold. The figure can be interpreted as giving the space of extended models that can be significantly disfavored with respect tow=−1 at a given accuracy. The results for the 3 benchmark models mentioned above (fluid-like, phantom or small departures fromw=−1) are summarized in Table 1. Instead, we can ask the question which precision needs to reached to support ΛCDM at a given level. This is shown in Table 2 for odds 20:1 and 150:1. We see that to rule out a fluid-like model, which also covers the parameter space expected for canonical scalar field dark energy, we need to reach a precision comparable to the one that the Euclid satellite is expected to attain.

Table 2: Required precision σ of the value ofw for future surveys in order to disfavor the three benchmark models againstw=−1 for two different strengths of evidence.

Model (∆+,∆−) Requiredσfor odds

>20 : 1 >150 : 1 Phantom (0,10) 0.4 5·10−2

Fluid-like (2/3,0) 3·10−2 _3·10−3

Small departures (0.01,0.01) 4·10−4 _5·10−5

By considering the modelM2we can also provide a direct link with the target DETF FoM: Let

us choose (fairly arbitrarily) a flat probability distribution for the prior, of width ∆w0 and ∆wa

in the dark-energy parameters, so that the value of the prior is 1/(∆w0∆wa) everywhere. Let us

assume that the likelihood is Gaussian in w0 and wa and centered on ΛCDM (i.e., the data fully

supports Λ as the dark energy).

As above, we need to distinguish different cases depending on the width of the prior. If you accept the argument of the previous section that we expect only a small deviation fromw=−1, and set a prior width of order 0.01 on both w0 and wa, then the posterior is dominated by the

prior, and the ratio will be of order 1 if the future data is compatible with w = −1. Since the precision of the experiment is comparable to the expected deviation, both ΛCDM and evolving dark energy are equally probable (as argued above and shown for modelM1 in Table 1), and we

row in Table 2).

However, one often considers a much wider range forw, for example the fluid-like model with w0 ∈ [−1/3,−1] and wa ∈ [−1,1] with equal probability (and neglecting some subtleties near

w=−1). If the likelihood is much narrower than the prior range, then the value of the normalized posterior at w=−1 will be 2/(2πp|Cov(w0, wa)|= FoM/π (since we excludedw < −1, else it

would half this value). The Bayes factor is then given by B01=

∆w0∆waFoM

π . (1.4.6)

For the prior given above, we end up with B01 ≈ 4FoM/(3π) ≈ 0.4FoM. In order to reach a

“decisive” Bayes factor, usually characterized as lnB >5 or B >150, we thus need a figure of merit exceeding 375. Demanding that Euclid achieve a FoM>500 places us, therefore, on the safe side and allows to reach the same conclusions (the ability to favor the ΛCDM expansion history decisivelyif the data is in full agreement withw=−1) under small variations of the prior as well.

To summarize, the most direct effect of dynamical dark energy is the modification of the expan- sion history. We used inflation as a dark-energy prototype to show that the current experimental bounds of w ≈ −1.0±0.1 are not yet sufficient to significantly favor a parameter-free ΛCDM expansion history: we showed that we need to reach a percent-level accuracy both to have any chance of observing a deviation ofwfrom−1 if the dark energy is similar to inflation, and because it is at this point that aw=−1 expansion history beings to be favored decisively for prior widths of order 1.

We do not expect to be able to improve much our knowledge with a lower-precision measurement ofw, unless dark energy is significantly different fromw=−1 either at late times or, for example, owing to a significant early-dark-energy component [946]. A large deviation would be the preferred situation for Euclid, as then we would be able to observe the evolution of dark energy rather than just a fixed state, which would be much more revealing. However, even if the expansion history matches that of ΛCDM to some arbitrary precision, this does not imply that the cosmological constant is accelerating the universe. Even on such configurations a large amount of freedom exists which can then only be tested by investigating the evolution of large-scale structure, to which we now turn.