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Is the Dynamics of Tracking Dark Energy Detectable?

We highlight the unexpected impact of nucleosynthesis and other early universe constraints on the detectability of tracking quintessence dynamics at late times, showing that such dynamics may well be invisible until the unveiling of the Stage-IV dark energy experiments (DUNE, JDEM, LSST, SKA). Nucleosynthesis forces |w'(0)|<0.2 for the models we consider and strongly limits potential deviations from LCDM. Surprisingly, the standard CPL parametrisation, w(z) = w_0 + w_a z/(1+z), cannot match the nucleosynthesis bound for minimally coupled tracking scalar fields. Given that such models are arguably the best-motivated alternatives to a cosmological constant these results may significantly impact future cosmological survey design and imply that dark energy may well be dynamical even if we do not detect any dynamics in the next decade.


I. INTRODUCTION
The quintessential enigma of modern cosmology is that of dark energy -the shadowy source of the accelerated expansion of the cosmos. Einstein's "greatest blunder"the cosmological constant, Λ, remains the most economical explanation for this observed acceleration, but more exciting alternatives could imply the existence of new types of matter, modifications of the Einstein equations or even violation of the Copernican principle.
Of these however, one can argue that only scalar fields are consistent with current constraints while simultaneously being theoretically well-founded. The more exotic alternatives either are claimed to have theoretical pathologies or are likely to be indistinguishable from the vanilla ΛCDM model which, given all current data, remains flavour of the month [1]. Models in the former category include K-essence [2] which may exhibit an unbounded adiabatic speed of sound [3] and the DGP model [4], which has some theoretical problems with ghosts (see [5] and references therein) while those in the latter class include f (R) modifications of gravity [6] and unified dark energy [7]. Of the scalar fields, arguably the best-motivated and most compelling are the scaling quintessence models [8,9,10]. We consider such scaling models with a redshift-dependent equation of state parameter w(z) = p/ρ that tracks the dominant energy density component of the cosmos (w = 1 3 , 0) until a redshift z = z t at which point it undergoes a transition to w < 0 which triggers the onset of acceleration.
Big Bang Nucleosynthesis (BBN) provides strong constraints on the energy density of dark energy during the radiation dominated era at a temperature of T ∼ 1MeV, implying that Ω DE (T ∼ 1MeV) < ǫ = 0.045 at 2σ [11]. If the scalar field has reached the scaling attractor solution at the time of BBN, this early universe constraint on the energy density of the dark energy will be carried forward until the field exists the scaling regime (at z t ). The time at which the attractor for the scalar field is approached depends on the details of the physics before the radiation dominated epoch. But in [10] it was shown that, for a typical inflationary model with the usual method of reheating, the field will approach the attractor long before nucleosynthesis. Therefore we assume that the field is already scaling with the background fluid at the time of BBN. In fact, our results are unchanged even if scaling only occurs by decoupling, since similar or better constraints on the dark energy density exist from the Cosmic Microwave Background (CMB), Ω DE (T ∼ 1eV) < 0.04 [12]. Constraints on w(z) from BBN taken together with recent data require that z t > 5 [13].
In the case of scaling dark energy models these constraints at early times are preserved by the subsequent evolution of the cosmos, implying that the limit Ω DE < ǫ = 0.045 holds for all redshifts greater than z t , the redshift at which the equation of state stops scaling and becomes negative. We find new implications for the magnitude of allowed deviations of these scaling models from Λ. To motivate our results consider first a toy, stepfunction model for w(z) in which w is constant for z < z t and w = 0 for z ≥ z t . Requiring Ω DE (z t ) < ǫ = 0.045 (the BBN constraint preserved by the scaling field) with w ≥ −1 and Ω DE = 0.7 today implies that z t > 2.6, an unexpectedly large number given that most cosmic probes in the next decade will be limited to z < 2. To achieve z t = 1 one requires instead w = −1.88, a value disfavoured by current observations [1]. Figure (2) shows that the early dark energy constraints on the scaling models we consider force the derivative of w(z) to be very small: |w ′ (0)| < 0.2 for all our models, significantly smaller than will be detectable in the next decade [15]. If w(0) < −0.9 the deviation of the Hubble rate from Λ is less than 3% and the deviation of the distance modulus is less than 0.04 mag. In addition, we find the surprising result that the standard CPL parametrisation [14] cannot be used to describe scaling minimally coupled scalar showing their approach to the BBN limits of ǫ = ΩDE(z = zBBN) = 0.045 for the polynomial parameterisation of w(z) and 3 4 ǫ for the double exponential potential. For comparison we also show the curves for Λ and the CPL [14] w(z) with the lowest asymptotic value of ΩDE in this model while still assuming w ≥ −1, showing its inability to match the BBN constraint and describe a canonical scalar field. Figure (2) shows the corresponding observational quantities for the scaling quintessence models.
fields (which automatically have w ≥ −1) and simultaneously match the nucleosynthesis bound. Throughout the paper we assume a flat universe with Ω m = 0.3 today and all results are valid for redshifts less than matterradiation equality. We start by describing the constraints on general models in Section II, we give the specific scaling models we consider in Section III and discuss the performance of standard parameterisations in describing these scaling fields in Section IV. We conclude in Section V.

II. GENERAL RESULTS
One can derive model-independent constraints on cosmological observables (we consider the Hubble rate, H(z), and distance modulus, µ(z)) for scaling models.
determines the redshift dependence of the dark energy density. The evolution of Ω DE as a function of redshift is illustrated for various models in Figure (1). From the above discussion, plus the scaling requirement that w(z ≥ z t ) = 0, one can show that where r = Ω DE /Ω m is evaluated today. Using this result implies the following general but stringent constraint: where the last equality arises from imposing ǫ = 0.045 and the upper bound from setting r = 0 as z t → ∞. This limit can clearly be seen in the top middle panel of Fig. (2). This robust result implies that detecting deviations from Λ at high redshift will be difficult using Hubble rate measurements alone since they are bound to be less than 2.3%. This does not, however, strongly constrain the behaviour of H DE /H Λ for z < z t , once the field has left the scaling regime. We will show in two classes of models that its maximum value is less than 5% and occurs around z ≃ 1. The latter result is good news for Baryon Acoustic Oscillation (BAO) surveys such as WiggleZ, BOSS and WFMOS [16] which will probe this range of redshifts. Similarly one can also place robust bounds on the deviation of the distance modulus, µ(z), from the Λ prediction. The quantity ∆µ ≡ µ DE (z) − µ Λ (z) is given by: where d L (z) is the corresponding model luminosity distance. If we assume that there exists a number α such that H(z)/H Λ (z) ≤ 1 + α 2 for all z (for example in the constraint on H discussed above A bound of α 2 = 0.025 gives |∆µ(z)| ≤ 0.054 mag -see the top row of Fig. (2). This is a conservative upper bound since we have shown that for z ≥ z t , α 2 ≤ ǫ/2. For the oscillating double exponential w(z) models (with −30 ≤ µ < 0) considered later, one has α 2 < 0.015 which yields the constraint |∆µ(z)| ≤ 0.032 mag. These general results do not constrain H(z) for z < z t , which instead requires a specific model for w(z). We now consider two classes of models which describe a wide range of scalar field dynamics.

A. Polynomial w(z) parametrisation
First we consider a quadratic parametrisation of the dark energy equation of state, w(z) [17]: We apply the constraint w(z) ≥ −1 since we want to describe minimally coupled scalar fields with canonical kinetic terms. The linear case with w 2 = 0 requires z t ≃ 6.2 to match BBN (for ǫ = 0.045) if w 0 = −1 and if we allow w 0 to be free the BBN constraint implies the correlation w 1 ≃ −0.4w 0 − 0.2 for the interesting region −1 < w 0 < −0.8. The other case we consider is w 0 = −1, w 2 = 0. Continuity at z = z t then implies The BBN constraint provides w 1 in terms of z t . The resulting family of curves and observables are shown in the top row of Fig. (2). We note that for z ≤ 1 the BBN constraint is so strong that all models have w(z) < −0.8, as shown in Fig. (2), and the largest deviation of H(z) from the Λ comparison model is about 2.7% with the largest deviation in the distance modulus only about 0.03 magnitudes (occurring at z = 2). This shows that if w ≃ −1 today we cannot expect significant deviations from Λ at any redshift and only Stage-IV experiments [15] are likely to detect dark energy dynamics with any real significance. However while current data favour w(0) ≃ −1 they are consistent with larger values. To study how this zeropoint affects our results we now consider simulations of a scalar field with a popular family of scalar field potentials, V (φ), which also allows us to examine the impact of oscillations in w(z).

B. Double Exponential Potential
A single exponential potential is well known to give early scaling [9] but cannot also lead to late-time acceleration. One well-studied way to combine the two is via the double exponential potential [18]: where late-time acceleration is induced if µ ≤ √ 2. The dynamics of φ is sensitive to the sign of µ, in that negative values of µ yield a potential that oscillates around the minimum of the field. This can clearly be seen in Fig. 2, for the potentials with −30 ≤ µ ≤ 0 We assume that the scalar field is in the scaling regime during radiation domination, and hence satisfying the BBN constraint requires λ ≥ 2/ √ ǫ ≥ 9.43 [9]. We choose λ = 9.43 to maximise deviations from Λ. Unlike a perfectly scaling model of the form we assumed in the previous section, Ω φ actually decreases in the transition to matter domination and we have Ω φ < 3/4 × ǫ = 0.034 during matter domination, leading to even more stringent results -see Fig. (1). The extra 3/4 factor is specific to the double exponential potential and is responsible for the reduction of the asymptotic values of H DE (z)/H ΛCDM , seen in the bottom middle panel of Fig. (2), relative to the predictions of Eq.(3). We choose M 1 = 10 −14 m pl and use Planck units where κ = 1.
We numerically solve the evolution equation for φ with V (φ) given by Eq. (8), together with the radiation and matter fluids. These are all coupled to gravity through the Friedmann equation For each value of µ we find M 2 such that Ω DE = 0.7 today which implies M 2 ∼ 10 −31 m pl for µ ∼ 1. The resulting w(z) curves for these models, together with the predicted observables (H(z), ∆µ(z)) are shown in the bottom row of Fig. (2). For negative µ -we study −30 ≤ µ < 0the potential has a global minimum and for z 0.2 the equation of state satisfies w φ ≤ −0.98. As a result all the negative µ models show tiny deviations from Λ: less than 1.5% for H(z) and less than 0.015 mag for ∆µ (see Fig. (2)). This will make detection extremely difficult even with the Stage-IV dark energy experiments such as DUNE, JDEM, LSST and SKA [15]. In contrast, positive values of µ (0 ≤ µ ≤ 1) simply modify the slope of the potential. They can yield values of w φ significantly different from −1 today, e.g. for µ ∼ 1 one finds w φ (0) ∼ −0.8 which is consistent (at about the 2σ level [1,13]) with current observations and which we therefore take as the upper bound for µ. The bottom row of Fig. (2) shows that the maximum allowed deviation for H(z) from H Λ (z) in this case is about 5%, peaking at z ∼ 1 with a maximum value of ∆µ ∼ 0.9 mag. Such a model will be detectable with Stage-III supernova surveys (at the 99.97% confidence level) and with the upcoming BAO experiments, since the maximum deviation in H(z) coincides with the redshift ranges in which they will operate, i.e. z ∼ 0.7 to 1.1. However, for values more consistent with the current best-fits, w φ (0) < −0.9 one finds much smaller deviations of 2.7% and 0.045 mag respectively for H(z) and ∆µ which again will require Stage-IV experiments for conclusive detection as can be seen in the right-hand panels of Fig. (2). Similar results will apply to other modifications of the exponential potential, e.g. [19].

IV. PERFORMANCE OF STANDARD PARAMETRISATIONS
The most widely used parametrisation for dark energy, the Chevallier-Polarski-Linder (CPL) parametrisation [14], is also the basis for the DETF figure of merit [15]. Surprisingly this parametrisation fails dramatically to meet the BBN constraint if we demand w(z) ≥ −1, Ω DE ∼ 0.7 today and w(z ≥ z t ) = 0 as before (for some z t ). This is clearly visible in the top curve of Fig. (1), which shows the lowest attainable value of Ω DE with w 0 = −1. In retrospect this is understandable since to reach the scaling value w = 0 for some z t requires w a > −w 0 . However in this case w(z) doesn't spend enough time at sufficiently negative values to force Ω DE (z) down to the BBN value. The least phantom value of w 0 that is able to satisfy the BBN constraint is w 0 = −1.3. In contrast the logarithmic expansion w(z) = w 0 + w 1 ln(1 + z) is able to match the BBN constraint with w(z) ≥ −1, but only for z t > 12.4.

V. CONCLUSIONS
Scaling field models are arguably the best-motivated alternatives to the cosmological constant. We have shown that the constraints on the energy density of the scalar field at the time of Big Bang Nucleosynthesis and decoupling strongly limit their allowed dynamics. If w today is not close to the maximum value allowed by current data then detection of dynamics will likely have to wait a decade for the Stage-IV DETF experiments. Of course, these strong conclusions are only true for scaling models and if one allows exotic phantom behaviour (w < −1) the conclusion is much more rosy. We discussed two specific families of these scaling models -imposing the BBN and CMB constraints on general scaling models is left for future work.
One might ask how the constraints discussed here will affect growth of structure. Since the growth in minimally coupled scalar fields smoothly approaches that of ΛCDM when w(z) → −1 (see e.g. Fig. 2 in [20]) our results show that the growth in models allowed by BBN and CMB constraints will be close to those in ΛCDM at z < 1, making detection difficult. To what extent there are deviations at higher redshift, and whether they will be detectable by future cluster or weak lensing surveys is again left as a subject for future study.
Finally we have shown that the standard CPL parametrisation, w(z) = w 0 + w a z/(1 + z), fails dramatically to match the BBN constraint when describing scaling fields which satisfy w ≥ −1. This is particularly important given that the CPL parametrisation is the ba-sis of the DETF figure of merit [15] which is now the de facto standard for the optimisation of future cosmological surveys, e.g. [21]. A concern therefore is that optimisations may be unwittingly biased away from scaling dark energy models. More work in this area is clearly needed to assess the implications for cosmological survey design, but it is clear that the current non-detection of dark energy dynamics should neither come as a surprise, nor should it discourage us from the hunt.