PARAMETERIZING AND MEASURING DARK ENERGY TRAJECTORIES FROM LATE INFLATONS

One of the greatest mysteries in physics is the nature of dark energy (DE) which drives the present-day cosmic acceleration, inferred to exist from supernovae (SNe) data (Riess et al. 1998; Perlmutter et al. 1999), and from a combination of cosmic microwave background (CMB) and large-scale structure (LSS) data (Bond & Jaffe 1999). Although there have been voluminous outpourings on possible theoretical explanations of DE, recently reviewed in Copeland et al. (2006), Kamionkowski (2007), Linder (2008a, 2008b), and Silvestri & Trodden (2009), we are far from consensus. An observational target is to determine if there are temporal (and spatial) variations beyond the simple constant Λ. Limits from the evolving data continue to roughly center on the cosmological constant case Λ, which could be significantly strengthened in near-future experiments—or ruled out. We explore the class of effective scalar field models for DE evolution in this paper, develop a three-parameter expression which accurately approximates the dynamical histories in most of those models, and determine current constraints and forecast future ones on these parameters.

The mean DE density is a fraction Ω_de of the mean total energy density,

$$\begin{equation} \rho _{\rm de} (\ln a)\equiv \rho _{\rm tot} \Omega _{\rm de}\,,\quad \ \ 3{M_{{\rm p}}}^2 H^2(a)= \rho _{\rm tot}(a)\,, \end{equation} \tag{ 1 }$$

which is itself related to the Hubble parameter H through the energy constraint equation of gravity theory as indicated. Ω_de(a) rises from a small fraction relative to the matter at high redshift to its current ∼0.7 value. Here ${M_{{\rm p}}}=1/\sqrt{8\pi G_N} =2.44\times 10^{18} \, \mbox{GeV}$, is the reduced Planck mass, G_N is Newton's gravitational constant, and ℏ and c are set to unity, as they are throughout this paper.

Much observational effort is being unleashed to determine as much as we can about the change of the trajectory ρ_de with expansion factor a, expressed through a logarithmic running with respect to ln a:

$$\begin{equation} {-}{1 \over 2} {d\ln \rho _{\rm de} \over d\ln a} \equiv \epsilon _{\rm de}(\ln a) \equiv {3\over 2}(1+w_{\rm de}) \equiv {3\over 2}{(\rho _{\rm de}+p_{\rm de})\over \rho _{\rm de}}. \end{equation} \tag{ 2 }$$

This ρ_de-run is interpreted as defining a phenomenological average pressure-to-density ratio, the DE equation of state (EOS). The total "acceleration factor," ≡ 1 + q, where the conventional "deceleration parameter" is $q\equiv -a\ddot{a}/\dot{a}^2$ = −dln(Ha)/dln a, is similarly related to the running of the total energy density:

$$\begin{equation} {-}{1 \over 2} {d\ln \rho _{\rm tot} \over d\ln a} \equiv \epsilon = \epsilon _{m}\Omega _m + \epsilon _{\rm de}\Omega _{\rm de},\quad \ \ \Omega _m + \Omega _{\rm de}=1\,. \end{equation} \tag{ 3 }$$

Equation (3) shows is the density-weighted sum of the acceleration factors for matter and DE. Matter here is everything but the DE. Its EOS has _m = 3/2 in the non-relativistic-matter-dominated phase (dark matter and baryons) and _m = 2 in the radiation-dominated phase.

We would like to use the data to constrain ρ_de(ln a) with as few prior assumptions on the nature of the trajectories as is feasible. However, such blind analyses are actually never truly blind, since ρ_de or w_de is expanded in a truncated basis of mode functions of one sort or another: there will necessarily be assumptions made on the smoothness associated with the order of the mode expansion and on the measure, i.e., prior probability, of the unknown coefficients multiplying the modes. The most relevant current data for constraining w_de, Type Ia Supernova (SNIa) compilations, extends back only about one e-folding in a, and probes a double integral of w_de, smoothing over irregularities. The consequence is that unless w_de was very wildly varying, only a few parameters are likely to be extractable no matter what expansion is made.

Low-order expansions include the oft-used but nonphysical cases of constant w₀ ≠ −1 and the two-parameter linear expansion in a (Chevallier & Polarski 2001; Linder 2003; Wang & Tegmark 2004; Upadhye et al. 2005; Liddle et al. 2006; Francis et al. 2007; Wang et al. 2007),

$$\begin{equation} w(a)=w_0+w_a (1-a) \, \end{equation} \tag{ 4 }$$

adopted by the Dark Energy Task Force (DETF; see Albrecht et al. 2006). The current observational data we use in this paper to constrain our more physically motivated parameterized trajectories are applied in Figure 1 to w₀ and w_a, assuming uniform uncorrelated priors on each. The area of the nearly elliptical 1σ error contour has been used to compare how current and proposed DE probes do relative to each other; its inverse defines the DETF "figure of merit" (FOM; see Albrecht et al. 2009).

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Constant w_de and Equation (4) can be considered to be the zeroth- and first-order polynomial expansion in the variable 1 − a. Why not redshift z (Upadhye et al. 2005; Linder & Huterer 2005) or the scale factor to some power (Linder & Huterer 2005) or ln a? Why pivot about a = 1?

Why is it only linear in a? Why not expand to higher order? Why not use localized spline mode functions, the simplest of which is a set of contiguous top hats (unity in a redshift band, zero outside)? These cases too have been much explored in the literature, both for current and forecast data. The method of principal component (parameter eigenmode) analysis takes orthogonal linear combinations of the mode functions and of their coefficients, rank-ordered by the error matrix eigenmodes (see, e.g., Huterer & Starkman 2003; Crittenden & Pogosian 2005). As expected only a few linear combinations of mode function coefficients are reasonably determined. Thus, another few-parameter approach expands w_de in many modes localized in redshift but uses only the lowest error linear combinations. At least at the linearized level, the few new parameters so introduced are uncorrelated. However, the eigenmodes are sensitive to the data chosen and the prior probabilities of the mode parameters. An alternative to the w_de expansion is a direct mode expansion in , considered at low order by Rapetti et al. (2007); since the _m part is known, this is an expansion in (_de − _m)Ω_de, in which case w_de becomes a derived trajectory; for w_de parameterizations it is (a) that is the derived trajectory.

The partially blind expansions of 1 + w_de are similar to the early universe inflation expansions of the scalar power spectrum ${\cal P}_s (\ln k)$ (e.g., J. R. Bond et al. 2010, in preparation), except in that case one is "running" in "resolution," ln k. However, there is an approximate relation between the comoving wavenumber k and the time when that specific k-wave went out of causal contact by crossing the instantaneous comoving "horizon" parameterized by Ha. This allows one to translate the power spectra ln k-trajectories into dynamical ln Ha-trajectories, ${\cal P}_s (\ln Ha)$. The lowest order is a uniform slope $n_s -1 = d\ln {\cal P}_s /\ln k$; next order is a running of that slope, rather like the DETF linear w_de expansion. Beyond that, one can go to higher order, e.g., by expanding in Chebyshev polynomials in ln k, or by using localized modes to determine the power spectrum in k bands. Generally, there may be tensor and isocurvature power spectra contributing to the signals, and these would have their own expansions.

In inflation theory, the tensor and scalar spectra are related to each other by an approximate consistency condition: both are derivable from parameterized acceleration factor trajectories defining the inflaton EOS, 1 + w_ϕ(a) = 2/3 (with zero Ω_m in early universe inflation). Thus, there is a very close analogy between phenomenological treatments of early and late universe inflation. However, there is a big difference: power spectra can be determined by over ∼10 e-foldings in k from CMB and LSS data on clustering, and hence over ∼10 e-foldings in a, whereas DE data probe little more than an e-folding of a. This means that higher order partially blind mode expansions are less likely to bear fruit in the late-inflaton case.

This arbitrariness in semi-blind expansions motivates our quest to find a theoretically motivated prior probability on general DE trajectories characterized by a few parameters, with the smoothness that is imposed following from physical models of their origin rather than from arbitrary restrictions on blind paths. Such a prior has not been much emphasized in DE physics, except on an individual-model basis. The theory space of models whose trajectories we might wish to range over include:

• 1.  
  quintessence, with the acceleration driven by a scalar field minimally coupled to gravity with an effective potential with small effective mass and a canonical kinetic energy (Ratra & Peebles 1988; Wetterich 1988; Frieman et al. 1995; Binétruy 1999; Barreiro et al. 2000; Brax & Martin 1999; Copeland et al. 2000; de La Macorra & Stephan-Otto 2001; Linder 2006; Huterer & Peiris 2007; Linder 2008c);
• 2.  
  k-essence, late inflation with a non-canonical scalar field kinetic energy as well as a potential (Armendariz-Picon et al. 2001; Malquarti et al. 2003; Malquarti et al. 2003; de Putter & Linder 2007);
• 3.  
  F(R, ϕ) models, with Lagrangians involving a function of the curvature scalar R or a generalized dilaton ϕ, that can be transformed through a conformal mapping into a theory with an effective scalar field, albeit with interesting fifth force couplings to the matter sector popping out in the transform (Capozziello & Fang 2002; Carroll et al. 2004; de La Cruz-Dombriz & Dobado 2006; Nojiri & Odintsov 2006; de Felice & Tsujikawa 2010; Sotiriou & Faraoni 2010); and
• 4.  
  phantom energy with w_de < −1, and negative effective kinetic energy, also with a potential (Copeland et al. 2006; Carroll et al. 2003; Caldwell 2002; Caldwell et al. 2003).

In this paper, we concentrate on the quintessence class of models, but allow trajectories that would arise from a huge range of potentials, with our fitting formula only deviating significantly when |1 + w_de|>0.5. A class of models that we do not try to include are ones with DE having damped oscillations before settling into a minimum Λ (Kaloper & Sorbo 2006; Dutta & Scherrer 2008).

We also extend our paths to include ones with w_de < −1, the phantom regime where the null energy condition (Carroll et al. 2003) is violated. This can be done by making the kinetic energy negative, at least over some range, but such models are ill-defined. There have been models proposed utilizing extra UV physics, such as with a Lorentz-violating cutoff (Cline et al. 2004) or with extra fields (Sergienko & Rubakov 2008). Regions with w_de < −1 are consistent with current observations, so a semi-blind phenomenology should embrace that possibility. It is straightforward to extend our quintessence fitting formula to the phantom region, so we do, but show results with and without imposing the theoretical prior that these are highly unlikely.

The evolution of w_de for a quintessence field ϕ depends on its potential V(ϕ) and its initial conditions. For a flat Friedmann–Robertson–Walker (FRW) universe with given present matter density ρ_m0 and DE density ρ_de0, any function w_de(a) satisfying 0 ⩽ 1 +  w_de ⩽ 2 can be reverse-engineered into a potential V(ϕ) and an initial field momentum, $\Pi _{\phi, \rm ini} \equiv \dot{\phi }_{\rm ini}$. (The initial field value ϕ_ini can be eliminated by a translation in V(ϕ) and the sign of $\Pi _{\phi,\rm ini}$ by a reflection in V(ϕ).) Given this one-to-one mapping between $\lbrace V(\phi), |\dot{\phi }_{\rm ini}|; \rho _{m0}\rbrace$ and {w(a), ρ_de0; ρ_m0}, we may think we should allow for all w_de trajectories. Generally, these will lead to very baroque potentials indeed, and unrealistic initial momenta. The quintessence models with specific potentials that have been proposed in the literature are characterized by a few parameters, and have a long period in which trajectories relax to attractor ones, leading to smooth trajectories in the observable range at low redshift. These are the simple class of potentials we are interested in here: our philosophy is to consider "simple potentials" rather than simple mathematical forms of w_de(a). The potentials which we quantitatively discuss in later sections include: power laws (Ratra & Peebles 1988; Steinhardt et al. 1999), exponentials (Ratra & Peebles 1988), double exponentials (Barreiro et al. 2000; Sen & Sethi 2002; Neupane 2004a, 2004b; Jarv et al. 2004), cosines from pseudo-Nambu Goldstone bosons (pnGB) (Frieman et al. 1995; Kaloper & Sorbo 2006), and supergravity (SUGRA) motivated models (Brax & Martin 1999). Our general method applies to a much broader class than these though: we find that the specific global form of the potential is not relevant because the relatively small motions on its surface in the observable range allow only minimal local shape characteristics to be determined, and these we can encode with the three key physical variables parameterizing our w_de. Even with the forecasts data to come, we can refine our determinations but not gain significantly more information—unless the entire framework is shown to be at variance with the data.

Quintessence models are often classified into tracking and thawing models (Caldwell & Linder 2005; Linder 2006, 2008c; Huterer & Peiris 2007; Sullivan et al. 2007; Scherrer & Sen 2008). Tracking models were first proposed to solve the coincidence problem, i.e., why the DE starts to dominate not very long after the epoch of matter–radiation equality (∼7.5 e-folds). However, a simple negative power-law tracking model that solves the coincidence problem predicts w_de ≳ −0.5 today, which is strongly disfavored by current observational data. In order to achieve the observationally favored w_de ≈ −1 today, one has to assume the potential changes at an energy scale close to the energy density at matter–radiation equality. The coincidence problem is hence converted to a fine-tuning problem. In this paper, we take the tracking models as the high-redshift limit and parameterize it with a "tracking parameter" ε_ϕ∞ that characterizes the attractor solution. The thawing models, where the scalar field is frozen at high redshift due to large Hubble friction, can be regarded as a special case of tracking models where the tracking parameter is zero. Because of the nature of tracking behavior, where all solutions regardless of initial conditions approach an attractor, we do not need an extra parameter to parameterize the initial field momentum.

In either tracking models or thawing models the scalar field has to be slowly rolling or moderately rolling at low redshift, so that the late-universe acceleration can be achieved. The main physical quantity that affects the dynamics of quintessence in the late-universe accelerating epoch is the slope of the potential. The field momentum is "damped" by Hubble friction, and the slope of the potential will determine how fast the field will be rolling down; actually, as we show here, it is the ϕ-gradient of ln V that matters, and this defines our second "slope parameter" ε_s.

It turns out that a two-parameter formula utilizing just ε_ϕ∞ and ε_s works very well, because the field rolls slowly at low redshift, indeed is almost frozen, so the trajectory does not explore changes in the slope of ln V. However, we want to extend our formula to cover the space of moderate-roll as well as slow-roll paths. Moreover, in cases in which ln V changes significantly, even slow-roll paths may explore the curvature of ln V as well as its ϕ-gradient. Accordingly, we expanded our formula to encompass such cases, introducing a third parameter ζ_s, which is related to the second ϕ-derivative of ln V, explicitly so in thawing models.

Of course, there is one more important fourth parameter, characterizing the late-inflaton energy scale, such as the current DE density ρ_de,0 at redshift 0. This is related to the current Hubble parameter H₀ ≡ 100 h kms⁻¹Mpc⁻¹, the present-day fractional matter density Ω_m0, and the current fractional DE density Ω_Λ ≡ Ω_de,0 = 1 − Ω_m0h²/h²: ρ_de,0 = 3M_p²H²₀Ω_Λ. With CMB data, the sound-crossing angular scale at recombination is actually used, as is the physical density parameter of the matter, Ω_m0h², which is typically derived from separately determined dark matter and baryonic physical density parameters, Ω_dm0h² and Ω_b0h². With any nontrivial w_de EOS, this depends upon the parameters of w_de as well as on one of h or the inter-related zero redshift density parameters. The important point here is that the prior measure is uniform in the sound-crossing angular scale at recombination and in Ω_dm0h² and Ω_b0h², not in h or the energy scale of late inflation. If the quantities are well measured, as they are, the specific prior does not matter very much.

What to use for the prior probability of the parameters of w_de, in which most are not well determined? The prior chosen does matter and has to be understood when assessing the meaning of the derived constraints. Usually a uniform prior in w₀ and w_a is used for the DETF form, Equation (4). The analog for our parameterization is to be uniform in the relatively well-determined ε_s, but we show what variations in the measure on it do, e.g., using $\sqrt{|\varepsilon _{s} |}$ or disallowing phantom trajectories, ε_s>0. The measure for each of the other two parameters is also taken to be uniform. (Our physics-based parameterization leads to an approximate "consistency relation" between w₀ and w_a, a strong prior relative to the usual uniform one.)

In Section 2, we manipulate the dynamic equations for quintessence and phantom fields to derive our approximate solution, w_de(a|ε_s, ε_ϕ∞, ζ_s, Ω_m0). We show how well this formula works in reproducing full trajectories computed for a variety of potentials. In Section 4, we describe our extension of the CosmoMC program package (Lewis & Bridle 2002) to treat our parameterized DE and the following updated cosmological data sets: SNIa, galaxy power spectra that probe LSS, weak lensing (WL), CMB, and Lyα forest (Lyα). The present-day data are used to constrain our dynamical w_de(a). In Section 5, we investigate how future observational cosmology surveys can sharpen the constraints on w_de(a|ε_s, ε_ϕ∞, ζ_s, Ω_m0). We restrict our attention to flat universes. We discuss our results in Section 6.

We assume the DE is the energy density of a quintessence (or phantom) field, with a canonical kinetic energy and an effective potential energy V(ϕ). This is derived from a Lagrangian density

$$\begin{equation} {\cal L}=\pm \frac{1}{2}\partial _\mu \phi \partial ^\mu \phi -V(\phi), \end{equation} \tag{ 5 }$$

where "+" is for quintessence ("−" for phantom). Throughout this paper, we adopt the (+, −, −, −) metric signature. For homogeneous fields the energy density, pressure, and EOS are

$$\begin{equation} \rho _\phi =\frac{1}{2}\Pi _\phi ^2+V(\phi),\quad \ \ p_\phi =\frac{1}{2}\Pi _\phi ^2-V(\phi), \end{equation} \tag{ 6 }$$

$$\begin{equation} 1+w_\phi \equiv \frac{\rho _\phi +p_\phi }{\rho _\phi } = \frac{\Pi _\phi ^2}{\rho _\phi },\quad \ \Pi _\phi \equiv {d\phi \over dt}. \end{equation} \tag{ 7 }$$

Since we identify the DE with an inflaton, we hereafter use ϕ as the subscript rather than "de." The field ϕ does not have to be a fundamental scalar—it could be an effective field, an order parameter associated with some collective combination of fields. A simple way to include the phantom field case is to change the kinetic energy sign to minus, although thereby making it a ghost field with unpalatable properties even if it is the most straightforward way to get 1 + w_de < 0.

The two scalar field equations, for the evolution of the field, $\dot{\phi }={\Pi }_\phi$, and of the field momentum, $\dot{\Pi }_\phi = -3H{\Pi }_\phi -\partial V/\partial \phi$, transform to

$$\begin{equation} \dot{\phi }/H: \sqrt{\epsilon _{\phi }\Omega _{\phi }} = {\pm } d\psi /d\ln a \,, \ \psi \equiv \phi /(\sqrt{2}{M_{{\rm p}}}), \end{equation} \tag{ 8 }$$

$$\begin{eqnarray} \dot{\Pi }_\phi /H: &&\ \frac{d^2\psi }{ (d\ln a)^2} + 3\left(1-\frac{\epsilon }{3}\right)\, \frac{d\psi }{ d\ln a} \nonumber\\ && = -{3\over 2} \frac{d\ln V}{d\psi }\Omega _\phi \left(1-\frac{\epsilon _{\phi }}{3} \right)\,. \end{eqnarray} \tag{ 9 }$$

The "±" sign in Equation (8) depends upon whether the field rolls down the potential to larger ψ (+) or smaller ψ (−) as the universe expands. For definiteness we take it to be positive. The second equation can be recast into a first-order equation for $d\ln \sqrt{\epsilon _{\phi }\Omega _{\phi }}/d\ln a$ which is implicitly used in what follows, we prefer to work instead with a first integral, the energy conservation Equation (2).

As long as Π_ϕ is strictly positive, ψ is as viable a time variable as ln a, although it only changes by $\sqrt{\epsilon _{\phi }\Omega _{\phi }}$ in an e-folding of a. Thus, along with trajectories in ln a, we could reconstruct the late-inflaton potential V(ϕ) as a function of ϕ and the energy density as a function of the field:

$$\begin{equation} \sqrt{\epsilon _V} \equiv -{1 \over 2} {d\ln V\over d\psi } = {\sqrt{\epsilon _{\phi }} \over \sqrt{\Omega _{\phi }} } \left(1+{{d\ln \epsilon _{\phi } /d\ln a} \over {6(1-\epsilon _{\phi } /3) }}\right)\,, \end{equation} \tag{ 10 }$$

$$\begin{eqnarray} \sqrt{\epsilon _E} &&\equiv -{1 \over 2} {d\ln \rho _{\phi }\over d\psi } = {\sqrt{\epsilon _{\phi }} \over \sqrt{\Omega _{\phi }} } \,. \end{eqnarray} \tag{ 11 }$$

These "constraint" equations require knowledge of both _ϕ and Ω_ϕ, but the latter runs according to

$$\begin{equation} \frac{1}{2}\frac{d\ln \big[\Omega _{\phi }^{-1}-1\big]}{d\ln a} = \epsilon _{\phi } - \epsilon _m \,, \end{equation} \tag{ 12 }$$

and hence is functionally determined. Equation (12) is obtained by taking the difference of Equations (2) and (3) and grouping all Ω_ϕ terms on the left-hand side.

For early universe inflation with a single inflaton, Ω_ϕ = 1, and _ϕ = _E = ≈ _V. The field momentum therefore obeys the relation Π_ϕ = −2M_p²∂H/∂ϕ, which is derived more generally from the momentum constraint equation of general relativity (Salopek & Bond 1990).

The field equations form a complete system if V(ψ) is known, but what we wish to do is to learn about V. So we follow the running of a different grouping of variables, namely the differential equations for the set of parameters $\sqrt{\epsilon _{\phi }}$ and $\sqrt{\epsilon _{V}\Omega _{\phi }}$, with a third equation for the running of Ω_ϕ, Equation (12):

$$\begin{equation} \frac{d\sqrt{\epsilon _{\phi }}}{d\ln a} = 3 (\sqrt{\epsilon _V\Omega _{\phi }} -\sqrt{\epsilon _{\phi }}) (1-\epsilon _{\phi }/3), \end{equation} \tag{ 13 }$$

$$\begin{equation} \frac{d\ln \sqrt{\epsilon _V\Omega _{\phi }}}{d\ln a} = (\epsilon _{\phi }-\epsilon _m)(1-\Omega _{\phi }) + 2\gamma \sqrt{\epsilon _{\phi }} \sqrt{\epsilon _V \Omega _{\phi }}. \end{equation} \tag{ 14 }$$

The reason for choosing Equation (14) for $\sqrt{\epsilon _V\Omega _{\phi }}$ rather than the simpler running equation in $\sqrt{\epsilon _V}$

$$\begin{equation} \frac{1}{2}d\ln \sqrt{\epsilon _V}/d\ln a = \gamma \sqrt{\epsilon _{\phi }} \sqrt{\epsilon _V \Omega _{\phi }}, \end{equation} \tag{ 15 }$$

is because we must allow for the possibility that the potential could be steep in the early universe, so _V may be large, even though the allowed steepness now is constrained. For tracking models where a high redshift attractor with constant w_ϕ exists, _VΩ_ϕ tends to a constant, and is nicely bounded, allowing better approximations.

(The ϕ → −ϕ symmetry allows us to fix the $\pm \sqrt{\epsilon _{\phi }}$ and $\pm \sqrt{\epsilon _V\Omega _{\phi }}$ ambiguity to the positive sign. For phantom energy, we use the negative sign, in Equations (13)–(15). In these equations, we have restricted ourselves to fields that always roll down for quintessence, or up for phantom. The interesting class of oscillating quintessence models (Kaloper & Sorbo 2006; Dutta & Scherrer 2008) are thus not considered.

These equations are not closed, but depend upon a potential shape parameter γ, defined by

$$\begin{equation} \gamma \equiv \frac{ \partial ^2 \ln V /\partial \psi ^2 }{ (\partial \ln V /\partial \psi)^2}. \end{equation} \tag{ 16 }$$

It is related to the effective mass squared in H² units through

$$\begin{equation} m_{\rm \phi,eff}^2 /H^2 = 6 (1+\gamma)\epsilon _V \Omega _{\phi } (1-\epsilon _{\phi }/3), \end{equation} \tag{ 17 }$$

Determining how γ evolves involves yet higher derivatives of ln V, ultimately an infinite hierarchy unless the hierarchy is closed by specific forms of V. However, although not closed in $\sqrt{\epsilon _V\Omega _{\phi }}$, the equations are conducive for finding an accurate approximate solution, which we express in terms of a new time variable,

$$\begin{equation} y\equiv \sqrt{\Omega _{\rm \phi app}} \,,\quad \ \ \Omega _{\rm \phi app} \equiv \frac{\rho _{\rm \phi eq}}{\rho _m(a)+\rho _{\rm \phi eq}}= \frac{(a/a_{\rm eq})^3}{1+ (a/a_{\rm eq})^3}, \end{equation} \tag{ 18 }$$

where the subscript "eq" defines variables at the redshift of matter–DE equality:

$$\begin{equation} a_{\rm eq}\equiv (\rho _{m0}/\rho _{\rm m,eq})^{1/3},\quad \ \, \rho _{\rm m,eq}=\rho _{\rm \phi eq}, \, \Omega _{\rm \phi eq}=\Omega _{\rm m, eq}=1/2. \end{equation} \tag{ 19 }$$

Thus, y is an approximation to $\sqrt{\Omega _{\phi }}$, pivoting about the expansion factor a_eq in the small _ϕ limit. By definition $y_{\rm eq}\equiv y\vert _{a=a_{\rm eq}} = 1/\sqrt{2}$.

We now show how working with these running variables leads to a one-parameter fitting formula, expressed in terms of

$$\begin{equation} \varepsilon _{s} \equiv \pm \epsilon _V\vert _{a=a_{\rm eq}}. \end{equation} \tag{ 20 }$$

The two-parameter fitting formula adds the asymptotic EOS factor

$$\begin{equation} {\varepsilon _{\phi \infty }}\equiv \pm \epsilon _V\Omega _{\phi }\vert _\infty. \end{equation} \tag{ 21 }$$

The minus sign in Equations (20) and (21) is a convenient way to extend the parameterization to cover the phantom case. Here the subscript ∞ refers to the a ≪ 1 limit. The existence of limit _VΩ_ϕ|_∞ is a consequence of the attractor in tracking models.³ In addition, if the initial field momentum is far from the attractor, it would add another variable, but the damping to the attractor goes as a⁻⁶ in _ϕ, and hence should be well established before we get to the observable regime for trajectories. For thawing models, where _V|_a≪1 → const and Ω_ϕ|_∞ = 0, the ε_ϕ∞ parameter is zero. In both tracking and thawing models, it can be shown that ${\varepsilon _{\phi \infty }}= \epsilon _\phi \vert _\infty \equiv {3\over 2}(1+w_\phi)|_\infty$, explaining why the symbol ε_ϕ∞ is used for this parameter. This is further discussed in Section 2.4.

The three-parameter form involves, in addition to these two, a parameter which is a curious relative finite difference of $d\sqrt{\epsilon _V\Omega _{\phi }}/dy$ about y_eq/2:

$$\begin{equation} \zeta _{s} \equiv \frac{d\sqrt{\epsilon _V\Omega _{\phi }}/dy\vert _{\rm eq} - d\sqrt{\epsilon _V\Omega _{\phi }}/dy\vert _{\infty }}{d\sqrt{\epsilon _V\Omega _{\phi }}/dy\vert _{\rm eq} + d\sqrt{\epsilon _V\Omega _{\phi }}/dy\vert _{\infty }}. \end{equation} \tag{ 22 }$$

The physical content of the "running parameter" ζ_s is more complicated than that of ε_s and ε_ϕ∞. It is related not only to the second derivative of ln V, through γ, thus extending the slope parameter ε_s to another order, but also to the field momentum. As we mentioned in Section 1, the small stretch of the potential surface over which the late inflaton moves in the observable range make it difficult to determine the second derivative of ln V from the data. In thawing models, for which the field momentum locally traces the slope of ln V, the dependence of ζ_s on the field momentum can be eliminated. However, if the field momentum is sufficiently small, w_ϕ does not respond to d²ln V/dϕ². We discuss these cases in Section 6.

What we demonstrate is that a relation linear in y

$$\begin{equation} \sqrt{\epsilon _V\Omega _{\phi }} \approx \sqrt{{\varepsilon _{\phi \infty }}} +(\sqrt{\varepsilon _{s}} - \sqrt{2{\varepsilon _{\phi \infty }}}) y \,, \end{equation} \tag{ 23 }$$

is a suitable approximation. It yields the two-parameter formula for _ϕ, which maintains the basic form linear in parameters,

$$\begin{equation} \sqrt{\epsilon _{\phi }} \approx \sqrt{{\varepsilon _{\phi \infty }}} + \left(\sqrt{\varepsilon _{s} } - \sqrt{2{\varepsilon _{\phi \infty }}} \, \right)F\left({a\over a_{\rm eq}}\right) \,, \end{equation} \tag{ 24 }$$

with

$$\begin{equation} F(x) \equiv \frac{\sqrt{1+x^3}}{x^{3/2}}-\frac{\ln \, [x^{3/2}+\sqrt{1+x^3}]}{x^3}. \end{equation} \tag{ 25 }$$

The one-parameter case has $\sqrt{{\varepsilon _{\phi \infty }}}$ set to zero, hence the simple $\sqrt{\epsilon _V\Omega _{\phi }} \approx \sqrt{\varepsilon _{s} \Omega _{\rm \phi app}}$ and _ϕ = ε_sF²(a/a_eq), which we regard as the logical physically motivated improvement to the conventional single-w₀ parameterization, 1 + w = (1 + w₀)F²(a/a_eq)/F²(1/a_eq).

The approximation for the three-parameter formula adds a quadratic correction in y:

$$\begin{equation} \sqrt{\epsilon _V\Omega _{\phi }} \approx \sqrt{{\varepsilon _{\phi \infty }}} + (\sqrt{\varepsilon _{s}} - \sqrt{2{\varepsilon _{\phi \infty }}})y \left[1- \zeta _{s} \left(1- {y \over y_{\rm eq}}\right)\right], \end{equation} \tag{ 26 }$$

and a more complex form for the DE trajectories,

$$\begin{equation} \sqrt{\epsilon _{\phi }} = \sqrt{{\varepsilon _{\phi \infty }}} +(\sqrt{\varepsilon _{s} }- \sqrt{2{\varepsilon _{\phi \infty }}})\left[ F\left({a\over a_{\rm eq}}\right) + \zeta _{s} F_2\left({a\over a_{\rm eq}}\right)\right] \,, \end{equation} \tag{ 27 }$$

with

$$\begin{equation} F_2 (x) \equiv \sqrt{2}\left[1-{\ln \left(1+x^3\right) \over x^3}\right] - F(x). \end{equation} \tag{ 28 }$$

Equation (27) gives the final three-parameter 1 + w_ϕ parameterization that will be used throughout the paper. To cover the phantom case, we put absolute values everywhere that ε_s and ε_ϕ∞ appears, and multiply 1 + w_de by the sign, sgn(ε_s) = sgn(ε_ϕ∞). With ζ_s fixed to be zero, the three-parameter formula becomes the two-parameter formula (24), in which again ε_ϕ∞ can be set to be zero to give the slow-roll thawing (one-parameter) parameterization (see Section 6.1). The details of derivation of these formulae are given in Sections 2.5 and 2.6.

The class of quintessence/phantom models where the field is slow-rolling in the late-universe accelerating phase is of primary interest. Qualitatively this implies _ϕ ≪ 1 at low redshift, but we would like to have a parameterization covering a larger prior space allowing for higher _ϕ and |dψ/dln a|, and let observations determine the allowed speed of the roll. Therefore we also include moderate-roll models in our parameterization, making sure our approximate formula covers well trajectories with values of |1 + w_ϕ| which extend up to 0.5 and _ϕ to 0.75 at low redshift.

At high redshift the properties of the potential are poorly constrained by observations, and we have a very large set of possibilities to contend with. Since the useful data for constraining DE are at the lower redshifts, what we really need is just a reasonable shutoff at high redshift. One way we tried in early versions of this work was just to cap _ϕ at some _ϕ∞ at a redshift well beyond the probe regime. Here we still utilize a cap as a parameter, but let physics be the guide to how it is implemented so that the _ϕ trajectories smoothly join higher to lower redshifts.

The way we choose to do this here is to restrict our attention to tracking and thawing models for which the asymptotic forms are easily parameterized. Tracking models have an early universe attractor which implies _ϕ indeed becomes a constant _ϕ∞, which is smaller than _m. If we apply this attractor to Equation (13), we obtain $\sqrt{\epsilon _{V}\Omega _{\phi }}$ is constant at high redshift, equaling our $\sqrt{{\varepsilon _{\phi \infty }}}$. By definition, thawing models have _ϕ∞ = 0.

Consider the two high redshift possibilities exist for the tracking models. One has _ϕ = _m, which, when combined with Equation (14), implies the potential structure parameter γ vanishes, hence one gets an exponential potential as an asymptotic solution for V, as discussed in Copeland et al. (1998), Liddle & Scherrer (1999), and Copeland et al. (2006). Another possibility has _ϕ < _m, hence from Equation (14) we obtain γ = (_m − _ϕ)/(2_ϕ) is a positive constant. Solving this equation for γ yields a negative power-law potential, V ∝ ψ^−1/γ. For both cases, we must have an asymptotically constant γ ⩾ 0 to give a constant

$$\begin{equation} {\varepsilon _{\phi \infty }}\rightarrow \epsilon _{m}/(1+2\gamma _{\infty }) \, \end{equation} \tag{ 29 }$$

where γ_∞ is the high redshift limit of the shape parameter Equation (16). This also shows why ε_ϕ∞/_m is actually a better parameter choice than ε_ϕ∞, since the ratio is conserved as the matter EOS changes.

The difference between tracking and thawing models is not only quantitative, but is also qualitative. For tracking models, the asymptotic limit $\sqrt{{\varepsilon _{\phi \infty }}}$ has a dual interpretation. One is that the shape of potential has to be properly chosen to have the asymptotic limit of the right-hand side of

$$\begin{equation} \sqrt{{\varepsilon _{\phi \infty }}} \rightarrow \sqrt{\epsilon _V\Omega _{\phi }}\vert _\infty = -\frac{1}{2} \frac{d\ln V}{d\psi }\vert _{\infty } \sqrt{{\Omega _\phi }_{\infty }}, \end{equation} \tag{ 30 }$$

existing. We already know how to choose the potential—it has to be asymptotically either an exponential or a negative power law. Another interpretation directly relates $\sqrt{{\varepsilon _{\phi \infty }}}$ to the property of the potential through Equation (29). In the thawing scenario, Equation (14) is trivial, and the shape parameter γ is no longer tied to the vanishing ε_ϕ∞, though Equation (30) still holds.

For phantom models the motivation for tracking solutions is questionable. We nevertheless allow for reciprocal _ϕ-trajectories as for quintessence, just flipping the sign to extend the phenomenology to 1 + w_ϕ < 0, as has become conventional in DE papers. What we do not do, however, is try to parameterize trajectories that cross _ϕ = 0, as is done in the DETF w₀–w_a phenomenology (see Figure 1).

In the slow-roll limit, ε_V does not vary much. As mentioned above, we use ε_s ≡ ±_V,eq evaluated at the equality of matter and DE to characterize the (average) slope of ln V at low redshift, and Ω_m0 or h as a way to encode the actual value of the potential V₀ at zero redshift. To model V(ϕ) at high redshift for both tracking models and thawing models, we use the interpretation (30) characterized by the "tracking parameter" ε_ϕ∞ = (_VΩ_ϕ)|_∞, which is bounded by the tracking condition 0 ⩽ ε_ϕ∞/_m ⩽ 1.

Equation (13) shows that to solve for _ϕ(a) one only needs to know $\sqrt{\epsilon _V \Omega _{\phi }}$ plus an initial condition, $\sqrt{{\varepsilon _{\phi \infty }}}$, but that too is just $\sqrt{\epsilon _V \Omega _{\phi }}$ in the a → 0 limit. We know as well $\sqrt{\epsilon _V \Omega _{\phi }}$ at a_eq, namely $\sqrt{\varepsilon _{s} /2}$. If the rolling is quite slow, _V will be nearly ε_s, and Ω_ϕ will be nearly Ω_ϕapp = y², hence

$$\begin{equation} \sqrt{\epsilon _V \Omega _{\phi }} \approx \sqrt{\varepsilon _{s} \Omega _{\rm \phi app}} = \sqrt{\varepsilon _{s} }y \, \end{equation} \tag{ 31 }$$

the one-parameter approximation. But we are also assuming we know the y = 0 boundary condition, $\sqrt{\epsilon _V \Omega _{\phi }}\vert _\infty$ as well as this y_eq value. If we make the simplest linear y relation through the two points, we get our first-order approximation, Equation (23), for $\sqrt{\epsilon _V \Omega _{\phi }}$.

To get the DE EOS w_ϕ, we need to integrate Equation (13), with our $\sqrt{\epsilon _V \Omega _{\phi }}$ approximation. To facilitate this, we make another approximation,

$$\begin{equation} (\sqrt{\epsilon _V\Omega _{\phi }} -\sqrt{\epsilon _{\phi }}) (1-\epsilon _{\phi }/3) \approx (\sqrt{\epsilon _V\Omega _{\phi }} -\sqrt{\epsilon _{\phi }}) \,, \end{equation} \tag{ 32 }$$

which is always a good one: at high redshift the tracking behavior enforces $\sqrt{\epsilon _V\Omega _{\phi }} -\sqrt{\epsilon _{\phi }} \rightarrow 0$ and at low redshift _ϕ/3 ≪ 1. The analytic solution for $\sqrt{\epsilon _{\phi }}$ retains the form linear in the two parameters, yielding Equation (24), and hence the w_ϕ approximation

$$\begin{equation} 1+w_{\phi }(a) \approx \frac{2}{3}\bigg [\sqrt{{\varepsilon _{\phi \infty }}} + \left(\sqrt{\varepsilon _{s} } - \sqrt{2{\varepsilon _{\phi \infty }}} \, \right) F\bigg (\frac{a}{a_{\rm eq}}\bigg)\bigg ]^2, \end{equation} \tag{ 33 }$$

where F is defined in Equation (25).

The three DE parameters a_eq, ε_ϕ∞, and ε_s are related to Ω_m0 through the constraint equation

$$\begin{equation} \left[1+\exp {\left(2\int ^1_{a_{\rm eq}}(\epsilon _{\phi }-\epsilon _m) \frac{da}{a}\right)}\right]^{-1} = 1- \Omega _{m0} \end{equation} \tag{ 34 }$$

obtained by integrating Equation (12) from a = a_eq, where by definition Ω_ϕ,eq = 1/2, to today, a = 1, hence Ω_m0 is actually quite a complex parameter, involving the entire w_ϕ(a; a_eq, ε_s, ε_ϕ∞)/a history, as of course is a_eq(Ω_m0, ε_s, ε_ϕ∞), which we treat as a derived parameter from the trajectories. The zeroth-order solution for a_eq(Ω_m0, ε_s, ε_ϕ∞) only depends upon Ω_m0:

$$\begin{equation} a_{\rm eq}\approx \left(\frac{\Omega _{m0}}{1-\Omega _{m0}} \right)^{1/3}. \end{equation} \tag{ 35 }$$

In conjunction with the approximate two-parameter w_ϕ, Equation (33), this a_eq completes our first approximation for DE dynamics.

The linear approximation of $\sqrt{\epsilon _V \Omega _{\phi }}(y)$ and the zeroth-order approximation for a_eq rely on the slow-roll assumption. For moderate-roll models (|1 + w_ϕ| ≳ 0.2), the two-parameter approximation is often not sufficiently accurate, with errors sometimes larger than 0.01. We now turn to the improved three-parameter fit to w_ϕ; we need to considerably refine a_eq as well to obtain the desired high accuracy.

The quadratic expansion of Equation (26) of$\sqrt{\epsilon _V \Omega _{\phi }}$ in y leads to Equation (27) for $\sqrt{\epsilon _{\phi }}$, in terms of the two functions F and F₂ of a/a_eq. Since the ζ_s term is a "correction term," we impose a measure restriction on its uniform prior by requiring |ζ_s| ≲ 1. Thus w_ϕ(a; a_eq, ε_s, ε_ϕ∞, ζ_s) follows, with the phantom paths covered by _ϕ,phantom(a; ε_s, ε_ϕ∞, ζ_s) = sgn(ε_s)_{ϕ,quintessence}(a; |ε_s|, |ε_ϕ∞|, ζ_s).

We also need to improve a_eq, using the constraint Equation (34). We do not actually need the exact solution, but just a good approximation that works for Ω_m0 ∼ 0.3. For example, the following fitting formula is sufficiently good (error ≲0.01) for 0.1 < Ω_m0 < 0.5:

$$\begin{equation} \ln a_{\rm eq}({\Omega _{m0}},\varepsilon _{s},{\varepsilon _{\phi \infty }},\zeta _{s}) = \frac{\ln \left[ {\Omega _{m0}}/(1-{\Omega _{m0}})\right]}{3-{\rm sgn\,} (\varepsilon _{s}) \delta }, \end{equation} \tag{ 36 }$$

where the correction to the index is

$$\begin{eqnarray} \delta & \equiv& \lbrace \sqrt{|{\varepsilon _{\phi \infty }}|}+ [0.91-0.78{\Omega _{m0}}\nonumber\\ && + (0.24-0.76{\Omega _{m0}})\zeta _{s} ] (\sqrt{|\varepsilon _{s} |}-\sqrt{2|{\varepsilon _{\phi \infty }}|})\rbrace ^2 \nonumber\\ && + [\sqrt{|{\varepsilon _{\phi \infty }}|}+(0.53-0.09\zeta _{s})(\sqrt{|\varepsilon _{s} |}-\sqrt{2|{\varepsilon _{\phi \infty }}|})]^2.\nonumber\\ \end{eqnarray} \tag{ 37 }$$

For each of the data sets used in this paper we either wrote a new module to calculate the likelihood or modified the CosmoMC likelihood code to include dynamic w models.

Cosmic microwave background. Our complete CMB data sets include WMAP 7 yr (Komatsu et al. 2010; Jarosik et al. 2010), ACT (The ACT Collaboration et al. 2010), BICEP (Chiang et al. 2010), QUaD (Castro et al. 2009), ACBAR (Reichardt et al. 2009; Kuo et al. 2007; Runyan et al. 2003; Goldstein et al. 2003), CBI (Sievers et al. 2007; Readhead et al. 2004a, 2004b; Pearson et al. 2003), BOOMERANG (Jones et al. 2006; Piacentini et al. 2006; Montroy et al. 2006), VSA (Dickinson et al. 2004), and MAXIMA (Hanany et al. 2000).

For high-resolution CMB experiments we need to take account of other sources of power beyond the primary CMB. We always include the CMB lensing contribution even though with current data its influence is not yet strongly detected (Reichardt et al. 2009). For most high-resolution data sets, radio sources have been subtracted and residual contributions have been marginalized over (The ACT Collaboration et al. 2010). However, other frequency-dependent sources will be lurking in the data, and they too should be marginalized over. We follow a well-worn path for treating the thermal Sunyaev–Zeldovich (SZ) secondary anisotropy (Sunyaev & Zeldovich 1972, 1980): we use a power template from a gas dynamical cosmological simulation (with heating only due to shocks) described in Bond et al. (2002, 2005) with an overall amplitude multiplier A_SZ, as was used in the CBI and ACBAR papers. This template does not differ by much from the more sophisticated ones obtained by Battaglia et al. (2010) that include cooling and feedback. Other SZ template choices with different shapes and amplitudes (Komatsu & Seljak 2002; Sehgal et al. 2010) have been made by the Wilkinson Microwave Anisotropy Probe (WMAP) and ACT and SPT, and as a robustness alternative by CBI. A key result is that as long as the "nuisance parameter" A_SZ is marginalized, other cosmological parameters do not vary much as the SZ template changes. We make no other use of A_SZ here, although it can encode the tension between the cosmology derived from the CMB primary anisotropy and the predictions for the SZ signal in that cosmology.

In addition to thermal SZ, there are a number of other sources: kinetic SZ with a power spectrum whose shape roughly looks like the thermal SZ one (Battaglia et al. 2010); and submillimeter dusty galaxy sources, which have clustering contributions in addition to Poisson fluctuations. Thermal SZ is a small effect at WMAP and Boomerang resolution. At CBI's 30 GHz, kinetic SZ is small compared with thermal SZ, but it is competitive at ACBAR, QUaD's, and ACT's 150 GHz and of course dominates at the ∼220 GHz thermal-SZ null, but the data are such that little would be added by separately modeling the frequency dependence and shape difference of it, so de facto it is bundled into the generic A_SZ of the frequency-scaled thermal-SZ template. We chose not to include the SPT data in our treatment because the influence of the submillimeter dusty galaxy sources should be simultaneously modeled, and where its band powers lie (beyond ℓ ∼ 2000), the primary SZ power is small.

Type Ia supernova. We use the Kessler et al. (2009) data set, which combines the SDSS-II SN samples with the data from the ESSENCE project (Miknaitis et al. 2007; Wood-Vasey et al. 2007), the Supernova Legacy Survey (SNLS; Astier et al. 2006), the Hubble Space Telescope (HST; Garnavich et al. 1998; Knop et al. 2003; Riess et al. 2004, 2007), and a compilation of nearby SN measurements. Two light curve fitting methods, MLCS2K2 and SALT-II, are employed in Kessler et al. (2009). The different assumptions about the nature of the SN color variation lead to a significant apparent discrepancy in w. For definiteness, we choose the SALT-II-fit results in this paper, but caution that if we had tried to assign a systematic error to account for the differences in the two methods the error bars we obtain would open up, and the mean values for 1 + w may not center as much around zero as we find.

Large-scale structure. The LSS data are for the power spectrum of the Sloan Digital Sky Survey Data Release 7 (SDSS-DR7) Luminous Red Galaxy (LRG) samples (Reid et al. 2010). We have modified the original LSS likelihood module to make it compatible with time-varying w models.

Weak lensing. Five WL data sets are used in this paper. The effective survey area $A_{\rm {eff}}$ and galaxy number density $n_{\rm {eff}}$ of each survey are listed in Table 1.

Table 1. Weak Lensing Data Sets

Data Sets	$A_{\rm{eff}}$ (deg2)	neff (arcmin−2)
COSMOSa	1.6	40
CFHTLS-wideb	22	12
GaBODSc	13	12.5
RCSd	53	8
VIRMOS-DESCARTe	8.5	15

Notes. ^aMassey et al. (2007); Lesgourgues et al. (2007). ^bHoekstra et al. (2006); Schimd et al. (2007). ^cHoekstra et al. (2002); Hoekstra et al. (2002). ^dHoekstra et al. (2002); Hoekstra et al. (2002). ^eVan Waerbeke et al. (2005); Schimd et al. (2007).

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For the COSMOS data, we use the CosmoMC plug-in written by Lesgourgues et al. (2007) with our modifications for dynamic DE models.

For the other four WL data sets we use the covariance matrices given by Benjamin et al. (2007). To calculate the likelihood we wrote a CosmoMC plug-in code. We take the best-fit parameters α, β, z₀ for n(z) ∝ (z/z₀)^αexp [ − (z/z₀)^β], and marginalize over z₀, assuming a Gaussian prior with a width such that the mean redshift z_m has an uncertainty of 0.03(1 + z_m). We have checked that further marginalizing over other n(z) parameters (α and β) has no significant impact (DeLope Amigo et al. 2009).

Lyα forest. Two Lyα data sets are applied: (1) the set from Viel et al. (2004) consisting of the LUQAS sample (Kim et al. 2004) and the data given in Croft et al. (2002), and (2) the SDSS Lyα data presented in McDonald et al. (2005) and McDonald et al. (2006). To calculate the likelihood we interpolate the χ² table in a three-dimensional parameter space, where the three parameters are amplitude, index, and the running of linear CDM power spectrum at pivot k = 0.9 h Mpc⁻¹.

Other constraints. We have also used in CosmoMC the following observational constraints: (1) the distance-ladder constraint on Hubble parameter, obtained by the HST Key Project (Riess et al. 2009); (2) constraints from the BBN: Ω_b0h² = 0.022 ± 0.002 (Gaussian prior) and Ω_Λ(z = 10¹⁰) < 0.2 (see Steigman 2006; O'Meara et al. 2006; Ferreira & Joyce 1997; Bean et al. 2001; Copeland et al. 2006); and (3) an isotopic constraint on the age of universe 10Gyr < age < 20Gyr (see e.g., Dauphas 2005). For the combined data sets, none of these add much. They are useful if we are looking at the impact of the various data sets in isolation on constraining our parameters.

Using our modified CosmoMC, we ran Markov Chain Monte Carlo (MCMC) calculations to determine the likelihood of cosmological parameters, which include the standard six, A_SZ and our DE parameters, w₀–w_a, for Figure 1 and the new ones ε_s, |ε_ϕ∞|/_m, and ζ_s for most of the rest. Here we use |ε_ϕ∞|/_m as a fundamental parameter to eliminate the dependence of ε_ϕ∞ on _m and the sign of ε_s, whereas ε_ϕ∞ adjusts as one evolves from a relativistic to non-relativistic matter EOS. The main results are summarized in Table 2. The basic six are Ω_b0h², proportional to the current physical density of baryons; Ω_c0h², the physical cold dark matter density; θ, the angle subtended by sound horizon at "last scattering" of the CMB, at z ∼ 1100; ln A_s, with A_s being the primordial scalar metric perturbation power evaluated at pivot wavenumber k = 0.002 Mpc⁻¹; n_s, the spectral index of primordial scalar metric perturbation; and τ, the reionization Compton depth. The measures on each of these variables are taken to be uniform. Derived parameters include z_re, the reionization redshift; age (Gyr), the age of universe in units of gigayears; σ₈, today's amplitude of the linear-extrapolated matter density perturbations over a top-hat spherical window with radius 8 h⁻¹ Mpc. And, of great importance for DE, Ω_m0; and H₀ in unit of km s⁻¹Mpc⁻¹, which set the overall energy scale of late inflatons.

Table 2. Cosmic Parameter Constraints: ΛCDM, w₀-CDM, w₀–w_a-CDM, ε_s–ε_ϕ∞–ζ_s-CDM

	ΛCDM	w = w0	w=w0+wa(1 − a)	Track+Thaw	Thaw
Current data: CMB+LSS+WL+SN1a+Lyα
Ωb0h2	0.0225+0.0004−0.0004	0.0225+0.0004−0.0004	0.0225+0.0004−0.0004	0.0225+0.0004−0.0004	0.0225+0.0004−0.0004
Ωc0h2	0.1173+0.0020−0.0019	0.1172+0.0024−0.0024	0.1177+0.0025−0.0023	0.1174+0.0029−0.0029	0.1175+0.0023−0.0022
θ	1.042+0.002−0.002	1.042+0.002−0.002	1.042+0.003−0.002	1.042+0.003−0.002	1.0417+0.0021−0.0021
τ	0.089+0.014−0.013	0.090+0.015−0.014	0.088+0.014−0.013	0.090+0.015−0.014	0.088+0.015−0.014
ns	0.96+0.01−0.01	0.96+0.01−0.01	0.96+0.01−0.01	0.96+0.01−0.01	0.958+0.011−0.011
ln(1010As)	3.24+0.03−0.03	3.24+0.03−0.03	3.24+0.03−0.03	3.24+0.03−0.03	3.239+0.033−0.033
$A_{\rm{SZ}}$	0.56+0.11−0.15	0.56+0.11−0.14	0.57+0.11−0.14	0.56+0.11−0.14	0.56+0.11−0.14
Ωm	0.292+0.011−0.010	0.294+0.012−0.012	0.293+0.012−0.012	0.293+0.015−0.014	0.293+0.013−0.011
σ8	0.844+0.015−0.016	0.841+0.026−0.026	0.847+0.026−0.026	0.844+0.035−0.036	0.844+0.024−0.023
zre	10.8+1.2−1.1	10.9+1.2−1.2	10.7+1.1−1.1	10.9+1.2−1.2	10.8+1.2−1.2
H0	69.2+1.0−1.0	69.0+1.4−1.4	69.2+1.4−1.4	69.0+1.9−1.8	69.1+1.4−1.4
w0	...	−0.99+0.05−0.06	−0.98+0.14−0.11	...	...
wa	...	...	−0.05+0.35−0.58	...	...
εs	...	...	...	0.00+0.18−0.17	−0.00+0.27−0.29
|εϕ∞|/m	...	...	...	0.00+0.21+0.58	...
ζs	...	...	...	n.c.	n.c.
Forecasted data: Planck2.5yr + low-z-BOSS + CHIME + Euclid-WL + JDEM-SN
Ωb0h2	0.02200+0.00007−0.00007	0.02200+0.00007−0.00007	0.02200+0.00007−0.00008	0.02200+0.00007−0.00008	0.02200+0.0007−0.0007
Ωc0h2	0.11282+0.00024−0.00023	0.11280+0.00027−0.00027	0.11282+0.00026−0.00029	0.1128+0.0003−0.0003	0.1128+0.0003−0.003
θ	1.0463+0.0002−0.0002	1.0463+0.0002−0.0002	1.0463+0.0003−0.0002	1.0463+0.0002−0.0002	1.0463+0.0002−0.0002
τ	0.090+0.003−0.003	0.090+0.004−0.004	0.090+0.004−0.004	0.090+0.005−0.005	0.090+0.004−0.004
ns	0.970+0.002−0.002	0.970+0.002−0.002	0.970+0.002−0.002	0.970+0.002−0.002	0.970+0.002−0.002
ln(1010As)	3.115+0.008−0.008	3.115+0.009−0.009	3.115+0.009−0.009	3.115+0.010−0.010	3.115+0.009−0.009
Ωm	0.260+0.001−0.001	0.261+0.002−0.002	0.260+0.003−0.003	0.2609+0.0022−0.0022	0.2605+0.0027−0.0024
σ8	0.7999+0.0016−0.0017	0.7994+0.0023−0.0025	0.800+0.003−0.003	0.7992+0.0027−0.0027	0.7996+0.0026−0.0029
zre	10.9+0.3−0.3	10.9+0.3−0.3	10.9+0.3−0.3	10.9+0.4−0.4	10.9+0.3−0.3
H0	72.0+0.1−0.1	71.9+0.3−0.3	72.0+0.4−0.4	71.85+0.37−0.29	71.94+0.34−0.36
w0	...	−1.00+0.01−0.01	−1.00+0.03−0.03	...	...
wa	...	...	0.01+0.08−0.08	...	...
εs	...	...	...	0.005+0.031−0.025	0.008+0.056−0.054
|εϕ∞|/m	...	...	...	0.000+0.034+0.093	...
ζs	...	...	...	n.c.	n.c.

Notes. Tracking + thawing models use w_de(a|ε_s, ε_ϕ∞, ζ_s) of Equation (27). Thawing enforces ε_ϕ∞ = 0. n.c. stands for "not constrained." H₀ has units km s⁻¹Mpc⁻¹. θ is in radians/100. _m is 3/2 (or 2) in the matter-dominated (or radiation-dominated) regime. All bounds are marginalized 1σ (68.3% CL) limits, except that for |ε_ϕ∞|/_m marginalized 1σ and 2σ (95% CL) upper bounds are given.

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We begin with our versions of the familiar contour plots that show how the various data sets combine to limit the size of the allowed parameter regions. Figures 6 and 7 show the best-fit cosmological parameters do depend on the specific subset of the data that is used. We find that most cosmological parameters are stable when we vary the choice of data sets. The data making the largest differences are the SDSS-DR7-LRG data driving Ω_m up and the Lyα data driving σ₈ up. We take SN+CMB as a "basis," for which Ω_m0 = 0.267^+0.019_−0.018 and σ₈ = 0.817^+0.023_−0.021. Adding LSS pushes Ω_m0 to 0.292^+0.011_−0.010; whether or not WL and Lyα are added or not is not that relevant, as Figure 6 shows. Lyα pushes σ₈ to 0.844^+0.015_−0.016. These drifts of best-fit values are at the ∼2σ, level in terms of the σ's derived from using all of the data sets, as is evident visually in Figure 6.

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Zoom In Zoom Out Reset image size

Figure 7 shows how the combination of complementary data sets constrains ε_s and Ω_m0, the two key parameters that determine low-redshift observables. The label "All" refers to all the data sets described in this section, and "CMB" refers to all CMB data sets, and so forth. In Figure 8, marginalized two-dimensional likelihood contours are shown for all our DE parameters. The left panel shows that the slope parameter ε_s and tracking parameter ε_ϕ∞ are both constrained. The constraint on ε_s limits how steep the potential could be at low redshift. The upper bound of ε_ϕ∞ indicates that the field cannot be rolling too fast at intermediate redshift z ∼ 1. The right panel shows that ζ_s is not constrained and is almost uncorrelated with ε_s. This is because the current observational data favor slow-roll (|1 + w_ϕ| ≪ 1), in which case the ζ_s correction in w_ϕ is very small. Another way to interpret this is that a slowly rolling field does not "feel" the curvature of potential. In Section 6, we will discuss the meaning and measurement of ζ_s in more detail.

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Because of the correlation between ε_s and ε_ϕ∞, the marginalized likelihood of ε_s depends on the prior of ε_ϕ∞. On the other hand, the constraint on ε_s will also depend on the prior on ε_s itself. For example, we can apply a flat prior on $d\ln V/d\phi |_{a=a_{\rm eq}}$, rather than a flat prior on the squared slope ε_s. Other priors we have tried are the "thawing prior" ε_ϕ∞ = 0, the "quintessence prior" ε_s>0, and "slow-roll thawing prior" ε_ϕ∞ = ζ_s = 0. The results are summarized in Table 3.

Table 3. Marginalized 68.3%, 95.4%, and 99.7% CL Constraints on ε_s under Different Prior Assumptions

Prior	Constraint
Flat prior on εs	εs = 0.00+0.18+0.39+0.72−0.17−0.44−0.82
Flat prior on $\sqrt{|\varepsilon _{s} |}$	εs = 0.00+0.09+0.27+0.46−0.07−0.28−0.52
Thawing prior εϕ∞ = 0	εs = 0.00 + 0.27+0.53+0.80−0.29−0.61−0.98
Slow-roll thawing εϕ∞ = ζs = 0	εs = −0.01+0.26+0.50+0.70−0.28−0.59−0.86
Quintessence εs>0	εs = 0.00+0.18+0.39+0.76

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In Figure 9, we show the reconstructed trajectories of the DE EOS, Hubble parameter, distance moduli, and the growth factor D of linear perturbations. The w_ϕ information has also been compressed into a few bands with errors, although that is only to guide the eye. The off-diagonal correlation matrix elements between bands is large, encoding the coherent nature of the trajectories. Also, the likelihood surface in all w_ϕ bands is decidedly non-Gaussian, and should be characterized for this information to be statistically useful in constraining models by itself. The other variables shown involve integrals of w_ϕ, and thus are not as sensitive to detailed rapid change aspects of w_ϕ, which, in any case, our late-inflaton models do not give. The bottom panels show how impressive the constraints on trajectory bundles should become with planned experiments, a subject to which we now turn.

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5.1.1. Planck CMB Simulation

Planck 2.5 years (five sky surveys) of multiple (CMB) channel data are used in the forecast, with the instrument characteristics for the channels used listed in Table 5 using Planck "Blue Book" detector sensitivities and the values given for the full width half-maxima (FWHMs).

Table 5. Planck Instrument Characteristics

Channel Frequency (GHz)	70	100	143
Resolutiona (arcmin)	14	10	7.1
Sensitivityb intensity (μK)	8.8	4.7	4.1
Sensitivity polarization (μK)	12.5	7.5	7.8

Notes. ^aFWHM assuming Gaussian beams. ^bThis is for 30 months of integration.

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For a nearly full-sky (we use f_sky = 0.75) CMB experiment, the likelihood ${\cal L}$ can be approximated with the following formula (Baumann et al. 2009):

$$\begin{eqnarray} -\!2 \ln {{\cal L}} &=& \sum _{l=l_{\rm min}}^{l_{\rm max}} (2l+1)f_{\rm sky} \left[ -3 + \frac{\hat{C}_l^{B\!B}}{C_l^{B\!B}} + \ln \left(\frac{C_l^{B\!B}}{\hat{C}_l^{B\!B}}\right) \right. \nonumber\\ && + \frac{\hat{C}_l^{T\!T}C_l^{E\!E} + \hat{C}_l^{E\!E}C_l^{T\!T} - 2\hat{C}_l^{T\!E}C_l^{T\!E}}{C_l^{T\!T}C_l^{E\!E}-\big(C_l^{T\!E}\big)^2} \nonumber\\ &&\left. +\ln {\left(\frac{C_l^{T\!T}C_l^{E\!E}-\big(C_l^{T\!E}\big)^2}{\hat{C}_l^{T\!T}\hat{C}_l^{E\!E}-\big(\hat{C}_l^{T\!E}\big)^2}\right)}\right], \end{eqnarray} \tag{ 39 }$$

where l_min = 3 and l_max = 2500 have been used in our calculation. Here, $\hat{C}_l$ is the observed (or simulated input) angular power spectrum and C_l is the theoretical power spectrum plus noise.

We use the model described in Dunkley et al. (2009) and Baumann et al. (2009) to propagate the effect of polarization foreground residuals into the estimated uncertainties on the cosmological parameters. For simplicity, only the dominating components in the frequency bands we are using, i.e., the synchrotron and dust signals, are considered in our simulation. The fraction of the residual power spectra are all assumed to be 5%.

5.1.2. JDEM SN Simulation

For the JDEM SN simulation, we use the model given by the DETF forecast (Albrecht et al. 2006), with roughly 2500 spectroscopic SN at 0.03 < z < 1.7 and 500 nearby samples. The apparent magnitude of SN is modeled as

$$\begin{eqnarray} m &=& M+5\log _{10}\left(\frac{d_L}{\mbox{Mpc}}\right)+25 \nonumber\\ && -\mu ^Lz-\mu ^Qz^2 - \mu ^S\delta _{{\rm near}}, \end{eqnarray} \tag{ 40 }$$

where δ_near is unity for the nearby samples and zero otherwise.

There are four nuisance parameters in this model. The SN absolute magnitude is expanded as a quadratic function M − μ^Lz − μ^Qz² to account for the possible redshift dependence of the SN peak luminosity, where M is a free parameter with a flat prior over −∞ < M < +∞; and for μ^L and μ^Q, Gaussian priors $\mu ^{L,Q}=0.00\pm 0.03/\sqrt{2}$ (the "pessimistic" case in the DETF forecast) are applied. Finally, given that the nearby samples are obtained from different projects, an offset μ^S is added to the nearby samples only. For μ^S we apply a Gaussian prior μ^S = 0.00 ± 0.01.

The intrinsic uncertainty in the SN absolute magnitude is assumed to be 0.1, to which an uncertainty due to a peculiar velocity 400 km s⁻¹ is quadratically added.

5.1.3. BAO Simulation

5.1.4. EUCLID Weak Lensing Simulation

We assume a WL survey with the following "Yellow Book" EUCLID specifications (Euclid Study Team 2009):

$$\begin{equation} f_{{\rm sky}}=0.5 \,,\quad \ \big\langle \gamma _{{\rm int}}^2 \big\rangle ^{1/2}=0.35 \,,\quad \bar{n}=40 \ \rm { galaxies}/ {\rm arcmin}^{2}, \end{equation} \tag{ 41 }$$

where 〈γ²_int〉^1/2 is the intrinsic galaxy ellipticity, and $\bar{n}$ is the average galaxy number density being observed. At high ℓ, the non-Gaussianity of the dark matter density field becomes important (Doré et al. 2009). For simplicity we use an ℓ-cutoff ℓ_max = 2500 to avoid modeling the high-ℓ non-Gaussianity, which will provide more cosmic information, hence, the WL constraints shown in this paper are conservative.

We use a fiducial galaxy distribution,

$$\begin{equation} n(z)\propto \left(\frac{z}{z_0}\right)^2\exp {\left[-\left(\frac{z}{z_0}\right)^{1.5}\right]} \ {\rm with} \ z_0=0.6. \end{equation} \tag{ 42 }$$

The galaxies are divided into four tomography bins, with the same number of galaxies in each redshift bin. The uncertainty in the median redshift in each redshift bin is assumed to be 0.004(1 + z_m), where z_m is the median redshift in that bin.

In the ideal case in which the galaxy redshift distribution function is perfectly known, the formula for calculating WL tomography observables and covariance matrices can be found in, e.g., Ma et al. (2006). In order to propagate the uncertainties of photo-z parameters onto the uncertainties of the cosmological parameters, we ran Monte Carlo simulations (with random Gaussian shift of the median redshift in each bin) to obtain the covariance matrices due to redshift uncertainties, which are then added to the ideal covariance matrices.

The constraints on cosmological parameters from future experiments are shown in Table 2 below those for current data. The future observations should improve the measurement of ε_s significantly—about five times better than the current best constraint. There is also a significant improvement on the upper bound of ε_ϕ∞, which together with the constraint on ε_s can be used to rule out many tracking models. The running parameter ζ_s remains unconstrained.

To compare different DE probes, in Figure 10 we plot the marginalized constraints on Ω_m0 and ε_s for different data sets. To break the degeneracy between DE parameters and other cosmological parameters, we apply the Planck-only CMB constraints as a prior on each of the non-CMB data sets.

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The forecasted BAO, WL, and SN results, when combined with the Planck prior, are all comparable. We note in particular that the ground-based BAO surveys deliver similar measurements at a fraction of the cost of the space experiments, although of course much useful collateral information will come from all of the probes.

The shrinkage in the allowed parameter space is visualized in Figure 9 through the decrease in area of the trajectory bundles from current to future data, showing trajectories of 1 + w_ϕ sampled down to the 1σ level. We also show the mean and standard deviation of 1 + w_ϕ at the center of four uniform bands in a. As mentioned above, the bands are highly correlated because of the coherence of the trajectories, a consequence of their physical origin. We apply such band analyses to early (as well as late) universe inflation that encodes such smoothing theory priors in future papers.

Figure 9 also plots $(1+z)^{-3/2}(H/H_0) = \sqrt{\rho _{\rm tot}a^3/\rho _{\rm tot,0}}$ trajectories, which are flat in the high-redshift matter-dominated regime. They are less spread out than the w bundle because H depends upon an integral of w. One of the observable quantities measured with SNe is the luminosity distance. What we plot is something more akin to relative magnitudes of standard candles, 5log₁₀(d_L/d_L;ref), in terms of d_L;ref, the luminosity distance of a reference ΛCDM model. For luminosity distance trajectories reconstructed with the current data we choose a reference model with Ω_m0 = 0.29, which the SDSS LSS data drove us to; for forecasts, we chose Ω_m0 = 0.26, what the other current data are more compatible with, for the reference. Because the SN data only measure the ratio of luminosity distances, for each trajectory we normalize d_L/d_L;ref to be unit in the low-redshift limit by varying H₀ in the reference model. The error bars shown are for the current SN data (Kessler et al. 2009), showing compatibility with ΛCDM, and, based upon the coherence of the quintessence-based prior and the large error bars, little flexibility in trying to fit the rise and fall of the data means.

The rightmost panels of Figure 9 show the linear growth factor for dark matter fluctuations relative to the expansion factor, (1 + z)D(z). The normalization is such that it is unity in the matter-dominated regime. It should be determined quite precisely for the quintessence prior with future data.

In slow-roll thawing models, to first order w(z) only depends on two physical quantities: one determining "when to roll down," quantified mostly by 1 − Ω_m0, and one determining "how fast to roll down," quantified by the slope of the potential, i.e., ε_s, since ε_ϕ∞ = 0, and ζ_s ≈ 0:

$$\begin{equation} 1+w_\phi \vert _{\rm slowroll, thawing} \approx \frac{2\varepsilon _{s} }{3} F^2\bigg (\frac{a}{a_{\rm eq}}\bigg)\,, \end{equation} \tag{ 43 }$$

where F is an analytical function given by Equation (25), and $a_{\rm eq}\vert _{\rm slowroll, thawing} \approx (\frac{{\Omega _{m0}}}{1-{\Omega _{m0}}}) ^{1/[3-1.08(1-{\Omega _{m0}})\varepsilon _{s} ]}$ to sufficient accuracy. We plot F²(x) in the left panel of Figure 11. It is zero in the small a regime and unity in the large a regime, and at a_eq is $F^2(1)=[\sqrt{2}-\ln (1+\sqrt{2})]^2 \approx 0.284$. The right panel of Figure 11 plots the derivative dF²/dx showing |dw/da| maximizes around a = a_eq (at redshift ∼0.4). A few examples of 1 + w(a) are plotted in Figure 12 to show the mapping from ε_s to w(a) trajectories. Three stages are evident in the significant time dependence of dw/da: the Hubble-frozen stage at a ≪ a_eq, where w has the asymptotic value −1; the thawing stage when the DE density is comparable to the matter component; and the future inflationary stage where DE dominates (a ≫ a_eq) and a future attractor with 1 + w → 2ε_s/3 is approached, agreeing with the well-known early-inflation solution 1 + w = 2_V/3.

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Even with such models, there is in principle another parameter: the initial field momentum is unlikely to be exactly on the ε_ϕ∞ = 0 attractor. In that case, $\dot{\phi }$ falls as a⁻³ until the attractor is reached, a "burn-in" phase. (Such burn-in phases are evident in Figure 2 for tracking models.) In early work on parameterizing late inflatons we added a parameter a_s to characterize when this drop toward the attractor occurred, and found it could be (marginally) constrained by the data to be small relative to the a-region over which the trajectories feel the DE-probing data. However, we decided to drop this extra parameter for this paper since the expectation of late-inflaton models is that the attractor would have been established long before the redshift range relevant for DE probes.

The three-parameter, two-parameter, and one-parameter fits all give about the same result for the statistics of ε_s derived from current data. Figure 13 illustrates why we find ε_s ≈ 0.0 ± 0.28. It shows the confrontation of banded SN data on distance moduli defined by

$$\begin{equation} \mu \equiv 25+ 5\log _{10}{\left(\frac{d_L}{\rm Mpc}\right)}, \end{equation} \tag{ 44 }$$

where d_L is the physical luminosity distance, with trajectories in μ with differing values of ε_s (for the h (or Ω_m0) values as described.) The current DE constraints are largely determined by SN, in conjunction with the Ω_mh²-fixing CMB.

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The parameterization (43) can be simply related to the phenomenological w₀–w_a parameterization by a first-order Taylor series expansion about a redshift zero pivot:

$$\begin{equation} 1+w_0\equiv \frac{2}{3} \varepsilon _{s} F^2\bigg (\frac{1}{a_{\rm eq}}\bigg), \end{equation} \tag{ 45 }$$

and

$$\begin{equation} w_a\equiv - \frac{2}{3} (\varepsilon _{s} /a_{\rm eq})\left. \frac{dF^2(x)}{dx}\right|_{x=1/a_{\rm eq}}. \end{equation} \tag{ 46 }$$

From formula (43), it follows that w₀ and w_a should satisfy the linear relation

$$\begin{equation} 1+w_0+w_a(1-a^*)=0 \, \end{equation} \tag{ 47 }$$

where

$$\begin{equation} a^*=1-\frac{\sqrt{1+a_{\rm eq}^3}-a_{\rm eq}^3 \ln \left(\frac{1+\sqrt{1+a_{\rm eq}^3}}{a_{\rm eq}^{3/2}} \right)}{6a_{\rm eq}^3 \left[ \ln \left(\frac{1+\sqrt{1+a_{\rm eq}^3}}{a_{\rm eq}^{3/2}}\right)-\frac{1}{\sqrt{1+a_{\rm eq}^3}} \right]} \end{equation} \tag{ 48 }$$

is roughly the scale factor where the field is unfrozen. Over the range of 0.17 < Ω_m < 0.5, this constraint Equation (47) can be well approximated by the formula

$$\begin{equation} 1+w_0+w_a \left(0.264+\frac{0.132}{{\Omega _{m0}}} \right)=0. \end{equation} \tag{ 49 }$$

Such a degeneracy line is plotted in Figure 1. Rather than a single w₀ parameter, constraining to this line based on a more realistic theoretical prior obviously makes for a more precise measurement of w. On the other hand, once the measurements of w₀ and w_a are both sufficiently accurate, the slow-roll thawing models could be falsified if they do not encompass part of the line. Obviously, though, it would be better to test deviations relative to true thawing trajectories rather than the ones fit by this perturbation expansion about redshift zero. One way to do this is to demonstrate that 1 + w_∞ is not compatible with zero, as we now discuss.

Similar w₀–w_a relations have appeared in Kallosh et al. (2003) for the linear potential and de Putter & Linder (2008) for the pnGB model. Our thorough analytic treatment here shows that such a relation: (1) is general for all slow-roll thawing models and (2) depends on Ω_m0.

The DETF w₀–w_a parameterization which is linear in a can be thought of as a parameterization in terms of w₀ and w_∞ = w_a + w₀ (or _de,0 and _de,∞ in the DE acceleration factor language). The upper panel w₀–w_a contour map in Figure 14 shows that it is only a minor adjustment of the w₀–w_a of Figure 1. If we said there is a strict dichotomy between trajectories that are quintessence and trajectories that are phantom, as has been the case in our treatment in this paper, only the two of the four quadrants of Figure 14 would be allowed. (In the w₀–w_a figures, the prior looks quite curious visually, in that the line w_a = −(1 + w₀) through the origin has only the region above it allowed in the 1 + w₀>0 regime, and only the region below it allowed in the 1 + w₀ < 0 regime.)

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To make this issue more concrete, we show in the lower panel of Figure 14 what happens for our physics-motivated two-parameter linear model, Equation (33), except that it is linear for $\pm \sqrt{\vert 1+w_{\rm de} \vert }$, and in the "time variable" F not in a. In terms of w_ϕ0 and w_ϕ∞, we have $\sqrt{\epsilon _{\phi }} = \sqrt{\epsilon _{\phi 0}} +(\sqrt{\epsilon _{\phi \infty }} - \sqrt{\epsilon _{\phi 0}})$ [1 − F(a/a_eq)/F(1/a_eq)]. The prior measure on 1 + w_ϕ0 and w_ϕ∞ is uniform as in the linear-a case. Once the prior that only two quadrants are allowed is imposed upon the linear-a case, it looks reasonably similar to the lower panel.

The determination of our various parameters depends upon an extended redshift regime with direct or indirect observational information. An error in a parameter is built up by a sum of deviations of the observational data from the observable constructed using the model for w_de, using the maximum likelihood formula. The data-sensitive redshift regime is quite extended, not concentrated around zero redshift for w₀, not concentrated at high redshift for w_∞ or ε_ϕ∞, and not concentrated at a_eq for ε_s. An illustration of the redshift reach of the ε_s parameter is Figure 13 for the SN observable, namely a magnitude difference. The specific redshift reach is very data dependent of course.

For slow-roll thawing quintessence models, the running parameter ζ_s can be related to the second derivative of ln V. Using the single-parameter approximation we can approximately calculate dϕ/dln a at a = a_eq. The result is

$$\begin{equation} \left.\frac{d\phi }{d\ln a}\right|_{a=a_{\rm eq}}= \left. \sqrt{2} {M_{{\rm p}}}\sqrt{\epsilon _{\phi } \Omega _{\phi }}\right|_{a=a_{\rm eq}} \approx 0.533 \sqrt{\varepsilon _{s} }{M_{{\rm p}}}. \end{equation} \tag{ 50 }$$

An immediate consequence is that the amount that ϕ rolls, at least at late time, is small compared with the Planck mass. In the slow-roll limit, to the zeroth order, Equation (26) can be reformulated as

$$\begin{equation} \frac{d\sqrt{2\varepsilon _V}}{dN}\vert _{a=a_{\rm eq}}\approx \frac{3}{2\sqrt{2}}\zeta _{s} \sqrt{\varepsilon _{s} }. \end{equation} \tag{ 51 }$$

Combining Equations (50) and (51), and that ${M_{{\rm p}}}^2d^2\ln V/d\phi ^2 = - d\sqrt{2\varepsilon _{s} }/ d\phi$, we obtain

$$\begin{equation} \zeta _{s} \approx - \frac{C{M_{{\rm p}}}^2}{2}\frac{d^2\ln V}{d\phi ^2} = - \frac{C}{4}\frac{d^2\ln V}{d\psi ^2} \, \end{equation} \tag{ 52 }$$

where

$$\begin{equation} C\equiv \frac{8}{3}\left(1-\frac{\ln {(1+\sqrt{2})}}{\sqrt{2}}\right) \approx 1.005. \end{equation} \tag{ 53 }$$

For phantom models C is negative.

We have shown in Section 5 that, for the fiducial ΛCDM model (with ε_s = ε_ϕ∞ = 0), the running parameter ζ_s cannot be measured. For a nearly flat potential, the field momentum is constrained to be very small, hence so is the change δϕ, so the second-order derivative of ln V is not probed. To demonstrate what it takes to probe ζ_s, we ran simulations of fiducial models with varying ε_s (0, 0.25, 0.5, and 0.75). The resulting forecasts for the constraints on ε_s and ζ_s are shown in Figure 15. We conclude that unless the true model has a large ε_s ≳ 0.5, which is disfavored by current observations at nearly the 2σ level, we will not be able to measure ζ_s. Thus if the second derivative of ln V is of order unity or less over the observable range, as it has been engineered to be in most quintessence models, its actual value will not be measurable. However, if |M_p²d²ln V/dϕ²| ≫ 1, the field would be oscillating. Though in their own right, oscillatory quintessence models are not considered in this work.

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The late-universe acceleration requires the field to be in a slow-roll or at most a moderate-roll at low redshift. In tracking models, the high redshift _ϕ constancy implies the kinetic energy density follows the potential energy density of the scalar field: Π²_ϕ/2V → ε_ϕ∞/(3 − ε_ϕ∞). The field could be fast-rolling in the early universe, with large ε_ϕ∞ and a steep potential, and indeed is the case for many tracking models. If the potential is very flat, field momenta fall as a⁻³, The rate of Hubble damping in the flat potential limit is $\dot{\phi }\propto a^{-3}$. However, as proposed in most tracking models, the potential is steep at high redshift, and gradually turns flat at low redshift. The actual damping rate itself is related to the field momentum, which again relies on how steep potential is at high redshift. The complicated self-regulated damping behavior of field momentum is encoded in the tracking parameter ε_ϕ∞ in Equation (27) (see Figure 16 where the damping of field momentum is visualized in w(z) space). At high redshift the approximation $\dot{\phi }\sim a^{-3}$ completely fails, but at low redshift the damping rate asymptotically approaches sxs $\dot{\phi }\sim a^{-3}$.

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Since DE is subdominant at high redshift, the observational probes there are not very constraining. In the three-parameter approximation, instead of an asymptotic ε_ϕ∞, we could use a moderate redshift pivot z_pivot ∼ 1, with variables $\varepsilon _{\phi \rm pivot}$, ε_s, and ζ_s, and the asymptotic regime now a controlled extrapolation. Indeed, using the method of Section 6.3 we could define everything in terms of pivots of _ϕ; for example, at z = 0 as we used in Section 6.3, and with a third about half way in between 0 and 1, such as at a_eq, which is about 0.7, so z_eq ∼ 0.4. Since all would be functions of our original three parameters and a_eq through the specific evaluations of F and F₂ at the pivots, the formula does not change, but the measure (prior on the parameters) would be different, namely to one uniform in the three $\varepsilon _{\phi \rm pivot}$s.

Whatever the specific parameters chosen, in all cases the lack of high-redshift constraining power means that our reconstructed w_ϕ(a) trajectories such as in Figure 9 should be regarded as low-redshift observational constraints extrapolated through theory-imposed priors on the smooth change in quintessence paths at high redshift. The tracking assumption that we have used in this paper provides one such extrapolation, but others may be considered. For these, we need to define a prior on $\dot{\phi }_{\rm pivot}$ and information on the damping rate of $\dot{\phi }$, and this could have an impact on the marginalized posterior likelihood of ε_s. We have presented an example of this, showing that whereas the flat prior 0 ⩽ |ε_ϕ∞| < _m gives ε_s = 0.00^+0.18_−0.17, the ε_ϕ∞ = 0 "thawing prior" gives ε_s = −0.00^+0.27_−0.29. We note though that because DE is so subdominant at higher redshift, observables such as the Hubble parameter H(z), the luminosity distance d_L(z), and the linear perturbation growth rate D(z) are insensitive to the high-z extrapolation of w_ϕ(z), as we showed in Figure 9. Thus, even for the SUGRA model shown in the middle panel of Figure 2 with a time-varying attractor which is not fit as well at high z, what the data say about it is confined to low z, and hence the results obtained with our formula apply to such models as well; in particular, the late-time potential steepness must be controlled according to the ε_s constraint for them to be viable.

Suppose that one has in mind a specific quintessence potential V(ϕ). How can our constraints, in particular on ε_s, be used to test viability? The first thing is to see what _V looks like at a function of ϕ. One does not know ϕ_eq of course, but it had better lie in the range for ε_s allowed by the data. The model would be ruled out if no ϕ has _V penetrating the allowed region. Otherwise, the field equation of motion has to be solved to get ϕ_eq and hence ε_s from its definition. Such a calculation is inevitable because quintessence models do not solve the fine-tuning problem. For tracking models, the low-redshift dynamics is usually designed (i.e., are fine-tuned) to deviate from an attractor. For thawing models, the initial value of ϕ_∞ (to which ϕ is frozen at high redshift) needs to be fine-tuned as well.

Although it seems unlikely that the community will abandon the DETF-sanctioned w₀–w_a linear a model and its FOM as a way to state results for current and future DE experiments, our easy-to-use ε_s constraint with its clear physics-based meaning could be quoted as well. The two-parameter case goes well beyond linear a with the same number of parameters. Because we learn little about the third parameter one might think that the three-parameter case has one parameter too many to bother with, but it is not really redundant since it helps to accurately pave the w_ϕ paths. To facilitate general-purpose use of this parameterization, we wrote a fortran90 module that calculates w_ϕ(a|ε_s, ε_ϕ∞, ζ_s, Ω_m0) (http://www.cita.utoronto.ca/~zqhuang/work/wphi.f90). The f90 code is well wrapped so that implementing it in CosmoMC is similar to what has to be done to install the w₀–w_a parameterization to deliver more physically meaningful MCMC results.

One has to integrate w_ϕ(z) once to get the DE density ρ_ϕ(z) and the Hubble parameter H(z). Observational data are often directly related to either H(z) or ρ_de(z), and not sensitive to w_de(z) variations. Thus, one may prefer to directly parameterize ρ_de(z) (Wang & Tegmark 2004; Alam et al. 2004) or H(z) (Sahni et al. 2003; Alam et al. 2004, 2007) on phenomenological grounds. A semi-blind expansion of H(z) or ρ_de(z), e.g., in polynomials or in bands, differs from one in w_de(z), since the prior measures on the coefficients are radically altered. By contrast, a model such as ours for w_de(z; ε_s, ε_ϕ∞, ζ_s) characterized by physical parameters is also a model for H(z; ε_s, ε_ϕ∞, ζ_s) or ρ_de(z; ε_s, ε_ϕ∞, ζ_s), obtained by integration.

Where the freedom does lie is in the prior measures imposed on the parameters, a few examples of which were discussed in Section 4.

For thawing models, our parameters ε_s and ζ_s are based on a local expansion of ln V(ϕ) at low redshift. One should obtain similar results by directly reconstructing the local V(ϕ). An attempt at local V(ϕ) reconstruction was done in Sahlén et al. (2005), where a polynomial expansion was used,

$$\begin{equation} V(\phi)=V_0+V_1\phi +V_2\phi ^2+\cdots \, \end{equation} \tag{ A6 }$$

with ϕ chosen to be zero at present. A result highlighted in that paper is that there is a strong degeneracy between V₁ and $\dot{\phi }_0$. We shall see that this degeneracy is actually a result of the prior, and not something that observational data are telling us, as we now show.

For simplicity we assume that Ω_m0 and h are known, and hence the parameter V₀ is fixed. Although V₁ is defined at the pivot a = 1, it is approximately proportional to $\sqrt{\varepsilon _{s} }$ since _V varies slowly. Because $\dot{\phi }_0$ is a function of 1 + w|_z=0, the degeneracy between V₁ and $\dot{\phi }_0$ is roughly the degeneracy between ε_s and _ϕ0, an expression of the low-redshift dynamics being mainly dependent upon the slope of ln V. This is explicitly shown in Figure 16.

Huterer & Peiris (2007) tried to do V(ϕ) reconstruction using what was then recent observational data. They used RG flow parameters _V, η_V, etc. expanded about an initial redshift z_start = 3, and evolved the field equations till z = 0. The constraint they find on the parameter _V|_z=3 is very weak. From our work, this is understandable because the high-redshift dynamics are determined by the tracking parameter rather than by ε_V. They also calculated the posterior probability of ε_V at redshift zero, which is much better constrained. Their result is consistent with what we have obtained for ε_s. Also the large uncertainty in η_V|_z=0 they find is similar to our result for ζ_s.

Recently Fernandez-Martinez & Verde (2008) have proposed Chebyshev expansions of V(z) or w_ϕ(z), a method we have applied to treat early universe inflation (J. R. Bond et al. 2010, in preparation), but as we have discussed in Section 1 there is de facto much less information in the ∼ one e-folding in a that DE probes cover than the ∼10 e-foldings in a that CMB+LSS power spectrum analyses cover.