Phenomenological parameterization of quintessence

C.Wetterich

REVIEW

Phenomenological parameterization of quintessence

C.Wetterich

Astrophysics · Submitted 11 Mar 2004 · 12 references

Pith's verdict: A clean three-parameter dark energy fit built on the logit of Omega_h, with closed-form H(z) and a built-in early dark energy knob; the real gap is that it is never fitted to data.

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arXiv IDastro-ph/0403289 (version 1)

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CitationC.Wetterich. "Phenomenological parameterization of quintessence." arXiv:astro-ph/0403289 (2004). https://pith.science/paper/VN3WQOMY

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The thing worth knowing: this is a well-built parameterization paper, and the trick is the variable, not the ansatz. Wetterich defines R = ln(Omega_h/(1-Omega_h)), which turns the Omega_h-w relation into dR/dy = 3w exactly (eq 3, matter domination only). Then R = R_0 + 3w_0 y/(1+by) gives a closed-form averaged equation of state (eq 12) and hence a closed-form H(z) (eq 11). That is the real convenience: no numerical integral for the luminosity distance, Omega_h automatically stays in (0,1), and b maps one-to-one onto an early dark energy fraction Omega_e (eqs 6-7). Compared with CPL, which is fine at low z and nonsense at last scattering, this is designed for the full 0 < z < 1100 range. The bridge w' = -2 w_0 b (eq 9) lets you translate to CPL fits, which is a nice touch. The potential reconstruction (eqs 15-20) is a genuine inversion, not a restatement of the input, and the monotonicity bound (eq 23) follows correctly from eq 13 and eq 21, though it constrains the ansatz rather than quintessence generally. The general bound (eq 24) is ansatz-independent, so the claim that near-Lambda models are hard to distinguish at low z holds up. What it does well: it is honest about scope. The text after eq 13 says outright the form is not general enough and cannot handle a sign change in w; footnote 2 flags the radiation correction; and there are two extended versions (alpha in eq 26, gamma in eq 30) for when three parameters are not enough. That is how a parameterization paper should be written. Soft spots, in order of size. First, there is no fit to any data, not one supernova sample, not the CMB shift parameter. The whole value proposition is that three parameters capture the data without bias, and that is asserted, not shown. A single best-fit figure would have settled it. Second, the abstract's reconstruction claim (abstract:s2) is a bit ahead of the body: the k=1 route (eq 17) does deliver V(phi) from (Omega_M, w_0, Omega_e), but the exponential-V route needs a kinetial k(phi), which is extra structure. Presentational, not wrong. On the stress-test note about eq 24: I think that is a misparse. The arXiv text flattens the fraction, but read with the leading minus on the whole fraction, w_up = -[1-w_0-(1+w_0)e^{6y}]/[1-w_0+(1+w_0)e^{6y}] is exactly the tanh solution: it gives w_0 at y=0, tends to 1 as y grows, and solves dw/dy = 3(1-w^2). If the minus sits inside the numerator it is a typo, but either way it is a side remark about distinguishability from Lambda, not load-bearing. Who should read it: anyone doing joint SN/CMB/BAO fits who wants a dark energy parameterization that survives at high z, and anyone fitting early dark energy. Worth a serious referee. The ask is modest, fit it to something, and the machinery is sound.

Referee report

Significance

If the three-parameter family adequately covers the trajectories of interest, a single closed-form function spans the whole observable range 0 ≤ z ≲ 1100 and maps directly onto a scalar potential, which is genuinely useful for joint supernova, CMB and structure-formation analyses and for comparing them on a common footing. The paper earns credit for several concrete deliverables: the closed-form H(z) and w̄_h(z) in Eqs. (11)-(12), the explicit inversion relating b to Ω_e in Eq. (7), the low-redshift relation w' = -2w_0 b in Eq. (9) (a falsifiable prediction tying the two deviation parameters together), and the monotonicity bound of Eq. (23), which is a real constraint on the parameter space rather than a restatement of the ansatz. The author also states the limitations of the ansatz honestly in the paragraph following Eq. (13), noting that the form is not general enough to describe a change of sign of w_h; that acknowledgment is appropriate and is not itself a defect.

Major objections (4)

[Eq. (24) and the bound w_h ≤ w_h^up] Eq. (24) does not solve the differential equation it is presented as solving. The stated solution w_h^up(y) = [-1-w_0-(1+w_0)e^{6y}]/[1-w_0+(1+w_0)e^{6y}] fails the initial condition: at y=0 it gives -(1+w_0) rather than w_0, and at w_0=-1 it gives 0 rather than -1. The correct solution of ∂w_h/∂y = 3(1-w_h^2) with w_h(0)=w_0 is w_h^up(y) = [(1+w_0)e^{6y}-(1-w_0)]/[(1+w_0)e^{6y}+(1-w_0)]. This matters because the sentence immediately following uses Eq. (24) to state a bound, w_h(z) ≤ w_h^up(z), for any monotonic potential; with the printed expression the 'bound' falls below -1 for w_0 > -1 and is therefore not a physical upper bound at all. The qualitative conclusion the author draws from it (that for 1+w_0 ≪ 1 one needs large z before w_h departs appreciably from -1, with the departure growing like e^{6y}) survives the correction, but the displayed equation and the bound derived from it must be fixed.
[Text following Eq. (7), identifying Ω_e with Ω_h^(ls) and Ω̄_h^(sf)] The paper states that 'for practical purposes within the three parameter parameterization Ω_e equals Ω_h^(ls) and Ω̄_h^(sf) and enters therefore very directly the CMB anisotropies or structure formation.' This identification rests on Eq. (1), which the author explicitly derives assuming that the non-dark-energy component is pressureless matter; at z ≈ 1100 radiation dominates the expansion, as footnote 2 concedes. The approximation is also quantitatively loose: from Eq. (4), R(∞) - R(y_ls) = 3w_0/[b(1+b y_ls)], which for b=1, y_ls ≈ 7 and w_0=-1 is -0.375 in the logarithm of the odds, i.e. roughly a 30% shift, so Ω_e = 0.02 corresponds to Ω_h(z=1100) ≈ 0.014 rather than 0.02. The claim that Ω_e equals the structure-formation weighted mean Ω̄_h^(sf) is a separate and stronger statement, since a weighted mean over an epoch is not the asymptotic value. The author should either restrict these identifications to the matter-dominated regime, or state the accuracy of the Ω_e ↔ Ω_h^(ls) map as a function of b and w_0, or use the extended parameterization of Eq. (30) with γ fixed by Ω_h^(ls) as suggested later in the paper.
[Eq. (4) and the recommendation to fit b, including negative b] The paper recommends quoting b rather than Ω_e for data fitting on the grounds that Eq. (4) 'also covers the case of negative b.' For b < 0, however, R(y) has a pole at y = -1/b, so Ω_h → 1 at the finite redshift z = e^{-1/b}-1. Unless |b| < 1/ln(1101) ≈ 0.14, this pole lies inside the observable range and the parameterization becomes unphysical there. Since the paper is recommending b as the fitting parameter for supernovae, this restriction should be stated explicitly, together with the corresponding prior or the statement that fits yielding b < -0.14 are outside the range where the ansatz can be applied to CMB or structure-formation data.
[Overall: absence of any confrontation with data or with a known quintessence model] The paper's case for the parameterization is that it is simple, covers the full redshift range, and maps onto a scalar potential; all of these are demonstrated algebraically. What is not demonstrated is that the three-parameter family actually reproduces the trajectories of quintessence models that people use. A short demonstration would substantially strengthen the paper: fit Eq. (4) to a standard Ratra-Peebles or exponential-potential scaling solution over 0 ≤ z ≤ 1100 and quote the residuals in w_h(z) or Ω_h(z), and, if possible, the corresponding shift in the reconstructed potential. Without such a check the reader has no way to judge how restrictive Eq. (4) is, and the abstract's claim that the potential 'can be reconstructed' remains a statement about the algebra rather than about the class of models the parameterization is meant to represent.

Minor objections (6)

[Abstract, sentence [abstract:s2]] The abstract states without qualification that the quintessence potential can be reconstructed from the three parameters. In the body this holds for the standard-kinetic-term route (k=1, Eq. 17), whereas the second route fixes V to the exponential form of Eq. (18) and absorbs the model dependence into a derived kinetial k(φ) (Eq. 19). A phrase such as 'for a standard kinetic term, or up to a kinetial k(φ)' in the abstract would match the body.
[Paragraph following Eq. (23)] The sentence 'we recall that w_0 < 1 or b < 0 are not compatible with scalar quintessence models (with T > 0)' appears to contain a typo: w_0 < 1 is satisfied by every accelerating model and cannot be the intended condition. Presumably w_0 > 1 is meant.
[Eq. (31) and the sentence defining γ] The text states that γ denotes the asymptotic value w_h(z→∞) = γ. From Eq. (31), bγ y(2+by)/(1+by)^2 → γ as y→∞, so the asymptotic value is w_0 + γ, not γ. Correspondingly, the statement that negative γ drives Ω_h → 0 requires γ < -w_0 rather than merely γ < 0.
[Eqs. (26)-(28)] The placement of the exponent α in Eq. (26) is hard to parse as printed, and the reader would benefit from an explicit statement that the α-extension satisfies Ω_h(0) = 1-Ω_M and reduces to Eq. (4) at α = 1, together with a one-line derivation of Eq. (28) from the y→∞ limit.
[Eqs. (8)-(9)] Eq. (9) relates w' of the Taylor expansion in z to b; it would be clearer to state that w' denotes dw_h/dz at z=0 and that the relation holds to first order in z, since the two expansions are in different variables (z versus y = ln(1+z)).
[References] Several reference details need correction: ref. [1] (Wetterich, Nucl. Phys. B302, 668) is dated 1998 but should be 1988; ref. [3] (Caldwell, Dave, Steinhardt) is cited as 'Phys. Rev. Lett. D59, 123504', but Physical Review Letters has no D series, and the paper is Phys. Rev. Lett. 80, 1582 (1998); ref. [8] (Linder) should read Phys. Rev. Lett. 90, 091301 (2003).

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The paper

The paper claims

A three-parameter formula is proposed to describe dark energy across the entire observable redshift range, including the possibility that some dark energy was already present before galaxies formed.

The reading

Dark energy is usually described by its present amount and its present equation of state, but those two numbers say nothing reliable about how much dark energy existed at last scattering or during structure formation. This paper proposes tracking a different quantity, $R(y) = 3w_0 y/(1+by)$ added to a constant fixed by today's matter fraction, where $y = 3w_0 y/(1+by)$ is shorthand for $3w_0 y/(1+by)$ with $y = 3w_0 y/(1+by)$; more plainly, the combination $R = 3w_0 y/(1+by)$ grows linearly at low redshift and saturates at high redshift. The saturation level is a third parameter, the early dark energy fraction $3w_0 y/(1+by)$, and the bending parameter $3w_0 y/(1+by)$ measures where the turnover happens. If the proposal holds, a single closed-form Hubble rate covers supernovae, the cosmic microwave background and structure formation at once, and each parameter set can be converted back into an actual scalar field potential and kinetic term.

What carries the argument

The carrier is the variable $R(y) = 3w_0 y/(1+by)$, the logit of the dark energy fraction, whose redshift derivative equals three times the equation of state whenever the non-dark-energy component is pressureless. Parameterizing $R$ rather than $3w_0 y/(1+by)$ or $3w_0 y/(1+by)$ directly keeps the dark energy fraction automatically between zero and one, makes the Hubble rate algebraic, and makes the map to a scalar potential explicit. The bending parameter $3w_0 y/(1+by)$ is the piece that distinguishes this from a constant-$3w_0 y/(1+by)$ model and connects the low-redshift equation of state to the early dark energy fraction through $3w_0 y/(1+by)$.

Extended reading notes

Core claim

The paper's claim is that a three-parameter family, written in terms of the present matter fraction $3w_0 y/(1+by)$, the present equation of state $3w_0 y/(1+by)$ and a bending parameter $3w_0 y/(1+by)$, captures quintessence with a time-varying equation of state over the whole range $0 < z < 1200$. The construction works through the variable $R(y) = 3w_0 y/(1+by)$, whose derivative is three times the equation of state, so that a simple ansatz for $R$ generates both $3w_0 y/(1+by)$ and $3w_0 y/(1+by)$ in closed form. The resulting Hubble rate is $3w_0 y/(1+by)$, with an averaged equation of state $3w_0 y/(1+by)$ that interpolates between $3w_0 y/(1+by)$ today and the early dark energy value at high redshift. Because the same three numbers fix the scalar field's kinetic and potential energies through $3w_0 y/(1+by)$ and $3w_0 y/(1+by)$, the parameterization is not merely a fitting function: each parameter set corresponds to a definite quintessence model, and the potential can be reconstructed, either with a standard kinetic term or as a standard exponential potential with a non-trivial kinetic function. The paper also derives a constraint: requiring the potential to be monotonic in the scalar field bounds the bending parameter, $3w_0 y/(1+by)$, which in turn implies that for $3w_0 y/(1+by)$ close to $-1$ the equation of state cannot depart appreciably from $-1$ at low redshift.

Load-bearing premise

The load-bearing premise is that one functional shape, $w_h(y) = w_0/(1+by)^2$, covers the quintessence models nature might actually use; the paper itself states that this form is not general enough to describe a change of sign of the equation of state, such as the positive value expected during radiation domination, and that corrections from radiation are left out of the main treatment.

Editorial extensions

If this is right

Supernova distances, CMB anisotropies and structure growth could be compared within one three-parameter family instead of three separate fitting functions, so a disagreement between redshift ranges becomes a statement about the parameters rather than about incompatible ansätze.
A measured early dark energy fraction fixes the bending parameter, which in turn predicts the low-redshift drift of the equation of state through $3w_0 y/(1+by)$, giving a consistency check between high- and low-redshift data.
Every accepted parameter triple yields a concrete scalar potential and kinetic function, so cosmological fits translate directly into statements about the shape of the dark energy potential.
The monotonic-potential bound implies that if $3w_0 y/(1+by)$ is confirmed near $-1$, any detectable variation of the equation of state must appear only at appreciable redshift, which sets a target for surveys rather than for local measurements.
The extended versions with an extra parameter $3w_0 y/(1+by)$ or $3w_0 y/(1+by)$ allow a sizeable early dark energy fraction to coexist with $3w_0 y/(1+by)$ near $-1$ when data from different epochs pull in different directions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The parameterization is a candidate bridge between two communities: cosmologists who fit $3w_0 y/(1+by)$ and $3w_0 y/(1+by)$ from data, and model builders who specify a potential. A useful test would be to take published potentials with tracking behaviour and ask how well a single triple reproduces them across the full redshift range.
Because the ansatz saturates at high redshift, it makes a sharp, falsifiable statement that the dark energy fraction at last scattering and the mean fraction during structure formation are numerically close; a detection of a large gap between those two numbers would point to the extended forms rather than the three-parameter core.
The monotonic-potential inequality is the kind of constraint that could be folded into Markov-chain fits as a prior, turning a statement about allowed scalar field models into a restriction on the region of parameter space that counts as quintessence at all.
The paper's own remark that the form cannot describe a sign change of the equation of state suggests an obvious extension: patching the ansatz across the radiation-to-matter transition, where the equation of state is expected to be positive, and checking whether the reconstructed potential remains monotonic.

Original abstract

We propose a simple phenomenological parameterization of quintessence with a time-varying equation of state. In particular, it accounts for the possibility of early dark energy. The quintessence potential can be reconstructed in terms of the present fraction in dark energy, the present equation of state and the amount of early dark energy.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We propose a simple phenomenological parameterization of quintessence with a time-varying equation of state. In particular, it accounts for the possibility of early dark energy. The quintessence potential can be reconstructed in terms of the present fraction in dark energy, the present equation of state and the amount of early dark energy.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose a simple phenomenological parameterization of quintessence with a time-varying equation of state.	contribution 0.99	0.93	0.29	asserted 0.97
s1	In particular, it accounts for the possibility of early dark energy.	contribution 0.92	0.56	0.10	hedged possibility 0.58
s2	The quintessence potential can be reconstructed in terms of the present fraction in dark energy, the present equation of state and the amount of early dark energy.	result 0.39	0.79	0.39	asserted 0.96

What kind of paper the classifier reports this is: theoretical result 0.54 · evidence from proof or derivation 0.79.

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arXiv listing abstract v1

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (12)

[1]

C. Wetterich, Nucl. Phys. B302, 668 (1998)

View on Pith 1998
[2]

B. Ratra, P. J. E. Peebles, Phys. Rev. D37, 3406 (1988)

View on Pith 1988
[3]

R. Caldwell, R. Dave, P. Steinhardt, Phys. Rev. Lett. D59,123504 (1999)

View on Pith 1999
[4]

M. Doran, M. Lilley, M. Schwindt, C. Wetterich, Ap. J. 559, 501 (2 001)

View on Pith
[5]

P. G. Ferreira, M. Joyce, Phys. Rev. D58, 023503 (1998)

View on Pith 1998
[6]

M. Doran, M. Schwindt, C. Wetterich, Phys. Rev. D64, 123520 (2001)

View on Pith 2001
[7]

M. Doran, C. Wetterich, Nucl. Phys. B124, 57 (2003)

View on Pith 2003
[8]

P. Astier, Phys. Lett. B500, 8 (2001) J. Weller, A. Albrecht, Phys. Rev. Lett. 86, 1939 (2001); Phys. Rev. D65, 103512 (2002) E. V. Linder, Phys. Rev. Lett. 90, 91301 (2003)

View on Pith 2001
[9]

G. Efstathiou, MNRAS 342, 810 (2000) B. F. Gerke, G. Efstathiou, astro-ph/0201336

View on Pith Pith review arXiv 2000
[10]

P. S. Corasaniti, E. J. Copeland, astro-ph/0205544

View on Pith Pith review arXiv
[11]

C. Wetterich, hep-ph/0302116, in “Strong and Electroweak M atter 2002”, Ed. M. G. Schmidt, p. 230, World Scientiﬁc, 2003

View on Pith Pith review arXiv 2002
[12]

A. Hebecker, C. Wetterich, Phys. Lett. B497, 281 (2001) 9

View on Pith 2001

Forward citations

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How the literature uses it

61 arXiv papers cite this in 67 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Compares with it 1 paper

Our parametrisation is similar to this in the sense that it directly relates $(t)$ to $(t)$, however our parametrisation does not rely on scale factor/redshift time, unlike Refs. [ref] .

The Dark Energy Cosmic Clock: A New Way to Parametrise the Equation of State · Introduction · 0.97

Cites it as background 34 papers

Group E modifies the pre-recombination expansion rate through a transient, non-radiative energy component: axion-like early dark energy (EDE) [ref] , whose many realizations and extensions [ref] and confrontations with data [ref] are reviewed in [ref] ; cold new early dark energy (NEDE) [ref] ; early modified gravity (EMG) [ref] as a representative model respecting Solar-System constraints within scalar-tensor theory of gravity [ref] ;

The $H_0$ World Cup. I. Summary of the baseline group stage results · Introduction · 0.96
Early Dark Energy models are a natural hypothesis of dark energy, see e.g., Refs. [ref] .

A double take on early and interacting dark energy from JWST · Early Dark Energy · 0.99
Among these possibilities, the presence of an early dark energy (EDE) component, proposed before the rise of the Hubble tension [ref] , has been actively studied in this context (see e.g. [ref] ).

Early dark energy constraints with late-time expansion marginalization · Introduction · 0.94

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

C. Wetterich, Nucl. Phys. B302, 668 (1998)

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