A classification of scalar field potentials with cosmological scaling solutions

Andrew R Liddle; Robert J Scherrer

REVIEW

A classification of scalar field potentials with cosmological scaling solutions

Andrew R Liddle, Robert J Scherrer

Astrophysics · Submitted 22 Sep 1998 · 15 references

Pith's verdict: A clean, checkable classification that genuinely adds a third branch, with a slightly overstated 'complete analysis' and a sign typo in Eq. (16) for negative α.

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CitationAndrew R Liddle, Robert J Scherrer. "A classification of scalar field potentials with cosmological scaling solutions." arXiv:astro-ph/9809272 (1998). https://pith.science/paper/YVWCZTJA

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Liddle and Scherrer set out to close a question: which scalar field potentials give exact power-law scaling of the field's energy density while a perfect fluid dominates? The answer is three families — exponential, negative power law, positive power law — and the third is new. The derivation is short and checkable: Eq. (7) pins the kinetic fraction at n/6, Eq. (10) gives φ̇ ∝ t^{-n/m}, and integrating forces either the exponential (Eq. 11) or the power law with α = 2n/(n-m) (Eqs. 13–15). I checked the algebra and the stability thresholds in Eq. (25) against the stated m=3 and m=4 values; they are consistent. Extending the Ratra–Peebles stability analysis to arbitrary m is a real, if modest, contribution. What the paper does well: it is honest about the limits of its own result. The new positive power-law branch turns out to be cosmologically useless — the field decays relative to the fluid, so it is not a dark energy candidate — and the authors say so plainly in Section III.B. The discussion of the Zlatev–Wang–Steinhardt potential in Section III.C is a fair critique: that potential moves the tuning into the shape of V rather than eliminating it. Where it is soft: the abstract's 'complete analysis' is a bit stronger than the evidence. The phase-plane analysis in Section II.C is numerical, shown for one value of m (m=4), and explicitly restricted to even integer α ('For positive α, the interesting case is when α is an even integer'). Eq. (25) is stated for general real α, and the local linearization about u=1 is valid for any α, but the global attractor claim is not demonstrated for non-even α, where φ^α is not defined for φ<0 and the u=-1 critical point does not exist. That is a gap in the 'complete' language, not in the classification itself. Separately, Eq. (16) looks sign-wrong for α<0: the physical force pushes φ away from zero, so the rescaled equation should carry +φ^{α-1}, not −φ^{α-1}. The negative-α stability conclusion is independently supported by Ratra–Peebles, so this is a typo-level issue, not a load-bearing one. The exhaustiveness claim is definitional: it holds for exact power-law scaling with fixed kinetic-to-potential ratio (Eq. 7), and excludes the oscillatory averaged scaling of Eq. (27). The authors acknowledge this in Sections III.B and IV, so it is a stated boundary, not a concealed one. Who should read it: anyone working on quintessence or tracker models who needs a reference for which potentials scale and how stable they are. It is a useful organizing paper, not a new dark energy model. Recommendation: send it to referees. It is solid, modest, and honestly bounded. The sign typo and the even-α restriction are worth a referee's attention but do not threaten the main result.

Referee report

Major objections (2)

[Section II.B, Eqs. (16)-(17) and Eq. (21)] For α<0, the physical potential V = V0 φ^α with V0>0 has dV/dφ < 0, so the rescaled equation of motion should carry +φ^{α-1}, not the -φ^{α-1} printed in Eq. (16). As written, Eq. (17) gives a complex A for α<0: for α=-6 and m=3, the product (2/(α-2))(6/m - α/(α-2)) is -0.3125, so A^{α-2} is negative and A is not real. The stability conclusion for α<0 is nonetheless correct (and agrees with Ratra-Peebles), because the sign errors in Eqs. (16) and (17) cancel in the linearized equation, but the derivation as printed is internally inconsistent and should be corrected. In addition, Eq. (21) has the bracket [6/m - α/(α-2)] while Eq. (20) has [α/(α-2) - 6/m]; one of these is a typo, and the eigenvalues in Eq. (22) are consistent with Eq. (20), so Eq. (21) should be corrected to match.
[Section II.C, Fig. 1, and Eq. (25)] The phase-plane analysis in Section II.C is explicitly restricted to even integer α ('For positive α, the interesting case is when α is an even integer, and then there are three critical points'). The global attractor claim—that generic initial conditions converge to the scaling solution—is therefore demonstrated only for even integer α, while Eq. (25) states the stability threshold for general real α. For non-even α, V ∝ φ^α is not defined for φ<0 (or requires |φ|), so the field cannot cross φ=0 and the phase-space topology differs from the even-α case. The local linearization about u=1 (Eqs. 22-23) is valid for any α, so the threshold itself is probably correct, but the abstract's 'complete analysis' and the global attractor language should be qualified, or the analysis extended to non-even α.

Minor objections (6)

[Abstract] The phrase 'complete analysis of exact solutions and their stability properties' is slightly overstated: the phase-plane analysis is restricted to even integer α (Section II.C), and the paper explicitly confines itself to exact solutions (Section IV). Consider softening the abstract to match the scope.
[Section IV] The authors acknowledge that approximate scaling solutions, such as the Zlatev-Wang-Steinhardt potential, are not classified. This is appropriate, but the abstract's 'all potentials' should be read as 'all potentials within the stated definition'; making that qualification explicit in the abstract would help.
[Reference [3]] Reference [3] is listed as a preprint (astro-ph/9807002). The paper's own analysis of the ZWS potential is independent, so this is not a problem for the central claim, but the citation should be updated if the work has since been published.
[Fig. 2] The dashed kinetic-energy line oscillates too sharply for the plotting resolution, as the caption notes. A zoomed inset or a smoothed envelope would make the figure more informative.
[Section III.B] The statement that the oscillating solution's scaling is independent of the dominant component's equation of state is correct, but it would help to state explicitly that this is precisely why the oscillating solution falls outside the paper's definition of a scaling solution.
[Section I] The paper uses 'tracker' for the m=n case. Since the literature sometimes uses 'tracker' more broadly for any attractor solution, a brief note on the terminology would avoid confusion.

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The paper

The paper claims

The paper sets out to prove that only three shapes of scalar-field potential can make the field's energy density fall as an exact power of the cosmic scale factor while ordinary matter or radiation dominates the expansion, and that the long

The reading

The paper asks a closing question: if a scalar field's energy density is to fall off as an exact power of the cosmic scale factor, $R^{-n}$, while a perfect fluid with its own power law $R^{-m}$ dominates the expansion, what shape must the field's potential have? The authors answer that only three shapes work — an exponential, a negative power law, and, claimed here as new, a positive power law — and for each they give the exact solution and a stability analysis. The question matters because a slowly rolling scalar field is the leading alternative to a bare cosmological constant, and a field whose density tracks the dominant component can stay negligible for a long time and only surface late, which is precisely the fine-tuning such models are meant to relieve. The new branch is the one worth noticing: with a positive power law the field's share of the total energy shrinks rather than grows, so the approximation that the field can be neglected in the expansion becomes better and better justified with time. The authors also map where each branch does and does not behave as an attractor, which decides whether the exact solutions are curiosities or the inevitable late-time fate of generic initial conditions.

What carries the argument

The load-bearing device is a change of variables that turns the scalar field equation into an autonomous phase plane. Because $ρ_φ propto R^{-n}$ forces the kinetic energy to be a fixed fraction $n/6$ of the total field energy, the field trajectory must go as $φ = A t^{1-n/m}$, and substituting that into the equation of motion forces the potential into the power-law form with exponent $α = 2n/(n-m)$. Setting $t = e^{τ}$ and $u = φ/φ_e$, where $φ_e$ is the exact singular solution, converts the nonlinear equation into a two-dimensional system with critical points at $u = -1, 0, +1$. Linearising about $u = ±1$ gives eigenvalues whose real parts are negative exactly when $2n - m - 6 < 0$, which is the stability condition quoted above; the $u = 0$ point, corresponding to the field sitting motionless at the bottom of the potential, turns out to be an unstable saddle, meaning a displaced field returns to the minimum along the scaling solution rather than by simple rest.

Extended reading notes

Core claim

The central claim is a classification theorem. Writing the fluid density as $ρ propto R^{-m}$ and demanding $ρ_φ propto R^{-n}$ with $ρ_φ ll ρ$, the authors show that the scalar field equation admits power-law solutions only for three potentials: $V(φ) propto exp(-λφ)$ when $n = m$ (the familiar exponential, where the field mimics the background fluid); $V(φ) propto φ^{α}$ with $α < 0$ when $n < m$ (the Ratra–Peebles case, where the field grows relative to the fluid); and $V(φ) propto φ^{α}$ with $α > 2$, equivalently $n = α m/(α-2) > m$, when the field decays relative to the fluid. The third class is the paper's new result. The authors then ask whether these particular solutions are attractors, and find an asymmetric answer: every negative power law is a stable attractor for all $α$, while a positive power law is an attractor only when $α > 2(6+m)/(6-m)$ — that is, $α > 6$ during matter domination and $α > 10$ during radiation domination, with the boundary case a marginally stable vortex rather than an attractor. For $0 < α ≤ 2$ no scaling occurs; the authors solve those cases exactly and find either runaway or oscillation.

Load-bearing premise

The classification is exhaustive only under the paper's own definition of a scaling solution — an exact power law with a fixed kinetic-to-potential energy ratio — and the authors show elsewhere in the paper that solutions scaling as a power law only after averaging over oscillations fall outside that definition.

Editorial extensions

If this is right

The search for potentials that give exact scaling closes: no shape outside the exponential, the negative power law, and the positive power law needs to be tried.
Negative power laws are attractors for every exponent, so a field on such a potential inevitably grows relative to the fluid and eventually drives a phase of intermediate inflation with $R propto exp[t^{4/(4-α)}]$.
Positive power laws with large $α$ offer a way to hide a field: starting from $ρ_φ gg ρ_{rad}$, the field redshifts almost like kinetic energy and then settles into a near-tracker with $ρ_φ propto R^{-(4+2/7)}$ for $α = 30$, arriving at $ρ_φ approx ρ_{rad}$ without fine-tuning the initial density.
The stability threshold $α > 2(6+m)/(6-m)$ makes the required steepness depend on the background equation of state, so the same potential can be an attractor in one era and not in another.
The borderline cases $α = 6$ (matter era) and $α = 10$ (radiation era) give closed phase-space loops rather than approach, so the field oscillates forever around the scaling solution without settling onto it.

Reading between the lines

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

The exhaustiveness is a statement about a definition as much as about physics: the paper requires an exact power law with a fixed kinetic-to-potential ratio, and the authors themselves derive in Eq. (27) an oscillation-averaged scaling $ρ_φ propto R^{-6α/(α+2)}$ that falls outside it, so a genuinely complete taxonomy would have to be indexed by the definition used.
The $α = 30$ example hints that the practical constraint is weaker than the theorem suggests: any potential steep at large field and shallow at small field can reproduce the drop-then-coast behaviour, so the exact-scaling classification may rule out less than it appears to.
Because the threshold $2(6+m)/(6-m)$ diverges as $m ightarrow 6$, a stiff-dominated early era would admit almost no positive power law as an attractor, which is a checkable statement about stiff-matter cosmologies.
Since the positive power-law branch drives $Ω_φ ightarrow 0$, it can hide a field but cannot by itself produce late acceleration; a working model would need to combine it with a second, growing branch or a feature in the potential.

Original abstract

An attractive method of obtaining an effective cosmological constant at the present epoch is through the potential energy of a scalar field. Considering models with a perfect fluid and a scalar field, we classify all potentials for which the scalar field energy density scales as a power-law of the scale factor when the perfect fluid density dominates. There are three possibilities. The first two are well known; the much-investigated exponential potentials have the scalar field mimicking the evolution of the perfect fluid, while for negative power-laws, introduced by Ratra and Peebles, the scalar field density grows relative to that of the fluid. The third possibility is a new one, where the potential is a positive power-law and the scalar field energy density decays relative to the perfect fluid. We provide a complete analysis of exact solutions and their stability properties, and investigate a range of possible cosmological applications.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.14 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.55 · compares with prior results 0.84 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

An attractive method of obtaining an effective cosmological constant at the present epoch is through the potential energy of a scalar field. Considering models with a perfect fluid and a scalar field, we classify all potentials for which the scalar field energy density scales as a power-law of the scale factor when the perfect fluid density dominates. There are three possibilities. The first two are well known; the much-investigated exponential potentials have the scalar field mimicking the evolution of the perfect fluid, while for negative power-laws, introduced by Ratra and Peebles, the scalar field density grows relative to that of the fluid. The third possibility is a new one, where the potential is a positive power-law and the scalar field energy density decays relative to the perfect fluid. We provide a complete analysis of exact solutions and their stability properties, and investigate a range of possible cosmological applications.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	An attractive method of obtaining an effective cosmological constant at the present epoch is through the potential energy of a scalar field.	background 1.00	0.07	0.03	asserted 0.76
s1	Considering models with a perfect fluid and a scalar field, we classify all potentials for which the scalar field energy density scales as a power-law of the scale factor when the perfect fluid density dominates.	contribution 0.96	0.93	0.19	asserted 0.95
s2	There are three possibilities.	result 0.97	0.70	0.08	asserted 0.98
s3	The first two are well known; the much-investigated exponential potentials have the scalar field mimicking the evolution of the perfect fluid, while for negative power-laws, introduced by Ratra and Peebles, the scalar field density grows relative to that of the fluid.	result 0.57	0.07	0.11	asserted 1.00
s4	The third possibility is a new one, where the potential is a positive power-law and the scalar field energy density decays relative to the perfect fluid.	result 0.90	0.83	0.32	asserted 1.00
s5	We provide a complete analysis of exact solutions and their stability properties, and investigate a range of possible cosmological applications.	contribution 0.97	0.95	0.14	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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T body sentences
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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

Figures

**Figure 1.** FIG. 1. Three phase planes for the radiation-dominated

**Figure 2.** FIG. 2. The evolution of the energy densities in a radia

Figures reproduced from arXiv:astro-ph/9809272 by the authors.

References

Works cited (15)

[1]

S. Perlmutter et al., Nature 391, 51 (1998); B. P. Schmidt et al., Report No. astro-ph/9805200, to appear, Astro- phys. J

View on Pith Pith review arXiv 1998
[2]

S. D. M. White, J. F. Navarro, A. E. Evrard and C. S. Frenk, Nature 366, 429 (1993)

View on Pith 1993
[3]

I. Zlatev, L. Wang and P. J. Steinhardt, Report No. astro- ph/9807002

View on Pith
[4]

F. Lucchin and S. Matarrese, Phys. Rev. D 32, 1316 (1985); J. J. Halliwell, Phys. Lett. B185, 341 (1987); A. B. Burd and J. D. Barrow, Nucl. Phys. B308, 929 (1988)

View on Pith 1985
[5]

C. Wetterich, Nucl. Phys. B302, 668 (1988)

View on Pith 1988
[6]

D. Wands, E. J. Copeland and A. R. Liddle, Ann. N. Y. Acad. Sci. 688, 647 (1993)

View on Pith 1993
[7]

P. G. Ferreira and M. Joyce, Phys. Rev. Lett. 79, 4740 (1997); Phys. Rev D 58, 023503 (1998)

View on Pith 1997
[8]

E. J. Copeland, A. R. Liddle and D. Wands, Phys. Rev. D 57, 4686 (1998)

View on Pith 1998
[9]

B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988)

View on Pith 1988
[10]

A. G. Muslimov, Class. Quant. Grav. 7, 231 (1990); J. D. Barrow, Phys. Lett. B 235, 40 (1990)

View on Pith 1990
[11]

L. Kofman, A. Linde and A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994), Phys. Rev. D 56, 3258 (1997); D. H. J. Chung, E. W. Kolb and A. Riotto, Report No. hep-ph/9802238

View on Pith Pith review arXiv 1994
[12]

M. S. Turner, Phys. Rev. D 28, 1243 (1983)

View on Pith 1983
[13]

B. Spokoiny, Phys. Lett. B 315, 40 (1993); M. Joyce and T. Prokope´ c, Phys. Rev. D 57, 6022 (1998)

View on Pith 1993
[14]

S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Chicago, University of Chicago Press (1939)

View on Pith 1939
[15]

L. Wang, private communication. 8

View on Pith

Forward citations

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How the literature uses it

70 arXiv papers cite this in 82 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

Papers citing it · 37 sentences without a confident label

Cites it as background 43 papers

Such solution, termed as scaling solution (see [ref] for a detailed classification of the potentials that lead to scaling solutions), is important in order to deal with the coincidence problem because, due to the attractor behavior of the scaling solution, if the scalar field is at the beginning of radiation in the basin of attraction of this scaling solution, it evolves as a radiation fluid.

Quintessential Inflation for Exponential Type Potentials: Scaling and Tracker Behavior · Introduction · 0.88
The role of symmetries in the context of the generation of new cosmological solution for the inflation was investigated in [ref] .

Scalar Field Cosmology -- Geometry of Dynamics · Symmetry group of cosmological dynamical systems · 0.88
Scaling solutions [ref] are characterized by a constant ratio between the energy density of the matter components and that of the scalar field.

Constraining cosmological scaling solutions of a Galileon field · Introduction · 0.98

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]

S. Perlmutter et al., Nature 391, 51 (1998); B. P. Schmidt et al., Report No. astro-ph/9805200, to appear, Astro- phys. J

View on Pith Pith review arXiv 1998

[2] [2]

S. D. M. White, J. F. Navarro, A. E. Evrard and C. S. Frenk, Nature 366, 429 (1993)

View on Pith 1993

[3] [3]

I. Zlatev, L. Wang and P. J. Steinhardt, Report No. astro- ph/9807002

View on Pith

[4] [4]

F. Lucchin and S. Matarrese, Phys. Rev. D 32, 1316 (1985); J. J. Halliwell, Phys. Lett. B185, 341 (1987); A. B. Burd and J. D. Barrow, Nucl. Phys. B308, 929 (1988)

View on Pith 1985

[5] [5]

C. Wetterich, Nucl. Phys. B302, 668 (1988)

View on Pith 1988

[6] [6]

D. Wands, E. J. Copeland and A. R. Liddle, Ann. N. Y. Acad. Sci. 688, 647 (1993)

View on Pith 1993

[7] [7]

P. G. Ferreira and M. Joyce, Phys. Rev. Lett. 79, 4740 (1997); Phys. Rev D 58, 023503 (1998)

View on Pith 1997

[8] [8]

E. J. Copeland, A. R. Liddle and D. Wands, Phys. Rev. D 57, 4686 (1998)

View on Pith 1998

[9] [9]

B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988)

View on Pith 1988

[10] [10]

A. G. Muslimov, Class. Quant. Grav. 7, 231 (1990); J. D. Barrow, Phys. Lett. B 235, 40 (1990)

View on Pith 1990

[11] [11]

L. Kofman, A. Linde and A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994), Phys. Rev. D 56, 3258 (1997); D. H. J. Chung, E. W. Kolb and A. Riotto, Report No. hep-ph/9802238

View on Pith Pith review arXiv 1994

[12] [12]

M. S. Turner, Phys. Rev. D 28, 1243 (1983)

View on Pith 1983

[13] [13]

B. Spokoiny, Phys. Lett. B 315, 40 (1993); M. Joyce and T. Prokope´ c, Phys. Rev. D 57, 6022 (1998)

View on Pith 1993

[14] [14]

S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Chicago, University of Chicago Press (1939)

View on Pith 1939

[15] [15]

L. Wang, private communication. 8

View on Pith