Accelerated expansion of the universe driven by tachyonic matter

T.Padmanabhan

REVIEW

Accelerated expansion of the universe driven by tachyonic matter

T.Padmanabhan

High Energy Physics - Theory · Submitted 18 Apr 2002 · 3 references

Pith's verdict: A short, correct reconstruction recipe; the dark-energy reading is explicitly conditional on string-theory input the paper does not supply.

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arXiv IDhep-th/0204150 (version 1)

Pith ID5I6RI3FA, Pith's permanent record of this paper and its review

CitationT.Padmanabhan. "Accelerated expansion of the universe driven by tachyonic matter." arXiv:hep-th/0204150 (2002). https://pith.science/paper/5I6RI3FA

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Editor's note

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The thing to know up front: this is a short recipe paper, and the recipe works. Padmanabhan takes the standard trick — trade a(t) for V(φ) — and applies it to the tachyonic Lagrangian L = −V√(1−∂φ∂φ). Eqs (16)–(17) follow in three lines from w = φ̇²−1 and ρ = V/√(1−φ̇²), and for a = t^n you get φ̇² = 2/(3n) and V ∝ (φ−φ0)^{-2}. I rederived it; it is right. The normal-scalar half (Eqs 6–11) is textbook and is presented as such, which is the correct framing given the abstract's own admission that one function can always be traded for another [abstract:s2]. What is genuinely new is the tachyonic half, the V ∝ φ^{-2} result, and the observation that in the small-φ̇ limit the tachyon reduces to a normal scalar with ψ = ∫√V dφ and an exponential potential — the same potential the normal-scalar route gives for power-law expansion. That mapping is clean and useful, and it is why people cite this paper. Where it is soft, in order. First, the physical claim — tachyon condensate as today's dark energy [abstract:s5] — rests entirely on whether string theory permits φ̇(∞) ≠ 1. The paper says so plainly in §3, works out both branches, and notes that φ̇(∞)=1 forces a ∝ t^{2/3}, pure dust, no acceleration. That is honest and it is the right way to leave it, but it does mean the abstract reads stronger than the body supports. Minor, but the abstract is what gets quoted. Second, the 'almost trivial' framing in §3 oversells the generality a little: the tachyon only covers w < 0 (Ḣ<0 and φ̇²<1), so it cannot reproduce radiation domination. The n > 2/3 condition encodes this but is not spelled out as a restriction. Third, the asymptotic analysis with e^{−Ct} corrections is fine but sketchy — it exists to argue the φ^{-2} tail dominates, and it does, but the reader takes the O() bookkeeping on faith. None of these are load-bearing failures. The math is right, the caveats are stated, and the paper does not pretend the reconstruction is a prediction. Who should read it: anyone doing tachyon cosmology, and anyone who wants a worked example of reverse-engineering a potential. It is a methods note, not a model. Peer review: yes, send it out. It is short, correct, and became a standard reference for a reason.

Referee report

Major objections (2)

[Abstract; §3, Eqs. (16)–(19)] The abstract states without qualification that 'it is always possible to find a scalar field potential which will reproduce a given evolution,' and §3 describes the tachyonic reconstruction as 'almost trivial.' For the tachyonic case this is not true for arbitrary a(t): Eq. (16) requires Ḣ < 0 and φ̇² < 1, i.e. −1 < w < 0, so a pure tachyon cannot reproduce radiation domination (w = 1/3, which would need φ̇² = 4/3) or matter domination (w = 0). The body encodes the restriction in the n > 2/3 condition and in the remark that Eq. (16) implies Ḣ < 0, but the abstract and the 'almost trivial' framing should be qualified to state the restricted range of w. A concrete check: attempting to reconstruct a(t) = t^{1/2} through Eq. (16) gives φ̇² = 4/3 > 1, outside the regime where the Lagrangian is defined.
[§3, Eqs. (22)–(25) and the discussion of φ̇(∞)] The paper correctly identifies that the physical interpretation hinges on whether string theory requires φ̇(∞) = 1 asymptotically, and states that the cosmological implications 'are very different' in the two cases. This is an acknowledged limitation rather than an error, but neither branch is developed: the n ≫ 1 accelerating branch is asserted to be attainable with V ∝ φ^{−2} 'without any restriction on A,' yet no argument or reference is given that the Sen rolling-tachyon solutions admit such an asymptotic velocity, and the alternative 'easy way out' of using the solutions at t ≳ t_Planck is stated without any estimate of the timescale problem the paper itself raises. A short quantitative statement — for example, what asymptotic form of V(φ) the cited Sen solutions actually produce, or a worked example in the n ≫ 1 branch — would materially strengthen the headline suggestion in abstract:s5.

Minor objections (5)

[§2, Eq. (12)] The definition f = β/(4+β) appears inconsistent with Eq. (13). The potential in Eq. (13), with leading exponent −β and sub-leading coefficient β²/6, follows from f = 4/(4+β); with f = β/(4+β) one instead obtains V ∝ φ^{−16/β}. Please check and correct the definition.
[§2, Eq. (13)] The second factor should be parenthesized explicitly as (1 − (β²/6)(M_Pl²/φ²)) to avoid the ambiguity of the printed form.
[§3, Eq. (25)] The integration constant is dropped; the relation should read H^{−1} = (3/2)∫φ̇² dt + const. This does not affect the asymptotic argument but should be stated.
[§3, Eqs. (18)–(19)] At n = 2/3 Eq. (19) is singular; the text defers to Eq. (25) and the φ̇(∞) = 1 discussion, but a one-line remark at Eq. (19) that the power-law ansatz breaks down there would help the reader.
[§2, Eqs. (8)–(9)] The generalization to a known matter density is stated without derivation; a sentence noting that Eqs. (8)–(9) reduce to Eqs. (6)–(7) as Q → 0 would make the correspondence transparent.

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The paper

The paper claims

Any chosen expansion history of the universe can be reproduced by a tachyonic scalar field with a suitably chosen potential, and for power-law expansion the required potential is a simple inverse-square of the field.

The reading

Cosmologists routinely invent a scalar field with a potential $V(φ)$ whenever the observed expansion of the universe refuses to match their expectations, and the author's starting observation is that this freedom is not innocent: because one function of time can always be traded for another, a potential can always be found that reproduces any prescribed $a(t)$. The paper turns that observation into an explicit recipe, first for an ordinary quintessence field and then for the tachyonic field with Lagrangian $L = -V(φ)[1 - ∂_aφ∂^aφ]^{1/2}$ that had recently emerged from string theory. In the tachyonic case the recipe is especially clean: the expansion rate alone fixes the field's velocity and its potential through two closed formulas, and a power-law universe $a = t^n$ corresponds to $V ∝ (φ-φ_0)^{-2}$. For $n > 1$ that same solution describes accelerated expansion, which is what makes the construction interesting: a tachyon condensate could in principle play the role now assigned to dark energy.

What carries the argument

The machinery is a reverse-engineering map from the expansion factor to the potential. Because a single source in a Friedmann universe leaves only two independent equations, specifying $a(t)$ over-determines nothing: the equation of state parameter $w(t) = -1 - (2/3)Ḣ/H^2$ is fixed by the geometry alone, and for the tachyon $w = φ̇^2 - 1$ then hands over $φ̇$, while the product of the energy density and pressure expressions hands over $V$. The map is invertible in both directions, which is what lets the author claim that a tachyonic potential and a normal scalar potential can drive identical cosmologies.

Extended reading notes

Core claim

The paper's central result is a pair of formulas that invert the Friedmann equations for a tachyonic condensate. Given any expansion history $a(t)$, the field's velocity is fixed by $φ̇^2 = -(2/3)(Ḣ/H^2)$ and the potential by $V = (3H^2/8πG)[1 + (2/3)Ḣ/H^2]^{1/2}$, so that no separate dynamical input is needed to construct a model. Applying this to $a = t^n$ yields $φ ∝ t$ and $V(φ) = (n/4πG)(1 - 2/3n)^{1/2}(φ-φ_0)^{-2}$, a potential that vanishes as the field grows, in line with the asymptotic behaviour string theory is expected to demand. The same procedure is carried out for a normal scalar field, where $a = t^n$ gives an exponential potential and $a ∝ exp(αt^f)$ gives a power-law potential with a correction term, recovering the standard inflationary and quintessence forms as special cases of one algorithm. The author also shows that when $φ̇^2 ≪ 1$ the tachyonic Lagrangian reduces to an ordinary scalar with a redefined field and potential, so the two recipes are two descriptions of one underlying freedom.

Load-bearing premise

The claim that a tachyon condensate can drive today's accelerated expansion rests on string theory permitting the field's late-time velocity to differ from one; if it must approach one, the same equations give a pressureless universe with no acceleration at all.

Editorial extensions

If this is right

For any desired cosmological history, including an accelerating late phase, a tachyonic potential can be written down explicitly rather than searched for.
A tachyon condensate with $V ∝ φ^{-2}$ produces power-law expansion $a = t^n$ with $n > 1$, so it can mimic a cosmological constant at late times.
In the small-velocity limit the tachyonic model maps onto an ordinary scalar with an exponential potential, unifying the tachyon and quintessence model-building programmes.
If string theory instead forces the asymptotic field velocity to unity, the same equations force $a ∝ t^{2/3}$, a pressureless dust universe with no acceleration.
The construction also supplies power-law inflation in the early universe, so the same potential can be read as an inflaton rather than as dark energy.

Reading between the lines

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

The recipe is a statement about the degeneracy of single-field models, not about which potential nature chose; any measurement of $w(z)$ that departs from the tachyon's fixed relation $w = φ̇^2 - 1$ would break the identification rather than merely re-tune it.
Because the map is invertible, the observational question becomes whether the tachyon's equation of state can be distinguished from a tuned quintessence field at all, which is a question about the time variation of $w$ rather than its present value.
The unresolved asymptotic condition on $φ̇$ is the pivot: settling it would convert this from a model-building exercise into a prediction, since $n = 2/3$ and $n > 1$ are separated by whether the field's late-time velocity reaches unity.

Original abstract

It is an accepted practice in cosmology to invoke a scalar field with potential $V(\phi)$ when observed evolution of the universe cannot be reconciled with theoretical prejudices. Since one function-degree-of-freedom in the expansion factor $a(t)$ can be traded off for the function $V(\phi)$, it is {\it always} possible to find a scalar field potential which will reproduce a given evolution. I provide a recipe for determining $V(\phi)$ from $a(t)$ in two cases:(i) Normal scalar field with Lagrangian ${\cal L} = (1/2)\partial_a\phi \partial^a\phi - V(\phi)$ used in quintessence/dark energy models. (ii) A tachyonic field with Lagrangian ${\cal L} = -V(\phi) [ 1- \partial_a\phi \partial^a\phi]^{1/2} $, motivated by recent string theoretic results. In the latter case, it is possible to have accelerated expansion of the universe during the late phase in certain cases. This suggests a string theory based interpretation of the current phase of the universe with tachyonic condensate acting as effective cosmological constant.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,149 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.22 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

It is an accepted practice in cosmology to invoke a scalar field with potential $V(\phi)$ when observed evolution of the universe cannot be reconciled with theoretical prejudices. Since one function-degree-of-freedom in the expansion factor $a(t)$ can be traded off for the function $V(\phi)$, it is {\it always} possible to find a scalar field potential which will reproduce a given evolution. I provide a recipe for determining $V(\phi)$ from $a(t)$ in two cases:(i) Normal scalar field with Lagrangian ${\cal L} = (1/2)\partial_a\phi \partial^a\phi - V(\phi)$ used in quintessence/dark energy models. (ii) A tachyonic field with Lagrangian ${\cal L} = -V(\phi) [ 1- \partial_a\phi \partial^a\phi]^{1/2} $, motivated by recent string theoretic results. In the latter case, it is possible to have accelerated expansion of the universe during the late phase in certain cases. This suggests a string theory based interpretation of the current phase of the universe with tachyonic condensate acting as effective cosmological constant.

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	It is an accepted practice in cosmology to invoke a scalar field with potential $V(\phi)$ when observed evolution of the universe cannot be reconciled with theoretical prejudices.	background 1.00	0.03	0.03	asserted 1.00
s1	Since one function-degree-of-freedom in the expansion factor $a(t)$ can be traded off for the function $V(\phi)$, it is {\it always} possible to find a scalar field potential which will reproduce a given evolution.	background 0.91	0.11	0.10	asserted 0.99
s2	I provide a recipe for determining $V(\phi)$ from $a(t)$ in two cases:(i) Normal scalar field with Lagrangian ${\cal L} = (1/2)\partial_a\phi \partial^a\phi - V(\phi)$ used in quintessence/dark energy models.	contribution 0.66	0.82	0.06	asserted 0.99
s3	(ii) A tachyonic field with Lagrangian ${\cal L} = -V(\phi) [ 1- \partial_a\phi \partial^a\phi]^{1/2} $, motivated by recent string theoretic results.	method 0.63	0.23	0.04	asserted 0.64
s4	In the latter case, it is possible to have accelerated expansion of the universe during the late phase in certain cases.	result 0.98	0.63	0.39	hedged possibility 0.99
s5	This suggests a string theory based interpretation of the current phase of the universe with tachyonic condensate acting as effective cosmological constant.	interpretation 1.00	0.28	0.66	hedged possibility 1.00

What kind of paper the classifier reports this is: method or tool 0.78 · evidence from proof or derivation 0.96.

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

not run

S sections
T body sentences
C citations
R cross-paper relations

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 (3)

[1]

A. Sen, arXiv: hep-th/0203211; arXiv: hep-th/0203265; arXiv:hep-th/0204143

View on Pith Pith review arXiv
[2]

G.W. Gibbons, arXiv:hep-th/0204008; M. Fairbairn and M.H. Tytgat, arXiv:hep-th/0204070; S. Mukohyama, arXiv:hep-th/0204084

View on Pith Pith review arXiv
[3]

B. Ratra, and P.J.E. Peebles, Phys. Rev. D 37, 3406 (1988); C. Wetterich, Nuclear Physics B 302, 668 (1988); P.G. Ferreira and M. Joyce, Phys. Rev. D 58, 023503 (1998); J. Frieman, C.T. Hill, A. Stebbins, and I. Waga, (1995) Phys. Rev. Lett. 75, 2077; P. Brax and J. Mar- tin Phys. Lett. B 468, 40 (1999); P. Brax and J. Mar- tin Phys. Rev. D 61, 103502 (20...

View on Pith 1988

Forward citations

Cited by 17 Pith papers

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Bayesian evidence from DESI BAO plus CMB and SN data favors the standard CPL evolving dark energy model over both simpler constant-w and more complex higher-order extensions.
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Power-law plateau inflation can sit inside the 1 sigma region of ACT DR6, but the reheating bound from gravitational waves forces the total number of e-folds N_k below about 62.
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Starobinsky inflation with a tuned Einstein-Gauss-Bonnet coupling can reproduce the ACT-enhanced scalar spectral index within 1σ.
A parametric model for dark energy
astro-ph 2008-05 conditional novelty 4.0 of 10

A new bounded parameterization of the dark energy equation of state, w(z) = w0 + w1 z(1+z)/(1+z^2), fits current supernova, BAO, CMB and H(z) data with w0 = -1.11 and w1 = 0.43, but is not benchmarked against the cosm...
Accelerated Universe from Modified Chaplygin Gas and Tachyonic Fluid
hep-th 2002-05 conditional novelty 4.0 of 10

A modified Chaplygin gas with equation of state p = A rho - B / rho^n interpolates from matter-like to de Sitter expansion and can be mapped onto a canonical scalar field and, approximately, onto a tachyon field.
Dark energy in Horndeski theories after GW170817: A review
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How later papers used it

80 later papers cite it

Evidence: some. 80 of the citing papers share no author with it, and 4 of those carry a confident label from Pith's citation classifier.

Pith read 8 of the 98 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 compared its own results with it.

Compares with it 1 paper

Recovered the power law solution for the scale factor with an inverse square potential in the opposite limit, matching this paper.

k-essence and tachyons in brane-worlds (2009) · 0.81 · paraphrased: the licence does not allow quoting

How the literature uses it

59 arXiv papers cite this in 70 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 · 17 sentences without a confident label

Builds on it 1 paper

We extend the correspondence between quintessence and tachyon models which has been extensively studied in [ref] .

Quintessence and tachyon dark energy models with a constant equation of state parameter · Introduction · 0.98

Uses its method or tool 1 paper

Recently, it was used the algorithm developed in Ref. [ref] to construct potential which can implement power law solution on FRW brane world in the presence of ordinary scalar field as well as tachyonic field [ref] .

k-essence and tachyons in brane-worlds · Introduction · 0.81

Compares with it 1 paper

In the opposite limit $→∞$, the potential takes the form $V∝ϕ^-2$ and we recover the power law solution for the scale factor with an inverse square potential [ref] , [ref] [ref] .

k-essence and tachyons in brane-worlds · k-essence in brane-world model · 0.81

Cites it as background 44 papers

The dark energy is treated as scalar field (tachyoinc or quintessence), among itself, the tachyonic scalar field arising from string theory [ref] (for different reasons in our context) has been widely used in literature [ref] .

Cosmic expansion driven by real scalar field for different forms of potential · Introduction · 0.97
Several workers have investigated the cosmological behaviour of the the tachyonic scalar field [ref] , and have found forms of potential for power law expansion and pure exponential expansion.

A tachyonic scalar field with mutually interacting components · Introduction · 0.90
Lately, efforts in the framework of k-essence have been directed toward model building using power law solutions which preserve [ref] , [ref] , [ref] , [ref] or violate the weak-energy condition [ref] .

k-essence and tachyons in brane-worlds · Introduction · 0.84

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]

A. Sen, arXiv: hep-th/0203211; arXiv: hep-th/0203265; arXiv:hep-th/0204143

View on Pith Pith review arXiv

[2] [2]

G.W. Gibbons, arXiv:hep-th/0204008; M. Fairbairn and M.H. Tytgat, arXiv:hep-th/0204070; S. Mukohyama, arXiv:hep-th/0204084

View on Pith Pith review arXiv

[3] [3]

B. Ratra, and P.J.E. Peebles, Phys. Rev. D 37, 3406 (1988); C. Wetterich, Nuclear Physics B 302, 668 (1988); P.G. Ferreira and M. Joyce, Phys. Rev. D 58, 023503 (1998); J. Frieman, C.T. Hill, A. Stebbins, and I. Waga, (1995) Phys. Rev. Lett. 75, 2077; P. Brax and J. Mar- tin Phys. Lett. B 468, 40 (1999); P. Brax and J. Mar- tin Phys. Rev. D 61, 103502 (20...

View on Pith 1988