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Scalar Speed Limits and Cosmology: Acceleration from D-cceleration

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Eva Silverstein, David Tong

High Energy Physics - Theory · Submitted 23 Oct 2003 · 39 references

Pith's verdict: A clean, explicit derivation of DBI-driven power-law inflation from a quadratic potential — the mechanism holds up, but the abstract's 'subPlanckian, no fine tuning' claim is undercut by the paper's own footnote.

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arXiv IDhep-th/0310221 (version 3)
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CitationDavid Tong, Eva Silverstein. "Scalar Speed Limits and Cosmology: Acceleration from D-cceleration." arXiv:hep-th/0310221 (2003). https://pith.science/paper/YDVUP7J6
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The paper

The paper claims

The paper argues that the Dirac-Born-Infeld kinetic term of a probe D3-brane falling down an AdS throat, coupled to four-dimensional gravity, produces slow-roll inflation from an ordinary quadratic mass term at sub-Planckian field values, a

The reading

In the strong-coupling limit of $N=4$ super Yang-Mills, the scalar fields that parametrize the moduli space cannot roll arbitrarily fast. Causality on the gravity side of the correspondence caps the velocity at $|dot varphi| le varphi^2/sqrt{lambda}$, so the field takes an infinite time, as measured by a boundary observer, to reach the origin. The paper's aim is to show that this cap, which comes from the Dirac-Born-Infeld kinetic term of a probe D3-brane descending the AdS throat, survives the coupling of the gauge theory to four-dimensional gravity and changes which cosmologies are available. The headline consequence is a mechanism for slow-roll inflation from an ordinary quadratic mass term at sub-Planckian field values: because the kinetic term shuts down as the field approaches the origin, the potential barely has to be flat, and acceleration in fact gets stronger as the mass grows. The authors also catalogue the non-inflating cosmologies of the same system, including a dust-like phase resembling tachyon matter, and check that particle production, string production, and back reaction stay under control at large 't Hooft coupling.

What carries the argument

The engine is the Dirac-Born-Infeld Lagrangian for a probe D3-brane moving radially in the AdS Poincare patch, written in field-theory variables with $f(varphi) = lambda/varphi^4$, together with the Lorentz-like factor $gamma = (1-f(varphi)|dotvarphi^2)^{-1/2}$ built from it. Expanding the square root gives the canonical kinetic term plus a series in $lambda|dotvarphi^2/varphi^4$ that becomes order one near the origin; this is what enforces the speed limit and what makes $gamma$ grow without bound at late times. The cosmological solutions are then obtained through the Hamilton-Jacobi trick of treating $varphi$ as the time variable, which turns the Friedmann equations into a first-order pair: an algebraic equation fixing $V(varphi)$ in terms of $H(varphi)$ and $H'(varphi)$, and a separate expression for $|dotvarphi$ in terms of $H'$ and $f$. That separation is what allows the authors to solve sequentially and read off the scale-factor exponent directly from the coefficient $h_1$.

Extended reading notes

Core claim

In Case B of Section 5.1 the authors take a potential $V = V_2varphi^2$ and an ansatz $H = h_1varphi$, solve the Hamilton-Jacobi form of the Friedmann equations, and find $varphi to sqrt{lambda}/t$ together with $a(t) propto t^{h_1sqrt{lambda}}$, where $h_1sqrt{lambda} = (1/3)(1+sqrt{1+3V_2lambda/(g_sM_p^2)})$. Inflation therefore requires $V_2lambda/(g_sM_p^2) > 1$, which is the reverse of the usual condition: a steeper potential accelerates more, because the speed limit rather than the flatness of $V$ supplies the slow roll. The same mechanism defuses the curvature coupling $Rvarphi^2$, since the factor $gamma = (1-lambda|dotvarphi^2/varphi^4)^{-1/2}$ that blows up at late times suppresses both the potential and curvature terms in the scalar equation of motion, leaving $M_p^2 + varphi^2$ as a small shift to the effective Planck mass when $varphi ll M_p$. Different potentials give different endpoints: a pure quartic, which is what an anti-D3-brane generates, yields $a sim t^{2/3}$ and a kinetic dust equation of state, while a negative $V_2$ permits a burst of acceleration that settles into a steady state with $a sim exp(-c/t^2)$. A toy capped throat with $f(varphi) = lambda/(varphi^2+mu^2)^2$ lets the field reach the origin in finite time, so it can oscillate and reheat.

Load-bearing premise

The inflationary window needs a quadratic mass term for $varphi$ large enough to satisfy $V_2lambda > g_sM_p^2$, and the paper inserts that term by hand: it argues that couplings to other sectors of a compactification can generate it while leaving the DBI kinetic terms intact, but states plainly that no explicit compactification example has been constructed, that tuning of Kaluza-Klein couplings may be required, and that the hard mass available from the cutoff geometry itself, $m_{DBI} sim mu/sqrt{lambda}$, is far too small to inflate.

Editorial extensions

If this is right

  • Inflationary model building in string compactifications need not hunt for flat directions: a quadratic mass term suffices, and the required condition is a lower bound on the mass, $V_2lambda > g_sM_p^2$, rather than an upper bound.
  • The number of e-foldings is $n sim h_1sqrt{lambda} sim msqrt{lambda}/(sqrt{g_s}M_p)$, so a large enough soft mass gives more than 100 e-foldings over the window $mu ll varphi ll M_p$.
  • Curvature corrections of the form $Rvarphi^2$ and $H^2varphi^2$, which obstruct other brane-inflation models, are suppressed by powers of $gamma^{-1}$ in this regime, so sub-Planckian inflation is protected.
  • An anti-D3-brane in the throat produces $a sim t^{2/3}$ with particle production suppressed by $1/sqrt{lambda}$ and $1/lambda^{1/4}$, a cleaner relative of tachyon matter with the same dust equation of state.
  • A capped throat lets the scalar pass through the origin at finite speed, so the field oscillates about the minimum afterwards and ordinary reheating becomes available.

Reading between the lines

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

  • The $gamma$-suppression argument is not specific to this brane: any modulus whose kinetic term descends from a warped DBI action should get the same protection from curvature couplings, which would make speed-limit inflation a generic feature of throat compactifications rather than a single model.
  • Because the velocity bound is causal and hard, the total field-space distance the scalar can cover is tied to the elapsed time; this suggests an observable relation between the tensor-to-scalar ratio and the field excursion that would distinguish the scenario from ordinary slow roll.
  • The construction points to a search criterion for explicit flux compactifications: look for sectors that generate a soft mass for a throat modulus without generating a comparable quartic coupling, since that combination is precisely what opens the accelerating window.
  • The non-Gaussianity constraint reported in the authors' own footnote suggests the sub-Planckian feature and the observed density-perturbation amplitude may be in tension; a dedicated $f_{NL}$ calculation for Case B would decide whether the mechanism survives contact with the CMB.
Original abstract

Causality on the gravity side of the AdS/CFT correspondence restricts motion on the moduli space of the N=4 super Yang Mills theory by imposing a speed limit on how fast the scalar field may roll. This effect can be traced to higher derivative operators arising from integrating out light degrees of freedom near the origin. In the strong coupling limit of the theory, the dynamics is well approximated by the Dirac-Born-Infeld Lagrangian for a probe D3-brane moving toward the horizon of the AdS Poincare patch, combined with an estimate of the (ultimately suppressed) rate of particle and string production in the system. We analyze the motion of a rolling scalar field explicitly in the strong coupling regime of the field theory, and extend the analysis to cosmological systems obtained by coupling this type of field theory to four dimensional gravity. This leads to a mechanism for slow roll inflation for a massive scalar at subPlanckian VEV without need for a flat potential (realizing a version of k-inflation in a microphysical framework). It also leads to a variety of novel FRW cosmologies, some of which are related to those obtained with tachyon matter.

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, 5,131 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.44 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

Causality on the gravity side of the AdS/CFT correspondence restricts motion on the moduli space of the N=4 super Yang Mills theory by imposing a speed limit on how fast the scalar field may roll. This effect can be traced to higher derivative operators arising from integrating out light degrees of freedom near the origin. In the strong coupling limit of the theory, the dynamics is well approximated by the Dirac-Born-Infeld Lagrangian for a probe D3-brane moving toward the horizon of the AdS Poincare patch, combined with an estimate of the (ultimately suppressed) rate of particle and string production in the system. We analyze the motion of a rolling scalar field explicitly in the strong coupling regime of the field theory, and extend the analysis to cosmological systems obtained by coupling this type of field theory to four dimensional gravity. This leads to a mechanism for slow roll inflation for a massive scalar at subPlanckian VEV without need for a flat potential (realizing a version of k-inflation in a microphysical framework). It also leads to a variety of novel FRW cosmologies, some of which are related to those obtained with tachyon matter.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Causality on the gravity side of the AdS/CFT correspondence restricts motion on the moduli space of the N=4 super Yang Mills theory by imposing a speed limit on how fast the scalar field may roll. background 0.84 0.17 0.23 asserted 0.99
s1 This effect can be traced to higher derivative operators arising from integrating out light degrees of freedom near the origin. interpretation 0.66 0.29 0.04 asserted 0.91
s2 In the strong coupling limit of the theory, the dynamics is well approximated by the Dirac-Born-Infeld Lagrangian for a probe D3-brane moving toward the horizon of the AdS Poincare patch, combined with an estimate of the (ultimately suppressed) rate of particle and string production in the system. method 0.95 0.25 0.07 asserted 0.89
s3 We analyze the motion of a rolling scalar field explicitly in the strong coupling regime of the field theory, and extend the analysis to cosmological systems obtained by coupling this type of field theory to four dimensional gravity. contribution 0.97 0.93 0.08 asserted 0.98
s4 This leads to a mechanism for slow roll inflation for a massive scalar at subPlanckian VEV without need for a flat potential (realizing a version of k-inflation in a microphysical framework). result 0.75 0.86 0.50 asserted 0.99
s5 It also leads to a variety of novel FRW cosmologies, some of which are related to those obtained with tachyon matter. result 0.93 0.82 0.12 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
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
How later papers used it

298 later papers cite it

Evidence: substantial. 294 of the citing papers share no author with it, and 20 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 500 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

Builds on it 1 of 5 papers with this label

  • Expanded the potential to first order and found that this case corresponds to a generalization of case A of this paper.

    Universality in generalized models of inflation (2016) · 0.85 · paraphrased: the licence does not allow quoting

Uses its method or tool 2 of 4 papers with this label

  • Used an approximate Hamilton-Jacobi analysis of the motion in a similar manner to that of this paper's original Silverstein-Tong calculation.

    Spinflation with Angular Potentials (2011) · 0.87 · paraphrased: the licence does not allow quoting

  • Used the Hamilton-Jacobi formalism, treating the scalar as a monotonic time variable and searching for solutions to first order equations, as in this paper.

    Brane Inflation is Attractive (2008) · 0.81 · paraphrased: the licence does not allow quoting

How the literature uses it

281 arXiv papers cite this in 469 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 · 265 sentences without a confident label

Builds on it 4 papers
  • We focus on a particular class of string inflation constituting a generic extension of the DBI inflation [ref] .

    DBI Galileon inflation in the light of Planck 2015 · Introduction · 0.90

  • Examples of this type include the extension of DBI inflation [ref] to the multi-field context.

    Automorphic inflation · Automorphic actions · 0.87

  • Building on this, more generically one obtains higher-dimension operators suppressed not by a hard mass parameter, but instead by the inflaton vev itself, $ϕ$ [ref] .

    String Cosmology: A Review · Inflation in quantum field theory · 0.84

Uses its method or tool 4 papers
  • The equation of motion for a homogenous $ϕ$, together with the Friedmann equation, is [equation] The energy density and pressure of the system are [equation] One can solve these equations in many cases of interest using the Hamilton-Jacobi formalism in which we treat $ϕ$ as our monotonic time variable and search for solutions $H=H(ϕ)$ to the following first order equations [ref] , [equation] This leads to a simple expression for the relativistic gamma-factor of the brane, [equation] from which we learn that in order to have the speed limiting effect e.g. $γ≫1$ we need both $f(ϕ)$ and $H'$ to be large (in Planck units).

    Brane Inflation is Attractive · D-brane Inflation · 0.81

  • This then allows an approximate Hamilton-Jacobi analysis of the motion in a similar manner to that of the original Silversten-Tong calculation [ref] .

    Spinflation with Angular Potentials · Inflationary analysis · 0.87

  • Homogeneous solutions for the above action coupled to gravity can be found [ref] using the Hamilton-Jacobi formalism.

    Constraining Running Non-Gaussianity · DBI Inflation: a case study · 0.81

Compares with it 14 papers
  • We find that the larger the inflaton mass, the greater the amount of inflation, as in the original DBI inflation model [ref] , with (roughly) $N_e∝m g_sM$.

    Spinflation · Numerical solutions · 0.93

  • The emphasis of Ref. [ref] is to compare the origin of this $η$-problem to that of the $h$-problem [ref] in DBI inflation [ref] .

    Strongly Coupled Inflaton · Introduction · 0.97

  • The same is true for the potential and friction terms in the case of standard DBI inflation in an AdS throat (see Ref. [ref] ).

    Statistical Anisotropy from Vector Curvaton in D-brane Inflation · Moving brane · 0.86

Cites it as background 152 papers
  • In the Dirac-Born-Infeld (DBI) inflation [ref] , we consider a D3-brane in which our Universe is confined to in the bulk as shown in figure [ref].

    Cosmological signatures of brane inflation · String theory and Dirac-Born-Infeld (DBI) inflation · 0.91

  • On the other hand, an alternative approach to the $η$ problem which gives up slow-roll was suggested in the idea of DBI inflation [ref] .

    Constraints on Wrapped DBI Inflation in a Warped Throat · Introduction · 0.85

  • Within this framework (where non-perturbative corrections play an important role, and a supersymmetry breaking sector is incorporated to generate positive vacuum energy) various plausible models of inflation using moduli fields have been suggested in the literature: “N-flation" [ref] , “Kähler moduli inflation" [ref] , “Inflating in a Better Racetrack" [ref] , and using brane moduli “KKLMMT" and related scenarios [ref] .

    Searching for Inflation in Simple String Theory Models: An Astrophysical Perspective · Can string theory describe inflation? · 0.88

Discussion

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