REVIEW
DBI Galileons in the Einstein Frame: Local Gravity and Cosmology
Cosmology and Nongalactic Astrophysics · Submitted 30 Oct 2012 · 110 references
Pith's verdict: A useful frame dictionary and an honest negative result on the DCDM model; the abstract's framing of two screening mechanisms under one roof is looser than the body, which says they need opposite signs of B.
T0 · AI review
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Referee report
The paper
The paper claims
A disformally coupled scalar is secretly a DBI Galileon, and that explains why it hides
The reading
What carries the argument
The load-bearing object is the disformal metric $g^{M}_{μν} = A(φ)g_{μν} + B(φ)φ_{,μ}φ_{,ν}$ and the transformation that moves its disformal part from the matter sector into the gravitational sector. Computing the Ricci scalar of the barred metric requires the connection difference $K^{α}_{μν} = Γ̄^{α}_{μν} - Γ^{α}_{μν}$, which for the simplified case $g^{M}_{μν} = g_{μν} + π_{,μ}π_{,ν}$ collapses to $K^{α}_{μν} = γ^2 π^{,α}π_{;μν}$ with $γ = 1/√{1 + π_{,α}π^{,α}}$. That compact form is what makes the curvature computation tractable and yields the quartic DBI Galileon. On the cosmology side, the machinery is the coupling factor $Q_0$ in the background equations: for a purely conformal coupling $Q_0^{(c)} = (A'/2A)ρ$, proportional to the coupled matter density, while for a purely disformal coupling $Q_0^{(d)}$ is proportional instead to the scalar field energy density $ρ_φ$ — a structurally different interaction, and one that produces explicit $k^2$ dependence in the perturbed coupling and hence scale-dependent growth.
Extended reading notes
Core claim
The central claim is an equivalence: a theory whose gravitational sector is the ordinary Einstein-Hilbert action, but whose matter Lagrangian is built from the disformal metric of Eq. (6), can be rewritten in a frame where the disformal factor is absorbed into the gravitational sector, and there the gravitational Lagrangian becomes the quartic DBI Galileon. The paper derives this explicitly, obtaining $L_{GF} = γ^{-1} R - γ[(□π)^2 - π_{;μν}π^{;μν}]$ with $G_4 = γ^{-1} = √{1 - 2X}$, matching the Horndeski $G_4$ structure and, in the small-$X$ limit, the quartic covariant Galileon. The authors note this expression was previously obtained in the brane-world context, and their contribution is to read it as the gravitational-sector image of a disformally coupled Einstein-Frame theory. From this they argue that disformal couplings inherit the Vainshtein mechanism, and they add a second, distinct screening channel: in the limit of high density and negligible pressure, the scalar field equation reduces to $¨φ ≈ -(B'/2B)φ̇^2 + A'(φ̇^2/A - 1/2B)$, which contains no reference to the local matter density and suppresses spatial gradients by powers of $p/ρ$. They also locate a Vainshtein radius $r_V = (2BS^2/A)^{1/4}$ where the static vacuum solution $φ = -S/r + φ_0$ breaks down, and observe that this requires $B<0$ whereas the disformal screening requires $B>0$, so the two effects may not coexist in the simplest models.
Load-bearing premise
The equivalence and the clean quartic DBI form both rest on the conformal and disformal factors depending only on the scalar field and not on its kinetic term, an assumption the authors state explicitly in footnote 6 and set aside as future work.
Editorial extensions
If this is right
- If the equivalence holds, disformally coupled dark energy models are not a new class of theory but a re-coordinatization of a known Horndeski sector, so results derived for DBI Galileons transfer directly to them and vice versa.
- The Vainshtein mechanism comes for free in these theories, which means a disformal coupling can act on cosmological scales while remaining invisible to Solar System tests.
- A second screening channel exists that does not rely on the field potential: in dense, non-relativistic matter the field equation decouples from the local density, so the field rolls homogeneously and forms no spatial gradients to source a fifth force.
- Disformal couplings generate scale-dependent growth and a scale-dependent bias between coupled dark matter and uncoupled baryons, which is absent in conformally coupled models and could in principle be distinguished observationally.
- The specific Disformally Coupled Dark Matter model with exponential couplings produces an effective gravitational constant too large at late times, an excessive integrated Sachs-Wolfe effect, and large-scale oscillatory features, and is excluded by linear-regime observations.
Reading between the lines
- If the disformal-to-Galileon map is exact, then the extensive existing literature on Vainshtein screening in Galileon and massive-gravity contexts could be re-read as statements about disformal couplings, which would shortcut a lot of independent derivation.
- The sign conflict between the two mechanisms — $B>0$ for disformal screening, $B<0$ for the Vainshtein radius — suggests the paper's two headline results may not be simultaneously available in a single model, which would make the 'two faces of one structure' framing weaker than it first appears.
- The pressure instability noted in Section IV A, where $Bp$ can flip the sign of the spatial derivative coefficient, looks like the sharpest handle for testing these models: relativistic environments such as the early universe or compact objects probe exactly the regime where the theory may go gradient-unstable.
- Because the disformal coupling responds to radiation and pressure while the conformal one responds to trace of energy-momentum, a disformal component coupled to photons or neutrinos would leave a CMB signature that is qualitatively different from any conformal model, and this seems like the cleanest observational discriminator.
Original abstract
It is shown that a disformally coupled theory in which the gravitational sector has the Einstein-Hilbert form is equivalent to a quartic DBI Galileon Lagrangian, possessing non-linear higher derivative interactions, and hence allowing for the Vainshtein effect. This Einstein Frame description considerably simplifies the dynamical equations and highlights the role of the different terms. The study of highly dense, non-relativistic environments within this description unravels the existence of a disformal screening mechanism, while the study of static vacuum configurations reveals the existence of a Vainshtein radius, at which the asymptotic solution breaks down. Disformal couplings to matter also allow the construction of Dark Energy models, which behave differently than conformally coupled ones and introduce new effects on the growth of Large Scale Structure over cosmological scales, on which the scalar force is not screened. We consider a simple Disformally Coupled Dark Matter model in detail, in which standard model particles follow geodesics of the gravitational metric and only Dark Matter is affected by the disformal scalar field. This particular model is not compatible with observations in the linearly perturbed regime. Nonetheless, disformally coupled theories offer enough freedom to construct realistic cosmological scenarios, which can be distinguished from the standard model through characteristic signatures.
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.
The abstract, sentence by sentence
It is shown that a disformally coupled theory in which the gravitational sector has the Einstein-Hilbert form is equivalent to a quartic DBI Galileon Lagrangian, possessing non-linear higher derivative interactions, and hence allowing for the Vainshtein effect. This Einstein Frame description considerably simplifies the dynamical equations and highlights the role of the different terms. The study of highly dense, non-relativistic environments within this description unravels the existence of a disformal screening mechanism, while the study of static vacuum configurations reveals the existence of a Vainshtein radius, at which the asymptotic solution breaks down. Disformal couplings to matter also allow the construction of Dark Energy models, which behave differently than conformally coupled ones and introduce new effects on the growth of Large Scale Structure over cosmological scales, on which the scalar force is not screened. We consider a simple Disformally Coupled Dark Matter model in detail, in which standard model particles follow geodesics of the gravitational metric and only Dark Matter is affected by the disformal scalar field. This particular model is not compatible with observations in the linearly perturbed regime. Nonetheless, disformally coupled theories offer enough freedom to construct realistic cosmological scenarios, which can be distinguished from the standard model through characteristic signatures.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | It is shown that a disformally coupled theory in which the gravitational sector has the Einstein-Hilbert form is equivalent to a quartic DBI Galileon Lagrangian, possessing non-linear higher derivative interactions, and hence allowing for the Vainshtein effect. | result 0.84 | 0.81 | 0.67 | asserted 1.00 |
| s1 | This Einstein Frame description considerably simplifies the dynamical equations and highlights the role of the different terms. | interpretation 0.70 | 0.62 | 0.08 | asserted 1.00 |
| s2 | The study of highly dense, non-relativistic environments within this description unravels the existence of a disformal screening mechanism, while the study of static vacuum configurations reveals the existence of a Vainshtein radius, at which the asymptotic solution breaks down. | result 0.98 | 0.86 | 0.19 | asserted 1.00 |
| s3 | Disformal couplings to matter also allow the construction of Dark Energy models, which behave differently than conformally coupled ones and introduce new effects on the growth of Large Scale Structure over cosmological scales, on which the scalar force is not screened. | result 0.44 | 0.38 | 0.18 | asserted 0.71 |
| s4 | We consider a simple Disformally Coupled Dark Matter model in detail, in which standard model particles follow geodesics of the gravitational metric and only Dark Matter is affected by the disformal scalar field. | contribution 0.85 | 0.84 | 0.07 | asserted 0.98 |
| s5 | This particular model is not compatible with observations in the linearly perturbed regime. | limitation 0.78 | 0.79 | 0.07 | asserted 1.00 |
| s6 | Nonetheless, disformally coupled theories offer enough freedom to construct realistic cosmological scenarios, which can be distinguished from the standard model through characteristic signatures. | interpretation 0.85 | 0.41 | 0.39 | asserted 0.50 |
What was examined, and what was not
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Figures
Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:1210.8016 by the authors.
References
How later papers used it
78 later papers cite it
1 later paper by other authors is listed below: 1 built on it.
Builds on it 1 paper
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Extended the formalism to disformally coupled dark energy, noting that this was studied briefly in this paper's Appendix C.
Towards Viable Cosmological Models of Disformal Theories of Gravity (2014) · 0.85 · paraphrased: the licence does not allow quoting
How the literature uses it
74 arXiv papers cite this in 131 sentences
Papers citing it · 74 sentences without a confident label
Builds on it 1 paper
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Here, we will extend the formalism to disformally coupled dark energy[Note that this was studied briefly in [ref] , Appendix C.].
Towards Viable Cosmological Models of Disformal Theories of Gravity · Phase Space Construction · 0.85
Uses its method or tool 1 paper
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Using the familiar conformal and extended conformal transformations [ref] , we obtain interaction terms for a more general class of models.
Interacting dark sector from Horndeski theories and beyond: Mapping fields and fluids · Introduction · 0.92
Compares with it 4 papers
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This expression for $δQ$ agrees with the equation given in Ref. [ref] for the case of disformally coupled pressureless fluid.
Probing the imprints of generalized interacting dark energy on the growth of perturbations · Appendix A2: Newtonian gauge · 0.97
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Furthermore, note that our screening is also distinct from the “disformal screening” mechanism proposed in [ref] (in which large $ρ$ means that $ϕ$ is independent of the local energy density), the “derivative chameleon” of [ref] (see also the discussion in [ref] ), and other kinetic screening mechanisms in which $X := (∇ϕ)^2$ becomes large (here, $ϕ/M_P$ and $(∇ϕ)^2/^4$ both remain small on all scales of interest).
Novel Screening with Two Bodies: Summing the ladder in disformal scalar-tensor theories · Introduction · 0.98
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We will furthermore compare features of the dark D–brane cosmological scenario to other dark energy theories with disformal couplings, such as the ones proposed in Refs. [ref] .
Dark D-Brane Cosmology: from background evolution to cosmological perturbations · Introduction · 0.97
Cites it as support 1 paper
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This feature has been verified in explicit calculations in both the Jordan and Einstein frames, see e.g. [ref] .
Non-Gaussianity in multi-sound-speed disformally coupled inflation · Introduction · 0.99
Cites it as background 40 papers
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For example, the simplest and commonly used action [ref] is given by [equation] where the gravitational sector is the Einstein-Hilbert action in terms of the metric $g$ plus and a scalar field $ϕ$,[In principle, one could choose any form for the gravitational sector [ref] .] the latter being responsible for a disformal coupling with matter fields $ψ_I$ through [equation] Under the cosmological assumption we adopt throughout this paper, this amounts to the replacement of the lapse function in the matter sector by [equation] One may wonder whether the system is still disformally invariant as a whole.
Cosmological disformal invariance · Gravity plus matter · 0.90
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The term proportional to $D$ is known as the disformal coupling and, in contrast to the first term—the conformal coupling—it has only recently begun to be studied in detail [ref] .
Solar System Constraints on Disformal Gravity Theories · Introduction · 0.96
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The evolutions for the background universe and the density perturbations for the disformal gravity theory obtained from the disformal transformation with $C = C(ϕ)$ and $D=D(ϕ)$ have been studied both in the Einstein frame [ref] and in the Jordan frame [ref] .
Vainshtein mechanism in general purely disformal gravity theory · Introduction · 0.92
Discussion
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