REVIEW
General Gauss-Bonnet brane cosmology
High Energy Physics - Theory · Submitted 18 Feb 2002 · 34 references
Pith's verdict: The Birkhoff theorem for Gauss-Bonnet bulk is clean and correctly derived; the cubic Friedmann equation is the interesting half but is the one a referee should actually check.
T0 · AI review
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Referee report
The paper
The paper claims
Adding the Gauss-Bonnet curvature-squared term to five-dimensional gravity does not destroy Birkhoff's staticity theorem: the general bulk solution with constant three-dimensional spatial curvature is the static Boulware-Deser black hole, a
The reading
What carries the argument
The load-bearing device is the pair of integrability conditions obtained from the combinations $E_{tt} + E_{zz} pm 2E_{tz} = 0$ after passing to light-cone coordinates $u = (t-z)/2$, $v = (t+z)/2$. In the pure Einstein-Hilbert case these read $B_{,uu} - 2nu_{,u}B_{,u} = 0$ and $B_{,vv} - 2nu_{,v}B_{,v} = 0$, are independent of the cosmological constant, and integrate to $B = B(U+V)$, $e^{2nu} = B' U' V'$ for arbitrary $U(u)$, $V(v)$ — which is precisely staticity after a conformal transformation. With the Gauss-Bonnet term included, the same combinations produce the same two conditions multiplied by the common factor $9B^{4/3}e^{2nu} + 36alpha kappa B^{2/3}e^{2nu} + 4alpha B_{,u}B_{,v}$, so the Gauss-Bonnet contribution factorizes out rather than spoiling integrability. The dichotomy — vanishing integrability condition (static) versus vanishing prefactor (Class I, fine-tuned) — is what generates the paper's two solution classes. The second piece of machinery is the junction-condition derivation at the brane, where the authors insist on integrating the field equations over an infinitesimal interval rather than matching Dirac distributions, because first derivatives of the metric appear squared and produce a removable discontinuity exactly at the brane's location.
Extended reading notes
Core claim
The central claim is a generalized Birkhoff theorem for five-dimensional Gauss-Bonnet gravity: for a spacetime whose three-dimensional spatial slices have constant curvature, in the presence of a bulk cosmological constant, the general solution of the field equations is locally static unless the Gauss-Bonnet coupling $alpha$ and the cosmological constant parameter $k^2$ satisfy $8alpha k^2 = 1$. In the static case the solution is the Boulware-Deser/Cai black hole, the Gauss-Bonnet extension of the AdS-Schwarzschild geometry, with two branches distinguished by the sign in front of the square root and only one of them physically viable. In the fine-tuned case a second, non-static class exists in which the metric function $B(t,z)$ is arbitrary and the conformal factor is fixed by $e^{2nu} = 4alpha(B_{,z}^2 - B_{,t}^2)/[9B^{2/3}(B^{2/3} + 4alpha kappa)]$; the authors note this class exists in any dimension $d$ under the analogous tuning $96alpha k^2/((d-1)(d-2)) = 1$. Because the tuning is impossible when the cosmological constant vanishes, Birkhoff's theorem holds unconditionally in that limit. The brane-cosmology consequence is that the Friedmann equation becomes a cubic polynomial in $H^2 + V(R)/R^2$, reducing to the known $H^2 propto rho^2$ law as $alpha to 0$ and, for the special bulk relation $8alpha k^2 = -3$ at late times, giving back $H^2 propto rho$ without a brane tension.
Load-bearing premise
The cubic Friedmann equation depends entirely on the junction conditions the authors obtain by integrating the field equations across an infinitesimally thin shell at the brane, a step they themselves flag as delicate because squared first derivatives of the metric create a removable discontinuity precisely where the brane sits.
Editorial extensions
If this is right
- Birkhoff's staticity theorem holds in five-dimensional Gauss-Bonnet gravity whenever the bulk cosmological constant is zero, or whenever $8alpha k^2 \neq 1$, so a brane observer has only one bulk degree of freedom, the brane trajectory, exactly as in the Einstein-Hilbert case.
- The unique static bulk solution is the Boulware-Deser/Cai black hole, whose two branches have different stability properties; the branch that reduces smoothly to the Kottler solution as $alpha \to 0$ is the one selected by the small-coupling limit of the Friedmann equation.
- The generalized Friedmann equation is cubic in $H^2 + V(R)/R^2$, so the three powers are expected to dominate in succession over cosmic history, giving epochs with different effective equations of state.
- For the tuned case $8alpha k^2 = 1$ with $mu = 0$, the Friedmann equation collapses to $H^2 = (k^2 rho/2)^{2/3} - 2k^2 - kappa/R^2$, a concrete alternative early-time expansion law.
- With a brane tension satisfying the modified Randall-Sundrum fine-tuning $T = [(1 - \sqrt{1 - 8alpha k^2})/alpha]^{1/2}(2 + \sqrt{1 - 8alpha k^2})$, late-time cosmology recovers $H^2 = m_{Pl}^{-2}\rho/3 - \kappa/R^2 + O(\mu/R^4)$, and the derived Planck mass satisfies $M^3 < m_{Pl}^2 k$, so the Gauss-Bonnet correction lowers the fundamental five-dimensional mass scale.
Reading between the lines
- If the cubic Friedmann equation is right, then the Gauss-Bonnet coupling gives a knob that changes the effective equation of state between early and late epochs without introducing new brane matter; a reader could ask whether any observation of the expansion history (nucleosynthesis, for instance) constrains $alpha$ independently of the Randall-Sundrum tension tuning.
- The same factorization trick should be tried for the next Lovelock densities in higher dimensions, where the integrability conditions would acquire more factors; the paper's own remark that the tuning generalizes to $96alpha k^2/((d-1)(d-2)) = 1$ suggests the staticity theorem is a feature of the whole Lovelock family rather than of Gauss-Bonnet alone.
- Because the Class I solution requires $alpha$ and $k^2$ to have opposite signs, a positive string-scale coupling forces a negative bulk cosmological constant there, which sits awkwardly with the late-time example that uses $8alpha k^2 = -3$ together with an AdS bulk; a careful reader would want the sign bookkeeping of the two examples reconciled.
- The claim that the unstable branch still yields ordinary late-time cosmology is the paper's most fragile-looking consequence, since it relies on expanding around a solution the authors elsewhere accept is classically unstable; a stability analysis of the brane trajectory on that branch would test whether the late-time agreement is physical.
Original abstract
We consider 5-dimensional spacetimes of constant 3-dimensional spatial curvature in the presence of a bulk cosmological constant. We find the general solution of such a configuration in the presence of a Gauss-Bonnet term. Two classes of non-trivial bulk solutions are found. The first class is valid only under a fine tuning relation between the Gauss-Bonnet coupling constant and the cosmological constant of the bulk spacetime. The second class of solutions are static and are the extensions of the AdS-Schwarzchild black holes. Hence in the absence of a cosmological constant or if the fine tuning relation is not true, the generalised Birkhoff's staticity theorem holds even in the presence of Gauss-Bonnet curvature terms. We examine the consequences in brane world cosmology obtaining the generalised Friedmann equations for a perfect fluid 3-brane and discuss how this modifies the usual scenario.
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
We consider 5-dimensional spacetimes of constant 3-dimensional spatial curvature in the presence of a bulk cosmological constant. We find the general solution of such a configuration in the presence of a Gauss-Bonnet term. Two classes of non-trivial bulk solutions are found. The first class is valid only under a fine tuning relation between the Gauss-Bonnet coupling constant and the cosmological constant of the bulk spacetime. The second class of solutions are static and are the extensions of the AdS-Schwarzchild black holes. Hence in the absence of a cosmological constant or if the fine tuning relation is not true, the generalised Birkhoff's staticity theorem holds even in the presence of Gauss-Bonnet curvature terms. We examine the consequences in brane world cosmology obtaining the generalised Friedmann equations for a perfect fluid 3-brane and discuss how this modifies the usual scenario.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider 5-dimensional spacetimes of constant 3-dimensional spatial curvature in the presence of a bulk cosmological constant. | background 0.71 | 0.38 | 0.03 | asserted 0.86 |
| s1 | We find the general solution of such a configuration in the presence of a Gauss-Bonnet term. | contribution 0.71 | 0.96 | 0.10 | asserted 1.00 |
| s2 | Two classes of non-trivial bulk solutions are found. | result 0.99 | 0.96 | 0.12 | asserted 1.00 |
| s3 | The first class is valid only under a fine tuning relation between the Gauss-Bonnet coupling constant and the cosmological constant of the bulk spacetime. | result 0.98 | 0.88 | 0.11 | asserted 1.00 |
| s4 | The second class of solutions are static and are the extensions of the AdS-Schwarzchild black holes. | result 1.00 | 0.91 | 0.10 | asserted 1.00 |
| s5 | Hence in the absence of a cosmological constant or if the fine tuning relation is not true, the generalised Birkhoff's staticity theorem holds even in the presence of Gauss-Bonnet curvature terms. | interpretation 0.71 | 0.80 | 0.34 | asserted 1.00 |
| s6 | We examine the consequences in brane world cosmology obtaining the generalised Friedmann equations for a perfect fluid 3-brane and discuss how this modifies the usual scenario. | contribution 0.92 | 0.94 | 0.25 | asserted 0.96 |
What was examined, and what was not
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References
How the literature uses it
48 arXiv papers cite this in 61 sentences
Papers citing it · 33 sentences without a confident label
Uses its method or tool 1 paper
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Using a modified version of Birkhoff's staticity theorem [ref] , we found all six dimensional solutions describing a de Sitter, flat or anti de Sitter braneworld of codimension 2.
Self-properties of codimension-2 braneworlds · Discussion and Conclusions · 0.88
Uses its data or values 1 paper
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In particular, for a Randall–Sundrum brane $θ=1$ [ref] , while for a brane in Gauss–Bonnet gravity $θ=-1$ [ref] .
Route to nonlocal cosmology · Setup · 0.80
Compares with it 2 papers
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Cette d�g�n�rescence correspond � celle d�j� pr�sente dans [ref] .
Black hole solutions of modified gravity theories · Classe I · 0.87
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A similar property was recently proved for metric theories which preserve the second-order character of field equations [ref] .
Non-Metric Gravity II: Spherically Symmetric Solution, Missing Mass and Redshifts of Quasars · Summary · 0.85
Cites it as background 23 papers
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There are two well-known 5-dimensional brane-world cosmologies, namely, the Randall-Sundrum (RS) brane-world cosmology [ref] (for a review, see [ref] , references therein) and the Gauss-Bonnet (GB) brane-world cosmology [ref] .
Higgs-Portal Dark Matter in Brane-World Cosmology · Introduction · 1.00
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Perhaps the most complete presentation of the theorem has been given in Ref. [ref] building on the work of [ref] .
Shock waves and Birkhoff's theorem in Lovelock gravity · introduction · 0.81
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In this work we examine general single-brane models, including a Gauss-Bonnet term in the bulk [ref] (see also [ref] for a Gauss-Bonnet term in conventional 4D cosmology).
Holographic Dark Energy in Braneworld Models with a Gauss-Bonnet Term in the Bulk. Interacting Behavior and the w =-1 Crossing · Introduction · 0.81
Discussion
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