REVIEW 3 major objections 4 minor 36 references
Gauss-Bonnet dynamical compactification scenarios and their ghosts in the tensor sectors
T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read The paper proves that every vacuum Einstein-Gauss-Bonnet compactification considered here is either not a background attractor or carries a ghostly tensor mode, and that the stable modified configuration is ruled out by the measured speed…
desk verdict A solid, internally consistent no-go result for Gauss-Bonnet dynamical compactifications, with a real but narrow algebraic risk in the appendix and a modest presentation overclaim about 'stable' flat solutions. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the quadratic transverse-traceless action for the two decoupled graviton tensor sectors of the product spacetime, built around the original compactification in which the radion—the scale factor $b(t)$ of the extra-dimensional sub-manifold—is constant. For each sector the action is written with dimensionless coefficients: $K$ and $\tilde K$ multiply the time-derivative (kinetic) terms, the $c_{phys}$ and $c_{extr}$ coefficients multiply the spatial-gradient terms, and the $M^2$ and $\tilde M^2$ coefficients act as mass-like terms. A mode is a ghost exactly when its kinetic coefficient is negative, so the no-ghost conditions are $K>0$ and $\tilde K>0$. The background attractor behaviour is governed by the $3\times 3$ matrix $M^\kappa_0$ for homogeneous perturbations of $H$, $b$ and $u=\dot b$, whose eigenvalues have the form $-3$ and $-3/2\pm\sqrt{\Pi/\Sigma}$; negative real parts mark an attractor. The proof works by intersecting these attractor intervals with the no-ghost intervals and finding no overlap, and the same machinery yields $c^2_{phys}=(1+dX_0)/(1+(d-1)X_0)$ for the modified flat scenario, which the GW170817 bound then rejects.
What would settle it
A concrete check would be to evaluate the quadratic tensor action at an attractor point such as $d=4$, $\kappa=1$, $B_0=1.5$, where the paper's formulas give negative $K$; if an independent rederivation finds $K>0$ after removing total derivatives, the ghost claim fails. A broader numerical search for any $(d,\kappa,B_0)$ obeying the background equations with simultaneously positive $K$ and $\tilde K$ plus negative real parts of the background eigenvalues would also settle the no-go.
Extended reading notes
Core claim
The paper's core discovery is that every vacuum compactified solution of Einstein-Gauss-Bonnet gravity in $d+4$ dimensions considered in this class either fails to be an attractor or carries ghostly tensor perturbations. The authors split transverse-traceless metric perturbations into four-dimensional tensor modes $h_{ij}$ and extra-dimensional tensor modes $H_{AB}$; these two sectors decouple at quadratic order, and their kinetic coefficients $K$ and $\tilde K$ decide ghost-freeness. Combining the attractor ranges for the homogeneous background with the no-ghost requirements $K>0$ and $\tilde K>0$ leaves no overlap: for $\kappa=-1$ one coefficient is always negative, and for $\kappa=1$ the no-ghost interval lies outside every attractor interval. Relaxing the constant-radion hypothesis for a flat extra-dimensional space gives stable no-ghost configurations for $X_0$ in $(-2/(d-2),-1/(d-1))$, but their physical gravitational-wave speed $c^2_{phys}=(1+dX_0)/(1+(d-1)X_0)$ requires $|X_0|\lesssim 10^{-15}/d$, which is incompatible with that range. The stable modified scenario must therefore be discarded.
Load-bearing premise
The no-go depends on the lengthy quadratic perturbation calculation being correct and on a negative kinetic coefficient truly meaning a ghost; the authors themselves flag this as a necessary-condition analysis, so an algebra slip or a gauge artifact would open a stable window.
Editorial extensions
If this is right
- If the paper is right, no vacuum Einstein-Gauss-Bonnet compactification with static extra dimensions can serve as a stable late-time cosmological attractor: wherever the homogeneous background is stable, a tensor ghost appears.
- The stable flat-extra-dimensional modification is ghost-free only for $X_0$ in $(-2/(d-2),-1/(d-1))$, but those values make $|c^2_{GW}-1|$ of order $|X_0|$, far above the $10^{-15}$ level, so the solution must be rejected.
- For curved extra dimensions, merely allowing the internal scale factor to vary while keeping de Sitter expansion produces no compactified solution, so rescuing the scenario needs a larger departure.
- Adding matter or scanning the full $\{b,H\}$ configuration space are the concrete routes the paper identifies for finding overlap between background stability and ghost-freeness.
Reading between the lines
- If the quadratic-action no-go survives a nonlinear analysis, it would put pressure on any vacuum higher-dimensional mechanism that relies on Gauss-Bonnet terms alone for spontaneous compactification; matter or higher Lovelock terms would then have to do the stabilizing.
- The same kinetic-coefficient test could be applied to cubic and higher Lovelock compactifications to see whether the attractor-ghost clash is a generic feature of Lovelock gravity rather than a Gauss-Bonnet accident.
- Future gravitational-wave experiments with speed bounds tighter than $10^{-15}$ would shrink the allowed $|X_0|$ window, making the tension with the no-ghost interval sharper and easier to falsify.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper analyzes cosmological compactification in Einstein-Gauss-Bonnet gravity in (d+4) dimensions, with a four-dimensional de Sitter sector and d extra dimensions. For the original scenario with static extra dimensions, it derives the background solutions, characterizes when the homogeneous background is an attractor, and computes the quadratic transverse-traceless tensor actions. The authors find that the parameter ranges where the background is an attractor never overlap with the ranges where both tensor kinetic coefficients K and Ktilde are positive, for d=3, d=4, and d>=5. They then relax the static-extra-dimension hypothesis and find a flat extra-dimensional solution with L/H0 = X0, and show that it can satisfy the no-ghost conditions in a finite interval X0 in (-2/(d-2), -1/(d-1)) while being a background attractor, but that the resulting gravitational wave speed c_phys^2 = (1+dX0)/(1+(d-1)X0) is incompatible with the GW170817 bound, requiring |X0| <= 10^-15/d. The paper concludes that no vacuum Einstein-Gauss-Bonnet compactification in this class is both stable and consistent with the gravitational wave speed constraint.
Significance. If the central computation is correct, this is a significant negative result: it rules out an appealing dynamical compactification mechanism in vacuum Einstein-Gauss-Bonnet gravity and shows that the minimal modification that avoids tensor ghosts is observationally excluded by GW170817. The paper is self-contained and the derivation is transparent, with the explicit d=4 eigenfunction calculation in Sec. IV providing a genuine independent check of the signs of the kinetic coefficients in that dimension. The manuscript does not rely on a fitted parameter or an assumed input to reach its conclusion, and the use of the external gravitational wave speed bound is legitimate. The main limitation is that the stability analysis is explicitly a necessary-condition analysis: only homogeneous background perturbations and transverse-traceless tensor perturbations are considered, and the conclusion that the original compactifications are 'inherently unstable' is stronger than what the quadratic TT analysis alone can strictly prove.
major comments (3)
- [Abstract and Sec. V] The abstract states that 'new and stable solutions are found' and Sec. VI calls the flat extra-dimensional configuration 'stable', but the analysis in Sec. V checks only two things: the homogeneous background perturbations (Eq. 5.6) and the transverse-traceless tensor sectors (Eqs. 5.7-5.9). No scalar, vector, or mixed inhomogeneous perturbations are analyzed, and no statement is made about nonlinear stability. Please either restrict the claim to 'background attractor and no tensor ghosts' or perform a full linear stability analysis before calling these solutions 'stable'.
- [Sec. III and App. B] The core no-go result rests on the signs of the kinetic coefficients K and Ktilde in Eqs. (3.14a) and (3.14d), which are obtained from the lengthy second-order Gauss-Bonnet variation (B5b) after projection onto the product background (B10). The d=4 eigenfunction calculation in Sec. IV independently confirms the signs for d=4, but for d=3 and d>=5 the non-overlap of the no-ghost ranges with the attractor ranges is established only through algebraic expressions such as Eqs. (3.16), (3.18), (3.19), and (3.21). Because a single sign error in the projection or in the contraction of (B5b) would reverse the main conclusion, I ask the authors to provide an independent machine-checked verification of K and Ktilde, or at least a more systematic derivation that makes the sign of every term transparent for all d.
- [Sec. VI] The concluding statement 'as we proved in this work, such configurations are inherently unstable' overstates the logical force of the analysis, as the authors themselves acknowledge before Sec. III B when they say the study gives necessary conditions only. Negative kinetic coefficients in the quadratic TT action are strong evidence of ghosts, but constraints or nonlinear couplings could in principle remove them, and the scalar and vector sectors have not been studied here. Please soften the conclusion to 'no background attractor can be free of tensor ghosts within this TT analysis' or complete the stability analysis.
minor comments (4)
- [Eq. (B13a)] There is a typesetting error in the expression for the background Ricci scalar: '12 H^2_0 + 6d H0,L0 d(d+1) L^2_0' should presumably read '12 H^2_0 + 6d H0 L0 + d(d+1) L^2_0'.
- [Eq. (B15b)] The mass term for H^2_AB in the second-order Gauss-Bonnet action has an unbalanced parenthesis: the expression beginning with '6 H^4_0 + ... + (d-1)(d-2)(d^2 - 15d - 4/4' appears to require a closing bracket before the final '] H^2_AB'. Please correct the typography.
- [Sec. V.B.2] The sentence 'When d = 2, the range of stability spans every negative number up to −1/2' is ambiguous; since the attractor condition requires X0 > -3/d = -3/2, the intended interval is apparently X0 in (-3/2, -1/2). Please state the interval explicitly.
- [Sec. III.B.3] In the ratio displayed below Eq. (3.19), the symbol 'ˆK' should probably be '\tilde K' (the same Ktilde used in Eq. 3.14d), to avoid confusion with the notation for the physical-sector coefficient K.
Circularity Check
No significant circularity (score 0): every no-go result is computed from the fixed action (2.1) via on-shell quadratic perturbations, and the sole external input (GW170817 speed bound) is applied as a consistency test, never fitted.
full rationale
The derivation chain is self-contained, and no load-bearing result reduces to its own inputs by construction. The background compactified solutions (2.10) are obtained by solving the equations of motion (A1) for the fixed action (2.1) under the compactification ansatz (2.6); no background parameter is fitted to the conclusions. The attractor ranges of Sec. III A are computed from the linearized first-order system (3.2)-(3.3) and are re-derived in the paper rather than imported: the text notes the analysis "has been performed in [7]" but then repeats it "taking a slightly different approach," with conclusions "naturally in full agreement" with [7]. The no-ghost conditions K > 0 and K~ > 0 (3.14) are read off the quadratic TT action (3.12)-(3.13), which comes from the second-order expansion detailed in App. B, evaluated on-shell with (2.10); the non-overlap of the no-ghost intervals with the attractor intervals for d = 3, 4, and d >= 5 is a computed algebraic statement about intervals, not an assumed one. The d = 4 eigenfunction treatment in Sec. IV is an explicit cross-check of the same expansion, matching (4.2)/(4.5) with (3.18); a cross-check is not a circular reduction. In the modified flat scenario of Sec. V, both the stability window (5.10) and the gravitational wave speed (5.11) are derived from the quadratic action (B16) of the same fixed theory, and the GW170817 bound (5.12) is an external observational constraint applied after the fact; no parameter is fitted to it. The paper explicitly labels the stability study a necessary-condition analysis (Sec. III B) and does not claim the full nonlinear theory is proven ghost-free; that honest limitation, like the reliance on the un-machine-checked App. B algebra, is a correctness risk (an algebra error would invalidate the signs of K and K~), not circularity. The references contain no self-citations by the present authors; citations to the Toporensky-group works provide context, and the one load-bearing prior result (background attractor behavior) is re-derived rather than cited as an unverified premise. None of the definitional, fitted-prediction, self-citation, uniqueness-import, ansatz-smuggle, or renaming patterns applies.
Assumptions & free parameters
assumptions (5)
- standard math Lovelock theorem: in more than four dimensions the Gauss-Bonnet combination is the unique quadratic Lovelock term and yields second-order equations.
- domain assumption The spacetime is a product M4 x Md with homogeneous, isotropic factor metrics, Eq. (2.3), and a d-dimensional internal space of constant curvature kappa.
- domain assumption Compactification limit (2.6): N tends to 1, H tends to H0 > 0, and b tends to b0 > 0.
- domain assumption Transverse-traceless perturbations hij and HAB decouple at quadratic order, and a negative kinetic coefficient implies a ghost that renders the theory unstable.
- domain assumption GW170817 bound: the gravitational wave speed squared differs from the speed of light squared by less than about 10^-15.
Cite this review
Pith. "Pith review of Gauss-Bonnet dynamical compactification scenarios and their ghosts in the tensor sectors." pith.science (2026). https://pith.science/paper/GRVECDZT
@misc{pith2026241116983,
author = {Pith},
title = {Pith review of: Gauss-Bonnet dynamical compactification scenarios and their ghosts in the tensor sectors},
year = {2026},
howpublished = {\url{https://pith.science/paper/GRVECDZT}},
note = {Machine review of arXiv:2411.16983}
}
abstract
In a cosmological context, the Einstein-Gauss-Bonnet theory contains, in $d+4$ dimensions, a dynamical compactification scenario in which the additional dimensions settle down to a configuration with a constant radion/scale factor. Sadly however this work demonstrates that such a quite appealing framework is plagued by instabilities, either from the background configuration's unsteadiness or the ghostly behaviors of the tensorial perturbations. New and stable solutions are found by relaxing one of the hypotheses defining the original compactification scenario. However, such configurations do not respect the current bounds on the speed of propagation of gravitational waves, and thus have to be discarded. Those results thus advocate for a comprehensive study of compactification scenarios in the Gauss-Bonnet framework, their stability, and the effects of matter inclusion.
Reference graph
Works this paper leans on
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[6]
III A 1, in the case d = 2, the background is never stable
Case d = 3 As it has been shown in Sec. III A 1, in the case d = 2, the background is never stable. Hence we start by considering d = 3, for which the background solution ( 2.10) reads β = 3B2 0 − 2κ 12(κ − B2
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[7]
, λ = 9 − 6B4 0 − 3 B2 0 (B2 0 − κ) . (3.15) The no-ghost condition thus read K = B4 0 − 4κ B2 0 + 2 B2 0(B2 0 − κ) and ~K = − 2 B2 0 − κ B2 0 − κ . (3.16) When κ = −1, the ratio of the two is always negative, which means that one of th e two conditions is never fulfilled. When κ = 1, they are simultaneously positive only the range B0 ∈] √ 2 − √ 2; 1[, whi...
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[1]
taking κ = 0, the system of equations trivially reduces to Λ = 3 H 2 0 and α = − 1 4H 2 0
Flat extra-dimensional sub-manifolds require fine tunin g Considering a flat extra-dimensional sub-manifold, i.e. taking κ = 0, the system of equations trivially reduces to Λ = 3 H 2 0 and α = − 1 4H 2 0 . (2.7) Therefore, the two parameters of the action are linked by αΛ = −3/4, which reveals a strong fine-tuning. Hence, we will not consider a flat extra-dim...
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[2]
Constant-curvature extra-dimensional sub-manifolds a llow dynamical compactification In the case of a curved extra-dimensional sub-manifold, κ can be normalized to be ±1. Denoting β ≡ α H2 0 , λ ≡ Λ H 2 0 and B0 ≡ b0H0 , (2.8) the system ( A1) reduces to λ = 3 + d(d − 1) κ 2B2 0 + d(d − 1)(d − 2)(d − 3) 2B4 0 β + 6d(d − 1) κ B2 0 β , (2.9a) λ = 6(1 + 2 β) ...
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[3]
Indeed, it simply comes µκ + = 3 2 [ √ 1 + 32 27(1 − κ B2 0)2 − 1 ]
When d = 2, the background solution is never an attractor As will be clear in the following, the case d = 2 is quite specific, so it is treated apart from the others. Indeed, it simply comes µκ + = 3 2 [ √ 1 + 32 27(1 − κ B2 0)2 − 1 ] . (3.5) Note that when κ = 1, the eigenvalue is not defined at B0 = 1, which is nothing but the value of B+ 0 , see Eq ( 2.1...
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[4]
When d ≥ 3 and κ = − 1, the background solution is always an attractor For extra-dimensional manifolds with negative curvatures, the po lynomial Σ (− ) has no real roots excepted 0, hence the eigenvalue µ(− ) + is well defined. Then, a study of the zeros of the derivative of the r atio Π (− )/Σ (− ) shows that it is a monotonic function for B0 > 0. Expandi...
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[5]
When d ≥ 3 and κ = 1, there exist ranges of the parameter space where the backgro und solution is an attractor For extra-dimensional manifolds with positive curvatures, the poly nomial Σ (+) has two strictly positive roots when d ≥ 5, ordered as B(Σ ,1) 0 < B (Σ ,2) 0 and presented in App. A 3. Then, the derivative of the ratio Π (+)/Σ (+) also has two st...
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[8]
Case d = 4 In the case of four extra dimensions, treated in more detail in Sec. IV, the background solution is β = − (B2 0 − κ)B2 0 4(B4 0 − 1) , λ = − 3(B2 0 − κ)(B4 0 − 3κ B2 0 + 1) B2 0 (B4 0 − 1) . (3.17) The no-ghost condition thus read K = (B2 0 − 5κ)(B2 0 − κ) B4 0 − 1 and ~K = − 2 B2 0(B2 0 − κ) B4 0 − 1 . (3.18) When κ = −1, they have opposite si...
Show all 36 references
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[9]
(3.19b) When κ = −1, the ratio K ˆK = − 3 B4 0 + 6(d − 1) B2 0 + (d − 1)(2d − 3) 3B2 0[2 B2 0 + (d − 2)] , (3.20) is obviously negative (as d > 2)
Case d ≥ 5 In the case of more than 4 extra dimensions, the no-ghost conditio ns read K = 6B4 0 − 12(d − 1)κ B2 0 + 2(2d − 3)(d − 1) 6B4 0 + 3(d − 1)(d − 4)κ B2 0 − (d − 1)(d − 2)(d − 3) , (3.19a) ~K = − 6B2 0[2B2 0 − (d − 2)κ] 6B4 0 + 3(d − 1)(d − 4)κ B2 0 − (d − 1)(d − 2)(d ...
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[10]
In the spirit of what was done in Sec
Attractor behavior of this solution Let us first study the attractor behavior of the previously found solution. In the spirit of what was done in Sec. III A, one can use H = H0 ( 1 + δH ) , L = H0 ( X0 + δX ) . (5.5) As the dynamical system ( A1) is redundant, the constraint eq...
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[11]
compensate
No-ghost condition As for the fate of the tensorial modes, using the machinery develo ped in Sec. III B and the explicit computation performed in App. B 3, the two perturbation decouple as δ(2)S = M 2 Pl 8 ∫ dt d3x ddx a3bd H 2 0 { L(2) phys [ hij ] + L(2) extr [ HAB ] } , (5....
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[12]
Solving further the system, it comes L0 = H0, β = −1/12, and λ = 5, which is a strongly tuned expanding solution, hence of no interest for this work
Case d = 2 In the case d = 2, the combination 1 + 12β 2 Ea − Eb H 2 0 + ( 1 + 4β + 8β L H0 ) C H 2 0 = ( 1 + 4β + 8β L H0 )( 6 + 12β − λ + 4(1 + 12β) L H0 ) = 0 (5.13) imposes that L is constant. Solving further the system, it comes L0 = H0, β = −1/12, and λ = 5, which is a st...
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[13]
( A1c), and inject it in the equation of motion for a, yielding Ea = 12 ( ˙L + L2 − H0 L )( β L2 + 4β H0 L + 1 + 4β 4 H 2 0 + β κ b2 ) = 0
Case d = 3 In the case d = 3, one can first solve for λ the constraint, Eq. ( A1c), and inject it in the equation of motion for a, yielding Ea = 12 ( ˙L + L2 − H0 L )( β L2 + 4β H0 L + 1 + 4β 4 H 2 0 + β κ b2 ) = 0 . (5.14) So there are two branches of solution. The first branch...
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[14]
As previously, we suppose a de Sitter expansion for the physical manif old, namely a ∝ exp(H0t), and impose L ⁄= 0 and κ ⁄= 0
Case d = 4 To illustrate the behavior of the system for higher than 3 extra dime nsions, let us study the d = 4 case. As previously, we suppose a de Sitter expansion for the physical manif old, namely a ∝ exp(H0t), and impose L ⁄= 0 and κ ⁄= 0. On considering an appropriate co...
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[15]
compensate
Case of higher dimensions For more than 4 extra dimensions, we need to “compensate” the b− 4(t) dependency. Excepted for the already treated case b(t) = b0, the only way of doing so would be to impose L ∝ b− n or ˙L ∝ b− 2. The first case yields b(t) ∝ (1 + t/τ0) 1 n and the se...
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[16]
Cosmological background equations of motion Injecting the metric ( 2.3) in the action ( 2.1) and varying it wrt. {N, a, b} gives the background equations of motion Ea = −Λ + d(d − 1) κ 2 b2 + 3H 2 + 2d HL + d(d + 1) 2 L2 + 2 ˙H N + d ˙L N + α [ d(d − 1)(d − 2)(d − 3) κ2 2 b4 +...
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[17]
Perturbations of the background solution Transforming it to a first-order system by defining u ≡ ˙b and setting the lapse function to unity, the dynamical system ( A1) can be linearly perturbed as in Eq. ( 3.2), where M κ 0 = 0 0 1 − d(d− 1) M21 3 B2 0 ∆ 0 − 2 M22 ∆ 0 − d(...
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[18]
Attractor behaviour of the κ = 1 and d ≥ 5 cases When d ≥ 5 and κ = 1, the eigenvalue µ(+) + has a strictly negative real part on two ranges of B0, B0 ∈ ] B(Π ,1) 0 ; B(Σ ,1) 0 [ and B0 ∈ ] B(Π ,2) 0 ; B(Σ ,2) 0 [ , (A4) where the upper bounds are the two strictly positive roo...
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( 3.10), it is easy to establish the correspon- dence hij = a2(t) hij , hAB = b2(t) HAB , (B2) all other components of hµν being null
Generic perturbation Let us perturb the line element as gµν dxµdxν = [ ¯gµν + hµν ] dxµdxν , (B1) Comparing with the definition of the two perturbations hij and HAB, Eq. ( 3.10), it is easy to establish the correspon- dence hij = a2(t) hij , hAB = b2(t) HAB , (B2) all other com...
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Projection onto the sub-manifolds with the original comp actification scenario Let us project the quadratic action onto each sub-manifold, using the compactification scenario ( 2.6). The back- ground Christoffel symbol is given by ¯Γ 0 ij = H0 a2 ˆδij , ¯Γ i 0j = H0 δi j , (B6a) ...
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For simplicity, we use the shorthand L0 ≡ H0X0
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