{"id":"d9cb500e-6b4c-4fa8-a5e6-c2a09dda647f","arxiv_id":"2411.19873","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"All flat expanding cosmologies in dRGT massive gravity are strongly coupled at the perturbation level, so they cannot serve as healthy backgrounds for cosmic acceleration.","lead":"Massive gravity, a candidate alternative to Einstein's theory, can host expanding universes only if its auxiliary fields have uneven spatial profiles. This paper sorts those universes into two families and shows that small ripples in both families are badly behaved, so they are unlikely to describe our cosmos.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The classification is conditional on existence of Stueckelberg/reference-metric realizations of the X ansatz; Appendix A only constructs Mixed-branch profiles for Minkowski f_ab under Det[∂iϕ^n]≠0, leaving the Λ-branch realization (and any-f_ab generality) unproven.","rationale":"The reader's weakest_assumption is exactly the existence of Stueckelberg configurations realizing the X ansatz, and my analysis of the manuscript confirms that this is the most load-bearing unproven premise. The paper's own Conclusions admit the gap, and Appendix A only partially closes it for Minkowski f_ab and the Mixed branch. I considered whether the perturbation analysis was more fragile, but the stated strong-coupling results follow from the algebraic structure of the quadratic Lagrangians and the branch conditions; the Λ-branch vector kinetic term K ∝ ˙H = 0 and scalar K ∝ ˙H = 0 are straightforward consequences once the branch exists. I also considered whether the factorization of the continuity equation could hide an algebraic error, but no concrete mistake emerged. The honest concern is existence, not algebra. Hence I agree with the CONDITIONAL verdict: the paper is a valuable classification conditional on a stated but unproven existence assumption, and the verdict should remain CONDITIONAL rather than being upgraded or rejected, because the authors have explicitly flagged the limitation and provided a partial construction.","tokens_in":22421,"tokens_out":2318,"duration_ms":18563,"concrete_test":"Attempt an explicit construction of a Λ-branch profile for Minkowski f_ab: solve the system (A.1) with G = 3/2 and F(t) arbitrary, and check whether any smooth ϕa(t,x) with Det[∂iϕ^n] ≠ 0 satisfies (A.1) and the dRGT Stueckelberg equations. If no solution exists even in a perturbative expansion around the homogeneous profile, then the Λ-branch is empty for Minkowski f_ab and the central claim must be weakened to the Mixed branch and/or to non-Minkowski f_ab. A simpler diagnostic is to repeat the Appendix A derivation without assuming Det[∂iϕ^n] ≠ 0 and check whether the Λ-branch condition G = 3/2 can be met when the spatial Stueckelberg gradients are degenerate.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that imposing homogeneity and isotropy on X alone yields dynamical flat FLRW solutions, with branches that exist for any reference metric satisfying consistency conditions. However, the only explicit Stueckelberg profile construction is in Appendix A, and it assumes f_ab = η_ab and Det[∂iϕ^n] ≠ 0. Section A.1's construction is not actually a Λ-branch profile at all: it uses generic F(t) and G(t) linked by Eq. (A.13), so it supports relations like (3.18) rather than the G = 3/2 branch. The paper nowhere exhibits profiles satisfying the Λ-branch condition G = 3/2 for Minkowski f_ab. The Conclusions explicitly admit that no simple configuration is provided for Minkowski f_ab. Since the Λ-branch cosmology is F-independent and the strongest physical conclusion (only tensors propagate, scalars/vectors strongly coupled) relies on that branch's existence, the unproven existence of Stueckelberg profiles realizing G = 3/2 is load-bearing. If no such profiles exist for any f_ab, the Λ-branch is empty and the claimed dynamical flat FLRW solutions reduce to the Mixed branch only. Moreover, homogeneity and isotropy of X is assumed, but the consistency conditions (3.5) are PDEs for ϕa given f_ab; without a solubility theorem or explicit families for general f_ab, the classification remains conditional. This is not an internal inconsistency, but it is an unproven existence premise that the paper itself flags in Sec. 2 and the Conclusions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a classification of spatially flat FLRW cosmologies in ghost-free (dRGT) massive gravity. The central move is to impose the FLRW symmetries on the building block X^μ_ν = (√(g^{-1}∂φ^a∂φ^b f_ab))^μ_ν rather than on the Stueckelberg fields or the reference metric, assuming X^0_0 = F(t), X^0_i = 0, X^i_j = G(t)δ^i_j (Eq. 3.4). Under this ansatz the Stueckelberg stress-energy tensor is a perfect fluid depending only on F and G, and the continuity equation factorizes so that every solution belongs either to a Λ-branch (G = 3/2 in the minimal model; a quadratic condition in the full theory) or to a Mixed branch (Ġ + H(G−F) = 0). For a Minkowski reference metric, the Mixed branch admits G = g/a solutions whose Stueckelberg sector behaves as a mixture of perfect fluids, and a linearly expanding solution; the Λ-branch acts as a cosmological constant. A unitary-gauge perturbation analysis yields explicit quadratic Lagrangians: on the Λ-branch the scalar and vector kinetic terms vanish, and on the Mixed branch the scalar sound speed vanishes in the large-k limit, i.e., strong coupling at quadratic level in both cases. The whole classification is conditional on the existence of Stueckelberg/reference-metric profiles realizing (3.4); Appendix A constructs explicit profile families for the Minkowski reference metric, and Appendix B extends the stability analysis in the presence of matter.","tokens_in":22769,"tokens_out":27038,"duration_ms":217251,"significance":"If the classification is complete and the profile-existence premise is met, the paper establishes a clean structural result: every flat FLRW-compatible X background in dRGT is strongly coupled in the scalar sector at quadratic level, with only tensor modes propagating on the Λ-branch. The factorization of the continuity equation (3.13)/(3.26) and the explicit quadratic Lagrangians in Sec. 4 are concrete and checkable; the branch conditions are parameter-free in the sense that no fitted constants enter the classification. The authors also deserve credit for flagging the existence assumption in Sec. 2 and the Conclusions and for noting in footnote 6 that the earlier self-accelerating solutions of [18] have non-diagonal X, which sharpens the definition of the ansatz. The confirmatory value is real: the strong coupling of the self-accelerating solution is consistent with refs. [19, 23-26]. The main risk to the paper's significance is that the strongest conclusions (the Λ-branch existence and the exact zero sound speed on the Mixed branch) rest on premises that are partially unproven or only asserted; these are the points addressed in the major comments.","major_comments":[{"comment":"The construction in Appendix A.1, despite its title, does not realize the Λ-branch. Equation (A.13) is precisely the Mixed-branch relation (3.18) (with σ = k σ_u/2), so the families (A.15) tie G(t) to a(t) through ∫ F dt′; they do not produce the Λ-branch condition G = 3/2. Inserting G = 3/2 into (A.13) would force a(t) ∝ |k/2 + σ_u∫F dt′|, an additional constraint absent from the Λ-branch background analysis of Sec. 3.2.1, and no alternative profile family with G = 3/2 is exhibited. The Conclusions explicitly state that no simple configuration is provided for Minkowski f_ab. The perturbation results of Secs. 4.2.1 and 4.3.1 (vanishing vector and scalar kinetic terms, strong coupling) concern precisely this branch, so the unproven existence of Λ-branch profiles realizing (3.4) is load-bearing for the central claim. I recommend either constructing such a family explicitly, noting that the known self-accelerating solutions of refs. [20–22] have non-diagonal X as explained in footnote 6, or reframing the Λ-branch claims as explicitly conditional on an existence assumption.","section":"App. A.1; Secs. 3.2.1, 4.2–4.3; Conclusions"},{"comment":"There is a sign error in the vector no-ghost condition. From Eq. (4.12), the large-k kinetic coefficient is K_Êi ≃ −aḢ/2, so requiring K_Êi > 0 implies Ḣ < 0, not \"Ḣ must be positive\" as stated in the text; the rewritten Lagrangian (4.13) has the same sign structure. The scalar sector gives the same requirement: K_E² from Eq. (4.20) behaves as −(a k²/2)Ḣ at large k, and Sec. 4.3 correctly concludes Ḣ < 0. The manuscript therefore assigns opposite viability conditions to the vector and scalar sectors on the same Mixed-branch background; one of the statements is inconsistent with the displayed formulas. Since the parameter-space constraints (4.15) and the claim that vectors propagate healthily depend on this sign, the authors should correct it and re-state the resulting stability conditions.","section":"Sec. 4.2.2, Eqs. (4.11)–(4.13)"},{"comment":"The central strong-coupling conclusion on the Mixed branch is the vanishing of the scalar sound speed, which follows from the assertion that the left-hand side of (4.24) is \"exactly saturated, with the LHS exactly zero\" on this background. The saturation is stated without derivation. Because c_s² = 0 is a main physical output of the paper, the explicit evaluation of (4.24) using the Mixed-branch background equations (3.12), (3.13), and (3.17)–(3.18), together with the full-theory counterparts, should be displayed or the required identities given so that the claimed cancellation is directly checkable by the reader.","section":"Sec. 4.3.2, Eqs. (4.22)–(4.24)"},{"comment":"The Introduction's claim that the existence of the solutions \"is actually guaranteed for any possible choice of f_ab and ϕa which satisfy a set of consistency conditions (in the form of complicated partial differential equations)\" overstates what is proven. As stated, the sentence is near-tautological: the nontrivial content is that the PDE system (3.5) admits solutions for a useful class of reference metrics. The paper provides such families for f_ab = η_ab in Appendix A under the assumptions φ̇⁰ ≠ 0 and Det[∂_iϕ^n] ≠ 0, but no existence statement or example is given for general f_ab, and the Λ-branch case is not covered even for Minkowski (Major Comment 1). The \"similar construction\" for arbitrary f_ab announced at the start of Appendix A is not carried out. The authors should qualify the generality claim accordingly, or supply an existence theorem or a detailed sketch for the PDE system.","section":"Sec. 3.1, Eq. (3.5); Introduction, p. 3; App. A"}],"minor_comments":[{"comment":"The interval in which the Stueckelberg energy density is positive should read g/2 < a < g, since positivity of −(2 − 3G + G²) in (3.16)/(3.20) requires 1 < G < 2 with G = g/a; the printed \"g²\" appears to be a typo.","section":"Sec. 3.2.2, after Eq. (3.20)"},{"comment":"The statement that tensor perturbations impose no constraints on the background addresses only ghost and gradient instabilities. The sign of the tensor mass m²_ĥ in (4.7) can change with the dRGT parameters and the branch, and a negative m²_ĥ corresponds to a tachyonic instability; a comment on the associated parameter constraints (e.g., of the Higuchi type on the Λ-branch) would make the stability discussion complete.","section":"Sec. 4.1, Eqs. (4.6)–(4.7)"},{"comment":"The derivation leading to (A.3) uses φ̇⁰ ≠ 0 and Det[∂_iϕ^n] ≠ 0, but these assumptions are not stated when the system (A.1) is introduced, and their validity for the constructed families in A.2 (in particular the A² = 1 case) is not checked. Please state the assumptions explicitly and verify them for each displayed family.","section":"Appendix A, Eqs. (A.1)–(A.4)"},{"comment":"There are numerous typographical and consistency issues: \"FLRW\" is typeset as \"FLR W\" in the abstract and several places; the Sec. 3.2.2 heading uses \"Mix-Branch\" instead of \"Mixed Branch\"; in Sec. 2 the phrase \"condition∂iϕ\" is missing \"= 0\"; the characterization of Ref. [20] in the Introduction as a spherically symmetric Stueckelberg configuration should be checked against the cited paper; and the bibliography contains odd renderings of non-ASCII author names (e.g., refs. [10], [21]). A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest and technically rich, and the main gaps are addressable in revision: the Λ-branch profile existence, the vector-sector sign slip, and the asserted c_s² = 0 saturation. The paper is likely to be of solid interest to the dRGT and massive-gravity community and fits the journal's scope. I would ask the authors to soften the Introduction's \"guaranteed for any possible choice of fab\" claim and to make explicit that the Λ-branch strong-coupling results are conditional unless explicit profiles realizing G = 3/2 are supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is worth taking seriously. The new idea is to impose FLRW symmetry directly on the square-root tensor X rather than on the Stueckelberg fields and reference metric. That gives a clean two-branch classification—the self-accelerating Λ branch and the Mixed branch—and the full-theory perturbation calculation shows both are strongly coupled at quadratic order: only tensors propagate on the Λ branch, and the scalar sound speed vanishes on the Mixed branch. If correct, this closes off healthy flat FLRW cosmologies in dRGT, which is important for dark energy model building. The derivation is mostly explicit: the factorization of the continuity equation into two factors is elegant, and the Lagrangians are given in enough detail that a patient referee can verify them.\n\nThe soft spots are real but not fatal. The paper assumes the existence of Stueckelberg profiles realizing the X ansatz and does not construct a Λ-branch profile for Minkowski reference; the Conclusions admit this and point to earlier work. More confusing, Appendix A is mislabeled: the construction in A.1 produces the relation a^2G^2 = λ1(∫F+σ)^2, which is precisely the Mixed-branch relation (3.18), not a Λ-branch profile with G=3/2. Section A.2 then treats the F=0 subcase. So the appendix's headings do not match the branch names in the main text, and the reader must reverse-engineer which profiles actually support which branch.\n\nSecond, the claim that the Mixed-branch scalar has exactly zero sound speed is stated without showing the algebra. Since this is the central pathology of that branch, a referee should ask for the derivation. Third, the Λ-branch strong-coupling statement in Sec. 4 is for the vacuum case (Hdot=0); the matter case is deferred to Appendix B, where the result is recovered. That is acceptable but should be flagged in the main text.\n\nThe stress-test worry that the Λ-branch might be empty is, I think, overstated: the self-accelerating branch is known in the literature with explicit profiles, so the branch is not vacuous. What the paper does not do is show that those known profiles satisfy the new consistency conditions. That is an exposition gap, not a load-bearing flaw. The core classification and the perturbation result deserve a serious referee; the paper needs revision for clarity. Send it to review.","headline":"A credible negative result for dRGT flat FLRW cosmologies, held back by a mislabeled appendix and an unproven zero-sound-speed claim that a referee should check.","tokens_in":23296,"tokens_out":8343,"would_cite":true,"duration_ms":78382,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.50.Kd","98.80.-k"],"model":"deepseek-v4-flash","headline":"This paper classifies the dynamical flat FLRW backgrounds of dRGT massive gravity by demanding that the square-root tensor itself be homogeneous and isotropic, and shows that on every such background scalar and vector perturbations are…","keywords":["massive gravity","dRGT","FLRW cosmology","Stueckelberg fields","square-root tensor","strong coupling","cosmological perturbations","self-accelerating solution"],"falsifier":"Find an explicit family of Stueckelberg fields and a reference metric that satisfies Eqs. (3.5) with $G=3/2$ and a Minkowski $f_{ab}=\\eta_{ab}$; existence would put the $\\Lambda$-branch on a concrete footing, while a no-go proof would empty that branch in the standard setup. Alternatively, compute the cubic action on the Mixed branch: a nonzero cubic vertex would confirm the strong-coupling diagnosis, whereas a vanishing cubic vertex would reopen the question of whether the scalar mode truly fails to propagate.","tokens_in":22167,"feed_emoji":"🌌","tokens_out":11959,"duration_ms":89595,"temperature":0.7,"pith_summary":"The paper asks what flat, spatially homogeneous and isotropic (FLRW) universes ghost-free dRGT massive gravity can actually support. Instead of guessing a profile for the Stueckelberg fields and a reference metric, it imposes the FLRW symmetries directly on the square-root tensor $X^\\mu{}_\\nu$ that builds the graviton potential. This yields two families of dynamical backgrounds: a self-accelerating branch in which the Stueckelberg sector behaves exactly like a cosmological constant, and a mixed branch in which it behaves like a mixture of perfect fluids, including a dust-like component in the full theory. The paper then shows that on both branches linear perturbations are pathological: scalar and vector modes are strongly coupled on the self-accelerating branch, and the scalar mode acquires zero sound speed on the mixed branch. The upshot is that none of the flat FLRW backgrounds admitted by this classification can support a healthy, standard cosmological perturbation theory.","feed_headline":"Strong coupling sinks every flat expanding universe in massive gravity","feed_subtitle":"Only tensors propagate on the Lambda branch; scalar sound speed vanishes on the Mixed branch, leaving neither viable.","key_machinery":"The carrying object is the square-root tensor $X^\\mu{}_\\nu = \\left(\\sqrt{g^{-1}\\partial\\phi^a\\partial\\phi^b f_{ab}}\\right)^\\mu{}_\\nu$, the single building block from which every dRGT potential is assembled via $K^\\mu{}_\\nu=\\delta^\\mu{}_\\nu-X^\\mu{}_\\nu$. Imposing on it the FLRW form $X^0{}_0=F(t)$, $X^0{}_i=0$, $X^i{}_j=G(t)\\delta^i{}_j$ reduces the background dynamics to two time-dependent functions $F$ and $G$ and to the constraint system $F^2=-\\dot\\phi^a\\dot\\phi^b f_{ab}$, $0=\\dot\\phi^a\\partial_i\\phi^b f_{ab}$, $G^2\\delta_{ij}=a^{-2}\\partial_i\\phi^a\\partial_j\\phi^b f_{ab}$. The continuity equation for the Stueckelberg sector factorizes into an algebraic branch selector (which fixes $G$ on the $\\Lambda$-branch) and the universal factor $\\dot{G}+H(G-F)$ that defines the Mixed branch; this factorization is what turns the search for cosmologies into a complete classification.","core_discovery":"On the paper's own terms, the central discovery is that requiring the square-root tensor to respect the symmetries of a flat FLRW universe, $X^0{}_0=F(t)$, $X^0{}_i=0$, $X^i{}_j=G(t)\\delta^i{}_j$, reorganizes dRGT massive gravity into a two-branch classification. In the $\\Lambda$-branch the value of $G$ is fixed (to $3/2$ in the minimal model, or to the roots $G_\\pm$ of a quadratic in the full theory), and the Stueckelberg stress-energy tensor becomes a cosmological constant with $w=-1$; in the Mixed branch the relation $\\dot{G}+H(G-F)=0$ holds, and for a Minkowski reference metric this leads to solutions with $F=0$, $G=g/a$ (a mixture of fluids with equations of state $w=-1,-2/3,-1/3$, plus $w=0$ in the full theory) and to solutions with $F\\neq 0$ in which $H\\propto 1/a$ and $\\ddot{a}=0$. The fields that realize these backgrounds must be inhomogeneous and/or anisotropic; homogeneous Stueckelberg fields with a Minkowski reference metric return the old no-go $\\dot{a}=0$. In the unitary-gauge perturbative expansion on these backgrounds, the quadratic Lagrangian has vanishing kinetic terms for the scalar and vector modes on the $\\Lambda$-branch, while on the Mixed branch the vector mode can propagate (subject to positivity and gradient-stability constraints) but the scalar mode has zero speed of sound in the sub-horizon limit; adding matter does not change this qualitative picture.","pith_inferences":["The symmetry-first method is not tied to a particular $f_{ab}$: the same $F,G$ classification could in principle be re-run for non-Minkowski reference metrics, non-flat spatial curvature, or Bianchi-type symmetries, and the paper does not rule out that new branches could appear there.","The paper gives no explicit Stueckelberg profile realizing the $\\Lambda$-branch when the reference metric is Minkowski; closing that gap is the minimal test of whether the self-accelerating branch actually exists in the standard setup.","A concrete nonlinear test would be to compute the cubic action on the Mixed branch: if the cubic vertices are nonzero, the zero-sound-speed scalar mode is genuinely strongly coupled, and if they vanish the diagnosis would change.","Because the backgrounds require inhomogeneous Stueckelberg fields, the effective perfect-fluid description hides a large internal configuration space; a natural next step is to check whether those hidden configurations introduce additional light degrees of freedom beyond the metric perturbation analysis."],"forward_implications":["On the $\\Lambda$-branch only tensor metric perturbations propagate; scalar and vector modes are frozen at quadratic order, so the self-accelerating background cannot seed structure formation in the standard way.","On the Mixed branch the scalar mode has vanishing speed of sound on sub-horizon scales, so scalar perturbations do not propagate linearly; vector modes can propagate only if $\\dot{H}>0$ and a gradient-stability inequality is satisfied.","Adding ordinary matter to either background does not cure the scalar/vector pathologies; in the large-$k$ limit the scalar mode still has zero sound speed.","The full dRGT theory adds a dust-like ($w=0$) component to the Stueckelberg fluid on the Mixed branch, opening a background-level channel for massive-gravity polarizations to mimic dark matter, but the perturbation results block using those backgrounds as a viable cosmological model.","For any flat FLRW background in this classification, the dRGT theory cannot reproduce the healthy linear perturbation theory of $\\Lambda$CDM; this places the cosmologies under strong-coupling control rather than under weak-field perturbative control."],"supporting_citations":[{"why":"Constructs the ghost-free dRGT action and potentials $U_i[\\mathcal K]$ whose background dynamics the paper analyzes.","marker":"[12]"},{"why":"Establishes the no-go result for Minkowski reference metric with homogeneous Stueckelberg fields ($\\dot a=0$) and the self-accelerating solution the $\\Lambda$-branch generalizes.","marker":"[18]"},{"why":"Shows that three of the five degrees of freedom are strongly coupled on the self-accelerating background, the result the present paper extends to the full classification.","marker":"[19]"},{"why":"Provides exact self-accelerating cosmologies with inhomogeneous Stueckelberg profiles for a Minkowski reference metric, the exemplar of the Mixed branch.","marker":"[20]"},{"why":"Proposes imposing the symmetries on the effective matter tensor / square-root combination, the approach the paper develops; its explicit profiles are noted to fall back to the no-go.","marker":"[27]"},{"why":"Derives parallel FRW branches in bigravity, which the paper cites as background for the same mixed-branch structure.","marker":"[29]"}],"fun_headline_variants":["Flat expanding universes in massive gravity are all strongly coupled","All flat FLRW cosmologies in massive gravity are strongly coupled","Strong coupling rules out every expanding universe in massive gravity","Massive gravity: every flat FLRW solution is strongly coupled","dRGT massive gravity: no healthy flat expanding cosmos"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification rests on the assumption that at least one actual choice of reference metric and Stueckelberg-field profile satisfies the consistency constraints (3.5) for each branch; the paper proves the consistency conditions are necessary but does not exhibit a $\\Lambda$-branch profile for a Minkowski reference metric, and its Mixed-branch construction assumes the spatial Jacobian $\\det(\\partial_i\\phi^n)$ is nonzero.","fun_headline_variants_meta":{"raw":{"variants":["Flat expanding universes in massive gravity are all strongly coupled","All flat FLRW cosmologies in massive gravity are strongly coupled","Strong coupling rules out every expanding universe in massive gravity","Massive gravity: every flat FLRW solution is strongly coupled","dRGT massive gravity: no healthy flat expanding cosmos"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00109,"raw_usage":{"total_tokens":4632,"prompt_tokens":1099,"completion_tokens":3533,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":715,"completion_tokens_details":{"reasoning_tokens":3451}},"tokens_in":715,"tokens_out":3533,"duration_ms":23909,"temperature":1.0,"reasoning_tokens":3451,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:43:37.525427+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find an explicit family of Stueckelberg fields and a reference metric that satisfies Eqs. (3.5) with $G=3/2$ and a Minkowski $f_{ab}=\\eta_{ab}$; existence would put the $\\Lambda$-branch on a concrete footing, while a no-go proof would empty that branch in the standard setup. Alternatively, compute the cubic action on the Mixed branch: a nonzero cubic vertex would confirm the strong-coupling diagnosis, whereas a vanishing cubic vertex would reopen the question of whether the scalar mode truly fails to propagate.","supporting_citations":[{"cited_title":"Cosmology and perturbations in massive gravity","cited_arxiv_id":"1206.3617","evidence_quote":"Shows that three of the five degrees of freedom are strongly coupled on the self-accelerating background, the result the present paper extends to the full classification."},{"cited_title":"Exact self-accelerating cosmologies in the ghost-free massive gravity -- the detailed derivation","cited_arxiv_id":"1207.3723","evidence_quote":"Provides exact self-accelerating cosmologies with inhomogeneous Stueckelberg profiles for a Minkowski reference metric, the exemplar of the Mixed branch."},{"cited_title":"Effective matter cosmologies of massive gravity i: non-physical fluids,","cited_arxiv_id":null,"evidence_quote":"Proposes imposing the symmetries on the effective matter tensor / square-root combination, the approach the paper develops; its explicit profiles are noted to fall back to the no-go."},{"cited_title":"Frw cosmology in ghost free massive gravity from bigravity,","cited_arxiv_id":null,"evidence_quote":"Derives parallel FRW branches in bigravity, which the paper cites as background for the same mixed-branch structure."}],"review_version":1}