{"id":"4e4f5800-da61-4f27-b6f9-489520e3f693","arxiv_id":"2505.18192","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Tensor perturbations in F(T,T_G) teleparallel gravity produce gravitational waves that propagate at the speed of light, with a modified amplitude.","lead":"This paper studies gravitational waves in a modified teleparallel gravity theory with a Gauss-Bonnet term. It finds the waves travel at exactly the speed of light, matching observations from the 2017 neutron star merger.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The luminal-speed claim rests on an unshown second-order action; the only supplied perturbative data contain a determinant factor error that, if propagated, would introduce a mass term.","rationale":"The paper's headline result, that F(T,T_G) teleparallel gravity has alpha_T = 0, is interesting and would be a meaningful extension of the f(T) result, but it is supported only by an asserted quadratic action. The authors do not display the expansion of T_G, which involves many contractions of contortion and would be the natural place for k^2/a^2 corrections or mass terms to appear. The only explicit perturbative quantities in Appendix A are inconsistent: the determinant of the given vierbein is a^3(1 - 1/8 h^2), not a^3(1 - 1/4 h^2); this is a concrete arithmetic error in the manuscript rather than a matter of interpretation. Because e multiplies F in the action, this error changes the quadratic action by a term proportional to the background F and h^2, which would contribute a mass to the GWPE. Even if the error is only a typo in the appendix, the published text provides no way to verify the central coefficient, the absence of a mass term, or the alpha_T = 0 statement. The f(T) limit F = -T is a useful control: if the computation is correct it must reduce to the standard luminal tensor action; if it does not, the error is substantive. For these reasons the paper should not be accepted without the full second-order expansion or a supplementary computation. Since the reader already set CONDITIONAL, our stress-test does not move the verdict, but it sharpens the condition: the authors must correct the determinant expansion and demonstrate the cancellation of h^2 terms explicitly.","tokens_in":13248,"tokens_out":15446,"duration_ms":140417,"concrete_test":"Use a computer algebra system (xAct/xPert, MapleGR, or equivalent) to expand e, T, and T_G to second order in the transverse-traceless h_ij using the vierbein (31) and definitions (11) and (15). Verify (i) whether det(e^a_alpha) = a^3(1 - (1/8)Tr(h^2)), (ii) whether the resulting action reproduces (33) with C_tensor = -F_T + 4H dot(F_TG) and no h^2 term. As a built-in control, set F = -T: the expansion must reproduce the known f(T) tensor action with luminal speed; any residual h^2 or modified (partial h)^2 coefficient invalidates Eq. (34).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Eq. (34), which follows only if the second-order action is exactly (33). The derivation of (33) is not shown: the paper states that inserting the perturbed vierbein into (11) and (15) yields (33), but Appendix A supplies only first-order torsion components and a second-order determinant that appears to be wrong. Direct evaluation of det(e^a_alpha) for the vierbein (31) gives e = a^3[1 - (1/8) Tr(h^2)] (with h^2 = h_{ij}h^{ij}), not the a^3[1 - (1/4)h^2] written in the appendix. Since the action is integral e F(T,T_G), the second-order piece e^(2) multiplies the background F(T,T_G); a spurious e^(2) leaves an uncancelled h^2 term unless T^(2) and T_G^(2) contain exactly compensating contributions. The appendix does not give T^(2) or T_G^(2), so the reader cannot check the claimed coefficient C_tensor = -F_T + 4H dot(F_TG) or the absence of a mass term. If the factor error propagates, Eq. (34) would acquire a k^0 h term and the alpha_T = 0 conclusion would fail. This is the single load-bearing step, and it is currently unverified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies tensor perturbations of a spatially flat FLRW background in F(T,T_G) teleparallel gravity. Its central claim is that the quadratic action for transverse-traceless tensor modes takes the form (33) with coefficient Ctensor = -F_T + 4H Fdot_TG. Varying this action gives the gravitational-wave propagation equation (34), whose k^2/a^2 term has coefficient unity, implying alpha_T = 0 and luminal gravitational-wave speed for the entire class of F(T,T_G) theories. The paper also relates the modified friction term to the GW/EM luminosity-distance ratio and concludes that the class is observationally consistent with the GW170817 speed bound. The derivation is based on a perturbed vierbein (31), but the crucial second-order action is asserted rather than derived; Appendix A supplies only partial perturbation data.","tokens_in":13638,"tokens_out":8608,"duration_ms":81312,"significance":"If the central calculation is correct, the result is significant: it would place the broad F(T,T_G) teleparallel family on the observationally safe side of the GW170817 constraint, in contrast to many curvature-based modified-gravity theories. The paper also gives a clear stability condition, Ctensor > 0, and a standard-siren relation for the modified amplitude, which are testable in principle. The result passes the consistency check of reducing to GR for F = -T and to the linear Gauss-Bonnet limit, both of which give Ctensor = 1. The main concern is that the load-bearing second-order action is not verified from the text, and one displayed intermediate expression appears to contain a determinant error. The contribution is timely and potentially important, but its validity cannot currently be assessed from the manuscript as written.","major_comments":[{"comment":"The derivation of the second-order action (33) is the load-bearing step and is not shown. The text states that inserting the perturbed relations into (11) and (15) and expanding the action yields (33), but Appendix A lists only torsion components up to first order (with some second-order corrections) and does not give T^(2) or T_G^(2). Since the action integral contains e F(T,T_G), the second-order part e^(2) multiplies the background F and must be cancelled by contributions from F_T T^(2) and F_TG T_G^(2). Without an explicit expansion of T and T_G to second order, the claimed coefficient Ctensor = -F_T + 4H Fdot_TG and the absence of a mass term are unsupported. Please provide the complete second-order expansion of e, T, and T_G, including the boundary terms discarded after integration by parts.","section":"III, Eq. (33), and Appendix A"},{"comment":"The stated determinant of the perturbed vierbein appears to be incorrect. For the spatial block a(delta_ij + (1/2)h_ij) with h traceless, direct evaluation gives e = a^3[1 - (1/8)h_ij h^ij + O(h^3)], not a^3[1 - (1/4)h^2_ij] as written. Since e^(2) contributes to the quadratic action, this factor-of-two discrepancy could change the h^2 term in the action. The paper needs to correct this expression and demonstrate explicitly how the h^2 term is cancelled in the passage to Eq. (33), or explain if a different convention for h^2_ij is being used.","section":"Appendix A"}],"minor_comments":[{"comment":"The abstract says the distance-duality relationship is derived, but Eq. (36) is quoted from Ref. [66] rather than derived in this work; please rephrase to 'employed' or 'applied'.","section":"Abstract and Section III"},{"comment":"The notation h^2_ij is ambiguous; if it means h_i^k h_kj, the trace should be written explicitly as h_ij h^ij or h_i^k h_k^i.","section":"Appendix A"},{"comment":"The phrase 'after making sure that on shell conditions are satisfied' is vague; the only condition needed to obtain (34) from (33) is the Euler-Lagrange equation, and any additional assumptions should be stated explicitly.","section":"Section III, Eq. (34)"},{"comment":"The name 'Friedmann-Lemaitre-Robertson-Walker' contains a character-encoding artifact ('Lemaître'); please correct it.","section":"Eq. (25)"}],"recommendation":"major_revision","confidential_remarks":"The core issue is verifiability: the paper's main conclusion depends on the unshown expansion leading to Eq. (33), and the appendix contains an apparent determinant error. I found no indication of circular reasoning or parameter fitting; the authors should be given the opportunity to supply the full second-order calculation. If the missing expansion cannot be provided, the central claim should be withdrawn."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the new thing: this is the first derivation of the cosmological tensor perturbation equation for F(T,T_G) teleparallel gravity. The result is that tensor modes propagate at luminal speed, with the amplitude modified through M_*^2 = kappa^{-2}(-F_T + 4H dot(F_TG)). That is a useful step for the teleparallel modified-gravity community, and the approach—perturbing the action rather than the field equations—is the right one. The GR limit (F=-T) and the linear Gauss-Bonnet limit (F=-T+alpha T_G) both give C_tensor=1, so the speed claim passes those checks.\n\nThe soft spot is load-bearing. Equation (33), the second-order action from which everything follows, is stated without showing the second-order expansion of T and T_G. Appendix A lists only first-order torsion components, and the one second-order quantity it does give—the determinant e = a^3[1 - (1/4)Tr(h^2)]—is wrong by a factor of two. From the paper's own perturbed vierbein, the determinant is a^3[1 - (1/8)Tr(h^2)]. Because e multiplies the background F in the action, an incorrect e^(2) leaves an uncancelled h^2 term unless T^(2) and T_G^(2) compensate exactly, and the paper gives no way to check that. So the coefficient C_tensor = -F_T + 4H dot(F_TG) and the claimed absence of a mass term are, as things stand, unverified. I want to be clear that this is not a refutation: the consistency checks pass and the structure of the claim is plausible. But a reader cannot reproduce the central computation from the text, and the mistaken determinant in the appendix suggests the perturbative data need care.\n\nTwo secondary points. The abstract says the paper derives the distance-duality relationship, but in the text Eq. (36) is simply quoted from Ezquiaga and Zumalacárregui. And the paper would benefit from a sentence explaining why the speed stays luminal in the teleparallel Gauss-Bonnet setting when the curvature-based f(R,G) case in Ref. [79] deviates; that contrast is the kind of thing the reader will immediately wonder about.\n\nThe citation pattern is fine, with proper credit to [40] for the Minkowski polarizations and [78] for f(T) gravity. No fitted parameters, no circular steps.\n\nThis paper is for the modified-gravity and teleparallel-cosmology subfield. It deserves a serious referee, but the referee should demand a major revision: the full second-order expansion of T and T_G, or a supplementary file, showing how (33) follows from (11) and (15), plus a corrected appendix. If the computation holds up, the alpha_T = 0 result is worth having.","headline":"First F(T,T_G) tensor perturbation analysis gives a plausible luminal-speed result, but the central action is unshown and the appendix determinant has a factor-two error.","tokens_in":14025,"tokens_out":5367,"would_cite":false,"duration_ms":45622,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C35","83D05","83F05"],"pacs":["04.30.-w","04.50.Kd"],"model":"deepseek-v4-flash","headline":"Tensor perturbations of flat FLRW in F(T,T_G) teleparallel Gauss-Bonnet gravity give alpha_T = 0: gravitational waves travel at the speed of light.","keywords":["teleparallel gravity","Gauss-Bonnet gravity","tensor perturbations","gravitational wave speed","GW170817","FLRW cosmology","luminosity distance","modified gravity"],"falsifier":"A direct symbolic computation of $T^{(2)}$ and $T_G^{(2)}$ from the perturbed vierbein (31), followed by integration of the action (24), would settle the claim: if any term $h_{ij}\\nabla^2 h^{ij}$ or $h_{ij}h^{ij}$ survives with a coefficient other than $a^{-2}C_{\\rm tensor}$, then $\\alpha_T \\neq 0$ or a mass term is present.","tokens_in":13018,"feed_emoji":"🌌","tokens_out":11089,"duration_ms":97153,"temperature":0.7,"pith_summary":"This paper asks how the gravitational-wave sector behaves in $F(T,T_G)$ teleparallel Gauss-Bonnet gravity, a class of modified gravity in which the geometry is carried by torsion rather than curvature and the action is a function of the torsion scalar and its Gauss-Bonnet counterpart. By perturbing a spatially flat FLRW universe with a transverse, traceless tensor mode and expanding the action to second order, the authors derive a propagation equation in which the $k^2/a^2$ term has coefficient exactly one, so the tensor excess speed is zero and gravitational waves travel at the speed of light. The amplitude of the waves is not standard: the effective Planck mass becomes $M_*^2 = \\kappa^{-2}(-F_T + 4H\\dot{F}_{T_G})$, and the ratio of gravitational-wave to electromagnetic luminosity distance is set by the running of this mass. The paper's main point of interest is that the whole class of $F(T,T_G)$ cosmologies is therefore consistent with the GW170817 bound on the gravitational-wave speed, which has ruled out many other modified-gravity proposals. It also notes that avoiding ghost instabilities requires the tensor coefficient $C_{\\rm tensor} = -F_T + 4H\\dot{F}_{T_G}$ to be positive.","feed_headline":"Teleparallel Gauss-Bonnet gravity keeps gravitational-wave speed at c","feed_subtitle":"Gravitational waves keep light speed, so the whole teleparallel Gauss-Bonnet class survives the GW170817 bound.","key_machinery":"The central object is the quadratic action of the tensor mode, obtained by perturbing the vierbein as $\\delta e^{\\hat{k}}_\\alpha = \\frac{a}{2}\\delta^{\\hat{i}}_\\alpha \\delta^{\\hat{k}\\hat{j}} h_{\\hat{i}\\hat{j}}$ and expanding the torsion scalar $T$ and its teleparallel Gauss-Bonnet counterpart $T_G$ (the torsion object equal to the Gauss-Bonnet invariant up to a total divergence) to second order in $h_{ij}$. The transverse-traceless gauge conditions remove unwanted terms, integration by parts brings the action into the form (33), and variation with respect to $h_{ij}$ produces the propagation equation. The load-bearing mechanism is a set of cancellations in the second-order expansion that leave the gradient term with the same $a^{-2}$ coefficient as in general relativity, rather than a rescaled speed or an extra $k^2 h^2$ mass term; this cancellation is what forces $\\alpha_T = 0$.","core_discovery":"The central claim is that tensor perturbations of flat FLRW in $F(T,T_G)$ teleparallel gravity yield the second-order action (33), $S_T^{(2)} = \\frac{1}{2\\kappa^2}\\int dt\\,d^3x\\, a^3 \\frac{1}{2}C_{\\rm tensor}[\\dot{h}_{ij}^2 - a^{-2}(\\nabla h_{ij})^2]$, with $C_{\\rm tensor} = -F_T + 4H\\dot{F}_{T_G}$. Varying this action gives the gravitational-wave propagation equation (34), whose momentum term is exactly $k^2/a^2$ with no extra mass term, so comparison with the standard parametrization gives $\\alpha_T = 0$ and $M_*^2 = \\kappa^{-2}(-F_T + 4H\\dot{F}_{T_G})$. Thus in these theories gravitational waves propagate at the speed of light, while the waveform amplitude is damped or enhanced depending on the time evolution of $M_*$; stability against ghosts requires $C_{\\rm tensor} > 0$. The authors also connect the amplitude modification to the gravitational-wave versus electromagnetic luminosity-distance ratio through the standard-siren relation for GW luminosity distance. In the curvature-based analogue $f(R,G)$, by contrast, the propagation speed deviates from $c$, which is what makes the teleparallel result distinctive.","pith_inferences":["A natural next step, not taken here, is to perturb the same action at scalar and vector order; tensor ghosts may be absent while scalar instabilities still rule out many $F(T,T_G)$ functional forms, so the ghost-free condition is necessary but not sufficient.","Because the gradient coefficient was checked only on a flat background, the same second-order computation on an open or closed FLRW metric could produce a curvature-dependent contribution to $\\alpha_T$; testing that would show whether the luminal result is tied to flatness.","Standard-siren datasets from future detectors could turn the amplitude modification into a direct test: a measured GW/EM distance ratio that tracks the predicted exponential integral would favour $F(T,T_G)$, while a null result would constrain the allowed running of $M_*$.","The method used here, perturbing the action rather than the field equations, can be ported directly to other teleparallel extensions such as $f(T,B)$, where the same cancellation question decides whether their tensor speed is also protected."],"forward_implications":["Every $F(T,T_G)$ model built on flat FLRW automatically satisfies the GW170817-GRB170817A limit on gravitational-wave speed, because $\\alpha_T = 0$ at the level of this calculation.","Gravitational-wave amplitudes in these theories carry a fingerprint: the running effective Planck mass $M_*^2 = \\kappa^{-2}(-F_T + 4H\\dot{F}_{T_G})$ rescales the waveform and makes the GW luminosity distance differ from the electromagnetic one by $\\exp\\!\\left(\\tfrac{1}{2}\\int \\frac{\\alpha_M}{1+z}\\,dz\\right)$.","Requiring $C_{\\rm tensor} > 0$ to avoid ghost-like tensor modes translates into a concrete inequality on $F$ and its derivatives, which can be used to reject candidate functional forms of $F(T,T_G)$.","A measured gravitational-wave speed exactly equal to $c$ cannot, by itself, distinguish $F(T,T_G)$ from general relativity; distinguishing signals must come from the amplitude channel or from scalar and vector perturbations."],"supporting_citations":[{"why":"Defines the teleparallel Gauss-Bonnet scalar $T_G$ and the $F(T,T_G)$ action and field equations that the paper perturbs.","marker":"[56]"},{"why":"Provides the flat FLRW background values and Friedmann equations used for $T = 6H^2$ and $T_G = 24H^2(\\dot H + H^2)$.","marker":"[55]"},{"why":"Supplies the gravitational-wave propagation framework and polarization-mode analysis for modified teleparallel theories that the calculation adopts.","marker":"[40]"},{"why":"Gives the vierbein perturbation scheme and Weitzenböck-gauge treatment for cosmological tensor perturbations in teleparallel gravity.","marker":"[34]"},{"why":"States the standard parametrized propagation equation with $\\alpha_M$ and $\\alpha_T$ against which equation (34) is matched.","marker":"[76]"},{"why":"The GW170817 detection that sets the observational bound on gravitational-wave speed used to frame the result.","marker":"[62]"},{"why":"The electromagnetic counterpart GRB170817A whose near-simultaneous arrival produces the tight speed constraint.","marker":"[63]"},{"why":"The $f(T)$ tensor perturbation result with luminal speed that the present work extends to $F(T,T_G)$.","marker":"[78]"},{"why":"The curvature-based $f(R,G)$ case in which gravitational-wave speed deviates from $c$, the contrast that makes the teleparallel result notable.","marker":"[79]"},{"why":"Derives the amplitude-ratio relation $h \\sim h_{\\rm GR}\\,e^{-\\frac12\\int \\alpha_M H\\,d\\eta}$ used to interpret the damping.","marker":"[44]"}],"fun_headline_variants":["Teleparallel Gauss-Bonnet keeps gravitational waves at light speed","Gravitational waves speed stays c in teleparallel Gauss-Bonnet theory","Teleparallel Gauss-Bonnet gravity passes GW speed constraint","No speed deviation for gravitational waves in teleparallel Gauss-Bonnet"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that $T$ and $T_G$ expand to second order in the tensor perturbation into exactly the quadratic action (33) with coefficient $C_{\\rm tensor}$ and no additional $k^2 h^2$ terms; the paper states this outcome but the appendix lists only first-order torsion components, so the expansion is not shown.","fun_headline_variants_meta":{"raw":{"variants":["Teleparallel Gauss-Bonnet keeps gravitational waves at light speed","Gravitational waves speed stays c in teleparallel Gauss-Bonnet theory","Teleparallel Gauss-Bonnet gravity passes GW speed constraint","No speed deviation for gravitational waves in teleparallel Gauss-Bonnet"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000194,"raw_usage":{"total_tokens":1350,"prompt_tokens":938,"completion_tokens":412,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":338}},"tokens_in":554,"tokens_out":412,"duration_ms":4311,"temperature":1.0,"reasoning_tokens":338,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:08:13.621844+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct symbolic computation of $T^{(2)}$ and $T_G^{(2)}$ from the perturbed vierbein (31), followed by integration of the action (24), would settle the claim: if any term $h_{ij}\\nabla^2 h^{ij}$ or $h_{ij}h^{ij}$ survives with a coefficient other than $a^{-2}C_{\\rm tensor}$, then $\\alpha_T \\neq 0$ or a mass term is present.","supporting_citations":[{"cited_title":"Anisotropic stress as a signature of nonstandard propagation of gravitational waves,","cited_arxiv_id":null,"evidence_quote":"States the standard parametrized propagation equation with $\\alpha_M$ and $\\alpha_T$ against which equation (34) is matched."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The $f(T)$ tensor perturbation result with luminal speed that the present work extends to $F(T,T_G)$."},{"cited_title":"Reviving horndeski theory using teleparallel gravity after gw170817,","cited_arxiv_id":null,"evidence_quote":"Derives the amplitude-ratio relation $h \\sim h_{\\rm GR}\\,e^{-\\frac12\\int \\alpha_M H\\,d\\eta}$ used to interpret the damping."}],"review_version":1}