{"id":"c0ec7735-e5fc-44f4-bd59-cc0321e122ad","arxiv_id":"2502.05135","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"In the Einstein-Hilbert truncation on de Sitter, the Lorentzian FRG flow exhibits a non-Gaussian UV fixed point for ζ=1/2 and ζ=1 gauges over restricted parameter ranges.","lead":"This paper computes how Newton's constant and the cosmological constant run with scale for quantum gravity on a de Sitter, exponentially expanding, universe, using a Lorentzian functional renormalization group. It reports evidence for a UV fixed point in the most common gauges, supporting the asymptotic safety scenario for gravity, though the central computation depends on an unpublished companion paper.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The harmonic-gauge UV fixed point is not checkable: the ζ=1 graviton propagator and all beta functions are deferred to an in-preparation companion, so the central evidence rests on unpublished formulas.","rationale":"The reader's weakest assumption identifies the truncation and LPA as the main risk, with the companion paper mentioned as an additional issue. I find the more concrete and decisive gap to be the unavailability of the beta functions and, for the harmonic gauges, of the graviton propagator itself. The central claim is a numerical fixed-point result; its verification requires the fixed-point equations. Since they are explicitly withheld and the ζ=1 propagator is not even given, no independent check is possible from the text. The regulator q_k ∝ 1/G_N,k makes this more than a stylistic issue, because ∂_k q_k carries anomalous-dimension terms whose inclusion cannot be audited without the beta functions. This is not a demonstrated error, so I do not move the verdict away from CONDITIONAL; rather, the conditional status is reinforced. The proposed test, requiring explicit beta functions and reproducible numerics, would settle whether the harmonic-gauge fixed point is real or an artifact of the unpublished formulas.","tokens_in":33245,"tokens_out":9967,"duration_ms":109086,"concrete_test":"Obtain from the authors the explicit beta functions (for example in an ancillary file) for ζ=1, ξ=1, derived from the general-gauge propagator (3.54)–(3.56) with the masses (4.7), including all ∂_k q_k anomalous-dimension contributions; independently recompute the fixed point (g_*, λ_*) and the critical exponents shown in Figs. 9–10. If the fixed point persists with Re θ > 0, the concern is resolved. As a minimal internal check, repeat the same computation for ζ=1/2, ξ=1 using the explicit propagator (3.55) and verify that the fixed point reproduces Figs. 1–2; a mismatch would indicate an error in the projection or in the handling of η.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. 3.3 states that 'the full expressions for the propagator in a general gauge can be found in Ref. [74]', and Sec. 4.3 says the beta functions are 'long and unwieldy ... we do not report them explicitly'. For the harmonic gauges ζ=1, which include the standard ξ=1 case and where Figs. 9–16 provide the claimed UV fixed point, neither the graviton propagator nor the resulting β_g and β_λ is given. The fixed-point equations are therefore not independently checkable from this paper. This is load-bearing because the existence of the non-Gaussian fixed point and the positivity of the real part of the critical exponents is a numerical outcome of precisely those omitted expressions; an algebraic error in the ζ=1 tensor propagator or in the O(H²) projection would move or destroy the fixed point. Moreover, the regulator q_k in Eq. (4.6) is proportional to 1/G_N,k, so ∂_k q_k contains anomalous-dimension terms proportional to η; with the beta functions omitted, it cannot be verified that these terms were consistently included. This does not demonstrate an error, but it leaves the central claim conditional on unpublished material.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a Lorentzian, state-dependent functional renormalization group for quantum gravity on de Sitter spacetime. It constructs Hadamard Feynman propagators for massive scalar, vector and tensor fields in the Bunch-Davies vacuum, including a two-parameter gauge family, and uses their coincidence limits to project the FRGE onto the Einstein-Hilbert truncation. The flow is computed for the gauges ζ=1/2 and ζ=1 as a function of the gauge parameter ξ and the Hadamard scale α=ℓk; the numerical beta-function analysis yields non-Gaussian UV fixed points and complex critical exponents for a range of parameters. The authors conclude that this provides evidence for asymptotic safety of Lorentzian Einstein gravity in de Sitter space.","tokens_in":33448,"tokens_out":5306,"duration_ms":54187,"significance":"If the computation is correct, this is a substantial step: it is the first covariant Lorentzian FRG calculation on a cosmological background in the Einstein-Hilbert truncation, it treats state dependence and Hadamard subtraction explicitly, and it makes concrete statements about gauge and scheme dependence. The paper supplies explicit propagator coincidence limits for the ζ=1/2 case in Eqs. (4.10) and (4.11), and the qualitative comparison with the Euclidean Einstein-Hilbert fixed point is useful. The strength of the claim, however, is limited by the fact that the harmonic-gauge (ζ=1) beta functions and the general-gauge tensor propagator are deferred to an unpublished companion paper, so the main numerical evidence for the headline claim cannot be independently checked from the manuscript alone.","major_comments":[{"comment":"The central harmonic-gauge claim, Figs. 9-16, is not checkable from the manuscript. The ζ=1 graviton propagator is not given: Sec. 3.3 states that 'The full expressions for the propagator in a general gauge can be found in Ref. [74]', and Sec. 4.4 does not display the corresponding β_g and β_λ functions. The existence of the non-Gaussian fixed point and the sign of the real part of the critical exponents are numerical outputs of exactly those omitted expressions; an algebraic error in the O(H²) projection or in the tensor propagator would move or destroy the fixed point. The ζ=1 propagator coincidence limits, or the resulting beta functions, must be included or made available as an ancillary file before the central claim can be assessed.","section":"Sec. 3.3 and Sec. 4.4"},{"comment":"Even for the ζ=1/2 case, after displaying the propagator inputs (4.10) and (4.11), the paper states that the β functions are 'long and unwieldy ... we do not report them explicitly'. The flow diagrams and the PDE-continued fixed-point curves in Figs. 1-8 are therefore not reproducible from the text. Since the fixed point and critical exponents are the main quantitative results, at least one explicit β_g,β_λ pair for a representative choice of (ξ,α), or a supplementary file with all beta functions, is required.","section":"Sec. 4.3"},{"comment":"The interacting propagator entering the FRGE (2.13) is taken to be the free Feynman propagator with shifted masses (4.7), i.e., the local potential approximation is applied to the tensor sector without a dedicated check. Because the UV fixed point may be an artifact of the Einstein-Hilbert truncation, the manuscript should provide at least a qualitative estimate of the size of R² or Weyl² contributions on the right-hand side, or explain more precisely why the LPA is expected to capture the tensor sector.","section":"Sec. 2.1 and Sec. 4.2"},{"comment":"The regulator q_k in Eq. (4.6) is proportional to 1/G_N,k, so k∂_k q_k contains anomalous-dimension terms proportional to η. Because the beta functions are not reported, the reader cannot verify that these terms were consistently included in the numerical solution. The manuscript should state explicitly how η enters the right-hand side of Eq. (2.13) for this regulator, or display at least one beta function in which this dependence can be seen.","section":"Sec. 4.2, Eq. (4.6)"}],"minor_comments":[{"comment":"The definition 'η = ∂k lngk−2' appears to omit the factor k in the derivative; if η is the standard anomalous dimension, it should read η = k∂_k ln g_k − 2.","section":"Sec. 4.3"},{"comment":"The constant c′ in the vector propagator (3.20) is set to zero 'for simplicity'; since it is a free parameter of the construction, its effect on the flow should be mentioned at least briefly.","section":"Sec. 3.2"},{"comment":"The comparison with causal dynamical triangulations is only qualitative; the sentence that the results 'could be compared' is vague and no quantitative connection is established.","section":"Sec. 5"}],"recommendation":"major_revision","confidential_remarks":"The main concern is publication timing and checkability: Ref. [74] is in preparation and contains formulas on which the central ζ=1 claim depends. I would not reject the paper, as the framework is credible and the ζ=1/2 expressions are explicit, but the manuscript should not be accepted until the omitted beta functions and the ζ=1 propagator data are available to the reader."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is the first FRG flow computation for the Einstein–Hilbert truncation on de Sitter in the covariant Lorentzian pAQFT framework, and it introduces a genuinely missing ingredient—massive graviton and ghost Hadamard propagators that other de Sitter calculations can reuse. The claimed UV fixed point in common gauges is plausible, and the state dependence is explicit rather than hidden.\n\nWhat the paper does well: the Hadamard-subtracted coincidence limits in Secs. 3.4 and 3.5 are displayed in usable form for scalars, vectors, and tensors to order H^2. For ζ=1/2 the integrated right-hand side of the flow is given explicitly in Eqs. (4.10)–(4.11), which is enough to derive the beta functions even if the paper does not print them. The discussion of the Higuchi bound as a restriction on the flow, and the numerical breakdown near λ=1/2, is honest. The paper also does not oversell: it flags the proximity of the fixed point to the asymptote and the lack of IR-complete trajectories.\n\nThe soft spot is real and load-bearing. The harmonic-gauge section, which carries the paper's main claim, is opaque: the ζ=1 graviton propagator is not given, the beta functions are not given, and the fixed points and critical exponents appear only in figures. The stress-test concern lands. For ζ=1 the central numerical result is computed from expressions in the in-preparation companion [74], and the reader cannot verify the fixed-point equations independently. An algebraic slip in the tensor propagator or in the O(H^2) projection would move or destroy the fixed point. Moreover, the regulator q_k contains 1/G_N,k, so ∂_k q_k generates anomalous-dimension terms proportional to η; without the beta functions one cannot check that those terms were consistently included. These are gaps rather than demonstrated errors, but they are exactly where the main claim lives.\n\nThe truncation dependence is also unresolved. The LPA identification—interacting propagator as the free massive propagator with masses dressed by k^2 and Λ_k—is a standard but untested assumption; higher-curvature operators could shift or remove the fixed point. The authors acknowledge this in the discussion, which is to their credit.\n\nWho this is for: people working on Lorentzian asymptotic safety, de Sitter QFT, and FRG methods. It deserves a serious referee, and I would send it to peer review rather than desk reject. For my own citation habits over the next year, though, I would wait until the companion appears or the beta functions are placed in a supplement; citing the fixed point without being able to check it is uncomfortable.","headline":"First de Sitter FRG flow in covariant Lorentzian asymptotic safety with genuinely reusable massive Hadamard propagators, but the harmonic-gauge fixed point the paper advertises rests on formulas parked in an unpublished companion.","tokens_in":34027,"tokens_out":2188,"would_cite":false,"duration_ms":23281,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"On de Sitter spacetime, the Lorentzian renormalization group flow of Einstein gravity possesses a non-Gaussian ultraviolet fixed point for the standard gauges, evidence for asymptotically safe quantum gravity.","keywords":["asymptotic safety","de Sitter spacetime","functional renormalization group","Lorentzian quantum gravity","de Sitter-invariant Hadamard state","Einstein-Hilbert truncation","Higuchi bound","Callan-Symanzik regulator"],"falsifier":"Extend the effective average action by an $R^2$ or Weyl-squared coupling and search for a simultaneous non-Gaussian fixed point of $(g,\\lambda,a_2)$ with the same state and regulator; if no such fixed point exists, or if the Einstein-Hilbert fixed point moves discontinuously, the reported UV completion is a truncation artifact rather than a property of the full theory.","tokens_in":1900,"feed_emoji":"🌌","tokens_out":2280,"duration_ms":101660,"temperature":0.7,"pith_summary":"This paper derives, for the first time, the functional renormalization group flow of Lorentzian Einstein gravity on a de Sitter background, in the Einstein-Hilbert truncation and in the de Sitter-invariant Hadamard vacuum state. It shows that, for the two most common gauge families, $\\zeta=\\frac12$ and harmonic gauges with $\\zeta=1$, the flow possesses a non-Gaussian ultraviolet fixed point over a range of the Hadamard-scale parameter $\\alpha$ and gauge parameter $\\xi$, with complex critical exponents whose real part is positive for standard choices such as $\\xi=1$, $\\alpha\\approx0.4$. The authors take this as evidence that asymptotically safe quantum gravity is realized in de Sitter spacetime, a key cosmological background, and that the state dependence inherent to Lorentzian quantum gravity can be handled covariantly. The computation relies on new massive scalar, vector, and tensor Feynman propagators in de Sitter that satisfy the Hadamard condition.","feed_headline":"De Sitter gravity gains a UV fixed point in Lorentzian flow","feed_subtitle":"First complete de Sitter RG flow in the Einstein-Hilbert truncation finds a non-Gaussian fixed point for common gauges.","key_machinery":"The engine is the Lorentzian functional renormalization group equation (2.13) with a mass-like Callan-Symanzik regulator, combined with the local potential approximation, in which the interacting propagator is taken to be the free Feynman propagator of massive fields with masses $m^2=k^2+2(3H^2-\\Lambda_k)$ and $M^2=k^2(3-2/\\xi\\zeta^2)+2(3H^2-\\Lambda_k)$. The right-hand side is the coincidence limit of the Hadamard-normal-ordered propagator of the graviton and ghost fields, evaluated with the newly derived de Sitter-invariant Hadamard propagators for massive scalars, vectors, and tensors. Hadamard subtraction removes the universal UV divergences and introduces the scale parameter $\\alpha=\\ell k$, while the vacuum state fixes the quantum-state dependence. Comparing coefficients of order $H^0$ and $H^2$ on both sides of the flow equation yields the $\\beta$ functions of $g_k=G_{N,k}k^2$ and $\\lambda_k=\\Lambda_k/k^2$.","core_discovery":"The paper's central claim is that the non-perturbative RG flow for Lorentzian quantum gravity in de Sitter space, evaluated in the Einstein-Hilbert truncation with the de Sitter-invariant Hadamard vacuum, admits a non-trivial UV fixed point $(g_*,\\lambda_*)$ for the gauges $\\zeta=\\frac12$ and $\\zeta=1$. For $\\xi=1$ and $\\alpha\\approx0.4$, the critical exponents are complex conjugates with positive real part, so the fixed point attracts the flow in the UV; the authors interpret this as evidence for the UV completion of gravity in de Sitter space. The flow is constrained by the Higuchi bound, which forces $\\lambda_k\\le\\frac12$ and creates an asymptote near which numerical evaluation becomes delicate. The fixed point's location depends on the Hadamard scale $\\alpha=\\ell k$, but the critical exponents vary only mildly and remain close to the known Euclidean results for the most common gauges.","pith_inferences":["If the fixed point survives higher-order truncations, one could search for a function $\\alpha(\\xi)$ that makes all critical exponents exactly gauge-independent; the figures in the paper suggest this is a concrete numerical project.","The Hadamard-subtraction technique could be applied to other homogeneous spacetimes, such as FLRW with a scalar source, connecting asymptotic safety directly to inflationary cosmology.","A potential observational consequence, if an IR fixed point is eventually identified, is a prediction for the scale dependence of the effective cosmological constant that could be compared with CMB and large-scale-structure data; the paper gestures at this but does not compute it.","A different choice of vacuum state could move the fixed point away from the Higuchi asymptote, making global UV-to-IR trajectories numerically accessible; testing this is a natural next step within the same framework."],"forward_implications":["If correct, asymptotic safety is realized in a Lorentzian, cosmological setting, not only in Euclidean signature, and a de Sitter-invariant Hadamard vacuum provides a viable state for the UV-complete graviton.","The Higuchi bound $\\lambda_k\\le\\frac12$ is a physical obstruction to global RG trajectories in this truncation; including matter fields or higher-order terms may lift it and allow IR-complete flows.","The state dependence of the flow means different vacua can change the phase diagram, so asymptotic safety must be assessed state by state in curved spacetimes.","The gauge and Hadamard-scale dependence of the fixed point can in principle be minimized by choosing $\\alpha=\\alpha(\\xi,\\zeta)$, yielding approximately scheme-independent critical exponents.","The same covariant Lorentzian flow equations can be applied to other backgrounds, such as anti-de Sitter or FLRW with a scalar field, and to gauge-invariant cosmological observables."],"supporting_citations":[{"why":"It derives the Lorentzian Wetterich equation in the algebraic QFT framework on which this paper's flow equation is based.","marker":"[24]"},{"why":"It is the first application of the Lorentzian RG flow to quantum gravity, supplying the background-independent contributions that this paper extends to the de Sitter background.","marker":"[26]"},{"why":"It is Reuter's functional renormalization group equation for quantum gravity, the starting point of the flow used here.","marker":"[14]"},{"why":"It is the companion paper that supplies the explicit general-gauge propagator formulas for massive gravitons and ghosts in de Sitter, on which the flow computation relies.","marker":"[74]"},{"why":"It constructs the covariant graviton two-point function in de Sitter by canonical quantization, providing a consistency check for the new propagators.","marker":"[69]"},{"why":"It classifies vacuum states in de Sitter and fixes the de Sitter-invariant Hadamard state selected in this paper.","marker":"[50]"},{"why":"It establishes Higuchi's forbidden mass range for spin-2 fields in de Sitter, which produces the $\\lambda_k\\le 1/2$ asymptote that shapes the flow.","marker":"[72]"},{"why":"It is the Euclidean Einstein-Hilbert truncation flow of Reuter and Saueressig, the reference result to which the Lorentzian fixed points and critical exponents are compared.","marker":"[10]"}],"fun_headline_variants":["Lorentzian de Sitter gravity reveals UV fixed point","De Sitter spacetime flow reaches UV fixed point","Quantum gravity in de Sitter gets UV fixed point","Asymptotic safety found in Lorentzian de Sitter","UV fixed point emerges in de Sitter gravity flow"],"cache_read_input_tokens":36096,"weakest_assumption_plain":"The flow is computed entirely in the Einstein-Hilbert truncation with the interacting propagator approximated by the free massive Feynman propagator; if curvature-squared or higher-order operators, or corrections beyond this local-potential approximation, contribute significantly, the ultraviolet fixed point could be an artifact of this truncation.","fun_headline_variants_meta":{"raw":{"variants":["Lorentzian de Sitter gravity reveals UV fixed point","De Sitter spacetime flow reaches UV fixed point","Quantum gravity in de Sitter gets UV fixed point","Asymptotic safety found in Lorentzian de Sitter","UV fixed point emerges in de Sitter gravity flow"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"total_tokens":1351,"prompt_tokens":842,"completion_tokens":509,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":458,"completion_tokens_details":{"reasoning_tokens":431}},"tokens_in":458,"tokens_out":509,"duration_ms":5034,"temperature":1.0,"reasoning_tokens":431,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:07:26.732750+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Extend the effective average action by an $R^2$ or Weyl-squared coupling and search for a simultaneous non-Gaussian fixed point of $(g,\\lambda,a_2)$ with the same state and regulator; if no such fixed point exists, or if the Einstein-Hilbert fixed point moves discontinuously, the reported UV completion is a truncation artifact rather than a property of the full theory.","supporting_citations":[],"review_version":1}