{"id":"d20e78cc-8f89-4c4c-a3a0-735e4ac9c374","arxiv_id":"1402.2059","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Inflationary models that fit the observed spectral index all predict a running of the tilt near -0.001, a value set mainly by the observed tilt and the assumed 50 to 60 e-folds.","lead":"The authors sort inflationary models into four families based on how the equation of state behaves at large numbers of e-foldings, and compute the running of the spectral index for each. They find that models matching Planck's spectral index all predict a running near minus one part in a thousand.","discovery_kind":"unclear","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"For the perturbative and logarithmic classes, α_s = (n_s−1)/N is an exact leading-order identity (Eqs. 19, 20, 35), so the abstract's 'Remarkably, we find' presents a kinematic consequence of fitting n_s ≈ 0.96 as an empirical discovery.","rationale":"The reader's weakest_assumption identifies the right soft spot: the clustering is largely set by the fit to n_s, not by the class structure. I would sharpen it slightly. For the perturbative and logarithmic classes, α_s = (n_s−1)/N is not merely 'roughly' true — it is an exact leading-order identity following from the 1/N expansion (Eqs. 19, 20, 35). The constant class gives α_s = 0, and the non-perturbative class gives α_s = −c r/4, even smaller. So all four classes satisfy |α_s| ≲ |n_s−1|/N by construction, and the observed value log₁₀|α_s| ≈ −3.2 is just −0.04/55. The paper is honest about this in Section IV, where it calls the observation 'purely phenomenological' and explains that models fitting Planck have Δn_s below about 0.01 over ΔN ≈ 10. The derivations themselves (Eqs. 8, 13, 19, 20, 30, 35) check out at leading order, and the classification is a genuine, if incremental, extension of Mukhanov 2013 and Roest 2013. The running calculation for all classes is new. The concern is therefore about framing, not correctness: the abstract's 'Remarkably, we find' oversells a kinematic consequence, and the claim that future permil-level probes could 'differentiate these models among themselves' is weaker than stated if α_s is largely fixed by n_s and N. A referee would reasonably ask for the abstract to be toned down and for the future-prospects sentence to be qualified. That is a minor revision, so CONDITIONAL is the right verdict, matching the reader's assessment. No code or data are provided, but the analysis is pencil-and-paper reproducible, and the paper's own Section III.C acknowledges the poor 1/N expansion for non-perturbative models, so that is not a new finding.","tokens_in":12711,"tokens_out":10094,"duration_ms":187163,"concrete_test":"For each model plotted in Fig. 3b, compute α_s exactly (e.g., Eq. 21 for chaotic inflation, Eq. 23 for inverse hilltop, Eq. 25 for Starobinsky, Eq. 33 for natural inflation) and compare to (n_s−1)/N at N = 55. If the two agree to within the half-decade spread shown in Fig. 3b for all perturbative and logarithmic models, the clustering is kinematic and the abstract's framing should be revised; if some class deviates by more than the spread, the clustering carries genuine class-dependent information.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim [abstract:s4] is that α_s clusters near −0.001 across all universality classes. For the perturbative class (Eqs. 19–20) and the logarithmic class (Eq. 35), the leading-order expressions give α_s = (n_s−1)/N exactly: p=1 gives n_s−1 = −(2ϵ_1+1)/N and α_s = −(2ϵ_1+1)/N²; p>1 gives n_s−1 = −p/N and α_s = −p/N²; logarithmic p=2 gives n_s−1 = −2/N and α_s = −2/N². The constant class has α_s = 0 (Eq. 16), and the non-perturbative class has α_s = −c r/4 (Eq. 30), even smaller. So for all four classes, |α_s| ≲ |n_s−1|/N by construction. With n_s ≈ 0.96 and N ≈ 55, this is ≈ −7×10⁻⁴, i.e., log₁₀|α_s| ≈ −3.2. The 'clustering' is therefore not a prediction of the class structure but a restatement of the fit to n_s. The paper acknowledges this in Section IV ('This is a purely phenomenological observation'), but the abstract's 'Remarkably, we find' does not. The consequence for the paper's stated motivation — that future permil-level probes 'could allow us to differentiate these models among themselves' — is also weakened: if α_s is fixed by n_s and N, measuring it more precisely mostly re-measures n_s and N rather than discriminating classes.","agreement_with_reader":"agree"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"The manuscript extends the large-N classification of inflationary models according to the behaviour of the equation-of-state parameter ϵ(N). Building on the perturbative 1/N^p case, the authors introduce constant, non-perturbative (ϵ ∝ e^{−2cN}), and logarithmic (ϵ ∝ ln^q N / N^p) classes, and express n_s, r, and α_s in terms of ϵ(N) through the 'N-formalism' of Eqs. (8) and (13). For each class they give representative potentials (power-law, chaotic, inverse hilltop, Starobinsky/T-models, hilltop, new, natural, Kähler moduli) and assess how well the leading large-N approximation reproduces the exact slow-roll result. Section IV then fixes each model's free parameter to match Planck's n_s and plots α_s against n_s in Fig. 3b, concluding that the running clusters near α_s ≈ −10^{−3} (log₁₀|α_s| ≈ −3.2) across all classes, with a stated spread of about half a decade. A short appendix treats k-inflation as a separate class with c_s ≠ 1.","tokens_in":13037,"tokens_out":10586,"duration_ms":187789,"significance":"The classification itself is a genuinely useful organizing framework: it gives a compact dictionary from asymptotic potential shape to large-N observables, and the leading-order expressions I checked (constant, p = 1, p > 1, non-perturbative, logarithmic) follow correctly from Eqs. (8) and (13). The paper also makes a falsifiable statement — that viable inflation predicts a running at the permil level, below Planck's sensitivity but within reach of proposed 21 cm or PRISM-class probes — which is the kind of concrete target that makes the classification worth having. The main caveat is that the headline clustering is closer to a kinematic consequence of fitting n_s than to an independent prediction of the class structure, as discussed below.","major_comments":[{"comment":"The abstract's 'Remarkably, we find that the tilt's runnings essentially cluster around the permil level' presents as an empirical discovery what is, at leading order, an identity. For the perturbative class, Eqs. (19)–(20) give α_s = (n_s − 1)/N exactly (p = 1: n_s − 1 = −(2ϵ₁ + 1)/N, α_s = −(2ϵ₁ + 1)/N²; p > 1: n_s − 1 = −p/N, α_s = −p/N²), and the logarithmic class, Eq. (35), gives the same relation. The constant class has α_s = 0 (Eq. 16) and the non-perturbative class has α_s = −c r/4 (Eq. 30), both even smaller. Hence |α_s| ≲ |n_s − 1|/N by construction for every class, and with n_s ≈ 0.96 and N ≈ 55 the value −7×10⁻⁴ follows without any model input beyond the fit. The authors acknowledge this in Section IV ('This is a purely phenomenological observation'), but the abstract and the Introduction do not. I recommend stating the identity α_s ≈ (n_s − 1)/N explicitly in Section IV and tempering the abstract's framing accordingly; the result remains worth reporting, but as a consistency relation rather than a surprise.","section":"Abstract, sentence 4; Section IV","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The claim that a permil-level measurement of α_s 'could allow us to differentiate these models among themselves' is not supported by the class structure. Since all four classes share α_s ≈ (n_s − 1)/N at leading order, a precise α_s measurement at fixed n_s primarily determines N rather than discriminating between classes; the spread visible in Fig. 3b arises from subleading terms and from scanning N over [50, 60], not from the leading class behaviour. The authors should either qualify this sentence or show explicitly which classes separate at permil precision and by how much, for example by quoting α_s at fixed N = 55 for each class.","section":"Section IV, final paragraph","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The text states that the clustering has 'a range of only half a decade', but the vertical axis of Fig. 3b spans log₁₀|α_s| from −4.0 to −2.6, i.e. about 1.4 decades, and the plotted models (including hybrid near the top) appear to cover more than half a decade. The quoted spread should be reconciled with the figure: state the actual range among models that survive the Planck n_s constraint, say whether hybrid inflation is included in that range, and give the numerical endpoints rather than a qualitative 'half a decade'.","section":"Section IV and Fig. 3b","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The sentence 'It is different from that of New inflation since α_s = −cr/4 is not satisfied in this case ∀N, since (ϵ′/ϵ)′ ≠ 0' is hard to parse. Eq. (30) presents α_s = −4cϵ₀e^{−2cN} = −cr/4 as the class prediction, so the intended point seems to be that natural inflation deviates from the pure exponential form at subleading order. Please rewrite this sentence to say so directly.","section":"Section III.C, paragraph on natural inflation","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"New inflation is described as 'almost ruled out' and then as lying 'just inside the 2σ contour allowed by Planck'. These two statements pull in opposite directions; please choose one characterization and state the margin explicitly.","section":"Section III.C, new inflation","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The non-perturbative row lists only the potential 1 − φ², but natural inflation with V ∝ 1 + cos(φ/μ) is also presented in Section III.C as a member of this class. Either add it to the table or note that the row shows a single representative.","section":"Table I, non-perturbative row","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The paper fixes each parameter to n_s = 0.96 but does not propagate the Planck uncertainty, n_s = 0.9603 ± 0.0073 (Eq. 1), into α_s. A one-line estimate, δα_s ≈ δn_s/N ≈ 1.3×10⁻⁴, would show that the clustering is robust against the observational error and would strengthen the section.","section":"Section IV","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The tensor running α_t = 2ϵ′ is defined but never discussed or evaluated for any class. Either comment briefly on its magnitude or drop it from the equation to avoid an unused definition.","section":"Eq. (13)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The claim that the four classes cover 'the large majority of inflationary models' is asserted rather than demonstrated. The paper gives a representative list, not a census; please soften the wording or justify it with a count of models in a standard compilation such as the Encyclopaedia Inflationaris cited as Ref. [4].","section":"Abstract, sentence 2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The displayed expression for the conformal time τ is difficult to parse as typeset, with nested integrals and exponents that do not read cleanly. Please check the typesetting and, if the expression is only meant to be schematic, say so.","section":"Eq. (10)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The appended note changes the emphasis of the paper (from Starobinsky-like to chaotic models) without altering the body text, which still describes chaotic p = 1 as 'almost ruled out'. For a journal version, either integrate the BICEP2 discussion into Sections I and IV or state clearly in the body which statements the note supersedes.","section":"Note added","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The classification and the leading-order algebra are sound and worth publishing. My reservation is that the abstract's headline ('Remarkably, we find') oversells a relation the body itself calls purely phenomenological; the authors are aware of the point, so this should be straightforward to fix by stating the identity α_s ≈ (n_s − 1)/N and adjusting the framing. The 'differentiate models' sentence in Section IV also needs qualification, since all classes share the same leading-order relation. I would be glad to see a revised version."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"The useful part of this paper is the classification, not the headline. Garcia-Bellido and Roest take the perturbative 1/N story of Mukhanov and Roest and add three more large-N behaviours: constant, non-perturbative, and logarithmic. The N-formalism write-up (Eqs. 8 and 13) is clean, the table is a handy reference, and the model-by-model mapping onto classes is done carefully. I checked the leading-order expressions for the constant, p=1, p>1, p=2, non-perturbative and logarithmic cases and they come out right. They are also honest where the expansion fails: Section III.C says plainly that for new and natural inflation the leading 1/N approximation is poor and the full expression is needed. The k-inflation appendix is a sensible addendum. That is real, if incremental, work, and it is the kind of thing I would point a student to when they ask which potentials land in which large-N bin. Now the soft spot, and it is the one the abstract leans on. For the perturbative and logarithmic classes, alpha_s = (n_s - 1)/N is an exact leading-order identity, not a consequence of the class structure: p=1 gives -(2eps_1+1)/N^2 against n_s-1 = -(2eps_1+1)/N, p>1 gives -p/N^2 against -p/N, and the logarithmic p=2 case is the same. The constant class has alpha_s = 0 and the non-perturbative class has alpha_s = -c r/4, smaller still. So |alpha_s| is bounded by |n_s-1|/N by construction, and with n_s = 0.96 and N around 55 you get log10|alpha_s| around -3.2 whether or not any universality class exists. The authors say as much in Section IV ('purely phenomenological observation') and give the Delta n_s argument, so they know. The abstract's 'Remarkably, we find' does not reflect that, and the closing claim that permil-level measurements 'could allow us to differentiate these models among themselves' is weaker than stated: if alpha_s is pinned by n_s and N, a better measurement mostly re-measures those two. Minor point: the BICEP2 framing has not aged well, though the classification stands without it. Recommendation: worth a referee report, with a request to soften the abstract and the discrimination claim. The physics is sound; the packaging oversells one equation.","headline":"A tidy, honest extension of the 1/N classification, but the headline 'clustering' of alpha_s is an algebraic identity dressed as a discovery.","tokens_in":13754,"tokens_out":1242,"would_cite":true,"duration_ms":23075,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["98.80.Cq","98.80.Es"],"model":"selfhosted/deepseek-v4.1-flash","headline":"Most inflation models that fit the measured spectral tilt predict a running of that tilt near one part in a thousand.","keywords":["inflation","spectral index running","slow-roll parameters","number of e-foldings","universality classes","cosmological perturbations","tensor-to-scalar ratio","primordial power spectrum"],"falsifier":"A measurement of the running that is robustly far from $-0.001$ — for instance a CMB-plus-large-scale-structure or 21 cm analysis settling near the Planck central value $alpha_s = -0.0134$ with small error bars — would break the claimed clustering, as would any model inside one of the four classes that fits $n_s$ and $r$ while predicting $|alpha_s|$ well above the permil level.","tokens_in":13975,"feed_emoji":"🌌","tokens_out":2604,"duration_ms":57511,"temperature":0.7,"pith_summary":"Inflation's observables are usually written as functions of $N$, the number of e-foldings between when a mode left the horizon and when inflation ended. Since $N$ is at least about 50, the authors argue that one should sort models by how the equation-of-state parameter $N$ behaves at large $N$, and they extend the earlier 1/N classification with three further classes: constant, exponentially falling (non-perturbative), and logarithmic. They then compute the running of the spectral tilt, $alpha_s$, for every class, after fixing each model's free parameter so that it reproduces the observed tilt $n_s = 0.96$ over $N$ between 50 and 60. The result they emphasise is that the running lands near $-0.001$ in every class, a spread of only about half a decade in $log_{10}|alpha_s|$, even though the tensor-to-scalar ratio $r$ ranges over several orders of magnitude across the same models. If that holds, no near-future CMB experiment short of a dedicated survey can separate these classes by their running, and the running is not a useful discriminator of inflation models.","feed_headline":"Viable inflation predicts a tilt running near one in a thousand","feed_subtitle":"Pin each model to the measured spectral tilt, and the running of that tilt lands at about -0.001 across four model classes.","key_machinery":"The carrying device is the 'N-formalism': the whole inflationary background is specified by the equation-of-state parameter $epsilon(N)$, with the Hubble slow-roll parameters recovered as $delta = epsilon + epsilon'/(2epsilon)$ and $xi = 3epsilon'/2 + epsilon^2 + (epsilon'/epsilon)'/2$, and the observables as $n_s - 1 = -2epsilon + epsilon'/epsilon$, $r = 16epsilon$, and $alpha_s = 2epsilon' - (epsilon'/epsilon)'$. Because everything reduces to one function of $N$, models can be sorted by the large-$N$ asymptotics of that function into four classes: constant ($epsilon = epsilon_0$), perturbative ($epsilon = epsilon_p/N^p$, with $p=1$ giving chaotic inflation and $p=2$ giving Starobinsky/T-models), non-perturbative ($epsilon = epsilon_0 e^{-2cN}$, covering new and natural inflation), and logarithmic ($epsilon = epsilon_p ln^q N / N^p$, covering Kähler moduli inflation). The classification is the paper's organising structure; the running calculation is then a short derivative away in each case.","core_discovery":"The paper's central claim is a clustering: take the models that survive the Planck constraint on the scalar tilt, fix their parameters so that $n_s = 0.96$ at $N$ between 50 and 60, and the running of the tilt comes out around $alpha_s = -0.001$ ($log_{10}|alpha_s|$ near $-3.2$) regardless of which large-$N$ universality class the model belongs to. This is stated in the abstract as the tilt's runnings clustering 'around the permil level' and displayed in the paper's Fig. 3b, which plots $n_s$ against $log_{10}(-alpha_s)$ for chaotic, Starobinsky, natural, brane, log, Kähler, T-model, moduli, hybrid and power-law cases. The contrast the authors draw is with the tensor-to-scalar ratio: in Fig. 3a the same models spread over several orders of magnitude in $r$, so the running is far more uniform than the amplitude of gravitational waves. The authors are careful to say this is a phenomenological observation rather than a theorem, and they explain it by noting that models inside the Planck contour vary the tilt by less than about $0.01$ across $Delta N$ near 10, which forces $alpha_s$ to be of order $(n_s-1)/N$.","pith_inferences":["The clustering can be read as a consistency check on the fitting procedure rather than a prediction of the class structure: if every model is pinned to the same tilt over the same narrow range of $N$, a running of order $(n_s-1)/N$ is close to guaranteed, so the half-decade spread is the informative part, not the central value.","The quoted Planck value $alpha_s = -0.0134 \\pm 0.0090$ is roughly an order of magnitude larger in magnitude than $-0.001$ while still consistent within about one and a half sigma; a future measurement settling near $-0.013$ would put the clustering under real pressure rather than merely refining it.","If the running really is near the permil level across classes, then a detection of a substantially larger running would be evidence for a model outside the four classes, which makes the running a background discriminator rather than a model discriminator."],"forward_implications":["Because $alpha_s$ sits near $-0.001$ for every class, measuring the running cannot by itself tell the universality classes apart; the tensor-to-scalar ratio remains the sharper discriminator.","Detecting a running at the permil level requires a probe with a much wider lever arm in scale than the CMB, which is why the authors point to 21 cm surveys such as the Square Kilometer Array.","Models in the non-perturbative class (new and natural inflation) are the exception where the leading large-$N$ approximation fails: the effective expansion parameter is $1/(cN)$, not $1/N$, so the full expression is needed even inside the Planck region.","Supergravity hybrid inflation, whose $epsilon(N)$ oscillates as $sin(kappa N/pi)$, falls outside the classification altogether, so the scheme covers most but not all models."],"supporting_citations":[{"why":"Supplies the Planck 2013 measurement of the spectral index and the running that the model parameters are fixed against.","marker":"[1]"},{"why":"The earlier classification of inflationary models by 1/N behaviour in Hubble slow-roll parameters that this paper extends.","marker":"[7]"},{"why":"The companion universality-class analysis in terms of potential slow-roll parameters, which the present paper deliberately departs from.","marker":"[8]"},{"why":"Starobinsky's model, the $p=2$ reference case whose $epsilon = 3/(4N^2)$ anchors the perturbative class.","marker":"[6]"},{"why":"Kallosh-Linde T-models, used as the generalisation within the $p=2$ class with an exact $epsilon(N)$ expression.","marker":"[16]"},{"why":"New inflation, one of the two worked examples of the non-perturbative class where the exponential form of $epsilon(N)$ is exact.","marker":"[18]"},{"why":"Natural inflation, the second non-perturbative example, used to show that $alpha_s = -c r/4$ does not hold identically.","marker":"[19]"},{"why":"Kähler moduli inflation, the model supplying the logarithmic class with $epsilon = c sqrt(ln N)/(2N^2)$.","marker":"[20]"},{"why":"The BICEP2 tensor-to-scalar ratio that motivates taking the large-$N$ expansion seriously and constrains the models in Fig. 3a.","marker":"[3]"}],"fun_headline_variants":["Four inflation classes, one running: tilt drift lands near -0.001","Inflation models cluster at a permil running rather than a spread","Running of the tilt stays near 0.001 across large-N classes","Tensor modes spread widely; the tilt's running does not","A phenomenological clustering at permil level, not a theorem"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The clustering is only as meaningful as the sample that produces it: the authors fix each model's free parameter to match the observed tilt over $N$ between 50 and 60, and they themselves call the resulting uniformity 'a purely phenomenological observation' rather than a consequence of the class structure.","fun_headline_variants_meta":{"raw":{"variants":["Four inflation classes, one running: tilt drift lands near -0.001","Inflation models cluster at a permil running rather than a spread","Running of the tilt stays near 0.001 across large-N classes","Tensor modes spread widely; the tilt's running does not","A phenomenological clustering at permil level, not a theorem"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":1377,"prompt_tokens":910,"completion_tokens":467,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":390}},"tokens_in":910,"tokens_out":467,"duration_ms":9103,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":390,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-10T08:28:33.311449+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"A measurement of the running that is robustly far from $-0.001$ — for instance a CMB-plus-large-scale-structure or 21 cm analysis settling near the Planck central value $alpha_s = -0.0134$ with small error bars — would break the claimed clustering, as would any model inside one of the four classes that fits $n_s$ and $r$ while predicting $|alpha_s|$ well above the permil level.","supporting_citations":[{"cited_title":"Planck 2013 results. XXII. Constraints on inflation","cited_arxiv_id":"1303.5082","evidence_quote":"Supplies the Planck 2013 measurement of the spectral index and the running that the model parameters are fixed against."},{"cited_title":"Quantum Cosmological Perturbations: Predictions and Observations","cited_arxiv_id":"1303.3925","evidence_quote":"The earlier classification of inflationary models by 1/N behaviour in Hubble slow-roll parameters that this paper extends."},{"cited_title":"Universality classes of inflation","cited_arxiv_id":"1309.1285","evidence_quote":"The companion universality-class analysis in terms of potential slow-roll parameters, which the present paper deliberately departs from."},{"cited_title":"Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations,","cited_arxiv_id":null,"evidence_quote":"Starobinsky's model, the $p=2$ reference case whose $epsilon = 3/(4N^2)$ anchors the perturbative class."},{"cited_title":"Universality Class in Conformal Inflation","cited_arxiv_id":"1306.5220","evidence_quote":"Kallosh-Linde T-models, used as the generalisation within the $p=2$ class with an exact $epsilon(N)$ expression."},{"cited_title":"A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,","cited_arxiv_id":null,"evidence_quote":"New inflation, one of the two worked examples of the non-perturbative class where the exponential form of $epsilon(N)$ is exact."},{"cited_title":"Natural inflation with pseudo - Nambu-Goldstone bosons,","cited_arxiv_id":null,"evidence_quote":"Natural inflation, the second non-perturbative example, used to show that $alpha_s = -c r/4$ does not hold identically."},{"cited_title":"Kahler Moduli Inflation","cited_arxiv_id":"hep-th/0509012","evidence_quote":"Kähler moduli inflation, the model supplying the logarithmic class with $epsilon = c sqrt(ln N)/(2N^2)$."},{"cited_title":"BICEP2 I: Detection Of B-mode Polarization at Degree Angular Scales","cited_arxiv_id":"1403.3985","evidence_quote":"The BICEP2 tensor-to-scalar ratio that motivates taking the large-$N$ expansion seriously and constrains the models in Fig. 3a."}],"review_version":1}