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Cited by 143Cosmology and Gravitation TheoriesNoncommutative and Quantum Gravity TheoriesGalaxies: Formation, Evolution, Phenomena

Conditions for large non-Gaussianity in two-field slow-roll inflation

Open access
Byrnes, Christian T, Choi, Ki-Young, Hall, Lisa M H·2008·Journal of Cosmology and Astroparticle Physics
DOI 10.1088/1475-7516/2008/10/008
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Abstract

We study the level of primordial non-Gaussianity in slow-roll two-field inflation. Using an analytic formula for the non-linear parameter f NL in the case of a sum or product separable potential, we find that it is possible to generate significant non-Gaussianity even during slow-roll inflation with Gaussian perturbations at Hubble exit. In this paper we give the general conditions for obtaining large non-Gaussianity and calculate the level of fine-tuning required to obtain this. We present explicit models in which the non-Gaussianity at the end of inflation can exceed the current observational bound of .

AI snapshot

Written from the paper’s full text.

TL;DR

The paper derives general conditions for large primordial non-Gaussianity in two-field slow-roll inflation with sum or product separable potentials, using analytic δN formulas. It shows significant \(f_{\rm NL}\) can arise even when field perturbations are Gaussian at Hubble exit, but only for finely tuned initial conditions and specific trajectory-angle evolution.

Key findings

  • Large non-Gaussianity can be generated during slow-roll inflation without violating slow roll; explicit models can exceed the current observational bound \(|f_{\rm NL}| \lesssim 100\). Example A, \(W=\frac12 e^{-\lambda\phi^2/M_P^2}m^2\chi^2\), gives \(f_{\rm NL}\sim 100\) for \(\lambda=0.04,0.05\) and \(\phi_*\sim 10^{-3}M_P\) or \(10^{-4}M_P\).
  • The general condition requires the background trajectory angle to grow by about two orders of magnitude or more during inflation; for typical slow-roll parameters, the initially subdominant field must satisfy \(\cos^2\theta_*=\epsilon^\phi_*/\epsilon_*\lesssim 10^{-7}\). If observations find \(f_{\rm NL}\sim 10\) with \(\eta\sim 0.01\), one needs \(f_p\) or \(g_p\sim 1000\).
  • A quadratic product potential \(W=\frac12 m^2\phi^2\chi^2\) cannot generate \(|f_{\rm NL}|>1\), because \(\sin^2\theta\) decreases during inflation.
  • For hybrid inflation, Table I lists examples with \(f_{\rm NL}=-42, 9.27, -11.1\), spectral index \(n_\zeta-1=0, 0.09, -0.02\), and tensor-to-scalar ratio \(r=0.006, 0.10, 0.026\); the first example has a scale-invariant spectrum and small \(r\).

Methods

  • Theoretical analysis of two-field slow-roll inflation using the δN formalism and analytic formulas for \(f_{\rm NL}\) in product and sum separable potentials, following Vernizzi–Wands and Choi–Hall–van de Bruck.
  • Derives conditions by analysing auxiliary functions over allowed trajectory angles \(\theta_*,\theta_e\), for product potentials \(W=U(\phi)V(\chi)\), sum potentials \(W=U(\phi)+V(\chi)\), and an extension of the product case to non-canonical kinetic terms.
  • Presents explicit examples: quadratic times exponential potential, quadratic product potential, and two-field hybrid inflation; for Example A, numerically solves the full equations of motion until \(\epsilon=1\) and compares the analytic evolution of \(f_{\rm NL}\). No observational data sample is used; the WMAP bound \(|f_{\rm NL}|\lesssim 100\) is referenced.

Limitations

  • The paper states that large non-Gaussianity always requires fine-tuning of initial conditions; for typical slow-roll parameters, \(\cos^2\theta_*\lesssim 10^{-7}\), and for product potentials the final trajectory angle must lie in a narrow range.
  • The paper states that if a field value is too small, quantum fluctuations can invalidate the classical trajectory; loop corrections may not be negligible and are not generally included; the analytic formula breaks down when slow roll breaks down, e.g. during the fast-roll stage in Example A.
  • The paper notes that the local \(f_{\rm NL}\) bound may not be applicable to the calculated \(f_{\rm NL}\) when multiple field directions contribute, and that non-canonical sum potentials are difficult to analyse, so only the product case is partially treated.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1088/1475-7516/2008/10/008

Cites

83 of 83 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

14 of 16 cited sources checked · 5 supported · 0 flagged · accuracy not yet measured

  • Interpretationreasoning
    § A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 10

    In Example A, the potential is dominated by the quadratic potential of the χ field, corrected by -1/2 λ m^2 φ^2 χ^2, which might be important for preheating.

    “In this slow-roll limit, the exponential function can be expanded so that the potential is dominated by the quadratic potential of χ fi eld, 1 2 m 2 χ 2 , and corrected by the integration between χ and ϕ fi eld, -1 2 λm 2 ϕ 2 χ 2 , which might be important for preheating [38, 39].”
    Quote in the paperNot a substantial claim2 of 2 sources checked
    • cites [38] Seery D, 2007, J. Cosmol. Astropart. Phys. source

      Section 5 discusses quadratic approximations to axion potentials, but does not describe Example A or a χ–ϕ interaction. Searches for the proposed interaction and preheating found no passage in our copy stating the claimed expansion or its significance for preheating. (LLM, conf=0.74)

      Experimental — accuracy not yet measured
    • cites [39] source

      The numerical investigation (p. 5) describes a two-field quadratic potential, but does not identify a χ-field potential dominated by ½m²χ². Searches for the exponential expansion, the −½λm²φ²χ² interaction, and preheating found no passage addressing those parts of the claim. (LLM, conf=0.74)

      Experimental — accuracy not yet measured
  • Findingown data
    § C. Hybrid inflation · p. 12

    We find a region with much larger non-Gaussianity than was found in [5].

    “We therefore find a region with much larger non-Gaussianity than was found in [5].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [5] Alabidi L, 2006, J. Cosmol. Astropart. Phys. source
  • Premisecitation
    § C. Hybrid inflation · p. 12

    The condition to neglect quantum fluctuations is |χ̇|π/H^2 > √3/2.

    “In order that we can neglect the effect of the quantum fluctuations, the condition we require on the background trajectory is | ˙ χ | π/H 2 > √ 3 / 2 [40].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [40] Yokoyama S, 2007, J. Cosmol. Astropart. Phys. source

      The background-equations section (p. 3) discusses neglecting the non-Gaussian contribution of initial field perturbations, not a condition for neglecting quantum fluctuations along a background trajectory. Searches for quantum fluctuations, χ̇, and √3/2 found no passage stating the claimed inequality; equations in our copy are not decoded. (LLM, conf=0.72)

      Experimental — accuracy not yet measured
  • Premisecitation
    § VI. NON-GAUSSIANITY WITH NON-CANONICAL KINETIC TERMS (PRODUCT POTENTIALS) · p. 13

    With a product potential, local non-Gaussianity is given by [21].

    “With a product potential, the local non-Gaussianity is given by [21]:”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [21] Enqvist K, 2005, J. Cosmol. Astropart. Phys. source

      Section 2 (p. 4) gives a non-linearity parameter for the curvaton model, and Section 3 (p. 5) specifies a nearly quadratic potential with a small correction. The copy does not establish that this expression applies to a product potential, so the potential specified in the citing sentence is unsupported. (LLM, conf=0.82)

      Experimental — accuracy not yet measured
  • Study design and conductown data
    § C. Hybrid inflation · p. 13

    For the explicit values in Table I, the loop correction does not dominate.

    “We have checked that for the explicit values given in the table the loop correction does not dominate (under the assumption that the same loop correction is dominant here as the one in [41]).”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [41] source

      The cited paper’s section on the approximate analytical expression for fNL gives example parameter values, but it does not identify them as Table I values or assess whether a loop correction dominates. Searches for Table I and loop-correction dominance found no passage addressing that check. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • The authors state this as an assumption: The same loop correction is dominant here as the one in [41].
  • May combine two claims. 17
  • Classified as probably not a substantial claim. 19
  • The quote only partly states this claim. 2

  • May combine two claims.

    Interpretationown data
    § C. Hybrid inflation · p. 12

    Our analysis of the parameter space is more extensive than [5] and allows for very small initial field values.

    “Our formula for f (4) NL is also consistent with the calculation of [5] but our analysis of the parameter space is more extensive and in particular it allows for very small initial field values.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [5] Alabidi L, 2006, J. Cosmol. Astropart. Phys. source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingcitation
    § C. Hybrid inflation · p. 12

    Our formula for f_NL^(4) is consistent with the calculation of [5].

    “Our formula for f (4) NL is also consistent with the calculation of [5] but our analysis of the parameter space is more extensive and in particular it allows for very small initial field values.”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [5] Alabidi L, 2006, J. Cosmol. Astropart. Phys. source

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § C. Hybrid inflation · p. 13

    In the 2-field hybrid model with the same potential as Eqn. (67), an unstable straight trajectory, and α and β both negative, for certain initial values the loop correction is dominant and can generate an observable f_NL.

    “They consider a 2-field hybrid model with the same potential as Eqn. (67) in the special case of an unstable straight trajectory along one of the axes, with α and β both negative. In this case, they find (for certain initial values) the loop correction is dominant and can generate an observable f NL .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [41] source

      Section III studies a two-field double-inflation model with a quadratic potential, not the specified hybrid model. Searches for a hybrid potential, negative α and β, an unstable straight trajectory, and a dominant loop correction found no passage addressing those conditions. The paper’s small final f_NL result concerns its different model, so it does not contradict the claim about the hybrid model. (LLM, conf=0.78)

      Experimental — accuracy not yet measured

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § C. Hybrid inflation · p. 13

    [41] has shown the first explicit example of an inflation model where higher-order terms cannot be neglected.

    “These 'higher order' terms are usually neglected, but [41] has shown the first explicit example of an inflation model where they cannot be neglected.”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [41] source

      Section III describes an explicit double-inflation model in which slow-roll conditions are temporarily violated. But the paper does not call it the first such example or show that the unspecified higher-order terms cannot be neglected; its Summary instead says the final non-Gaussianity remains suppressed. (LLM, conf=0.78)

      Experimental — accuracy not yet measured

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § C. Hybrid inflation · p. 12

    Cogollo et al. have calculated the effect of the loop correction to the primordial power spectrum and bispectrum.

    “Very recently Cogollo et al. have calculated the effect of the loop correction to the primordial power spectrum and bispectrum [41].”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [41] source

      The Introduction and the section on the nonlinear parameter discuss primordial non-Gaussianity, the power spectrum, and the bispectrum, but do not identify a calculation by Cogollo et al. Searches for Cogollo and for loop corrections to the power spectrum and bispectrum found no passage addressing the attributed claim in our copy. (LLM, conf=0.72)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Findingnone given
    § A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 10

    Around the region for large |f_NL|, the curvature perturbation of the φ-field becomes comparable to, and then dominates, that of the χ-field.

    “Note that, around the region for large | f NL | , the curvature perturbation of ϕ -field becomes comparable to, and then dominates, that of the χ -field.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § IV. LARGE NON-GAUSSIANITY IN SUM POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) + V ( χ ) · p. 8

    There are two regions with potentially large non-Gaussianity for a sum separable potential: Region C with u≪1 and cos2θ*≪1, and Region D with v≪1 and sin2θ*≪1.

    “Once again there are two regions with potentially large non-Gaussianity: C u ≪ 1, and cos 2 θ ∗ ≪ 1 D v ≪ 1, and sin 2 θ ∗ ≪ 1.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data
    § A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 10

    The analytic constraints differ slightly from the plots because of the value of ηφφ used.

    “We note that the analytic constraints differ slightly from the plots, due to the value of η ϕϕ used.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: In the analytics, ηϕϕ ∼ -2λ is assumed.

  • May combine two claims.

    Findingown data
    § III. LARGE NON-GAUSSIANITY IN PRODUCT POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) V ( χ ) · p. 4

    The function jp(θ*, θe) satisfies 0 ≤ jp ≤ 1, so the j-term can never lead to significant values of f(4)NL.

    “We find that the function j p ( θ ∗ , θ e ) satisfies 0 ≤ j p ≤ 1 and therefore the 'j'-term can never lead to significant values of f (4) NL .”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data
    § C. Hybrid inflation · p. 12

    Table I gives explicit examples of values of α, β, φ*, χ* that lead to large non-Gaussianity.

    “In Table I, we give some explicit examples of values of α, β, ϕ ∗ and χ ∗ which lead to a large non-Gaussianity.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingreasoning
    § A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 10

    In Example A, constraints on the initial values of the field φ can be obtained from Eqn. (36).

    “From Eqn. (36), the constraints on the initial values of the field ϕ can be obtained:”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given
    § III. LARGE NON-GAUSSIANITY IN PRODUCT POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) V ( χ ) · p. 6

    The spectral index is changed by the presence of ̃r but remains slow-roll suppressed.

    “The spectral index is changed by the presence of ˜ r , but remains slow roll suppressed.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given
    § A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 10

    The full formula Eqn. (44) is almost the same as Eqn. (63).

    “using the full formula Eqn. (44), which is almost same as Eqn. (63).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 10

    For λ=0.04, f_NL^(4) can be very large when φ* ∼ 10^-3 M_P; for λ=0.05, when φ* ∼ 10^-4 M_P.

    “Here it can be seen that f (4) NL can be very large when ϕ ∗ ∼ 10 -3 M P ( λ = 0 . 04) or ϕ ∗ ∼ 10 -4 M P ( λ = 0 . 05).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § I. INTRODUCTION · p. 2

    In the models considered, non-Gaussianity is large over many e-foldings of inflation and all slow-roll parameters remain smaller than unity.

    “In the models we consider the non-Gaussianity is large over many e -foldings of inflation and all of the slow roll parameters remain smaller than unity.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § IV. LARGE NON-GAUSSIANITY IN SUM POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) + V ( χ ) · p. 8

    The function gs has exactly the same form as gp for the product potential in the region where it is large, so it is large in exactly the same areas.

    “The function g s has exactly the same form as the function g p for the product potential in the region where it is”
    • Also stated in: § IV. LARGE NON-GAUSSIANITY IN SUM POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) + V ( χ ) · p. 8 · plain
      “and large, see Eqn. (28) replacing sin 2 θ e with v . Hence it is large in exactly the same areas, and for g s ≳ G s we require”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § III. LARGE NON-GAUSSIANITY IN PRODUCT POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) V ( χ ) · p. 6

    The spectrum of curvature perturbations can be dominated by fluctuations in the slow-rolling field φ or χ depending on ̃r, which can be smaller or bigger than unity.

    “The spectrum of curvature perturbations can be dominated by the fluctuations in the slow-rolling field ϕ or χ depending on ˜ r , which can be smaller or bigger than unity.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § VII. CONCLUSION · p. 15

    Large non-Gaussianity in the hybrid model is possible either if the η's have opposite sign or if both η's have the same sign.

    “We find this is possible either if the η 's have the opposite sign (so that inflation takes place near a saddle point) or when both of the η 's have the same sign.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingnone given
    § VI. NON-GAUSSIANITY WITH NON-CANONICAL KINETIC TERMS (PRODUCT POTENTIALS) · p. 14

    If X=0, then α=1 and Fp=fp, Gp=gp, Hp=hp and Jp=jp.

    “If X = 0, then α = 1 and F p = f p , G p = g p , H p = h p and J p = j p .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 11

    Reheating is expected to take place in almost the same way as in normal chaotic inflation in this regime.

    “We therefore expect reheating to take place in almost the same way as it would in normal chaotic inflation.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

    Study design and conductown data
    § C. Hybrid inflation · p. 12

    The product potential formalism gives the same results as the sum potential analysis.

    “We have checked that this gives the same results.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § C. Hybrid inflation · p. 12

    The product potential formalism W=W0 exp(αφ^2/M_P^2) exp(βχ^2/M_P^2) is equivalent to the sum potential Eqn. (67) in the limit of vacuum domination.

    “This is equivalent to the sum potential Eqn. (67) in the limit of vacuum domination ( | α | ϕ 2 /M 2 P ≪ 1 and | β | χ 2 /M 2 P ≪ 1).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § IV. LARGE NON-GAUSSIANITY IN SUM POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) + V ( χ ) · p. 9

    When Ve≃V*, which can be true in many models because sin2θ*≃(Vχ/Uφ)^2≪1, the potential V is extremely flat at least initially.

    “We may analyse this equation for the specific case when V e ≃ V ∗ , which can be true in many models because sin 2 θ ∗ ≃ ( V χ /U ϕ ) 2 ≪ 1, so at least initially the potential V is extremely flat.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Interpretationreasoning
    § A. Condition for Large Non-Gaussianity in Sum Potentials · p. 9

    To get large non-Gaussianity in sum potentials, one needs a constraint on the function gs for Region C and Region D.

    “From the previous section, it is clear that, to get large non-Gaussianity, we need a constraint on the function g s , for Region C,”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 11

    In the regime of large non-Gaussianity, the background trajectory must be almost entirely in the one-field direction.

    “We note, for reference, that in this regime of large non-Gaussianity, the background trajectory must be at almost entirely in the one-field direction.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Study design and conductown data
    § A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 10

    During the fast-roll stage around N=58, slow-roll breaks down and the analytic formula is not valid.

    “However during this fast-roll stage, slow-roll breaks down and our analytic formula is not valid any more.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationnone given
    § B. Direct Observation of f (4) NL and Fine-Tuning · p. 7

    Eqns. (36) and (37) encode the level of fine-tuning required on the initial and final slow-roll parameters to obtain an observable level of non-Gaussianity.

    “Eqns. (36) and (37) encode the level of fine-tuning required on the initial and final slow-roll parameters to obtain an observable level of non-Gaussianity.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: |ηϕϕ| ≫ cos2θe |ηχχ - 4ǫe|.
    • The authors state this as an assumption: |ηχχ| ≫ sin2θe |ηϕϕ - 4ǫe|.

  • Classified as probably not a substantial claim.

    Interpretationnone given
    § VI. NON-GAUSSIANITY WITH NON-CANONICAL KINETIC TERMS (PRODUCT POTENTIALS) · p. 13

    For a sum potential, non-canonical modifications are non-trivial and difficult to analyse.

    “In the case of the sum potential, the modifications are non-trivial and are difficult to analyse.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationnone given
    § VI. NON-GAUSSIANITY WITH NON-CANONICAL KINETIC TERMS (PRODUCT POTENTIALS) · p. 13

    For a product potential, previous analysis can be partially used.

    “When the potential is of product form, however, we may at least partially use the previous analysis.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § VII. CONCLUSION · p. 14

    For a sum separable potential, the initial background trajectory must be similarly fine-tuned to be nearly parallel to one of the inflaton axes, but the tuning at the end of inflation is not written so simply.

    “The analysis for a sum separable potential is very similar but more complicated to analyse. The initial background trajectory must be similarly fine-tuned to be nearly parallel to one of the axes of the inflaton fields, but the tuning at the end of inflation is not written so simply.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § VII. CONCLUSION · p. 14

    This requirement means the field trajectory is almost exactly parallel to the χ axis initially and typically requires a finely tuned initial condition.

    “This means that the field trajectory is almost exactly parallel to the χ axes initially and typically this requires a finely tuned initial condition.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § VI. NON-GAUSSIANITY WITH NON-CANONICAL KINETIC TERMS (PRODUCT POTENTIALS) · p. 13

    Our findings for product potential relate also to sum potentials.

    “We therefore concentrate only on a product potential, but our findings relate also to sum potentials.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given
    § VI. NON-GAUSSIANITY WITH NON-CANONICAL KINETIC TERMS (PRODUCT POTENTIALS) · p. 14

    We require θ* ∼ 0, π/2 as before.

    “Note, though, that we require θ ∗ ∼ 0 , π 2 as before.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § VII. CONCLUSION · p. 15

    The large scale loop correction may not be negligible.

    “There is also the possibility that the large scale loop correction which we have not generally considered in this paper may not be negligible.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § VII. CONCLUSION · p. 14

    When the angle of the background trajectory grows sufficiently, one of the fields must dominate throughout inflation, yet the remaining field cannot remain fully insignificant.

    “When the angle of the background trajectory grows sufficiently (in relative terms), we still need one of the fields to dominate throughout inflation, yet the remaining field can not remain full insignificant.”
    Quote in the paperSubstantial claim
35 of 35

  • The authors state this as an assumption.

    All slow-roll parameters are less than unity during inflation.

    “We consider slow-roll inflation, during which all of the slow-roll parameters defined below are less than unity.”

  • The authors state this as an assumption.

    |ηχχ| ≫ sin2θe |ηeϕϕ - 4ǫe|.

    “Here we assumed that | η χχ |≫ sin 2 θ e ∣ ∣ η e ϕϕ -4 ǫ e ∣ ∣ .”

  • The authors state this as an assumption.

    |ηϕϕ| ≫ cos2θe |ηχχ - 4ǫe|.

    “where we have assumed that | η ϕϕ |≫ cos 2 θ e | η χχ -4 ǫ e | .”

  • The authors state this as an assumption.

    |ηχχ| ≫ sin2θe |ηϕϕ - 4ǫe|.

    “where we have assumed that | η χχ |≫ sin 2 θ e | η ϕϕ -4 ǫ e | .”

  • The authors state this as an assumption.

    In the analytics, ηϕϕ ∼ -2λ is assumed.

    “In the analytics, we assume η ϕϕ ∼ -2 λ , but use the exact form (Eqn. (60)) in the plots.”

  • The authors state this as an assumption.

    Inflation ends abruptly by another waterfall field not written in the potential.

    “Here we assume that inflation ends abruptly by another waterfall field which we don't write down in the potential above.”

  • The authors state this as an assumption.

    The classical drift down the potential, 3Hχ̇ ≃ -Wχ = -W0βχ, is assumed.

    “If the value becomes too small then the motion of the χ fi eld will become dominated by quantum fluctuations rather than the classical drift down the potential, 3 H ˙ χ ≃ -W χ = -W 0 βχ , which we have assumed.”

  • The authors state this as an assumption.

    The same loop correction is dominant here as the one in [41].

    “under the assumption that the same loop correction is dominant here as the one in [41]”

  • The authors state this as an assumption.

    The tree-level calculation of the model is assumed in Eqn. (70).

    “which is assumed in Eqn. (70).”
9 of 9

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Product-potential plot cites the sum-potential formula

    Minor

    The text says the product-potential results in Fig. 3 use Eqn. (44), while Section IV identifies Eqn. (44) as the expression being rewritten for a sum potential.

  • First hybrid example has the opposite sign of the stated f NL

    Minor

    The first row of Table I gives f NL = -42, but the discussion describes that example as f NL ≃ 50. The approximate magnitude is similar, but the signs disagree.

2 of 2

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. Starobinsky A A, 1985, Pis. Zh. Eksp. Teor. Fiz., 42, 124
  2. Starobinsky A A, 1985, JETP Lett., 42, 152

Cited by

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Related

Authors
Byrnes, C. T., Choi, K.-Y., & Hall, L. M. H.
Affiliations
  • Heidelberg University (DE)
  • Instituto de Física Teórica (ES)
  • University of Sheffield (GB)
ORCID
Venue
Journal of Cosmology and Astroparticle Physics
Publisher
IOP Publishing
Published
2008-10-08
Volume
2008
Issue
10
Pages
008
ISSN
  • 1475-7508 (ISSN-L)
  • 1475-7516
DOI
10.1088/1475-7516/2008/10/008
OpenAlex
W2027344882
Semantic Scholar
a524191f4db58ff6c82f010251c44c0bf5d854c8
Cited by
143
References
73
Citations by year
2025: 1 · 2024: 2 · 2023: 1 · 2022: 1 · 2021: 4 · 2020: 5 · 2019: 3 · 2018: 7 · 2017: 2 · 2016: 4 · 2015: 4 · 2014: 9 · 2013: 18 · 2012: 17
FWCI
12.66
Type
journal article
Language
en
Open access
Locations
  • Journal of Cosmology and Astroparticle Physics (publishedVersion)
  • arXiv (Cornell University) (acceptedVersion)
  • DIGITAL.CSIC (Spanish National Research Council (CSIC)) (submittedVersion, other-oa)
  • Figshare (submittedVersion, cc-by)
  • PUB – Publications at Bielefeld University (Bielefeld University) (submittedVersion)
  • Publikationen an der Universität Bielefeld (Universität Bielefeld) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Galaxies: Formation, Evolution, Phenomena
Keywords
primordial non-Gaussianity, two-field inflation, slow-roll inflation, separable potentials, fine-tuning, Gaussian perturbation

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-23.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-23.

Cited sources are checked by gpt-6-sol.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

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