CoChat
Sign Up
Cited by 274Cosmology and Gravitation TheoriesNoncommutative and Quantum Gravity TheoriesGalaxies: Formation, Evolution, Phenomena

Non-Gaussianities in two-field inflation

Open access
Vernizzi, Filippo, Wands, David·2006·Journal of Cosmology and Astroparticle Physics
DOI 10.1088/1475-7516/2006/05/019
Open this paper in CoChat — free accountOpen PDF

Abstract

We study the bispectrum of the curvature perturbation on uniform energy density hypersurfaces in models of inflation with two scalar fields evolving simultaneously. In the case of a separable potential, it is possible to compute the curvature perturbation up to second order in the perturbations, generated on large scales due to the presence of non-adiabatic perturbations, by employing the δ N -formalism, in the slow-roll approximation. In this case, we provide an analytic formula for the nonlinear parameter f NL . We apply this formula to double inflation with two massive fields, showing that it does not generate significant non-Gaussianity; the nonlinear parameter at the end of inflation is slow-roll suppressed. Finally, we develop a numerical method for generic two-field models of inflation, which allows us to go beyond the slow-roll approximation and confirms our analytic results for double inflation.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper computes the bispectrum and nonlinear parameter \(f_{\rm NL}\) for two-field inflation using the \(\delta N\)-formalism up to second order, providing an analytic formula for separable potentials in slow roll. It finds that double quadratic inflation does not generate significant non-Gaussianity, with \(f_{\rm NL}\) slow-roll suppressed and numerically confirmed.

Key findings

  • For two-field inflation with a separable potential, the authors derive an analytic expression for the nonlinear parameter \(f_{\rm NL}\), Eq. (68), in terms of the potential and slow-roll parameters.
  • In double quadratic inflation with two massive fields, they conclude that non-Gaussianity is slow-roll suppressed: \(|f_{\rm NL}|\ll 1\).
  • For the specific numerical example with mass ratio \(R=1/9\) and initial condition \(\phi=\chi=13\,m_P\), at the end of inflation they find \(-(6/5)f^{(4)}_{\rm NL}=0.008\), \(-(6/5)f^{(3)}_{\rm NL}=0.008\), and \(-(6/5)f_{\rm NL}=0.016\simeq 1/60\).
  • Their numerical calculation agrees remarkably well with the analytic results and confirms that nonlinear evolution on super-Hubble scales during double quadratic inflation does not produce detectable non-Gaussianity; they disagree with Rigopoulos et al. and agree with Alabidi and Lyth.

Methods

  • Uses the \(\delta N\)-formalism to compute the curvature perturbation \(\zeta\) on uniform density hypersurfaces up to second order in perturbations, assuming slow roll at Hubble exit.
  • Specializes to two scalar fields with canonical kinetic terms and a separable potential \(W=U(\phi)+V(\chi)\), deriving analytic expressions for the power spectrum, spectral index, and \(f_{\rm NL}\).
  • Develops a numerical method that solves the full background field and Friedmann equations and computes first and second derivatives of \(N\) by finite differences, allowing slow roll to be relaxed after Hubble exit; applies it to double quadratic inflation.

Limitations

  • The analytic formula assumes slow-roll conditions for all fields from Hubble exit until the end of inflation; the numerical method still assumes slow roll at Hubble exit.
  • The analytic treatment is restricted to two fields with a separable potential, though the numerical method is stated to be extendable to any two-field model.
  • The authors state they are unable to explain the discrepancy with Rigopoulos et al., who found large non-Gaussianity in the same double quadratic model.
  • They leave to future work the investigation of models where the quantity \(A\), and thus \(f_{\rm NL}\), can be large.

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

Integrity signals

Registry

Crossref · 10.1088/1475-7516/2006/05/019

Cites

88 of 88 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

40 of 49 cited sources checked · 15 supported · 1 flagged · accuracy not yet measured

  • Premisecitation
    § 2 Non-Gaussian perturbations in multi-field inflation · p. 5

    The δN-formalism is equivalent to integrating the evolution of ζ on super-Hubble scales from Hubble exit until tc.

    “In particular, the δN -formalism is equivalent to integrate the evolution of ζ on super-Hubble scales from Hubble exit until t c .”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [20] source

      The paper’s discussion of Eqs. (19), (23), and (32) addresses the evolution and conservation of ζ on super-Hubble scales. It does not state that the δN formalism is equivalent to integrating that evolution from Hubble exit to tc. (LLM, conf=0.80)

      Experimental — accuracy not yet measured
  • Interpretationcitation
    § 3.3 Curvature perturbation and non-Gaussianity during inflation · p. 15

    The adiabatic nonlinear parameter can be expressed as a consistency relation between observables.

    “Note that, using Eqs. (62) and (28) for the adiabatic case, this can be expressed as a consistency relation between observables [5],”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [5] Maldacena J, 2003, J. High Energy Phys. source
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.
  • Premisecitation
    § 1 Introduction · p. 2

    Non-adiabatic perturbations produced during multi-field inflation can generate detectable non-Gaussianity in the density field after inflation, in models such as the curvaton scenario or reheating.

    “Furthermore, non-adiabatic perturbations produced during multi-field inflation can certainly generate detectable non-Gaussianity in the density field after inflation, in models such as the curvaton scenario [10] (see also [11]) or reheating [12, 13, 14, 15].”
    Quote in the paperSubstantial claim5 of 6 sources checked
    • cites [10] source
    • cites [11] source

      The Abstract and Conclusions support the central point: non-Gaussian isocurvature perturbations in multifield inflation can transfer to adiabatic perturbations and produce significant primordial non-Gaussianity. But Section III says the paper's transfer mechanism is effective only during inflation; it does not establish the curvaton or post-inflation reheating examples attached to this citation. (LLM, conf=0.88)

      Experimental — accuracy not yet measured
    • cites [12] source

      The introduction (p. 1) says multifield inflation can produce non-Gaussian isocurvature fluctuations that are transferred to curvature fluctuations, and discusses prospects for detecting primordial non-Gaussianity. But the paper’s models transfer those fluctuations during inflation; it does not establish the cited curvaton or reheating examples as mechanisms producing non-Gaussianity after inflation. (LLM, conf=0.87)

      Experimental — accuracy not yet measured
    • cites [13] source
    • cites [14] source
    • cites [15] Barnaby N Cline J M, 2006

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Premisecitation
    § 2.1 Two-point statistics · p. 5

    For multi-field inflation, the scalar spectral index at lowest order in slow-roll is given by Eq. (15).

    “For multi-field inflation, the expression for the scalar spectral index has been given, for instance, in [31, 1]. At lowest order in slow-roll one has [2]”
    Quote in the paperNot a substantial claim2 of 3 sources checked
    • cites [31] Malik K A, 2005, J. Cosmol. Astropart. Phys. source

      The Introduction mentions calculating the spectral index, but gives no expression for it. Section 8 applies slow roll to a two-field model to calculate the second-order curvature perturbation, not the lowest-order scalar spectral index. A search for “spectral index” found no formula in our copy that could verify the attribution to Eq. (15). (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • cites [1] source
    • cites [2] Liddle A R, 2000, Cosmological Inflation and Large-scale Structure source
  • Premisecitation
    § 2.2 Three-point statistics · p. 7

    During slow-roll inflation, the field three-point correlation functions do not vanish.

    “However, Seery and Lidsey [35] have found that during slow-roll inflation the field three-point correlation functions do not vanish and read”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [35] source

      Section 4 (p. 6) discusses scalar-field perturbations and the spectral index under slow-roll conditions, but does not address whether field three-point correlations vanish. Searches for three-point correlations and the bispectrum found no passage supporting the claim. The citing sentence names Seery and Lidsey, but that does not establish a source from this paper's text. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • The authors state this as an assumption: All k_i are of the same order of magnitude so that they cross the Hubble radius approximately at the same time.
  • May combine two claims. 22
  • Classified as probably not a substantial claim. 9
  • The quote only partly states this claim. 3

  • May combine two claims.

    Interpretationcitation
    § 2.2 Three-point statistics · p. 8

    The second term in Eq. (26) is momentum independent and local in real space, because it is due to the evolution of nonlinearities outside the Hubble radius during inflation.

    “is momentum independent and local in real space, because it is due to the evolution of nonlinearities outside the Hubble radius during inflation.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [20] source

      In the discussion following Eq. (6), the paper says that, for single-field inflation, non-Gaussianities in the conserved comoving curvature perturbation can only be generated inside the Hubble radius and are conserved outside it. This is contrary to the claim's explanation that the term arises from evolving nonlinearities outside the Hubble radius. The paper also does not identify a momentum-independent, real-space-local second term. (LLM, conf=0.86)

      Checked by a reasoning model — may be wrong.

      Experimental — accuracy not yet measured
    • The authors state this as an assumption: The connected part of the four-point correlation functions is negligible.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 1 Introduction · p. 2

    Slow-roll models of inflation generically predict an almost scale-invariant, almost Gaussian distribution of primordial density perturbations.

    “In particular, slow-roll models of inflation generically predict an almost scale-invariant, almost Gaussian distribution of primordial density perturbations [1, 2].”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [1] source
    • cites [2] Liddle A R, 2000, Cosmological Inflation and Large-scale Structure source

  • Classified as probably not a substantial claim.

    Premisecitation
    § 2 Non-Gaussian perturbations in multi-field inflation · p. 4

    According to the δN-formalism, ζ evaluated at some time tc is equivalent, on large scales, to the perturbation of the number of e-foldings N from an initial flat hypersurface at t* to a final comoving or uniform density hypersurface at tc.

    “According to the so called δN -formalism [30, 31, 32], ζ , evaluated at some time t = t c , is equivalent, on large scales , to the perturbation of the number of e -foldings N ( t c , t ∗ , x ) from an initial fl at hypersurface at t = t ∗ , to a final comoving - or, equivalently, uniform density - hypersurface at t = t c .”
    Quote in the paperSubstantial claim3 of 3 sources checked
    • cites [30] source

      In “Two definitions of the curvature perturbation” (p. 2), the paper defines ζ on uniform-density slices and says these practically coincide with comoving slices. In “Single-component inflation” (p. 7), it relates ζ to a difference in e-folds between slicings. The available text does not state the claim’s specific construction from an initial flat hypersurface at t* to a final uniform-density hypersurface at tc. (LLM, conf=0.82)

      Experimental — accuracy not yet measured
    • cites [31] Malik K A, 2005, J. Cosmol. Astropart. Phys. source

      The introduction says the ΔN formalism relates the comoving curvature perturbation to a perturbation in the number of e-foldings. The discussion around Eq. (4.8) says comoving and uniform-density curvature perturbations coincide on large scales. Our copy does not specify the claimed initial flat and final comoving or uniform-density hypersurfaces for that e-folding calculation. (LLM, conf=0.85)

      Experimental — accuracy not yet measured
    • cites [32] source

      The Separate Universe Approach (p. 4) relates the large-scale curvature perturbation to differences in integrated expansion between hypersurfaces, supporting the δN part of the claim. But the paper says comoving and uniform-density hypersurfaces coincide for adiabatic perturbations (p. 6) and need not coincide when non-adiabatic pressure perturbations are present (p. 7). The citing sentence omits that qualification when calling the final hypersurfaces equivalent. (LLM, conf=0.88)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Interpretationcitation
    § 2.2 Three-point statistics · p. 8

    The first term on the right hand side of Eq. (26) is momentum dependent and comes from the three-point correlation functions of the fields, computed by quantizing the perturbations inside the Hubble radius during inflation.

    “The first term on the right hand side of Eq. (26) is momentum dependent and comes from the three-point correlation functions of the fields, Eq. (21), computed by quantizing the perturbations inside the Hubble radius during inflation [35, 37].”
    Quote in the paperNot a substantial claim2 of 2 sources checked
    • cites [35] source

      The Introduction (p. 2) describes scalar-field vacuum fluctuations on scales smaller than the Hubble radius, and §4 (p. 6) discusses the momentum dependence of the perturbation spectrum. Our copy does not connect either point to a three-point correlation function or show that such a function produces the cited first term. (LLM, conf=0.82)

      Experimental — accuracy not yet measured
    • cites [37] source

      The Introduction describes vacuum fluctuations of light fields during inflation, and Section II discusses conserving second-order perturbations produced during inflation. Neither passage addresses the claimed momentum-dependent term, its origin in field three-point correlations, or their calculation by quantizing perturbations inside the Hubble radius; searches for those terms found no supporting passage in our copy. (LLM, conf=0.74)

      Experimental — accuracy not yet measured
    • The authors state this as an assumption: All k_i are of the same order of magnitude so that they cross the Hubble radius approximately at the same time.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 2 Non-Gaussian perturbations in multi-field inflation · p. 4

    The definition of N as the volume expansion rate of the t=const hypersurface holds also at second or higher order in cosmological perturbations.

    “This definition is not restricted to linear theory but holds also at second or higher order in the cosmological perturbations [32, 33, 18, 21].”
    Quote in the paperSubstantial claim3 of 4 sources checked
    • cites [32] source

      The rate-of-change discussion (p. 3) defines an expansion rate with respect to coordinate time for constant-time hypersurfaces, and the separate-universe discussion (p. 4) defines the integrated expansion N. But the paper frames its treatment in linear perturbation theory and says in the Conclusions (p. 8) that the separate-universe description applies as long as linear perturbation theory is valid. It does not establish that this definition holds at second or higher order. (LLM, conf=0.86)

      Experimental — accuracy not yet measured
    • cites [33] Starobinsky A A, 1985, Pis. Zh. Eksp. Teor. Fiz.

      Not checked in the free run.

    • cites [18] source

      Section II (p. 4) defines a local scale factor from the spatial-metric determinant, and Section III (p. 6) uses that determinant to construct a nonlinear curvature perturbation. This supports a nonlinear volume-expansion interpretation in the paper’s long-wavelength approximation, but not an unrestricted claim about perturbations at all scales. The paper also calls its symbol N the lapse function, not the volume expansion rate (Section II, p. 2). (LLM, conf=0.83)

      Experimental — accuracy not yet measured
    • cites [21] source

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation
    § 1 Introduction · p. 2

    The size of the bispectrum during single-field, slow-roll inflation is related to the spectral tilt of the power spectrum and is constrained to be small.

    “For instance, it is known that the size of the bispectrum during single-field, slow-roll inflation is related to the spectral tilt of the power spectrum and is thus constrained to be small [5].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [5] Maldacena J, 2003, J. High Energy Phys. source

  • Classified as probably not a substantial claim.

    Premisecitation
    § 1 Introduction · p. 2

    Rigopoulos et al. used numerical simulations of the stochastic dynamics on large scales and found significant non-Gaussianity even in simple two-field models.

    “Rigopoulos et al. [22] have used numerical simulations of the stochastic dynamics on large scales and found significant non-Gaussianity even in simple two-field models.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [22] source

      The discussion on p. 4 says only that non-Gaussianity can evolve on large scales when more than one field is present. The references on p. 4 name Rigopoulos and Shellard, but our copy does not say they ran numerical simulations of stochastic dynamics or found significant non-Gaussianity in two-field models (searched: numerical, stochastic, two-field). (LLM, conf=0.76)

      Experimental — accuracy not yet measured

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

    Premisecitation
    § 1 Introduction · p. 2

    Inflationary models with higher derivative operators, such as ghost inflation or inflation based on the Dirac-Born-Infeld action, can produce higher non-Gaussianity, possibly detectable.

    “On the other hand, inflationary models with higher derivative operators, such as ghost inflation or inflation based on the Dirac-Born-Infeld action, can produce higher non-Gaussianity [6, 7, 8, 9], possibly detectable.”
    Quote in the paperNot a substantial claim4 of 4 sources checked
    • cites [6] Arkani-Hamed N, 2004, J. Cosmol. Astropart. Phys. source
    • cites [7] source
    • cites [8] Seery D, 2005, J. Cosmol. Astropart. Phys. source
    • cites [9] Creminelli P, 2003, J. Cosmol. Astropart. Phys. source

      The Abstract says higher-dimension operators, including a higher-derivative interaction, can strongly enhance non-Gaussianity. The Conclusions say the predicted effect is near the threshold of detectability. Our copy does not identify ghost inflation or Dirac–Born–Infeld inflation as examples, so those specifics are not supported by this source. (LLM, conf=0.86)

      Experimental — accuracy not yet measured

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

    Findingcitation
    § 6 Conclusion · p. 26

    The paper's conclusion that f_NL is small for the specific double quadratic inflation model investigated in Sec. 5 is in qualitative agreement with an improved numerical calculation by the authors of [22].

    “Note added: After completing this work we have learnt [50] that our conclusion that f NL is small for the specific double quadratic inflation model investigated in Sec. 5 is in qualitative agreement with an improved numerical calculation by the authors of [22].”
    Quote in the paperNot a substantial claim2 of 2 sources checked
    • cites [50] source

      The Introduction (p. 2) and Double Inflation section (p. 3) describe a two-field inflation model, but the copy does not report an f_NL calculation or a comparison with an improved numerical calculation. Searches for f_NL and non-Gaussianity found no passage addressing that claimed agreement. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • cites [22] source

      The Conclusion (p. 4) discusses the evolution of non-Gaussianity during inflation, but gives no result for f_NL in a double quadratic model. Searches for the model, Sec. 5, and the improved numerical calculation found no passage addressing the claimed agreement. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • The authors state this as an assumption: The potential is separable into the sum of two potentials, each of which is dependent on only one of the two fields.
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.

  • May combine two claims.

    Findingown data
    § 6 Conclusion · p. 25

    The paper explicitly calculated in this model the non-Gaussianity due to the three-point function for the field perturbations at Hubble exit, using the result of Seery and Lidsey [8], and showed that it is small, as previously argued by Lyth and Zaballa [37].

    “However, we have explicitly calculated in this model the non-Gaussianity due to the three-point function for the field perturbations at Hubble exit, using the result of Seery and Lidsey [8], and shown that it is small, as previously argued by Lyth and Zaballa [37].”
    Quote in the paperSubstantial claim2 of 2 sources checked
    • cites [8] Seery D, 2005, J. Cosmol. Astropart. Phys. source
    • cites [37] source

      Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.

      Experimental — accuracy not yet measured
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.

  • May combine two claims.

    Premisecitation
    § 6 Conclusion · p. 25

    The formalism used by Rigopoulos et al. differs from the δN-formalism in that it takes into account the sub-Hubble evolution of field perturbations via stochastic terms and integrates explicitly the coupled evolution of the adiabatic and entropy super-Hubble perturbations.

    “The formalism used by Rigopoulos et al. differs from the δN -formalism in two aspects: First, it takes into account the sub-Hubble evolution of the field perturbations via stochastic terms. Second, it integrates explicitly the coupled evolution of the adiabatic and entropy super-Hubble perturbations.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [22] source

      The paper identifies Rigopoulos and Shellard in its references (p. 4) and notes a related long-wavelength result (p. 2), but does not describe their formalism as using stochastic terms or explicitly integrating coupled adiabatic and entropy perturbations. Its discussion of evolution at all scales concerns the authors’ own formulation (p. 4). Searches for those distinctive features did not find support for the claimed comparison with δN. (LLM, conf=0.76)

      Experimental — accuracy not yet measured

  • The quote only partly states this claim.

    Findingreasoning
    § 3.3 Curvature perturbation and non-Gaussianity during inflation · p. 14

    In the limit tc→t*, the spectral index for purely adiabatic perturbations is given by Eq. (61).

    “Note that in the limit t c → t ∗ , using Eq. (59), one finds the spectral index for purely adiabatic perturbations,”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The potential of the fields is separable into the sum of two potentials each dependent on only one of the two fields.
    • The authors state this as an assumption: The slow-roll conditions are satisfied for both fields during inflation.

  • May combine two claims.

    Findingreasoning
    § 3.3 Curvature perturbation and non-Gaussianity during inflation · p. 13

    The power spectrum of the curvature perturbation in two-field slow-roll inflation can never be less than the power spectrum derived by considering only adiabatic perturbations restricted to the background trajectory at fixed C; there is an additional contribution due to non-adiabatic perturbations at Hubble exit.

    “Note that this can never be less than the power spectrum derived by considering only adiabatic perturbations restricted to the background trajectory (at fixed C ). In multi-field inflation there is an additional contribution to the power spectrum due to non-adiabatic perturbations at Hubble exit. Thus one has P ζ ≥ P ζ | C”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The potential of the fields is separable into the sum of two potentials each dependent on only one of the two fields.
    • The authors state this as an assumption: The slow-roll conditions are satisfied for both fields during inflation.

  • May combine two claims.

    Findingreasoning
    § 5.2 Numerical analysis · p. 24

    Consequently, the power spectrum and the spectral index grow monotonically, following the growth of ζ.

    “Consequently, the power spectrum and the spectral index grow monotonically, following the growth of ζ .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 5.1 Slow-roll analysis · p. 18

    In the case where fields have settled to their minima, the dependence on the field values at t = t_c in Eq. (57) disappears and Eqs. (58), (61), (64) and (68) can be rewritten using the potential (75) and Eq. (76) with φ_c = χ_c = 0.

    “In this case the dependence on the field values at t = t c in Eq. (57) disappears and we can simply rewrite Eqs. (58), (61), (64) and (68) by using the potential (75) and Eq. (76) with φ c = χ c = 0, as”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The potential is separable into the sum of two potentials, each of which is dependent on only one of the two fields.
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.

  • May combine two claims.

    Findingreasoning
    § 5.1 Slow-roll analysis · p. 18

    After inflation, the fields have settled to their minima, which implies Z_c = 0 and A = 0.

    “However, after inflation, the fields have settled to their minima, which implies Z c = 0 and also A = 0.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The potential is separable into the sum of two potentials, each of which is dependent on only one of the two fields.
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.

  • May combine two claims.

    Findingown data
    § 5.2 Numerical analysis · p. 24

    For the specific model at the end of inflation (ǫ_e ≃ 1), in the limit f(k1,k2,k3)=0 (k1≪k2≈k3), -(6/5)f_NL^(4) = 0.008 and -(6/5)f_NL^(3) = 0.008, which coincide with the analytical result -(6/5)f_NL = 0.016 ≃ 1/60.

    “One can compute the value of f NL at the end of inflation, for ǫ e ≃ 1, in the limit where f ( k 1 , k 2 , k 3 ) = 0 ( k 1 ≪ k 2 ≈ k 3 ). One finds -(6 / 5) f (4) NL = 0 . 008 and -(6 / 5) f (3) NL = 0 . 008 which coincide with the analytical result of Eq. (83), -(6 / 5) f NL = 0 . 016 ≃ 1 / 60.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 5.2 Numerical analysis · p. 24

    After a few oscillations, the heavy field energy density is red-shifted and the light field dominates the universe while ζ becomes constant in time again.

    “After few oscillations, the heavy field energy density is red-shifted and the light field dominates the universe while ζ becomes constant in time again.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § 3.1 Slow-roll dynamics of background fields · p. 10

    If two fields are present, the phase-space becomes two-dimensional and there is an infinite number of possible classical trajectories in field space; the values of the two fields at the end of inflation will in general depend on the choice of trajectory.

    “However, if two fields are present, the phase-space becomes two-dimensional and there is an infinite number of possible classical trajectories in field space. The values of the two fields at the end of inflation will in general depend on the choice of trajectory.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [38] Zaballa I Rodriguez Y Lyth D H, 2006

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Premisecitation
    § 3.1 Slow-roll dynamics of background fields · p. 10

    In single-field inflation, the slow-roll solution forms a one-dimensional phase space, so once the inflationary attractor has been reached, there is a unique trajectory; the end of inflation takes place at a fixed value of the inflaton field corresponding to a fixed energy density.

    “In single field inflation the slow-roll solution forms a one-dimensional phase space. This means that once the inflationary attractor has been reached, there is a unique trajectory. In particular, the end of inflation takes place at a fixed value of the inflaton field which in turn corresponds to a fixed energy density.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [38] Zaballa I Rodriguez Y Lyth D H, 2006

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationreasoning
    § 3.2 Perturbed expansion during slow-roll · p. 12

    In two-field models, large-scale perturbations are not in general adiabatic and ζ evolves during inflation.

    “In two-field models, however, the large-scale perturbations are not in general adiabatic and ζ evolves during inflation.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The potential of the fields is separable into the sum of two potentials each dependent on only one of the two fields.
    • The authors state this as an assumption: The slow-roll conditions are satisfied for both fields during inflation.

  • May combine two claims.

    Interpretationreasoning
    § 4 Perturbations after inflation · p. 16

    In the case where one field has stabilized before the end of inflation, A = 0 at the end of inflation and the nonlinear parameter f_NL^(4) will be small.

    “Note that in this case A = 0 at the end of inflation and the nonlinear parameter f (4) NL will be small.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The potential is separable into the sum of two potentials, each of which is dependent on only one of the two fields.
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.

  • May combine two claims.

    Interpretationreasoning
    § 4 Perturbations after inflation · p. 17

    In a hybrid-type limit where the false vacuum dominates the self-interaction energy of the slowly-rolling fields, [δρ/ρ]_e is negligible and δN(t_c,t_e) ≪ δN(t_e,t_*).

    “In a hybrid-type limit, where the false vacuum dominates the self-interaction energy of the slowly-rolling fields, we can take the effective potential to be almost completely flat (consistent with the slow-roll approximation) so that [ δρ/ρ ] e is negligible and δN ( t c , t e ) ≪ δN ( t e , t ∗ ).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning
    § 4 Perturbations after inflation · p. 16

    In single field inflation, large-scale perturbations are adiabatic and the curvature perturbation on uniform density hypersurfaces remains constant both during and after inflation, independent of the detailed physics at the end of inflation.

    “In single field inflation large-scale perturbations are adiabatic and thus the curvature perturbation on uniform density hypersurfaces remains constant both during and after inflation, independent of the detailed physics occurring at the end of inflation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 4 Perturbations after inflation · p. 16

    If one of the two fields has stabilized before the end of inflation, so that the end of inflation is dominated by a single field, then Z_c becomes constant, and the power spectrum, scalar spectral index and non-linear parameters after the end of inflation are given by Eqs. (58), (61), (64) and (68) with Z_c = U_c = const.

    “If one of the two fields - for instance φ - has stabilized before the end of inflation, so that the end of inflation is dominated by a single field χ , then Z c becomes constant, and the power spectrum, the scalar spectral index and the non-linear parameters after the end of inflation are simply given by Eqs. (58), (61), (64) and (68) with Z c = U c = const.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The potential is separable into the sum of two potentials, each of which is dependent on only one of the two fields.
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 2 Non-Gaussian perturbations in multi-field inflation · p. 3

    The curvature perturbation on uniform energy density hypersurfaces, ζ, is conserved on large scales for adiabatic perturbations.

    “This quantity is widely used, especially because it is conserved, on large scales, for adiabatic perturbations [29].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [29] source

  • May combine two claims.

    Interpretationreasoning
    § 5.2 Numerical analysis · p. 24

    The isocurvature perturbations in the massive field are suppressed at the end of inflation, and hence the calculated nonlinear parameter is the primordial f_NL constrained by observations, unless a curvaton-type mechanism alters the large-scale curvature perturbation after inflation.

    “The isocurvature perturbations in the massive field are suppressed at the end of inflation and hence the nonlinear parameter that we have calculated is indeed the primordial f NL constrained by observations, unless we have some curvaton-type mechanism which alters the large-scale curvature perturbation after inflation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 6 Conclusion · p. 25

    The contribution to f_NL coming from the super-Hubble evolution should be equally taken into account by both formalisms, and the paper is unable to explain the discrepancy between the results.

    “Thus, the contribution to f NL coming from the super-Hubble evolution should be equally taken into account by both formalisms and we are unable to explain the discrepancy between the results.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 6 Conclusion · p. 25

    Including the dependence of φ_c and χ_c upon the field values at Hubble exit considerably complicates the analysis but allows following perturbations during inflation.

    “Including the dependence of φ c and χ c upon the field values at Hubble exit considerably complicates our analysis, but allows to follow perturbations during inflation.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Study design and conductown data
    § 6 Conclusion · p. 25

    The numerical method based on the δN-formalism, which allows relaxing the slow-roll assumption after Hubble exit, confirms the analytic results.

    “This method, which can be extended to any two-field model, confirms our analytic results.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.
    • The authors state this as an assumption: The δN-formalism assumes that the fields are in slow-roll at Hubble exit.

  • Classified as probably not a substantial claim.

    Interpretationnone given
    § 6 Conclusion · p. 25

    The possibility of producing a large non-Gaussianity is left open in models of two-field inflation where the mass of the isocurvature field orthogonal to the classical trajectory becomes larger than the Hubble rate and both the first slow-roll parameters of the two fields do not vanish at the end of inflation.

    “However, the possibility of producing a large non-Gaussianity is left open in models of two-field inflation where the mass of the isocurvature field orthogonal to the classical trajectory becomes larger than the Hubble rate and both the first slow-roll parameters of the two fields do not vanish at the end of inflation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 6 Conclusion · p. 24

    A large f_NL after inflation requires a large η_ss at the end of inflation, although a large η_ss does not necessarily imply large non-Gaussianity.

    “In particular, we have shown that a large f NL after inflation requires a large η ss - defined in Eq. (67) at the end of inflation, although a large η ss does not necessarily imply large non-Gaussianity.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The potential is separable into the sum of two potentials, each of which is dependent on only one of the two fields.
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.

  • The quote only partly states this claim.

    Study design and conductown data
    § 5.2 Numerical analysis · p. 23

    Assuming slow-roll from t_* to t_e is well justified for the specific case R = 1/9 and initial condition φ = χ = 13 m_P.

    “One can see that assuming slow-roll from t ∗ to t e is well justified.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.
33 of 33

  • The authors state this as an assumption.

    The model consists of minimally coupled scalar fields with canonical kinetic terms and potential W.

    “For simplicity, we will consider a model of inflation driven by a set of minimally coupled scalar fields with canonical kinetic terms and potential W , described by the action”

  • The authors state this as an assumption.

    Scalar field perturbations are uncorrelated stochastic variables at early times, with a scale-invariant spectrum of massless scalar fields in de-Sitter space.

    “Let us now take the scalar field perturbations δϕ I as uncorrelated stochastic variables at early times, with scale invariant spectrum of massless scalar fields in de-Sitter space.”

  • The authors state this as an assumption.

    Correlation functions higher than the four-point are neglected.

    “where a star denotes the convolution and we have neglected correlation functions higher than the four-point (see [35, 34]).”

  • The authors state this as an assumption.

    All k_i are of the same order of magnitude so that they cross the Hubble radius approximately at the same time.

    “where k t = k 1 + k 2 + k 3 , and it is assumed that all the k i are of the same order of magnitude so that they cross the Hubble radius approximately at the same time.”

  • The authors state this as an assumption.

    The connected part of the four-point correlation functions is negligible.

    “To evaluate the second line of Eq. (20), as in [35] we assume that the connected part of the four-point correlation functions is negligible and we make use of Wick's theorem to reduce the four-point functions to products of two-point functions.”

  • The authors state this as an assumption.

    The potential of the fields is separable into the sum of two potentials each dependent on only one of the two fields.

    “We assume that the potential of the fields is separable into the sum of two potentials each of which is dependent on only one of the two fields,”

  • The authors state this as an assumption.

    The slow-roll conditions are satisfied for both fields during inflation.

    “Here it is assumed that the slow-roll conditions are satisfied for both fields during inflation.”

  • The authors state this as an assumption.

    U' ≥ 0 and V' ≥ 0.

    “Note that throughout this paper, for simplicity, we will assume U ′ ≥ 0 and V ′ ≥ 0 so that we may eliminate first derivatives of the potential in favor of the slow-roll parameters √ ǫ φ and √ ǫ χ .”

  • The authors state this as an assumption.

    The potential is separable into the sum of two potentials, each of which is dependent on only one of the two fields.

    “We have then specialized to the two-field case, and assumed that the potential is separable into the sum of two potentials, each of which is dependent on only one of the two fields.”

  • The authors state this as an assumption.

    Slow-roll conditions are satisfied by all fields from Hubble exit until the end of inflation, and slow-roll is assumed all along the inflationary evolution from t_* to t_c.

    “Furthermore, in deriving Eqs. (58), (61), (64), (68) and the background evolution relation (79), we have assumed slow-roll all along the inflationary evolution, from t ∗ to t c .”

  • The authors state this as an assumption.

    The δN-formalism assumes that the fields are in slow-roll at Hubble exit.

    “The δN -formalism assumes that the fields are in slow-roll at Hubble exit, Eq. (8) (see [48] for a development of the δN -formalism applicable to more general situations).”
11 of 11

Cites

Flagged first, then by citations

Research works printed but not linked: 5

  1. Starobinsky A A, 1985, Pis. Zh. Eksp. Teor. Fiz., 42, 124
  2. Starobinsky A A, 1985, JETP Lett., 42, 152
  3. Lee H C, 2005, J. Cosmol. Astropart. Phys., 2005, 004
  4. Press W H, 1992, Numerical Recipes in FORTRAN: The Art of Scientific Computing
  5. 10.1103/physrevd.63.0630xx · DOI

Cited by

Sorted by citations

Authors
Vernizzi, F., & Wands, D.
Affiliations
  • University of Helsinki (FI)
  • University of Portsmouth (GB)
ORCID
Vernizzi, Filippo
Venue
Journal of Cosmology and Astroparticle Physics
Publisher
IOP Publishing
Published
2006-05-26
Volume
2006
Issue
05
Pages
019-019
ISSN
  • 1475-7508 (ISSN-L)
  • 1475-7516
DOI
10.1088/1475-7516/2006/05/019
OpenAlex
W2079415417
Semantic Scholar
a811ff0416098de3bd4c280dd3a692bdb4b7c4b0
Cited by
274
References
44
Citations by year
2026: 1 · 2025: 6 · 2024: 4 · 2023: 5 · 2022: 5 · 2021: 3 · 2020: 5 · 2019: 4 · 2018: 11 · 2017: 10 · 2016: 14 · 2015: 17 · 2014: 20 · 2013: 19 · 2012: 30
FWCI
13.24
Type
journal article
Language
en
Open access
Locations
  • Journal of Cosmology and Astroparticle Physics (publishedVersion)
  • arXiv (Cornell University) (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
two-field inflation, non-Gaussianity, bispectrum, curvature perturbations, non-adiabatic perturbations, delta N formalism, slow-roll approximation, double inflation, massive fields, separable potentials, primordial perturbations

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.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review