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Conditions for large non-Gaussianity in two-field slow-roll inflation
Open accessAbstract
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 .
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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.
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Claims the paper makes
32 Own interpretation · 30 No basis given · 18 Own result · 5 Couldn’t confirm · 3 Supported · 3 Partly supported · 3 Own method · 1 Not checked yet · accuracy not yet measured
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“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].”
What the sources say
- Could not verifycites [38] Seery D, 2007, J. Cosmol. Astropart. Phys. sourceExperimental — accuracy not yet measured
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)
- Could not verifycites [39] sourceExperimental — accuracy not yet measured
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)
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In the paper
“We therefore find a region with much larger non-Gaussianity than was found in [5].”
What the sources say
- Supportedcites [5] Alabidi L, 2006, J. Cosmol. Astropart. Phys. source
Section 7, “Results for the non-Gaussianity” (page unavailable), reports a region with 1 < f_NL < 2.48, providing the earlier result used for comparison. The citing authors’ claim that their own region has much larger non-Gaussianity is their conclusion, not a finding attributed to this source. (LLM, conf=0.90)
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In the paper
“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].”
What the sources say
- Could not verifycites [40] Yokoyama S, 2007, J. Cosmol. Astropart. Phys. sourceExperimental — accuracy not yet measured
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)
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In the paper
“With a product potential, the local non-Gaussianity is given by [21]:”
What the sources say
- Partly supportedcites [21] Enqvist K, 2005, J. Cosmol. Astropart. Phys. sourceExperimental — accuracy not yet measured
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)
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In the paper
“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]).”
What the sources say
- Could not verifycites [41] sourceExperimental — accuracy not yet measured
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)
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- The authors state this as an assumption: The same loop correction is dominant here as the one in [41].
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In the paper
“Generally, the scalar fields need not only couple through the potential, but may also couple kinetically [44, 45, 46, 47].”
What the sources say
- Not matchedcites [44] Starobinsky A A, 1985, JETP Lett.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Supportedcites [45] source
Section 3 (p. 5) describes a scalar-field action with a field-space metric, while §3.2 (p. 5) explicitly includes the term h_ab φ̇ᵃφ̇ᵇ. Together with the scalar potential discussed in §3.1 (p. 5), this supports the distinction between potential and kinetic coupling. (LLM, conf=0.86)
- Supportedcites [46] source
The Introduction (p. 2) defines a metric on the multi-scalar field space and says the simplifying choice of a trivial metric can be generalized to a non-trivial one. Section 2.2 (p. 8) describes that generalization, supporting the possibility of kinetic coupling between fields rather than coupling only through the potential. (LLM, conf=0.87)
- Could not verifycites [47] Lyth D H, 2005, J. Cosmol. Astropart. Phys. sourceExperimental — accuracy not yet measured
Section 3.1 mentions scalar fields only as an example for which negligible anisotropic stress can be verified; section 2.3 treats matter as a perfect fluid. Searches for kinetic coupling and coupling through a potential found no passage in our copy addressing how scalar fields couple. (LLM, conf=0.73)
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In the paper
“For | f NL | to be larger than unity, the function g p ( θ ∗ , θ e ) must be bigger than the inverse of the slow-roll parameters in the square bracket.”
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- The authors state this as an assumption: |ηχχ| ≫ sin2θe |ηeϕϕ - 4ǫe|.
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“Specifically, the slow-roll parameters and the value of f p or g p must be such to give | f (4) NL | ≳ 1 in Eqn. (26).”
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“This condition also leads to a large growth of the subdominant slow-roll parameter.”
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“We can see that | f (4) NL | increases sharply around N = 30.”
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“Around N = 58, the χ -field begins to roll so fast that cos 2 θ decreases and | f (4) NL | increases once more.”
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“In this scenario, v ∼ sin 2 θ e W e /W ∗ and hence”
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“Hence sin 2 θ decreases during inflation and this model cannot generate | f NL | > 1.”
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“We note that the dominant slow-roll parameters η ϕϕ and η χχ are constant in this model.”
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- The authors state this as an assumption: Inflation ends abruptly by another waterfall field not written in the potential.
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“As in Eqn. (30), we find exact upper limits on the parameters:”
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“The main point to note is that these prefactors are large (and can be very large) in two regions:”
- Also stated in: § III. LARGE NON-GAUSSIANITY IN PRODUCT POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) V ( χ ) · p. 5 · plain
“Each of these regions describes one of the fields dominating over the other in kinetic energy.”
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- Also stated in: § III. LARGE NON-GAUSSIANITY IN PRODUCT POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) V ( χ ) · p. 5 · plain
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“The first example in the Table I shows that it is possible to have f NL ≃ 50 and a scale invariant spectrum.”
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“The tensor-to-scalar ratio is also small in this example.”
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“Finally, we can approximate f (4) NL in Region D as”
- Also stated in: § IV. LARGE NON-GAUSSIANITY IN SUM POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) + V ( χ ) · p. 9 · plain
“where we have used u ≃ 1 and cos 2 θ ∗ ≃ 1.”
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- Also stated in: § IV. LARGE NON-GAUSSIANITY IN SUM POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) + V ( χ ) · p. 9 · plain
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“The first term of f (4) NL in Eqn. (49) is always smaller than unity, since 0 ≤ j s ≤ 1.”
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“Then f (4) NL begins to decrease around N = 45 as cos 4 θ e becomes bigger than cos 2 θ ∗ .”
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“We note from Table I that we require a very small value of χ ∗ in Region D.”
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“Indeed, the form of power spectrum and spectral index in this large non-Gaussianity region, are same as that of product potential, Eqn. (32) and (33).”
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“As an example, if observations find f (4) NL ∼ 10, for standard order of slow-roll parameters ( η ∼ 0 . 01), one requires f p or g p ∼ 1000.”
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“and secondly we see that sin 2 θ must typically grow by at least two orders of magnitude during inflation, since we where require”
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“In the special type of potential when V e ≃ V ∗ , such as hybrid inflation, v ∼ sin 2 θ e W e /W ∗ and hence”
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“Firstly, from ∂g p /∂ (sin 2 θ e ) = 0 and the definition of g p , the initial and final angle of the trajectory are constrained:”
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“we find that it is possible to generate significant non-Gaussianity even during slow-roll inflation with Gaussian perturbations at Hubble exit”
- Also stated in: § I. INTRODUCTION · p. 1 · boosted
“We show that it is possible to generate an extremely large non-Gaussianity during slow-roll inflation even when the field perturbations at Hubble exit are Gaussian.”
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- Also stated in: § I. INTRODUCTION · p. 1 · boosted
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“We present explicit models in which the non-Gaussianity at the end of inflation can exceed the current observational bound of | f NL | ≲ 100.”
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“We temporarily analyse this second region, Region D, in which v ≪ 1 , sin 2 θ ∗ ≪ 1 and g s can be large.”
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“In both these regions, these prefactors can account for a significant level of f (4) NL , despite the relevant terms in Eqn. (26) having slow-roll factors.”
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“In this range, the non linearity parameter is well approximated as”
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- The authors state this as an assumption: In the analytics, ηϕϕ ∼ -2λ is assumed.
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“One is a product potential driven by a field with a quadratic potential, which ends inflation when it approaches the minimum of the potential. If the second field with an exponential potential has a sufficiently small initial value then for certain values of the model parameter, λ , this field grows by the right amount to generate a large non-Gaussianity, f NL ∼ 100.”
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“However we have also found that if one of the field values is too small then quantum fluctuations may perturb the background trajectory to an extent that the classical trajectory in field space is no longer valid.”
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“Using P ζ ≃ 10 -10 , H 2 ≃ W 0 and Eqn. (42) the condition that the classical trajectory is valid can be rewritten as (1 + ˜ r ) β 2 χ 2 ∗ / ( α 2 ϕ 2 ∗ ) > 6 P ζ , where ˜ r = sin 4 θ e / sin 2 θ ∗ .”
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“We also consider the sum separable model of hybrid inflation. We provide a few specific choices of the model parameters, which are effectively η ϕϕ and η χχ , for a suitably large ratio of the initial field values this model generates a large non-Gaussianity.”
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“From this definition, it is clear that allowing α = 0, the range of θ e for which we can obtain large f (4) NL opens up.”
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“We also note that the symmetry between two fields is broken, due to b ( ϕ ), which is apparent in Figs. 6.”
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“We have presented two explicit models where a large non-Gaussianity can be generated during slow roll inflation.”
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“Furthermore, when X > 0 (or b e > b ∗ ), we no longer require cos 2 θ e ≪ 1.”
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In the paper
“We consider under which conditions a general model can generate a large non-Gaussianity and show that this always requires some fine tuning of initial conditions.”
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In the paper
“Hence sin 2 θ decreases during inflation and this model cannot generate | f NL | > 1.”
- Also stated in: § B. Quadratic product potential · p. 11 · plain
“cannot generate a large non-Gaussianity during inflation.”
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- Also stated in: § B. Quadratic product potential · p. 11 · plain
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In the paper
“We note, however, that the tensor-to-scalar ratio can only be suppressed in multiple field inflation, since the tensor power spectrum is unchanged from the single field value and P ζ is enhanced.”
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In the paper
“Ultimately, the two-field system must have one field which dominates the evolution throughout. This corresponds to being in Region A or B (depending on which field dominates). Hence, one field must dominate the evolution almost fully at the beginning, while the other field gains a relatively large percentage of kinetic energy by the end of inflation.”
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In the paper
“Since inflation ends when the χ fi eld rolls close to its minimum, i.e. when ǫ = 1 ≃ ǫ χ , the only way to generate large non-Gaussianity is to start with small field of ϕ (which corresponds to Region A).”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“This can be understood from the evolution of two fields: from this time, the velocity of ϕ (or cos 2 θ ) increases (as seen in the third and fourth plot of Fig. 4) to satisfy the condition Eqn. (36) to obtain large non-Gaussianity.”
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In the paper
“Note, however, that single-field inflation (i.e. with one field static) will not lead to a large value of non-Gaussianity, since these fractions vanish exactly for ˙ χ e = 0 or ˙ ϕ e = 0 ( θ e = 0 or π / 2 respectively).”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“However this is only possible for specific values of the model parameters and initial conditions.”
- Also stated in: § B. Direct Observation of f (4) NL and Fine-Tuning · p. 7 · plain
“This requires very special initial values for the fields.”
plainconditional- Asserted: passes — The authors put it forward as true.
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- Also stated in: § B. Direct Observation of f (4) NL and Fine-Tuning · p. 7 · plain
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In the paper
“Noting that in Region D sin θ = βχ/ ( αϕ ), from Eqn. (68) we require N ( α -β ) > 1 so that sin 2 θ grows significantly during inflation.”
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In the paper
“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.”
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In the paper
“However this does provide a further restriction on the allowed parameter space of the 'tree level' calculation of the model, which is assumed in Eqn. (70).”
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In the paper
“However they can still be significant if the coefficient to the term given by the derivative of N is extremely large, e.g. [43].”
What the sources say
- Not checked yetcites [43] Starobinsky A A, 1985, Pis. Zh. Eksp. Teor. Fiz.
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“We have shown that it is possible to generate a large level of non-Gaussianity during inflation without violating slow roll and when the inflaton field perturbations are Gaussian at Hubble exit.”
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In the paper
“We have checked that this is satisfied for all of the examples given in the Table I, but it does provide a significant constraint on the total parameter space.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“We have also shown how the constraints on generating a large non-Gaussianity may be eased if we generalise the inflaton model to allow a non-canonical kinetic term.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“We have checked that this is satisfied for all of the examples given in the Table I, but it does provide a significant constraint on the total parameter space.”
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In the paper
“The general conditions for generating a large non-Gaussianity show that the inflaton potential must have a specific shape so that the angle of the background trajectory can grow by about two orders of magnitude or more during inflation.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“In the case of a product separable potential we then also require that the final value of the angle of the background trajectory lies in a narrow range much greater than the initial value (in relative terms) but still nearly parallel to the χ axis.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“When the kinetic energies are non-canonical, the local non-Gaussianity is altered from the previous sections.”
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In the paper
“For typical values of the slow roll parameters, we require that initially the subdominant field, say ϕ , must satisfy cos 2 θ ∗ = ǫ ∗ ϕ /ǫ ∗ ≲ 10 -7 .”
plain- Asserted: passes — The authors put it forward as true.
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- It may state two separate claims as one. 17
- Probably not a substantial claim. 19
- The quote only partly states this claim. 2
Every source the paper cites for this claim was checked and supports it.
In the paper
“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.”
What the sources say
- Supportedcites [5] Alabidi L, 2006, J. Cosmol. Astropart. Phys. source
The citation points to Alabidi’s calculation of f_NL, not to the citing authors’ claim about the scope of their own analysis. Section 7 reports a full parameter-space calculation of f_NL and examines initial field values from 0 to 1, including values below 0.1. The comparative assessment that the citing authors’ analysis is more extensive is their own framing. (LLM, conf=0.86)
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“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.”
What the sources say
- Supportedcites [5] Alabidi L, 2006, J. Cosmol. Astropart. Phys. source
Section 4 derives an expression for f_NL in equation (16), so the cited paper contains the calculation to which the citing sentence refers. The sentence’s comparison of parameter-space analyses is the citing authors’ own claim, not part of what this citation is offered to establish. (LLM, conf=0.86)
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Supported
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“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 .”
What the sources say
- Could not verifycites [41] sourceExperimental — accuracy not yet measured
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)
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Could not verify
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“These 'higher order' terms are usually neglected, but [41] has shown the first explicit example of an inflation model where they cannot be neglected.”
What the sources say
- Partly supportedcites [41] sourceExperimental — accuracy not yet measured
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)
boosted- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Partly supported
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Very recently Cogollo et al. have calculated the effect of the loop correction to the primordial power spectrum and bispectrum [41].”
What the sources say
- Could not verifycites [41] sourceExperimental — accuracy not yet measured
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)
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Could not verify
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“We note that the analytic constraints differ slightly from the plots, due to the value of η ϕϕ used.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- The authors state this as an assumption: In the analytics, ηϕϕ ∼ -2λ is assumed.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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 .”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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In the paper
“In Table I, we give some explicit examples of values of α, β, ϕ ∗ and χ ∗ which lead to a large non-Gaussianity.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“From Eqn. (36), the constraints on the initial values of the field ϕ can be obtained:”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The spectral index is changed by the presence of ˜ r , but remains slow roll suppressed.”
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“using the full formula Eqn. (44), which is almost same as Eqn. (63).”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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).”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § IV. LARGE NON-GAUSSIANITY IN SUM POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) + V ( χ ) · p. 8 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
plainconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“If X = 0, then α = 1 and F p = f p , G p = g p , H p = h p and J p = j p .”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We therefore expect reheating to take place in almost the same way as it would in normal chaotic inflation.”
hedged- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“We have checked that this gives the same results.”
boosted- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The quote only partly states this claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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).”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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,”
boosteddirect- Asserted: passes — The authors put it forward as true.
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The quote only partly states this claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“However during this fast-roll stage, slow-roll breaks down and our analytic formula is not valid any more.”
plaindirect- Asserted: passes — The authors put it forward as true.
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- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- The authors state this as an assumption: |ηϕϕ| ≫ cos2θe |ηχχ - 4ǫe|.
- The authors state this as an assumption: |ηχχ| ≫ sin2θe |ηϕϕ - 4ǫe|.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the case of the sum potential, the modifications are non-trivial and are difficult to analyse.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When the potential is of product form, however, we may at least partially use the previous analysis.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This means that the field trajectory is almost exactly parallel to the χ axes initially and typically this requires a finely tuned initial condition.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We therefore concentrate only on a product potential, but our findings relate also to sum potentials.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Note, though, that we require θ ∗ ∼ 0 , π 2 as before.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“There is also the possibility that the large scale loop correction which we have not generally considered in this paper may not be negligible.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
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).”
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.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ A. Quadratic times exponential potential, W ( ϕ,χ ) = 1 2 e -λϕ 2 /M 2 P m 2 χ 2 · p. 10
“The contour plots of f (4) NL for two values of λ ( λ = 0 . 04 , 0 . 05) are given in Fig. 3 using the full formula Eqn. (44), which is almost same as Eqn. (63).”
§ IV. LARGE NON-GAUSSIANITY IN SUM POTENTIALS, W ( ϕ,χ ) = U ( ϕ ) + V ( χ ) · p. 8
“Similar to the analysis of a product potential, it is possible to re-write Eqn. (44):”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ C. Hybrid inflation · p. 12
“The first example in the Table I shows that it is possible to have f NL ≃ 50 and a scale invariant spectrum.”
§ C. Hybrid inflation · p. 12
“| 0.018 | -0.018 | 1 | 0.00018 | -42 | 0 | 0.006 |”
Citations
Cites
Non-gaussian features of primordial fluctuations in single field inflationary models
Maldacena, J. (2003) · Journal of High Energy Physics · doi:10.1088/1126-6708/2003/05/013 · Cited by 2,906
Towards the theory of reheating after inflation
Kofman, L. A., Linde, A. D., & Starobinsky, A. A. (1997) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.56.3258 · Cited by 1,987
Kofman, L. A., Linde, A. D., & Starobinsky, A. A. (1994) · Physical Review Letters · doi:10.1103/physrevlett.73.3195 · Cited by 1,792
Generating the curvature perturbation without an inflaton
Lyth, D. H., & Wands, D. (2002) · Physics Letters B · doi:10.1016/s0370-2693(01)01366-1 · Cited by 1,455
Non-Gaussianity from inflation: theory and observations
Bartolo, N., Komatsu, E., Matarrese, S., & Riotto, A. W. (2004) · Physics Reports · doi:10.1016/j.physrep.2004.08.022 · Cited by 1,144
DBI in the sky: Non-Gaussianity from inflation with a speed limit
Alishahiha, M., Silverstein, E., & Tong, D. (2004) · Physical Review D · doi:10.1103/physrevd.70.123505 · Cited by 1,064
Acoustic signatures in the primary microwave background bispectrum
Komatsu, E., & Spergel, D. N. (2001) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.63.063002 · Cited by 1,062
Sasaki, M., & Stewart, E. D. (1996) · Progress of Theoretical Physics · doi:10.1143/ptp.95.71 · Cited by 982
Observational signatures and non-Gaussianities of general single-field inflation
Chen, X., Huang, M.-X., Shamit Kachru, A., & Shiu, G. (2007) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2007/01/002 · Cited by 957
Effects of cosmological moduli fields on cosmic microwave background
Moroi, T., & Takahashi, T. (2001) · Physics Letters B · doi:10.1016/s0370-2693(01)01295-3 · Cited by 938
Research works printed but not linked: 2
- Starobinsky A A, 1985, Pis. Zh. Eksp. Teor. Fiz., 42, 124
- Starobinsky A A, 1985, JETP Lett., 42, 152
Cited by
Sorted by citations
Cosmology and fundamental physics with the Euclid satellite
The Euclid Theory Working Group, Amendola, L., Appleby, S. A., Avgoustidis, A., Bacon, D., Baker, T., Baldi, M., Bartolo, N., Blanchard, A., Bonvin, C., et al. (2018) · Living Reviews in Relativity · doi:10.1007/s41114-017-0010-3 · Cited by 1,096
Cosmology and Fundamental Physics with the Euclid Satellite
The Euclid Theory Working Group, Amendola, L., Appleby, S., Bacon, D., Baker, T., Baldi, M., Bartolo, N., Blanchard, A., Bonvin, C., Borgani, S., et al. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-6 · Cited by 899
Barnaby, N., Pajer, E., & Peloso, M. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.85.023525 · Cited by 367
Phenomenology of a pseudo-scalar inflaton: naturally large nongaussianity
Barnaby, N., Namba, R., & Peloso, M. (2011) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2011/04/009 · Cited by 270
Shape of primordial non-Gaussianity and the CMB bispectrum
Fergusson, J. R., & Shellard, E. P. S. (2009) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.80.043510 · Cited by 219
Metadata
- Authors
- Byrnes, C. T., Choi, K.-Y., & Hall, L. M. H.
- Affiliations
- Heidelberg University (DE)
- Instituto de Física Teórica (ES)
- University of Sheffield (GB)
- 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
- 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
- green
- arxiv.org
- 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 › Nuclear and High Energy 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.
A language model drew the claims from the paper on 2026-10-23, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-23 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (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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