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Review of Local Non‐Gaussianity from Multifield Inflation
Open accessAbstract
We review models which generate a large non‐Gaussianity of the local form. We first briefly consider three models which generate the non‐Gaussianity either at or after the end of inflation; the curvaton scenario, modulated (p)reheating, and an inhomogeneous end of inflation. We then focus on ways of generating the non‐Gaussianity during inflation. We derive general conditions which a product or sum separable potential must satisfy in order to generate a large local bispectrum during slow‐roll inflation. As an application, we consider two‐field hybrid inflation. We then derive a formalism not based on slow roll which can be applied to models in which the slow‐roll parameters become large before inflation ends. An exactly soluble two‐field model is given in which this happens. Finally, we also consider further non‐Gaussian observables, a scale dependence of fNL and the trispectrum.
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TL;DR
This review surveys multifield inflation models that can generate large local-type primordial non-Gaussianity, covering curvaton, modulated (p)reheating, inhomogeneous end of inflation, and during-inflation mechanisms. It derives conditions for large \(f_{\rm NL}\) in separable slow-roll potentials, applies them to two-field hybrid inflation, and extends the analysis beyond slow roll with an exactly soluble two-field model.
Key findings
- Local non-Gaussianity can be generated at or after inflation by the curvaton scenario, modulated (p)reheating, and an inhomogeneous end of inflation; during slow-roll multifield inflation it can also be large if the trajectory is almost entirely along one field but the orthogonal field grows by at least an order of magnitude toward the end.
- For a product separable potential, large \(|f_{\rm NL}|\) requires conditions such as \(\sin^2\theta_* \sin^4\theta_e (1/\sin^2\theta_e G_p -1)\); a quadratic-times-exponential toy model gives \(|f_{\rm NL}|\sim 30\) with \(\phi_*\sim 10^{-3}M_P\) for \(\lambda=0.05\) and \(\chi_*=16M_P\).
- Two-field hybrid inflation can yield large \(f_{\rm NL}\) for positive, negative, or mixed \(\eta\) parameters; Table 1’s first example (\(\eta_{\phi\phi}=0.04\), \(\eta_{\chi\chi}=-0.04\), \(\phi_*=1\), \(\chi_*=6.8\times10^{-5}\), \(r=1\)) gives \(f_{\rm NL}=-123\), \(\tau_{\rm NL}=4.4\times10^4\), \(g_{\rm NL}=-33\), \(n_\zeta-1=0\), and \(r=0.006\), showing \(|f_{\rm NL}|>100\) with a scale-invariant spectrum is possible.
- Beyond slow roll, an exact two-field solution with \(\alpha=-100\), \(\beta=20\), \(m=6\), \(n=6/5\), and \(N_{\rm tot}=60\) gives \(f_{\rm NL}\approx58\) from the simplified formula and \(\approx53\) from the full formula, with \(\gamma_\phi=29\) and \(\gamma_\chi=-13\) at the end; the review also notes \(\tau_{\rm NL}\ge(6f_{\rm NL}/5)^2\), WMAP5 bounds \(-7.4<g_{\rm NL}/10^5<8.2\) and \(-0.6<\tau_{\rm NL}/10^4<3.3\) at 95% CL, and scale dependence satisfying \(-4(\eta_{\phi\phi}-\eta_{\chi\chi})<n_{f_{\rm NL}}<0\).
Methods
- Review and analytic derivation using the \(\delta N\) formalism, with definitions of the power spectrum, bispectrum, \(f_{\rm NL}\), trispectrum parameters \(\tau_{\rm NL}\) and \(g_{\rm NL}\), and the scale dependence \(n_{f_{\rm NL}}\).
- Slow-roll analysis for sum- or product-separable potentials, deriving general conditions for large local \(f_{\rm NL}\), then applying them to two-field hybrid inflation including waterfall-field end-of-inflation effects.
- Beyond slow roll, uses first-order Hamilton-Jacobi formalism with a separable Hubble parameter and presents an exact two-field exponential-potential solution, comparing simplified and full \(f_{\rm NL}\) formulas.
Limitations
- Stated: the paper considers only adiabatic primordial perturbations and neglects secondary non-Gaussianities generated at later times; further evolution after inflation in the hybrid model is model-dependent and “has not been considered in depth”; the exact solution’s potential needs modification after inflation for reheating; initial conditions require fine-tuning.
- Stated: the curvaton treatment uses a sudden-decay approximation, though it agrees well with full numerical studies; modulated instant preheating models are allowed only in a small corner of parameter space and for special initial conditions.
- Evident from the text: as a review, it synthesizes prior work and presents no new observational data; numerical examples depend on chosen parameter values and on analytic approximations such as slow roll, separable potentials, or separable Hubble parameter.
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75 Own interpretation · 43 Sources not found · 35 No basis given · 16 Own result · 2 Own method
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“In this region where ϕ χ , | f NL | > 1 is fulfilled by the condition sin 2 θ * sin 4 θ ⎛ ⎜ ⎝ 5 η χχ 6sin 2 θ -1 ⎞ ⎟ ⎠,(33)”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4. Hybrid Inflation with Two Inflaton Fields
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“In the case of two-field quadratic inflation with two massive fields, an analytic calculation is possible with slow-roll conditions, and this shows that the nonlinear parameter is small [98,99].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 3. Non-Gaussianity during Slow-Roll Inflation
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- Not matchedcites [98]
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“For the separable potential by product large nonlinear parameter | f NL | 1 is possible when (there is a symmetric region of large non-Gaussianity under the exchange of the two fields).”
§ Review of Local Non-Gaussianity from Multifield Inflation > 3.2. Conditions for Generating a Large f NL .
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“Therefore, the large f NL is obtained for small r χ .”
§ Review of Local Non-Gaussianity from Multifield Inflation > 2.1. Curvaton Scenario.
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“For a product separable potential, W(ϕ, χ) = U(ϕ)V (χ), we can find the number of e-foldings analytically in the slowroll limit [107],”
§ Review of Local Non-Gaussianity from Multifield Inflation > 3.1. General Formulas.
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“However, all of the above models of modulated instant preheating are only allowed, even at the linear level in perturbations, for a small corner of parameter space and even then only for a special choice of initial conditions [85].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 2.2. Modulated (P)reheating.
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“This implies three inequalities on θ sin 2 θ * < 1 3 5 6 2 3 4 4 η χχ 2 , sin 2 θ < 5 6 η χχ , sin 2 θ sin 2 θ * > 24 5 1 η χχ .(35)”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4. Hybrid Inflation with Two Inflaton Fields
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“The sign of f NL is determined by the sign of η χχ .”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.1. Simplified Formula for the Observables When f NL Is Large.
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“we notice that in this example γ φ and γ χ can become much larger than unity at the end of inflation if |α| and/or |β| are much greater than one.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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“In particular, we provide analytical equations that express the nonlinearity parameter f NL , in terms of quantities that generalise the slow-roll parameters during inflation.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5. Multiple-Field Inflation without Slow Roll
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“The numerical calculation confirms the analytical prediction that it is possible to generate a narrow spike of large non-Gaussianity while the inflaton trajectory turns a corner, but the non-Gaussianity decays quickly after the corner [99,100].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 3. Non-Gaussianity during Slow-Roll Inflation
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- Not matchedcites [99]
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“We note that the dominant slow-roll parameters η ϕϕ and η χχ are constants during inflation in the vacuum dominated regime and that they are much larger than the slow-roll parameters ϕ and χ throughout inflation.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4. Hybrid Inflation with Two Inflaton Fields
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“For this potential, considering the condition for large non-Gaussianity, (26), large f NL is generated when the ϕ at horizon exit is small.One example is found to give | f NL | ∼ 30 with ϕ * ∼ 10 -3 M P for λ = 0.05 and χ * = 16M P .”
§ Review of Local Non-Gaussianity from Multifield Inflation > 3.2. Conditions for Generating a Large f NL .
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“Table 1 : Table showing some initial conditions for the hybrid inflation model that lead to large levels of non-Gaussianity.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2. Effect of the Waterfall Field and Further Evolution after Inflation.
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“For many values of the initial parameters when |η ϕϕ /η χχ | / = 1, the magnitude of f NL decreases compared to Table 1, but the reverse can also happen, an example of this where | f NL | grows by more than an order of magnitude is shown in the final row of Tables 1 and 2.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2. Effect of the Waterfall Field and Further Evolution after Inflation.
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“So, we have a red spectral index in agreement with present day observations [126] and in [112] it was also shown that the tensor-to-scalar ratio for this model is negligible.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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- Not matchedcites [126]
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“We also see that it is possible to generate a large non-Gaussianity during slow roll with η ϕϕ and η χχ both positive or both negative, or when one is positive and the other negative corresponding to a saddle point.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.1. Simplified Formula for the Observables When f NL Is Large.
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“Notice that the solution becomes singular at late times, when the scale factor vanishes and the field values diverge.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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“At the end of inflation, the parameters are γ φ = 29, γ χ = -13 and this shows that the slow-roll approximation has been significantly broken”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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“From Figure 2, one can see that the trajectory is straight for much of inflation but it turn near the end of inflation (during the last e-folding before inflation ends), and it is during this time that f NL grows larger than unity.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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“In the final example in Table 1, we give an explicit example of parameter values which give rise to an f NL which is probably too small to be detected with Planck but with a very large trispectrum through τ NL > 10 3 that should be detectable at a high significance.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1.1. Trispectrum in Two-Field Hybrid Inflation.
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“For both cases, the spectral index of f NL satisfies -4 η ϕϕ -η χχ < n fNL < 0,(77) for any value of r and hence f NL will be smaller on small scales.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
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“We have shown, at least in the case of a separable potential, that the trajectory is required to be almost entirely along the direction of one field but that the orthogonal field must become more important towards the end of inflation, and hence the inflationary trajectory must curve.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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“We note that if m 1 then from ( 60) n 1, so the trispectrum through g NL will give the dominant signal of non-Gaussianity through a large, negative g NL .”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1.1. Trispectrum in Two-Field Hybrid Inflation.
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“Our expression for f NL in (66) depends on the initial values through p and q, and this dependence drops out at leading order in α.Hence, f NL is independent of the number of e-foldings and scaleindependent.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
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“In the regime where |g NL | and τ NL are greater than unity, they are given by”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1.1. Trispectrum in Two-Field Hybrid Inflation.
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“The main conditions which must be satisfied is that the difference of the two η parameters must not be too small, η ϕϕ -η χχ ∼ 0.1, and the value of the χ field must be very subdominant to that of the ϕ field initially (or vice versa).”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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“For some parameter choices, this leads to a strong break down of slow roll before the end of inflation, which may give rise to a large non-Gaussianity.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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“We have shown how this can be done in the context a separable Hubble parameter instead of a separable potential and this leads to an exact expression for (the local part of) f NL in these models.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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“For our choice of parameters, we notice that f NL is still increasing at the end of inflation, see Figure 1, but for other choices it may start to decrease before inflation ends, see [115].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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- Not matchedcites [115]
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“As an explicit example, an exact two-field solution with an exponential potential was given.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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“It follows from (71) that τ NL > (6 f NL /5) 2 , so τ NL may be large and provide an extra observable parameter for this model.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1.1. Trispectrum in Two-Field Hybrid Inflation.
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“As an explicit case, we take α = -100, β = 20, m = 6, n = 6/5 and the initial conditions to satisfy (62) with N tot = 60.Then, independently of the values of A 1 and B 1 , we find f NL 58 from the simplified formula (66), and f NL 53 from the full formula (53).”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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In the paper
“The conditions can be satisfied for any possible combination of positive and negative η parameters, so the potential can be bowl-shaped, a hill-top, or have a saddle point.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
plain- Asserted: passes — The authors put it forward as true.
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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
“This shows that the fields roll much more quickly during the final stage of inflation, and the trajectory curves near the end.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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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
“All of these models generate the large non-Gaussianity after horizon exit, such as after reheating, at the end of inflation, at the phase transition, or during inflation after horizon exit and involves the perturbation of the nonadiabatic mode.Therefore, the non-Gaussianity of these models is of the local type which is distinguishable from other shapes of non-Gaussianity”
§ Review of Local Non-Gaussianity from Multifield Inflation > 1. Introduction
plaindirect- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Non-Guassianity of the local form can also be generated in the ekpyrotic scenario [50,51], although in the simplest case f NL is large and negative [52], which is observationally ruled out.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 2. Summary of Popular Models Generating a Large Local Non-Gaussianity
What the sources say
- Not matchedcites [50]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [51]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [52]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
- Not matched
In the paper
“It is well known that single-field inflation does not lead to a detectably large non-Gaussianity, in fact f NL is suppressed by slowroll parameters [13].”
- Also stated in: § 3. Non-Gaussianity during Slow-Roll Inflation · boosted
“It is well known that the single field slow-roll inflation with canonical kinetic terms generates a nonlinear parameter of the order of the spectral tilt of the spectrum and thus too small to be observed [13].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 1. Introduction
What the sources say
- Not matchedcites [13]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosteddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § 3. Non-Gaussianity during Slow-Roll Inflation · boosted
In the paper
“The current limit on the local type of the nonlinearity parameter f NL from seven years of WMAP data [2] is -10 < f NL < 74 at the 95% confidence level.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 1. Introduction
What the sources say
- Not matchedcites [2]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Observably large non-Gaussianity can be obtained by breaking the slow-roll conditions during inflation [14], using extended kinetic terms [15][16][17][18][19][20][21][22][23][24][25], see also the reviews [26,27], or going beyond models of single-field inflation [11,[28][29][30][31][32][33][34]”
§ Review of Local Non-Gaussianity from Multifield Inflation > 1. Introduction
What the sources say
- Not matchedcites [14]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [15]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [11]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- 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.
- Not matched
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 non-Gaussianity can also be generated at the end of inflation since the perturbation at the end of inflation is not required to be Gaussian.Thus, we expect that the observable large non-Gaussianity can be generated in this process.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 2.3. Inhomogeneous End of Inflation.
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“Using an analytic formula for the nonlinear parameter f NL in the case of a sum or product separable potential, it is possible to generate significant non-Gaussianity even during slow-roll inflation with Gaussian perturbations at Hubble exit and the general conditions for it can be written explicitly [94].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 3.2. Conditions for Generating a Large f NL .
What the sources say
- Not matchedcites [94]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Not matched
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 amplitude of f NL depends exponentially on the difference of the slow-roll parameters, η ϕϕ -η χχ , which we require to be positive to be in the branch of large non-Gaussianity where sin 2 θ 1, while the spectral index depends on a weighted sum of the slow-roll parameters, so it is possible to have a large non-Gaussianity and a scale invariant spectrum.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.1. Simplified Formula for the Observables When f NL Is Large.
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- The authors state this as an assumption: Inflation ends abruptly by a waterfall field which is heavy during inflation and hence does not affect the dynamics during inflation.
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, large non-Gaussianity can be realised in either of two regions cos 2 θ ≡ φ2 φ2 + χ2 ϕ ϕ + χ 1, or sin 2 θ ≡ χ2 φ2 + χ2 χ ϕ + χ 1.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4. Hybrid Inflation with Two Inflaton Fields
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- The authors state this as an assumption: Inflation ends abruptly by a waterfall field which is heavy during inflation and hence does not affect the dynamics during inflation.
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 because when r χ = 1, the curvaton energy density dominates over the radiation component before decay, then ζ is effectively the curvature perturbation of a single field (curvaton), ζ χ , and therefore non-Gaussianity f NL = f NL,χ = -5/4.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 2.1. Curvaton Scenario.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
- The authors state this as an assumption: The curvaton field decays suddenly on the H = Γ uniform energy-density hypersurface.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Recently, an analysis based on the general analytic formula was done by Byrnes et al. [94,95], and they showed that it is possible to generate observable large non-Gaussianity even during slow-roll multifield inflation models and explicitly wrote the conditions as well as concrete examples”
§ Review of Local Non-Gaussianity from Multifield Inflation > 3. Non-Gaussianity during Slow-Roll Inflation
What the sources say
- Not matchedcites [94]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [95]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Not matched
In the paper
“If the family of trajectories is not perpendicular to the surface on which inflation ends then there is a possibility that inflation ends on a slice which is not of uniform density and the generation of the curvature perturbation at the end of inflation is expected [34]”
§ Review of Local Non-Gaussianity from Multifield Inflation > 2.3. Inhomogeneous End of Inflation.
What the sources say
- Not matchedcites [34]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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.
It may state two separate claims as one.
- Not matched
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
“For large f NL , the substantial increase in sin θ is necessary between horizon exit and the end of inflation, which corresponds to the curve of the trajectory in the field space during the slow-roll inflation.However, the background trajectory must be almost entirely in the one-field direction.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 3.2. Conditions for Generating a Large f NL .
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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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
“In general, we cannot just ignore the curvature perturbation generated during inflation.Furthermore, that is also affected by the end of inflation.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 2.3. Inhomogeneous End of Inflation.
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
“It is in this regime that the two ways of writing the potential given above are equivalent (at leading order in slow roll), so we can treat the potential as being both sum and product separable.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4. Hybrid Inflation with Two Inflaton Fields
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- The authors state this as an assumption: Inflation ends abruptly by a waterfall field which is heavy during inflation and hence does not affect the dynamics during inflation.
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 reiterate that the formula for f NL is exact (apart from neglecting a slow-roll suppressed contribution due to the non-Gaussianity of the fields at Hubble exit) and is not based on a slow-roll expansion.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5. Multiple-Field Inflation without Slow Roll
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- The authors state this as an assumption: The quantities δ and γ are much smaller than unity at Hubble exit t=t*, so that the δN formalism can be used.
In the paper
“In the special case where the waterfall field is also light during inflation, Barnaby and Cline [110] have shown there is the possibility of generating a large non-Gaussianity during preheating for certain parameter values.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2.3. Further Evolution after Inflation.
What the sources say
- Not matchedcites [110]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedconditional- 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: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
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 this case, the potential does not have a minimum, but inflation still ends through H growing larger than unity, and we can trust our results in this regime.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one. 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
“Further evolution will depend on the details of reheating in a model dependent way.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2.3. Further Evolution after Inflation.
plainconditional- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“A potential with a similar form may be motivated in string theory, in the context of Kähler moduli inflation [118][119][120][121][122][123][124][125].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
What the sources say
- Not matchedcites [118]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [119]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [120]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [121]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [122]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [123]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [124]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [125]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedconditional- 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: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“It has also been shown for the hybrid potential we are considering that this can be used to generate a large amount of non-Gaussianity, for certain parameters values and fine tuning of the parameters [90,93].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2. Effect of the Waterfall Field and Further Evolution after Inflation.
What the sources say
- Not matchedcites [90]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [93]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plainconditional- 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: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
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 main idea on which our arguments are based is to focus on inflationary multifield trajectories in which the Hubble rate, and not the potential, is separable.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5. Multiple-Field Inflation without Slow Roll
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
In the paper
“We have checked in [95] that the correction from the extra expansion which occurs from the surface on which inflation ends up to a surface of uniform energy density is small, and therefore one can still use the δN formalism in this case.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2. Effect of the Waterfall Field and Further Evolution after Inflation.
What the sources say
- Not matchedcites [95]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosted- 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.
- Not matched
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
“However, in this case, inflation does not end abruptly when the waterfall field is destabilised so this is not the scenario we have considered in this paper.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2.3. Further Evolution after Inflation.
plain- 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.
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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.
In the paper
“The results for this model were verified using a novel calculational method of momentum transport equations in [109].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.1. Simplified Formula for the Observables When f NL Is Large.
What the sources say
- Not matchedcites [109]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- The authors state this as an assumption: Inflation ends abruptly by a waterfall field which is heavy during inflation and hence does not affect the dynamics during inflation.
- Not matched
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 potential would need modifications which apply after inflation in order for reheating to take place and these modifications may provide a minimum for the potential.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
hedgedconditional- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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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. It may state two separate claims as one.
In the paper
“Very few exact solutions in multiple-field inflation are known [117].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
What the sources say
- Not matchedcites [117]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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.
- Not matched
In the paper
“This is possible even if there is only one inflaton field and the waterfall field present.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2.3. Further Evolution after Inflation.
What the sources say
- Not matchedcites [110]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedconditional- 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: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
In the paper
“A recent claim that even a heavy waterfall field may lead to a scale invariant local non-Gaussianity has been withdrawn [111].”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2.3. Further Evolution after Inflation.
What the sources say
- Not matchedcites [111]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- 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: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
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 singularity occurs well after inflation ends, and we will not need to discuss it in our analysis.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
boosted- 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.
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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
“However, if we consider the role of the waterfall field, then after the waterfall field is destabilised, there may be a further evolution of the primordial curvature perturbation, which will lead to a change of the observable parameters.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2.3. Further Evolution after Inflation.
hedgedconditional- 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
“Although δ * and γ * are necessarily small, the quantity A is not suppressed by these parameters and can assume large values during inflation, enhancing f NL .”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5. Multiple-Field Inflation without Slow Roll
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: The quantities δ and γ are much smaller than unity at Hubble exit t=t*, so that the δN formalism can be used.
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 corresponds to a break down in slow roll, although H remains smaller than unity during the inflationary era by definition.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
plaincorrelational- Asserted: unclear — It is unclear whether 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: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one. The quote only partly states this claim.
In the paper
“We note in agreement with [99] that if one of the fields has reached a minimum so φ = 0 or χ = 0 at the end of inflation then A H = 0 and the non-Gaussianity at the end of inflation will be very small.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5. Multiple-Field Inflation without Slow Roll
What the sources say
- Not matchedcites [99]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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.
- Not matched
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 also possible in the exact solution which we presented in Section 5 and we consider this next.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1.1. Trispectrum in Two-Field Hybrid Inflation.
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: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“In detail, we see from (4.4) and (4.24) in [93] that their formulas for f NL does not depend on N or on any quantities evaluated at Hubble exit.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
What the sources say
- Not matchedcites [93]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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.
It may state two separate claims as one.
- Not matched
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
“For these three scenarios, the light field which generates the primordial curvature perturbation after or at the end of inflation can be treated as a test field which does not affect the inflationary dynamics.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“and the trispectrum nonlinearity parameters can be calculated using [98,106,127]”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1. Trispectrum.
What the sources say
- Not matchedcites [98]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [106]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [127]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
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 see that g NL is subdominant to f NL and hence will not provide a competitive observational signature.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1.1. Trispectrum in Two-Field Hybrid Inflation.
plaindirect- 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
“In the curvaton and modulated reheating scenarios, this conversion occurs after the end of inflation, while in the inhomogeneous end of inflation scenario, this conversion occurs on the non-uniform energy density hypersurface on which inflation ends.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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 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
“However, it is usually assumed that τ NL ∼ f 2 NL since both arise from second derivatives in the δN formalism.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1.1. Trispectrum in Two-Field Hybrid Inflation.
hedgedcorrelational- 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: passes — The paper gives a reason why it holds.
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
“This observational sensitivity to n fNL is about a factor of two larger than the current preferred value of the power spectrums spectral index and may therefore provide an interesting extra constraint on non-Gaussian models.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
hedgedconditional- 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
“In the example of two-field hybrid inflation, f NL has a scale dependence both because of the exponential term in f NL , (41), and because r will vary through the change of the initial value of sin 2 θ * .”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
plaindirect- 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
“We can also see in agreement with the statement at the beginning of this subsection that in this case, if the mass of the two fields are equal, then the two fields have the same scale dependence and n fNL = 0.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
plaindirect- 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
“Alternatively, the trispectrum through a large g NL might give the first observational signature, as is possible in self-interacting curvaton models or the exact solution with an exponential potential.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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.
It may state two separate claims as one.
- Rests on an assumption: The trispectrum (four-point function) depends on two nonlinearity parameters.
In the paper
“For this, we require that r 1, although in practice if we make it too small it may no longer be possible to satisfy a constraint on a minimum possible field velocity consistent with a classical slow-roll trajectory, as discussed in [95]”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1.1. Trispectrum in Two-Field Hybrid Inflation.
What the sources say
- Not matchedcites [95]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plainconditional- 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.
- Not matched
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
“Hence, we see that the trispectrum depends on two nonlinearity parameters (as opposed to one, f NL , for the bispectrum), and they may be observationally distinguishable since they are prefactors of terms with different shape dependences (68).”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1. Trispectrum.
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.
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
“One similarity that this hybrid inflation model has together with the quadratic curvaton scenario is that in both cases the initially subdominant (approximately isocurvature) field χ has the ratio δχ/χ ∼ ζ χ approximately constant and the field fluctuations do not become more non-Gaussian with time.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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.
It may state two separate claims as one.
In the paper
“However, it was shown in [143] that provided one takes the derivative while keeping the ratio of the three k vectors fixed then n fNL is independent of the shape of the triangle described by the three k vectors, which makes this a well defined quantity.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
What the sources say
- Not matchedcites [143]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosted- 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.
- Not matched
In the paper
“Observational prospects for this quantity were considered in [144], who showed that the Planck satellite is sensitive to n fNL 0.1, assuming a fiducial value of f NL = 50.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
What the sources say
- Not matchedcites [144]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
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
“However, the amount also depends on r, and when this is large, then the χ field is almost solely responsible for generating ζ at both first and second order and n fNL is suppressed.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
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 stress that in the regime where |α|, |β| 1 the slow-roll parameters will necessarily become much greater than unity by the end of inflation, which may correspond to |γ ss | 1: this is one reason by which f NL can become large.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one.
In the paper
“This inequality between τ NL and f NL is true in general [75], and equality is reached whenever a single field direction during inflation generates the primordial curvature perturbation.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.1.1. Trispectrum in Two-Field Hybrid Inflation.
What the sources say
- Not matchedcites [75]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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.
- Not matched
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
“Our formalism is at least in principle suitable to study the evolution of non-Gaussianity after inflation, when the parameter H becomes larger than unity”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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. 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
“If the current observational hints (which are not statistically significant) that f NL ∼ 40 turn out to be true, then both the bispectrum and the trispectrum should be large enough for Planck to detect.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
plainconditional- 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.
- Rests on an assumption: The trispectrum (four-point function) depends on two nonlinearity parameters.
- Rests on an assumption: In general, τ_NL ≥ (6 f_NL /5)^2.
- Rests on an assumption: Planck will be able to detect the bispectrum and trispectrum if they are large enough.
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
“If f NL is detected, it will also be possible to either constrain or detect a scale dependence of this parameter.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
hedgedconditional- 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: passes — The paper gives a reason why it holds.
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 stress that this formula provide only the dominant contributions to f NL .It is valid in the case |α|, |β| > R 1, and m, n < R 2 .”
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
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.
- The authors state this as an assumption: The quantities m and n are not too big; they satisfy m,n < R2.
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 absolute terms, the change to the angle of the Advances in Astronomy background trajectory is small (compared to a trajectory which turns by a right angle during inflation), but in relative terms it must grow by at least an order of magnitude.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
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.
In the paper
“In the specific cases, they considered to generate a large non-Gaussianity the non-Gaussianity was generated purely at the end of inflation and f NL is scale-independent.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 6.2. Scale Dependence of f NL .
What the sources say
- Not matchedcites [93]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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.
- Not matched
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
“It is then a model, dependent question whether there will be further evolution to the observables during reheating in this model, this deserves further attention.”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
plain- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.
The authors state this as an assumption.
Inflation ends abruptly by a waterfall field which is heavy during inflation and hence does not affect the dynamics during inflation.
“We assume that inflation ends abruptly by a waterfall field which is heavy during inflation and hence does not affect the dynamics during inflation.”
The authors state this as an assumption.
The curvaton field decays suddenly on the H = Γ uniform energy-density hypersurface.
“Assuming the sudden decay of the curvaton field on the H = Γ uniform energydensity hypersurface, that is, when the local Hubble rate equals the decay rate for the curvaton, leads to a nonlinear relation between the local curvaton density and the radiation density before and after the decay”
The authors state this as an assumption.
The transition to the radiation epoch at the end of inflation is quick, so the dynamics of the waterfall field can be neglected.
“So far, we have assumed a quick transition to the radiation epoch at the end of inflation, thereby neglecting the dynamics of the waterfall field.”
The authors state this as an assumption.
The quantities δ and γ are much smaller than unity at Hubble exit t=t*, so that the δN formalism can be used.
“We assume that the quantities δ and γ are much smaller than unity at Hubble exit t = t * , in order that we can use the δN formalism.”
The authors state this as an assumption.
The quantities m and n are not too big; they satisfy m,n < R2.
“We assume that the quantities m and n are not too big.Namely, they satisfy the inequality m, n R 2 .”
The authors state this as an assumption.
The fields have a Gaussian distribution at Hubble exit.
“assuming that the fields have a Gaussian distribution at Hubble exit”
The authors state this as an assumption.
The local model of non-Gaussianity is assumed when interpreting the WMAP constraint.
“assuming the local model of non-Gaussianity”
The paper relies on this without support.
The trispectrum (four-point function) depends on two nonlinearity parameters.
“The trispectrum (four-point function) depends on two nonlinearity parameters.”
The paper relies on this without support.
In general, τ_NL ≥ (6 f_NL /5)^2.
“In general, τ NL ≥ (6 f NL /5) 2 .”
The paper relies on this without support.
Planck will be able to detect the bispectrum and trispectrum if they are large enough.
“then both the bispectrum and the trispectrum should be large enough for Planck to detect.”
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
§ Review of Local Non-Gaussianity from Multifield Inflation > 5.1. Exact Solution.
“f NL 53 from the full formula (53).”
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
“Inflation ends when H = 1; for this example f NL 59 at that time.”
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
§ Review of Local Non-Gaussianity from Multifield Inflation > 7. Conclusions
“| 0.02 | -0.04 | 1 | 0.00037 | 1 | 3.5 | 0 .02 | 0.026 |”
§ Review of Local Non-Gaussianity from Multifield Inflation > 4.2. Effect of the Waterfall Field and Further Evolution after Inflation.
“0.02 -0.04 1 0.00037 1 -11.1 357 -2.6 -0.02 0.002”
Citations
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Non-gaussian features of primordial fluctuations in single field inflationary models
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Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution
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Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution
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Planck 2015 results: XVII. Constraints on primordial non-Gaussianity
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Metadata
- Authors
- Byrnes, C. T., & Choi, K.-Y.
- Affiliations
- Bielefeld University (DE)
- Pusan National University (KR)
- Corresponding author
- Choi, Ki-Young
- Venue
- Advances in Astronomy
- Publisher
- Wiley
- Published
- 2010-01
- Volume
- 2010
- Issue
- 1
- ISSN
- 1687-7969 (ISSN-L)
- 1687-7977
- OpenAlex
- W2120328039
- Semantic Scholar
- 98df8ab56191b74076cc80e705ab8bc9f2686a29
- Cited by
- 178
- References
- 163
- Citations by year
- 2026: 5 · 2025: 9 · 2024: 7 · 2023: 7 · 2022: 7 · 2021: 7 · 2020: 10 · 2019: 3 · 2018: 15 · 2017: 6 · 2016: 9 · 2015: 7 · 2014: 14 · 2013: 20 · 2012: 17
- FWCI
- 12.88
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Locations
- Advances in Astronomy (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- PUB – Publications at Bielefeld University (Bielefeld University) (submittedVersion, other-oa)
- Publikationen an der Universität Bielefeld (Universität Bielefeld) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- Korea Research Foundation (KRF-2008-341-C00008)
- 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 › Nuclear and High Energy Physics › Dark Matter and Cosmic Phenomena
- Keywords
- local non-Gaussianity, multi-field inflation, curvaton, bispectrum, slow-roll inflation, separable potentials, trispectrum, primordial non-Gaussianity
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