Review of Local Non‐Gaussianity from Multifield Inflation

May combine two claims.

Numerical calculation confirms that a narrow spike of large non-Gaussianity can be generated while the inflaton trajectory turns a corner, but it decays quickly after the corner.

“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].”

• cites [99]Not matched

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May combine two claims.

In the vacuum-dominated regime, the dominant slow-roll parameters ηφφ and ηχχ are constants during inflation and are much larger than the slow-roll parameters φ and χ throughout inflation.

“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.”

• AssumptionThe 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.

May combine two claims.

For the quadratic times exponential potential, large fNL is generated when φ at horizon exit is small; one example gives |fNL| ∼ 30 with φ* ∼ 10^-3 MP for λ=0.05 and χ*=16MP.

“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 .”

• cites [94]Not matched

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

Table 1 shows initial conditions for the hybrid inflation model that lead to large levels of non-Gaussianity.

“Table 1 : Table showing some initial conditions for the hybrid inflation model that lead to large levels of non-Gaussianity.”

May combine two claims.

For many values of the initial parameters when |ηφφ/ηχχ| != 1, the magnitude of fNL decreases compared to Table 1, but the reverse can also happen; an example where |fNL| grows by more than an order of magnitude is shown in the final row of Tables 1 and 2.

“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.”

• AssumptionThe 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.

May combine two claims.

We have a red spectral index in agreement with present day observations, and the tensor-to-scalar ratio for this model is negligible.

“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.”

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May combine two claims.

It is possible to generate large non-Gaussianity during slow roll with ηφφ and ηχχ both positive or both negative, or when one is positive and the other negative corresponding to a saddle point.

“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.”

• AssumptionThe 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.

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

The solution becomes singular at late times, when the scale factor vanishes and the field values diverge.

“Notice that the solution becomes singular at late times, when the scale factor vanishes and the field values diverge.”

May combine two claims.

At the end of inflation, the parameters are γφ=29, γχ=-13, showing that the slow-roll approximation has been significantly broken.

“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”

May combine two claims.

From Figure 2, the trajectory is straight for much of inflation but turns near the end of inflation (during the last e-folding before inflation ends), and it is during this time that fNL grows larger than unity.

“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.”

May combine two claims.

In the final example in Table 1, we give an explicit example of parameter values which give rise to an fNL which is probably too small to be detected with Planck but with a very large trispectrum through τNL > 10^3 that should be detectable at a high significance.

“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.”

May combine two claims.

For both cases, the spectral index of fNL satisfies -4(ηφφ - ηχχ) < n_fNL < 0 for any value of r, and hence fNL will be smaller on small scales.

“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.”

May combine two claims.

At least in the case of a separable potential, the trajectory is required to be almost entirely along the direction of one field but the orthogonal field must become more important towards the end of inflation, and hence the inflationary trajectory must curve.

“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.”

May combine two claims.

If m << 1 then from (60) n >> 1, so the trispectrum through gNL will give the dominant signal of non-Gaussianity through a large, negative gNL.

“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 .”

May combine two claims.

Our expression for fNL in (66) depends on the initial values through p and q, and this dependence drops out at leading order in α; hence fNL is independent of the number of e-foldings and scale-independent.

“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.”

The quote only partly states this claim.

In the regime where |gNL| and τNL are greater than unity, they are given by equations (70) and (71).

“In the regime where |g NL | and τ NL are greater than unity, they are given by”

May combine two claims.

The main conditions which must be satisfied are that the difference of the two η parameters must not be too small, ηφφ - ηχχ ∼ 0.1, and the value of the χ field must be very subdominant to that of the φ field initially (or vice versa).

“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).”

May combine two claims.

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.

“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.”

May combine two claims.

We have shown how this can be done in the context of a separable Hubble parameter instead of a separable potential, and this leads to an exact expression for the local part of fNL in these models.

“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.”

May combine two claims.

For our choice of parameters, fNL is still increasing at the end of inflation, but for other choices it may start to decrease before inflation ends.

“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].”

• cites [115]Not matched

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

As an explicit example, an exact two-field solution with an exponential potential was given.

“As an explicit example, an exact two-field solution with an exponential potential was given.”

May combine two claims.

It follows from (71) that τNL > (6 fNL/5)^2, so τNL may be large and provide an extra observable parameter for this model.

“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.”

May combine two claims.

For α=-100, β=20, m=6, n=6/5 and initial conditions satisfying (62) with Ntot=60, independently of A1 and B1, fNL≈58 from simplified formula (66) and fNL≈53 from full formula (53).

“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).”

May combine two claims.

The conditions required to generate a large non-Gaussianity in two-field hybrid inflation 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.

“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.”

May combine two claims.

For the parameters given after (66), the fields roll much more quickly during the final stage of inflation, and the trajectory curves near the end.

“This shows that the fields roll much more quickly during the final stage of inflation, and the trajectory curves near the end.”

May combine two claims.

The models reviewed in Section 2 generate large non-Gaussianity after horizon exit, and therefore their non-Gaussianity is of local type.

“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”

May combine two claims.

Local-form non-Gaussianity can also be generated in the ekpyrotic scenario, but in the simplest case fNL is large and negative and observationally ruled out.

“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.”

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May combine two claims.

Single-field inflation does not lead to a detectably large non-Gaussianity; fNL is suppressed by slow-roll parameters.

“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].”

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

The current limit on local fNL from seven years of WMAP data is -10 < fNL < 74 at the 95% confidence level.

“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.”

• cites [2]Not matched

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May combine two claims.

Observably large non-Gaussianity can be obtained by breaking slow-roll conditions during inflation, using extended kinetic terms, or going beyond single-field inflation.

“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]”

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May combine two claims.

Observable large non-Gaussianity can be generated at the end of inflation because the perturbation at the end of inflation is not required to be Gaussian.

“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.”

May combine two claims.

Using an analytic formula for fNL for sum or product separable potentials, significant non-Gaussianity can be generated even during slow-roll inflation with Gaussian perturbations at Hubble exit, and general conditions can be written explicitly.

“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].”

• cites [94]Not matched

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May combine two claims.

The amplitude of fNL depends exponentially on ηφφ - ηχχ, which must be positive for the large non-Gaussianity branch where sin^2 θ ≫ 1; the spectral index depends on a weighted sum of slow-roll parameters, so large non-Gaussianity and a scale-invariant spectrum are possible.

“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.”

• AssumptionThe 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.

May combine two claims.

In two-field hybrid inflation, large non-Gaussianity can be realised in either of two regions: cos^2 θ ≫ 1 or sin^2 θ ≫ 1.

“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.”

• AssumptionThe 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.

May combine two claims.

When rχ = 1, the curvaton dominates before decay, ζ is effectively the single-field curvaton perturbation, and fNL = -5/4.

“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.”

• AssumptionThe authors state this as an assumption: The curvaton field decays suddenly on the H = Γ uniform energy-density hypersurface.

May combine two claims.

Byrnes et al. showed that observable large non-Gaussianity can be generated even during slow-roll multifield inflation and wrote the conditions and examples.

“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”

• cites [94]Not matched

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• cites [95]Not matched

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May combine two claims.

If the family of inflationary trajectories is not perpendicular to the surface on which inflation ends, inflation can end on a non-uniform-density slice and curvature perturbation is generated at the end of inflation.

“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]”

• cites [34]Not matched

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May combine two claims.

For large fNL, a substantial increase in sin θ is necessary between horizon exit and the end of inflation, while the background trajectory must be almost entirely in the one-field direction.

“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.”

May combine two claims.

In general, the curvature perturbation generated during inflation cannot be ignored and is also affected by the end of inflation.

“In general, we cannot just ignore the curvature perturbation generated during inflation.Furthermore, that is also affected by the end of inflation.”

May combine two claims.

In the vacuum-dominated regime of two-field hybrid inflation, the two ways of writing the potential are equivalent at leading order in slow roll, so the potential can be treated as both sum and product separable.

“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.”

• AssumptionThe 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.

May combine two claims.

The formula for fNL 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.

“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.”

• AssumptionThe 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.

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

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.

“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.”

• cites [110]Not matched

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May combine two claims. The quote only partly states this claim.

In the regime where |α| and |β| are both large, the potential does not have a minimum, but inflation still ends through ε growing larger than unity, and we can trust our results in this regime.

“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.”

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

Further evolution after inflation will depend on the details of reheating in a model dependent way.

“Further evolution will depend on the details of reheating in a model dependent way.”

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

A potential with a similar form may be motivated in string theory, in the context of Kähler moduli inflation.

“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].”

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

For the hybrid potential considered, it has been shown that a large amount of non-Gaussianity can be generated for certain parameter values and fine tuning of the parameters.

“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].”

• cites [90]Not matched

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• cites [93]Not matched

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Classified as probably not a substantial claim.

The main idea of the method is to focus on inflationary multifield trajectories in which the Hubble rate, and not the potential, is separable.

“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.”

May combine two claims.

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, so one can still use the δN formalism in this case.

“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.”

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

In the preheating scenario, inflation does not end abruptly when the waterfall field is destabilised, so it is not the scenario considered in this 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.”

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

The results for this model were verified using a novel calculational method of momentum transport equations.

“The results for this model were verified using a novel calculational method of momentum transport equations in [109].”

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• AssumptionThe 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.

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

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.

“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.”

Classified as probably not a substantial claim.

Very few exact solutions in multiple-field inflation are known.

“Very few exact solutions in multiple-field inflation are known [117].”

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

This preheating non-Gaussianity is possible even if there is only one inflaton field and the waterfall field present.

“This is possible even if there is only one inflaton field and the waterfall field present.”

• cites [110]Not matched

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

A recent claim that even a heavy waterfall field may lead to a scale invariant local non-Gaussianity has been withdrawn.

“A recent claim that even a heavy waterfall field may lead to a scale invariant local non-Gaussianity has been withdrawn [111].”

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

This singularity occurs well after inflation ends.

“This singularity occurs well after inflation ends, and we will not need to discuss it in our analysis.”

May combine two claims.

If the role of the waterfall field is considered, after the waterfall field is destabilised there may be further evolution of the primordial curvature perturbation, which will lead to a change of the observable parameters.

“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.”

May combine two claims.

Although δ* and γ* are necessarily small, the quantity A is not suppressed by these parameters and can assume large values during inflation, enhancing fNL.

“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 .”

• AssumptionThe 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.

May combine two claims. The quote only partly states this claim.

This corresponds to a break down in slow roll, although ε remains smaller than unity during the inflationary era by definition.

“This corresponds to a break down in slow roll, although H remains smaller than unity during the inflationary era by definition.”

May combine two claims.

If one of the fields has reached a minimum so φdot=0 or χdot=0 at the end of inflation then A_H=0 and the non-Gaussianity at the end of inflation will be very small.

“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.”

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

This is also possible in the exact solution presented in Section 5.

“This is also possible in the exact solution which we presented in Section 5 and we consider this next.”

May combine two claims.

In [93], their formulas for fNL do not depend on N or on any quantities evaluated at Hubble exit.

“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.”

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

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.

“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.”

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

The trispectrum nonlinearity parameters can be calculated using equation (69).

“and the trispectrum nonlinearity parameters can be calculated using [98,106,127]”

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May combine two claims.

gNL is subdominant to fNL and hence will not provide a competitive observational signature.

“We see that g NL is subdominant to f NL and hence will not provide a competitive observational signature.”

May combine two claims.

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.

“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.”

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

It is usually assumed that τNL ∼ fNL^2 since both arise from second derivatives in the δN formalism.

“However, it is usually assumed that τ NL ∼ f 2 NL since both arise from second derivatives in the δN formalism.”

May combine two claims.

This observational sensitivity to n_fNL is about a factor of two larger than the current preferred value of the power spectrum's spectral index and may therefore provide an interesting extra constraint on non-Gaussian models.

“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.”

May combine two claims.

In the example of two-field hybrid inflation, fNL has a scale dependence both because of the exponential term in fNL, (41), and because r will vary through the change of the initial value of sin^2 θ*.

“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 * .”

May combine two claims.

If the mass of the two fields are equal, then the two fields have the same scale dependence and n_fNL = 0.

“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.”

May combine two claims.

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.

“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.”

• AssumptionRests on an assumption: The trispectrum (four-point function) depends on two nonlinearity parameters.

May combine two claims.

For this, we require 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.

“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]”

• cites [95]Not matched

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

May combine two claims.

The trispectrum depends on two nonlinearity parameters (as opposed to one, fNL, for the bispectrum), and they may be observationally distinguishable since they are prefactors of terms with different shape dependences.

“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).”

May combine two claims.

In both the hybrid inflation model and the quadratic curvaton scenario, the initially subdominant (approximately isocurvature) field χ has the ratio δχ/χ ∼ ζχ approximately constant and the field fluctuations do not become more non-Gaussian with time.

“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.”

May combine two claims.

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.

“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.”

• cites [143]Not matched

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

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

Observational prospects for n_fNL were considered in [144], who showed that the Planck satellite is sensitive to n_fNL ~ 0.1, assuming a fiducial value of fNL = 50.

“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.”

• cites [144]Not matched

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

May combine two claims.

The amount of scale dependence also depends on r, and when r is large, the χ field is almost solely responsible for generating ζ at both first and second order and n_fNL is suppressed.

“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.”

May combine two claims.

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 |γss| >> 1; this is one reason by which fNL can become large.

“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.”

May combine two claims.

The inequality τNL > (6 fNL/5)^2 is true in general, and equality is reached whenever a single field direction during inflation generates the primordial curvature perturbation.

“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.”

• cites [75]Not matched

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

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). The quote only partly states this claim.

Our formalism is at least in principle suitable to study the evolution of non-Gaussianity after inflation, when the parameter ε becomes larger than unity.

“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”

May combine two claims.

If the current observational hints that f_NL is approximately 40 are true, then both the bispectrum and the trispectrum should be large enough for Planck to detect.

“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.”

• AssumptionRests on an assumption: The trispectrum (four-point function) depends on two nonlinearity parameters.
• AssumptionRests on an assumption: In general, τ_NL ≥ (6 f_NL /5)^2.
• AssumptionRests on an assumption: Planck will be able to detect the bispectrum and trispectrum if they are large enough.

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

If f_NL is detected, it will also be possible to either constrain or detect a scale dependence of this parameter.

“If f NL is detected, it will also be possible to either constrain or detect a scale dependence of this parameter.”

May combine two claims.

The formula (66) provides only the dominant contributions to fNL and is valid in the case |α|, |β| > R1, and m, n < R2.

“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 .”

• AssumptionThe authors state this as an assumption: The quantities m and n are not too big; they satisfy m,n < R2.

May combine two claims.

In absolute terms, the change to the angle of the background trajectory is small, but in relative terms it must grow by at least an order of magnitude.

“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.”

May combine two claims.

In the specific cases considered in [93] to generate a large non-Gaussianity, the non-Gaussianity was generated purely at the end of inflation and fNL is scale-independent.

“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.”

• cites [93]Not matched

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

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

It is a model dependent question whether there will be further evolution to the observables during reheating in this model.

“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.”