Spectral running and non-Gaussianity from slow-roll inflation in generalized two-field models
Open accessAbstract
Theories beyond the standard model such as string theory motivate low energy effective field theories with several scalar fields which are not only coupled through a potential but also through their kinetic terms. For such theories we derive the general formulae for the running of the spectral indices for the adiabatic, isocurvature and correlation spectra in the case of two-field inflation. We also compute the expected non-Gaussianity in such models for specific forms of the potentials. We find that the coupling has little impact on the level of non-Gaussianity during inflation.
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Written from the paper’s full text.
TL;DR
The paper derives general slow-roll formulas for the running of adiabatic, isocurvature, and correlation spectral indices in generalized two-field inflation with kinetic coupling, and computes non-Gaussianity for product and sum potentials. It finds that the kinetic coupling has little impact on the level of non-Gaussianity during inflation, with \(f_{\rm NL}\sim 1/N\) and unobservable in the slow-roll examples considered.
Key findings
- General expressions are obtained for the running spectral indices \(\alpha_\zeta\), \(\alpha_S\), and \(\alpha_C\); \(\alpha_S\) is independent of the correlation angle \(\Delta\), meaning it is fixed at horizon crossing and does not change thereafter.
- The first non-Gaussianity term is approximated as \(-6/5 f_{\rm NL}^{(3)} \sim r/16\), too small to be observable; for Jordan-Brans-Dicke product potentials the second term gives \(-6/5 f_{\rm NL}^{(4)} \approx 1/(2N)\), with small coupling corrections such as \(\beta \phi/NM_P\) or \(\beta \phi^2/NM_P^2\).
- For sum quadratic potentials, numerical results give \(f_{\rm NL}\sim 1/N\), and the authors conclude that \(\beta\) has little impact on the level of non-linearity.
- Enough inflation in the JBD examples requires \(\beta<0.05\) for \(\phi_*,\chi_*<50M_P\) in model 1, and \(\beta<0.0005\) in model 2; analytical slow-roll approximations agree with numerical results within 5%.
Methods
- The authors study a two-field action with a non-canonical kinetic term \(e^{2b(\phi)}\partial_\mu\chi\partial^\mu\chi\), derive background and perturbation equations in slow roll, and use the transfer-matrix formalism for adiabatic/isocurvature perturbations.
- Non-Gaussianity is computed using the \(\delta N\) formalism for separable product potentials \(W=U(\phi)V(\chi)\) and sum potentials \(W=U(\phi)+V(\chi)\).
- Specific models include Jordan-Brans-Dicke and Brans-Dicke-type product potentials with quartic or quadratic \(\chi\) potentials, plus sum quadratic potentials; numerical solutions of the background and Friedmann equations are obtained without assuming slow roll on a \(\phi\)-\(\chi\) grid.
Limitations
- Stated: \(\alpha_C\) diverges at \(\Delta\to\pi/2\), but this divergence is not observable; no analytical approximation was found for models 3 and 4, so numerical results are given for simple sum-potential models.
- Evident: the analysis is restricted to two scalar fields, slow-roll inflation, and specific separable potentials; the numerical grid uses initial values \(\phi_i\) and \(\chi_i\) with \(\dot\phi_i=\dot\chi_i=0\), and no full observational likelihood or parameter-estimation analysis is performed.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
9 of 11 cited sources checked · 1 supported · 0 flagged · accuracy not yet measured
- Premisecitation
In the presence of multiple fields during inflation, the curvature perturbation ζ does not remain constant and varies because non-vanishing isocurvature perturbations act as a source term for the change of ζ.
“However, in the presence of multiple fields during the inflationary epoch, ζ does not remain constant and varies since the non-vanishing isocurvature perturbations act as a source term for the change of ζ [5,6].”
Quote in the paperSubstantial claim2 of 2 sources checked- cites [5] source
The perturbed-field-equations section (p. 7) says entropy perturbations can cause ζ to evolve on super-Hubble scales in a two-field model. But it also says ζ remains constant for adiabatic modes, and the Conclusions (p. 15) identify two-field regimes where it remains constant. The claim overstates this possibility as an inevitable consequence of multiple fields. (LLM, conf=0.88)
Experimental — accuracy not yet measured - cites [6] source
- Findingcitation
The paper derives general formulae for the running of the spectral indices for the adiabatic, isocurvature and correlation spectra in the case of two-field inflation.
“For such theories we derive the general formulae for the running of the spectral indices for the adiabatic, isocurvature and correlation spectra in the case of two-field inflation.”
- Also stated in: § JCAP02(2007)029 · plain
“We obtain the general formulae for the running of the spectral indices (adiabatic, non-adiabatic and correlated), following [27] and [28].”
Quote in the paperSubstantial claim1 of 2 sources checked- cites [27] Barnaby N Cline J M, 2006
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [28] source
Section IV (p. 5) gives spectral indices for adiabatic and isocurvature perturbations in an exact two-field inflationary solution. It does not provide general formulae for the running of those indices or for a correlation spectrum. (LLM, conf=0.86)
Experimental — accuracy not yet measured
- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- Also stated in: § JCAP02(2007)029 · plain
- Interpretationcitation
The first term -6/5 f(3)_NL is too small to be observable in CMB experiments.
“which is certainly too small to be observable in CMB experiments.Also see [39].”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [39] Seery D, 2005, J. Cosmol. Astropart. Phys. source
Section 2.3 (p. 10) identifies the first term as the momentum-dependent contribution, and the Conclusions (p. 21) say non-Gaussianity arising from microphysics is typically small when fluctuations are close to scale-invariance. The paper does not establish that this term is too small to be observable in CMB experiments; it also says the term can be important when the superhorizon contribution is absent. (LLM, conf=0.82)
Experimental — accuracy not yet measured
- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- Findingnone given
The coupling has little impact on the level of non-Gaussianity during inflation.
“We find that the coupling has little impact on the level of non-Gaussianity during inflation.”
Quote in the paperSubstantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- Findingreasoning
When b_phi = 0, the running spectral indices are given by θ, Δ and 9 slow-roll parameters.
“However, considering the component fields (σ, s), H, T SS and ξ 2 sss do not appear in the running of spectral indices, which reduces the parameters to 11. Therefore the running spectral indices are given by: θ, Δ and 9 slow-roll parameters.”
Quote in the paperSubstantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- Findingreasoning
The running spectral index of isocurvature perturbation α_S is independent of Δ.
“We note that the running spectral index of isocurvature perturbation, α S , is independent of Δ, which means that the α S is determined at horizon crossing and does not change thereafter.”
Quote in the paperSubstantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- Findingown data
Equation (4.2) is valid for non-canonical kinetic terms.
“In appendix C, we have shown equation (4.2) is valid for non-canonical kinetic terms.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- Findingreasoning
The running of the spectral index is affected by the presence of the coupling.
“The running is affected by the presence of the coupling and therefore future experiments constraining the running of the spectral index will give vital information about the existence of fields which couple non-trivially to the field(s) driving inflation.”
Quote in the paperSubstantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- Findingown data
The paper computes the expected non-Gaussianity in such models for specific forms of the potentials.
“We also compute the expected non-Gaussianity in such models for specific forms of the potentials.”
- Also stated in: § JCAP02(2007)029 · plain
“Furthermore, we derive the general formulae for non-Gaussianity and apply them to some specific models.”
Quote in the paperSubstantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- Also stated in: § JCAP02(2007)029 · plain
- Findingnone given
For sum potentials W=U+V=1/2 m_φ^2 φ^2 + 1/2 m_χ^2 χ^2, the slow-roll parameters simplify greatly and are given by equations (6.2)-(6.4).
“If we define the potential W (ϕ, χ) = U(ϕ) + V (χ) = 1 2 m 2 ϕ ϕ 2 + 1 2 m 2 χ χ 2 , (6 .1) then the slow-roll parameters simplify greatly and are given by”
Quote in the paperSubstantial claim - Findingreasoning
In the very small β limit, equation (5.18) can be approximated by equation (5.19).
“In this very small limit, we can approximate equation (5.18):”
Quote in the paperSubstantial claim - Findingown data
Numerically, f_NL is of the order 1/N even with non-canonical couplings.
“We find numerically that the f NL is of that order of magnitude, even with non-canonical couplings.”
Quote in the paperSubstantial claim - Findingreasoning
In order to obtain enough inflation, for φ*, χ* < 50 M_P, β < 0.0005 is required.
“In order to obtain enough inflation, for ϕ * , χ * < 50M P , then we require β < 0.0005.”
Quote in the paperSubstantial claim - Findingreasoning
In the limit 8β^2 N << 1, equation (5.10) gives *n/4N < 0.004n, taking N ≈ 60.
“From this fact, 8β 2 N 1 and the limit of equation (5.10) gives * n/4N < 0.004n, where we have taken N ≈ 60.”
Quote in the paperNot a substantial claim - Findingnone given
The paper derived general formulae for the running of the spectral indices for adiabatic and entropic perturbations as well as for the cross-correlation spectrum.
“We have derived the general formulae for the running of the spectral indices for adiabatic and entropic perturbations as well as for the cross-correlation spectrum.”
Quote in the paperSubstantial claim - Findingown data
The calculated spectral indices and runnings for several example models are within present observational ranges.
“We have calculated the spectral indices and runnings for several example models.The results are within present observational ranges.”
Quote in the paperSubstantial claim - Findingnone given
The final term in equation (5.20) is directly due to b_φφ=0.
“The final term is directly due to b ϕϕ = 0.”
Quote in the paperNot a substantial claim - Findingreasoning
β < 0.05 is required to obtain enough e-foldings for φ*, χ* < 50 M_P.
“We find that β < 0.05 is required to obtain enough e-foldings for ϕ * , χ * < 50M P .”
Quote in the paperSubstantial claim - Findingreasoning
Using the δN-formalism and specializing to product and sum potentials, the paper derived general expressions for the non-linearity parameter f_NL during inflation.
“In addition, using the δN-formalism and specializing to the product potential (W (ϕ, χ) = U(ϕ)V (χ)) and the sum potential (W (ϕ, χ) = U(ϕ) + V (χ)) we have derived the general expressions for the non-linearity parameter f NL during inflation.”
Quote in the paperSubstantial claim - Findingreasoning
Using the approximation in equation (5.19), f_NL can be written as equation (5.21).
“Using the approximation in equation (5.19), we find”
Quote in the paperSubstantial claim - Interpretationreasoning
Future experiments constraining the running of the spectral index will give vital information about the existence of fields that couple non-trivially to the field(s) driving inflation.
“The running is affected by the presence of the coupling and therefore future experiments constraining the running of the spectral index will give vital information about the existence of fields which couple non-trivially to the field(s) driving inflation.”
Quote in the paperSubstantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- Interpretationnone given
When the coupling is quadratic in the φ field (as in equation (5.16)), the non-linearity is enhanced by a term βφ^2/N M_P^2.
“When the coupling is quadratic in the ϕ field (as in equation (5.16)), the non-linearity is enhanced by a term βϕ 2 /N M 2 P .”
Quote in the paperNot a substantial claim - Interpretationreasoning
Due to the quadratic coupling, the potential is steeper than that of traditional Brans-Dicke models, so a much smaller β is required to give enough e-folding number.
“Due to the quadratic coupling, the potential is steeper than that of traditional Brans-Dicke models, so that a much smaller β is required to give enough e-folding number.”
Quote in the paperSubstantial claim - Interpretationown data
Therefore, β has little impact on the level of non-linearity.
“Therefore we conclude that β has little impact on the level of non-linearity.”
Quote in the paperSubstantial claim
- Classified as probably not a substantial claim. 13
- May combine two claims. 23
- The quote only partly states this claim. 1
Classified as probably not a substantial claim.
PremisecitationThe spectral indices are given in terms of slow-roll parameters as calculated in [28].
“The spectral indices are best written in terms of slow-roll parameters.They have been calculated in [28] and are given by”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [28] source
Section IV (pp. 5–6) calculates spectral indices for an exact inflationary solution and presents them as functions of p and tan θ. The passages do not give the indices in terms of slow-roll parameters, so that part of the attribution is unsupported by our copy. (LLM, conf=0.83)
Experimental — accuracy not yet measured
- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationCurrent observations limit the non-linearity parameter to |f_NL| < 100.
“Current observations limit the non-linearity parameter, |f NL | < 100 [30].”
Quote in the paperNot a substantial claim1 of 1 sources checked- cites [30] source
The paper discusses linear perturbations (Section II, p. 2) and observational implications of correlated perturbations (Introduction, p. 1), but the passages do not give a bound on the non-linearity parameter. Searches for f_NL, non-Gaussianity, and the value 100 found no passage stating |f_NL| < 100. (LLM, conf=0.76)
Experimental — accuracy not yet measured
May combine two claims.
FindingreasoningWhen b(φ) = 0, the calculation of f(4)_NL simplifies and the result is identical to that found in [38].
“When b(ϕ) = 0, this calculation of f (4) NL simplifies, since B S = C S = G = H = 0, α u = α v = 1, (4.29) and the result is identical to that found in [38].”
Quote in the paperNot a substantial claim1 of 1 sources checked- cites [38] source
The bispectrum discussion (p. 2) says the paper calculates f_NL, but our copy does not decode the resulting formula. Searches for b(φ) and the named terms did not find the b(φ)=0 simplification, so the claimed identity between the two results cannot be checked from our copy. (LLM, conf=0.70)
Experimental — accuracy not yet measured
- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationSimilar cases with canonical kinetic terms were previously studied in [38], where they were shown not to generate significant non-Gaussianity.
“Similar cases with canonical kinetic terms were previously studied in [38], where it was shown that they do not generate significant non-Gaussianity.”
Quote in the paperNot a substantial claim1 of 1 sources checked- cites [38] source
The two-component inflation-model example (p. 3) finds non-Gaussianity too small to observe, supporting the claim for that case. But another two-component model (p. 4) can yield |f_NL| > 1 under specified conditions. The paper therefore does not support the unqualified statement that similar cases do not generate significant non-Gaussianity; the passages also do not identify the models by their kinetic terms. (LLM, conf=0.84)
Experimental — accuracy not yet measured
- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
Classified as probably not a substantial claim.
PremisecitationIn previous work [38], for theories with canonical kinetic terms and a sum potential, f_NL ≈ 1/N.
“In previous work, [38], for theories with canonical kinetic terms and a sum potential, f NL ≈ 1/N .”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [38] source
In “A two-component inflation model” (p. 3), the paper gives N ∝ |θc/θ| and f_NL ≃ |θ/θc|, implying f_NL ∝ 1/N for that particular model. It does not establish the claimed result generally for theories with canonical kinetic terms and a sum potential; its other two-component example (p. 4) instead discusses conditions yielding |f_NL| > 1. (LLM, conf=0.82)
Experimental — accuracy not yet measured
May combine two claims.
FindingreasoningIn the slow-roll limit and on large scales, the evolutions of curvature and isocurvature perturbations can be written in terms of slow-roll parameters as ζ = BHS and Ṡ = -γHS.
“In the slow-roll limit and on large scales, we find the evolutions of curvature and isocurvature perturbation can be written in terms of slow-roll parameters, ζ BHS, Ṡ -γHS, (2.18)”
Quote in the paperSubstantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
May combine two claims.
Findingnone givenIn the small β limit, *_φ < *_χ and the last two terms in equation (5.14) are negligible compared to the first two terms.
“It can be noted that in this small β limit, * ϕ < * χ and the last two terms in this equation are negligible compared to the first two terms.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
FindingreasoningFor models with a sum of two quadratic potentials, due to the complexity of the analytic formula for f_NL, it is difficult to find an approximate equation.
“We also considered models with a sum of two quadratic potentials.Due to the complexity of the analytic formula for f NL , it is difficult to find an approximate equation.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
FindingreasoningIn this limit of β, equation (5.10) can be approximated by equation (5.11).
“In this limit of β, we can approximate equation (5.10) further:”
Quote in the paperNot a substantial claimMay combine two claims.
FindingreasoningIn the JBD example with quadratic and quartic potentials, the slow-roll parameter for field φ is given by the constant coupling parameter β, and the slow-roll condition for φ implies β itself has to be small.
“One of the specific examples we have considered is the Jordan-Brans-Dicke theory with quadratic and quartic potentials.In this case, the slow-roll parameter for the field ϕ is given by the (constant) coupling parameter β, defined in equation (5.9).The slow-roll condition for the field ϕ implies that β itself has to be small.”
Quote in the paperNot a substantial claimMay combine two claims.
FindingreasoningFor V(χ)=λχ^n in JBD theory, using e = e_φ + e_χ = 1, N can be analytically solved in the slow-roll limit, yielding equation (5.10).
“It is possible to consider the case V (χ) = λχ n .In this JBD theory, and using the fact that at the end of inflation e = e ϕ + e χ = 1, we can analytically solve N in the slow-roll”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown dataThe paper provides numerical results for simple models and showed that f_NL is unobservable for the slow-roll limit.
“We therefore provide numerical results for simple models and showed that f NL 1 and is unobservable for the slow-roll limit.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Findingown dataThe results for model 4 are very similar to those of model 3 and are therefore not shown.
“The results for model 4 are very similar to those of model 3 and are therefore not shown.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingnone givenModels 2a and 2b have a reflection symmetry about φ (θ→-θ) and/or χ (θ→π-θ).
“Models 2a and 2b have a reflection symmetry about ϕ (θ → -θ) and/or χ (θ → π-θ).”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
FindingreasoningWhen b(φ) = 0, α = 1 and b = 0, restoring the symmetry between φ and χ in equation (4.10).
“When b(ϕ) = 0, then α = 1 and b = 0, which means that the symmetry between ϕ and χ in equation (4.10) is restored.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
May combine two claims.
FindingreasoningAll first- and second-order slow-roll parameters are proportional to and 2 respectively, making the primordial spectral indices proportional to 2 in the lowest order.
“Hence, all the first-and second-order slow-roll parameters are proportional to and 2 respectively which make the primordial spectral indices proportional to 2 in the lowest order.”
Quote in the paperSubstantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- The authors state this as an assumption: In the Jordan-Brans-Dicke theory considered, β is a positive constant.
Classified as probably not a substantial claim.
FindingreasoningUsing equation (5.11), f_NL can be written in terms of the required e-folding number in the small β limit, yielding equation (5.15).
“Using equation (5.11), f NL can be written in terms of the required e-folding number in the small β limit (required for enough inflation)”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningFor this sum potential, it is no longer possible to directly relate the parameters to and θ; however, if the ratio of masses is r=m_χ^2/m_φ^2, the slow-roll parameters can be rewritten in terms of , β and re^{-2b} only, as in equations (6.6)-(6.7).
“It is no longer possible to directly relate the parameters to and θ.However, if we represent the ratio of masses by r = m 2 χ m 2 ϕ , (6.5) then the slow-roll parameters can be rewritten in terms of , β and re -2b only:”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone givenThe result for model 1b is almost identical to that for model 1a.
“The result for model 1b is almost identical and is not shown.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- The authors state this as an assumption: In the Jordan-Brans-Dicke theory considered, β is a positive constant.
May combine two claims.
FindingreasoningFor model 1a, B (defined in equation (2.19)) is identically zero due to cancelling slow-roll parameters, so from equation (2.18) the curvature perturbation remains constant after horizon crossing (ζ=0) for this JBD model with quartic potential.
“As an aside, it can be noted that, for the specific case of model 1a, B (defined in equation (2.19)) is identically zero, due to a cancelling of the slow-roll parameters.This means that from equation (2.18) the curvature perturbation remains constant after horizon crossing ( ζ = 0) for this JBD model with quartic potential.”
Quote in the paperNot a substantial claimMay combine two claims.
FindingreasoningWe see the symmetry in figure 2 around the point (θ=π/2, Δ=π/2); after inflation, Δ is no longer dependent on the fields φ and χ, so this symmetry is not necessarily valid any more.
“For this reason, we see the symmetry in figure 2 around the point (θ = π/2, Δ = π/2).However after inflation, Δ is no longer dependent on the fields ϕ and χ, thus this symmetry is not necessarily valid any more.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Findingnone givenModel 2b is almost identical to model 2a, so its numerical results are not shown.
“Model 2b is almost identical to model 2a and the numerical results are not shown.”
Quote in the paperNot a substantial claimMay combine two claims.
FindingreasoningIn this case, an analytical formula for f_NL during inflation is given by equation (5.14) and is of order 1/N, where N is the total number of e-foldings.
“In this case, an analytical formula for f NL during inflation is given by equation (5.14) and is of order 1/N , where N is the total number of e-foldings.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoningα_S is determined at horizon crossing and does not change thereafter.
“We note that the running spectral index of isocurvature perturbation, α S , is independent of Δ, which means that the α S is determined at horizon crossing and does not change thereafter.”
Quote in the paperSubstantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationA perfect CMB experiment cannot hope to detect |f_NL| < 3.
“A perfect CMB experiment cannot hope to detect |f NL | < 3 [31].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [31] Creminelli P Senatore L Zaldarriaga M Tegmark M, 2006
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
InterpretationreasoningAs Δ → π/2, T_ζS is non-zero but very small, and the correlation amplitude would be small and therefore unobservable.
“As Δ → π/2, T ζS is non-zero, but very small.In this limit, the correlation amplitude would be also small and therefore unobservable.”
Quote in the paperNot a substantial claimThe quote only partly states this claim.
Study design and conductown dataThe analytical slow-roll approximations given in equations (5.14) and (5.20) were seen to agree within 5% with the numerical results.
“The analytical slow-roll approximations given in equations (5.14) and (5.20) were seen to agree within 5%.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Study design and conductown dataThe background equations of motion and the Friedmann equation were solved numerically without assuming slow-roll, in order to verify the analytical results.
“In order to verify our analytical results of the previous section, we numerically solved the background equations of motion, equation (2.1) and the Friedmann equation, equation (2.2), not assuming slow-roll.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoningθ is calculated at horizon crossing and depends on β and the initial values of φ and χ, whereas Δ represents evolution after horizon crossing and depends on the late-time evolution of the Universe.
“We note that θ is calculated at horizon crossing and depends on β and the initial values of ϕ and χ, whereas Δ represents the evolution after horizon crossing and depends on the late time evolution of the Universe.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: During inflation, the fields follow the slow-roll limit.
- The authors state this as an assumption: In the Jordan-Brans-Dicke theory considered, β is a positive constant.
May combine two claims.
InterpretationreasoningAs in model 1, in order to achieve enough inflation, the parameters are required to be small and hence -6/5 f_NL^(4) ≈ 1/2N.
“As in model 1, in order to achieve enough inflation, the parameters are required to be small and hence -6 5 f (4) NL ≈ 1/2N.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningFor the correlated spectral running α_C, there is a divergence at Δ → π/2, but this divergence is not observable.
“For the correlated spectral running, α C , there is an divergence at Δ → π/2.This divergence is not observable.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationreasoningAt Δ = π/2 (T_ζS = 0), there is no evolution in P_ζ and the amplitude of the correlation spectrum is zero, so it is impossible to define a spectral index or running at this point.
“From equation (2.23), it can be seen that, for Δ = π/2 (T ζS = 0), there is no evolution in P ζ and the amplitude of the correlation spectrum is zero.It is therefore impossible to define a spectral index or running at this point.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationnone givenNeither of these enhancement terms is significant, and in the slow-roll limit f_NL is not much enhanced by the coupling.
“Neither of these terms are significant and in the slow-roll limit, f NL is not much enhanced by the coupling.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningBecause the sign of tan Δ is determined by the sign of α in equation (2.20) and α is proportional to sin 2θ for the potential in equation (5.1), reflection of φ or χ changes the sign of tan Δ before the end of inflation.
“However since the sign of tan Δ is determined by the sign of α in equation (2.20) and α is proportional to sin 2θ for the potential in equation (5.1),The reflection of ϕ or χ changes also the sign of tan Δ before the end of inflation.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningIn order to achieve enough inflation, the parameters are required to be small and hence -6/5 f_NL^(4) ≈ 1/2N, which is unobservable.
“It is clear that, in order to achieve enough inflation, the parameters are required to be small and hence -6 5 f (4) NL ≈ 1/2N, which is unobservable.”
Quote in the paperSubstantial claim
The authors state this as an assumption.
During inflation, the fields follow the slow-roll limit.
“During inflation, the fields are assumed to follow the slow-roll limit, φ = σ cos θ -W ϕ 3H , χ = σ sin θe -b -W χ 3H e -2b , (2.16) H 2 (ϕ, χ) 1 3M 2 P W (ϕ, χ).”
The authors state this as an assumption.
In the Jordan-Brans-Dicke theory considered, β is a positive constant.
“where we assume β is a positive constant.”
The authors state this as an assumption.
β is positive.
“Assuming β to be positive, the effect of the coupling is”
Cites
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“The first term, which we call −(6) 5f (3) NL, can be written as [31], −(6) 5 f (3) NL = r 16 (1 + f), (4.”
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Separable and non-separable multi-field inflation and large non-Gaussianity
Mazumdar, A., & Wang, L. (2012) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2012/09/005 · Cited by 30 · see citing works
Non-linear curvature perturbation in multi-field inflation models with non-minimal coupling
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Non-Gaussianities of single field inflation with nonminimal coupling
Qiu, T., & Yang, K.-C. (2011) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.83.084022 · Cited by 30 · see citing works
Note on non-Gaussianities in two-field inflation
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Singular cosmological evolution using canonical and ghost scalar fields
Nojiri, S., Odintsov, S. D., Oikonomou, V. K., & Saridakis, E. N. (2015) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2015/9/044 · Cited by 29 · see citing works
The squeezed limit of the bispectrum in multi-field inflation
Kenton, Z., & Mulryne, D. J. (2015) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2015/10/018 · Cited by 28 · see citing works
General formula for the running of local fNL
Byrnes, C. T., & Gong, J.-O. (2012) · Physics Letters B · doi:10.1016/j.physletb.2012.11.052 · Cited by 26 · see citing works
Large trispectrum in two-field slow-roll inflation
Elliston, J., Alabidi, L., Huston, I., et al. (2012) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2012/09/001 · Cited by 25 · see citing works
Bispectra from two-field inflation using the long-wavelength formalism
Tzavara, E., & Tent, B. V. (2011) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2011/06/026 · Cited by 25 · see citing works
First-order formalism for twinlike models with several real scalar fields
Bazeia, D., Lobão, A. S., Losano, L., et al. (2014) · The European Physical Journal C · doi:10.1140/epjc/s10052-014-2755-0 · Cited by 24 · see citing works
Multi-field effects in a simple extension ofR2inflation
Mori, T., Kohri, K., & WHITE, J. W. (2017) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2017/10/044 · Cited by 24 · see citing works
“Finite” non-Gaussianities and tensor–scalar ratio in large volume Swiss-cheese compactifications
Misra, A., & Shukla, P. (2008) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2008.10.022 · Cited by 24 · see citing works
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Seven lessons from manyfield inflation in random potentials
Dias, M., Frazer, J., & Marsh, M. C. D. (2018) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2018/01/036 · Cited by 23 · see citing works
Adiabaticity and the fate of non-Gaussianities: The trispectrum and beyond
Meyers, J., & Sivanandam, N. (2011) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.84.063522 · Cited by 21 · see citing works
On cross-correlations between curvature and isocurvature perturbations during inflation
Gao, X. (2010) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2010/02/019 · Cited by 21 · see citing works
Large non-Gaussianity in non-minimal inflation
Gong, J.-O., & Lee, H. M. (2011) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2011/11/040 · Cited by 21 · see citing works
Ellis, J. R., Gonzalo, T. E., Harz, J., & Wei-chih Huang, A. (2015) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2015/03/039 · Cited by 20 · see citing works
Constraints on neutrino density and velocity isocurvature modes from WMAP-9 data
Савелайнен, М., Väliviita, J., Walia, P., et al. (2013) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.88.063010 · Cited by 19 · see citing works
Cosmological observables in multi-field inflation with a non-flat field space
Gao, X. Q., Li, T., & Shukla, P. (2014) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2014/10/008 · Cited by 19 · see citing works
Primordial curvature perturbation during and at the end of multi-field inflation
Choi, K.-Y., Kim, S. A., & Kyae, B. (2012) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2012.04.004 · Cited by 18 · see citing works
Self-citationTesting supersymmetric Higgs inflation with non-Gaussianity
Kawai, S., & Kim, J. (2015) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.91.045021 · Cited by 18 · see citing works
Seven lessons from manyfield inflation in random potentials
Dias, M., Frazer, J., & Marsh, M. C. D. (2018) · DESY (CERN, DESY, Fermilab, IHEP, and SLAC) · doi:10.17863/cam.24831 · Cited by 17 · see citing works
On non-Gaussianities in two-field poly-instanton inflation
Gao, X., & Shukla, P. (2013) · Journal of High Energy Physics · doi:10.1007/jhep03(2013)061 · Cited by 16 · see citing works
Magnetogenesis and the primordial non-gaussianityPreprint
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Magnetogenesis and the primordial non-gaussianity
Seery, D. (2009) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2009/08/018 · Cited by 15 · see citing works
Primordial perturbations in multi-scalar inflation
Abedi, H., & Abbassi, A. M. (2017) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2017/07/049 · Cited by 14 · see citing works
Multi-fluid cosmology in Einstein gravity: analytical solutions
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On Non-Gaussianities in Two-Field Poly-Instanton InflationPreprint
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Cosmology from the kinetic polarized Sunyaev Zel'dovich effect
Hotinli, S. C., Holder, G. P., Johnson, M. C., et al. (2022) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2022/10/026 · Cited by 13 · see citing works
Exploring two-field inflation in the Wess-Zumino model
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Local scale-dependent non-gaussian curvature perturbations at cubic order
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Modulated reheating by a curvaton
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Self-citationSupergravity, α-attractors and primordial non-Gaussianity
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Observational constraints on two-field warm inflation
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Large tensor spectrum of BICEP2 in the natural supersymmetric hybrid model
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Non-Gaussianity in two-field inflation beyond the slow-roll approximation
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Non-Gaussianity and gravitational wave background in curvaton with a double well potential
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- Authors
- Choi, K.-Y., Hall, L. M. H., & Bruck, C. V. D.
- Affiliations
- University of Sheffield (GB)
- Venue
- Journal of Cosmology and Astroparticle Physics
- Publisher
- IOP Publishing
- Published
- 2007-02-23
- Volume
- 2007
- Issue
- 02
- Pages
- 029-029
- ISSN
- 1475-7508 (ISSN-L)
- 1475-7516
- OpenAlex
- W2036888309
- Semantic Scholar
- b9b2149a2245f92040133211a22883cae608cc24
- Cited by
- 98
- References
- 48
- Citations by year
- 2026: 1 · 2025: 2 · 2023: 2 · 2022: 2 · 2021: 3 · 2020: 2 · 2019: 2 · 2018: 8 · 2017: 6 · 2016: 3 · 2015: 7 · 2014: 6 · 2013: 8 · 2012: 9
- FWCI
- 5.20
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of Cosmology and Astroparticle Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- two-field inflation, slow-roll inflation, running of the spectral index, non-Gaussianity, isocurvature perturbations, adiabatic perturbations, correlation spectra, scalar field coupling, kinetic coupling, string cosmology, effective field theory
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