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On Cross-correlations between Curvature and Isocurvature Perturbations during Inflation
High Energy Physics - Theory · Submitted 27 Aug 2009
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Moreover, we find the cross-correlation, which was previously shown to be first-order in slow-roll parameter, can be enhanced when the sound speed of isocurvature perturbation is much smaller than that of the curvature perturbation.
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The abstract, sentence by sentence
We investigate the effects of couplings between curvature and isocurvature perturbations before and around horizon-crossings during cosmological inflation. We consider a generalized two-field inflation model, in which the non-canonical kinetic term allows us arbitrary sound speeds of curvature and isocurvature perturbations. By using the field-theoretical perturbative analysis, we calculate the cross-spectrum between curvature and isocurvature perturbations and the corrections to curvature and isocurvature power spectra due to the presence of couplings between them. Our analysis confirms previous results that the cross-correlations are generated and amplified when perturbations cross the horizons. Moreover, we find the cross-correlation, which was previously shown to be first-order in slow-roll parameter, can be enhanced when the sound speed of isocurvature perturbation is much smaller than that of the curvature perturbation. This is because in this case the isocurvature perturbation exits its horizon much earlier than the curvature perturbation and acts as a nearly constant source on the curvature perturbation.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the effects of couplings between curvature and isocurvature perturbations before and around horizon-crossings during cosmological inflation. | contribution 0.90 | 0.87 | 0.05 | unclear 0.57 |
| s1 | We consider a generalized two-field inflation model, in which the non-canonical kinetic term allows us arbitrary sound speeds of curvature and isocurvature perturbations. | method 0.87 | 0.64 | 0.03 | asserted 0.96 |
| s2 | By using the field-theoretical perturbative analysis, we calculate the cross-spectrum between curvature and isocurvature perturbations and the corrections to curvature and isocurvature power spectra due to the presence of couplings between them. | method 0.76 | 0.96 | 0.09 | asserted 0.99 |
| s3 | Our analysis confirms previous results that the cross-correlations are generated and amplified when perturbations cross the horizons. | result 0.96 | 0.32 | 0.22 | asserted 1.00 |
| s4 | Moreover, we find the cross-correlation, which was previously shown to be first-order in slow-roll parameter, can be enhanced when the sound speed of isocurvature perturbation is much smaller than that of the curvature perturbation. | result 1.00 | 0.94 | 0.47 | asserted 0.99 |
| s5 | This is because in this case the isocurvature perturbation exits its horizon much earlier than the curvature perturbation and acts as a nearly constant source on the curvature perturbation. | interpretation 1.00 | 0.70 | 0.11 | asserted 1.00 |
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The paper
Original abstract
We investigate the effects of couplings between curvature and isocurvature perturbations before and around horizon-crossings during cosmological inflation. We consider a generalized two-field inflation model, in which the non-canonical kinetic term allows us arbitrary sound speeds of curvature and isocurvature perturbations. By using the field-theoretical perturbative analysis, we calculate the cross-spectrum between curvature and isocurvature perturbations and the corrections to curvature and isocurvature power spectra due to the presence of couplings between them. Our analysis confirms previous results that the cross-correlations are generated and amplified when perturbations cross the horizons. Moreover, we find the cross-correlation, which was previously shown to be first-order in slow-roll parameter, can be enhanced when the sound speed of isocurvature perturbation is much smaller than that of the curvature perturbation. This is because in this case the isocurvature perturbation exits its horizon much earlier than the curvature perturbation and acts as a nearly constant source on the curvature perturbation.
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References
How the literature uses it
16 arXiv papers cite this in 23 sentences
Papers citing it · 10 sentences without a confident label
Uses its method or tool 3 papers
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Assuming for simplicity that the field is approximately in a circular motion with a small and constant angular velocity, one can use the in-in formulism [ref] to calculate the 2-point function of the curvature perturbation by quantizing the curvature and isocurvature perturbations as free fields, with the coupling terms as interaction vertices [ref] .
Curvature Perturbation Spectrum in Two-field Inflation with a Turning Trajectory · Introduction · 0.99
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We use the field-theoretical perturbative method [ref] to calculate the influence of the new field.
Cosmological perturbations in the spatially covariant gravity with a dynamical lapse function · Using in-in formalism for the correction to spectra · 1.00
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To see, we compute the correction to $_$ due to $^I_(2)$ using the commutator form of the in-in formalism [ref] in a similar fashion as in [ref] .
Correlation of isocurvature perturbation and non-Gaussianity · B. Computation of the correction to the leading order $_$ · 0.97
Compares with it 1 paper
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We also assume that the trajectory is straight before and after the turn[This differs from a few works [ref] , where the trajectory follows a circle with constant angular velocity (which we refer to as “constant turn").].
Influence of heavy modes on perturbations in multiple field inflation · Introduction · 0.92
Cites it as support 1 paper
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As was shown in [ref] , the cross-correlations between adiabatic/entropic modes would be enhanced by a small $/$ ratio.].
On primordial trispectrum from exchanging scalar modes in general multiple field inflationary models · Linear Perturbations · 0.97
Cites it as background 5 papers
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When the turning rate is relatively small, the correlations between adiabatic and isocurbature modes have been studied perturbatively [ref] .
Coupling structure of multi-field primordial perturbations · Introduction and motivation · 0.83
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Some others put their stake on signatures that cannot appear in the simplest single-field model, such as a large deviation from the Gaussian distribution in the CMB temperature fluctuations.[As a partial list, see [ref] and references therein for various models and recent development along this direction.]
Note on Non-Gaussianities in Two-field Inflation · Introduction · 0.82
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Light scalar fields, with masses much smaller than the Hubble expansion rate $H$, are known to produce potentially large levels of entropy perturbations during inflation [ref] that could lead to (1) non-adiabatic features in the Cosmic Microwave Background (CMB) [ref] , and (2) the super-horizon evolution of curvature perturbations [ref] .
Cumulative effects in inflation with ultra-light entropy modes · Introduction · 0.94
What it cites · 89 sentences without a confident label
Contests 3 papers
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However, one can verify that, for models with different adiabatic/entropy speeds of sound, the “diagonalization" cannot be done easily and thus the treatments in [ref] cannot be viewed as good approximation in this case.
Curvature and isocurvature perturbations in two-field inflation · Conclusion and Discussion · contest 0.87
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However, one can verify that, for models with different adiabatic/entropy speeds of sound, the “diagonalization" cannot be done easily and thus the treatments in [ref] cannot be viewed as good approximation in this case.
Oscillations During Inflation and the Cosmological Density Perturbations · Conclusion and Discussion · contest 0.87
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However, one can verify that, for models with different adiabatic/entropy speeds of sound, the “diagonalization" cannot be done easily and thus the treatments in [ref] cannot be viewed as good approximation in this case.
Curvature and isocurvature perturbations from two-field inflation in a slow-roll expansion · Conclusion and Discussion · contest 0.87
Builds on 3 papers
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Thus our calculations can be viewed as generalization of previous results in [ref] .
Curvature and isocurvature perturbations from two-field inflation in a slow-roll expansion · Around Horizon-crossings · 0.95
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Thus our calculations can be viewed as generalization of previous results in [ref] .
Curvature and isocurvature perturbations in two-field inflation · Around Horizon-crossings · 0.95
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Our work can be viewed as generalization of the analysis in [ref] to a general class of two-field models with non-canonical kinetic terms and arbitrary speeds of sound for adiabatic and entropy modes, $c_a ≠c_e$.
Oscillations During Inflation and the Cosmological Density Perturbations · Introduction · 1.00
Uses the method or tool of 5 papers
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After instantaneous adiabatic/entropy modes decomposition, up to total derivative terms, the second-order action for the scalar perturbations takes the form [ref] S_2≡∫dtd^3xa^3[1/2K_mnQ̇^mQ̇^n-δ_mn1/2a^2∂_iQ^m∂_iQ^n+Ξ_mnQ̇^mQ^n-1/2M_mnQ^mQ^n] , with K_mn ≡δ_mn + σ̇^2 P_,IK JL e^I_σ e^K_n e^J_σ e^L_m , = δ_mn + 1/c_a^2-1 δ_σm δ_σn + 1/c_e^2 - 1 δ_mn- δ_σm δ_σn , Ξ_mn ≡𝒩_IJe_m^Ie_n^J+(P_⟨IJ⟩+2P_⟨IK⟩⟨JL⟩X^KL)e_m^Iė_n^J , -M_mn = -ℳ_IJe_m^Ie_n^J+2𝒩_IJė_m^Ie_n^J+(P_⟨IJ⟩+2P_⟨IK⟩⟨JL⟩X^KL)ė_m^Iė_n^J , (m,n=σ,s) where 𝒩_IJ ≡ϕ̇^KP_⟨KI⟩,J-2/HX^KLX^NMP_⟨NI⟩⟨KL⟩P_⟨MJ⟩ , -ℳ_IJ ≡P_,IJ+(X^MNP_⟨MN⟩+2X^MNX^PQP_⟨MN⟩⟨PQ⟩-3H^2)X^KL/H^2P_⟨KI⟩P_⟨LJ⟩ +1/H(P_,I-2X^MNP_⟨MN⟩,I)P_⟨KJ⟩ϕ̇^K+1/a^3d/dt(a^3/HX^KLP_⟨KI⟩P_⟨LJ⟩) .
Primordial fluctuations and non-Gaussianities in multi-field DBI inflation · Linear Perturbations · 0.84
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After instantaneous adiabatic/entropy modes decomposition, up to total derivative terms, the second-order action for the scalar perturbations takes the form [ref] S_2≡∫dtd^3xa^3[1/2K_mnQ̇^mQ̇^n-δ_mn1/2a^2∂_iQ^m∂_iQ^n+Ξ_mnQ̇^mQ^n-1/2M_mnQ^mQ^n] , with K_mn ≡δ_mn + σ̇^2 P_,IK JL e^I_σ e^K_n e^J_σ e^L_m , = δ_mn + 1/c_a^2-1 δ_σm δ_σn + 1/c_e^2 - 1 δ_mn- δ_σm δ_σn , Ξ_mn ≡𝒩_IJe_m^Ie_n^J+(P_⟨IJ⟩+2P_⟨IK⟩⟨JL⟩X^KL)e_m^Iė_n^J , -M_mn = -ℳ_IJe_m^Ie_n^J+2𝒩_IJė_m^Ie_n^J+(P_⟨IJ⟩+2P_⟨IK⟩⟨JL⟩X^KL)ė_m^Iė_n^J , (m,n=σ,s) where 𝒩_IJ ≡ϕ̇^KP_⟨KI⟩,J-2/HX^KLX^NMP_⟨NI⟩⟨KL⟩P_⟨MJ⟩ , -ℳ_IJ ≡P_,IJ+(X^MNP_⟨MN⟩+2X^MNX^PQP_⟨MN⟩⟨PQ⟩-3H^2)X^KL/H^2P_⟨KI⟩P_⟨LJ⟩ +1/H(P_,I-2X^MNP_⟨MN⟩,I)P_⟨KJ⟩ϕ̇^K+1/a^3d/dt(a^3/HX^KLP_⟨KI⟩P_⟨LJ⟩) .
Primordial perturbations and non-Gaussianities in DBI and general multi-field inflation · Linear Perturbations · 0.84
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After instantaneous adiabatic/entropy modes decomposition, up to total derivative terms, the second-order action for the scalar perturbations takes the form [ref] S_2≡∫dtd^3xa^3[1/2K_mnQ̇^mQ̇^n-δ_mn1/2a^2∂_iQ^m∂_iQ^n+Ξ_mnQ̇^mQ^n-1/2M_mnQ^mQ^n] , with K_mn ≡δ_mn + σ̇^2 P_,IK JL e^I_σ e^K_n e^J_σ e^L_m , = δ_mn + 1/c_a^2-1 δ_σm δ_σn + 1/c_e^2 - 1 δ_mn- δ_σm δ_σn , Ξ_mn ≡𝒩_IJe_m^Ie_n^J+(P_⟨IJ⟩+2P_⟨IK⟩⟨JL⟩X^KL)e_m^Iė_n^J , -M_mn = -ℳ_IJe_m^Ie_n^J+2𝒩_IJė_m^Ie_n^J+(P_⟨IJ⟩+2P_⟨IK⟩⟨JL⟩X^KL)ė_m^Iė_n^J , (m,n=σ,s) where 𝒩_IJ ≡ϕ̇^KP_⟨KI⟩,J-2/HX^KLX^NMP_⟨NI⟩⟨KL⟩P_⟨MJ⟩ , -ℳ_IJ ≡P_,IJ+(X^MNP_⟨MN⟩+2X^MNX^PQP_⟨MN⟩⟨PQ⟩-3H^2)X^KL/H^2P_⟨KI⟩P_⟨LJ⟩ +1/H(P_,I-2X^MNP_⟨MN⟩,I)P_⟨KJ⟩ϕ̇^K+1/a^3d/dt(a^3/HX^KLP_⟨KI⟩P_⟨LJ⟩) .
Non-gaussianity from the bispectrum in general multiple field inflation · Linear Perturbations · 0.84
Compares with 3 papers
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The cross-power spectrum around adiabatic horizon-crossing is approximately[([ref]) can be compared with (e.g.) Eq.(37) in [ref] .
Curvature and isocurvature perturbations from two-field inflation in a slow-roll expansion · Around Horizon-crossings · 0.97
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Actually one may verify that when setting $c_a = c_e$, ([ref]) reduces to Eq.(79) in [ref] .
Curvature and isocurvature perturbations in two-field inflation · Cross-correlations · 0.96
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Our goal is to show that the field-theoretical perturbative approach taken in this note is essentially equivalent to previous “diagonalization" method taken in [ref] .
Oscillations During Inflation and the Cosmological Density Perturbations · Comparison with Previous Results with $c_a = c_e$ · 0.99
Cites as support 3 papers
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This confirms previous result that cross-correlations among different perturbation modes are always negligible when modes are deep in side their respective sound-horizons [ref] , since there the system reduces to a collection of weakly-coupled oscillators.
Curvature and isocurvature perturbations in two-field inflation · Cross-correlations · 0.97
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This confirms previous result that cross-correlations among different perturbation modes are always negligible when modes are deep in side their respective sound-horizons [ref] , since there the system reduces to a collection of weakly-coupled oscillators.
Oscillations During Inflation and the Cosmological Density Perturbations · Cross-correlations · 0.97
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This confirms previous result that cross-correlations among different perturbation modes are always negligible when modes are deep in side their respective sound-horizons [ref] , since there the system reduces to a collection of weakly-coupled oscillators.
Curvature and isocurvature perturbations from two-field inflation in a slow-roll expansion · Cross-correlations · 0.97
Cites as background 44 papers
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Previous investigations of the cross-correlations are based on diagonalizting the coupled equations of motion [ref] .
Curvature and isocurvature perturbations from two-field inflation in a slow-roll expansion · Conclusion and Discussion · 0.81
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Previous investigations of the cross-correlations are based on diagonalizting the coupled equations of motion [ref] .
Oscillations During Inflation and the Cosmological Density Perturbations · Conclusion and Discussion · 0.81
-
Previous investigations of the cross-correlations are based on diagonalizting the coupled equations of motion [ref] .
Curvature and isocurvature perturbations in two-field inflation · Conclusion and Discussion · 0.81
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