A short note on the curvature perturbation at second order

Christopherson, Adam J.; Nalson, Ellie; Malik, Karim A.

doi:10.1088/0264-9381/32/7/075005

Cited by 8 publications

(15 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) δ c . We find that equation (3.3) in [7] agrees with (51). 17 Equation (51) has also been given by Carrilho and Malik (2015) [5] (see equation (3.3)).…”

Section: Examplessupporting

confidence: 81%

“…The only gauge that is commonly used that does not fulfil this condition is the synchronous gauge, which is useful for treating dust models. 7 However, the synchronous gauge is not a fully fixed gauge and the natural way to completely fixing this gauge, and thereby relate quantities to physical observables, is to relate it to the total matter gauge, which does obey the above conditions, see Appendix B.7 in [26]. As a consequence of the above spatial gauge fixing, the remaining gauge freedom is described by gauge fields to second order restricted to be of the form (1)…”

Section: Equations (15) Yieldmentioning

confidence: 99%

See 1 more Smart Citation

Second order cosmological perturbations: simplified gauge change formulas

Uggla

Wainwright

2019

Class. Quantum Grav.

View full text Add to dashboard Cite

In this paper we present a new formulation of the change of gauge formulas in second order cosmological perturbation theory which unifies and simplifies known results. Our approach is based on defining new second order scalar perturbation variables by adding a multiple of the square of the corresponding first order variables to each second order variable. A bonus is that these new perturbation variables are of broader significance in that they also simplify the analysis of second order scalar perturbations in the super-horizon regime in a number of ways, and lead to new conserved quantities. * Electronic address: claes.uggla@kau.se † Electronic address: jwainwri@uwaterloo.ca 1 See, for example, Bartolo et al (2010) [3] and Tram et al (2016) [24].2 Other papers that have been influential in developing and applying second order cosmological perturbation theory but do not emphasize change of gauge formulas are Bartolo et al (2004) [1] and Nakamura (2007) [20]. Examples of recent papers that use change of gauge formulas at second order are Malik (2005) [15], Christopherson et al (

show abstract

“…(1) δ c . We find that equation (3.3) in [7] agrees with (51). 17 Equation (51) has also been given by Carrilho and Malik (2015) [5] (see equation (3.3)).…”

Section: Examplessupporting

confidence: 81%

Section: Equations (15) Yieldmentioning

confidence: 99%

Second order cosmological perturbations: simplified gauge change formulas

Uggla

Wainwright

2019

Class. Quantum Grav.

View full text Add to dashboard Cite

show abstract

“…While the final version of this paper was being prepared, a preprint by Christopherson, Nalson & Malik appeared in which the second-order gauge transformation was given explicitly for a scalar field model [5]. We comment on the relation between our results in §4.…”

Section: Introductionmentioning

confidence: 89%

“…For a mode of wavenumber k this requires k/aH ≪ 1, making spatial gradients negligible. We verify that the two approaches give equivalent answers and clarify some issues regarding nonlocal terms which appear in the perturbation theory expressions.Our final expressions will be used in forthcoming papers which describe numerical calculation of the two-and three-point functions in multiple-field inflation.While the final version of this paper was being prepared, a preprint by Christopherson, Nalson & Malik appeared in which the second-order gauge transformation was given explicitly for a scalar field model [5]. We comment on the relation between our results in §4.Notation.-We work in units where c = = 1.…”

mentioning

confidence: 99%

The curvature perturbation at second order

Dias

Elliston

Frazer

et al. 2015

J. Cosmol. Astropart. Phys.

View full text Add to dashboard Cite

We give an explicit relation, up to second-order terms, between scalar-field fluctuations defined on spatially-flat slices and the curvature perturbation on uniform-density slices. This expression is a necessary ingredient for calculating observable quantities at second-order and beyond in multiple-field inflation. We show that traditional cosmological perturbation theory and the 'separate universe' approach yield equivalent expressions for superhorizon wavenumbers, and in particular that all nonlocal terms can be eliminated from the perturbation-theory expressions. IntroductionAccording to our current ideas, structure in the universe was seeded by quantum fluctuations which were amplified during an inflationary epoch. During inflation these fluctuations dominate the variation in energy density from place to place and therefore generate a gravitational response which can be probed by cosmological observations. Inflationary amplification is believed to occur for any sufficiently light degree of freedom, in the sense that its mass m was substantially less than the Hubble rate H while scales of interest were being carried beyond the horizon. Models motivated by modern concepts in high-energy physics often invoke many light fields, and therefore can be tested only if we have an understanding of their effects. The literature surrounding calculations of the inflationary density perturbation is now very mature-often with agreement on subtle effects to second-or even third-order in perturbation theory-which allows these effects to be predicted in some detail. But despite this maturity it is remarkable that no completely explicit formula has been given for the uniform-density gauge curvature perturbation in an inflationary model with an arbitrary number of fields. 1 A formula of this type would give the next-order term in the classic result ζ = −φ α δφ α /2M 2 P Hǫ which has long been known at first order. It is a key element in computing non-Gaussian signatures in the statistics of the inflationary density perturbation. Here and below, ǫ ≡ −Ḣ/H 2 is the usual slow-roll parameter and δφ α labels the species of light fields.In this paper we supply the missing formula, valid for an arbitrary number of canonical fields and without using the slow-roll approximation. We perform the calculation using two independent methods: traditional 'cosmological perturbation theory', which is an expansion in the amplitude of small fluctuations around a Robertson-Walker background, and the 'separate universe approach', which is an expansion in the amplitude of gradients of the perturbations. In practice (although not required in principle), separate-universe calculations often invoke a second expansion in the amplitude of the fluctuations, after which the two methods should agree for any Fourier mode much larger than the cosmological horizon. For a mode of wavenumber k this requires k/aH ≪ 1, making spatial gradients negligible. We verify that the two approaches give equivalent answers and clarify some issues regarding nonlocal terms which ...

show abstract

“…Write the background Einstein tensor in the form(0) G η η = −3H 2 , (0) G i i = 3H 2 (1 − 2q) 5. This excludes the synchronous gauge (for a recent work using the synchronous gauge, see e.g [8]…”

mentioning

confidence: 99%

Dynamics of cosmological perturbations at first and second order

Uggla

Wainwright

2018

Phys. Rev. D

View full text Add to dashboard Cite

In this paper we give five gauge-invariant systems of governing equations for first and second order scalar perturbations of flat Friedmann-Lemaître universes that are minimal in the sense that they contain no redundant equations or variables. We normalize the variables so that they are dimensionless, which leads to systems of equations that are simple and ready-to-use. We compare the properties and utility of the different systems. For example, they serve as a starting point for finding explicit solutions for two benchmark problems in cosmological perturbation theory at second order: adiabatic perturbations in the super-horizon regime (the long wavelength limit) and perturbations of ΛCDM universes. However, our framework has much wider applicability and serves as a reference for future work in the field. * Electronic address: claes.uggla@kau.se † Electronic address: jwainwri@uwaterloo.ca 1 See, for example, Bartolo et al (2010) [3] and Tram et al (2016) [20].

show abstract

A short note on the curvature perturbation at second order

Cited by 8 publications

References 33 publications

Second order cosmological perturbations: simplified gauge change formulas

Second order cosmological perturbations: simplified gauge change formulas

The curvature perturbation at second order

Dynamics of cosmological perturbations at first and second order

Contact Info

Product

Resources

About