### Abstract

We consider next-to-leading-order (one-loop) nonlinear corrections to the bispectrum and skewness of cosmological density fluctuations induced by gravitational evolution, focusing on the case of Gaussian initial conditions and scale-free initial power spectra, P(k) ∝ k^{n}. As has been established by comparison with numerical simulations, leading order (tree-level) perturbation theory describes these quantities at the largest scales. The one-loop perturbation theory provides a tool to probe the transition to the nonlinear regime on smaller scales. In this work, we find that, as a function of spectral index n, the one-loop bispectrum follows a pattern analogous to that of the one-loop power spectrum, which shows a change in behavior at a "critical index" n_{c} ≈ 1.4, where nonlinear corrections vanish. The tree-level perturbation theory predicts a characteristic dependence of the bispectrum on the shape of the triangle defined by its arguments. For n ≲ n_{c}, one-loop corrections increase this configuration dependence of the leading order contribution; for n ≳ n_{c}, one-loop corrections tend to cancel the configuration dependence of the tree-level bispectrum, in agreement with known results from n = -1 numerical simulations. A similar situation is shown to hold for the Zeldovich approximation, where n_{c} ≈ -1.75. We obtain explicit analytic expressions for the one-loop bispectrum for n = -2 initial power spectra, for both the exact dynamics of gravitational instability and the Zeldovich approximation. We also compute the skewness factor, including local averaging of the density field, for n = -2: S_{3}(R) = 4.02 + 3.83σ_{G}^{2}(R) for Gaussian smoothing and S_{3}(R) = 3.86 + 3.18σ_{TH}^{2}(R) for top-hat smoothing, where σ^{2}(R) is the variance of the density field fluctuations smoothed over a window of radius R. A comparison with fully nonlinear numerical simulations implies that, for n < -1, the one-loop perturbation theory can extend our understanding of nonlinear clustering down to scales where the transition to the stable clustering regime begins.

Original language | English (US) |
---|---|

Pages (from-to) | 1-17 |

Number of pages | 17 |

Journal | Astrophysical Journal |

Volume | 487 |

Issue number | 1 PART I |

DOIs | |

State | Published - 1997 |

### Fingerprint

### Keywords

- Cosmology: theory
- Large-scale structure of the universe
- Methods: numerical

### ASJC Scopus subject areas

- Space and Planetary Science

### Cite this

*Astrophysical Journal*,

*487*(1 PART I), 1-17. https://doi.org/10.1086/304578

**Cosmological perturbations : Entering the nonlinear regime.** / Scoccimarro, Román.

Research output: Contribution to journal › Article

*Astrophysical Journal*, vol. 487, no. 1 PART I, pp. 1-17. https://doi.org/10.1086/304578

}

TY - JOUR

T1 - Cosmological perturbations

T2 - Entering the nonlinear regime

AU - Scoccimarro, Román

PY - 1997

Y1 - 1997

N2 - We consider next-to-leading-order (one-loop) nonlinear corrections to the bispectrum and skewness of cosmological density fluctuations induced by gravitational evolution, focusing on the case of Gaussian initial conditions and scale-free initial power spectra, P(k) ∝ kn. As has been established by comparison with numerical simulations, leading order (tree-level) perturbation theory describes these quantities at the largest scales. The one-loop perturbation theory provides a tool to probe the transition to the nonlinear regime on smaller scales. In this work, we find that, as a function of spectral index n, the one-loop bispectrum follows a pattern analogous to that of the one-loop power spectrum, which shows a change in behavior at a "critical index" nc ≈ 1.4, where nonlinear corrections vanish. The tree-level perturbation theory predicts a characteristic dependence of the bispectrum on the shape of the triangle defined by its arguments. For n ≲ nc, one-loop corrections increase this configuration dependence of the leading order contribution; for n ≳ nc, one-loop corrections tend to cancel the configuration dependence of the tree-level bispectrum, in agreement with known results from n = -1 numerical simulations. A similar situation is shown to hold for the Zeldovich approximation, where nc ≈ -1.75. We obtain explicit analytic expressions for the one-loop bispectrum for n = -2 initial power spectra, for both the exact dynamics of gravitational instability and the Zeldovich approximation. We also compute the skewness factor, including local averaging of the density field, for n = -2: S3(R) = 4.02 + 3.83σG2(R) for Gaussian smoothing and S3(R) = 3.86 + 3.18σTH2(R) for top-hat smoothing, where σ2(R) is the variance of the density field fluctuations smoothed over a window of radius R. A comparison with fully nonlinear numerical simulations implies that, for n < -1, the one-loop perturbation theory can extend our understanding of nonlinear clustering down to scales where the transition to the stable clustering regime begins.

AB - We consider next-to-leading-order (one-loop) nonlinear corrections to the bispectrum and skewness of cosmological density fluctuations induced by gravitational evolution, focusing on the case of Gaussian initial conditions and scale-free initial power spectra, P(k) ∝ kn. As has been established by comparison with numerical simulations, leading order (tree-level) perturbation theory describes these quantities at the largest scales. The one-loop perturbation theory provides a tool to probe the transition to the nonlinear regime on smaller scales. In this work, we find that, as a function of spectral index n, the one-loop bispectrum follows a pattern analogous to that of the one-loop power spectrum, which shows a change in behavior at a "critical index" nc ≈ 1.4, where nonlinear corrections vanish. The tree-level perturbation theory predicts a characteristic dependence of the bispectrum on the shape of the triangle defined by its arguments. For n ≲ nc, one-loop corrections increase this configuration dependence of the leading order contribution; for n ≳ nc, one-loop corrections tend to cancel the configuration dependence of the tree-level bispectrum, in agreement with known results from n = -1 numerical simulations. A similar situation is shown to hold for the Zeldovich approximation, where nc ≈ -1.75. We obtain explicit analytic expressions for the one-loop bispectrum for n = -2 initial power spectra, for both the exact dynamics of gravitational instability and the Zeldovich approximation. We also compute the skewness factor, including local averaging of the density field, for n = -2: S3(R) = 4.02 + 3.83σG2(R) for Gaussian smoothing and S3(R) = 3.86 + 3.18σTH2(R) for top-hat smoothing, where σ2(R) is the variance of the density field fluctuations smoothed over a window of radius R. A comparison with fully nonlinear numerical simulations implies that, for n < -1, the one-loop perturbation theory can extend our understanding of nonlinear clustering down to scales where the transition to the stable clustering regime begins.

KW - Cosmology: theory

KW - Large-scale structure of the universe

KW - Methods: numerical

UR - http://www.scopus.com/inward/record.url?scp=21944455285&partnerID=8YFLogxK

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U2 - 10.1086/304578

DO - 10.1086/304578

M3 - Article

VL - 487

SP - 1

EP - 17

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 1 PART I

ER -