## Final Report Summary - AGILE (Perturbative Approaches to Gravitational Instability and Lensing in Cosmology)

The project has as main goal the development and testing of analytical predictions for the statistical properties of the matter and galaxy distributions at large scales in the observable Universe, typically described in terms of their spatial correlation functions. Such broad topic is at the center of current cosmological investigations, since measurements of the galaxy and weak lensing power spectrum or two-point correlation functions in future surveys is expected to provide valuable information on the mechanism responsible for the current accelerated expansion of the Universe, as well as on the initial conditions and on the physics of the early Universe.

In this context, the AGILE project presents three characterising aspects. In the first place, it focuses on analytical predictions in nonlinear perturbation theory (PT), with a specific attention to the lastest developments of Renormalised PT (Crocce & Scoccimarro, 2006; Bernardeau et al., 2008) and the Time-RG approach (Pietroni, 2008). Second, it extends the predictions and analysis for two-point correlation functions to higher-order correlators, a direct result of the nonlinear evolution and therefore a fundamental test of our theoretical models. In the third place, a specific goal of the project is to assess possible observable consequences of departures from the standard cosmological model, with the cases of primordial non-Gaussianity and quintessence models as primary examples.

A key aspect of the investigations of the AGILE project consists, in fact, in showing that nonlinear corrections in LSS observables should indeed be seen as an opportunity to test physics beyond the standard model, rather than a problem in the reconstruction of the underlying linear theory quantities.

Following the timeline proposed for the project, a couple of papers published during the first year of the project dealt with the problem of the effects of primordial non-Gaussianity on matter correlators on intermediate scales. Sefusatti et al (2010) provides a detailed comparison of the matter bispectrum measured in N-body simulations with both Gaussian and non-Gaussian initial conditions with predictions in Eulerian PT. Such comparison involves all measurable configurations down to the mildly nonlinear regime from redshift z = 0 to z = 2 and shows a representative choice of triangular configurations sets. The theoretical predictions include up to 1-loop corrections in PT. The agreement between the numerical and theoretical resutls is remarkable even at intermediate scales and low redshift. The effect of local non-Gaussian initial conditions on nonlinear corrections is remarkable, particularly for squeezed configurations. The non-Gaussian correction for generic triangles is expected to lead to interesting limits on non-Gaussian parameters from weak lensing observations.

As the next step in the project proposal, Sefusatti et al (2011) present a complete study of the effects of primordial non-Gaussianity on the power spectrum and bispectrum of biased populations. This work has direct relevance for the interpretation of large-scales redshfit surveys observations. By comparing with full measurements of the halo power spectrum and bispectrum in N-body simulations with non-Gaussian initial conditions it shown that a reasonable agreement is provided by a model characterised by several, distinct and non-trivial corrections both to the underlying matter correlators and to halo bispectrum. In particular it highlights the significant potentiality of higher-order correlation functions to test the initial conditions of the Universe.

In an effort to increase the accuracy of the predictions for matter correlators, Bernardeau, Crocce & Sefusatti (2010) extends previous results in Renormalised PT (RPT) to the case of non-Gaussian initial conditions, by showing that the reordering scheme of the perturbative corrections based on multipoint propagators retain its validity when higher-order initial correlators are non-vanishing. As for the Gaussian case, it is possible to derive an explicit expression for the high-momentum limit of the propagators, still characterised by an exponential cut-off. We find the correction to the large-k limit of the propagator due to non-Gaussianity to be independent of the initial bispectrum with a dependence, instead, on the kurtosis of the one-dimensional initial displacement field.

On a side, Jeong, Schmidt & Sefusatti (2011) studies some novel corrections due non-Gaussian initial conditions to weak lensing observables. Estimators for weak lensing observables such as shear and convergence generally have nonlinear corrections, which, in principle, make weak lensing power spectra sensitive to primordial non-Gaussianity. These contributions for weak lensing autocorrelation and cross-correlation power spectra, are quantitatively evaluated.

In addition, at the beginning of the project, an invited review by the principal investigator has benn published on the broad topic of primordial non-Gaussianity and higher-order correlation functions (Liguori, Sefusatti, Fergusson & Shellard, 2010).

A different departure from the standard LambdaCDM scenario is considered in Sefusatti & Verniszi (2011) which studies structure formation in the presence of a quintessence component with zero speed of sound in the framework of Eulerian PT. It is shown that for such dark energy model quintessence and dark matter can be studied as a unique fluid in terms of the total energy density contrast and the common velocity. The clustering of quintessence is responsible for a rapid evolution of the growth rate at low redshifts, and modifies the standard relation between the velocity divergence and the growth factor. For the total fluid, exact solutions are derived for the linear growth function in integral forms as in the LambdaCDM case. Also studied are second oreder solution in PT and the corresponding tree-level bispectra. The reduced bispectrum, in particular, receives sensible modifications only in the clustering case and can potentially be used to detect or rule out the model.

As a further development of these first results, D'Amico & Sefusatti (2011) extends the recently proposed Time-RG method to provide accurate predictions for the density power spectrum in models of clustering quintessence. It is shown that quintessence perturbations induce small corrections to the nonlinear evolution of power spectrum contrasting with the large effect of a vanishing speed of sound on the linear growth function at low redshift. For this reason, models with the same normalisation of the linear density power spectrum can present significantly different nonlinear corrections depending on the value of the sound speed. The relation between linear and nonlinear growth of structures should be properly taken into account in constraining models with inhomogeneous dark energy

In this context, the AGILE project presents three characterising aspects. In the first place, it focuses on analytical predictions in nonlinear perturbation theory (PT), with a specific attention to the lastest developments of Renormalised PT (Crocce & Scoccimarro, 2006; Bernardeau et al., 2008) and the Time-RG approach (Pietroni, 2008). Second, it extends the predictions and analysis for two-point correlation functions to higher-order correlators, a direct result of the nonlinear evolution and therefore a fundamental test of our theoretical models. In the third place, a specific goal of the project is to assess possible observable consequences of departures from the standard cosmological model, with the cases of primordial non-Gaussianity and quintessence models as primary examples.

A key aspect of the investigations of the AGILE project consists, in fact, in showing that nonlinear corrections in LSS observables should indeed be seen as an opportunity to test physics beyond the standard model, rather than a problem in the reconstruction of the underlying linear theory quantities.

Following the timeline proposed for the project, a couple of papers published during the first year of the project dealt with the problem of the effects of primordial non-Gaussianity on matter correlators on intermediate scales. Sefusatti et al (2010) provides a detailed comparison of the matter bispectrum measured in N-body simulations with both Gaussian and non-Gaussian initial conditions with predictions in Eulerian PT. Such comparison involves all measurable configurations down to the mildly nonlinear regime from redshift z = 0 to z = 2 and shows a representative choice of triangular configurations sets. The theoretical predictions include up to 1-loop corrections in PT. The agreement between the numerical and theoretical resutls is remarkable even at intermediate scales and low redshift. The effect of local non-Gaussian initial conditions on nonlinear corrections is remarkable, particularly for squeezed configurations. The non-Gaussian correction for generic triangles is expected to lead to interesting limits on non-Gaussian parameters from weak lensing observations.

As the next step in the project proposal, Sefusatti et al (2011) present a complete study of the effects of primordial non-Gaussianity on the power spectrum and bispectrum of biased populations. This work has direct relevance for the interpretation of large-scales redshfit surveys observations. By comparing with full measurements of the halo power spectrum and bispectrum in N-body simulations with non-Gaussian initial conditions it shown that a reasonable agreement is provided by a model characterised by several, distinct and non-trivial corrections both to the underlying matter correlators and to halo bispectrum. In particular it highlights the significant potentiality of higher-order correlation functions to test the initial conditions of the Universe.

In an effort to increase the accuracy of the predictions for matter correlators, Bernardeau, Crocce & Sefusatti (2010) extends previous results in Renormalised PT (RPT) to the case of non-Gaussian initial conditions, by showing that the reordering scheme of the perturbative corrections based on multipoint propagators retain its validity when higher-order initial correlators are non-vanishing. As for the Gaussian case, it is possible to derive an explicit expression for the high-momentum limit of the propagators, still characterised by an exponential cut-off. We find the correction to the large-k limit of the propagator due to non-Gaussianity to be independent of the initial bispectrum with a dependence, instead, on the kurtosis of the one-dimensional initial displacement field.

On a side, Jeong, Schmidt & Sefusatti (2011) studies some novel corrections due non-Gaussian initial conditions to weak lensing observables. Estimators for weak lensing observables such as shear and convergence generally have nonlinear corrections, which, in principle, make weak lensing power spectra sensitive to primordial non-Gaussianity. These contributions for weak lensing autocorrelation and cross-correlation power spectra, are quantitatively evaluated.

In addition, at the beginning of the project, an invited review by the principal investigator has benn published on the broad topic of primordial non-Gaussianity and higher-order correlation functions (Liguori, Sefusatti, Fergusson & Shellard, 2010).

A different departure from the standard LambdaCDM scenario is considered in Sefusatti & Verniszi (2011) which studies structure formation in the presence of a quintessence component with zero speed of sound in the framework of Eulerian PT. It is shown that for such dark energy model quintessence and dark matter can be studied as a unique fluid in terms of the total energy density contrast and the common velocity. The clustering of quintessence is responsible for a rapid evolution of the growth rate at low redshifts, and modifies the standard relation between the velocity divergence and the growth factor. For the total fluid, exact solutions are derived for the linear growth function in integral forms as in the LambdaCDM case. Also studied are second oreder solution in PT and the corresponding tree-level bispectra. The reduced bispectrum, in particular, receives sensible modifications only in the clustering case and can potentially be used to detect or rule out the model.

As a further development of these first results, D'Amico & Sefusatti (2011) extends the recently proposed Time-RG method to provide accurate predictions for the density power spectrum in models of clustering quintessence. It is shown that quintessence perturbations induce small corrections to the nonlinear evolution of power spectrum contrasting with the large effect of a vanishing speed of sound on the linear growth function at low redshift. For this reason, models with the same normalisation of the linear density power spectrum can present significantly different nonlinear corrections depending on the value of the sound speed. The relation between linear and nonlinear growth of structures should be properly taken into account in constraining models with inhomogeneous dark energy