Structure formation with massive neutrinos: going beyond linear theory

TL;DR

This work addresses the non-linear corrections to the matter power spectrum in cosmologies with massive neutrinos by developing a hybrid two-fluid framework that is matched to the full Boltzmann solution at high redshift and evolved with a time-flow non-linear scheme. The neutrino component is treated as a fluid with an effective pressure, incorporating the scale-dependent free-streaming length and ensuring momentum conservation, which is essential for correct decoupling of short- and long-wavelength modes. The authors demonstrate that non-linear neutrino dynamics are necessary to avoid spurious large-scale behavior that arises when neutrinos are treated linearly, and they quantify the impact of full scale dependence in the bispectrum on neutrino-involving spectra. The two-fluid approach yields percent-level accuracy for the CDM/baryon spectrum in the quasi-linear regime and provides a solid, momentum-conserving basis for future higher-order perturbative calculations and resummation techniques in the presence of massive neutrinos.

Abstract

We compute non-linear corrections to the matter power spectrum taking the time- and scale-dependent free-streaming length of neutrinos into account. We adopt a hybrid scheme that matches the full Boltzmann hierarchy to an effective two-fluid description at an intermediate redshift. The non-linearities in the neutrino component are taken into account by using an extension of the time-flow framework. We point out that this remedies a spurious behaviour that occurs when neglecting non-linear terms for neutrinos. This behaviour is related to how efficiently short modes decouple from long modes and can be traced back to the violation of momentum conservation if neutrinos are treated linearly. Furthermore, we compare our results at next to leading order to various other methods and quantify the accuracy of the fluid description. Due to the correct decoupling behaviour of short modes, the two-fluid scheme is a suitable starting point to compute higher orders in perturbations or for resummation methods.

Figures (11)

• Figure 1: The soft one-loop contribution to the propagator.
• Figure 2: Illustration of the screening of short-scale fluctuations on large scales. When treating neutrino and CDM/baryon perturbations on an equal footing, momentum conservation ensures that the leading contribution at large distances is the quadrupole moment. Within approximation schemes that treat neutrinos and CDM/baryons differently (e.g. neglecting non-linearities for the former) potentially a spurious dipole contribution can be generated which upsets the $k^2$-suppression of non-linear corrections to the matter power spectrum at small $k$.
• Figure 3: Linear solution for three neutrino species with $M_\nu=\sum m_\nu=0.21\,$eV as function of redshift for $k=0.1 h/$Mpc. The solid blue line corresponds to $\delta_{cb}$, and the solid red line to $\delta_{\nu}$. The dashed lines show the velocity divergence $-\theta_{cb}/{\cal H}$ (red) and $-\theta_\nu/{\cal H}$ (blue). The dots are taken from the solution of the Boltzmann equations obtained with CLASS.
• Figure 4: Linear solution for three neutrino species with $M_\nu=\sum m_\nu=0.21\,$eV as function of $k$ for $z=0$. The solutions are normalized to the corresponding solution of the Boltzmann equations obtained from CLASS. The assignment of the lines is as in Fig. \ref{['fig:lin1']}. The deviations for small $k$ are not related to the fluid approximation, but signal the influence of (gauge-dependent) relativistic corrections to the Newtonian limit.
• Figure 5: Nonlinear corrections to the density power spectra $\Delta P_{ab}=P_{ab}-P_{ab,lin}$ for cold/baryonic matter $P_{cb,cb}$ (blue), for neutrinos $P_{\nu,\nu}$ (red), and for the cross correlation $P_{cb,\nu}$ (magenta) at redshift $z=0$ (left panel) and $z=3$ (right panel). The black line shows the linear matter power spectrum. Solid lines correspond to wavenumbers for which $\Delta P>0$, and dashed to $\Delta P<0$. Dotted lines show the corresponding results obtained when taking the full scale dependence for the bispectrum into account.