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Register a new userFully relativistic treatment of light neutrinos in $N$-body simulations
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Cosmological $N$-body simulations are typically purely run with particles using Newtonian equations of motion. However, such simulations can be made fully consistent with general relativity using a well-defined prescription. Here, we extend the formalism previously developed for $Lambda$CDM cosmologies with massless neutrinos to include the effects of massive, but light neutrinos. We have implemented the method in two different $N$-body codes, CONCEPT and PKDGRAV, and demonstrate that they produce consistent results. We furthermore show that we can recover all appropriate limits, including the full GR solution in linear perturbation theory at the per mille level of precision.
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In the next decade, cosmological surveys will have the statistical power to detect the absolute neutrino mass scale. N-body simulations of large-scale structure formation play a central role in interpreting data from such surveys. Yet these simulations are Newtonian in nature. We provide a quantitative study of the limitations to treating neutrinos, implemented as N-body particles, in N-body codes, focusing on the error introduced by neglecting special relativistic effects. Special relativistic effects are potentially important due to the large thermal velocities of neutrino particles in the simulation box. We derive a self-consistent theory of linear perturbations in Newtonian and non-relativistic neutrinos and use this to demonstrate that N-body simulations overestimate the neutrino free-streaming scale, and cause errors in the matter power spectrum that depend on the initial redshift of the simulations. For $z_{i} lesssim 100$, and neutrino masses within the currently allowed range, this error is $lesssim 0.5%$, though represents an up to $sim 10%$ correction to the shape of the neutrino-induced suppression to the cold dark matter power spectrum. We argue that the simulations accurately model non-linear clustering of neutrinos so that the error is confined to linear scales.
The N-body gauge allows the introduction of relativistic effects in Newtonian cosmological simulations. Here we extend this framework to general Horndeski gravity theories, and investigate the relativistic effects that the scalar field introduces in the matter power spectrum at intermediate and large scales. In particular, we show that the kineticity function at these scales enhances the amplitude of the signal of contributions coming from the extra degree of freedom. Using the Quasi-Static Approximation (QSA), we separate modified gravity effects into two parts: one that only affects small-scale physics, and one that is due to relativistic effects. This allows our formalism to be readily implemented in modified gravity N-body codes in a straightforward manner, e.g., relativistic effects can be included as an additional linear density field in simulations. We identify the emergence of gravity acoustic oscillations (GAOs) in the matter power spectrum at large scales, $k sim 10^{-3}-10^{-2}$ Mpc$^{-1}$. GAO features have a purely relativistic origin, coming from the dynamical nature of the scalar field. GAOs may be enhanced to detectable levels by the rapid evolution of the dark energy sound horizon in certain modified gravity models and can be seen as a new test of gravity at scales probed by future galaxy and intensity-mapping surveys.
We present a new method for generating initial conditions for numerical cosmological simulations in which massive neutrinos are treated as an extra set of N-body (collisionless) particles. It allows us to accurately follow the density field for both Cold Dark Matter (CDM) and neutrinos at both high and low redshifts. At high redshifts, the new method is able to reduce the shot noise in the neutrino power spectrum by a factor of more than $10^7$ compared to previous methods, where the power spectrum was dominated by shot noise at all scales. We find that our new approach also helps to reduce the noise on the total matter power spectrum on large scales, whereas on small scales the results agree with previous simulations. Our new method also allows for a systematic study of clustering of the low velocity tail of the distribution function of neutrinos. This method also allows for the study of the evolution of the overall velocity distribution as a function of the environment determined by the CDM field.
Initial conditions for (Newtonian) cosmological N-body simulations are usually set by re-scaling the present-day power spectrum obtained from linear (relativistic) Boltzmann codes to the desired initial redshift of the simulation. This back-scaling method can account for the effect of inhomogeneous residual thermal radiation at early times, which is absent in the Newtonian simulations. We analyse this procedure from a fully relativistic perspective, employing the recently-proposed Newtonian motion gauge framework. We find that N-body simulations for LambdaCDM cosmology starting from back-scaled initial conditions can be self-consistently embedded in a relativistic space-time with first-order metric potentials calculated using a linear Boltzmann code. This space-time coincides with a simple N-body gauge for z<50 for all observable modes. Care must be taken, however, when simulating non-standard cosmologies. As an example, we analyse the back-scaling method in a cosmology with decaying dark matter, and show that metric perturbations become large at early times in the back-scaling approach, indicating a breakdown of the perturbative description. We suggest a suitable forwards approach for such cases.
We discuss the relation between the output of Newtonian N-body simulations on scales that approach or exceed the particle horizon to the description of General Relativity. At leading order, the Zeldovich approximation is correct on large scales, coinciding with the General Relativistic result. At second order in the initial metric potential, the trajectories of particles deviate from the second order Newtonian result and hence the validity of 2LPT initial conditions should be reassessed when used in very large simulations. We also advocate using the expression for the synchronous gauge density as a well behaved measure of density fluctuations on such scales.
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