The Effective Field Theory of Dark Matter and Structure Formation: Semi-Analytical Results

Abstract

Complimenting recent work on the effective field theory of cosmological large scale structures, here we present detailed approximate analytical results and further pedagogical understanding of the method. We start from the collisionless Boltzmann equation and integrate out short modes of a dark matter/dark energy dominated universe (LambdaCDM) whose matter is comprised of massive particles as used in cosmological simulations. This establishes a long distance effective fluid, valid for length scales larger than the non-linear scale ~ 10 Mpc, and provides the complete description of large scale structure formation. Extracting the time dependence, we derive recursion relations that encode the perturbative solution. This is exact for the matter dominated era and quite accurate in LambdaCDM also. The effective fluid is characterized by physical parameters, including sound speed and viscosity. These two fluid parameters play a degenerate role with each other and lead to a relative correction from standard perturbation theory of the form ~ 10^{-6}c^2k^2/H^2. Starting from the linear theory, we calculate corrections to cosmological observables, such as the baryon-acoustic-oscillation peak, which we compute semi-analytically at one-loop order. Due to the non-zero fluid parameters, the predictions of the effective field theory agree with observation much more accurately than standard perturbation theory and we explain why. We also discuss corrections from treating dark matter as interacting or wave-like and other issues.

Figures (9)

• Figure 1: Vertex for interaction between long-long mode coupling or long-short mode coupling.
• Figure 2: Linear power spectrum of density fluctuations $P_L(k)$ computed from CAMB, with $n_s=0.96$, $z=0$, $\Omega_m=0.226$, $\Omega_k=0$. The plot for $k<k_{eq}$ shows the approximate scale invariance of the spectrum.
• Figure 3: Linear standard deviation of density fluctuations $\Delta_\delta(k)$ computed from CAMB, with $n_s=0.96$, $z=0$, $\Omega_m=0.226$, $\Omega_k=0$. The plot indicates that the evolution is perturbative for small $k$ and non-perturbative for high $k$.
• Figure 4: Linear density correlation function $\xi(r)$ computed from CAMB in a $\Lambda$CDM universe, with $n_s=0.96$, $z=0$, $\Omega_m=0.226$, $\Omega_k=0$. This clearly shows the baryon-acoustic-oscillation peak at $r\sim 120$ [Mpc/h].
• Figure 5: Curl of velocity ${\bf w}$ (normalized to some initial value) in a $k$-mode as a function of time in a matter dominated era in the linear approximation. Blue is for $c_{sv}^2>0$ with re-summation and red is for $c_{sv}^2=0$. This shows the decay in vorticity over time at this order of analysis.