Dark matter clustering: A simple renormalization group approach

I compute a renormalization group (RG) improvement to the standard beyond-linear-order Eulerian perturbation theory (PT) calculation of the power spectrum of large-scale density fluctuations in the Universe. At $z=0$, for a power spectrum matching current observations, lowest order RGPT appears to be as accurate as one can test using existing numerical simulation-calibrated fitting formulas out to at least $k\simeq 0.3h{\mathrm{Mpc}}^{-1}$; although inaccuracy is guaranteed at some level by approximations in the calculation (which can be improved in the future). In contrast, standard PT breaks down virtually as soon as beyond-linear corrections become non-negligible, on scales even larger than $k=0.1h{\mathrm{Mpc}}^{-1}$. This extension in range of validity could substantially enhance the usefulness of PT for interpreting baryonic acoustic oscillation surveys aimed at probing dark energy, for example. I show that the predicted power spectrum converges at high $k$ to a power law with index given by the fixed-point solution of the RG equation. I discuss many possible future directions for this line of work. The basic calculation of this paper should be easily understandable without any prior knowledge of RG methods, while a rich background of mathematical physics literature exists for the interested reader.

Figure 1

Thick lines show $P(k)/P_L(k)$ at $z=0$ for the Peacock and Dodds (PD96) fitting formula [71] (black, solid), 3’s HALOFIT formula (blue, long-dashed), the RG-improved PT of this paper (red, short-dashed), and standard PT (green, dotted). Thin lines show the other cases divided by the prediction of PD96. (a) and (b) are similar except for the axis scales. Note that standard PT was used to calibrate HALOFIT at $k<0.1h{\mathrm{Mpc}}^{-1}$.

Figure 2

As Fig. 1 except at $z\sim 1$. We label this and subsequent $z>0$ figures by the redshift at which the amplitude of the linear theory power in a flat $\Lambda \mathrm{CDM}$ model with ${\Omega }_m=0.281$ matches the amplitude in the calculation. The actual redshift in the EdS model used for the calculation is somewhat lower (e.g., for equal $z=0$ linear power, $z=1$ in the realistic $\Lambda \mathrm{CDM}$ model corresponds to $z=0.61$ in the EdS model).

Figure 3

As Fig. 1 except at $z\sim 3$ ($z=2.08$ in the EdS model).

Figure 4

As Fig. 1 except at $z\sim 6$ ($z=4.37$ in the EdS model).