Efficient implementation of the time renormalization group

The time renormalization group (TRG) is an effective method for accurate calculations of the matter power spectrum at the scale of the first baryonic acoustic oscillations. By using a particular variable transformation in the TRG formalism, we can reduce the 2D integral in the source term of the equations of motion for the power spectrum into a series of 1D integrals. The shape of the integrand allows us to precompute only 13 antiderivatives numerically, which can then be reused when evaluating the outer integral. While this introduces a few challenges to keep numerical noise under control, we find that the computation time for nonlinear corrections to the matter power spectrum decreases by a factor of 50. This opens up the possibility to use TRG for mass production as in Markov chain Monte Carlo methods. A fortran code demonstrating this new algorithm is publicly available.

Figure 1

All 14 integrands (in arbitrary units) after carrying out the $p$ integration at different values for $k$. As $k$ increases, the integrands approach a function with a vertical asymptote and a singularity at $k$.

Figure 2

The cutoff wave number $k_{\max }$ at $z=0$ barely depends on the number of sampling points when performing the $q$ integration.

Figure 3

The nonlinear power spectrum computed by trgfast at $z=0$ compared to the one computed by Class. Both agree up to 1%.

Figure 4

The nonlinear power spectrum computed by trgfast at $z=0$ compared to the one computed by copter. The power spectra agree up to 3.5%.

Figure 5

A comparison of the power spectrum at $z=0$ obtained from different sources divided by a linear no-wiggle power spectrum from [5]. The sources are coyote (blue, thick), trgfast with only 12 independent equations (TRG-12, red), copter (green, dashed), trgfast with all 14 equations (TRG-14, purple), Class (black, dashed), halofit (orange, dashed), and the linear power spectrum (gray). The bottom panel is simply a zoomed-in view on the BAO scale.

Figure 6

Comparison of the trgfast result with $\mathrm{\Lambda }\mathrm{CDM}$ $N$-body data [38] for different redshifts. Each power spectrum has been divided by a no-wiggle spectrum from [5]. Agreement is good for redshift $z=1$ and higher, then some discrepancy is noticeable just as in Fig. 7 of [28]. Note the difference in the scale of the $y$ direction for some panels.

Figure 7

These curves indicate at which wave number $k_{\%}$ the power spectra from trgfast and $\mathrm{\Lambda }\mathrm{CDM}$ $N$-body simulations [38] start to deviate by more than 1%, 2%, and 3% for each redshift. We left out the case for $z=0.5$ because the power spectra coincidentally agree too well to have a meaningful value for $k_{\%}$ here.

Figure 8

The relative difference of the nonlinear power spectrum obtained from trgfast and from CoDECS at $z=0$ in the coupled quintessence case. The difference is comparable to the $\mathrm{\Lambda }\mathrm{CDM}$ case.

Figure 9

The relative difference of the growth function obtained from trgfast and from CoDECS. The discrepancy is less than 2% for all relevant redshifts.