Supplemental Material to “Low frequency tail of gravitational

wave spectra from hydromagnetic turbulence”

by Ramkishor Sharma & Axel Brandenburg

Abstract

In Figs. 3 and 4 of the original version of our paper https://arxiv.org/abs/2206.00055, we have shown the time evolution of |T̃(k)| and | ˙h̃(k)| for five different k values of the wave vector (k, 0, 0). In this Supplemental Material, we provide the time evolution of the same quantities for wave vectors (0, k, 0) and (0, 0, k). The overall behavior of the phases is similar to that for (k, 0, 0).

Time evolution of stress and the derivative of gravitational wave strain at selected wave vectors

As discussed in Section II.B of the paper, the time evolutions of |T̃(k)| and | ˙h̃(k)| for a particular wave vector can be very different than the collective effect of all the wave vectors of the same length. Furthermore, arg(T̃) and arg( ˙h̃) remain approximately constant for some time and start evolving more strongly after that. Here, we provide the time time evolutions of |T̃(k)|, | ˙h̃(k)|, arg(T̃), and arg( ˙h̃) for two more wave vectors (0, k, 0) and (0, 0, k) in Figures 1 and 2, respectively. Figures 1 and 2 are for the helical case (HEL) and Figures 1 and 2 are the corresponding figures for the nonhelical case (NHEL).

Conclusion

As is evident from these figures, the over behavior of arg(T̃) and arg( ˙h̃) is similar as for the wave vector (k, 0, 0) shown in Figure 3 and 4 of the paper for the helical and non helical case, respectively.

Figure 1: Modulus and phase of T̃(k, t) and ˙ h̃(k, t) for the helical case for k = (0, k, 0) with k = 0.3 (orange), 0.4 (red), 0.5 (green), 0.6 (blue), and 0.7 (black). The inset shows the phase with a linear abscissa.

Figure 2: Same as Figure 3, but for the wave vector k = (0, 0, k). Figure 3: Same as Figure 1, but for the nonhelical case Figure 4: Same as Figure 2, but for the nonhelical case.

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