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On the analogy between gravitationally driven

turbulence and classical turbulence

Nils T. Bassea

aElsas v¨ag 23, 423 38 Torslanda, Sweden

April 17, 2020

Abstract

Power spectra of simulated density ﬂuctuations generated by gravitational

fragmentation are compared to corresponding expressions from classical tur-

bulence.

1. Introduction

A recent paper [1] includes power spectra of simulated density ﬂuctuations

generated by gravitational fragmentation. We have used [2] to extract the

power spectrum for the scale factor a= 200, so there are inaccuracies in our

postprocessing.

Previously, we have studied large-scale cosmological power spectra and

attempted to place them in the context of classical turbulence using the

example of turbulence measured in fusion plasmas [3, 4].

2. Power spectra

We treat power spectra Pas a function of wavenumber ratio kratio ≡

k/khorizon. For classical turbulence, the 3D Kolmogorov scaling is:

P(kratio)3D∝E(kratio)3D

k2

ratio

∝k−11/3

ratio ,(1)

where Eis energy.

For large wavenumbers, energy is dissipated and becomes an exponential

function:

Email address: nils.basse@npb.dk (Nils T. Basse)

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P(kratio)dissipation ∝exp(−n×kratio),(2)

where nis a constant.

Results are summarized in Fig. 1:

The Kolmogorov scaling agrees with the simulated power spectrum over

roughly an order of magnitude of intermediate wavenumber ratios.

The exponential ﬁt is applied for wavenumber ratios larger than 10;

here, n= 1.14.

100101

k/khorizon

10-20

10-15

10-10

10-5

100

P(k) [a.u.]

PRL 124, 061301 (2020)

3D Kolgomorov cascade

Exponential fit

Threshold for exponential fit

Figure 1: Simulated power spectrum (blue) with Kolmogorov scaling (black) and exponen-

tial ﬁt (red) versus wavenumber ratio. The vertical magenta line indicates the wavenumber

ratio used for the exponential ﬁt: Only wavenumber ratios larger than the threshold value

are used.

3. Dimensionality of cosmological turbulence

Findings in this work indicate that density ﬂuctuations based on gravita-

tional fragmentation appear to be 3D. However, our previous work has shown

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that large-scale cosmological power spectra are most likely closer related to

2D.

Thus, we speculate that early cosmological turbulence was 3D and during

expansion transitioned to 2D [5]. This departure from isotropy might be

related to the anisotropy of cosmic acceleration [6].

4. Conclusions

We have shown that the simulations of density ﬂuctuations based on

gravitational fragmentation can be analysed as classical turbulence. Based

on this, we argue that an analogy exists between ﬂuctuations observed in

cosmology and classical turbulence.

References

[1] Musoke N, Hotchkiss S and Easther R. Lighting the dark: Evolution of

the postinﬂationary universe. Phys. Rev. Lett. 2020;124:061301.

[2] Plot Digitizer 2020. http://plotdigitizer.sourceforge.net/

[3] Basse NP. Density ﬂuctuations on mm and Mpc scales. Physics Letters

A 2005;340:456-460.

[4] Basse NP. A study of multiscale density ﬂuctuation measurements. IEEE

Transactions on Plasma Science 2008;36:458-461.

[5] Dickau JJ. Fractal cosmology. Chaos, Solitons and Fractals

2009;41:2103-2105.

[6] Colin J, Mohayaee R, Rameez M and Sarkar S. Evidence for anisotropy

of cosmic acceleration. Astronomy & Astrophysics 2019;631:L13.

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