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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
2
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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