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Kinetic field theory: Non-linear cosmic power spectra in the mean-field approximation

by Matthias Bartelmann, Johannes Dombrowski, Sara Konrad, Elena Kozlikin, Robert Lilow, Carsten Littek, Christophe Pixius, Felix Fabis

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Submission summary

Authors (as registered SciPost users): Matthias Bartelmann · Sara Konrad · Elena Kozlikin
Submission information
Preprint Link: scipost_202104_00024v1  (pdf)
Date accepted: 2021-06-18
Date submitted: 2021-04-21 11:56
Submitted by: Bartelmann, Matthias
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
Approach: Theoretical


We use the recently developed Kinetic Field Theory (KFT) for cosmic structure formation to show how non-linear power spectra for cosmic density fluctuations can be calculated in a mean-field approximation to the particle interactions. Our main result is a simple, closed and analytic, approximate expression for this power spectrum. This expression has two parameters characterising non-linear structure growth which can be calibrated within KFT itself. Using this self-calibration, the non-linear power spectrum agrees with results obtained from numerical simulations to within typically $\lesssim10\,\%$ up to wave numbers $k\lesssim10\,h\,\mathrm{Mpc}^{-1}$ at redshift $z = 0$. Adjusting the two parameters to optimise agreement with numerical simulations, the relative difference to numerical results shrinks to typically $\lesssim 5\,\%$. As part of the derivation of our mean-field approximation, we show that the effective interaction potential between dark-matter particles relative to Zel'dovich trajectories is sourced by non-linear cosmic density fluctuations only, and is approximately of Yukawa rather than Newtonian shape.

Author comments upon resubmission

Dear Editors,

thank you very much for making the referees' reports available to us, and sorry for the tedious procedure of finding referees for this paper! We also thank the referees for carefully reviewing our manuscript and for their constructive comments. In response to the reports, we have quite extensively revised our paper as detailed below. We hope that it is now acceptable for publication in SciPost Physics.

Best regards,

Matthias Bartelmann
on behalf of the authors

List of changes

Response to the referee's reports

Since the second referee (in temporal order) does not request any changes, we focus here on the first referee's report. We go through its individual items and list our response and the changes applied to the manuscript.

Referee: "The authors refer to their method as KFT: however I do not see anything truely "kinetic", in the statistical physics sense of the word, in their work. No mention of distribution function neither of the Boltzmann equation. In fact they use fluid equations such as Burger's model to describe various phenomena. I would ask the authors to either relate to the established kinetic theory in statphys or remove the term.

Response: In order to clarify this issue and define in which sense KFT is a kinetic theory, we have added this second paragraph to Sect. 1:

"Unlike other approaches to kinetic theory, KFT is not based on a phase-space density function subject to the Liouville or Boltzmann equations. It is a kinetic theory in the sense that it describes the joint evolution of a particle ensemble while intentionally integrating over microscopic information. KFT defines the initial state of a particle ensemble by covering phase-space with a probability distribution. It samples this distribution with discrete particles and follows their Hamiltonian phase-space trajectories in time. Each trajectory thus carries an initial occupation probability through phase space. The set of all particle trajectories is the field that KFT operates on. Statistical properties of the particle ensemble can be derived from the evolved bundle of trajectories. A discussion of how KFT relates to the BBKS hierarchy of kinetic theory can be found in [Phys. Rev. E 91 (2015) f2120]."

Referee: They claim that the interaction between particles, past Zeldovich crossing, is Yukawa rather than Newton. I ask the authors to tone this down: the claim is that the Newtonian interaction can be "modeled" by Yukawa potential after orbit crossing. They are not suggesting that we run cosmological simulations and rewrite our perturbation theory using Yukawa potential!! I ask the authors to rewrite the text concerning this point.

Response: We are now explaining in detail how the Yukawa-shaped interaction potential occurs, clarifying that it is of statistical nature. We further describe the meaning of the Yukawa scale k_0 and its relevance for non-linear structure formation. We have added the following paragraphs after Fig. 1:

"It is important to emphasize that expression (12) for the particle-particle interaction potential v is statistical in the sense that this potential describes the interaction between particles in an ensemble average. Thus, v(q) is cannot be taken as the interaction potential between two individual particles separated by q, but must be understood as the ensemble-averaged potential between such particles. The derivation leading to (12) shows that v depends on the spatial correlations within the particle ensemble.

The Yukawa scale k_0 separates non-linear from linear scales. On larger scales, the density contrast evolves linearly. The gravitational interaction on such scales is already captured by the Zel'dovich approximation and thus needs to be removed from the interaction potential. On smaller scales, trajectories deviate from the inertial Zel'dovich trajectories under the influence of that part of the gravitational interaction neglected by the Zel'dovich approximation.''

Referee: Yukawa coupling: when used in particle physics has a proper well-defined parameter in terms of fundamental constants and masses of particles. There is no such a things here: the Yukawa coupling scale is rather adhoc, it is time-dependent (which the authors just ignore) and itself a function of density fluctuation, etc. This is related to the general problem in cosmology that we lack a proper "perturbation parameter". I ask the authors to address these issues and to justify the fact that they are ignoring the time-dependence of their "Yukawa scale".

Response: Part of this comment is now covered by our reply to the previous item. To further clarify the meaning of the Yukawa scale and our approximation of setting it constant, we extended the last paragraph of Sect. 2 as follows:

"Concluding this section, we wish to point out that the shape (13) of the transition from the linear to the non-linear shape of the power spectrum is a convenient fitting function that may be improved. Thus, the Yukawa form of the interaction potential is a convenient approximation, but not a fundamental result. The time dependence of the Yukawa scale $k_0$ is sufficiently weak for us to neglect it for simplicity in our mean-field approach."

Referee: Burger Turbulence. The authors refer to Burger Turbulence as Navier-Stokes, I ask the authors to correct this in the text, as Burger turbulence is just a model of potential turbulence and has little to do with Navier-Stokes equation. I find the use of Burger turbulence magical rather than scientific! To search for a damping scale they use Burgers, also modified slightly because in Burgers there are shocks and so no smoothed-out structures. This is more a demonstration of the failure of their Yukawa modeling of the particle interaction than anything else. Indeed if Yukawa potential could model the particle interactions properly, there would be no need to then invoke yet another model (fluid and not kinetic) to model again the particle interactions in the multistream regions.

Response: We now clarify in the text that Burgers' equation is used in our context in the same way as it has been used previously in cosmology, i.e. as an effective model to suppress the re-expansion of cosmic structures. Also, we now emphasize that we use Burgers' equation for the sole purpose of suppressing the re-expansion of cosmic structures in our mean-field approximation of the interaction term, without reference to the possibility of modelling turbulence with it. We have added a further paragraph to this effect after Eq. (43) which also includes relevant references:

"We should emphasize that we are introducing Burgers' instead of the Zel'dovich approximation here with the same motivation as it has been used in cosmology before, beginning with [MNRAS 236 (1989) 260], which has been to suppress the re-expansion of cosmic structures in the Zel'dovich approximation; cf.\ [MNRAS 247 (1990) 260] for an early application to large-scale structures, and [Phys. U. 55 (2012) 223] for a review. The only purpose we pursue here with Burgers' approximation is to reduce the amount of damping in the KFT interaction term. Since this damping comes out too strong in the Zel'dovich approximation, lowering it by Burgers' approximation seems appropriate."

Referee: Indeed Burger turbulence has been used extensively years ago to model large-scale structure of the Universe and was refered to as ``adhesion model''. Since the purpose of this work is to model nonlinear power spectrum, then I ask the authors to also compare their power spectrum to those from the adhesion model, and indeed other existing models of nonlinear power spectrum. I ask the authors to provide plot on which they show the PS from simulations, from their model and also adhesion model, truncated ZA model, models of Scoccimarro et al and Taruya et al, Ostriker et al etc. …

Response: Understandable as this request by the referee is, it would be quite (and, as we believe, unnecessarily) difficult to satisfy it. This is because, first of all, we are not using Burgers' equation or the Zel'dovich approximation to model the non-linear power spectrum. Instead, we model the power spectrum from the evolution of the KFT generating functional, in which the unperturbed or inertial particle trajectories are modelled as Zel'dovich trajectories, but which importantly includes the interaction term between particles to quantify the deviation of the non-linear power spectrum from its Zel'dovich shape.

We use Burgers' approximation exclusively to reduce the unphysical small-scale suppression of cosmic structures in the interaction term. Where non-linear power spectra are derived in the literature from the Zel'dovich or adhesion approximations or improved versions thereof, they start deviating from the numerically modelled power spectra near k >~ 0.2 h/Mpc for redshifts z >~ 0.35-0.5, while our non-linear power spectrum agrees well with numerical simulations up to k <~ 10 h/Mpc at z = 0. To address this comment of the referee, we have added the following paragraph to the conclusions, including several references which are interesting in this context:

"Compared to approaches based on the Zel'dovich approximation and its second-order or truncated variants, or on Burgers' equation, KFT provides a systematic procedure to include particle interactions at different levels of approximation. This is reflected by the comparison of non-linear power spectra obtained from Zel'dovich-based approximation schemes with numerical results; e.g. Fig. 7 in [Phys. Rev. D 80 (2009) l3503] or Fig. 9 in [Phys. Rev. D 86 (2012) j3528]. There, substantial deviations typically set in at k >~ 0.2 h/Mpc for redshifts z <~ 0.35-0.5. The mean-field approach together with the simplifying assumptions introduced in this paper allow to calculate the non-linear power spectrum very quickly for k <~ 10 h/Mpc at z = 0. In forthcoming papers, we will relax these assumptions and put the derivation of the mean-field term on a more rigourous foundation."

Referee: The authors test their model against simulations and show that indeed their model provides a good agreement for the nonlinear PS. However they need use simulations to set their Yukawa scale and viscosity parameters in the first place! Indeed their free parameters are best determined by requiring their model to match the simulations. This problem need be addressed.

Response: Concerning this point, we respectfully disagree with the referee. In fact, our Fig. 2 shows that KFT allows to determine its only two parameters nu and k_0 in such a way that the non-linear power spectrum is already in good agreement with numerical simulations, without calibrating them by means of such simulations. For further clarification, we have added the following paragraph to the end of Sect. 4:

"We are not suggesting that we should ultimately calibrate the viscosity and the Yukawa scale by numerical simulations. Rather, a more rigorous future derivation of the power spectrum and higher-order spectra from KFT should explain the values of nu and k_0 more accurately than we have done here. Instead, we are showing Fig. 3 to emphasize that small variations of these two parameters about the values suggested by KFT itself lead to a result which resembles the shape of the numerically simulated, non-linear power spectrum very well, despite the simplicity of our mean-field ansatz."

Referee: Apart from this, I find it rather disappointing that the numerical simulations provide a final test of the "kinetic" model here. Indeed the real interest of using kinetic theory is that they can go beyond numerical simulations which are limited by resolution. Kinetic theory could model structures that are often missed in numerical simulations because of numerical artifacts. In this resepect kinetic theory remains superior to present numerical simulations and only a full simulation of Vlasov-Poisson (VP) equation would provide the ultimate model of nonlinear evolution. Since VP simulation are non-existent, and shall most probably not emerge in the forseeable future because of the huge computational resources they require, a clever kinetic theory-based model could provide a guideline in the meantime. However disappointingly this point is completely missed out by the authors. Authors need discuss this problem, namely can they model structures 'caustics, streams, that are often missed in the simulations due to lack of resolution?

Response: We agree that KFT should ultimately be able to test the results of numerical simulations rather than the reverse. We have not reached this level yet, but believe that the present paper shows how the mean-field approximation within KFT reproduces numerical results with minimal effort and two parameters that can be calibrated by KFT itself. To clarify this point, we have added the following text to the conclusions:

"We thus view this paper as a report on ongoing work, not as a final answer. The main advantages of KFT compared to other analytic approaches to large-scale structure formation, in particular Effective Field Theory [JHEP 9 (2012) 82, Phys. Rev. D 89 (2014) d3521, JCAP 5 (2014) 22, JCAP 7 (2014) 57], are that KFT avoids the shell-crossing problem, thus allowing to enter quite deeply into the non-linear regime, and has a minimum of parameters which can be determined from the theory itself."

Published as SciPost Phys. 10, 153 (2021)

Reports on this Submission

Anonymous Report 2 on 2021-6-9 (Invited Report)


The journal acceptance criteria are met and I recommend the paper for publication.

  • validity: good
  • significance: good
  • originality: high
  • clarity: ok
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2021-4-22 (Invited Report)


I believe that the authors have provided a very extensive a convincing reply to all the points raised by the referees. The extra text added in response to the referees points have made the paper quality even more robust in terms of completeness and scientific rigor.


I believe the paper is now ready for publication in this journal in its present form.

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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