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Lattice Simulations of Non-minimally Coupled Scalar Fields in the Jordan Frame

by Daniel G. Figueroa, Adrien Florio, Toby Opferkuch, and Ben A. Stefanek

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Submission summary

Authors (as registered SciPost users): Toby Opferkuch
Submission information
Preprint Link: scipost_202209_00065v1  (pdf)
Date submitted: 2022-09-29 22:46
Submitted by: Opferkuch, Toby
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Phenomenology
Approaches: Theoretical, Computational, Phenomenological


The presence of scalar fields with non-minimal gravitational interactions of the form $\xi |\phi|^2 R$ may have important implications for the physics of the early universe. We propose a procedure to solve the dynamics of non-minimally coupled scalar fields directly in the Jordan frame, where the non-minimal couplings are maintained explicitly. Our algorithm can be applied to lattice simulations that include minimally coupled fields and an arbitrary number of non-minimally coupled scalars, with the expansion of the universe sourced by all fields present. This includes situations when the dynamics become fully inhomogeneous, fully non-linear (due to e.g. backreaction or mode rescattering effects), and/or when the expansion of the universe is dominated by non-minimally coupled species. As an example, we study geometric preheating with a non-minimally coupled scalar spectator field when the inflaton oscillates following the end of inflation.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2022-12-23 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202209_00065v1, delivered 2022-12-23, doi: 10.21468/SciPost.Report.6370


1. Carefully articulated formalism---detailed, reproducible, and clear
2. Validation and comparison of nonlinear simulation results with linearized solutions
3. Insightful interpretation of the underlying physics in terms of the Jordan-frame field equations
4. A useful reference to accompany public software


1. Inadequate discussion of prior work, chiefly previous numerical simulations of similar models
2. Lacking motivation and technical justification for the Jordan-frame approach compared to the Einstein frame
3. New results are not especially novel


The authors present a numerical procedure to perform 3D, nonlinear simulations of scalar field theories nonminimally coupled to gravity. They choose to work in the Jordan frame, where the scalars' kinetic structure remains canonical. They provide a proof-of-concept by simulating geometric preheating after inflation and establishing consistency with linearized calculations (where applicable). Their exposition is clear, well-written, and reproducible and will serve as a useful accompaniment to the software implementation they plan to make publicly available.

The main shortcoming of this paper as a submission to SciPost Physics is its lack of novelty, given that numerous papers have studied very similar models in detail using similar numerical methods. The main technical requirement for this class of models---the use of nonsymplectic integration algorithms---is supported by several other public software packages, even if the specific model the authors consider has not been implemented. In my view, this submission's composition and strengths, as well as the authors' intention to make their own implementation publicly available, make it very well suited for SciPost Physics Codebases and I would recommend publication there after a satisfactory revision.

Requested changes

The main revisions I request are the following:

1) Though the authors' introduction provides a thorough overview of the relevance and motivation for nonminimally coupled scalar fields, the current submission is most lacking in its discussion of its contributions relative to existing literature. In particular, numerous papers have studied numerous variations of the geometric preheating model---e.g., the authors' Refs. [37-40], and in particular, their Ref. [40] and [a] and [b] (which the authors do not currently reference, linked below) perform numerical simulations of highly similar models, generalized to multifield inflation cases where both scalars are nonminimally coupled. Since the authors' aim is to present (and make publicly available) a numerical scheme for solving such systems, a more thorough comparison of their simulation results to those of existing work is warranted.


2) Since the authors advertise the use of the Jordan frame as a positive feature of their scheme, I would have liked to see more substantive discussion of its benefits. While it's convenient to avoid the need to perform the conformal transformation to the Einstein frame, I might expect CosmoLattice (with its symbolic capabilities) to be able to automate that process. As such, it would be valuable to know if the Jordan frame offers other advantages---if, say, the equations are more numerically stable or less computationally expensive. In addition, the authors should note that 3D, nonlinear simulations of nonminimally coupled scalars in the Jordan frame were performed before: [c] considered a preheating of a nonminimally coupled inflaton in the Jordan frame and solved the same equation as the authors' Eqn. 21 (c.f. Eqn 12 in [c]).


In addition, I have the follow more specific/technical questions and comments:

3) Though utilizing the trace of the Einstein equations provides a slick means to specify the background evolution, the dependence of $p_\phi$ (Eqn. 18) on $a''$ is only algebraic. The authors might note that rearranging Eqns. 13 and 14 to isolate derivatives of the scale factor yields a result consistent with Eqn 21.

4) Figure 3, which presents some of the most important results of the simulations (and comparisons to the linear analysis), is not discussed nor even referenced in the main text. This should be amended. I also have two questions regarding it:
a) The $N = 2$ lines in the left panel curiously depict an apparent increase in spectral structure in the nonlinear simulations compare to the linearized results. The typical first effect of nonlinearities is to wash out any particular resonance structure that arises in the linear regime. On the other hand, in the linear regime such oscillations in wavenumber could simply be phase offsets due to evaluating the spectra at slightly different times. It would be interesting if the authors could determine whether this discrepancy has a nontrivial cause.
b) The simulation results in the right panel exhibit a very broad and flat spectrum out to the UV, but the results appear to be truncated. The authors should display the full spectra to enable evaluating the extent of validity of these simulations. Though I expect their main qualitative point---that self-interactions quench resonance---to be insensitive to resolution effects, the authors should be careful and upfront about assessing convergence (especially so that their submission can serve as a guide for others to properly and effectively use their software).

5) The authors should specify the physical size of the simulation volume and the timestep size used in all simulations.

6) To validate the authors' evolution scheme for FLRW expansion for this class of models, and since Runge-Kutta methods typically incur numerical dissipation error (affecting the satisfaction of conservation laws), the authors should report the performance of their numerical scheme in terms of the degree of violation of the Friedmann constraint (i.e., Eqn. 13).

7) The vertical limits of the left panel of Figure 4 truncate the spikes in the Ricci scalar for large $\xi$. If these indeed take values too large to fit into the axes limits, the authors should quote the peak value in the caption or main body (and, if available, the expected scaling with $\xi$). Likewise for the left panel of Figure 5.

8) It would be helpful if the captions of Figures 4 and 5 specified that the potential $V(\phi) = 0$.

9) Have the authors tested the adaptive time stepping routine they describe in Appendix C? To my knowledge, though use of low-storage Runge-Kutta methods is fairly common, adaptive routines have been little used in 3D simulations and it would be interesting to explore their utility for, e.g., the authors' model. Do the typical choices for timesteps yield, say, percent-level accuracy when compared to results using adaptive stepping? Do the adaptive routines provide substantial savings in simulation runtime? If possible, providing guidance for future users on best practice and potential pitfalls would be valuable (especially for this submission as a SciPost Physics Codebases publication).

10) At the top of page 17, I suspect the phrase "one need to solve (almost) all of the $k^{(i)}$ coefficients" is meant to say "one needs to *store*", and there appears to be a stray comma in equation C11.

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

Author:  Toby Opferkuch  on 2023-02-22  [id 3394]

(in reply to Report 2 on 2022-12-23)

Thank you for your careful reading of our manuscript. Please see attached pdf for our reply to your comments.



Anonymous Report 1 on 2022-12-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202209_00065v1, delivered 2022-12-06, doi: 10.21468/SciPost.Report.6270


This is the referee report to the manuscript entitled “Lattice Simulations of Non-minimally Coupled Scalar Fields in the Jordan Frame” written by Daniel G. Figueroa, Adrien Florio, Toby Opferkuch and Ben A. Stefanek.

In this manuscript, the authors study numerical lattice simulations of scalar field non-minimally coupled to the gravity in the early universe which is motivated by the renormalization theory in the curved spacetime and the cosmological inflation model.
In particular, the authors newly formulate an algorithm for the lattice simulation in the Jordan frame where the non-minimal coupling is explicitly maintained in the Lagrangian. This is in contrast to most of the previous studies in which the conformal transformation of the metric is performed in order to make the non-minimal coupling term absent in the Lagrangian.

First, the authors have carefully derived the self-contained system of evolution equations for both the non-minimally coupled scalar field and the background expansion of the universe in the Jordan frame. Then, the authors give a formulation for the numerical lattice simulation by discretizing the spacetime variables and replacing spatial derivatives of the scalar field with the finite deferences. The formulation is clearly written in the main text and the technical detail is sufficiently supplemented in the appendices.

In addition, the authors focus on the geometric preheating model as an explicit application of their formulation. Starting from the analysis of the linear evolution of the scalar field sourced by the inflationary fluctuations, which sets the initial values for the lattice simulation, the lattice simulation is performed to follow the subsequent non-linear evolution of the system. Resultant power spectra of the non-minimally coupled scalar field computed by lattice simulations are carefully compared with the results of the linear analysis and the non-linear effect is clearly shown. The evolution of the energy density of the system is computed as well and the authors discuss how the reheating proceeds in this model depending on the non-minimal coupling parameter and the inflaton potential.

Finally, the manuscript contains clear summary and conclusions. The authors also mention their plan to provide publicly available code based on the algorithm presented in the main text. It will allow anyone to reproduce all of the results in this manuscript.

On the whole, the discussion is clear without any ambiguities. In particular, this work opens a new avenue for simulating the non-minimal coupling model and also it will become useful reference especially for users of publicly available CosmoLattice code provided by authors themselves. Thus, I think this work satisfies the acceptance criteria and desrves the publication in SciPost Physics.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
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