Cosmic Structure Formation in the Non-linear Regime: Beyond Gaussian Statistics and Standard Cosmologies

Alex Gough

REVIEW 4 major objections 4 minor 89 references

Cosmic Structure Formation in the Non-linear Regime: Beyond Gaussian Statistics and Standard Cosmologies

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This thesis argues that the one-point probability distribution of the cosmic matter density, predicted by large deviations theory from Gaussian initial conditions and spherical collapse, stays accurate in modified gravity and dynamical…

desk verdict A solid, well-written PhD thesis that packages four published papers and one preprint; no new results beyond those papers, but a clear and honest entry point to LDT-based matter PDFs. read the letter →

arxiv 2411.16500 v1 pith:GC6TJCO3 submitted 2024-11-25 astro-ph.CO

Alex Gough This is my paper · ORCID

classification astro-ph.CO

keywords largedeviationstheorymatterdensityPDFmodifiedgravitydarkenergycosmicstructureformationwaveone-pointstatisticsphase-spacedynamics

topics Dark Energy

open problems Dark Energy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The thesis advances a practical claim: the smoothed matter density PDF is a survey-ready statistic, not just a theoretical curiosity. Using large deviations theory, it predicts the density PDF in modified gravity and evolving dark energy models to within a few percent, and Fisher forecasts show the PDF halves dark-energy parameter uncertainties and boosts modified-gravity detection significance by up to six times relative to the power spectrum alone. A second thread derives one-point PDF covariances from the joint two-cell PDF, recovering super-sample covariance that simulation boxes miss. A third thread uses a wavefunction forward model of dark matter to encode full phase-space dynamics beyond a perfect fluid, showing that interference patterns decompose into classical multi-stream trajectories and that caustics obey universal scaling. Together these results position non-Gaussian one-point statistics as a complement to two-point analyses for current and upcoming surveys.

What carries the argument

The load-bearing object is the large-deviations decay-rate function $\psi_\rho(\rho) = \tfrac{1}{2}\,\delta_L(\rho)^2\,\sigma_L^2(R,z)/\sigma_L^2(R\rho^{1/3},z)$, built from the Gaussian rate function for linear densities and contracted through the spherical-collapse mapping $F:\delta_L\mapsto\rho$; the log-density $\mu=\ln\rho$ is used as the transformed variable to extend convexity, and the PDF is recovered through a saddle-point approximation to the inverse Laplace transform. Its three ingredients are the linear variance $\sigma_L^2$, the spherical-collapse mapping (parametrised as $(1-\delta_L/\nu_{\mathrm{SC}})^{-\nu_{\mathrm{SC}}}$ in Einstein-de Sitter), and the non-linear log-density variance $\sigma^2_{\mathrm{NL},\ln\rho}$. For the dynamics chapters, the central mechanism is the Schrödinger wavefunction forward model whose stationary-phase 'unweaving' separates the wavefunction into stream components, with catastrophe theory providing universal scaling near caustics.

What would settle it

Take an f(R) or w0waCDM cosmology, compute the LDT PDF using only the lognormal-rescaled non-linear variance, and compare it against N-body measurements on $10\,h^{-1}\,\mathrm{Mpc}$ spheres at $z=0$ and $z=1$; if residuals exceed a few percent within two log-density standard deviations, the variance input is inadequate.

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Extended reading notes

Core claim

The central claim is that non-Gaussian one-point statistics of the cosmic density field can be predicted accurately enough to serve as cosmological probes. For the matter PDF, the large-deviations construction—Gaussian linear density, spherical-collapse mapping, and linear variance—extends to modified gravity and dynamical dark energy with two substitutions: the Einstein-de Sitter spherical collapse mapping rescaled by the ratio of linear variances, and a lognormal rescaling for the non-linear log-density variance. The thesis validates this against N-body simulations for f(R) gravity, DGP gravity, and w0waCDM on $10\,h^{-1}\,\mathrm{Mpc}$ spheres, and shows that the PDF adds information to the matter power spectrum in Fisher forecasts. For dynamics, it claims a wavefunction forward model encodes the full Vlasov phase-space behaviour beyond the perfect-fluid closure, with interference patterns unwoven into streams and universal scaling near caustics. For covariances, it claims the joint two-cell PDF predicts the one-point PDF covariance, including density-dependent clustering and super-sample covariance.

Load-bearing premise

The prediction stands or falls on the non-linear variance of the log-density, which the theory does not derive from first principles: it must be measured from the same simulations being compared or approximated by a lognormal rescaling calibrated to a fiducial cosmology.

Editorial extensions

If this is right

The matter PDF can be combined with the power spectrum in Fisher forecasts for Euclid-like survey volumes, halving the uncertainty on evolving dark energy parameters.
The PDF increases the detection significance of departures from general relativity by up to six times compared with the power spectrum alone in models like F6 and DGPw.
One-point PDF covariances, including super-sample covariance, can be obtained analytically from effective two-point PDF models, correcting a limitation of simulation-based covariances.
The wave-based forward model captures multi-streaming phase-space dynamics that a perfect-fluid closure misses, and near caustics its statistics display universal scaling.
Density statistics from wave dark matter forward models separate initial-condition effects from dynamical effects, sharpening predictions for wavelike dark matter candidates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same two-cell covariance machinery that predicts PDF covariances could be extended to build covariance matrices for counts-in-cells and weak-lensing PDFs in real surveys, where simulation-only covariances are prohibitively expensive.
Editorial inference: if the universal caustic scalings of the wave model hold generally, they give an analytic handle on interference statistics in ultralight axion dark matter without resolving the full wavefunction numerically.
Editorial inference: the weakest link in the PDF programme is the non-linear variance input; a first-principles calibration predicting this variance from the linear power spectrum would make the PDF a fully predictive cosmological probe, but this thesis does not establish such a calibration.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, here's my read. This is essentially a thesis by publication. Chapters 5-8 are versions of Cataneo et al. 2022, Gough & Uhlemann 2022a/2022b/2024, and Uhlemann et al. 2023. If you have followed that literature, you will not find new equations here. The value is in the packaging and the pedagogy: the background chapters and the LDT formalism chapter are unusually clear, and the appendices contain real derivations, such as matching the spherical collapse index to skewness, the minimal tree model, and the Wigner equation. The pyLDT code is public, and the claims are checked against N-body simulations, including f(R) and DGP suites.

The central scientific claim—that the matter PDF can be accurately predicted in modified gravity and dynamical dark energy and adds constraining power beyond the power spectrum—is defensible, but it is not fully first-principles. The non-linear variance of the log-density, which controls the PDF width, is either measured from the same simulations or approximated by Eq (5.23), a lognormal rescaling calibrated to a fiducial cosmology. The thesis is honest about this: Section 4.3.4 calls it a free parameter or measured input. That means the 'prediction' inherits calibration. The claim is still useful for survey forecasts, but it should be stated as a calibrated model, not a derivation. The same applies to the spherical collapse rescaling in Eq (5.20); the thesis says it is justified a posteriori.

I also have a moderate concern about the Fisher forecasts: the covariance matrix is assumed cosmology-independent and taken from Quijote. That is a conventional choice, but for modified gravity it is an assumption, and Section 5.6 acknowledges the resulting uncertainty only in passing.

What is genuinely good: the LDT derivation in Chapter 4 recovers the tree-level skewness and is presented more carefully than in the original papers; the covariance work in Chapter 6 is a nice extension; and the wave-based forward model chapters are honest about their limitations. The thesis is a useful compendium for a graduate student entering this area.

My verdict: as a journal submission, I would send it to review, because the underlying work is peer-reviewed and clearly presented, but I would make the authors specify what this version adds. I would not cite the thesis itself; I would cite the underlying papers. Bring it to reading group if someone wants to learn LDT.

Referee Report

4 major / 4 minor

Summary. This thesis (arXiv:2411.16500) develops and tests non-Gaussian one-point statistics of the cosmic matter field, with large deviations theory (LDT) as the central tool. Chapter 5 extends the LDT matter PDF to extended cosmologies — Hu-Sawicki f(R) gravity, nDGP, and w0waCDM — using modified linear growth, an Einstein-de Sitter spherical-collapse mapping rescaled by linear variances, and a lognormal rescaling of the fiducial non-linear log-density variance; the resulting PDF is compared with dedicated N-body simulations and used in Fisher forecasts for a Euclid-like survey. Chapter 6 derives covariance matrices of the one-point PDF from a two-cell joint PDF, including super-sample covariance and analytic bias functions in a minimal tree model. Chapters 7 and 8 develop a Schrödinger-Poisson forward model with an effective Planck constant as an alternative closure of the dark-matter Vlasov hierarchy, analysing interference, caustics, and density statistics. The thesis is compiled from four published papers and one preprint and is explicit about the author's contributions.

Significance. If the central claims are borne out, the LDT matter PDF is a survey-ready statistic that is sensitive to modified gravity and evolving dark energy in ways complementary to the power spectrum, and the wave-based model provides a useful tool for multi-streaming dynamics and fuzzy dark matter statistics. The thesis has concrete strengths: the pyLDT code is released, comparisons are made to dedicated simulations (including multiple realisations for f(R)), the Fisher forecasts are transparent about scale cuts and covariance assumptions, and several technical derivations are placed in appendices, making the work reproducible. The main limitation is that the headline PDF prediction depends on calibrated ingredients, so the 'accurately predicted' claim is conditional until those ingredients are independently validated or the claim is appropriately qualified.

major comments (4)

[Section 5.3.3, Eq. (5.23)] The width and normalisation of the LDT matter PDF are controlled by the non-linear log-density variance sigma^2_NL,ln(rho). In this chapter that variance is either measured from the very simulation suites used for validation, or obtained from Eq. (5.23), which rescales the fiducial cosmology's measured variance by the ratio of linear variances. The quoted 0.2-1% accuracy of Eq. (5.23) is an a posteriori calibration on the same DGP/f(R)/w0waCDM runs used to produce Figures 5.2-5.3. Therefore the agreement in those figures, and the Fisher forecasts in Sections 5.6.2-5.6.3, do not by themselves establish a first-principles prediction. I ask the authors to validate Eq. (5.23) on an independent simulation suite or with an independently calibrated model for sigma^2_NL,ln(rho), and to qualify the abstract claim accordingly, or to show that the Fisher results are insensitive to plausible errors in this input.
[Section 5.3.2, Eq. (5.20)] The spherical-collapse ingredient is also asserted rather than derived. The approximation delta_ext_L(rho) approximately (sigma_L^Lambda / sigma_L^ext) delta_EdS_L(rho) is described in the text as justified a posteriori by comparison with simulations. For f(R) gravity specifically, mass conservation and shell crossing are violated, so no exact mapping is available; the validation is indirect, via the reduced cumulants S3 and S4 in Eq. (5.22), rather than a direct test of the mapping. Because delta_L(rho) enters the rate function in Eq. (4.19), an error in this mapping propagates directly into the PDF shape. A sensitivity test (for example, using the exact DGP mapping for DGP, or an explicitly screened f(R) mapping) would show whether the claimed agreement is robust; if the mapping is only valid in the mildly non-linear regime, that limitation should be stated in the abstract.
[Abstract item 3 and Chapter 7] The claim that the wave-based forward model 'can encode the full phase-space dynamics beyond a perfect fluid' is stronger than what is demonstrated. The Schrödinger-Poisson model with an effective Planck constant is an alternative closure ansatz; the thesis shows that it reduces to Zel'dovich trajectories in the stationary-phase limit, produces interference patterns, and yields universal scaling near caustics, but it does not establish equivalence to the full Vlasov hierarchy. The Wigner representation in Appendix D.5 is a formal rewriting of the Schrödinger equation, not a proof that the closure captures all multi-stream moments. I recommend either adding a comparison against a full Vlasov-Poisson solution in a multi-streaming configuration beyond the simple examples used, or softening the 'full phase-space dynamics' wording to 'a wave-mechanical closure reproducing several multi-streaming effects'.
[Section 5.6.1] The Fisher forecasts assume that the joint covariance of the PDF and power spectrum is independent of cosmology and gravity theory and is equal to the Quijote LambdaCDM covariance. The text acknowledges this and argues that the induced error is small because sigma8 changes by only 1.6% (F6) and 3.8% (DGPw). This is a reasonable first approximation, but the headline improvements (up to a factor of six in detection significance) are conditional on it. I would like to see a robustness check in which the covariance is rescaled by the sigma8-induced variance change, or a simple two-parameter covariance model is varied, demonstrating that the predicted complementarity is not an artifact of using the LambdaCDM covariance.

minor comments (4)

[Section 5.6 heading] The heading 'Forecasting constrating power with the Fisher formalism' contains a typo; it should read 'constraining power'.
[Section 2.4] The statement that dark energy refers to 'any component with equation of state w < 1/3' should read w < -1/3; the current inequality would include ordinary matter, which does not accelerate the expansion.
[Equations (4.29) and (5.18)] The square-root symbols in these equations render as '/radicaltp/radicalvertex' artifacts in the present version; the final typeset version must use proper radical notation.
[Section 5.6.1] The sentence '...potentially complemented with predictions for effects induced by variations in the local mean density (Jamie' is incomplete; the accompanying citation appears to be cut off and should be restored.

Circularity Check

2 steps flagged · score 6.0 of 10

Central PDF 'prediction' inputs the simulated non-linear variance: Eq. (5.23) merely rescales a fiducial simulation measurement, so the headline agreement is partly inherited from the validation suite.

fitted input called prediction [Section 4.3.4; Section 5.3.3 Eq. (5.23); Section 5.4.1 footnote 5]
"The non-linear variance can either be viewed as a free parameter of this model, or can be measured from simulations. ... An effective approximation approximates the log-density non-linear variance at some arbitrary cosmology in terms of the linear variance and the non-linear variance at some fiducial cosmology ... σ2_lnρ(R,z )≃ ln[1+σ2_L(R,z )]/ln[1+σ2_L,fid(R,z )] σ2_lnρ,fid(R,z ). ... Because the linear theory normalisation cancels out the rate function, knowledge of σ8 is irrelevant for the LDT predictions when measurements of the variance of the simulated density field are available."

The LDT rate function for the matter PDF uses the non-linear variance of the log-density as a denominator and as the scale that sets the exponential width, so this single input controls the width and normalization of the predicted PDF. When that variance is measured from the same simulations used for validation, the agreement in the headline PDF comparisons is forced for the width rather than predicted from first principles. Equation (5.23) only transfers a fiducial measured value by ratios of linear variances; it does not derive the non-linear variance from linear theory. The abstract's claim that the PDF is 'accurately predicted' is therefore partly inherited from simulation calibration.
ansatz smuggled in via citation [Section 5.3.2 Eq. (5.20)]
"We find that by neglecting non-linear screening mechanisms, in the mildly non-linear regime (R ≳ 10 h−1 Mpc) any modified gravity and dark energy effect on the spherical collapse/expansion can be accurately captured by the following approximation δext_L(ρ,z )≈ σΛ_L(Rρ1/3,z )/σext_L(Rρ1/3,z )δEdS_L(ρ). ... This approximation was already argued in the dark energy case in Codis et al. (2016a), and is justified a posteriori by comparison to simulations in Cataneo et al. (2022)."

The extended-cosmology spherical-collapse mapping is not derived from the modified-gravity field equations; it is the Einstein-de Sitter mapping rescaled by linear-variance ratios, and its only stated justification is a posteriori agreement with the simulation suites used in the same PDF validation. Because spherical collapse sets the density-dependent tilt and skewness of the predicted PDF, this component of the 'prediction' is calibrated rather than predicted. The validation is attributed to Cataneo et al. (2022), a paper with overlapping authorship, making the support partly self-citational.

full rationale

Most of the thesis is not circular. Chapters 6-8 develop covariance predictions from joint two-cell PDFs and wave-mechanics statistics from independent Schrödinger/LPT and catastrophe-theory calculations, with no fitted input of the kind that drives the headline PDF claim. The circularity is concentrated in Chapter 5: the LDT matter PDF is a shape-generating formalism whose three inputs are listed in Section 4.3.4, and for extended cosmologies two of those inputs—the non-linear variance and the effective spherical-collapse mapping—are either measured from the validating simulations or approximated by equations (5.23) and (5.20) whose accuracy is established a posteriori on those same simulations. The PDF width is therefore not predicted from first principles, and the spherical-collapse density dependence is an EdS ansatz rescaled by linear theory. The shape and skewness information beyond the width is not simply a fit, which keeps the score at 6 rather than 8-10, but the abstract's unqualified 'accurately predicted' overstates the first-principles content of the central demonstration. The Fisher forecasts inherit this calibration and do not add independent circularity.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central claims rest on several calibrated inputs. The most important is the non-linear log-density variance, which is either measured from simulations or approximated by a lognormal rescaling. The extended-cosmology spherical collapse mapping and the effective Planck constant in the wave model are additional modeling choices. No new physical particles or forces are postulated.

free parameters (3)

Non-linear variance of log-density sigma^2_NL,ln(rho)(R,z) = Measured from simulations; or approximated by Eq (5.23)
Input to the LDT PDF construction. When measured from the same simulations used for validation, the comparison is partially calibrated; the lognormal approximation is calibrated on a fiducial cosmology.
Effective Planck constant hbar_PPT in wave forward model = Chosen for numerical resolution; Appendix E tests a larger value
The Schrodinger-Poisson forward model requires an effective wave scale; statistics such as interference features and skewness depend on this choice.
Spherical collapse index nu_SC = 21/13
The parametrized EdS collapse mapping Eq (3.92) uses nu_SC = 21/13 to match tree-order SPT skewness; not fitted to target data, but chosen to reproduce perturbation theory.

assumptions (8)

standard math Gaertner-Ellis theorem, Varadhan's theorem, and the contraction principle hold for the matter density field.
Used without proof to construct rate functions; Section 4.2.2.
domain assumption Initial matter density fluctuations are Gaussian.
The LDT construction starts from the Gaussian linear density PDF; Section 4.3.2.
domain assumption The dominant dynamical mapping from linear to non-linear densities for one-point statistics is spherical collapse.
Assumed after Valageas (2002) and used to contract the rate function; Section 4.3.2.
domain assumption Newtonian gravity and the Vlasov-Poisson equations describe structure formation on the scales considered.
Used throughout Part III; Section 3.2.
ad hoc to paper In modified gravity and dark energy cosmologies, the spherical collapse mapping can be approximated by rescaling the EdS collapse by the ratio of linear variances, Eq (5.20).
Justified a posteriori by simulation comparison, not derived from the modified field equations.
ad hoc to paper The non-linear log-density variance at an arbitrary cosmology is related to the fiducial one by Eq (5.23).
Lognormal rescaling calibrated on a fiducial cosmology; accuracy stated as 0.2-1%.
domain assumption The covariance of PDF and power spectrum from Quijote simulations is independent of cosmology and theory of gravity.
Used in Fisher forecasts, Section 5.6.1; acknowledged as an approximation.
ad hoc to paper The Schrodinger-Poisson system with an effective Planck constant provides a valid closure of the CDM Vlasov hierarchy.
Wave forward model used in Chapters 7-8 to represent multi-streaming CDM; dependence on hbar_PPT is tested but not derived from first principles.

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Cite this review

Pith. "Pith review of Cosmic Structure Formation in the Non-linear Regime: Beyond Gaussian Statistics and Standard Cosmologies." pith.science (2026). https://pith.science/paper/GC6TJCO3

@misc{pith2026241116500,
  author       = {Pith},
  title        = {Pith review of: Cosmic Structure Formation in the Non-linear Regime: Beyond Gaussian Statistics and Standard Cosmologies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GC6TJCO3}},
  note         = {Machine review of arXiv:2411.16500}
}

read the original abstract

The cosmic large scale structure encodes the formation and evolution of a weblike network of dark matter and galaxies within the Universe. The cosmological information is wrapped up in non-Gaussian statistics requiring characterisation beyond two-point correlations. Accurate modelling of these non-Gaussian statistics and the underlying non-linear dynamics of gravitational collapse are key to extracting maximal information from ongoing and upcoming cosmological surveys. This thesis centres on questions relating to clustering statistics, dynamics, and fundamental physics: A. How can we efficiently characterise the statistics of the late time matter field? B. How can we capture the non-linear phase-space dynamics of gravitational collapse? C. How do changes to fundamental physics impact those clustering statistics and dynamics? Specifically we present four aspects addressing these questions: 1. We demonstrate the probability distribution function (PDF) of the matter density can be accurately predicted in modified gravity and dynamical dark energy models, and that it provides good complementarity to standard two-point analyses for detecting these features. 2. We demonstrate the joint PDF of densities in two cells can be used to predict the covariance of the one-point PDF in simple clustering models, providing estimates of the density dependent clustering and super-sample covariance missed in cosmological simulations. 3. We use a wave-based forward model of dark matter to demonstrate its capability to encode the full phase-space dynamics beyond a perfect fluid and determine certain universal scaling features in such models. 4. Using the wave dark matter forward model we analyse one-point statistics to complement existing analytic and numerical approaches in studying fundamentally wavelike dark matter.

Figures

Figures reproduced from arXiv: 2411.16500 by the authors.

**Figure 1.1.** Map of the temperature fluctuations in the CMB measured by the Planck satellite (Planck Collaboration et al., 2020a). The relative size of these fluctuations is 10−5 and their distribution is measured to be very close to Gaussian. These tiny fluctuations eventually grow to become the structures in the Universe we see in maps of galaxies such as in [PITH_FULL_IMAGE:figures/full_fig_p016_1_1.png] view at source ↗

**Figure 1.2.** Map of galaxy positions from the 2MASS Extended Source Catalogue with the Point Source Catalogue view of the Milky Way in the centre (Jarrett, 2004) [PITH_FULL_IMAGE:figures/full_fig_p016_1_2.png] view at source ↗

**Figure 1.3.** Main themes of research in this thesis. Some of the main observables from cosmological surveys of large-scale structure are the positions and shapes of galaxies. These can be used to construct various [PITH_FULL_IMAGE:figures/full_fig_p017_1_3.png] view at source ↗

Figures from the paper (53 more)

**Figure 3.1.** Figure 3.1: The phase space evolution and density of a one-dimensional N-body simulation of 8192 particles in an Einstein-de Sitter background. The phase-space velocity variable here is v = a −3/2p/m where p is the standard canonical momentum. Particles are coloured according to…

**Figure 3.2.** Figure 3.2: The linear matter power spectrum probed by a variety of different cosmic probes, by ESA and the Planck Collaboration (Planck Collaboration et al., 2020a). The turn over in the spectrum occurs at roughly keq, with low k (large scales) scaling as k ns ∼ k and high k (s…

**Figure 3.3.** Figure 3.3: , where regions which expand more end up less dense in Eulerian space [PITH_FULL_IMAGE:figures/full_fig_p045_3_3.png]

**Figure 3.4.** Figure 3.4: The 2 point correlation function or power spectrum expanded to 1 loop order in perturbation theory. We use the shorthand δi = δ(xi), so subscripts indicate the spatial point, while superscripts represent the perturbative order. The three 1-loop diagrams here are of o…

**Figure 3.5.** Figure 3.5: Comparison of the analytic parametrisation and the exact spherical collapse mapping between linear and non-linear densities before shell crossing. We use the spherical collapse index νSC = 21/13 to match the skewness from SPT. The lower panel shows the relative error…

**Figure 3.6.** Figure 3.6: The tree diagrams for the 3- and 4-point correlation functions in a hierarchical model. The there are two reduced correlation functions Q4 corresponding to the two topologies of the diagrams. We use the shorthand δi = δ(xi), ξij = ξ(xi , xj ) and perm. refers to perm…

**Figure 4.1.** Figure 4.1: (Left) Distribution of the sample mean of n independent random variables drawn from the same exponential distribution with mean µ = 1. As the number of events grows the sample mean concentrates around the distribution mean. In the vicinity of the mean the distributio…

**Figure 4.2.** Figure 4.2: The PDF of the normalised cosmic density, measured in 10 h −1 Mpc spheres at redshifts z = 0–3. The data points are measured from the Quijote simulations (Villaescusa-Navarro et al., 2020). Solid lines show the LDT model for the matter PDF. Dashed lines show a lognor…

**Figure 4.3.** Figure 4.3: Schematic diagram for the main theorems of Large Deviations Theory as applied to cosmology. The Gärtner-Ellis Theorem is equation (4.10), Varadhan’s theorem equation (4.13), contraction principle (4.14). Beginning from knowledge of the PDF of the linear density PDF, …

**Figure 5.1.** Figure 5.1: Mapping between the final density ρ (normalised by the mean density) and the initial linearly-scaled density fluctuation δL for a spherical top-hat perturbation. (Upper panel) The curves show the density evolution for different background models. For ΛCDM (blue) and …

**Figure 5.2.** Figure 5.2: Comparison of the matter PDF in R = 10 h −1 Mpc spheres in ΛCDM, f(R), and DGP cosmologies. The cosmological parameters are chosen such that the clustering amplitude σ8 is the same in all cases at redshift 0, rather than matching the initial amplitude As. This normal…

**Figure 5.3.** Figure 5.3: Matter PDF in spheres of radius R = 10 Mpc/h at z = 0 (blue) and z = 1 (green) for ΛCDM (dashed) and modified gravity (solid). (Left panel) Data points are the simulation measurements from a single realisation (triangles for ΛCDM and squares for DGPm) and lines repre…

**Figure 5.4.** Figure 5.4: Marginalised Fisher forecast constraints on DGP (left) and f(R) (right) gravity. We use an external prior on ns and Ωb (as described in the text) for the DGPw and F6 fiducial cosmologies. Contours correspond to the matter PDF at 3 scales and 3 redshifts (green), the …

**Figure 5.5.** Figure 5.5: Comparison of PDF differences divided by the error on the PDF as estimated from the Quijote simulations. Line style indicates the redshift, while colour indicates parameter being deviated. The vertical lines represent the region used at each redshift to construct the…

**Figure 5.6.** Figure 5.6: Fisher forecast constraints on {Ωm, σ8, w0, wa} (marginalised over {Ωb, ns} using the external prior described in the text) for the w0CDM model around the fiducial Quijote ΛCDM cosmology. Contours correspond to the matter PDF at 3 scales and 3 redshifts (green), the …

**Figure 5.7.** Figure 5.7: Derivatives of the matter PDF in an evolving dark energy universe. The dependence of the matter PDF on Ωm is easily distinguished from the others by its distinct skewness (see [PITH_FULL_IMAGE:figures/full_fig_p091_5_7.png]

**Figure 6.1.** Figure 6.1: Schematic depiction of the different areas/volumes which are discussed and their relation to one another. The largest is the entire survey or simulation box, which may be broken into smaller patches or subboxes/subvolumes. The covariance of the PDF of the densities i…

**Figure 6.2.** Figure 6.2: Distance distributions within a cube and a square survey volume/area, as given by equations (6.8) & (6.9). Vertical lines are shown at √ 2 (the maximal separation in a square survey) and √ 3 (the maximal separation in a cubic survey). where ∆i is the i th bin width, …

**Figure 6.3.** Figure 6.3: In the presence of fluctuations larger than the survey/simulation box size, the mean overdensity δ can vary from its global mean value of 0. The SSC effect has been modelled previously in the context of the matter power spectrum (Takada and Hu, 2013; Barreira and Sch…

**Figure 6.4.** Figure 6.4: Covariance matrices between the bins of the matter PDF with differing degrees of overlap cells. These are measured from in R = 10 Mpc/h spheres at z = 1 from 1000 realisations of the Quijote simulations. The dotted lines indicate the location of the peak of the mean …

**Figure 6.5.** Figure 6.5: Scaling of the average of powers of the two-point cell correlation function compared to the variance of δ across 1000 realisations in the Quijote simulations smoothed on R = 10 Mpc/h at z = 0. the distribution of separations as in equation (6.6) cov(PG(δ1)PG(δ2)) = X…

**Figure 6.6.** Figure 6.6: Comparison of the PDF covariance matrix contributions for z = 0, R = 10 Mpc/h as measured from the 1 (Gpc/h) 3 Quijote simulations and the super-sample covariance contribution constructed from the DC mode following equation (6.40). The upper triangle shows the super-…

**Figure 6.7.** Figure 6.7: Comparison of the theoretically predicted sphere bias from LDT (blue solid), the lognormal approximation (blue dashed) and the measurement in one realisation of the fiducial cosmology of Quijote (blue data points) at radius R = 10 Mpc/h, spaced by 20 Mpc/h at redshif…

**Figure 6.8.** Figure 6.8: (Upper panel) A realisation of a Rayleigh-Lévy flight with 2.5 × 105 steps and periodic boundary conditions in a 3D box, together with the 2D projections. The box side length is set to 100, and small-scale regularisation length ℓ0 = 0.03. The slope parameter is α = 1…

**Figure 6.9.** Figure 6.9: Diagrammatic representation of the bias functions. The function ϕ0 is the generating function of all trees within one cell, represented with the hatched circle. The higher order functions ϕn generate all trees in one cell with n external lines. We begin by calculatin…

**Figure 6.10.** Figure 6.10: The one-point PDF of the matter density in the minimal tree model, as given by equation (6.57), on linear (left) and log-log (right) scales. need be integrated are the exponential part of the inverse Laplace transform against powers of ϕt . The lowest order bias fun…

**Figure 6.11.** Figure 6.11: A comparison of the first two leading order bias functions from the minimal tree model (6.58), b1,t (left panel) and b2,t (right panel), for different variances (red and blue lines) approaching the small-variance limit (6.59) (yellow dashed) and the Gaussian expecta…

**Figure 7.1.** Figure 7.1: Phase-space sheet describing the evolution of a Fourier mode under the Zel’dovich approximation described by the displacement mapping x(q, a) = q − a sin(q). 2 For times a > 1, the phase-space sheet becomes triple valued in velocity. Projecting this phase-space sheet…

**Figure 7.2.** Figure 7.2: shows the evolution of a wavefunction corresponding the same Zel’dovich initial conditions as in [PITH_FULL_IMAGE:figures/full_fig_p131_7_2.png]

**Figure 7.3.** Figure 7.3: Features of the wavefunction ψ (with ℏ = 0.05) with Zel’dovich initial conditions at time a = 1.75. (Upper panel) The density of the wavefunction ψ. The dashed line corresponds to the classical Zel’dovich density at the same time. The wavefunction follows this classi…

**Figure 7.4.** Figure 7.4: Comparison of the stationary phase approximation (SPA) and the numerical evolution of ψ (ini)(q) = exp(i cos(q)/ℏ). (Upper panel) The full spacetime evolution of this wavefunction, with ℏ = 0.01. (Lower panel) The density profile post shell crossing of this wavefunct…

**Figure 7.5.** Figure 7.5: The polar angle of the normalised wavefunctions (ℏ = 0.05) split according to each stationary point of the function ζ in equation (7.30). Each of these stationary points corresponds to one of the classical Zel’dovich trajectories. Outside the cusp there is a single s…

**Figure 7.6.** Figure 7.6: The structure of stationary points q∗ which solve ∇qζ(q∗; x, a) = 0 (equation (7.31)) for different values of x and a. The blue circles show the distribution of q∗ in the complex-q plane. Beginning at point (I) in the single-stream (white) region, two solutions are c…

**Figure 7.7.** Figure 7.7: The Husimi phase-space distributions for each of the individual wavefunctions associated with stationary points of ζ post shell crossing (ℏ = 0.01). We see that this stationary phase decomposition naturally dissects the classical Zel’dovich phase-space sheet (dotted …

**Figure 7.8.** Figure 7.8: Comparison of classical Zel’dovich trajectories to the stationary phase approximation for a wavefunction (with ℏ = 0.01) evolving under the free Schrödinger equation. The stationary phase wavefunction is the proper analogue to this classical system, having the exact …

**Figure 7.9.** Figure 7.9: The unwrapped phase (upper panel) and density (lower panel) of ψ at times past shell crossing when defects occur (with ℏ = 0.05). The spatial phase discontinuity is always ±π, as indicated by the horizontal lines, which are spaced apart by π. These phase discontinuit…

**Figure 7.10.** Figure 7.10: Location of branch points in the evolution of the free wavefunction (ℏ = 0.05). The branch points predicted by the stationary phase approximation (SPA) are shown as red crosses, and accurately capture the position of the interior branch points from the full wavefunc…

**Figure 7.11.** Figure 7.11: Illustration of an inadmissible spacetime loop for the Poincaré-Cartan invariant. Shown are the phase-space sheet for the system (top, as in [PITH_FULL_IMAGE:figures/full_fig_p148_7_11.png]

**Figure 7.12.** Figure 7.12: (Left) The phases of individual streams of the wavefunction model (with ℏ = 0.05) past shell crossing, at the time of the first interior phase jump. We show the phase of the individual SPA stream wavefunctions, ϕ SPA i (which are equal to the Eulerian velocity poten…

**Figure 7.13.** Figure 7.13: The spacetime evolution of the polar angle of the full stationary phase wavefunction (with ℏ = 0.05, (left)), the “average phase” associated with the mean velocity (middle), and the discontinuous “hidden phase” (right). The left panel also shows the spatial and temp…

**Figure 7.14.** Figure 7.14: The scalar velocity dispersion σ associated with the wavefunction ψ (for ℏ = 0.05) calculated by equation (7.25) using the hidden density ρ hid = [PITH_FULL_IMAGE:figures/full_fig_p152_7_14.png]

**Figure 7.15.** Figure 7.15: The cusp catastrophe ucusp(C1, C2; ν = 1) from equation (7.55) in control parameter space, coloured according to the same domain colouring as the wavefunction. This is numerically calculated using the contour shifting technique described in Section 7.2. This Figure …

**Figure 7.16.** Figure 7.16: Density of the wavefunction with Zel’dovich initial conditions for ℏ = 0.05. This demonstrates how the peak density, as well as the delay of shell crossing and the spatial width of the wavefunction at the cusp point scale with ℏ, related to the singularity and fring…

**Figure 8.1.** Figure 8.1: Projected 0.125 Mpc/h thick slice of the dark matter density field at z = 4.0. All plots are shown on the same logarithmic colour scale. These simulations have a box size of 128 Mpc/h and a grid resolution of 0.125 Mpc/h. The LPT runs traced the positions of (1024)3 …

**Figure 8.2.** Figure 8.2: Four way comparison of the power spectrum from LPT (CDM) and PPT (wave) perturbative simulations with either FDM or CDM initial conditions in a L = 128 Mpc/h box. The target linear power spectrum for CDM/FDM is shown as the grey lines. The lower plot shows the ratio …

**Figure 8.3.** Figure 8.3: shows the PDF of the rareness of the density ν = δ/σ and the log density µ = log(1 + δ) while varying the evolution dynamics and the initial conditions in spheres of radius R = 1 h −1 Mpc (8 times the grid scale). As changing the initial conditions changes the final …

**Figure 8.4.** Figure 8.4: (Top panel) Reduced skewness as a function of variance at z = 4, measured on top-hat smoothing scales 1–10 h −1 Mpc, log-spaced in R. Paler lines/open symbols show the results from 2nd order PT. The error bars are the standard deviation of the measured quantities acr…

**Figure 8.5.** Figure 8.5: Classification of cosmic web elements via critical points of the smoothed density field. The directions/colours of the arrows indicate the sign of the eigenvalues λ of the Hessian matrix Hij evaluated at the critical point. Outwards/orange arrows correspond to λ > 0 …

**Figure 8.6.** Figure 8.6: (Left panel) Total number of critical points in a density field smoothed in R = 1 h −1 Mpc spheres. (Right panels) Total number of critical points at z = 4 as a function of top-hat smoothing scale R. The lower panel shows the ratio to the number of critical points in…

**Figure 8.7.** Figure 8.7: shows these ratios as measured on R = 1 h −1 Mpc together with the Gaussian random field values. We see that generally the fully classical (LPT + CDM ICs) systems are the closer to the Gaussian ratios, with either FDM ICs or wave dynamics introducing higher deviation…

**Figure 8.8.** Figure 8.8: Evolution of the fraction of critical points on R = 1 h −1 Mpc of a given type. The horizontal grey lines show the fraction for a Gaussian random field. Both wave dynamics (orange) and FDM initial conditions (dashed lines) move these fractions further from their Gaus…

**Figure 8.9.** Figure 8.9: The PDFs of the overdensity of different critical points of the fully classical (LPT + CDM initial conditions) density field at z = 4 when smoothed on R = 1 Mpc/h. Each individual PDF is normalised. The PDF of the entire matter field (black dashed), and of all critic…

**Figure 8.10.** Figure 8.10: The PDFs of the density found in different critical point environments, linearly interpolated between bins. The vertical lines show the median value of δ for that cosmology, the grey vertical line divides under- and over-dense environments. Environments with more sh…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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work page Pith review arXiv 2019