REVIEW
Renormalized Cosmological Perturbation Theory
Astrophysics · Submitted 15 Sep 2005 · 25 references
Pith's verdict: A genuinely useful framework paper whose abstract promises more ordering than the body actually demonstrates.
T0 · AI review
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Referee report
The paper
The paper claims
Splitting the loop expansion into two kinds of diagrams makes perturbation theory behave like a series again, with each term dominating at its own scale.
The reading
What carries the argument
The nonlinear propagator $G(k)$, defined in Eq. (36) as the functional derivative of the final field with respect to the initial field, averaged over initial conditions. Its physical reading is the memory of a mode: at large scales it reduces to the linear growth factor, and at small scales the interactions erase the initial condition and it decays. In the Zel'dovich approximation this decay is exactly Gaussian, $G_delta = a, e^{-k^2 sigma_v^2/2}$, with $sigma_v^2$ the linear velocity dispersion, and the decay length $sigma_v^{-1} approx 0.16, h,$Mpc$^{-1}$ sets the scale where individual modes stop evolving linearly. The machinery that makes the resummation work is the Dyson equation, Eq. (44), which turns an infinite class of propagator diagrams into one integral equation, and its power-spectrum counterpart, Eq. (48), which separates the one-power-spectrum diagrams (propagator-dressed linear power) from the mode-coupling kernel $Phi$.
Extended reading notes
Core claim
The central claim, stated in the abstract and argued through the body of the paper, is that a well-defined renormalized perturbation theory follows once the nonlinear propagator is resummed: each term in the remaining mode-coupling series dominates at some characteristic scale and is subdominant otherwise, unlike standard perturbation theory where different loop corrections can become comparable in the nonlinear regime. The mechanism is a reorganization of the diagrammatic series. The nonlinear propagator $G_{ab}(k;eta) = langle delta Psi_a(k,eta) / delta phi_b(k') rangle$ is defined as the ensemble-averaged response of the final density and velocity-divergence fields to a variation in the initial conditions, and it obeys a Dyson equation, Eq. (44), built from a kernel $Sigma$ of principal-path-irreducible diagrams. The power spectrum then obeys Eq. (48), written as a term proportional to $G_{ac}(k) u_c P_0(k) u_d G_{bd}(-k)$ plus a mode-coupling integral of a kernel $Phi$ sandwiched between two nonlinear propagators. The authors show that $Sigma$ and $Phi$ themselves can be re-expanded in terms of nonlinear propagators, nonlinear power spectra, and a renormalized vertex, so that the whole tower is self-consistent up to the loop orders they check.
Load-bearing premise
The load-bearing premise is that the clean ordering seen in the Zel'dovich approximation, where the resummed propagator is exactly Gaussian and every reorganized loop term is positive, carries over to exact gravitational dynamics; the paper itself defers that exact-dynamics comparison to a companion paper.
Editorial extensions
If this is right
- A controlled expansion for the matter power spectrum in the weakly nonlinear regime becomes available, in which the size of the next term is a diagnostic of where the expansion can be trusted rather than an unknown.
- The nonlinear propagator provides a directly measurable quantity: it is the cross-correlation between final and initial density fields normalized by the initial power, so its predicted decay can be checked against simulations mode by mode.
- The reorganized power spectrum takes a two-part form, a suppressed linear spectrum plus a mode-coupling piece, which maps onto the two-halo and one-halo terms of the halo model, giving the halo-model decomposition a perturbative counterpart.
- In the halo-model reading, propagator renormalization corresponds to a scale-dependent linear bias $b_1(k)$ built from the mass function and halo profiles, Eq. (54), so the suppression of the propagator has a bias interpretation.
- Because the first term in the reorganized series carries a cutoff at the scale where the linear spectrum ends, propagator renormalization alone cannot generate power beyond that scale; genuine power generation at small scales lives entirely in the mode-coupling terms.
Reading between the lines
- If the ordering seen in the Zel'dovich approximation survives for exact dynamics, then the ratio of successive reorganized loop terms gives an internal error bar for perturbation-theory predictions at a given wavenumber, which would let weak-lensing and galaxy-survey analyses quote a theory uncertainty without external calibration.
- The propagator's decay scale is a natural marker for where the single-stream description itself is likely to be strained, since the authors note that vorticity power matches divergence power at roughly the same scale where the nonlinear power spectrum turns over.
- The correspondence with the halo model suggests a two-way transfer: measured scale-dependent halo bias could be used as an empirical input for the propagator, and conversely the propagator's predicted shape is a target that halo-model prescriptions could be tested against.
- The same time-ordered diagrammatic bookkeeping could be applied to biased tracers or redshift-space distortions, where the memory of initial conditions is degraded by additional physics and the propagator would carry that degradation explicitly.
Original abstract
We develop a new formalism to study nonlinear evolution in the growth of large-scale structure, by following the dynamics of gravitational clustering as it builds up in time. This approach is conveniently represented by Feynman diagrams constructed in terms of three objects: the initial conditions (e.g. perturbation spectrum), the vertex (describing non-linearities) and the propagator (describing linear evolution). We show that loop corrections to the linear power spectrum organize themselves into two classes of diagrams: one corresponding to mode-coupling effects, the other to a renormalization of the propagator. Resummation of the latter gives rise to a quantity that measures the memory of perturbations to initial conditions as a function of scale. As a result of this, we show that a well-defined (renormalized) perturbation theory follows, in the sense that each term in the remaining mode-coupling series dominates at some characteristic scale and is subdominant otherwise. This is unlike standard perturbation theory, where different loop corrections can become of the same magnitude in the nonlinear regime. In companion papers we compare the resummation of the propagator with numerical simulations, and apply these results to the calculation of the nonlinear power spectrum. Remarkably, the expressions in renormalized perturbation theory can be written in a way that closely resembles the halo model.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We develop a new formalism to study nonlinear evolution in the growth of large-scale structure, by following the dynamics of gravitational clustering as it builds up in time. This approach is conveniently represented by Feynman diagrams constructed in terms of three objects: the initial conditions (e.g. perturbation spectrum), the vertex (describing non-linearities) and the propagator (describing linear evolution). We show that loop corrections to the linear power spectrum organize themselves into two classes of diagrams: one corresponding to mode-coupling effects, the other to a renormalization of the propagator. Resummation of the latter gives rise to a quantity that measures the memory of perturbations to initial conditions as a function of scale. As a result of this, we show that a well-defined (renormalized) perturbation theory follows, in the sense that each term in the remaining mode-coupling series dominates at some characteristic scale and is subdominant otherwise. This is unlike standard perturbation theory, where different loop corrections can become of the same magnitude in the nonlinear regime. In companion papers we compare the resummation of the propagator with numerical simulations, and apply these results to the calculation of the nonlinear power spectrum. Remarkably, the expressions in renormalized perturbation theory can be written in a way that closely resembles the halo model.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We develop a new formalism to study nonlinear evolution in the growth of large-scale structure, by following the dynamics of gravitational clustering as it builds up in time. | contribution 0.94 | 0.95 | 0.07 | asserted 0.99 |
| s1 | This approach is conveniently represented by Feynman diagrams constructed in terms of three objects: the initial conditions (e.g. perturbation spectrum), the vertex (describing non-linearities) and the propagator (describing linear evolution). | method 0.96 | 0.55 | 0.04 | asserted 0.98 |
| s2 | We show that loop corrections to the linear power spectrum organize themselves into two classes of diagrams: one corresponding to mode-coupling effects, the other to a renormalization of the propagator. | result 0.87 | 0.94 | 0.13 | asserted 1.00 |
| s3 | Resummation of the latter gives rise to a quantity that measures the memory of perturbations to initial conditions as a function of scale. | result 0.83 | 0.86 | 0.11 | asserted 1.00 |
| s4 | As a result of this, we show that a well-defined (renormalized) perturbation theory follows, in the sense that each term in the remaining mode-coupling series dominates at some characteristic scale and is subdominant otherwise. | result 0.93 | 0.95 | 0.66 | asserted 1.00 |
| s5 | This is unlike standard perturbation theory, where different loop corrections can become of the same magnitude in the nonlinear regime. | interpretation 0.71 | 0.34 | 0.12 | asserted 1.00 |
| s6 | In companion papers we compare the resummation of the propagator with numerical simulations, and apply these results to the calculation of the nonlinear power spectrum. | future work 0.97 | 0.44 | 0.04 | asserted 0.84 |
| s7 | Remarkably, the expressions in renormalized perturbation theory can be written in a way that closely resembles the halo model. | interpretation 0.58 | 0.82 | 0.07 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
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Figures
Figures reproduced from arXiv:astro-ph/0509418 by the authors.
References
How later papers used it
212 later papers cite it
5 later papers by other authors are listed below: 1 built on it and 4 used its method or tool.
Builds on it 1 paper
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Generalized this paper's matrix based RPT formalism to the problem of dealing with N-component fluids, each evolving from a distinct set of initial conditions and contributing to the matter-density and velocity fields.
Cosmological perturbation theory for baryons and dark matter I: one-loop corrections in the RPT framework (2009) · 1.00 · paraphrased: the licence does not allow quoting
Uses its method or tool 4 of 47 papers with this label
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Used equations from this paper and another work to bring them into a given form, with indices a, b, c taking values 1, 2.
Non-linear Matter Spectrum for a Variable Equation of State (2010) · 0.87 · paraphrased: the licence does not allow quoting
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Calculated the nonlinear matter power spectrum using the Galilean-invariant renormalized perturbation theory of this paper.
Large-scale redshift space distortions in modified gravity theories (2018) · 0.88 · paraphrased: the licence does not allow quoting
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Used the diagrammatic language discussed in this paper and in another work to organize the perturbative expansion.
Nonlinear Power Spectrum from Resummed Perturbation Theory: a Leap Beyond the BAO Scale (2012) · 0.86 · paraphrased: the licence does not allow quoting
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Used equations from this paper and another work to bring them into a given form, with indices a, b, c taking values 1 to 4.
Non-linear Matter Spectra in Coupled Quintessence (2009) · 0.86 · paraphrased: the licence does not allow quoting
How the literature uses it
154 arXiv papers cite this in 273 sentences
Papers citing it · 143 sentences without a confident label
Contests it 3 papers
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At least for LCDM[Note that there is a wrong minus sign in eq (2.39) of [ref] ] [equation] Then, indeed, all $P_n$ are positive, as shown in [ref] .
Divergence of Perturbation Theory in Large Scale Structures · (non-)Convergence of RPT for the correlation function · contest 0.85
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From the perspective of the EFTofLSS, the previous approaches like SPT and relatives (see for example [ref] or [ref] ) are missing important terms in the equations of motion.
The Effective Field Theory of Large Scale Structures at Two Loops · Introduction · contest 0.78
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The most generic case of scale dependent growth factors is precluded to the approaches such as [ref] , while it can be properly treated here.
Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations · Comparison to other approaches · contest 0.74
Builds on it 1 paper
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In this paper we have generalized the matrix based RPT formalism of [ref] to the problem of dealing with $N$-component fluids, each of which evolves from a distinct set of initial conditions and contributes to the matter-density and velocity fields.
Cosmological perturbation theory for baryons and dark matter I: one-loop corrections in the RPT framework · Summary and Conclusions · 1.00
Uses its method or tool 27 papers
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([ref]), ([ref]) in the form [ref] [equation] The indices $a,b,c$ take values $1,2$.
Non-linear Matter Spectrum for a Variable Equation of State · Formalism · 0.87
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The initial conditions are set up at redshift $z_i = 59$ using the second-order Lagrangian perturbation theory [ref] implemented by [ref] and then parallelized in [ref] .
Separate universe approach to evaluate nonlinear matter power spectrum for non-flat $\Lambda$CDM model · $N$-body simulations · 1.00
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Hence, the $*$-symbol in our equation stands for the multiplication of the power spectrum with $G̃(k)$ before transforming it into a correlation function. $G̃(k)$ is defined as [equation] with the property [equation] The damping scale $k_*$ can be calculated from linear theory [ref] by [equation] where $P_lin(k)$ is again the linear power spectrum. $Λ$CDM predicts a value of $k_* ≃0.17h$Mpc$^-1$.
The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant · Large-scale correlation function · 0.90
Compares with it 7 papers
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In this appendix we simply compare the predictions at redshift $z=0$ and $z=1$ for the $Λ$CDM cosmology assumed in our results, with N-body simulations [ref] as well as with predictions in RPT [ref] [We thank Vincent Desjacques and Martin Crocce for providing us, respectively, with the power spectrum measurements and the RPT predictions.].
The nonlinear power spectrum in clustering quintessence cosmologies · On the accuracy and validity of the Time-RG approach · 0.88
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Considering the limit of a pressureless fluid, corresponding to $c_s→0$, we find the propagator to be [equation] which is identical to the result of [ref] .
Resummed Kinetic Field Theory: Using Mesoscopic Particle Hydrodynamics to Describe Baryonic Matter in a Cosmological Framework · Solution of the Linearized Equations · 0.97
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[ref] , and also with the one, closer in spirit, of ref.
Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations · Introduction · 0.99
Cites it as support 2 papers
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Recent developments in the description of matter nonlinearities in Perturbation Theory (see [ref] ) can justify even our more optimistic choice of $k_max=0.15$ at $z=0$, also in the case of non-Gaussian initial conditions [ref] .
Effects and Detectability of Quasi-Single Field Inflation in the Large-Scale Structure and Cosmic Microwave Background · Fisher matrix · 0.96
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The major problem of SPT is that higher order perturbative corrections to the power spectrum do not lead to significant improvements on mildly-nonlinear scales [ref] .
Non-linear perturbation theory extension of the Boltzmann code CLASS · Brief Overview of Perturbation Theory · 0.97
Cites it as background 73 papers
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The power spectrum [defined by $⟨δ() δ^*(')⟩= (2 π)^3 P(k) δ_D()$] follows immediately [ref] : P_Zel(k) = ^- k^2 σ_ψ^2∫d^3 q ^i k · [^k^i k^j C_ij() -1 ], where $C_ij() ≡⟨ψ_i(0) ψ_j()⟩$ is the correlation tensor of the displacement field and is given by C_ij()= ∫d^3 k/(2 π)^3 ^-i · k^i k^j/k^4P_L(k).
Perturbative interaction approach to cosmological structure formation · Summary of known results · 0.83
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This gravitational clustering is typically studied using a perturbative framework, known commonly as standard perturbation theory (SPT), and its extensions [ref] - however, these results are valid only in the quasilinear regime and tend to break down at smaller length scales.
Consistency Relations in Effective Field Theory · Introduction · 0.94
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The renormalized perturbation theory (RPT) [ref] is a representative pioneering work, and since it appeared, various approaches have been subsequently proposed [ref] .
Next-to-leading resummation of cosmological perturbations via the Lagrangian picture: 2-loop correction in real and redshift spaces · Introduction · 0.91
Discussion
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