Lagrangian perturbation theory for modified gravity

arXiv:1705.10719 · astro-ph.CO, gr-qc · Submitted 2017-05-30 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: We're looking at the paper: "Lagrangian perturbation theory for modified gravity".

Jocelyn: Here's the gist of the paper. The gist The authors present a formalism to compute Lagrangian displacement fields for a wide range of cosmologies in the context of perturbation theory up to third order, emphasizing theories with scale dependent gravitational strengths such as chameleons, and they discuss qualitative features arising from nonlinearities like force screening and frame-lagging terms Theoretical Framework The work develops a Lagrangian Perturbation Theory (LPT) theory applicable to a wide range of modified gravity theories, essentially those that can be written as a scalar tensor gravity theory The formalism is suitable for Horndeski models, which share the same range of validity that share several investigations in SPT that came after to the pioneering work of Ref.

60: The basic requirement is that the theory can be recasted as a scalar tensor gravity at linear order, all non linear contributions to the Klein-Gordon equation, including possible kinetic terms, can be written as sources and treated perturbatively Key Features of LPT in Modified Gravity The formalism introduces two qualitative features in the non-linear regime: one is that "nonlinearities lead to a screening of the force mediated by the scalar field" The second is "a consequence of the transformation of the Klein-Gordon equation from Eulerian to Lagrangian coordinates, producing frame-lagging terms that are important especially at large scales, and if not considered, the theory does not reduce to the ΛCDM model in that limit" These frame-lagging terms are non-negligible and necessary for recovering ΛCDM at sufficiently large scales where the fifth force mediated by the scalar field is essentially zero Perturbation Theory Orders The paper develops LPT theory up to third order, which is necessary because "their solutions lead to the first corrections to the power spectrum (the 1-loop) and correlation function" The third order Lagrangian displacement field is given by a complex expression involving various sources like screening, self-interaction, and frame-lagging terms Second Order Results The second order computation yields growth functions such as D(2)(k1, k2) and D(2)(k1, k2) which depend on three numbers that can be chosen as k1, k2 and k12 such that alternatively we can write D¯(2)(k1, k2) = D¯(2)(k12, k1, k2) The second order Lagrangian displacement kernels are derived from the first order fields given in Eqs.

Jocelyn: (48) and (51) Third Order Results and Statistics The third order equation of motion is given by Eq.

Vera: (70), which involves numerous terms including frame-lagging contributions, screening, and self-interaction terms The third order Lagrangian displacement kernel is expressed as L(3)i(k1, k2, k3) = i k i / k 2 D(3)(k1, k2)D+(k2)D+(k3)> Power Spectra and Divergences The paper computes the power spectrum and correlation function using different resummation schemes, including Convolution Lagrangian Perturbation Theory (CLPT) and Lagrangian Resummation Theory (LRT) The analysis shows that the Q functions have UV divergences for n ≥ 1/2 and the R functions for n ≥ −1, which are the same UV divergences that have the P22 and P13 pieces of the SPT power spectrum, respectively In general, IR divergences may show up both at p → 0 and p → k The formalism presented here is capable to capture this physical fact, mathematically through cancellations provided by the frame-lagging contributions The analysis suggests that the schemes of SPT and CLPT are IR-safe in MG Furthermore, the Q and R functions have the same k-dependence at large scales in MG and GR The analysis of the Q3 function shows that its large scale limit is Q3(k → 0) = k 4 / (10π2) Z ∞ 0 dp P squared L(p) p squared How it works The Lagrangian displacement velocity is defined as a momentum average of particles’ velocities over the ensemble, and for CDM scenarios, we can safely neglect velocity dispersions and the identity S = Ψ follows The matter perturbation is given by −δ(x) = J(q) − 1 / J(q) Second Order Computation The second order Lagrangian displacement to second order is given by kiΨi(2)(k) = i 2 Z k12=k 3/7 D¯(2)a (k1, k2) − D¯(2)b (k1, k j) / (k j) squared The growth functions are given by D(2)(a)(k1, k 2) = T −ˆ A(k)-1A(k)D+(k1)D+(k2) Third Order Computation The equation of motion to third order is given by (T −ˆ A(k))

Ψ(3)i,i: (k) = S1 + · · · + S10 The third order Lagrangian displacement kernel is L(3)i(k1, k2, k3) = i k i / k 2 D(3)(k1, k2)D+(k2)D+(k3)> Statistical Analysis The Q functions are constructed from the Lagrangian displacements up to third order and are used to compute the power spectrum and correlation function The power spectrum of Eq.

Title and authors: Vera: So we're looking at this paper, "Lagrangian perturbation theory for modified gravity," and the authors are Aviles and Cervantes-Cota. It basically sets up a way to compute how matter moves in these modified gravity theories using Lagrangian displacement fields, specifically up to third order.

Jocelyn: That sounds really technical, Vera. What does that actually mean for someone listening who isn't a cosmologist? Is this just more math about gravity?

Subrahmanyan: Not at all. It’s about providing a framework to study how structure forms in these modified gravity theories, which are often those with scale-dependent gravitational strengths, like chameleon models, using perturbation theory up to the third order.

Vera: Exactly. It lets us look at things beyond the simple linear descriptions and see how things behave when you add some non-linear effects into these more complex gravity theories.

Jocelyn: So it’s about moving past just seeing how gravity works in a simple vacuum, right? How do we actually model the motion of galaxies and clusters in these modified models?

Subrahmanyan: Right. It lets us get closer to what N-body simulations are trying to figure out, but with an analytical tool that can handle the complexity of those more exotic theories.

The paper's summary: Vera: So the authors summarize it by saying they develop a Lagrangian Perturbation Theory, or LPT, which is tailored for these modified gravity theories and is applicable to Horndeski models too.

Jocelyn: Horndeski models sound like a lot of terms. What’s the main point they’re making about this formalism?

Subrahmanyan: The main point is that they can treat all the non-linear contributions to the Klein-Gordon equation, including kinetic terms and other stuff, as sources that you can treat perturbatively. They’re aiming for a solid analytical tool for structure formation.

Vera: And they highlight two key qualitative features in the non-linear regime. First, they say nonlinearities lead to a screening of the force mediated by the scalar field. That’s important because it tells us how gravity might change when you look at these theories on cosmological scales.

Jocelyn: Screening of the force... so it means that in some regions, this extra gravitational effect gets turned off, right?

Subrahmanyan: That's what they mean. And the second feature is something a bit more subtle—it’s a consequence of changing the Klein-Gordon equation from Eulerian coordinates to Lagrangian coordinates. This produces frame-lagging terms that are crucial because if you ignore them, the theory doesn't actually reduce to our standard Lambda CDM model at large scales.

The paper's improvements: Vera: The improvement they suggest is the use of this LPT formalism up to third order, which is necessary because their solutions lead to the first corrections to both the power spectrum and correlation function, specifically what they call the one-loop corrections <ref:1705.10719#pg1>.

Jocelyn: Third order sounds heavy. What does that actually give us in terms of results? Is it just more complicated math?

Subrahmanyan: It gives us the third order Lagrangian displacement field, which is expressed as a complex expression involving various sources like screening, self-interaction, and those frame-lagging terms we talked about earlier. It’s a necessary step to get the first corrections to the statistics.

Vera: And then they go on to compute second order results for growth functions, D(two)(k1, k2), which depend on three numbers k1, k2 and k12 that they can write in an alternative way <ref:1705.10719#pg1>.

Jocelyn: Three numbers... that’s a lot of parameters to track. How do you actually use those results to see something new?

Subrahmanyan: The second order Lagrangian displacement kernels are derived from the first order fields in equations forty-eight and fifty-one which then feed into those growth functions. It’s a systematic way to build up the description of structure formation layer by layer.

Conclusion: Vera: So to wrap up, this paper shows that by using this formalism, they can capture the physical fact that nonlinearities lead to a screening of the force mediated by the scalar field through mathematical cancellations provided by those frame-lagging contributions.

Jocelyn: That’s a big claim about how these different parts of the math balance each other out to give us a physically sensible result. And they also suggest that these perturbation theory schemes, SPT and CLPT, are IR-safe in modified gravity.

Subrahmanyan: That IR safety is important because it means that the methods we use to compute things like power spectra and correlation functions are actually valid even in the infrared limit where things get very large scales. Furthermore, they found that the Q and R functions have the same k-dependence at large scales in modified gravity and GR.

Vera: And they give us a specific result for the Q3 function at k approaching zero, stating that its limit is proportional to k to the fourth power times a sum involving an integral over p squared times L(p) times p squared.

Jocelyn: So, it sounds like they’ve provided a clear mathematical pathway forward for studying these modified gravity models using these perturbation theories. Where do we go from here?

Subrahmanyan: We need to connect these analytical results with observational data from surveys like those Vera works on or the time-domain observations Jocelyn studies to see if we can measure those corrections.

Departamento de Física, Instituto Nacional de Investigaciones Nucleares · Consejo Nacional de Ciencia y Tecnología

astro-ph.CO, gr-qc

Submitted: 2017-05-30

Updated: 2026-10-19

Comments: 32 pages, 10 figures. v3: Matches published version. v4: Typos corrected

Journal ref: Phys. Rev. D 96, 123526 (2017)

DOI: 10.1103/PhysRevD.96.123526

Code: https://github.com/martinjameswhite/CLEFT

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 76/100

The gist: The gist The authors present a formalism to compute Lagrangian displacement fields for a wide range of cosmologies in the context of perturbation theory up to third order, emphasizing theories with

Key concepts

Lagrangian Perturbation Theory (LPT): This is a theoretical framework used to compute how matter particles move in space under gravity, focusing on the displacement of those particles. It is applied here to modified gravity theories, allowing researchers to study structure formation beyond simple linear approximations.
Frame-lagging terms: These are extra terms that appear when transforming equations from Eulerian (fixed position) to Lagrangian (particle motion) coordinates. They are important because they ensure that the theory correctly reduces to standard General Relativity at very large cosmological scales, where the fifth force mediated by modified gravity should vanish.
Force Screening: This is a non-linear effect where the presence of matter changes how gravity behaves. In this context, it means that when gravity is mediated by a scalar field (as in chameleon theories), the force becomes weaker or 'screened' in high-density regions, which helps explain why modified gravity effects are not seen everywhere.
Power Spectra and Resummation Schemes: These are mathematical tools used to quantify the distribution of matter fluctuations in the universe. The paper uses methods like Convolution Lagrangian Perturbation Theory (CLPT) and Lagrangian Resummation Theory (LRT) to handle complex calculations, helping to determine if the perturbation theory results are reliable.

Summary

The gist The authors present a formalism to compute Lagrangian displacement fields for a wide range of cosmologies in the context of perturbation theory up to third order, emphasizing theories with scale dependent gravitational strengths such as chameleons, and they discuss qualitative features arising from nonlinearities like force screening and frame-lagging terms

Theoretical Framework

The work develops a Lagrangian Perturbation Theory (LPT) theory applicable to a wide range of modified gravity theories, essentially those that can be written as a scalar tensor gravity theory The formalism is suitable for Horndeski models, which share the same range of validity that share several investigations in SPT that came after to the pioneering work of Ref. [60] The basic requirement is that the theory can be recasted as a scalar tensor gravity at linear order, all non linear contributions to the Klein-Gordon equation, including possible kinetic terms, can be written as sources and treated perturbatively

Key Features of LPT in Modified Gravity

The formalism introduces two qualitative features in the non-linear regime: one is that nonlinearities lead to a screening of the force mediated by the scalar field The second is "a consequence of the transformation of the Klein-Gordon equation from Eulerian to Lagrangian coordinates, producing frame-lagging terms that are important especially at large scales, and if not considered, the theory does not reduce to the ΛCDM model in that limit" These frame-lagging terms are non-negligible and necessary for recovering ΛCDM at sufficiently large scales where the fifth force mediated by the scalar field is essentially zero

Perturbation Theory Orders

The paper develops LPT theory up to third order, which is necessary because their solutions lead to the first corrections to the power spectrum (the 1-loop) and correlation function The third order Lagrangian displacement field is given by a complex expression involving various sources like screening, self-interaction, and frame-lagging terms

Second Order Results

The second order computation yields growth functions such as D(2)(k1, k2) and D(2)(k1, k2) which depend on three numbers that can be chosen as k1, k2 and k12 such that alternatively we can write D¯(2)(k1, k2) = D¯(2)(k12, k1, k2) The second order Lagrangian displacement kernels are derived from the first order fields given in Eqs. (48) and (51)

Third Order Results and Statistics

The third order equation of motion is given by Eq. (70), which involves numerous terms including frame-lagging contributions, screening, and self-interaction terms The third order Lagrangian displacement kernel is expressed as L(3)i(k1, k2, k3) = i k i / k 2 D(3)(k1, k2)D+(k2)D+(k3)>

Power Spectra and Divergences

The paper computes the power spectrum and correlation function using different resummation schemes, including Convolution Lagrangian Perturbation Theory (CLPT) and Lagrangian Resummation Theory (LRT) The analysis shows that the Q functions have UV divergences for n ≥ 1/2 and the R functions for n ≥ −1, which are the same UV divergences that have the P22 and P13 pieces of the SPT power spectrum, respectively In general, IR divergences may show up both at p → 0 and p → k

Conclusion

The formalism presented here is capable to capture this physical fact, mathematically through cancellations provided by the frame-lagging contributions The analysis suggests that the schemes of SPT and CLPT are IR-safe in MG Furthermore, the Q and R functions have the same k-dependence at large scales in MG and GR The analysis of the Q3 function shows that its large scale limit is Q3(k → 0) = k 4 / (10π2) Z ∞ 0 dp P squared L(p) p squared

How it works

The Lagrangian displacement velocity is defined as a momentum average of particles’ velocities over the ensemble, and for CDM scenarios, we can safely neglect velocity dispersions and the identity S = Ψ follows The matter perturbation is given by −δ(x) = J(q) − 1 / J(q)

Second Order Computation

The second order Lagrangian displacement to second order is given by kiΨi(2)(k) = i 2 Z k12=k 3/7 D¯(2)a (k1, k2) − D¯(2)b (k1, k j) / (k j) squared The growth functions are given by D(2)(a)(k1, k 2) = T −ˆ A(k)-1A(k)D+(k1)D+(k2)

Third Order Computation

The equation of motion to third order is given by (T −ˆ A(k)) Ψ(3)i,i = S1 + · · · + S10 The third order Lagrangian displacement kernel is L(3)i(k1, k2, k3) = i k i / k 2 D(3)(k1, k2)D+(k2)D+(k3)>

Statistical Analysis

The Q functions are constructed from the Lagrangian displacements up to third order and are used to compute the power spectrum and correlation function The power spectrum of Eq. (A29) coincides with the SPT power spectrum in the ΛCDM case The LRT power spectrum is written as PLRT(k) = e − k 2σ 2L PL(k) + NL, where NL denotes the loop integrals

Final Result Summary

The analysis shows that in the limit k → 0, D(3)symm(0, −p, p) = 0 as it should be the case since double squeezed configurations reduce to the 1-dimensional collapse in the limit of k → 0 The Q1 function at k → 0 is CQ1 = CQ1 / (4k 4/15π2 Z ∞ 0 dp P squared L(p) p 2) The analysis suggests that the schemes of SPT and CLPT are IR-safe in MG The analysis of the Q3 function shows that its large scale limit is Q3(k → 0) = k 4 / (10π2) Z ∞ 0 dp P squared L(p) p squared

How it works

The frame-lagging term of Eq.

Improvements for AI systems

Compute Lagrangian Displacement Fields Up to Third Order for Modified Gravity Theories: The improved system can compute Lagrangian displacement fields for a wide range of cosmologies in the context of perturbation theory up to third order. This allows for the computation of higher-order corrections to cosmological statistics beyond what is feasible with standard Eulerian perturbation theory.
Model Scale-Dependent Gravitational Strengths Accurately: The improved system can accommodate the case of theories with scale dependent gravitational strengths, such as chameleons, by treating the Klein-Gordon equation in Lagrangian coordinates, which produces frame-lagging terms that are important especially at large scales. This enables the accurate modeling of screening mechanisms like the chameleon or kinetic/Vainshtein mechanisms.
Analyze Non-Linear Regime Features: The system can identify two qualitative features [arising in the non-linear regime], One, as is well known, is that nonlinearities lead to a screening of the force mediated by the scalar field. This capability allows for a rigorous assessment of how nonlinearities affect gravity at cosmological scales.
Identify Frame-Lagging Terms: The system can explicitly calculate frame-lagging terms, which are described as important especially at large scales, and if not considered, the theory does not reduce to the ΛCDM model in that limit. This ensures that the theoretical framework correctly recovers GR at sufficiently large scales.
Compute Power Spectra and Correlation Functions via Resummation Schemes: The system can compute these quantities by using different resummation schemes, such as Convolution Lagrangian Perturbation Theory (CLPT) and the Lagrangian Resummation Theory (LRT), to study structure formation in modified gravity models like the Hu-Sawicky f(R) model.
Assess IR and UV Divergences: The improved system can discuss the IR divergences of these formalisms and analyze them in matter and Lagrangian displacement statistics, suggesting that schemes like SPT and CLPT are IR-safe in MG. This provides a rigorous check on the validity of perturbation theory.
Understand Growth Function Behavior: The system can determine how growth functions behave under different configurations, noting that for the F4 model, the leading term that survives in Eq. (93) is of order O((k/p) 4), and showing that the large scales modes of Q1 and Q2 functions receive negligible contributions from small scales. This guides the selection of appropriate perturbative methods based on scale.
Compute Lagrangian Displacement Power Spectra: The system can compute the relevant quantities, such as Q1(k), Q2(k), and R1(k), R2(k), which are used to construct matter power spectrum and bispectra, enabling detailed studies of non-linear structure formation statistics.

Abstract

We present a formalism to compute Lagrangian displacement fields for a wide range of cosmologies in the context of perturbation theory up to third order. We emphasize the case of theories with scale dependent gravitational strengths, such as chameleons, but our formalism can be accommodated to other modified gravity theories. In the non-linear regime two qualitative features arise. One, as is well known, is that nonlinearities lead to a screening of the force mediated by the scalar field. The second is a consequence of the transformation of the Klein-Gordon equation from Eulerian to Lagrangian coordinates, producing frame-lagging terms that are important especially at large scales, and if not considered, the theory does not reduce to the Λ CDM model in that limit. We apply our formalism to compute the 1-loop power spectrum and the correlation function in f(R) gravity by using different resummation schemes. We further discuss the IR divergences of these formalisms.