Convolved Lagrangian Perturbation Theory (CLPT)

• CLPT is a resummed configuration-space approach that models nonlinear large-scale structure using a Lagrangian displacement framework.
• It systematically integrates displacement cumulants to predict the two-point correlation, power spectrum, and velocity statistics in both real and redshift space.
• CLPT extends the Zel'dovich approximation by incorporating bias and modified gravity to achieve percent-level accuracy in quasi-linear clustering analyses.

Convolved Lagrangian Perturbation Theory (CLPT) is a resummed, configuration-space approach to nonlinear large-scale structure in cosmology, constructed from the Lagrangian description of fluid elements and their displacements. By systematically integrating cumulants of the Lagrangian displacement field, CLPT provides predictive models for the two-point correlation function, power spectrum, and velocity statistics of both matter and biased tracers, in real and redshift space. The formalism is a controlled expansion that incorporates large-scale flows nonperturbatively (via convolution resummation), includes one-loop (cubic) corrections, and readily accommodates Lagrangian bias and extensions to modified gravity theories and cosmologies with massive neutrinos. CLPT underpins quasi-analytical pipelines for next-generation large-scale structure surveys, providing accuracy at the percent level in the mildly nonlinear regime for both clustering and redshift-space distortions.

1. Lagrangian Framework and Displacement Expansion

CLPT operates within the Lagrangian perturbation theory (LPT) paradigm, labeling each fluid element by its initial position $q$ and tracking its Eulerian position $x(q, t) = q + \Psi(q, t)$, where $\Psi$ is the displacement field encoding structure formation. The displacement is expanded in powers of the initial (linear) density field:

$\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$

with $\Psi^{(1)}$ representing the Zel'dovich approximation and higher-order $\Psi^{(n)}$ correcting for nonlinear effects (Wang et al., 2013, Carlson et al., 2012, Vlah et al., 2014). The Eulerian density is then obtained by mass conservation:

$$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$

and for biased tracers, a local Lagrangian bias function $F[\delta_{m,R}(q)]$ is included:

$$1 + \delta(x) = \int d^3q\, F[\delta_{m,R}(q)] \, \delta^D[x - q - \Psi(q)],$$

where $\delta_{m,R}$ is the smoothed initial density (Wang et al., 2013).

2. Cumulant Expansion and Convolution Structure

The core of CLPT is the cumulant expansion of the characteristic function of the pairwise displacement, $x(q, t) = q + \Psi(q, t)$0. The two-point function is cast as

$x(q, t) = q + \Psi(q, t)$1

where $x(q, t) = q + \Psi(q, t)$2 and $x(q, t) = q + \Psi(q, t)$3 is the Fourier transform of $x(q, t) = q + \Psi(q, t)$4 (Wang et al., 2013, Carlson et al., 2012).

The cumulant expansion organizes the expectation $x(q, t) = q + \Psi(q, t)$5 as

$x(q, t) = q + \Psi(q, t)$6

where $x(q, t) = q + \Psi(q, t)$7 and $x(q, t) = q + \Psi(q, t)$8 denotes connected cumulants (Wang et al., 2013, Vlah et al., 2014). At one-loop CLPT, terms quadratic and cubic in the displacements and densities are kept.

Upon integrating out the $x(q, t) = q + \Psi(q, t)$9's and performing the Gaussian $\Psi$0-integral, the real-space correlation reduces to a single convolution:

$\Psi$1

with

$\Psi$2

where $\Psi$3, $\Psi$4, etc. are cumulants of $\Psi$5, $\Psi$6, $\Psi$7, and dots denote higher-order terms (Wang et al., 2013, Carlson et al., 2012).

3. Velocity Statistics and Redshift-Space Distortions

CLPT predicts not only density statistics but also velocity moments, essential for modeling redshift-space distortions (RSD). The generating function is augmented by a term linear in the pairwise velocity:

$\Psi$8

with the mean pairwise velocity given by

$\Psi$9

where $\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$0 is obtained by differentiation of the exponentiated cumulant expansion with respect to $\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$1, then integrating over $\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$2's and $\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$3 (Wang et al., 2013). The pairwise velocity dispersion, $\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$4, is similarly calculated, and the decomposition into parallel and perpendicular components follows.

Embedding the CLPT-predicted $\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$5, $\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$6, and $\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$7 into the Gaussian Streaming Model (GSM), the redshift-space two-point correlation function is:

$\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$8

with $\Psi(q) = \Psi^{(1)}(q) + \Psi^{(2)}(q) + \cdots,$9, $\Psi^{(1)}$0. Multipole moments are constructed via projection onto Legendre polynomials (Wang et al., 2013).

4. Biased Tracers and Extensions

Biased tracers are treated by introducing a local bias $\Psi^{(1)}$1 in the Lagrangian framework, with bias parameters $\Psi^{(1)}$2. The functional dependence propagates through the cumulant expansion, resulting in nonlinear bias contributions to all orders in the final convolution kernel $\Psi^{(1)}$3 (Carlson et al., 2012, Valogiannis et al., 2019). This structure naturally extends to cross-correlations between different biased tracers via separate bias parameter sets (Wang et al., 2013).

The CLPT construction is also applicable to marked correlation functions, where objects are weighted by environment-dependent marks $\Psi^{(1)}$4. The marked correlation function is expressed as a double convolution in the CLPT formalism, with perturbative approximations accurate at the sub-percent level on quasi-linear scales (Aviles et al., 2019).

In cosmologies beyond general relativity (GR), such as chameleon $\Psi^{(1)}$5 or Vainshtein nDGP models, CLPT incorporates scale- and environment-dependent growth factors and bias parameters obtained by an extended peak-background split framework (Valogiannis et al., 2019). For massive neutrino cosmologies, scale-dependent displacement kernels and frame-lagging terms ensure UV- and Galilean-invariant results, with percent-level accuracy down to $\Psi^{(1)}$6 Mpc/h (Aviles et al., 2020).

5. Numerical Implementation, Accuracy, and Deficiencies

Owing to rotational symmetry, the $\Psi^{(1)}$7-space integrals in real- and redshift-space clustering are efficiently reduced to one-dimensional Hankel transforms, typically computed using FFTLog or quadrature methods (Wang et al., 2013, Vlah et al., 2014). The cumulant ingredients ($\Psi^{(1)}$8, $\Psi^{(1)}$9, etc.) are built from integrals over the linear (or resummed) power spectrum.

CLPT achieves percent-level accuracy for the real-space correlation function and RSD multipoles down to $\Psi^{(n)}$0 Mpc/h for massive halos, as confirmed against $\Psi^{(n)}$1-body simulations for a wide range of masses and tracers (Wang et al., 2013, Carlson et al., 2012). The same pipeline applies for cross-correlations with appropriate bias parameters.

A major limitation of CLPT is the excessive contribution from small-scale displacements at one-loop order, leading to overestimated smoothing of high-$\Psi^{(n)}$2 density fluctuations and underestimation of small-scale power (missing the "one-halo" term). On smaller scales, CLPT over-damps the correlation function and does worse than the Zel'dovich approximation at $\Psi^{(n)}$3 Mpc/h (Vlah et al., 2014). The empirical "CLPTs" modification damps the input displacement spectrum above a nonlinear scale ($\Psi^{(n)}$4), partially restoring small-scale clustering but still requiring an additive non-perturbative or EFT term for full accuracy (Vlah et al., 2014).

Quantity	Scale of Agreement with Simulations	Limiting Factor
$\Psi^{(n)}$5 (real)	$\Psi^{(n)}$6--$\Psi^{(n)}$7, $\Psi^{(n)}$8--$\Psi^{(n)}$9 Mpc/h	Overdamped at $$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$0--$$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$1 Mpc/h
$$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$2 (monopole)	$$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$3, $$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$4 Mpc/h	Velocity dispersion modeling
$$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$5 (power spectrum)	$$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$6, $$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$7--$$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$8 Mpc$$1 + \delta(x) = \int d^3q \, \delta^D[x - q - \Psi(q)],$$9	Fast power decay at higher $F[\delta_{m,R}(q)]$0 (no one-halo term)
Marked statistics	$F[\delta_{m,R}(q)]$1, $F[\delta_{m,R}(q)]$2 Mpc/h	Fitting mark and bias degeneracies

6. Comparative and Interpretive Structure

CLPT extends and improves upon the Zel'dovich approximation by exponentiating the Gaussian two-point displacement cumulant to all orders, then systematically including higher cumulants (one-loop: three-point terms) (Vlah et al., 2014). For unbiased matter, truncating the expansion reproduces the Zel'dovich limit exactly. Compared to standard Eulerian perturbation theory (SPT), CLPT more effectively resums large-scale flows, which is crucial for accurate configuration-space BAO modeling.

The dependency of small-scale accuracy on underlying dark matter or tracer microphysics (e.g., halo exclusion, velocity dispersion, or nonlocal bias) suggests that a full effective field theory or halo-model extension is required for space scales $F[\delta_{m,R}(q)]$3 Mpc/h and Fourier modes $F[\delta_{m,R}(q)]$4 h/Mpc$F[\delta_{m,R}(q)]$5 (Vlah et al., 2014). Within its regime of validity, CLPT provides a tractable analytic instrument for interpreting quasi-linear scale clustering in current and future survey data, as well as for testing gravity and dark sector physics (Wang et al., 2013, Valogiannis et al., 2019, Aviles et al., 2020).