Two-Loop EFT Power Spectrum

• Two-Loop EFT Power Spectrum is a renormalized framework for modeling nonlinear matter clustering by expanding the power spectrum up to next-to-next-to-leading order in k/kₙₗ.
• It organizes contributions from linear theory, one-loop, two-loop terms, counterterms, and stochastic components while ensuring IR safety and cutoff independence.
• Advanced numerical strategies, including IR-safe integrands, FFT-PT, and COBRA methods, enable fast and precise cosmological inference and simulation matching.

Searching arXiv for the primary and related two-loop EFTofLSS papers to ground the article in current literature. The two-loop EFT power spectrum is the Effective Field Theory of Large Scale Structure description of the equal-time matter power spectrum carried through next-to-next-to-leading order in the expansion in $k/k_{\rm nl}$, where $k_{\rm nl}$ is the nonlinear scale. In this framework, the observable is organized as a sum of linear theory, one-loop and two-loop standard-perturbation-theory contributions, EFT counterterms, and stochastic terms, with renormalization enforcing cutoff independence and with infrared cancellations made explicit at the integrand level for equal-time correlators (Carrasco et al., 2013). In practice, the subject encompasses the original IR-safe reformulation of the two-loop matter integral, the renormalized EFTofLSS prediction for dark matter, later precision comparisons to simulations, exact-time and massive-neutrino generalizations, and recent fast-evaluation schemes and survey-level applications (Carrasco et al., 2013).

1. Perturbative definition and EFT organization

In the EFTofLSS, dark-matter correlators on large scales are computed in a controlled, manifestly convergent perturbative expansion whose small parameter is $k/k_{\rm nl}$. For equal-time correlators at $k\ll k_{\rm nl}$, the matter power spectrum is expanded schematically as

$$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$

where $P_{11}$ is the linear spectrum, the loop terms are the one- and two-loop SPT contributions evaluated with a UV regulator, $P_{\rm ct}$ collects EFT counterterms from the effective stress tensor, and $P_{\rm stoch}$ denotes stochastic contributions analytic in $k$ at low wavenumber (Carrasco et al., 2013).

This organization differs from standard perturbation theory because loop integrals are renormalized by local counterterms. After renormalization, the expansion is a convergent series in powers of $k/k_{\rm nl}$ rather than a formally divergent loop expansion. In the dark-matter case, the two-loop matter calculation was formulated as a renormalized extension of the one-loop EFTofLSS, with the leading counterterm proportional to $k_{\rm nl}$0 and with additional analytic structures becoming relevant depending on the renormalization scheme and target accuracy (Carrasco et al., 2013).

A later precision comparison against simulations emphasized that, at two-loop order, it is necessary to include three counterterms: a linear term in the over density, $k_{\rm nl}$1, a quadratic term, $k_{\rm nl}$2, and a higher derivative term, $k_{\rm nl}$3. In that study, after the inclusion of these three terms, the EFTofLSS at two-loop order matched simulation data up to $k_{\rm nl}$4 at $k_{\rm nl}$5, up to $k_{\rm nl}$6 at $k_{\rm nl}$7, and up to $k_{\rm nl}$8 at $k_{\rm nl}$9 (Foreman et al., 2015).

2. Standard perturbation-theory building blocks and two-loop diagrams

SPT expresses the density contrast as a series in the linear field with symmetrized kernels $k/k_{\rm nl}$0. At equal time,

$k/k_{\rm nl}$1

with the kernels obeying the standard recurrence relations and $k/k_{\rm nl}$2 (Carrasco et al., 2013). The explicit second-order kernel is

$k/k_{\rm nl}$3

with $k/k_{\rm nl}$4 (Carrasco et al., 2013).

At one loop, the SPT contribution is

$k/k_{\rm nl}$5

with

$k/k_{\rm nl}$6

$k/k_{\rm nl}$7

At two loops, the conventional decomposition is into $k/k_{\rm nl}$8, $k/k_{\rm nl}$9, $k\ll k_{\rm nl}$0, and $k\ll k_{\rm nl}$1, or equivalently into $k\ll k_{\rm nl}$2 after symmetries are used (Carrasco et al., 2013).

Using the conventions of the original IR-safe two-loop paper, the explicit two-loop terms are

$k\ll k_{\rm nl}$3

$k\ll k_{\rm nl}$4

$k\ll k_{\rm nl}$5

$k\ll k_{\rm nl}$6

The symmetry factors $k\ll k_{\rm nl}$7 are part of the definition in that formulation (Carrasco et al., 2013).

A related line of work generalized these kernels beyond Einstein–de Sitter time factorization. In $k\ll k_{\rm nl}$8CDM, the density and velocity kernels can be written in a separable form in time and momenta at each perturbative order, with time-dependent coefficients satisfying recursive differential equations. That construction yields exact two-loop dark-matter density and velocity power spectra in $k\ll k_{\rm nl}$9CDM, and shows that the difference between exact $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$0CDM and the quasi-EdS approximation can reach several percent at $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$1 for $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$2 (Fasiello et al., 2022).

3. Infrared structure and the IR-safe single-integrand construction

A central issue in the two-loop power spectrum is that individual SPT diagrams contain IR divergences even though equal-time correlators are IR finite after summation. Physically, long-wavelength modes generate coherent bulk flows that displace small-scale structures without changing equal-time statistics, so the leading IR sensitivity cancels as a consequence of the equivalence principle or Galilean invariance (Carrasco et al., 2013).

At one loop, the soft regions arise from $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$3 and from $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$4 in $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$5. After mapping both soft regions to $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$6 and symmetrizing over $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$7, the IR pieces cancel at the integrand level. At two loops, the same logic is more intricate because soft limits occur for $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$8, $$P(k)=P_{11}(k)+P_{1\text{-loop}}(k)+P_{2\text{-loop}}(k)+P_{\rm ct}(k)+P_{\rm stoch}(k),$$9, and composite momenta becoming soft. The original construction showed explicitly that, after suitable rearrangements, the leading double-soft asymptotics of the reordered integrands satisfy

$P_{11}$0

$P_{11}$1

$P_{11}$2

$P_{11}$3

so that the sum cancels exactly, $P_{11}$4 (Carrasco et al., 2013).

The main result is a global single-integral representation whose integrand is manifestly IR-safe before integration. In compact notation,

$P_{11}$5

$P_{11}$6

and the symmetrized IR-safe spectrum is

$P_{11}$7

$P_{11}$8

The step functions map all soft regions to a common boundary, the ordering $P_{11}$9 avoids double counting, and the parity symmetrization removes odd angular pieces (Carrasco et al., 2013).

With this construction, all leading double-soft terms cancel locally, all single-soft terms cancel between the appropriate subsets of diagrams, and the surviving small-$P_{\rm ct}$0 behavior scales as $P_{\rm ct}$1 or $P_{\rm ct}$2, which is integrable for $P_{\rm ct}$3. This is the precise sense in which the two-loop matter integrand is IR-safe (Carrasco et al., 2013).

Later work retained this IR-safe organization as a standard ingredient. For example, the weak-lensing two-loop EFTofLSS implementation explicitly states that it uses standard IR-safe integrands together with IR resummation, and the redshift-space two-loop calculation follows an analogous logic for the matter multipoles in redshift space (Saraivanov et al., 13 Mar 2026).

4. Renormalization, counterterms, and scaling structure

The EFT interpretation of the two-loop power spectrum is that loops are evaluated with a UV cutoff and their UV-sensitive parts are absorbed by counterterms. For the equal-time matter spectrum up to two loops, the basic structures include a leading speed-of-sound counterterm proportional to $P_{\rm ct}$4,

$P_{\rm ct}$5

higher-derivative terms such as $P_{\rm ct}$6, possible analytic counterterms proportional to $P_{\rm ct}$7 times one-loop shapes, and stochastic contributions analytic in $P_{\rm ct}$8 (Carrasco et al., 2013).

In scaling universes with $P_{\rm ct}$9, power counting gives

$P_{\rm stoch}$0

For $P_{\rm stoch}$1, the two-loop term is linearly UV divergent and the divergence scales as $P_{\rm stoch}$2, which is canceled by the $P_{\rm stoch}$3 counterterm. For $P_{\rm stoch}$4, the two-loop piece is UV finite and the residual IR sensitivity vanishes as $P_{\rm stoch}$5 (Carrasco et al., 2013).

The original two-loop EFTofLSS dark-matter paper took a particularly economical viewpoint: at that order in $P_{\rm stoch}$6, it argued that there is only one counterterm in the EFTofLSS that must be included for the matter power spectrum, namely the $P_{\rm stoch}$7 operator, although that operator contributes both at tree level and inside one-loop diagrams. In that setup, the two-loop prediction matched the nonlinear matter power spectrum at redshift zero to percent accuracy up to $P_{\rm stoch}$8 with one unknown coefficient fit to observations (Carrasco et al., 2013).

Subsequent precision studies emphasized a broader renormalized basis. The comparison to high-precision simulations concluded that the two-loop matter power spectrum requires a linear counterterm, a quadratic counterterm, and a higher-derivative counterterm to achieve UV-insensitive fits over the quoted $P_{\rm stoch}$9-ranges (Foreman et al., 2015). In a recent operator-level treatment for biased tracers, the matter limit was recovered by identifying the leading matter-sector counterterm with the renormalized $k$0 operator, and by embedding the $k$1 and $k$2 pieces into the next-to-leading-gradient basis (Bakx et al., 30 Jun 2026).

This suggests a scheme dependence in how one chooses the operator basis and organizes finite pieces, while preserving the same renormalized observable. A plausible implication is that the phrase “the two-loop EFT power spectrum” refers not to a unique counterterm basis, but to a class of renormalized descriptions related by basis choices and matching prescriptions, provided cutoff independence and symmetry constraints are maintained.

5. Numerical strategies and fast-evaluation methods

The IR-safe single-integrand representation was introduced primarily to make the two-loop calculation numerically reliable. In that formulation one aligns $k$3 with the $k$4-axis, parameterizes the loop variables by $k$5, imposes the step-function domain restrictions, and integrates the symmetrized six-dimensional integrand with adaptive Monte Carlo or quasi-Monte Carlo algorithms. Because the cancellation has already occurred at the integrand level, there are no large compensating contributions from distinct regions, and convergence is markedly faster and more stable than summing individually divergent diagrams (Carrasco et al., 2013).

An alternative strategy is FFT-PT, which reduces the two-loop SPT integrals to low-dimensional radial integrals by switching between Fourier and configuration space, integrating over orientations analytically, and expressing the result as finite sums of one-dimensional Hankel transforms. In that method, many contributions can be evaluated with one-dimensional FFTs rather than five-dimensional Monte Carlo integrals, and EFT counterterms can then be added algebraically on top of the SPT result (Schmittfull et al., 2016).

Recent work has focused on cosmological inference speed. One approach uses symbolic regression to emulate separately the IR-resummed building blocks entering the one- and two-loop EFTofLSS spectrum, leaving the counterterm time dependence unspecified so that different counterterm parametrizations can be fit a posteriori. That emulator suite reports errors of better than $k$6 within the EFT-valid $k$7-range and computational evaluation below $k$8 on a single core (Farakou et al., 7 Nov 2025).

Another approach uses the COBRA method to evaluate the two-loop EFT matter power spectrum in $k$9 millisecond at $k/k_{\rm nl}$0 precision on one CPU for arbitrary redshifts and on scales where perturbation theory applies. In an application to the Dark Sky simulation at $k/k_{\rm nl}$1, it found that the two-loop EFT can provide unbiased cosmological constraints on $k/k_{\rm nl}$2, $k/k_{\rm nl}$3 and $k/k_{\rm nl}$4 using scales up to $k/k_{\rm nl}$5, compared to $k/k_{\rm nl}$6 for the one-loop EFT, with a Figure of Merit increase by a factor $k/k_{\rm nl}$7 (Bakx et al., 1 Aug 2025).

A further efficiency-oriented development expands the linear spectrum around a fixed reference cosmology in a basis of propagator-like functions, precomputes cosmology-independent loop tensors with local UV subtractions and local IR cancellations, and then reconstructs the two-loop prediction by contracting these tensors with cosmology-dependent coefficients. That construction publicly releases precomputed integrals for the renormalized two-loop dark-matter power spectrum and states that eight EFT counterterms, including generated vorticity, are sufficient at that order for lensing-galaxy-signal analyses (Anastasiou et al., 5 Sep 2025).

6. Extensions, applications, and related generalizations

The two-loop EFT power spectrum has been extended in several directions beyond the equal-time real-space dark-matter case. One line of development incorporates exact $k/k_{\rm nl}$8CDM time evolution and massive neutrinos. In a two-fluid treatment of CDM+baryons and neutrinos, the two-loop density and velocity power spectra were computed with time- and scale-dependent growth, and comparison to Quijote simulations found percent agreement for the CDM+baryon density spectrum up to $k/k_{\rm nl}$9 at two loops for $k_{\rm nl}$00 (Garny et al., 2022).

Another extension concerns redshift space. The two-loop power spectrum in redshift space includes all nonlinearities from the redshift-space mapping and real-space dynamics up to fifth order, together with suitable EFT corrections and IR resummation. In that framework, the monopole and quadrupole agree with Quijote at percent level up to $k_{\rm nl}$01 at two-loop order, compared to $k_{\rm nl}$02 at one loop (Taule et al., 2023).

For biased tracers, two-loop renormalization becomes an operator-level problem involving fifth-order galaxy bias, next-to-leading-gradient operators, and stochastic renormalization of products of operators at coincidence. One recent treatment gives a complete two-loop power spectrum of biased tracers in real space, verifies that all UV limits are absorbed by the counterterms considered, and derives renormalization-group equations showing pronounced running of higher-gradient bias coefficients (Bakx et al., 30 Jun 2026). A separate galaxy-power-spectrum analysis formulated in a scheme compatible with CLASS-PT states that the two-loop galaxy power spectrum requires 21 additional EFT parameters per galaxy sample, achieves per mille-level agreement with the PT Challenge simulation at $k_{\rm nl}$03 up to $k_{\rm nl}$04, and improves the $k_{\rm nl}$05 error bar by about $k_{\rm nl}$06 relative to one loop (Ivanov, 29 Jun 2026).

A major contemporary application is weak lensing. A recent Roman-like cosmic-shear forecast implements a two-loop EFTofLSS matter spectrum with IR resummation, continuous-in-redshift counterterms calibrated to Euclid Emulator 2, PCA compression of nuisance directions, hydrodynamical-simulation-based priors, and neural-network emulation of the loop integrals. For $k_{\rm nl}$07, it forecasts unbiased $k_{\rm nl}$08 constraints with relative errors of about $k_{\rm nl}$09 for $k_{\rm nl}$10 and $k_{\rm nl}$11 for $k_{\rm nl}$12, while extending the usable scale range beyond the one-loop EFT in the presence of baryonic effects (Saraivanov et al., 13 Mar 2026).

Taken together, these developments establish the two-loop EFT power spectrum as a renormalized, IR-controlled description of mildly nonlinear clustering rather than merely a higher-order SPT correction. The original matter calculation provided the IR-safe integrand and the equal-time renormalization logic (Carrasco et al., 2013); later work showed how that structure can be embedded into simulation-calibrated dark-matter predictions, generalized to exact-time evolution, biased tracers, redshift space, and lensing observables, and accelerated to inference-grade runtimes (Foreman et al., 2015).