Parity-Even Cubic Weyl Operator

• The parity-even cubic Weyl operator is defined as the unique cubic contraction of the Weyl tensor that introduces higher-derivative corrections in four-dimensional gravitational effective field theory.
• It serves as a cutoff-suppressed modification in both black-hole EFT and inflationary perturbation theory, probing finite-size effects and inducing specific static tidal responses or primordial graviton bispectra.
• Its application leads to distinctive observational signatures, such as fixed-quadrupole benchmarks in Schwarzschild black holes and even-parity selection rules in the CMB bispectrum.

Searching arXiv for Cano's Teukolsky-based Love number analysis and related context. arxiv_search.query({"search_query":"au:Cano black holes Love numbers Teukolsky cubic Weyl", "start": 0, "max_results": 10}) arxiv_search.query({"search_query":"all:Teukolsky Love numbers cubic Weyl Cano", "start": 0, "max_results": 10}) arxiv_search.query({"search_query":"all:\"Weyl\" AND all:\"Love number\" AND au:Cano", "start": 0, "max_results": 10}) The parity-even cubic Weyl operator is the cubic contraction of the Weyl tensor,

$$C^3 \equiv C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu},$$

also written as $W^3$ in cosmological applications. In the four-dimensional gravitational effective field theory considered for Schwarzschild black holes, it is the unique parity-even cubic Weyl invariant, while in inflationary perturbation theory it appears as a parity-conserving higher-derivative graviton interaction with distinctive bispectral selection rules (Silva, 15 Jun 2026, Shiraishi et al., 2011). Across these settings, the operator is introduced as a cutoff-suppressed correction to Einstein gravity and serves as a probe of finite-size effects, higher-curvature response, and tensor non-Gaussianity.

1. Algebraic definition and symmetry properties

In the black-hole EFT formulation, the bulk action is

$$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$

with cutoff $\Lambda$ and Wilson coefficient $\lambda_e$ (Silva, 15 Jun 2026). The same cubic structure appears in the cosmological literature as

$$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$

where the Weyl tensor is

$$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$

(Shiraishi et al., 2011).

By construction, $W^3$ is parity-even. In the cosmological treatment it is explicitly described as parity-conserving and is contrasted with the parity-odd operator $\tilde W W^2$, whose observational imprint occupies complementary CMB multipole sectors (Shiraishi et al., 2011). This separation is structural rather than conventional: the parity assignment follows from the tensor contraction itself.

2. Effective-field-theory role and perturbative control

For Schwarzschild black holes, the relevant small parameter is

$$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$

so the cubic Weyl correction is treated perturbatively around the general-relativistic background (Silva, 15 Jun 2026). The background metric ansatz to $W^3$0 is

$W^3$1

with

$W^3$2

In the inflationary setting, the same operator is introduced on an exact de Sitter background with conformal time $W^3$3, and its strength is suppressed by $W^3$4 together with a time-dependent coupling $W^3$5 (Shiraishi et al., 2011). The normalization differs from the black-hole EFT normalization. This suggests that the operator’s physical interpretation is stable across applications, while the precise power of the cutoff and the coupling convention are context-dependent.

The EFT logic is similar in both cases: $W^3$6 encodes higher-derivative corrections beyond the Einstein-Hilbert term, and the perturbative expansion isolates its leading observable consequences. In the black-hole problem those consequences appear in static tidal response; in the cosmological problem they appear in a graviton three-point function.

3. Static even-parity quadrupole sector on Schwarzschild

The black-hole analysis focuses on static even-parity $W^3$7 perturbations in Regge-Wheeler gauge,

$W^3$8

$W^3$9

with first-order $$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$0 expansions

$$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$1

$$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$2

Here $$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$3 and $$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$4 satisfy the $$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$5 Zerilli/Regge-Wheeler static equations in pure GR (Silva, 15 Jun 2026).

Substituting these fields into the bulk action and expanding to quadratic order in the perturbation amplitude $$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$6 and to first order in $$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$7 yields a one-dimensional reduced radial Lagrangian,

$$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$8

The three pieces have distinct origins: the GR quadratic action on Schwarzschild, the correction induced by the $$S_{\rm bulk}=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,\Bigl[R+\lambda_e\Lambda^{-4}\, C_{\mu\nu}{}^{\rho\sigma}C_{\rho\sigma}{}^{\alpha\beta}C_{\alpha\beta}{}^{\mu\nu}\Bigr],$$9 change in the background, and the Weyl-cubic term evaluated on the unperturbed background but to quadratic order in the GR tidal perturbation (Silva, 15 Jun 2026).

A key technical feature is that higher-order radial derivatives, up to fourth order, appear only at $\Lambda$0. They are then eliminated by order reduction using the zeroth-order GR tidal equations. With the generalized Euler-Lagrange operator

$\Lambda$1

the $\Lambda$2 field equations become a linear inhomogeneous system for $\Lambda$3. One equation is purely algebraic in $\Lambda$4, fixing

$\Lambda$5

After substitution, the remaining system closes on the two-vector

$\Lambda$6

which satisfies

$\Lambda$7

with $\Lambda$8 and $\Lambda$9 rational functions of $\lambda_e$0 (Silva, 15 Jun 2026). The explicit radial integrand is described as bulky and was generated with Mathematica; the full expressions for $\lambda_e$1, $\lambda_e$2, and the cubic invariant on Schwarzschild are provided in the supplemental Mathematica notebook.

4. Boundary conditions and the fixed-quadrupole response benchmark

Near the horizon, with $\lambda_e$3, the residue data of the homogeneous system are

$\lambda_e$4

so the two local exponents are $\lambda_e$5 (Silva, 15 Jun 2026). Regularity on the future horizon removes the $\lambda_e$6 mode and leaves one free homogeneous datum,

$\lambda_e$7

At asymptotic infinity, a Laurent expansion,

$\lambda_e$8

reveals a free growing-branch constant proportional to $\lambda_e$9, denoted $$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$0, with $$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$1. The large-$$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$2 behavior is

$$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$3

$$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$4

$$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$5

The $$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$6 terms renormalize the applied tidal field, while the $$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$7 terms define the induced response. Pure GR has $$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$8 but no decaying branch (Silva, 15 Jun 2026).

Matching the horizon and asymptotic expansions gives the degeneracy relation

$$L_{W^3}(\tau)=\frac{f(\tau)}{\Lambda^2}\, W^{\mu\nu}{}_{\rho\sigma}W^{\rho\sigma}{}_{\alpha\beta}W^{\alpha\beta}{}_{\mu\nu}, \qquad f(\tau)=\Bigl(\frac{\tau}{\tau_*}\Bigr)^A,$$9

The remaining horizon datum therefore only shifts the growing solution. Imposing the no-tidal-renormalization convention,

$$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$0

removes this ambiguity and fixes the decaying branch: $$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$1 Restoring the metric-level definitions gives

$$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$2

Calibrating the spatial sector at fixed $$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$3 against the associated-Legendre branches $$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$4 and $$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$5, and using

$$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$6

the fixed-quadrupole response amplitude becomes

$$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$7

Equivalently, the scalar fixed-$$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$8 quotient is

$$W_{\mu\nu\rho\sigma}\equiv R_{\mu\nu\rho\sigma} -\frac{1}{2}\bigl(g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr) +\frac{R}{6}g_{\mu[\rho}g_{\sigma]\nu}$$9

(Silva, 15 Jun 2026). In the terminology of that work, these are the primary metric-sector results at $W^3$0.

5. Relation to Love numbers and interpretive caveats

A central interpretive issue is that the scalar quantity $W^3$1 is not, by itself, the analytically continued, gauge-invariant electric Love number (Silva, 15 Jun 2026). At fixed integer $W^3$2, the separation between physical response, tidal-field redefinition, and gauge artifacts is not unique. In the metric derivation, this ambiguity is encoded in the freedom $W^3$3 or, equivalently, $W^3$4, which reintroduces a pure growing solution unless a convention is imposed.

The canonical extraction of a gauge-invariant Love number $W^3$5 requires three further steps: working in a gauge-invariant master-variable framework, such as Teukolsky; performing an analytic continuation in $W^3$6, $W^3$7, and isolating the finite, non-running part as $W^3$8; and matching normalizations carefully (Silva, 15 Jun 2026). A Teukolsky-based analysis cited there finds, for the same parity-even $W^3$9 operator,

$\tilde W W^2$0

The same source notes that a rough identification $\tilde W W^2$1, obtained by setting $\tilde W W^2$2, would suggest

$\tilde W W^2$3

which differs from the canonical $\tilde W W^2$4. The mismatch is expected because the fixed-$\tilde W W^2$5 metric quotient and the analytically continued gauge-invariant Love number are different observables.

Within this framework, the parity-even cubic Weyl operator breaks the exact GR cancellation that makes the static $\tilde W W^2$6 Love numbers of a Schwarzschild black hole vanish, and it induces a genuine decaying response at $\tilde W W^2$7 (Silva, 15 Jun 2026). The correction enters at relative order $\tilde W W^2$8 and remains small so long as $\tilde W W^2$9.

6. Primordial graviton bispectra and CMB signatures

In the cosmological application, the parity-even cubic Weyl operator generates a graviton interaction Hamiltonian obtained from

$$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$0

after inserting the all-orders expansion of the metric and retaining cubic terms in the transverse-traceless graviton field $$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$1 (Shiraishi et al., 2011). The graviton bispectrum is then computed at tree level with the in-in formalism,

$$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$2

The resulting three-point function is parity-even and, in the notation of the inflationary analysis, scales as

$$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$3

where $$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$4 is the reduced shape function (Shiraishi et al., 2011). In exact de Sitter, no slow-roll suppression appears. The operator therefore produces a purely parity-even primordial bispectrum even in that limit.

Projected into the CMB, the reduced bispectrum obeys a parity-even selection rule: $$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$5 In particular, the temperature bispectrum $$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$6 is nonzero only when $$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$7 is even, i.e. $$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$8 even (Shiraishi et al., 2011). By contrast, the parity-odd operator $$\epsilon_e\equiv \lambda_e(\Lambda r_s)^{-4}\ll 1, \qquad r_s=2GM,$$9 contributes only when $W^3$00 is odd. This cleanly separates parity-even and parity-odd cubic Weyl interactions at the level of CMB multipole configurations.

At large scales, the temperature bispectrum from $W^3$01 is estimated numerically as

$W^3$02

and comparison with an equilateral-type nonlinearity bound $W^3$03 yields

$W^3$04

assuming $W^3$05 (Shiraishi et al., 2011). In this setting, the parity-even cubic Weyl operator functions as a higher-derivative source of tensor non-Gaussianity with a specific angular-parity signature.

7. Scope, significance, and common points of confusion

The operator’s significance depends on which observable is being extracted. In the black-hole problem, the central result is a fixed-quadrupole metric-sector benchmark,

$W^3$06

obtained after removing tidal renormalization by convention (Silva, 15 Jun 2026). In the cosmological problem, the same parity-even cubic Weyl structure is important because it produces a parity-even primordial graviton bispectrum and contributes only to even $W^3$07 CMB bispectra (Shiraishi et al., 2011).

A common misunderstanding is to treat all “Love-number-like” quotients extracted at fixed $W^3$08 as gauge-invariant tidal Love numbers. The black-hole analysis explicitly rejects that identification: $W^3$09 is a scalar fixed-$W^3$10 conversion of a metric branch ratio and not, by itself, the analytically continued gauge-invariant electric Love number (Silva, 15 Jun 2026). Another possible confusion is to equate “parity-even” with the absence of observationally distinctive parity structure. The CMB analysis shows the opposite: parity conservation forces the signal into even total multipole configurations, while the parity-odd cubic Weyl interaction occupies the odd sector (Shiraishi et al., 2011).

Taken together, these results establish the parity-even cubic Weyl operator as a concrete higher-curvature deformation with two sharply defined uses. In black-hole EFT it provides a reproducible metric-space benchmark for static tidal response. In primordial cosmology it provides a parity-conserving source of graviton non-Gaussianity with specific CMB selection rules.