Scalar, Vector, and Tensor Modes

• Scalar, vector, and tensor modes are distinct classes of metric perturbations that represent density fluctuations, rotational vorticity, and gravitational radiation, respectively.
• They are mathematically extracted via harmonic decomposition on symmetric backgrounds, employing scalar, vector, and tensor harmonics to isolate each mode.
• These modes underpin analyses in CMB observations, gravitational-wave detection, and tests of modified gravity through their unique spectral and polarization signatures.

Scalar, vector, and tensor modes constitute the fundamental decomposition of perturbations in spacetime and matter fields on a homogeneous and isotropic cosmological or gravitational background. This classification underpins the analysis of cosmic microwave background (CMB) anisotropies, gravitational wave polarizations, turbulence-driven GW sources, gauge theories of gravity, and quantized perturbative general relativity. Each mode transforms irreducibly under spatial rotations: scalars (spin-0), vectors (spin-1), and tensors (spin-2). The distinction has both formal (harmonic analysis, spectral methods) and physical (observational signatures, polarization content) consequences across theoretical and experimental cosmology, modified gravity, and numerical relativity.

1. Definitions and Mathematical Decomposition

In the perturbative analysis of a (3+1)-dimensional spacetime with background metric $\bar{g}_{\mu\nu}$, any symmetric tensor perturbation $h_{\mu\nu}$ may be decomposed according to representations of SO(3) symmetry on spatial hypersurfaces. Explicitly, in a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) background:

• Scalars: Perturbations constructed from scalar functions; associated with density/curvature fluctuations.
• Vectors: Transverse ($\partial^i V_i = 0$), divergence-free spatial vectors; describe vorticity.
• Tensors: Transverse-traceless symmetric, rank-2 tensors ($\partial^i h_{ij}=0$, $h^i{}_i=0$); encode gravitational radiative degrees of freedom.

In Fourier space, the helicity eigenstate labeling generalizes this:

• $\lambda=0$: scalar, $\lambda=\pm1$: vector, $\lambda=\pm2$: tensor (Shiraishi et al., 2010).

On S³ and Euclidean space, the decomposition is realized through scalar, vector, and tensor harmonic eigenfunctions of the Laplace-Beltrami operator. In spherical geometry, completeness and orthonormality relations for these harmonics enable spectral representation and mode extraction (Lindblom et al., 2017, Dai et al., 2012).

2. Physical Interpretation and Observational Signatures

Physically, the modes correspond to distinct observable effects:

• Scalar modes: Govern density perturbations, source CMB temperature (Sachs–Wolfe and Doppler effects) and E-mode polarization.
• Vector modes: Typically decay in standard inflation but can be sourced by topological defects or primordial magnetic fields; contribute to both E and B CMB polarizations.
• Tensor modes: Primordial gravitational waves; dominant source of B-mode CMB polarization and main content of GWs in standard General Relativity (Shiraishi et al., 2010, Collaboration et al., 2018).

General metric theories of gravity allow up to six possible GW polarizations (Collaboration et al., 2018, Liu et al., 2019, Lai et al., 31 May 2024):

• Two tensor (“plus” and “cross”)
• Two vector (“x” and “y”/shear)
• Two scalar (breathing and longitudinal)

The presence, absence, or mixing of these modes offers a probe of the fundamental symmetries and couplings in gravitational theory. LIGO-Virgo-KAGRA constraints place strong bounds on the vector and scalar GW energy densities (Collaboration et al., 2018). In generalized theories (Einstein-æther, generalized Proca, etc.), up to five or six independent polarizations are possible, depending on parameter space and background fields (Lai et al., 31 May 2024, Liu et al., 2019).

3. Harmonic and Spectral Representations: S³ and Angular Momentum Bases

Harmonic decomposition on compact manifolds such as S³ leverages eigenfunctions of the Laplacian for each mode:

• Scalar harmonics: $Y^{k\ell m}$ (eigenvalues $-k(k+2)/R_3^2$)
• Vector harmonics: Three classes distinguished by eigenvalue and divergence conditions; includes both longitudinal (gradients of scalar harmonics) and transverse modes.
• Tensor harmonics: Sixfold classification, including trace, divergence-free and transverse-traceless parts (Lindblom et al., 2017).

Total Angular Momentum (TAM) waves provide a decomposition in eigenstates of both the Laplacian and the $J^2$ operator. For vector and tensor fields, derivative operators applied to scalar TAM waves generate all linearly independent vector and tensor TAM waves (longitudinal, E/B, TE/TB, etc.). The construction facilitates projection onto spherical harmonic multipoles relevant for CMB and lensing analyses (Dai et al., 2012).

These harmonics support spectral (pseudo-spectral, multi-cube) numerical methods. Exponential convergence is observed for smooth fields with computational efficiency up to $k \sim$ several tens (Lindblom et al., 2017).

4. Local Extraction and Mode Coupling in Cosmological Perturbation Theory

The canonical SVT decomposition is inherently nonlocal due to boundary conditions (Green's function integrals). However, for trace-free rank-2 tensors $X_{ab}$, local differential operators can extract scalar and vector components:

• Scalar part: $(\nabla\cdot\nabla\cdot X)$
• Vector part: $\varepsilon_{abc} D^b(D^d X_d{}^c)$
• Tensor part (new result): $$[-\nabla^2+2K+2\,\mathrm{dis}\,\mathrm{div}]\bigl(\mathrm{curl}\,X\bigr)$$

These constructions commute with the Laplacian and enable gauge-invariant identification at the second and higher order, crucial for non-linear cosmological calculations (Clarkson et al., 2011). At second order, even in standard GR, quadratic source terms (scalar–tensor and tensor–tensor products) generate vector and tensor modes out of scalars, enforcing a fundamentally mixed, gauge-invariant evolution at high perturbative order (Zhang et al., 2017).

In anisotropic or matter-rich backgrounds, the mode mixing can be irreducible even at linear order, as emphasized by explicit derivation in (Dolgov et al., 2023). Ricci-curvature terms in the background and nontrivial energy-momentum sources generically couple scalar and tensor propagation, precluding strict separability regardless of gauge.

5. Mode Content in Modified Gravity and Cosmological Theories

Scalar–vector–tensor (SVT) gravity theories generalize the minimal gravitational action by including one or more of scalar fields, massive vector fields (generalized Proca), or both. The generic linear perturbative content in parity-invariant theories consists of:

• Two tensor degrees of freedom (dynamical, transverse-traceless metric)
• Two vector degrees of freedom (dynamical, transverse vector)
• Two scalar degrees of freedom (often a mix of longitudinal vector and metric scalars)

Covariant quadratic action analysis yields ghost-free and Laplacian stability conditions for each mode based on kinetic coefficients $(q_t,\,q_v,\,q_s)$ and (sound) speeds $(c_t,\,c_v,\,c_{s,1/2})$ (Kase et al., 2018, Heisenberg et al., 2018). Observational propagation speed constraints (as from GW170817) force $c_t^2 \rightarrow 1$, imposing severe restrictions on operator coefficients.

In Einstein–vector theories with arbitrary constant background vectors, the full space of accessible GW polarizations and their degree of mixing (tensor, vector, scalar) depends on mass terms, curvature couplings, and background anisotropy (Lai et al., 31 May 2024). Observational data (e.g., pulsar timing arrays) favor models with standard tensor and possibly breathing scalar modes at luminal speeds, but strongly suppress vector and longitudinal components.

In scalar–tensor–vector gravity (STVG/MOG), field equations admit five GW polarization states (2 tensor, 2 vector, 1 scalar), with explicit signatures in test-mass motion and energy flux (Liu et al., 2019). The polarization taxonomy is manifest in both harmonic gauge analysis and in the Landau–Lifshitz pseudotensor formulation for stress–energy.

6. Applications: CMB Bispectrum, Gravitational-Wave Background, and Turbulence

The SVT decomposition is critical in the computation of CMB anisotropy observables and GW backgrounds:

• In the full-sky CMB bispectrum, multipole coefficients are expanded in spin-weighted spherical harmonics and helicity eigenstates. The bispectrum expression involves Wigner-3j and 9j symbols and convolution integrals over primordial SVT-mode shape functions (Shiraishi et al., 2010). For instance, for a scalar–scalar–tensor bispectrum, the reduced bispectrum scales as $$b_{\ell\ell\ell} \sim \ell^{-4}\,8 \times 10^{-18}|g_{tss}|$$.
• In the stochastic gravitational-wave background search, distinct overlap reduction functions probe the contribution of tensor, vector, and scalar polarizations. Non-detections set direct upper limits: $\Omega_0^T < 5.6 \times 10^{-8}$, $\Omega_0^V < 6.4 \times 10^{-8}$, $\Omega_0^S < 1.1 \times 10^{-7}$ at 25 Hz (Collaboration et al., 2018).
• For turbulence-driven GW sources, the SVT content of the source stress tensor determines GW production efficiency. Acoustic turbulence (dominant in S and V modes) can out-produce GW amplitude relative to vortical or magnetic turbulence despite small T-mode fraction, due to temporal coherence of acoustic modes matched to GW phase velocity (Brandenburg et al., 2021).

7. Quantum and Nonlinear Extensions

In quantum gravity models such as loop quantum cosmology, holonomy corrections introduce a unified quantum parameter $\Omega$, modifying the Mukhanov–Sasaki equations for all scalar, vector, and tensor perturbations. The quantum-corrected algebra of constraints is anomaly-free only if the deformation is identical across all SVT sectors (Cailleteau et al., 2012).

Nonlinear effects (mode coupling) and higher-derivative corrections (Heisenberg–Euler, parity-violating terms) can generate or mix SVT modes, with physical implications ranging from B-mode polarization features to modified GW waveforms and energy spectra (Dolgov et al., 2023, Brandenburg et al., 2021).

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Scalar, vector, and tensor modes thus serve as the backbone of perturbative gravitational theory, encode the physical content of CMB and GW observables, and provide a rigorous harmonic framework for the analysis of diverse theories and phenomena in gravitational physics and cosmology. The precise identification, decomposition, and dynamical evolution of these modes—along with their allowed mixings and observational signatures—continue to guide both fundamental theoretical development and high-precision experimental cosmology.