Principal Spectra & Time-Dependent Kernels

• Principal spectra and time-dependent kernels are analytical tools that decompose complex response functions into eigenmodes and dynamical memory components in quantum and cosmological systems.
• The framework employs minimal basis sets and systematic kernel expansion to ensure computational accuracy and compliance with physical and symmetry constraints.
• Applications include predicting optical excitations, electron–hole interactions, and nonlinear structure formation by isolating spectral features in various many-body contexts.

Principal spectra and time-dependent kernels are foundational constructs in the calculation and analysis of response functions in quantum many-body, electronic, and cosmological systems. These concepts provide systematic frameworks to represent complex dynamic interactions—such as optical excitations, electronic correlations, and nonlinear structure formation—by decomposing physical responses into time-dependent basis functions and coupling matrices, capturing both spectral content and dynamical memory effects.

1. Conceptual Foundations: Principal Spectra and Kernel Decomposition

Principal spectra correspond to the eigenvalues and eigenmodes obtained from the spectral decomposition of two-point response functions—such as the current–current response $\chi_{\alpha\beta}(\mathbf r, \mathbf r'; \omega)$ or the density–density response $\chi(x, x', \omega)$—at each frequency $\omega$. The time-dependent kernel (exchange-correlation kernel $f_{xc}$, or generalized kernel in perturbative expansions) encapsulates the system’s many-body correlations and temporal memory, governing the evolution and coupling of principal spectral modes.

In quantum electronic systems, these kernels enter the Dyson-like equations relating noninteracting and interacting response functions: $$\chi = \chi_0 + \chi_0 \left[ u + f_{xc} \right] \chi,$$ where $u$ is the bare Coulomb interaction (Entwistle et al., 2018). In cosmological perturbation theory, perturbative expansions yield density/velocity kernels that factor into time-dependent growth spectra and wavenumber-dependent basis functions (Hartmeier et al., 2023).

2. Spectral Decomposition: Mathematical Structure

The decomposition of response functions into principal spectra is formulated through generalized eigenvalue problems: $$\int d^3 r' \; \chi_{\alpha\beta}(\mathbf r, \mathbf r'; \omega) \; \varphi_{n, \beta}(\mathbf r'; \omega) = \lambda_n(\omega)\; \varphi_{n, \alpha}(\mathbf r; \omega),$$ with $\lambda_n(\omega)$ as the spectral weights (principal spectra) and $\varphi_n$ as spectral modes (Gatti, 2010). For the exchange-correlation kernel, one may perform an analogous expansion: $$f_{xc}(x, x', \omega) = \sum_n \lambda_n(\omega) \psi_n(x) \psi_n^*(x'),$$ where, for frequencies below the first excitation, typically a single term dominates ($n=1$), with higher modes contributing above threshold (Entwistle et al., 2018).

In cosmological frameworks, non-linear kernels at order $n$ are formally written as a sum of terms, each a product of a time-dependent growth factor $d_{n,i}^X(\eta)$—the “principal spectrum” for each term—and an algebraic basis function $h_{n,i}^X(\{\mathbf{k}_i\})$: $$F_n(\{\mathbf{k}_i\};\eta) =\sum_{i=1}^{N_n^F}d_{n,i}^F(\eta)\, h_{n,i}^F(\{\mathbf{k}_i\}),$$ where $\eta = \ln D_1$ is the log-linear growth parameter (Hartmeier et al., 2023).

3. Frequency Dependence and Memory Effects of Kernels

Time-dependent kernels can display complex dynamical properties, including frequency dependence (memory effects) and nonlocal spatial couplings. For frequencies below the first excitation, the exact exchange-correlation kernel $f_{xc}$ in TDDFT shows nearly adiabatic behavior—weak dependence on $\omega$, negligible imaginary part, and a smooth, nonlocal spatial profile resembling the negative Coulomb interaction (Entwistle et al., 2018). At higher frequencies, $f_{xc}$ becomes strongly frequency-dependent, with sharp features in the imaginary part and increased spatial complexity reflecting additional nodal excitations.

In TDCDFT, the tensor kernel $f_{xc,\alpha\beta}$ can be represented in operator form involving functional derivatives of the noninteracting response and screened Coulomb interaction, thus encoding both band-gap opening and long-range excitonic effects (Gatti, 2010): $$f_{xc}^{(2)} = - \chi^{0\,-1} \left( \frac{\delta \chi^0}{\delta j} \right) W \left( \frac{\delta \chi^0}{\delta j} \right) \chi^{0\,-1},$$ With the long-wavelength limit collapsing to a universal form proportional to $-\alpha(\omega)/\omega^2$.

In cosmology, kernels factorize into time-dependent spectra and kinematic basis elements across arbitrary expansion histories, directly capturing redshift-dependent memory behavior (Hartmeier et al., 2023).

4. Minimal Bases and UV-Safe Kernel Construction

A crucial advance in cosmological perturbation theory is the identification of minimal bases—a set of irreducible wavenumber-dependent basis functions $h_{n,i}^X$ coupled to time-dependent growth spectra $d_{n,i}^X(\eta)$—dramatically simplifying numerical evaluation and ensuring explicit momentum-conservation scaling. For example, the reduced basis for the density kernel has $N_2^F=2$, $N_3^F=4$, $N_4^F=11$, $N_5^F=39$ terms at each order. Each basis function individually satisfies the expected scaling with $k, q$ due to momentum conservation, making loop integrals UV-finite term-by-term without large cancellations (Hartmeier et al., 2023).

The table below exemplifies the growth in minimal basis size:

Expansion order $n$	$N^F_n$ (density basis)	$N^G_n$ (velocity basis)
2	2	2
3	4	5
4	11	14
5	39	47

This minimal construction absorbs most UV corrections into effective field theory (EFT) counterterms, with the non-decaying “$\beta$”-functions for the EFT description directly computed from the growth spectra.

5. Role of Kernels in Predicting Principal Spectral Features

Exchange-correlation and time-dependent kernels fundamentally determine principal spectral features such as excitation energies and oscillator strengths. In practice, the screening-dependent hybrid functional (DDH) kernel built using a model dielectric function closely tracks the accuracy of full BSE-GW calculations for optical excitonic peaks (deviations below 0.02 eV in position and 10% in oscillator strength across multiple materials) (Tal et al., 2020):

Material	TD-DDH $E_1$ (eV)	$f_1$ (TD-DDH)	BSE-GW $E_1$ (eV)	$f_1$ (BSE-GW)	$\Delta E$ (meV)	$\Delta f$ (%)
Si	3.385	0.29	3.380	0.28	+5	+4
Diamond	7.061	0.12	7.055	0.11	+6	+9
SiC	5.019	0.15	5.010	0.14	+9	+7
Ar	11.62	0.04	11.60	0.03	+20	+33
NaCl	9.050	0.07	9.035	0.06	+15	+17
MgO	7.500	0.09	7.485	0.08	+15	+13

In TDDFT and TDCDFT, principal spectra in the noninteracting system are modified by the kernel, which is responsible for shifting excitation peak positions, sharpening resonance line shapes, and binding electron–hole pairs. In cosmology, the minimal kernel basis allows analytic EFT matching and precise loop corrections, absorbing up to 90% of deviations from traditional approximations (Hartmeier et al., 2023).

6. Design Principles and Practical Approximations

The design of effective time-dependent kernels is guided by analyzing principal spectra. By projecting the kernel onto dominant principal modes, researchers can systematically identify critical mode–mode couplings, tune long-range parameters, and extend the complexity of the kernel only when additional resonances or memory effects become relevant (Gatti, 2010). For example, in electronic TDDFT, a simple nonlocal kernel resembling $-u(x,x')$ suffices below the first excitation; beyond this regime, additional frequency-dependent corrections may be incorporated in a reduced “principal-component” basis (Entwistle et al., 2018).

In cosmological perturbation theory, the minimal basis choice enforces all required symmetry constraints, simplifies numerical loop evaluation, and ensures that residual “exact-time-dependence” effects remain at the percent level for realistic models (Hartmeier et al., 2023).

7. Outlook: Systematic Kernel Expansion and Spectral Accuracy

Current methodologies enable systematic improvements in kernel construction—either by expanding the principal spectra basis or refining the approximation to the screened interaction and Green’s functions. These approaches guarantee controlled progress toward exact many-body or cosmological response. A plausible implication is that as more principal modes are included, or time-dependent kernel structures are generalized, one can interpolate continuously between simple adiabatic kernels and fully memory-dependent treatments for high-frequency, high-resolution spectra (Gatti, 2010, Entwistle et al., 2018, Hartmeier et al., 2023). This systematic strategy balances spectral accuracy, computational tractability, and physical transparency across diverse domains.