The observer power spectrum for lightcone statistics, integrated relativistic observables and wide angle effects

Technical Summary: The Observer Power Spectrum for Lightcone Statistics

Problem Statement
The statistical description of large-scale structure is conventionally performed using power spectra in Fourier space. On spatial hypersurfaces at fixed time, statistical homogeneity ensures that Fourier modes with different wavevectors are uncorrelated, resulting in a diagonal power spectrum $P(k)$. However, cosmological observables are measured on the observer's past lightcone, not on a single spatial slice. This introduces three complications that break translational invariance in the standard source-space Fourier transform:

1. Radial Evolution: Growth factors, bias, and selection functions vary with distance.
2. Wide-Angle Geometry: Galaxies in a pair possess different lines of sight.
3. Integrated Effects: Phenomena like lensing magnification depend on the gravitational potential along the entire photon path, not just at the source.

Consequently, standard Fourier transforms over source positions on a lightcone yield non-diagonal spectra where different modes are correlated (mode-mixing). This necessitates the use of various distinct two-point statistics (e.g., mixed-space spectra $\bar{P}(k, d)$, non-diagonal lightcone spectra $\tilde{P}(k_1, k_2)$, or angular spectra $C_\ell$), each representing a different projection of the same underlying lightcone correlations. The paper identifies a lack of a unified framework that separates the intrinsic statistics of the observable from the geometric projection imposed by the lightcone and specific estimators.

Methodology: The Observer Power Spectrum
The authors propose a new definition of the power spectrum, termed the Observer Power Spectrum $P^{(O)}$, which restores diagonal structure by changing the variable of Fourier transformation.

• Definition: Instead of transforming over source positions $x_s$ on a single lightcone, the authors define the Observer Fourier Transform (OFT) by integrating over observer positions $x_0$ on a spatial hypersurface at fixed conformal time $\eta_0$, while holding the light-ray coordinates (radial distance $\chi$ and direction $\hat{n}$) fixed.
  $$X^{(O)}(k; \chi, \hat{n}) \equiv \int d^3x_0 e^{-i k \cdot x_0} X(x_0; \chi, \hat{n})$$
• Diagonalization: Because the integration domain is a full spatial hypersurface, statistical homogeneity is preserved by construction. This guarantees that the two-point correlation of the OFT modes is diagonal in $k$:
  $$\langle X^{(O)}(k_1; \chi_1, \hat{n}_1) X^{(O)}(k_2; \chi_2, \hat{n}_2) \rangle = (2\pi)^3 \delta_D^{(3)}(k_1 + k_2) P^{(O)}(k_1; \chi_1, \chi_2, \hat{n}_1, \hat{n}_2)$$
• Structure:
  • For local fields (e.g., density), $P^{(O)}$ is the product of the unequal-time spatial power spectrum and a geometric phase factor $e^{ik(\mu_1 \chi_1 - \mu_2 \chi_2)}$ accounting for the spatial offset between observer and source.
  • For integrated fields (e.g., lensing), $P^{(O)}$ factorizes into line-of-sight transfer functions acting independently on each leg, proportional to the intrinsic potential power spectrum $P_\Phi(k)$ at the same $k$. Crucially, this avoids the convolutions over Fourier modes found in source-space transforms.

Key Contributions and Results

1. Unified Framework for Lightcone Statistics: The paper demonstrates that all standard lightcone statistics are merely integral transforms (projections) of the single diagonal object $P^{(O)}$.

   • The lightcone correlation function $\xi_{LC}$ is the momentum-space integral of $P^{(O)}$.
   • The mixed-space spectrum $\bar{P}(k, d)$ and the non-diagonal lightcone spectrum $\tilde{P}(k_1, k_2)$ are recovered by projecting $P^{(O)}$ where the external labels ($\chi, \hat{n}$) become functions of the integration variables, thereby generating the mode-mixing.
   • Angular power spectra $C_\ell$ are obtained by projecting the diagonal multipoles of $P^{(O)}$.
   • The Yamamoto estimator expectation value is shown to be a windowed projection of $P^{(O)}$, valid for both wide-angle and integrated regimes without approximation.

2. Relativistic Galaxy Number Counts: The authors explicitly construct the relativistic kernel for the observed galaxy number count fluctuation $\Delta$, incorporating:

   • Newtonian density and Kaiser redshift-space distortions (RSD).
   • Doppler terms (peculiar velocities).
   • Integrated effects: Lensing magnification, time-delay, and the Integrated Sachs-Wolfe (ISW) effect.
   • They derive the explicit Legendre multipole decomposition of these kernels, showing how the full radial phase $e^{ik\mu\chi}$ leads to contributions across all multipoles $\ell$, rather than a finite truncation found in the plane-parallel limit.

3. Wide-Angle and Odd Parity Effects: The formalism naturally captures wide-angle effects without perturbative expansions in $s/d$.

   • It explicitly shows that odd multipoles (e.g., dipole, octupole) arise from the combination of wide-angle geometry and radial evolution (unequal redshifts), as well as from relativistic Doppler terms.
   • In the plane-parallel limit with equal redshifts, odd multipoles vanish, recovering standard results. However, for unequal redshifts or wide angles, the spectrum acquires a non-vanishing imaginary part, reflecting the breaking of symmetry.

4. Higher-Order Statistics: The construction extends directly to the observer bispectrum and $N$-point functions. Statistical homogeneity ensures a single momentum-conserving delta function for the external wavevectors, with the lightcone geometry encoded in external labels and per-leg kernels.

5. Observer Terms: The paper clarifies the treatment of observer-position contributions (e.g., the observer's velocity in the Doppler term). In the observer spectrum, these appear as boundary terms lacking the oscillatory radial phase $e^{ik\mu\chi}$. In a single realization (a specific survey), these act as coherent global modes (dipoles/monopoles) rather than fluctuating clustering signals, often absorbed into the mean density or subtracted by estimators.

Significance and Claims
The paper claims that the Observer Power Spectrum provides a "common Fourier-space description" of lightcone observables where the role of statistical homogeneity remains explicit. Its primary significance lies in:

• Separation of Physics and Geometry: It separates the intrinsic statistics of the observable from the projection effects (wide-angle, radial evolution, line-of-sight integration) imposed by the lightcone geometry.
• Validation Tool: It offers a natural theoretical object for validating relativistic effects against cosmological simulations (e.g., $N$-body codes). In simulations, one can measure $P^{(O)}$ by correlating lightcone rays from an ensemble of observers without reference to a survey window, before projecting to standard statistics. This avoids the difficulties of validating non-diagonal, mode-mixed spectra.
• Exactness: The definition is exact for the observable under consideration, extending beyond linear theory to nonlinear observables and higher-order statistics without requiring the approximations often necessary in source-space Fourier analyses.

The authors position this work as a foundational step, noting that the next logical extension is to apply this construction to the observer bispectrum and to validate the formalism against lightcone measurements in numerical simulations.