Anisotropic and Polarized Backgrounds

An anisotropic stochastic background carries a sky map, and a polarized background carries a $2\times2$ coherency matrix at each frequency and direction. Intensity and circular polarization are scalar and pseudoscalar sky fields, while linear polarization depends on the local plus-cross screen and therefore has spin weight four. This page defines that field-level decomposition and its isotropic, unpolarized limit; search pipelines, upper limits, and current detections are excluded.

Helpful background. Einstein Equations supplies tensor radiation. Waves, Green Functions, and Geometric Optics supplies random Fourier fields on the sphere. Gravitational Radiation and Waveforms supplies spin-weight and Stokes conventions. Relativistic Matter and Compact Objects supplies unresolved source populations. Relativistic Cosmology supplies primordial transfer and large-scale anisotropy.

A sky-resolved covariance

Let $\widetilde h_A(f,\boldsymbol n)$ be a positive-frequency plane-wave amplitude associated with propagation direction $\boldsymbol n$. Normalize the sphere delta function by

$$\int d\Omega_{\boldsymbol n}\, \delta^2(\boldsymbol n,\boldsymbol n')=1.$$

For a stationary background that is direction-local at second order, define

$$\left\langle \widetilde h_A(f,\boldsymbol n) \widetilde h_B(f',\boldsymbol n')^* \right\rangle =\frac12\delta(f-f') \delta^2(\boldsymbol n,\boldsymbol n') J_{AB}(f,\boldsymbol n).$$

$J$ is Hermitian and positive semidefinite. More general backgrounds can correlate different directions; then $J$ becomes a two-direction kernel rather than a local sky map. The sky and polarization decompositions are developed in Romano and Cornish 2017, §§7–8, PDF.

Stokes fields on the sphere

Use the same convention as the polarization chapter:

$$J=\frac12 \begin{pmatrix} I+Q&U+iV\\ U-iV&I-Q \end{pmatrix}.$$

At every $(f,\boldsymbol n)$,

$$I\ge0, \qquad I^2\ge Q^2+U^2+V^2.$$

$I$ is directional intensity, $V$ is directional circular polarization in the declared propagation and Fourier convention, and $Q,U$ are linear-polarization coordinates in the local screen. Normalize the total one-sided field spectrum by

$$S_h(f)=\int d\Omega_{\boldsymbol n}\,I(f,\boldsymbol n).$$

Thus the map units are $\mathrm{Hz}^{-1}\,\mathrm{sr}^{-1}$ when solid angle is displayed explicitly.

Screen rotations and spin four

Rotating the polarization screen by $\psi(\boldsymbol n)$ sends

$$I' = I, \qquad V'=V, \qquad (Q+iU)'=e^{4i\psi}(Q+iU).$$

The doubled tensor angle appears twice in the quadratic coherency, producing spin four. Therefore $Q$ and $U$ cannot each be expanded as ordinary scalar maps without fixing a discontinuous global screen. Instead use

$$\begin{aligned} I(f,\boldsymbol n)&=\sum_{\ell m}a^I_{\ell m}(f)Y_{\ell m}(\boldsymbol n),\\ V(f,\boldsymbol n)&=\sum_{\ell m}a^V_{\ell m}(f)Y_{\ell m}(\boldsymbol n),\\ Q+iU&=\sum_{\ell m}a^{+4}_{\ell m}(f) \,{}_{+4}Y_{\ell m}(\boldsymbol n),\\ Q-iU&=\sum_{\ell m}a^{-4}_{\ell m}(f) \,{}_{-4}Y_{\ell m}(\boldsymbol n). \end{aligned}$$

Linear-polarization multipoles begin at $\ell=4$. Combining the two spin sectors yields parity-even $E$-type and parity-odd $B$-type linear polarization, with exact signs fixed by the chosen harmonic convention.

Reality, Hermiticity, and parity checks

Since the four Stokes fields are real, scalar coefficients obey

$$a^X_{\ell,-m}=(-1)^m a^{X*}_{\ell m}, \qquad X\in\{I,V\}.$$

Conjugacy of $Q+iU$ and $Q-iU$, together with

$${}_{s}Y_{\ell m}^* =(-1)^{s+m}{}_{-s}Y_{\ell,-m},$$

gives

$$a^{-4}_{\ell m} =(-1)^m a^{+4*}_{\ell,-m}.$$

These relations halve the apparent complex degrees of freedom. Under spatial parity, intensity and $E$-type linear polarization are even, whereas circular handedness and $B$-type linear polarization are odd, once the antipodal screen convention is fixed. A claimed parity violation must therefore survive a change of screen basis and propagation-direction convention.

Isotropic and unpolarized reduction

An isotropic, unpolarized tensor background has

$$I(f,\boldsymbol n)=\frac{S_h(f)}{4\pi}, \qquad Q=U=V=0.$$

Only the intensity monopole remains:

$$a^I_{00}=\frac{S_h}{\sqrt{4\pi}}.$$

The coherency matrix is proportional to the identity at every direction, so both tensor polarizations carry equal power. Substitution into the detector response recovers the standard isotropic overlap formulation of Allen and Romano 1999, §§II–IV, PDF. Isotropy alone does not imply unpolarized radiation, and Gaussianity is a separate assumption again.

From sky fields to channel correlations

For ideal responses $R_I^A(f,\boldsymbol n)$, define a one-sided signal cross-spectrum by

$$\left\langle\widetilde s_I(f)\widetilde s_J(f')^*\right\rangle =\frac12\delta(f-f')C_{IJ}(f).$$

Substitution of the sky covariance then gives

$$C_{IJ}(f)=\int d\Omega_{\boldsymbol n}\, R_I^A J_{AB}R_J^{B*}.$$

The factor $1/2$ already occurs in the definition of the one-sided covariance; inserting it again in this response integral would halve the predicted channel variance. For $I=J$, positive semidefiniteness of $J_{AB}$ makes $C_{II}\ge0$, which is a useful normalization and conjugation check.

This linear map mixes angular and polarization moments according to baseline geometry and frequency. Some multipoles can be poorly conditioned or exactly invisible to a given network. That is an instrument-identifiability statement, not absence of the sky field. Calibration, noise covariance, estimators, and empirical conclusions belong to Gravitational-Wave Astronomy.

Limitations

Direction-local covariance is not appropriate for every coherent primordial state. A two-point map also does not capture intermittency, non-Gaussian bursts, or higher-order phase correlations. Finite observation time replaces ideal frequency delta functions by spectral windows, and peculiar motion mixes frequency and angular multipoles. Each extension should begin from the covariance definition rather than adjusting a final overlap formula.

Exercises

1. Normalize the isotropic limit. Integrate the isotropic intensity over the sky and compute its monopole coefficient.

Solution

Because the sphere has area $4\pi$,

$$\int d\Omega\,\frac{S_h}{4\pi}=S_h.$$

Using $Y_{00}=1/\sqrt{4\pi}$,

$$a^I_{00}=\int d\Omega\,IY_{00}^* =\frac{S_h}{\sqrt{4\pi}}.$$

All $\ell\gt0$ coefficients vanish by spherical-harmonic orthogonality. With $Q=U=V=0$, $J=(I/2)\mathbf1$, confirming equal plus and cross power.

This reduction checks both normalizations at once: angular integration recovers the total field spectrum, while the matrix trace recovers the directional intensity. No antenna average has entered either identity.

2. Check basis covariance. Show that $Q^2+U^2$ and the positivity bound are unchanged by a screen rotation.

Solution

The complex combination receives only a phase:

$$Q'+iU'=e^{4i\psi}(Q+iU).$$

Taking the modulus gives $Q'^2+U'^2=Q^2+U^2$. Since $I$ and $V$ are also unchanged, both sides of $I^2\ge Q^2+U^2+V^2$ are invariant. Positivity is therefore a property of the physical coherency matrix, not of one screen convention.

References

• Allen, Bruce, and Joseph D. Romano. “Detecting a Stochastic Background of Gravitational Radiation: Signal Processing Strategies and Sensitivities.” Physical Review D 59 (1999): 102001. DOI. Open PDF.
• Romano, Joseph D., and Neil J. Cornish. “Detection Methods for Stochastic Gravitational-Wave Backgrounds: A Unified Treatment.” Living Reviews in Relativity 20 (2017): 2. DOI. Open PDF.