Notation and conventions
This chapter collects the assumptions and symbols used throughout these notes. It is reference material; nothing here is derived, and a reader starting from 1 Newtonian galaxy clustering: bias as a response and redshift-space distortions can return to it as needed.
Standing assumptions
Unless stated otherwise:
the background is a spatially flat FLRW spacetime;
matter is pressureless and follows geodesics;
metric and light-cone observables are treated to linear order in 2 Gauge freedom and covariant number counts and 3 Cosmic rulers, cosmic clocks, and the observed density;
the bias discussion assumes Gaussian, adiabatic initial conditions and general relativity;
the metric signature is \((-,+,+,+)\);
\(\eta\) is conformal time, a prime is \(\partial_\eta\), \(\mathcal H=a'/a\), and \(H=\mathcal H/a\);
a tilde denotes a quantity inferred directly from observed redshift and direction using the background relation, for example \(\widetilde a=(1+\widetilde z)^{-1}\) and \(\widetilde{\boldsymbol{x}}=\widetilde\chi\hat{\boldsymbol{n}}\); it never denotes a gauge-transformed field, for which we write \(Q\mapsto Q+\Delta_\xi Q\);
\(\delta_g\) denotes a rest-frame galaxy density perturbation, while \(\Delta_g\) or \(\delta_g^{\mathrm{obs}}\) denotes an observed number-count perturbation.
We use \(\bar n_g\) for the mean physical number density and define
\[ b_e \equiv \frac{\mathrm d\ln(a^3\bar n_g)}{\mathrm d\ln a}. \]
Thus \(b_e=0\) for a conserved comoving population.
Two conventions that are easy to trip over
The conformal-Newtonian metric in Eq. 2.12 is written with \(g_{ij}=a^2(1+2\Phi)\delta_{ij}\). This is the opposite sign to the widespread convention \(g_{ij}=a^2(1-2\Phi_{\mathrm{common}})\delta_{ij}\). The local weak-field metric of Eq. 5.1 uses the common local convention instead, because that subsection works in local physical coordinates rather than in the conformal chart. Both choices are stated where they are used.
The Riemann convention is fixed in Eq. 4.1. The Manasse–Misner paper places the first covariant index before the contravariant one, which flips the sign of the all-lowered tensor relative to ours. Appendix D: Manasse–Misner index order and the quadratic algebra carries out the translation explicitly.
Symbols
The tables below list where each symbol is defined rather than restating its meaning in full.
Background and kinematics
| Symbol | Meaning |
|---|---|
| \(a\), \(\eta\), \(\tau\) | scale factor, conformal time, proper time |
| \(\mathcal H\), \(H\) | conformal and physical Hubble rates, \(H=\mathcal H/a\) |
| \(\chi\), \(\widetilde\chi\) | comoving distance, and the distance inferred from \(\widetilde z\) |
| \(D(a)\), \(f\) | linear growth factor and growth rate, Eq. 1.2 |
| \(\Omega_m(a)\) | matter density parameter at scale factor \(a\) |
| \(\boldsymbol{v}\), \(\theta\) | peculiar velocity and its divergence, Eq. 1.3 |
| \(\hat{\boldsymbol{n}}\), \(\mu\) | observed line of sight, and \(\mu=\hat{\boldsymbol{k}}\cdot\hat{\boldsymbol{n}}\) |
Metric perturbations and gauge
| Symbol | Meaning |
|---|---|
| \(A\), \(B\), \(D\), \(E\) | scalar metric perturbations, Eq. 2.7 and Eq. 2.8 |
| \(\Psi\), \(\Phi\) | conformal-Newtonian potentials, Eq. 2.12 |
| \(h_{ij}\) | spatial metric perturbation; \(h_\parallel=\hat n^i\hat n^jh_{ij}\) |
| \(\xi^\mu=(T,\partial^iL)\) | scalar gauge generator, Eq. 2.2 |
| \(\Delta x^\mu\) | displacement of the true source event from the inferred one, Eq. 2.22 |
| \(\Delta\ln a\) | scale-factor perturbation at fixed observed redshift, Eq. 2.24 |
| \(I_A\) | integrated lapse along the source worldline, Eq. 2.31 |
Appendix C: Gauge-transformation table tabulates how each of these transforms.
Tracers and bias
| Symbol | Meaning |
|---|---|
| \(n_g\), \(\bar n_g\) | physical number density of the sample and its mean |
| \(\delta_g\), \(\delta_g^{\mathrm{det}}\), \(\epsilon\) | tracer perturbation, its deterministic part, and the stochastic residual, Eq. 1.5 to Eq. 1.8 |
| \(\mathcal O_L^A\) | basis of long-wavelength local gravitational operators |
| \(\mathcal R^{(N)}\) | functional response kernel, Eq. 1.9 |
| \(b_1\), \(b_2\), \(b_{K^2}\), \(b_{\nabla^2\delta}\) | bias coefficients, Eq. 1.19 |
| \(b_e\) | evolution bias, defined above |
| \(s\) | magnification slope of a threshold sample, Eq. 3.41 |
| \(R_*\), \(k_*\) | formation scale and its inverse, Eq. 1.14 |
| \(K_{ij}\) | dimensionless trace-free tidal field, Eq. 1.18 |
Response normalizations are compared in Appendix A: Response conventions and functional derivatives.
Counts on the light cone
| Symbol | Meaning |
|---|---|
| \(j^\mu\), \(u_g^\mu\) | galaxy number current and four-velocity, Eq. 2.45 |
| \(\boldsymbol{\varepsilon}\) | spacetime volume form, Eq. 2.44 |
| \(\boldsymbol{\mathcal J}\) | current three-form, Eq. 2.46 |
| \(\mathsf X\), \(\mathsf X^*\) | light-cone embedding and its pullback, Eq. 2.49 |
| \(\widetilde{\boldsymbol{\omega}}\) | apparent volume form of an observed cell, Eq. 2.66 |
| \(\delta_g^{\mathrm{obs}}\) | observed number-count perturbation, Eq. 2.69 |
| \(\delta_g^{\mathrm{or}}\), \(\delta_g^{\mathrm{pt}}\) | source density at the observed-redshift and proper-time references, Eq. 3.29 and Eq. 3.35 |
Pullback identities are collected in Appendix B: Pullback and Lie-derivative identities.
Clocks and rulers
| Symbol | Meaning |
|---|---|
| \(\mathcal T\) | cosmic clock, Eq. 2.37 |
| \(P_{ij}\) | screen projector, Eq. 3.1 |
| \(\mathcal C\) | longitudinal ruler distortion, Eq. 3.16 |
| \(\mathcal B_i\) | mixed ruler distortion, Eq. 3.17 |
| \(\mathcal A_{ij}\) | transverse ruler distortion, Eq. 3.18 |
| \(\mathcal M\) | magnification, the trace of \(\mathcal A_{ij}\), Eq. 3.22 |
| \(\gamma_{ij}^{\mathrm{ruler}}\) | ruler shear, Eq. 3.23 |
| \(\widehat\kappa\) | coordinate convergence, Eq. 3.21 |
Local frames and tides
| Symbol | Meaning |
|---|---|
| \(e_{\hat a}\) | parallel-transported orthonormal tetrad, Eq. 4.13 |
| \(X^{\hat a}=(T,X^i)\) | Fermi normal coordinates, Eq. 4.17 |
| \(\mathcal G\) | the central geodesic |
| \(R^\rho{}_{\sigma\mu\nu}\) | Riemann tensor in the convention of Eq. 4.1 |
| \(\mathcal R_{ij}\) | electric part of the Riemann tensor, Eq. 4.52 |
| \(\mathcal E_{ij}^S\), \(\mathcal E_{ij}^{\mathrm{GW}}\) | scalar and tensor trace-free tides, Eq. 5.7 and Eq. 5.23 |
| \(S_{ij}^I\), \(\gamma_{ab}^I\) | intrinsic shape and its screen projection, Eq. 5.12 and Eq. 5.14 |
| \(R_E\) | temporal shape-response kernel, Eq. 5.11 |
| \(h_{ij}^{\mathrm{TT}}\) | transverse-traceless metric perturbation, Eq. 5.15 |
Index conventions
Greek indices \(\mu,\nu,\ldots\) run over spacetime and Latin indices \(i,j,\ldots\) over space. Indices \(a,b\) on screen-projected quantities run over the two directions transverse to the line of sight.
A hat marks a component referred to the orthonormal tetrad carried by a freely falling observer, equivalently to the Fermi normal coordinates built from that tetrad, as used throughout 4 Fermi normal coordinates: the Manasse–Misner construction and 5 Local tides, intrinsic alignment, and gravitational waves. Frame directions are \(\hat a,\hat b,\hat c,\hat d\) for all four and \(\hat 0\) for the time direction, which is never written bare. Latin \(i,j,k,l,m\) are the three spatial frame directions; these are frame indices by definition, so the hat would carry no information and is not written. An unhatted Greek index means an arbitrary chart on spacetime and nothing more, with \(\mu'\) available for a second chart.
The rule has one visible consequence worth anticipating. 5 Local tides, intrinsic alignment, and gravitational waves writes \(R_{0i0j}\) with a bare time index where it works inside the weak-field chart of Eq. 5.1 or the transverse-traceless chart of Eq. 5.15, and \(R_{\hat 0i\hat 0j}\) where the same curvature is referred to the observer’s frame. Reading the hat is enough to tell which is meant.
Angle brackets denote the symmetric trace-free part, \(\mathcal R_{\langle ij\rangle}\) as in Eq. 4.53, and round brackets on a pair of indices denote symmetrization. Square brackets around an operator, as in \([\delta^2]\) of Eq. 1.19, denote a renormalized composite operator rather than a commutator.
A bold symbol is a spatial three-vector, \(\boldsymbol{x}\) or \(\boldsymbol{k}\), while the same letter with an index is its component. A bold upright symbol is a differential form, as in \(\boldsymbol{\mathcal J}\) and \(\boldsymbol{\varepsilon}\).