1 Newtonian galaxy clustering: bias as a response and redshift-space distortions
Where we are
The Preface described a chain of maps running from an action to a catalogue. This chapter takes the last two links of that chain and computes them in the approximation where the subtleties of general relativity are invisible: scales well inside the horizon, weak fields, and a distant observer. Nothing here requires the machinery of the two chapters that follow, and that is precisely why it comes first. Working the problem once without covariance shows which of its features are physics and which were assumptions hidden by the notation.
The Newtonian calculation provides the baseline against which the relativistic construction will be compared. It already contains two logically distinct maps. First, gravitational evolution produces a matter field and galaxy formation converts that field into a tracer field. Second, the radial velocity converts the tracer field in real space into a field in redshift space. The first map is encoded by bias coefficients; the second is a coordinate Jacobian. Keeping them separate will make the general-relativistic extension transparent.
Two results carry forward. The first is a definition rather than a formula: galaxy bias is the functional response of a selected population to its long-wavelength environment, and the familiar coefficients \(b_1,b_2,b_{K^2}\) are derivatives of that response. Writing bias this way, rather than as a polynomial ansatz, is what allows the coefficients to survive the move to a relativistic setting, because a response can be defined at fixed local proper time whereas a polynomial in a coordinate-dependent field cannot. The second is the Kaiser formula, which shows that anisotropy in a measured power spectrum can be manufactured entirely by the choice of radial coordinate.
By the end of the chapter we will have an explicit list of what the Newtonian derivation left unstated. That list is the agenda for Chapter 2 and Chapter 3.
Figure placeholder. The two maps of this chapter, drawn as a single left-to-right diagram: matter density on a constant-time slice, then the tracer field produced by the formation response, then the redshift-space field produced by the radial coordinate Jacobian, with the perturbative object responsible for each arrow labelled.
1.1 Matter as a Newtonian fluid
We work in comoving coordinates \(\boldsymbol{x}\), with peculiar velocity \(\boldsymbol{v}=\mathrm d\boldsymbol{x}/\mathrm d\eta\). The density contrast \(\delta=(\rho_m-\bar\rho_m)/\bar\rho_m\) is defined on a constant-conformal-time hypersurface. On scales well inside the horizon and for nonrelativistic matter, the continuity, Euler, and Poisson equations close the evolution of \(\delta\), \(\boldsymbol{v}\), and the Newtonian potential \(\Phi\):
In comoving coordinates, pressureless matter obeys
\[ \begin{aligned} \delta' + \boldsymbol{\nabla}\!\cdot\!\big[(1+\delta)\boldsymbol{v}\big] &= 0,\\ \boldsymbol{v}' + \mathcal H\boldsymbol{v}+ (\boldsymbol{v}\!\cdot\!\boldsymbol{\nabla})\boldsymbol{v}&= -\boldsymbol{\nabla}\Phi,\\ \nabla^2\Phi &= 4\pi G a^2\bar\rho_m\delta =\frac32\mathcal H^2\Omega_m(a)\delta. \end{aligned} \tag{1.1}\]
Here \(\Omega_m=\Omega_m(a)\). The continuity equation is number conservation for matter elements, the Euler equation is their geodesic equation in the weak-field limit, and the Poisson equation relates the local gravitational curvature to the density source. The nonlinear terms transport density and velocity along the flow. Dropping products of perturbations gives a scale-independent growing mode for pressureless matter in standard general relativity. At linear order,
\[ \delta''+\mathcal H\delta'-\frac32\mathcal H^2\Omega_m(a)\delta=0, \qquad \delta(\boldsymbol{k},a)=D(a)\delta(\boldsymbol{k},a_{\mathrm{ini}}), \qquad f\equiv\frac{\mathrm d\ln D}{\mathrm d\ln a}. \tag{1.2}\]
With
\[ \delta(\boldsymbol{x})=\int\frac{\mathrm d^3k}{(2\pi)^3} \mathrm e^{\mathrm i\boldsymbol{k}\cdot\boldsymbol{x}}\delta(\boldsymbol{k}), \]
the linear continuity equation gives
\[ v_i(\boldsymbol{k})=\mathrm i\mathcal Hf\frac{k_i}{k^2}\delta(\boldsymbol{k}), \qquad \theta\equiv\boldsymbol{\nabla}\!\cdot\!\boldsymbol{v}=-\mathcal Hf\delta. \tag{1.3}\]
The velocity relation follows from \(\delta'+\theta=0\) and \(\delta'=\mathcal Hf\delta\). It contains one inverse power of \(k\) relative to the density. Consequently a line-of-sight velocity itself is important on large scales, while a velocity gradient is of the same perturbative order as the density. This distinction will reappear when relativistic redshift terms are organized by powers of \(\mathcal H/k\). The equal-time matter power spectrum is
\[ \left\langle \delta(\boldsymbol{k})\delta(\boldsymbol{k}')\right\rangle =(2\pi)^3\delta_D(\boldsymbol{k}+\boldsymbol{k}')P_{mm}(k). \tag{1.4}\]
This is a theoretical field on a chosen time slice. A galaxy catalogue measures neither this field nor this time slice directly.
The power spectrum in Eq. 1.4 is therefore not yet an observable galaxy power spectrum. Even before accounting for light propagation, the objects in a catalogue are selected tracers rather than randomly chosen matter particles. Their abundance can retain information about the long-wavelength environment in which they formed.
1.2 The tracer abundance as a functional of long-wavelength fields
Let \(n_g(\boldsymbol{x},\eta)\) be the physical number density of a chosen tracer sample and let \(\bar n_g(\eta)\) be its ensemble mean. We define
\[ 1+\delta_g(\boldsymbol{x},\eta) \equiv\frac{n_g(\boldsymbol{x},\eta)}{\bar n_g(\eta)}. \tag{1.5}\]
The definition of the sample is part of the definition of \(n_g\): mass threshold, luminosity cut, color selection, and any other source property must be held fixed when a response is taken. Two samples occupying the same matter distribution need not have the same bias coefficients. The statement \(\delta_g=b_1\delta\) is not a microscopic identity. It is the leading term in the response of the statistically averaged tracer abundance to a slowly varying gravitational environment.
It is tempting to read \(\delta_g=b_1\delta+\tfrac12b_2\delta^2+\cdots\) as a Taylor series of one field in another, fitted after the fact. It is not. Each coefficient is a derivative of the conditional mean abundance with respect to a long-wavelength gravitational operator, holding the physical definition of the sample fixed. The distinction has consequences that appear immediately: a response can be evaluated in any frame in which the long-wavelength environment is specified, so it survives the passage to a relativistic calculation, whereas a polynomial in a coordinate-dependent density does not. It also tells us which coefficients can exist at all, since the allowed operators are fixed by symmetry rather than by convenience.
Split the initial conditions, or equivalently the evolved fields, into long and short modes. Denote by \(\mathcal O_L^A\) a complete set of long-wavelength local gravitational observables; the label \(A\) distinguishes density, tidal tensors, their convective derivatives, and higher-derivative operators. Holding the long fields fixed and averaging over short modes defines a conditional mean
\[ \bar n_g[\mathcal O_L](x) \equiv \left\langle n_g(x)\right\rangle_{S\,|\,\mathcal O_L}. \tag{1.6}\]
The conditional average does not assume that individual galaxies form deterministically. It integrates over the short modes and over unresolved formation histories while retaining the specified long-wavelength background. The result is the part of the abundance that is predictable from the long fields. Fluctuations around this conditional mean are stochastic from the viewpoint of the long-wavelength theory. The deterministic tracer response is the functional
\[ \delta_g^{\mathrm{det}}[\mathcal O_L](x) \equiv\frac{\bar n_g[\mathcal O_L](x)-\bar n_g(x^0)}{\bar n_g(x^0)}. \tag{1.7}\]
The residual
\[ \epsilon(x)\equiv\delta_g(x)-\delta_g^{\mathrm{det}}[\mathcal O_L](x) \tag{1.8}\]
is stochastic with respect to the long fields. Its correlations are analytic in \(k^2\) on scales much larger than the tracer formation scale, subject to number- or momentum-conservation constraints appropriate to the tracer.
The functional Taylor coefficients are the response kernels
\[ \begin{aligned} \mathcal R^{(N)}_{A_1\cdots A_N} (x;x_1,\ldots,x_N) \equiv{}& \frac{1}{\bar n_g(x^0)} \left. \frac{\delta^N\bar n_g[\mathcal O_L](x)} {\delta\mathcal O_L^{A_1}(x_1)\cdots \delta\mathcal O_L^{A_N}(x_N)} \right|_{\mathcal O_L=0}. \end{aligned} \tag{1.9}\]
The kernel of order \(N\) answers a specific question: if the long-wavelength operator \(\mathcal O_L^{A_1}\) is perturbed at the spacetime point \(x_1\), and \(\mathcal O_L^{A_2}\) at \(x_2\), and so on, by how much does the mean abundance at \(x\) change? Because the perturbation is applied at arbitrary points, the kernel is a distribution over the whole past of \(x\) rather than a number. Appendix A records the alternative normalization in which the derivatives are taken of \(\ln\bar n_g\) instead, which agrees with this one at first order and differs beyond it. Assembling the kernels into a functional Taylor series gives
\[ \begin{aligned} \delta_g^{\mathrm{det}}(x) ={}&\sum_{N=1}^{\infty}\frac{1}{N!} \int\mathrm d^4x_1\cdots\mathrm d^4x_N\, \mathcal R^{(N)}_{A_1\cdots A_N} (x;x_1,\ldots,x_N) \prod_{a=1}^{N}\mathcal O_L^{A_a}(x_a). \end{aligned} \tag{1.10}\]
Eq. 1.9 is a definition. The bias expansion follows after imposing causality, the equivalence principle, rotational invariance, and locality on the response kernels.
At this stage the response kernel is allowed to connect the tracer at \(x\) to long fields at arbitrary spacetime points. The functional Taylor series is therefore more general than a local polynomial such as \(b_1\delta+b_2\delta^2/2\). The familiar operator expansion is obtained only after the physical support and tensor structure of the kernels have been restricted. The normalization by \(\bar n_g\) makes each kernel a fractional abundance response.
1.2.1 Causality, spatial locality, and time nonlocality
Galaxy formation has a finite spatial range and a finite duration. These two facts lead to different approximations. For wavelengths much longer than the formation scale \(R_*\), the spatial dependence can be expanded in derivatives. The formation time, however, need not be short compared with a Hubble time, so there is no analogous reason to replace the entire temporal history by an instantaneous response.
The galaxy observed at \((\boldsymbol{x},\eta)\) is built from matter elements whose trajectory is
\[ \boldsymbol{x}_{\mathrm{fl}}(\eta';\boldsymbol{x},\eta) =\boldsymbol{x}-\int_{\eta'}^{\eta}\mathrm d\eta''\, \boldsymbol{v}\bigl(\boldsymbol{x}_{\mathrm{fl}}(\eta''),\eta''\bigr). \tag{1.11}\]
The spacetime picture of Sec. 2.5 of1 is local over a spatial scale \(R_*\) but generally nonlocal over a formation time of order the Hubble time. At leading order in the spatial derivative expansion, the support of the response kernel lies on the past fluid trajectory:
\[ \begin{aligned} \mathcal R^{(N)}_{A_1\cdots A_N} (x;x_1,\ldots,x_N) \simeq{}& \prod_{a=1}^{N} \left[ \Theta(\eta-\eta_a) \delta_D^{(3)}\!\left( \boldsymbol{x}_a-\boldsymbol{x}_{\mathrm{fl}}(\eta_a;\boldsymbol{x},\eta) \right) \right]\\ &\times b_{A_1\cdots A_N} (\eta;\eta_1,\ldots,\eta_N) +\mathcal O(R_*^2\nabla^2). \end{aligned} \tag{1.12}\]
The spatial Dirac distributions in Eq. 1.12 are the leading term of the long-wavelength expansion; they do not state that the microscopic formation process occurs at a mathematical point. They state that, after coarse graining, all long fields are evaluated along the worldline of the same matter element. The remaining kernel \(b_{A_1\cdots A_N}(\eta;\eta_1,\ldots,\eta_N)\) retains the memory of when those fields acted. Substitution into Eq. 1.10 yields the explicitly time-nonlocal form
\[ \begin{aligned} \delta_g^{\mathrm{det}}(\boldsymbol{x},\eta) ={}&\sum_{N=1}^{\infty}\frac{1}{N!} \int^{\eta}\mathrm d\eta_1\cdots\mathrm d\eta_N\, b_{A_1\cdots A_N}(\eta;\eta_1,\ldots,\eta_N)\\ &\hspace{3em}\times \prod_{a=1}^{N} \mathcal O_L^{A_a} \bigl(\boldsymbol{x}_{\mathrm{fl}}(\eta_a),\eta_a\bigr) +\mathcal O(R_*^2\nabla^2). \end{aligned} \tag{1.13}\]
This equation is local in the sense relevant to the equivalence principle: the long fields are evaluated in the neighborhood of the same freely falling fluid element. It is not local in time.
Evaluating the operators on the fluid trajectory is also required by advection. A galaxy observed at \(\boldsymbol{x}\) did not in general occupy the same comoving coordinate at earlier times. Replacing \(\boldsymbol{x}_{\mathrm{fl}}(\eta_a)\) by the final Eulerian position would omit the transport of the long-wavelength environment and would generate an incomplete operator basis at nonlinear order.
The first spatial correction is generated by expanding the response around the trajectory. Isotropy eliminates a one-derivative scalar, so the leading scalar correction has the form
\[ \delta_g\supset b_{\nabla^2\delta} \frac{\nabla^2\delta}{k_*^2}, \qquad k_*\sim R_*^{-1}. \tag{1.14}\]
In Fourier space this gives an analytic correction proportional to \(k^2/k_*^2\).
Higher-derivative bias is therefore controlled by the ratio of the observed wavelength to the physical size of the formation region. It is distinct from loop corrections generated by nonlinear evolution: both can scale as powers of \(k\), but they represent different short-distance information and carry independent coefficients.
1.2.2 The homogeneous response and the usual bias parameters
A spatially homogeneous long density perturbation \(\Delta\) is the simplest background. The conditional mean abundance is then an ordinary function \(\bar n_g(\Delta,\eta)\), and Eq. 1.9 reduces to
\[ \boxed{ b_N(\eta) \equiv \frac{1}{\bar n_g(0,\eta)} \left. \frac{\partial^N\bar n_g(\Delta,\eta)}{\partial\Delta^N} \right|_{\Delta=0} } \tag{1.15}\]
with
\[ \frac{\bar n_g(\Delta)}{\bar n_g(0)}-1 =b_1\Delta+\frac{b_2}{2}\Delta^2+\cdots. \tag{1.16}\]
This is the separate-universe response definition for isotropic density bias. A tidal bias coefficient is obtained instead from a homogeneous anisotropic background and cannot be inferred from derivatives with respect to the scalar \(\Delta\) alone.
The separate-universe language is useful because a sufficiently long isotropic mode changes the local expansion history rather than producing an appreciable spatial gradient across the formation region. Measuring how the abundance changes in that modified background gives the same large-scale coefficient that appears in correlation functions, provided the response is defined at fixed local proper time and with the same physical sample selection.
The derivative in Eq. 1.15 is taken at fixed physical definition of the sample and at fixed local proper time. Holding coordinate time, observed redshift, or global scale factor fixed gives a different derivative. This distinction is hidden in Newtonian notation and will become explicit in Lectures 2–3.
1.2.3 Local gravitational operators
In a freely falling frame a constant potential and a uniform acceleration can be removed. That statement is the Newtonian shadow of the equivalence principle, and Chapter 4 will derive its exact form by constructing coordinates in which the connection vanishes along a whole worldline. What survives the removal is the second derivative of the potential, so to leading order in derivatives the local gravitational data are contained in the Hessian,
\[ \partial_i\partial_j\Phi =\frac13\delta_{ij}\nabla^2\Phi +\left(\partial_i\partial_j-\frac13\delta_{ij}\nabla^2\right)\Phi. \tag{1.17}\]
Using Poisson’s equation, its trace is the density. Define the dimensionless trace-free tidal field
\[ K_{ij} \equiv \left( \frac{\partial_i\partial_j}{\nabla^2} -\frac13\delta_{ij} \right)\delta, \qquad K^i{}_i=0. \tag{1.18}\]
A scalar tracer abundance cannot respond linearly to \(K_{ij}\) in an isotropic ensemble. The first scalar tidal invariant is \(K_{ij}K^{ij}\). A shape tensor can respond linearly to \(K_{ij}\); this is the response-theory origin of intrinsic alignment developed in Chapter 5.
The restriction is a statement about representations of the rotation group. A scalar abundance has no free spatial index with which to contract one trace-free rank-two tensor, whereas a shape observable already transforms as such a tensor. The same long-wavelength tide can therefore enter scalar clustering only at quadratic order but enter intrinsic shape at linear order.
Perturbative time evolution allows the time integrals in Eq. 1.13 to be reorganized into a basis of operators evaluated at the final time. Through second order and at leading derivative order,
\[ \delta_g =b_1\delta +\frac{b_2}{2}[\delta^2] +b_{K^2}[K_{ij}K^{ij}] +b_{\nabla^2\delta}\frac{\nabla^2\delta}{k_*^2} +\epsilon+\cdots. \tag{1.19}\]
The brackets denote renormalized composite operators. Operationally, the renormalized coefficients are the large-scale responses defined above; they do not depend on an arbitrary smoothing scale used to construct the bare operators.
For example, the bare field \(\delta^2\) contains a nonzero mean and short-mode contributions that can mimic lower-order operators. Renormalization subtracts those pieces so that a coefficient such as \(b_2\) measures a response to a long background rather than an artifact of how the density was smoothed. This is why the response definition and the operator basis should be specified together.
At tree level and for \(k\ll k_*\),
\[ \begin{aligned} P_{gm}(k)&=b_1P_{mm}(k)+\mathcal O(k^2P_{mm}),\\ P_{gg}(k)&=b_1^2P_{mm}(k)+P_{\epsilon\epsilon}^{\{0\}} +\mathcal O(k^2P_{mm},k^2P_{\epsilon\epsilon}). \end{aligned} \tag{1.20}\]
The response coefficient \(b_1\) is a property of the selected population, not a prediction of gravity alone.
The constant term \(P_{\epsilon\epsilon}^{\{0\}}\) is the leading large-scale stochastic contribution. It need not equal the Poisson value \(1/\bar n_g\): exclusion, satellite occupation, conservation laws, and sample construction can change it. The tracer–matter cross spectrum does not contain this leading white auto-power; under the usual large-scale constraints on stochastic cross-correlations, it isolates the linear response \(b_1\) more directly than the tracer auto-spectrum.
The functional derivative specifies what a bias parameter means. The local operator expansion specifies how the response can appear in long-wavelength correlation functions. The two statements are related but not interchangeable: a list of allowed operators without a response definition leaves the coefficients scheme-dependent, while a response to only a homogeneous density does not determine tidal or higher-derivative bias.
1.3 The real-to-redshift-space map
A spectroscopic catalogue labels a source by its observed redshift rather than by its real-space radial coordinate. In the Newtonian approximation the peculiar velocity is the only perturbation to this label. Converting the Doppler shift into a radial distance with the background relation produces a coordinate map; number conservation then supplies its Jacobian.
The linear result derived in this section is due to2, who pointed out that the coherent infall of matter towards overdense regions compresses structures along the line of sight and therefore amplifies the measured clustering in a way that depends on orientation. The observation turned what had looked like a nuisance into a measurement: the amplitude of the anisotropy is set by the growth rate \(f\), so redshift-space distortion became one of the standard probes of the growth of structure. The relativistic completion in Chapter 3 shows that the Kaiser term is one entry in a longer list, and identifies which of the remaining terms are suppressed on subhorizon scales.
Adopt the distant-observer approximation with a fixed line of sight \(\hat{\boldsymbol{n}}\). To leading Newtonian order, the Doppler perturbation to redshift is interpreted through the background distance-redshift relation as the map
\[ \widetilde{\boldsymbol{x}}=\boldsymbol{x}+u_\parallel(\boldsymbol{x})\hat{\boldsymbol{n}}, \qquad u_\parallel\equiv\frac{v_\parallel}{\mathcal H}, \qquad v_\parallel\equiv\hat{\boldsymbol{n}}\cdot\boldsymbol{v}. \tag{1.21}\]
The Jacobian matrix is
\[ \frac{\partial\widetilde x^i}{\partial x^j} =\delta^i{}_j+\hat n^i\partial_j u_\parallel. \tag{1.22}\]
The perturbation to the identity is a rank-one matrix, the outer product of the fixed line of sight with the gradient of the radial displacement. This is a direct consequence of the plane-parallel approximation: only the radial coordinate is relabelled, so only one row of the deformation is populated. The determinant of such a matrix is elementary. The matrix-determinant lemma, \(\det(I+\boldsymbol{a}\boldsymbol{b}^{\mathsf T})=1+\boldsymbol{b}\!\cdot\!\boldsymbol{a}\), gives
\[ J_s\equiv\det\left(\frac{\partial\widetilde x^i}{\partial x^j}\right) =1+\partial_\parallel u_\parallel =1+\frac{1}{\mathcal H}\partial_\parallel v_\parallel. \tag{1.23}\]
Number conservation in a one-to-one region of the map implies
\[ [1+\delta_g^s(\widetilde{\boldsymbol{x}})]\,\mathrm d^3\widetilde x =[1+\delta_g(\boldsymbol{x})]\,\mathrm d^3x, \tag{1.24}\]
and therefore
\[ 1+\delta_g^s(\widetilde{\boldsymbol{x}}) =\frac{1+\delta_g(\boldsymbol{x})} {1+\mathcal H^{-1}\partial_\parallel v_\parallel(\boldsymbol{x})}. \tag{1.25}\]
Even in the plane-parallel approximation, Eq. 1.25 contains two nonlinear effects: the determinant in the denominator and the fact that the real-space fields must be evaluated at the inverse-mapped position \(\boldsymbol{x}(\widetilde{\boldsymbol{x}})\). Linear theory keeps only the first variation of each factor and may set the two positions equal inside a first-order field. The argument shift \(\boldsymbol{x}(\widetilde{\boldsymbol{x}})\) is essential beyond linear order. At linear order,
\[ \delta_g^s =\delta_g-\frac{1}{\mathcal H}\partial_\parallel v_\parallel. \tag{1.26}\]
Using Eq. 1.3,
\[ \partial_\parallel v_\parallel(\boldsymbol{k}) =-\mathcal Hf\mu^2\delta(\boldsymbol{k}), \qquad \mu\equiv\hat{\boldsymbol{k}}\cdot\hat{\boldsymbol{n}}. \tag{1.27}\]
At leading bias order,
\[ \boxed{ \delta_g^s(\boldsymbol{k}) =\bigl(b_1+f\mu^2\bigr)\delta(\boldsymbol{k}) } \tag{1.28}\]
and
\[ P_{gg}^s(k,\mu)=\bigl(b_1+f\mu^2\bigr)^2P_L(k). \tag{1.29}\]
The \(\mu\) dependence is not an intrinsic anisotropy of the underlying matter statistics. It is generated by using the observer’s line of sight to define the radial coordinate. Modes parallel to the line of sight receive the full velocity-gradient contribution; transverse modes do not. It is conventional to compress this angular dependence into Legendre multipoles,
\[ P_\ell(k)=\frac{2\ell+1}{2} \int_{-1}^{1}\mathrm d\mu\, P_{gg}^s(k,\mu)\,\mathcal L_\ell(\mu), \tag{1.30}\]
where \(\mathcal L_\ell\) is a Legendre polynomial. At linear order only \(\ell=0,2,4\) are nonzero: The even multipoles are
\[ \begin{aligned} P_0&=\left(b_1^2+\frac23b_1f+\frac15f^2\right)P_L, \\ P_2&=\left(\frac43b_1f+\frac47f^2\right)P_L, \\ P_4&=\frac{8}{35}f^2P_L. \end{aligned} \tag{1.31}\]
The monopole mixes the overall clustering amplitude with growth, while the quadrupole and hexadecapole isolate the line-of-sight velocity response. In practice nonlinear velocities, wide-angle geometry, and relativistic light-cone terms modify this simple pattern, but the Kaiser result identifies the dominant short-distance limit.
1.4 What the Newtonian formula has suppressed
The two terms in Eq. 1.26 already have different origins:
\[ \delta_m \xrightarrow{\text{formation response}} \delta_g \xrightarrow{\text{radial coordinate Jacobian}} \delta_g^s. \tag{1.32}\]
The Newtonian calculation leaves implicit
the time hypersurface on which the functional derivative defining \(b_1\) is taken;
that a real catalogue is a flux of galaxy worldlines through the observer’s past light cone, not a density on one Euclidean time slice;
gravitational redshift, transverse deflection, area distortion, and observer terms;
the difference between constant coordinate time, constant observed redshift, and constant source proper time.
The next lecture makes the slicing problem explicit, derives the redshift displacement and the cosmic clock, and replaces the Euclidean number-conservation statement Eq. 1.24 by a covariant pullback of the galaxy current three-form. Chapter 3 then supplies the spatial ruler determinant and assembles the temporal and spatial pieces into the observed density.
The Newtonian formula is therefore not discarded. It will be recovered by taking the subhorizon, weak-field, plane-parallel limit of the covariant observable. The purpose of the relativistic treatment is to state precisely which source density and which volume Jacobian enter before that limit is taken.
Summary
Key ideas. Galaxy bias is the functional response of a selected tracer population to its long-wavelength environment, Eq. 1.9, not a polynomial fitted to the matter field. Two consequences follow. The allowed operators are fixed by symmetry, causality, and the equivalence principle rather than by convenience, which is why a scalar abundance can respond to \(K_{ij}K^{ij}\) but not to \(K_{ij}\). And the response is local in space over the formation scale \(R_*\) while remaining nonlocal in time over a Hubble time, so the operators must be evaluated along the past fluid trajectory. Separately, redshift-space anisotropy is not a property of the matter statistics; it is manufactured by using the observer’s line of sight to define the radial coordinate.
Main results.
- The functional Taylor expansion Eq. 1.10 and its local, time-nonlocal reduction Eq. 1.13.
- The separate-universe definition of the density bias coefficients, Eq. 1.15, taken at fixed local proper time and fixed physical sample definition.
- The renormalized operator basis Eq. 1.19, and the tree-level spectra Eq. 1.20.
- The Kaiser field and power spectrum, Eq. 1.28 and Eq. 1.29, with multipoles Eq. 1.31.
What the next chapter builds on. The Newtonian derivation left four things unstated, listed above Eq. 1.32: the time hypersurface on which the response is defined, the fact that a catalogue is a flux through the past light cone rather than a density on a Euclidean slice, the omission of gravitational redshift and propagation effects, and the difference between coordinate time, observed redshift, and source proper time. Chapter 2 makes the slicing problem explicit, derives the redshift displacement and the cosmic clock, and replaces Eq. 1.24 by the pullback of a covariant current three-form. Chapter 3 then supplies the spatial ruler determinant and assembles the temporal and spatial pieces into the observed density.