2 Gauge freedom and covariant number counts
Where we are
Chapter 1 ended with a list of things the Newtonian derivation had left unstated. Three of the four concern time: which hypersurface the bias response is defined on, which hypersurface a catalogue bin corresponds to, and how the two differ. The fourth is structural. A catalogue is not a density on a Euclidean time slice; it is a set of worldlines crossing the observer’s past light cone. The Newtonian statement \([1+\delta_g^s(\widetilde{\boldsymbol{x}})]\,\mathrm d^3\widetilde x=[1+\delta_g(\boldsymbol{x})]\,\mathrm d^3x\) presumes both a preferred simultaneity and a nondegenerate three-volume. Neither exists on a null surface.
General relativity introduces coordinate freedom, but the observational problem is not solved by selecting a preferred gauge. The catalogue count can be defined exactly as a geometric flux through the observer’s past light cone. Perturbation theory is then an expansion of this exact object. Gauge-dependent density, redshift, and displacement terms appear because the exact object has been split into background and perturbation, not because the measured count is ambiguous.
The chapter is arranged in two halves, in the order in which a working calculation meets the difficulties rather than in the order in which they are best explained.
The first half is perturbative and concrete. It defines a gauge transformation and shows immediately that a density perturbation has no meaning until a slicing is named. It then works out how the metric perturbations transform, how the observed redshift determines the displacement of the true source event from the fiducial one, and how the proper time elapsed at the source differs from the time the fiducial cosmology assigns to that redshift. The dimensionless field measuring the mismatch is the cosmic clock \(\mathcal T\), and it is gauge invariant. By the end of this half the reader has a working formalism and one nagging observation: several manifestly gauge-dependent quantities have combined into invariant ones, and nothing so far explains why.
The second half explains why. The object that counts worldlines through any three-surface, null or spacelike, is a differential three-form, and integrating it requires no metric on the surface at all. Written that way the count could not have depended on coordinates, and the cancellations of the first half become forced rather than fortunate. Along the way the light-cone geometry is put in a frame adapted to the observation, which separates the radial direction along the ray from the two transverse screen directions and shows in advance which distortions can change a volume.
What this chapter does not supply is the spatial part of the volume distortion itself. That is the business of Chapter 3.
Figure placeholder. Illustration of the pullback. On the left, a bundle of galaxy worldlines piercing a three-surface embedded in spacetime; on the right, the same bundle represented on the observed-coordinate manifold \((\widetilde z,\widetilde\theta,\widetilde\phi)\), with the embedding \(\mathsf X\) drawn as the arrow between them and the three tangent vectors \(\partial X^\mu/\partial y^a\) marked on the surface.
2.1 Gauge transformations and the slicing of a density perturbation
A gauge transformation in cosmological perturbation theory is a relabelling of spacetime events,
\[ x^\mu\mapsto x^\mu+\xi^\mu. \tag{2.1}\]
The transformation is passive. No field is moved and no physical configuration is altered; only the coordinates assigned to events change. Throughout these notes, a tilde is reserved for quantities inferred from the observed labels \((\widetilde z,\hat{\boldsymbol n})\) using the background cosmology. Gauge transformations are therefore written with \(\mapsto\) rather than by decorating the transformed field with another tilde.
In the scalar sector the generator is
\[ \xi^\mu=(T,\partial^iL), \tag{2.2}\]
where \(T\) shifts the time slicing and \(L\) relabels positions within a slice. Almost all of the difficulty addressed in this chapter comes from the first of these.
The reason is that a perturbation is not a property of a single event. Splitting a field into a background and a perturbation requires a background, the background is a function of time alone, and reading off “the time” of an event requires a slicing. Change the slicing and the perturbation changes, even though nothing physical has moved. The density makes this concrete.
The word “density” can obscure the actual comparison. The scalar \(\rho(x)\) at one event is invariant, but a perturbation compares it with a homogeneous reference value \(\bar\rho\) evaluated at a second, background event. A time slicing supplies the rule that pairs these events. Changing the slicing changes the pair and hence changes the perturbation.
The value of a physical scalar at a specified event is unchanged by a coordinate relabelling. The split
\[ \rho(\eta,\boldsymbol{x})=\bar\rho(\eta)+\delta\rho(\eta,\boldsymbol{x}) \tag{2.3}\]
compares the physical density with the background density at the same coordinate time. Under the time shift generated by \(T\), its perturbative representation changes as
\[ \delta\rho\mapsto\delta\rho-\bar\rho'T. \tag{2.4}\]
For pressureless matter, \(\bar\rho_m'=-3\mathcal H\bar\rho_m\), so
\[ \boxed{ \delta_m\mapsto\delta_m+3\mathcal HT }. \tag{2.5}\]
For a tracer with mean physical number density \(\bar n_g\),
\[ \delta_g \mapsto\delta_g-\frac{\bar n_g'}{\bar n_g}T. \tag{2.6}\]
These formulas do not say that density is unphysical. They say that the perturbation requires a rule for identifying simultaneous events.
The required rule depends on the physical question. Galaxy formation is naturally compared at equal local proper time, whereas a catalogue bin is constructed at equal observed redshift. The difference between these two hypersurfaces is measurable and is precisely the cosmic-clock perturbation introduced below.
2.2 Metric perturbations and scalar gauge transformations
To connect the geometric argument with standard cosmological perturbation variables, we now write the most general linear scalar perturbation of a flat FLRW metric. The four scalar functions \(A,B,D,E\) are related by two scalar coordinate freedoms; the Einstein and matter equations further constrain the dynamical combinations. A choice of gauge imposes two coordinate conditions; it does not select a different spacetime.
Write the linearly perturbed flat FLRW metric as
\[ \mathrm ds^2=a^2(\eta) \left[-(1+2A)\mathrm d\eta^2-2B_i\mathrm d\eta\mathrm dx^i +(\delta_{ij}+h_{ij})\mathrm dx^i\mathrm dx^j\right]. \tag{2.7}\]
In the scalar sector,
\[ B_i=\partial_iB, \qquad h_{ij}=2D\delta_{ij}+2\partial_i\partial_jE. \tag{2.8}\]
Under the generator Eq. 2.2, linear metric perturbations obey
\[ \delta g_{\mu\nu} \mapsto\delta g_{\mu\nu}-\mathcal L_\xi\bar g_{\mu\nu}. \tag{2.9}\]
Evaluating the Lie derivative gives
\[ \begin{aligned} A&\mapsto A-\mathcal HT-T',\\ B&\mapsto B+L'-T,\\ D&\mapsto D-\mathcal HT,\\ E&\mapsto E-L. \end{aligned} \tag{2.10}\]
If \(u^i=a^{-1}\partial^iv\) at linear order,
\[ v\mapsto v+L'. \tag{2.11}\]
Table C.1 collects these transformations together with those of the displacement, clock, and density variables introduced later in the chapter, in a single passive convention. It is worth consulting before comparing any formula here with one taken from another source, because the sign of the generator and the sign of the spatial potential both vary across the literature.
The systematic construction of combinations of metric and matter perturbations that are unchanged under Eq. 2.1 is due to1. Before that work, disagreements between calculations performed in different gauges were a recurring source of confusion, and spurious growing modes in synchronous gauge were a familiar hazard. Bardeen’s variables made it possible to state which parts of a perturbation carry physical information without first committing to coordinates. The construction in this chapter reaches the same conclusion from the opposite direction: rather than building invariants out of the fields, it defines the measurement covariantly and lets the invariance follow.
We use two common gauges for checks:
synchronous-comoving gauge: \(A=B=v=0\);
conformal-Newtonian gauge: \(B=E=0\), \(A=\Psi\), and \(D=\Phi\) in
\[ \mathrm ds^2=a^2\left[-(1+2\Psi)\mathrm d\eta^2 +(1+2\Phi)\delta_{ij}\mathrm dx^i\mathrm dx^j\right]. \tag{2.12}\]
The spatial-potential sign in Eq. 2.12 is opposite to the convention \(g_{ij}=a^2(1-2\Phi_{\mathrm{common}})\delta_{ij}\).
Synchronous-comoving and conformal-Newtonian gauges are useful because different terms simplify in each. Agreement of the final observable between them is a check, but gauge invariance is already guaranteed by the pullback construction. One should not interpret a term that vanishes in one gauge as a physically absent mechanism unless the complete observable combination vanishes.
2.3 Ray tracing from the observer to the source
The gauge discussion above identifies the problem: the coordinate position of a source is not an observable. The operational solution is not to choose a preferred coordinate system, but to formulate the observation as an initial-value problem whose boundary data are fixed in the observer’s local frame. The observer measures a photon energy and a direction, and the metric then determines which source event produced that photon.
Let the observation event be \(x_o\) and let the observer carry an orthonormal tetrad \(e_{\hat a}^{\ \mu}\) with
\[ e_{\hat0}^{\ \mu}=u_o^\mu, \qquad g_{\mu\nu}e_{\hat a}^{\ \mu}e_{\hat b}^{\ \nu} =\eta_{\hat a\hat b}. \tag{2.13}\]
The measured line of sight is the unit spatial vector
\[ \hat n^\mu=\hat n^{\hat i}e_{\hat i}^{\ \mu}, \qquad u_o\cdot\hat n=0, \qquad \hat n\cdot\hat n=1, \tag{2.14}\]
where \(\hat{\boldsymbol n}\) points from the observer toward the source. We take \(k^\mu\) to be the future-directed photon wavevector, so at the observer
\[ \boxed{ k_o^\mu=E_o\left(u_o^\mu-\hat n^\mu\right) }, \qquad E_o\equiv-u_{o\mu}k_o^\mu>0. \tag{2.15}\]
The overall normalization \(E_o\) only fixes the affine parameter and may be set to unity. The relative spatial components of \(k_o^\mu\) are fixed by the measured direction. Thus the local observation supplies both the endpoint and all initial data needed for the null geodesic.
The photon trajectory is obtained from
\[ \frac{\mathrm dx^\mu}{\mathrm d\lambda}=k^\mu, \qquad \frac{\mathrm Dk^\mu}{\mathrm d\lambda}=0, \tag{2.16}\]
or, in coordinates,
\[ \frac{\mathrm dk^\mu}{\mathrm d\lambda} =-\Gamma^\mu_{\alpha\beta}k^\alpha k^\beta. \tag{2.17}\]
Starting from \((x_o,k_o)\), one integrates this system backward along the ray. Equivalently, one keeps \(k^\mu\) future directed and integrates toward decreasing affine parameter. The source affine parameter \(\lambda_s\) is not specified in advance. It is selected by the measured redshift through
\[ 1+\widetilde z =\frac{-u_{s\mu}k_s^\mu}{-u_{o\mu}k_o^\mu} =\frac{E_s}{E_o}. \tag{2.18}\]
The first point on the chosen source congruence for which Eq. 2.18 holds is the physical emission event,
\[ x_s^\mu(\widetilde z,\hat{\boldsymbol n}) =x^\mu(\lambda_s). \tag{2.19}\]
The procedure can be summarized as
\[ \boxed{ (x_o,u_o,E_o,\hat{\boldsymbol n}) \xrightarrow{\ \mathrm Dk/\mathrm d\lambda=0\ } (x_s,k_s) \xrightarrow{\ E_s/E_o=1+\widetilde z\ } x_s(\widetilde z,\hat{\boldsymbol n}) }. \tag{2.20}\]
This is the operational content behind the displacement field used below. The background FLRW model assigns a fiducial event \(\widetilde x^\mu(\widetilde z,\hat{\boldsymbol n})\) to the same observed labels, while the geodesic integration gives the actual event. Their coordinate difference is \(\Delta x^\mu=x_s^\mu-\widetilde x^\mu\). Under a gauge transformation both coordinate descriptions change, but the observer data, the null ray, and the physical source event do not. The transformation of \(\Delta x^\mu\) is therefore forced to compensate the transformation of the fields evaluated at the fiducial event.
At linear order the exact boundary-value problem is expanded about the background ray. The perturbation to the photon energy gives \(\Delta\ln a\), while the integrated perturbations to the trajectory give \(\Delta x_\parallel\) and \(\Delta x_\perp^i\). The formulas in the next section are therefore not independent projection corrections. They are different components of one ray-tracing solution with boundary conditions fixed at the observer.
A cosmological observable is not obtained by choosing a source coordinate and propagating a photon toward the observer. The measured energy and direction are fixed at the observer, the null geodesic is integrated backward, and the observed redshift selects the emission event. This ordering is what ensures that source, propagation, and observer terms belong to one prediction.
2.4 Observed redshift and the displacement field
The background map assigns a spacetime event to each observed pair \((\widetilde z,\hat{\boldsymbol{n}})\). Perturbations change both the emission time and the spatial position of the actual source relative to that fiducial event. The displacement field \(\Delta x^\mu\) records this difference. Its time component is conveniently expressed as a perturbation to the emission scale factor.
The observed redshift and direction define the fiducial event
\[ \widetilde x^\mu =\bigl(\eta_0-\widetilde\chi, \widetilde\chi\hat{\boldsymbol{n}}\bigr), \qquad \widetilde a=(1+\widetilde z)^{-1}. \tag{2.21}\]
The physical source event is
\[ x^\mu=\widetilde x^\mu+\Delta x^\mu. \tag{2.22}\]
The observed labels \((\widetilde z,\hat{\boldsymbol n})\) are measured quantities and are therefore unchanged by a relabelling of coordinates, so the fiducial event \(\widetilde x^\mu\) carries no gauge dependence. All of it is absorbed by the displacement, which transforms as
\[ \Delta x^{\mu}\mapsto\Delta x^\mu+\xi^\mu. \tag{2.23}\]
This sign, opposite to the one carried by field perturbations at a fixed coordinate point, is what makes the observed count invariant; Section 2.11 shows that the two are the two halves of a single geometric statement.
Define
\[ \Delta\ln a \equiv\ln\frac{a(x^0)}{\widetilde a}. \tag{2.24}\]
Thus \(\Delta\ln a\) is evaluated at fixed measured redshift; it is not an independent fluctuation of the observed redshift. It answers the question: by how much does the background scale factor at the actual emission event differ from the scale factor that the fiducial FLRW model assigns to the measured redshift? Since the observed labels are held fixed while \(\Delta x^0\) transforms as in Eq. 2.23,
\[ \Delta\ln a\mapsto\Delta\ln a+\mathcal HT. \tag{2.25}\]
Operationally, redshift is the photon-energy ratio
\[ 1+\widetilde z =\frac{(u_\mu k^\mu)_s}{(u_\mu k^\mu)_o}. \tag{2.26}\]
Linearizing the source and observer energies and integrating the perturbed null geodesic gives, along the unperturbed ray,
\[ \begin{aligned} \Delta\ln a={}&A_o-A_s+v_{\parallel s}-v_{\parallel o} +\int_0^{\widetilde\chi}\mathrm d\chi \left[-A'+\frac12h_\parallel'+B_\parallel'\right]\\ &-H_0\int_0^{t_o}\mathrm dt\, A\bigl(\boldsymbol{0},\bar\eta(t)\bigr). \end{aligned} \tag{2.27}\]
The last term converts fixed observer coordinate time into fixed observer proper time. In conformal-Newtonian gauge,
\[ \begin{aligned} (\Delta\ln a)_{\mathrm{cN}}={}& \Psi_o-\Psi_s+v_{\parallel s}-v_{\parallel o} +\int_0^{\widetilde\chi}\mathrm d\chi\,(\Phi'-\Psi')\\ &-H_0\int_0^{t_o}\mathrm dt\, \Psi\bigl(\boldsymbol{0},\bar\eta(t)\bigr). \end{aligned} \tag{2.28}\]
The individual endpoint, line-of-sight, and observer contributions are gauge-dependent. Their combination is fixed by Eq. 2.26.
The endpoint terms describe the gravitational and Doppler shifts at source and observer. The line-of-sight integral describes the change in photon energy as the metric evolves along the ray. The final observer term enforces that the observation is made at fixed observer proper time rather than at fixed coordinate time. Dropping observer terms can leave spurious monopole or dipole contributions and can spoil gauge invariance even when the source terms are otherwise correct.
2.5 Source proper time at fixed observed redshift: the cosmic clock
The bias expansion of Chapter 1 compares local tracer populations at equal proper time. A catalogue, by contrast, compares sources at equal measured redshift. The quantity required for the bias relation is therefore not the coordinate time of the source, but the proper time accumulated along its worldline relative to the background event assigned to the same observed redshift.
Let \(\Sigma_{\mathrm{ini}}\) be a physically specified early hypersurface from which this proper time is measured. We set the zero of \(t_F\) on \(\Sigma_{\mathrm{ini}}\). Along a nonrelativistic source worldline \(\boldsymbol{x}_{\mathrm{fl}}(\eta)\),
\[ \mathrm dt_F=a(1+A)\mathrm d\eta \tag{2.29}\]
to first order; terms involving the peculiar velocity enter only quadratically. Thus
\[ t_F(\eta,\boldsymbol{x}) =\bar t(\eta)+I_A(\eta,\boldsymbol{x}), \tag{2.30}\]
where
\[ \begin{aligned} \bar t(\eta) &\equiv\int_{\eta_{\mathrm{ini}}}^{\eta} \mathrm d\eta'\,a(\eta'),\\ I_A(\eta,\boldsymbol{x}) &\equiv\int_{\eta_{\mathrm{ini}}}^{\eta} \mathrm d\eta'\,a(\eta') A\bigl(\boldsymbol{x}_{\mathrm{fl}}(\eta'),\eta'\bigr). \end{aligned} \tag{2.31}\]
At linear order, replacing \(\boldsymbol{x}_{\mathrm{fl}}(\eta')\) by the unperturbed spatial position inside the already first-order field \(A\) changes the integral only at second order.
The lower boundary needs one explicit convention. We use \(\Sigma_{\mathrm{ini}}\) as the common physical reference surface and anchor the time relabelling there by requiring
\[ (aT)_{\mathrm{ini}}=0. \tag{2.32}\]
Equivalently, the two coordinate descriptions use the same zero of \(t_F\) on \(\Sigma_{\mathrm{ini}}\). This is a boundary convention for the gauge generator, not a dynamical assumption. With an arbitrary lower-boundary convention, the transformation of \(I_A\) retains an explicit \((aT)_{\mathrm{ini}}\) term. Indeed, from Eq. 2.10,
\[ aA\mapsto aA-a\mathcal HT-aT' =aA-(aT)', \tag{2.33}\]
and hence
\[ I_A\mapsto I_A-aT+(aT)_{\mathrm{ini}} =I_A-aT \qquad ((aT)_{\mathrm{ini}}=0). \tag{2.34}\]
We can now define the clock operationally. At fixed observed labels \((\widetilde z,\hat{\boldsymbol n})\), the background cosmology assigns the fiducial event \(\widetilde x^\mu\) and the background proper time \(\bar t(\widetilde a)\). The actual source event is \(x_s^\mu=\widetilde x^\mu+\Delta x^\mu\). Their proper-time difference is
\[ \Delta t_F(\widetilde z,\hat{\boldsymbol n}) \equiv t_F(x_s)-\bar t(\widetilde a). \tag{2.35}\]
Expanding about the fiducial event gives
\[ \begin{aligned} \Delta t_F &=I_A(\widetilde x)+a\Delta x^0\\ &=I_A+\frac{\Delta\ln a}{H}, \end{aligned} \tag{2.36}\]
where the second line uses \(\Delta\ln a=\mathcal H\Delta x^0\) and \(\mathcal H=aH\). All background quantities in these equations are evaluated at \(\widetilde a=(1+\widetilde z)^{-1}\). The dimensionless cosmic clock of2 is therefore
\[ \boxed{ \mathcal T(\widetilde z,\hat{\boldsymbol{n}}) \equiv H\Delta t_F =H I_A+\Delta\ln a }. \tag{2.37}\]
The meaning is direct: \(\mathcal T/H\) is the perturbation in the local cosmic age of the source event selected at fixed observed redshift. It is not an additional force or propagation effect. It records that the observed-redshift hypersurface and the equal-proper-time hypersurface intersect the source congruence at different events. This is why any locally defined population whose mean abundance evolves with proper time receives a clock contribution.
The gauge invariance is now transparent. Using Eq. 2.34 and Eq. 2.25,
\[ \mathcal T \mapsto H(I_A-aT) +\Delta\ln a+\mathcal HT =\mathcal T. \tag{2.38}\]
2.6 Constant-redshift and constant-proper-time slicings
The two slicings are useful bookkeeping choices for the same physical configuration. In constant-observed-redshift gauge the source event has no time displacement relative to the background redshift label, so the entire mismatch appears as a proper-time perturbation. In constant-proper-time gauge the local source clock is unperturbed, so the same mismatch appears in \(\Delta\ln a\). Neither gauge is preferred by the observable.
A time shift
\[ T_z=-\frac{\Delta\ln a}{\mathcal H} \tag{2.39}\]
sends \(\Delta\ln a\mapsto0\). Denoting quantities in this slicing by a subscript \(z\),
\[ (\Delta\ln a)_z=0, \qquad H(I_A)_z=\mathcal T. \]
Coordinate time then labels the constant-observed-redshift hypersurface, while the entire mismatch appears as a source proper-time perturbation.
A time shift
\[ T_{\mathrm{pt}}=\frac{I_A}{a} \tag{2.40}\]
sends \(I_A\mapsto0\). In the constant-proper-time slicing,
\[ (I_A)_{\mathrm{pt}}=0, \qquad (\Delta\ln a)_{\mathrm{pt}}=\mathcal T. \tag{2.41}\]
Coordinate time then agrees with source proper time, while the entire mismatch appears in the relation between emission time and observed redshift.
For conserved matter,
\[ \begin{aligned} \delta_m^{\mathrm{or}}&\equiv\delta_m-3\Delta\ln a,\\ \delta_m^{\mathrm{pt}}&\equiv\delta_m+3H I_A, \end{aligned} \tag{2.42}\]
are separately gauge-invariant and satisfy
\[ \boxed{ \delta_m^{\mathrm{or}}=\delta_m^{\mathrm{pt}}-3\mathcal T }. \tag{2.43}\]
The bias response of Chapter 1 is naturally defined at fixed source proper time, so \(\delta_m^{\mathrm{pt}}\) is the density entering the relativistic bias relation. The catalogue, however, is binned at fixed observed redshift. Chapter 3 combines the clock with spatial ruler distortions to connect the two.
Eq. 2.43 isolates the mechanism behind the density gauge issue. The difference between the two gauge-invariant density perturbations is itself a gauge-invariant clock.
This completes the perturbative half of the chapter, and it is worth pausing on what has happened. Individually, \(\delta_m\), \(\Delta\ln a\), and \(I_A\) are all gauge dependent. Yet \(\mathcal T\), \(\delta_m^{\mathrm{or}}\), and \(\delta_m^{\mathrm{pt}}\) are not, and the invariance was verified in each case by substituting the transformation rules and watching terms cancel. Nothing in that procedure explains why the cancellations occur, and a calculation of the full number count involves many more terms of the same kind. The remainder of the chapter shows that the cancellations are not coincidences to be checked one at a time. They follow from the fact that the count was a single geometric object before it was split.
2.7 Galaxy number as the flux of a current three-form
A galaxy catalogue counts worldlines crossing a three-dimensional surface. The natural covariant object is therefore not a scalar density alone but a three-form that can be integrated over that surface. Contracting the number current with the spacetime volume form supplies exactly such an object and automatically includes the relative orientation between the current and the surface.
Let \(M\) be an oriented Lorentzian spacetime with volume form
\[ \boldsymbol{\varepsilon} \equiv\sqrt{-g}\, \mathrm dx^0\wedge\mathrm dx^1\wedge\mathrm dx^2\wedge\mathrm dx^3, \qquad \epsilon_{0123}=+1. \tag{2.44}\]
The galaxy current is
\[ j=j^\mu\partial_\mu, \qquad j^\mu=n_g u_g^\mu, \qquad u_g^\mu u_{g\mu}=-1, \tag{2.45}\]
where \(n_g\) is the physical number density measured in the galaxy rest frame. Contracting the four-volume form with the current gives the current three-form
\[ \boxed{ \boldsymbol{\mathcal J}\equiv\iota_j\boldsymbol{\varepsilon}=\star j^\flat } \tag{2.46}\]
with coordinate expression
\[ \boldsymbol{\mathcal J} =\frac{\sqrt{-g}}{3!}\, \epsilon_{\mu\nu\rho\sigma}j^\mu \mathrm dx^\nu\wedge\mathrm dx^\rho\wedge\mathrm dx^\sigma. \tag{2.47}\]
This is the covariant replacement for \((1+\delta_g)\mathrm d^3x\) in the Newtonian number-conservation argument.
At a given event, \(\boldsymbol{\mathcal J}\) contains both the rest-frame number density and the velocity with which the worldlines pierce the chosen hypersurface. These pieces should not be separated before the hypersurface is specified. On a surface orthogonal to \(u_g^\mu\) the velocity factor reduces to unity; on a tilted or null surface it is part of the measured flux.
Cartan’s identity gives its exterior derivative:
\[ \begin{aligned} \mathrm d\boldsymbol{\mathcal J} &=\mathrm d(\iota_j\boldsymbol{\varepsilon}) =\mathcal L_j\boldsymbol{\varepsilon}-\iota_j\mathrm d\boldsymbol{\varepsilon}\\ &=\left(\nabla_\mu j^\mu\right)\boldsymbol{\varepsilon}. \end{aligned} \tag{2.48}\]
Thus \(\nabla_\mu j^\mu=0\) is equivalent to \(\mathrm d\boldsymbol{\mathcal J}=0\). If two hypersurfaces bound a four-volume without galaxy creation, destruction, or flux through the side boundary, Stokes’ theorem gives the same total count on both. The functorial identities used here, and Cartan’s formula in particular, are collected in Appendix B; every step in this section is one of those identities, with no input from the field equations.
For an evolving tracer population one may instead have \(\nabla_\mu j^\mu\neq0\) because objects enter or leave the selected sample. The three-form still defines the count on any one hypersurface; nonconservation then describes physical source evolution between hypersurfaces. The mean effect of this evolution is later encoded by \(b_e\) rather than being folded into the geometric volume distortion.
2.8 Pullback to a three-dimensional hypersurface
The embedding \(\mathsf X\) specifies which spacetime event corresponds to each coordinate point \(y^a\) on the three-manifold \(\Sigma\). The pullback \(\mathsf X^*\) converts a spacetime form into a form that can be integrated using the coordinates \(y^a\). No coordinate system on spacetime is preferred by this operation; changing spacetime coordinates changes the components of both the form and the embedding derivatives while leaving the pulled-back form unchanged.
Let \(\Sigma\) be a spacelike or null three-dimensional hypersurface, represented by an embedding
\[ \mathsf X:\Sigma\longrightarrow M, \qquad y^a\longmapsto X^\mu(y^1,y^2,y^3). \tag{2.49}\]
The number of galaxy worldlines crossing \(\Sigma\) is
\[ \boxed{ N(\Sigma)=\int_\Sigma\mathsf X^*\boldsymbol{\mathcal J} }. \tag{2.50}\]
This formula is meaningful for a null hypersurface even though the induced metric on a null hypersurface is degenerate and does not define a Riemannian three-volume.
This point is central for cosmology. A constant-time slice has a natural proper three-volume, but the past light cone does not. Trying to define the count as “density times the induced light-cone volume” therefore introduces an unnecessary difficulty. The current three-form already has the correct degree and can be pulled back directly.
The pullback acts on the coordinate one-forms as
\[ \mathsf X^*(\mathrm dx^\mu) =\mathrm d(X^\mu(y)) =\frac{\partial X^\mu}{\partial y^a}\mathrm dy^a. \tag{2.51}\]
Substituting into Eq. 2.47 and collecting the coefficient of \(\mathrm dy^1\wedge\mathrm dy^2\wedge\mathrm dy^3\) gives
\[ \boxed{ \mathsf X^*\boldsymbol{\mathcal J} =\sqrt{-g}\, \epsilon_{\mu\nu\rho\sigma}j^\mu \frac{\partial X^\nu}{\partial y^1} \frac{\partial X^\rho}{\partial y^2} \frac{\partial X^\sigma}{\partial y^3} \mathrm dy^1\wedge\mathrm dy^2\wedge\mathrm dy^3 } \tag{2.52}\]
for the orientation chosen in Eq. 2.44. Reversing the orientation of \(\Sigma\) reverses the sign of the form; the physical count uses the orientation for which the integrand is positive.
The antisymmetric contraction in Eq. 2.52 is the covariant analogue of a Jacobian determinant. The three tangent vectors \(\partial X^\mu/\partial y^a\) span the hypersurface, while \(j^\mu\) supplies the fourth direction. Their oriented four-volume measures how many worldlines cross the coordinate cell \(\mathrm d^3y\).
2.8.1 Spacelike check
Suppose \(\Sigma\) is spacelike with future-directed unit normal \(n^\mu\) and induced volume element \(\mathrm d\Sigma\). The pullback reduces to
\[ \mathsf X^*\boldsymbol{\mathcal J}=-(j\cdot n)\mathrm d\Sigma. \tag{2.53}\]
On a hypersurface orthogonal to the galaxy four-velocity, \(n^\mu=u_g^\mu\), and hence
\[ -(j\cdot u_g)\mathrm d\Sigma=n_g\mathrm d\Sigma. \tag{2.54}\]
This recovers the rest-frame number density. On the past light cone there is no unit normal, but Eq. 2.52 remains valid without modification.
The spacelike check also shows what changes when the surface is not orthogonal to the galaxy flow: the factor \(-(j\cdot n)\) contains a Lorentz projection between the worldlines and the surface normal. The null case is the continuous limit in which a unit normal ceases to exist, while the differential-form expression remains regular.
2.9 Radial and tangential structure on the light cone
Eq. 2.52 is correct but opaque. A single antisymmetric contraction of four indices hides the fact that the light cone has a preferred direction, namely the one along which the photons travel, and that the remaining two directions are the observer’s screen. Adapting the frame to that split, following3, makes the structure of the count visible before any perturbation is introduced, and it explains in advance why only two of the six ruler distortions of Chapter 3 can change a volume.
Let \(k^\mu\) be tangent to the null generators of \(\Sigma_{\tau_o}\), that is, to the photon trajectories reaching the observer. Complete it to a null tetrad by choosing a second null vector \(l^\mu\) and a complex vector \(m^\mu\) spanning the two directions transverse to the ray:
\[ k\cdot k=l\cdot l=m\cdot m=0, \qquad k\cdot l=-1, \qquad m\cdot\bar m=1, \tag{2.55}\]
with \(k\cdot m=l\cdot m=0\). The complex vector is a repackaging of a real orthonormal screen basis,
\[ \begin{aligned} m^\mu&=\frac{1}{\sqrt2}\left(\hat e_1^\mu+\mathrm i\,\hat e_2^\mu\right),\\ \mathrm d\zeta&=\frac{1}{\sqrt2}\left(\theta^1+\mathrm i\,\theta^2\right). \end{aligned} \tag{2.56}\]
Here \(\hat e_1,\hat e_2\) are the real orthonormal screen vectors and \(\theta^1,\theta^2\) their dual one-forms. They are the same transverse pair used to write the deformation matrix in Section 3.5.
The tangent space of the light cone at a point is spanned by \(k\), \(m\), and \(\bar m\). A null surface is degenerate precisely because \(k\) is simultaneously tangent and normal, which is what obstructed defining an induced three-volume in Section 2.8. The current three-form does not care. Evaluating it on the three tangent directions and expanding the current in the tetrad basis,
\[ j^\mu=-(j\cdot l)k^\mu-(j\cdot k)l^\mu +(j\cdot\bar m)m^\mu+(j\cdot m)\bar m^\mu, \tag{2.57}\]
every term except the one proportional to \(l^\mu\) repeats a slot already occupied and is annihilated by the antisymmetry. The count therefore sees the current only through the single scalar
\[ n_{g,\mathrm{lc}}\equiv k\cdot j=n_g\,(k\cdot u_g), \tag{2.58}\]
which is the density coefficient associated with the chosen affine parametrization of the light cone. By itself it rescales when the affine parameter is rescaled; the invariant object is the product with the affine measure. Parametrizing the null generators by an affine parameter \(\lambda\) with \(k^\mu=\mathrm dx^\mu/\mathrm d\lambda\), the combination \(-(k\cdot u_g)\mathrm d\lambda\) is exactly the proper length increment along the beam measured in the galaxy rest frame,
\[ \mathrm d\ell\equiv-k^\mu u_{g\mu}\,\mathrm d\lambda>0, \tag{2.59}\]
so the factor \(k\cdot u_g\) in Eq. 2.58 is not an additional weight. It is what converts an affine parameter into a physical length. The transverse factor is the screen area form
\[ \boldsymbol\sigma \equiv\mathrm i\,\mathrm d\zeta\wedge\mathrm d\bar\zeta =\theta^1\wedge\theta^2. \tag{2.60}\]
Using Eq. 2.56, the equality is a two-line wedge-product check: the complex packaging is bookkeeping, and \(\boldsymbol\sigma\) is the ordinary proper area element of the screen. Combining it with the radial proper-length form gives
\[ \boxed{ \mathsf X^*\boldsymbol{\mathcal J} =n_g\,\mathrm d\ell\wedge\boldsymbol\sigma }. \tag{2.61}\]
Equivalently, on the observer’s past light cone,
\[ \boxed{ N(\Sigma_{\tau_o}) =\int_{\Sigma_{\tau_o}} n_g\,\mathrm d\ell\wedge\boldsymbol\sigma }. \tag{2.62}\]
As with Eq. 2.52, the orientation is fixed so that the integrand is positive.
Eq. 2.61 is the result. The count is a rest-frame number density multiplied by a radial proper length and a transverse proper area, with no metric on the null surface required anywhere.
Three things follow, and each is worth stating before the perturbative machinery starts.
The first is that only one component of the current survives. Of the four terms in Eq. 2.57 the pullback keeps only \(k\cdot j\). The factor \(-(k\cdot u_g)\) is the photon energy measured in the galaxy rest frame, but after Eq. 2.59 it belongs to the radial proper-length element rather than supplying a second weight. The same contraction also appears in the redshift ratio Eq. 2.26, so Doppler physics and radial number counting share one geometric ingredient.
The second is that the measure factorizes. The physical volume through which the worldlines pass is a radial proper length multiplied by a transverse proper area, and these are separately meaningful because the ray direction is singled out by the observation itself. This is the geometric origin of the split into longitudinal and transverse ruler distortions in Chapter 3.
The third is a prediction about which distortions can matter. A perturbation deforms the triad \((\hat e_\parallel,\hat e_1,\hat e_2)\), and the product in Eq. 2.61 responds to the determinant of that deformation. A radial stretch changes \(\mathrm d\ell\) and a transverse stretch changes \(\boldsymbol\sigma\), so both contribute at first order. Tilting the radial leg against the screen changes neither factor at first order, because a determinant is insensitive to off-diagonal entries until second order. This is exactly the statement, derived by other means in Eq. 3.28, that \(\mathcal B_i\) is absent from the linear volume Jacobian.
The tetrad is not needed to obtain any result in these notes; Eq. 2.52 already contains everything. What it buys is bookkeeping. Once the count is written as the rest-frame density \(n_g\) times a radial proper-length element \(\mathrm d\ell\) times a transverse proper-area element \(\boldsymbol\sigma\), each later contribution can be assigned to one of these three slots. The contraction \(k\cdot u_g\) is already contained in \(\mathrm d\ell\) and must not be counted again as a separate weight. Terms that seem to proliferate in a component calculation are often the same slot written in different variables.
3 carries this construction to fully nonlinear order. The factorization survives, and the ruler distortions become the entries of an exact three by three matrix whose definition assumes only that the rulers are infinitesimal, not that the perturbations are small. Its determinant reproduces the third statement above: the mixed entries appear there only through \(\mathcal B^2\) and \(\mathcal B_m\mathcal B_{\bar m}\), never linearly. The treatment below is the first order of that expansion.
3 evaluates the spatial determinant at fixed source proper time and then multiplies by a separate clock Jacobian to convert to the observed redshift. That is the factored, comoving-ruler bookkeeping described in Eq. 3.33, not the fixed-physical-ruler convention these notes use. The two agree, but the individual \(\mathcal C\) and \(\mathcal M\) differ between them by terms proportional to \(\mathcal T\). See the convention check in Section 3.6 before transcribing any formula.
2.10 Observed coordinates and the exact inferred density
The observer does not assign arbitrary coordinates to the light cone. A source is located by quantities measured at the detector: redshift and an angular direction, with the observation made at a specified observer proper time. These labels form a three-dimensional observed-coordinate manifold. A fiducial background cosmology is used only to assign an apparent spatial volume to a small cell in these labels.
The insistence that cosmological data live on the observer’s past light cone, and that a cosmological model should be formulated in terms of what is measurable there, was developed systematically by4 and collaborators under the name observational cosmology. The programme asked which spacetime geometries are consistent with data on a single null cone, and it made explicit that the natural coordinates for the problem are the ones an observer actually reports. The construction in this section is a perturbative instance of that viewpoint: \((\widetilde z,\widetilde\theta,\widetilde\phi)\) are observer labels, and the fiducial cosmology enters only through the volume form assigned to a cell of them.
Fix the observer’s proper time \(\tau_o\). On a regular patch of the past light cone \(\Sigma_{\tau_o}\), use the directly observed coordinates
\[ y^a=(\widetilde z,\widetilde\theta,\widetilde\phi). \tag{2.63}\]
The null geodesic and source worldline determine the physical embedding
\[ \mathsf X_{\tau_o}: (\widetilde z,\widetilde\theta,\widetilde\phi) \longmapsto X^\mu(\widetilde z,\widetilde\theta,\widetilde\phi;\tau_o). \tag{2.64}\]
The fiducial FLRW relation defines
\[ \widetilde a=(1+\widetilde z)^{-1}, \qquad \widetilde\chi(\widetilde z) =\int_0^{\widetilde z}\frac{\mathrm dz}{H(z)}. \tag{2.65}\]
The apparent physical volume form assigned to an observed cell is
\[ \widetilde{\boldsymbol{\omega}} \equiv \frac{\widetilde a^3\widetilde\chi^2}{H(\widetilde z)} \sin\widetilde\theta\, \mathrm d\widetilde z\wedge\mathrm d\widetilde\theta\wedge\mathrm d\widetilde\phi. \tag{2.66}\]
This is a form on the observed-coordinate manifold, not an induced volume form of the null hypersurface.
Changing the fiducial background changes the numerical volume assigned to a fixed observed cell and therefore changes the inferred overdensity field in the same way that an Alcock–Paczynski remapping changes a catalogue coordinate system. It does not change the integer number of sources in the cell. The physical count form and the fiducial volume form must therefore be kept conceptually distinct.
Define the inferred density \(n_g^{\mathrm{obs}}\) by the equality of three-forms
\[ \mathsf X_{\tau_o}^*\boldsymbol{\mathcal J} \equiv n_g^{\mathrm{obs}}(\widetilde z,\hat{\boldsymbol{n}})\, \widetilde{\boldsymbol{\omega}}. \tag{2.67}\]
Using Eq. 2.52,
\[ \begin{aligned} n_g^{\mathrm{obs}} ={}& \frac{H(\widetilde z)} {\widetilde a^3\widetilde\chi^2\sin\widetilde\theta} \sqrt{-g}\, \epsilon_{\mu\nu\rho\sigma}n_g u_g^\mu \frac{\partial X^\nu}{\partial\widetilde z} \frac{\partial X^\rho}{\partial\widetilde\theta} \frac{\partial X^\sigma}{\partial\widetilde\phi}. \end{aligned} \tag{2.68}\]
The orientation is understood to make the right-hand side positive. Eq. 2.68 is nonperturbative and coordinate-independent, although its displayed components refer to a coordinate chart on spacetime.
Coordinate independence follows because both sides of Eq. 2.67 are three-forms on the same observed manifold. Their ratio is a scalar function of \((\widetilde z,\hat{\boldsymbol{n}})\). The metric determinant, four-velocity, and embedding derivatives in Eq. 2.68 are individually coordinate-dependent components, but the full antisymmetric contraction is not.
The observed overdensity is then
\[ 1+\delta_g^{\mathrm{obs}}(\widetilde z,\hat{\boldsymbol{n}}) \equiv\frac{n_g^{\mathrm{obs}}(\widetilde z,\hat{\boldsymbol{n}})}{\bar n_g(\widetilde z)}. \tag{2.69}\]
A selection function can be included by replacing \(j^\mu\) with the current of the selected population or by multiplying the pulled-back count by a response to measured flux, size, color, and other observables.
This separation is useful because source selection has two aspects. A rest-frame cut changes the tracer population and hence its bias coefficients. A cut on an observed quantity can also respond to propagation effects such as magnification. The latter is an additional observable response and should not be absorbed into the rest-frame density bias.
2.11 Linearizing a pullback
Perturbing the observed count requires varying two ingredients at once. The physical current form differs from its FLRW value, and the event selected by a fixed observed redshift and angle is displaced from the background event. Keeping only one of these variations produces a gauge-dependent result. The Lie derivative provides a compact way to include the displacement of the embedding.
The exact expression becomes the usual perturbative observable after expanding both the current form and the light-cone embedding. Let
\[ \boldsymbol{\mathcal J}=\overline{\boldsymbol{\mathcal J}}+\delta\boldsymbol{\mathcal J}, \qquad X^\mu(y)=\overline X^\mu(y)+\Delta x^\mu(y). \tag{2.70}\]
To derive the linear expansion cleanly, extend \(\Delta x^\mu\) off the background image of \(\overline{\mathsf X}\) and let \(\varphi_\lambda\) be its flow. Locally,
\[ \mathsf X=\varphi_1\circ\overline{\mathsf X}. \tag{2.71}\]
For any differential form \(\alpha\),
\[ \left.\frac{\mathrm d}{\mathrm d\lambda}\varphi_\lambda^*\alpha \right|_{\lambda=0} =\mathcal L_{\Delta x}\alpha. \tag{2.72}\]
Therefore
\[ \begin{aligned} \mathsf X^*\boldsymbol{\mathcal J} &=\overline{\mathsf X}^*\varphi_1^* \left(\overline{\boldsymbol{\mathcal J}}+\delta\boldsymbol{\mathcal J}\right)\\ &=\overline{\mathsf X}^* \left[ \overline{\boldsymbol{\mathcal J}} +\delta\boldsymbol{\mathcal J} +\mathcal L_{\Delta x}\overline{\boldsymbol{\mathcal J}} \right] +\mathcal O(2). \end{aligned} \tag{2.73}\]
The extension of \(\Delta x^\mu\) away from the background light cone is only a device for defining a flow. At linear order the pulled-back result depends on the displacement along the embedded surface and on the background form evaluated there. Physically, the Lie-derivative term includes both the change in the background density along the displacement and the deformation of the coordinate cell. The two perturbative pieces have distinct meanings: \(\delta\boldsymbol{\mathcal J}\) changes the physical current at a fixed background event, while \(\mathcal L_{\Delta x}\overline{\boldsymbol{\mathcal J}}\) moves the source event selected by fixed observed coordinates.
Now apply the passive coordinate transformation Eq. 2.1, with which this chapter opened. The perturbation of a covariant form transforms as
\[ \delta\boldsymbol{\mathcal J} \mapsto\delta\boldsymbol{\mathcal J} -\mathcal L_\xi\overline{\boldsymbol{\mathcal J}}, \tag{2.74}\]
while the coordinate displacement of the same physical source event obeys \(\Delta x\mapsto\Delta x+\xi\). Consequently,
\[ \begin{aligned} &\bigl(\delta\boldsymbol{\mathcal J} -\mathcal L_\xi\overline{\boldsymbol{\mathcal J}}\bigr) +\mathcal L_{\Delta x+\xi}\overline{\boldsymbol{\mathcal J}}\\ &\qquad=\delta\boldsymbol{\mathcal J} +\mathcal L_{\Delta x}\overline{\boldsymbol{\mathcal J}}. \end{aligned} \tag{2.75}\]
This is the geometric origin of the cancellation among source, redshift, volume, propagation, and observer terms in a gauge-invariant number-count formula.
A passive gauge transformation relabels the same source event. The perturbation of the current at a fixed coordinate point changes by \(-\mathcal L_\xi\overline{\boldsymbol{\mathcal J}}\), while the coordinate displacement to the physical event changes by \(+\xi^\mu\). The opposite signs are not an accidental feature of a particular formula; they express the fact that changing the field at a fixed point and moving the point are complementary parts of the same pullback.
The gauge-invariant object is not a specially chosen density perturbation. It is the pulled-back current form divided by a volume form defined from observed coordinates. Gauge-dependent density and displacement perturbations are the two parts of the linear expansion of that one geometric object.
In the usual presentation one computes many separate contributions to the number count, finds that each is gauge dependent, and then observes that the total is not. The cancellation looks like a fortunate accident, and checking it is laborious. Eq. 2.75 shows that it is neither. The count was defined as the integral of one geometric object over one surface, so it could not have depended on coordinates; splitting it into a field perturbation and a displacement is a choice made by the calculator, and the two pieces must therefore transform oppositely. The practical consequence is a reliable bookkeeping rule. Any term that appears in \(\delta\boldsymbol{\mathcal J}\) has a partner in \(\mathcal L_{\Delta x}\overline{\boldsymbol{\mathcal J}}\), and a calculation that drops observer terms, or evaluates the source at the fiducial rather than the displaced event, has broken the pairing rather than found a new effect.
Looking back at the first half of the chapter with Eq. 2.75 in hand, the invariance of \(\mathcal T\) in Eq. 2.38 is no longer a computation that happened to work. It is one instance of the general statement, restricted to the temporal factor of the count. The same will be true of the ruler distortions constructed in Chapter 3: each is a ratio of a measured quantity to an inferred one, and each is invariant for the same structural reason.
Summary
Key ideas. A density perturbation is meaningless until a slicing is named, because it compares the physical density at one event with a background value at another, and a gauge transformation changes which events are paired. The mismatch between the two slicings that matter, source proper time and observed redshift, is itself gauge invariant and is the cosmic clock \(\mathcal T\). Behind every such invariance is one structural fact: a catalogue counts worldlines crossing a surface, the object that does this covariantly is the current three-form \(\boldsymbol{\mathcal J}=\iota_j\boldsymbol{\varepsilon}\), and a three-form can be pulled back to any three-manifold, including a past light cone that carries no induced volume element. The count is therefore the integral of one geometric object, and splitting it into a field perturbation \(\delta\boldsymbol{\mathcal J}\) and a displacement \(\mathcal L_{\Delta x}\overline{\boldsymbol{\mathcal J}}\) is a bookkeeping choice whose two halves must transform oppositely. Adapting the frame to the observation factorizes the count into a radial leg and a transverse area, which is why only two of the six ruler distortions can change a volume at linear order.
Main results.
- The passive gauge map Eq. 2.1 with scalar generator Eq. 2.2, and the resulting density transformations Eq. 2.5 and Eq. 2.6.
- The metric perturbation transformations Eq. 2.10 and Eq. 2.11, tabulated in Table C.1.
- The observer-to-source initial-value problem Eq. 2.20, which fixes the photon momentum locally at the observer and uses the measured redshift to select the emission event.
- The displacement field Eq. 2.22, its transformation Eq. 2.23, and the redshift perturbation Eq. 2.27, including the observer term that enforces fixed observer proper time.
- The cosmic clock Eq. 2.37, its invariance Eq. 2.38, and the slicing relation Eq. 2.43 between \(\delta_m^{\mathrm{or}}\) and \(\delta_m^{\mathrm{pt}}\).
- The current three-form Eq. 2.46, its conservation property Eq. 2.48, and the count as a pullback integral Eq. 2.50.
- The coordinate expression for the pullback, Eq. 2.52, valid on null and spacelike surfaces alike, and the spacelike check Eq. 2.54.
- The null-tetrad factorization Eq. 2.61 and count Eq. 2.62, separating the observable into rest-frame density, radial proper length, and transverse proper area without double-counting \(k\cdot u_g\).
- The exact inferred density Eq. 2.68, defined by the ratio of the pulled-back count form to the fiducial volume form Eq. 2.66.
- The linearized pullback Eq. 2.73 and the cancellation Eq. 2.75.
What the next chapter builds on. The clock \(\mathcal T\) supplies the temporal half of the map from a source density to an observed count. Chapter 3 constructs the spatial half by comparing a physical ruler at the source with the separation inferred from observed labels, decomposes the result into six distortions, and takes the determinant to obtain the volume Jacobian. Combining that determinant with \(\mathcal T\) and the proper-time bias relation of Chapter 1 produces the central result of Part I.