3 Cosmic rulers, cosmic clocks, and the observed density
Where we are
Chapter 2 established two things. The observed count is the pullback of a current three-form to the observed-coordinate manifold, which makes it gauge invariant by construction. And the source proper time at which a bias response is defined differs from the background time assigned to the observed redshift, by the gauge-invariant amount \(H^{-1}\mathcal T\). What remains is the spatial counterpart of that second statement: the volume of the cell into which the sources are binned is also not what the fiducial cosmology says it is.
This chapter supplies that piece and then assembles the result. The method is the same one used for the clock. Define a quantity operationally, in this case the ratio of a physical separation measured in the source rest frame to the separation an observer infers from redshifts and angles, and the ratio is automatically invariant even though every term in it is not.
A cosmic ruler is any small physical separation whose rest-frame length or statistical scale is known or modeled. The observer infers a separation from redshift and angle using a fiducial FLRW relation. Metric perturbations, velocities, and photon propagation change the relation between the physical and inferred separations. At linear order these changes form six independent ruler distortions, and their determinant gives the spatial volume part of the number-count Jacobian.
The six distortions are organized by how they sit relative to the line of sight: one longitudinal scalar \(\mathcal C\), a two-component transverse vector \(\mathcal B_i\), and a symmetric two-dimensional tensor \(\mathcal A_{ij}\) whose trace \(\mathcal M\) is the magnification and whose trace-free part is a shear. Only \(\mathcal C\) and \(\mathcal M\) can enter a scalar volume at linear order, and the chapter closes by combining them with the clock and the proper-time bias relation into
\[ \delta_g^{\mathrm{obs}} =\delta_g^{\mathrm{pt}}+(b_e-3)\mathcal T-\mathcal C-\mathcal M, \]
which is Eq. 3.38 below and the central result of Part I. The Kaiser field of Chapter 1 is its subhorizon limit.
One warning is worth stating in advance, because it is the most common way to get this calculation wrong. The bookkeeping used here fixes physical rulers, so the ruler determinant already delivers the physical volume at the source event. An additional transported-volume factor \(1+3\mathcal T\) does not belong on top of it. The alternative convention, in which such a factor is explicit, requires correspondingly shifted ruler distortions. Both are correct; mixing them double counts the clock.
Figure placeholder. The screen basis at an observed position. A short segment of the past light cone with the line of sight \(\hat{\boldsymbol n}\), the two transverse screen directions, and three infinitesimal rulers drawn along them, annotated to show which distortion (\(\mathcal C\), \(\mathcal B_i\), \(\mathcal A_{ij}\)) deforms which pair of legs.
3.1 A physical ruler and its inferred separation
The construction is local at the source in the sense that the two endpoints are taken infinitesimally close compared with the scale over which the background and perturbations vary. It is nevertheless an observed light-cone quantity because the endpoint coordinates are inferred from photons reaching the observer. The screen projector separates radial and angular directions defined by the observed line of sight.
At an observed position \((\widetilde z,\hat{\boldsymbol{n}})\), define
\[ P_{ij}=\delta_{ij}-\hat n_i\hat n_j, \qquad V_\parallel=\hat n_iV^i, \qquad V_\perp^i=P^i{}_jV^j. \tag{3.1}\]
The derivatives used below are
\[ \partial_\parallel=\hat n^i\partial_i, \qquad \partial_{\perp i}=P_i{}^j\partial_j, \qquad \partial_{\widetilde\chi}=\partial_\parallel-\partial_\eta \tag{3.2}\]
along the background past light cone at fixed observed direction.
For two nearby observed source positions, the fiducial FLRW separation is
\[ \widetilde r^2 =\widetilde a^2\delta_{ij} \delta\widetilde x^i\delta\widetilde x^j, \qquad \widetilde r_c^2 \equiv\delta_{ij}\delta\widetilde x^i\delta\widetilde x^j. \tag{3.3}\]
A physical ruler is defined by simultaneous separation in the local source rest frame:
\[ \boxed{ r_0^2=(g_{\mu\nu}+u_\mu u_\nu) \delta x^\mu\delta x^\nu }. \tag{3.4}\]
This definition is covariant. The projector \(g_{\mu\nu}+u_\mu u_\nu\) specifies the equal-proper-time comparison that is implicit in any physical length measurement.
Without the velocity projector, two endpoints that are simultaneous in the observer’s background coordinates need not be simultaneous in the source rest frame. The \(u_\mu u_\nu\) term removes the component of the separation along the source four-velocity and leaves the spatial length measured by a locally comoving observer.
The actual and inferred source positions satisfy
\[ x^\mu(\widetilde x)=\widetilde x^\mu+\Delta x^\mu(\widetilde x). \tag{3.5}\]
For endpoints separated by \(\delta\widetilde x^\mu\),
\[ \begin{aligned} \delta x^\mu &=x^\mu(\widetilde x+\delta\widetilde x) -x^\mu(\widetilde x)\\ &=\delta\widetilde x^\mu +\delta\widetilde x^\alpha \partial_\alpha\Delta x^\mu +\mathcal O(\delta\widetilde x^2). \end{aligned} \tag{3.6}\]
The two inferred endpoints lie on the background light cone, so
\[ \delta\widetilde x^0=-\delta\widetilde x_\parallel. \tag{3.7}\]
It follows that
\[ \begin{aligned} \delta\widetilde x^\alpha\partial_\alpha &=-\delta\widetilde x_\parallel\partial_\eta +(\delta\widetilde x_\parallel\hat n^i +\delta\widetilde x_\perp^i)\partial_i\\ &=\delta\widetilde x_\parallel\partial_{\widetilde\chi} +\delta\widetilde x_\perp^i\partial_{\perp i}. \end{aligned} \tag{3.8}\]
Eq. 3.8 is the differential form of the light-cone constraint. A radial change in the inferred source position changes both the spatial radius and the emission time, whereas a transverse change moves the endpoint on the screen at fixed background distance. Derivatives of \(\Delta x^\mu\) with respect to these directions are the local deformation tensor of the observed-to-physical map.
3.2 Expansion of the rest-frame length
The purpose of the following algebra is to express the covariant length Eq. 3.4 entirely in terms of perturbations evaluated at the fiducial observed position. There are three sources of the difference: the scale factor at emission is shifted, the local metric and velocity alter the rest-frame projector, and the displacement field changes the endpoint separation.
For the metric convention of Eq. 2.7, the source four-velocity is
\[ u^\mu=a^{-1}(1-A,v^i), \qquad u_\mu=a(-1-A,v_i-B_i) \tag{3.9}\]
to linear order. Hence
\[ \begin{aligned} (g+uu)_{00}&=0,\\ (g+uu)_{0i}&=-a^2v_i,\\ (g+uu)_{ij}&=a^2(\delta_{ij}+h_{ij}). \end{aligned} \tag{3.10}\]
The cancellation of \(A\) and \(B_i\) in these projector components is not a gauge choice. It follows from using the physical source four-velocity in the equal-proper-time projector. The lapse and shift still affect the observable through the redshift and displacement fields, but they do not appear as independent local length perturbations. The lapse and shift cancel from the local spatial projector except through the physical velocity. Substitution into Eq. 3.4 gives
\[ r_0^2=a^2 \left[(\delta_{ij}+h_{ij})\delta x^i\delta x^j -2v_i\delta x^0\delta x^i\right]. \tag{3.11}\]
At the source event,
\[ a(x^0)=\widetilde a(1+\Delta\ln a). \tag{3.12}\]
Use Eq. 3.6, Eq. 3.7, and Eq. 3.8 and keep one perturbation. The result is
\[ \begin{aligned} \frac{r_0^2-\widetilde r^2}{\widetilde a^2} ={}&2\Delta\ln a\, \delta_{ij}\delta\widetilde x^i\delta\widetilde x^j +h_{ij}\delta\widetilde x^i\delta\widetilde x^j +2v_i\delta\widetilde x_\parallel\delta\widetilde x^i\\ &+2\delta_{ij}\delta\widetilde x^i \left( \delta\widetilde x_\parallel\partial_{\widetilde\chi} +\delta\widetilde x_\perp^k\partial_{\perp k} \right)\Delta x^j. \end{aligned} \tag{3.13}\]
Because \(r_0=\widetilde r+\mathcal O(1)\),
\[ \frac{\widetilde r-r_0}{\widetilde r} =-\frac{r_0^2-\widetilde r^2}{2\widetilde r^2}. \tag{3.14}\]
Eq. 3.13 is the algebraic source of all six linear ruler observables.
Each term has a direct origin. The first rescales all lengths because the actual emission scale factor differs from the fiducial one. The second is the local spatial metric distortion. The third enforces simultaneity in the moving source frame. The final term is the gradient of the light-cone displacement and contains radial redshift distortions, transverse deflection, and their mixed components.
3.3 Longitudinal, mixed, and transverse distortions
A symmetric deformation of three-dimensional separations has six components. Relative to the observed line of sight they decompose into one longitudinal scalar \(\mathcal C\), a two-component transverse vector \(\mathcal B_i\), and a symmetric two-dimensional tensor \(\mathcal A_{ij}\) with three components. This decomposition is kinematic and does not assume scalar perturbations; vector and tensor metric perturbations fit into the same formulas.
Decompose the orientation dependence as1
\[ \frac{\widetilde r-r_0}{\widetilde r} =\mathcal C\frac{(\delta\widetilde x_\parallel)^2}{\widetilde r_c^2} +\mathcal B_i \frac{\delta\widetilde x_\parallel\delta\widetilde x_\perp^i} {\widetilde r_c^2} +\mathcal A_{ij} \frac{\delta\widetilde x_\perp^i\delta\widetilde x_\perp^j} {\widetilde r_c^2}. \tag{3.15}\]
Collecting the coefficient of \((\delta\widetilde x_\parallel)^2\) in Eq. 3.13 gives
\[ \boxed{ \mathcal C=-\Delta\ln a-\frac12h_\parallel-v_\parallel -\partial_{\widetilde\chi}\Delta x_\parallel }. \tag{3.16}\]
The mixed coefficient is
\[ \boxed{ \mathcal B_i =-P_i{}^jh_{jk}\hat n^k-v_{\perp i} -\hat n^k\partial_{\perp i}\Delta x_k -\partial_{\widetilde\chi}\Delta x_{\perp i} }. \tag{3.17}\]
The transverse tensor is
\[ \boxed{ \mathcal A_{ij} =-\Delta\ln a\,P_{ij} -\frac12P_i{}^kP_j{}^lh_{kl} -\frac12\left( P_j{}^k\partial_{\perp i} +P_i{}^k\partial_{\perp j} \right)\Delta x_k }. \tag{3.18}\]
Since \(\Delta x_i=\hat n_i\Delta x_\parallel+\Delta x_{\perp i}\) and the line-of-sight basis varies across the sky,
\[ \mathcal A_{ij} =-\Delta\ln a\,P_{ij} -\frac12P_i{}^kP_j{}^lh_{kl} -\partial_{\perp(i}\Delta x_{\perp j)} -\frac{\Delta x_\parallel}{\widetilde\chi}P_{ij}. \tag{3.19}\]
The last term is the change in transverse physical scale caused by a radial displacement.
The three observables have different operational meanings. \(\mathcal C\) changes a ruler aligned with the line of sight and contains the relativistic completion of redshift-space distortion. \(\mathcal B_i\) mixes a radial and a transverse leg and describes a skewing of the inferred coordinate grid. \(\mathcal A_{ij}\) acts within the screen and contains both isotropic magnification and spin-two shear.
The complete combinations \(\mathcal C\), \(\mathcal B_i\), and \(\mathcal A_{ij}\) are gauge-invariant because both \(r_0\) and \(\widetilde r\) are operationally defined. Their individual metric, velocity, and displacement pieces are not.
This is another instance of the pullback logic. A gauge transformation changes the coordinate displacement and the local metric evaluated at the fiducial endpoint, but it cannot change the ratio of the measured physical ruler to the ruler inferred from fixed observed labels.
\(\mathcal T\), \(\mathcal C\), \(\mathcal B_i\), and \(\mathcal A_{ij}\) are not convenient groupings of metric perturbations chosen to make an answer come out invariant. Each is defined as a comparison between two things an observer can in principle obtain: a physical duration or length in the source rest frame, and the duration or length that the fiducial cosmology assigns to the same pair of observed labels. Their invariance is a consequence of that definition, not a property to be checked afterwards. The practical payoff is that they can be reasoned about independently of the perturbation variables. One can ask what \(\mathcal M\) does to a survey without first choosing a gauge, and one can identify a term as physical or as bookkeeping by asking which measurement it changes. The same logic, applied to a local rather than a light-cone measurement, produces the Fermi frame of Chapter 4.
3.4 Magnification and ruler shear
The transverse tensor is naturally separated into a trace and a trace-free part. The trace changes the apparent area of a small source and therefore affects angular diameter distance and number-count dilution. The trace-free part changes shape at fixed area and is the relativistic ruler shear.
The trace and trace-free parts of the transverse distortion are
\[ \begin{aligned} \mathcal M&\equiv P^{ij}\mathcal A_{ij},\\ \gamma_{ij}^{\mathrm{ruler}} &\equiv\mathcal A_{ij}-\frac12P_{ij}\mathcal M. \end{aligned} \tag{3.20}\]
Define the coordinate convergence
\[ \widehat\kappa \equiv-\frac12\partial_{\perp i}\Delta x_\perp^i. \tag{3.21}\]
Taking the trace of Eq. 3.19 gives
\[ \boxed{ \mathcal M =-2\Delta\ln a -\frac12(h^i{}_i-h_\parallel) +2\widehat\kappa -2\frac{\Delta x_\parallel}{\widetilde\chi} }. \tag{3.22}\]
The trace-free part is
\[ \begin{aligned} \gamma_{ij}^{\mathrm{ruler}} ={}&-\frac12 \left(P_i{}^kP_j{}^l-\frac12P_{ij}P^{kl}\right)h_{kl}\\ &-\partial_{\perp(i}\Delta x_{\perp j)} -P_{ij}\widehat\kappa. \end{aligned} \tag{3.23}\]
The coordinate convergence \(\widehat\kappa\) is not an observable by itself. Local metric terms and radial/scale-factor perturbations complete the observable.
In the standard weak-lensing limit the additional terms are suppressed or cancel against endpoint conventions, leaving the familiar line-of-sight convergence. On horizon scales or in a general gauge, however, identifying \(\widehat\kappa\) alone with magnification omits pieces required by the operational distance measurement.
The angular- and luminosity-distance perturbations are
\[ \frac{\Delta D_A}{D_A} =\frac{\Delta D_L}{D_L} =-\frac12\mathcal M, \tag{3.24}\]
where the equality follows from distance duality. In the subhorizon weak-lensing limit, \(\mathcal M\to2\kappa\).
3.5 The ruler determinant
Number counts require the volume change rather than the distortion of one chosen ruler. Three independent infinitesimal separation vectors span a small rest-frame volume. The determinant of their linear map gives the ratio between that physical volume and the volume inferred from observed coordinates.
Choose an orthonormal apparent basis \((\hat e_\parallel,\hat e_1,\hat e_2)\) and write the unit orientation of a ruler as \(\hat q^a\). The fractional length distortion has the form
\[ \frac{\widetilde r-r_0}{\widetilde r} =\mathsf D_{ab}\hat q^a\hat q^b, \tag{3.25}\]
with symmetric part
\[ \mathsf D= \begin{pmatrix} \mathcal C& \mathcal B_1/2 & \mathcal B_2/2\\ \mathcal B_1/2 & \mathcal A_{11} & \mathcal A_{12}\\ \mathcal B_2/2 & \mathcal A_{12} & \mathcal A_{22} \end{pmatrix}. \tag{3.26}\]
An unobservable infinitesimal rotation may be added to the vector map, but it neither changes ruler lengths nor contributes to the determinant at linear order. Thus one may write
\[ r_0^a=(\delta^a{}_b-\mathsf D^a{}_b)\widetilde r^b \tag{3.27}\]
for the volume calculation. Three independent infinitesimal rulers span volumes related by
\[ \begin{aligned} \frac{V_0}{\widetilde V} &=\det(I-\mathsf D)\\ &=1-\operatorname{tr}\mathsf D+\mathcal O(2)\\ &=1-\mathcal C-\mathcal M+\mathcal O(2). \end{aligned} \tag{3.28}\]
The mixed ruler \(\mathcal B_i\) cannot enter a scalar volume at linear order. It first appears quadratically through the determinant.
This follows from rotational structure as well as from the determinant. A single transverse vector cannot form a scalar without another transverse vector. At linear order the volume responds only to the radial dilation and the transverse trace, \(\mathcal C+\mathcal M\).
3.6 Adding the clock and the source density
The current three-form combines the physical source density with the physical volume element through which the source worldlines pass. For the ruler convention used above, the determinant in Eq. 3.28 already maps an observed coordinate cell to the rest-frame physical volume at the actual source event. This point is important: the ruler was defined with the source projector \(g_{\mu\nu}+u_\mu u_\nu\), so the volume obtained from \(\mathcal C\) and \(\mathcal M\) is not a background volume at the fiducial time. It is the local volume measured on the source rest-frame hypersurface at the emission event.
Let \(\delta_g^{\mathrm{or}}\) denote the rest-frame physical density perturbation evaluated at the source event but compared with the background mean at the time assigned to the observed redshift:
\[ n_g(x_s)=\bar n_g(\widetilde a) \left[1+\delta_g^{\mathrm{or}}(\widetilde z,\hat{\boldsymbol{n}})\right]. \tag{3.29}\]
Since \(V_0/\widetilde V=1-\mathcal C-\mathcal M\) is already the physical rest-frame volume ratio at the source event, the number of sources in the observed cell is
\[ \begin{aligned} 1+\delta_g^{\mathrm{obs}} &=(1+\delta_g^{\mathrm{or}}) (1-\mathcal C-\mathcal M)+\mathcal O(2). \end{aligned} \tag{3.30}\]
Thus, for the fixed-physical-ruler convention of Eq. 3.4,
\[ \boxed{ \delta_g^{\mathrm{obs}} =\delta_g^{\mathrm{or}}-\mathcal C-\mathcal M }. \tag{3.31}\]
There is a true background identity
\[ \frac{V_0(t_F)}{V_0(\bar t(\widetilde a))} =1+3\mathcal T \tag{3.32}\]
for a physical volume transported with the background source congruence between two nearby proper times. It should not be multiplied into Eq. 3.30 when \(\mathcal C\) and \(\mathcal M\) are the fixed-physical-ruler distortions derived above. Doing so would count the same background volume expansion twice. Equivalently, one may introduce evolving or comoving rulers and keep an explicit \(1+3\mathcal T\) factor; then the ruler distortions themselves must be shifted by the corresponding time-evolution terms. The invariant number count is the same, but the bookkeeping differs.
In the notation of Schmidt–Jeong cosmic rulers, an evolving ruler has extra terms proportional to \(\mathrm d\ln r_0/\mathrm d\ln a\) in the longitudinal and transverse distortions. With the sign conventions of this chapter, for a comoving ruler these shifts are
\[ \mathcal C_{\mathrm{com}}=\mathcal C+\mathcal T, \qquad \mathcal A_{ij}^{\mathrm{com}}=\mathcal A_{ij}+\mathcal T P_{ij}, \qquad \mathcal M_{\mathrm{com}}=\mathcal M+2\mathcal T, \tag{3.33}\]
so that, to linear order,
\[ (1+3\mathcal T) (1-\mathcal C_{\mathrm{com}}-\mathcal M_{\mathrm{com}}) =1-\mathcal C-\mathcal M. \tag{3.34}\]
This is the convention used in factored current-form presentations such as2. The present notes keep the fixed-physical-ruler convention because it follows directly from Eq. 3.4 and from the determinant in Section 3.5.
The bias response is naturally defined at fixed source proper time. Write
\[ n_g(x_s)=\bar n_g(a_F) [1+\delta_g^{\mathrm{pt}}(x_s)]. \tag{3.35}\]
Since \(\ln(a_F/\widetilde a)=\mathcal T\),
\[ \begin{aligned} \bar n_g(a_F) &=\bar n_g(\widetilde a) \left[1+\frac{\mathrm d\ln\bar n_g}{\mathrm d\ln a}\mathcal T\right], \end{aligned} \tag{3.36}\]
and hence
\[ \delta_g^{\mathrm{or}} =\delta_g^{\mathrm{pt}} +\frac{\mathrm d\ln\bar n_g}{\mathrm d\ln a}\mathcal T =\delta_g^{\mathrm{pt}}+(b_e-3)\mathcal T. \tag{3.37}\]
The combination \(b_e-3\) is the logarithmic derivative of the physical mean density, while \(b_e\) is the derivative of the comoving mean abundance. Substitution into Eq. 3.31 gives
\[ \boxed{ \delta_g^{\mathrm{obs}}(\widetilde z,\hat{\boldsymbol{n}}) =\delta_g^{\mathrm{pt}}+(b_e-3)\mathcal T-\mathcal C-\mathcal M } \tag{3.38}\]
for a volume-limited sample at linear order. At leading bias order,
\[ \delta_g^{\mathrm{pt}}=b_1\delta_m^{\mathrm{pt}}, \tag{3.39}\]
so
\[ \boxed{ \delta_g^{\mathrm{obs}} =b_1\delta_m^{\mathrm{pt}}+(b_e-3)\mathcal T-\mathcal C-\mathcal M }. \tag{3.40}\]
The gauge issue is resolved by specifying all objects operationally: the source density at fixed proper time, the clock relating that time to the observed redshift, and the radial and transverse ruler distortions of the observed volume element.
For a conserved comoving population, \(b_e=0\), so the physical mean density evolves as \(a^{-3}\) and the clock contribution is \(-3\mathcal T\). This term is not cancelled by an additional transported-volume factor in the present convention, because the physical volume at the source event has already been supplied by the ruler determinant. For a population whose comoving abundance evolves, \(b_e\) adds the genuine selection-history response.
Eq. 3.38 uses fixed physical rulers. If instead one factors the current-form Jacobian into a clock factor and a spatial determinant at constant proper time, the spatial determinant must be built from the corresponding evolving-ruler distortions, as in Eq. 3.33. Mixing the fixed-physical-ruler \(\mathcal C,\mathcal M\) with an extra \(1+3\mathcal T\) factor double counts the clock.
Eq. 3.38 is the geometric count for a volume-limited population. Flux, color, size, or other cuts define additional responses of the selected current. They must be included separately and are not part of the rest-frame bias relation.
3.7 Flux selection and the lensing coefficient
For a threshold sample, define
\[ s\equiv\frac{\mathrm d\log_{10}\bar N(<m)}{\mathrm dm}. \tag{3.41}\]
In the weak-lensing limit, the transverse area increase gives the dilution \(-2\kappa\) through \(-\mathcal M\). Magnification raises the flux of sources across the threshold and contributes \(+5s\kappa\). Therefore
\[ \Delta_g^{\mathrm{lens}}=(5s-2)\kappa. \tag{3.42}\]
The coefficient is the sum of a geometric area Jacobian and a sample-selection response.
The two contributions should be distinguished when generalizing the sample definition. The \(-2\kappa\) term is present for any count per observed solid angle because the same sources are spread over a changed apparent area. The \(5s\kappa\) term depends on how rapidly the selected abundance changes at the flux threshold and vanishes for a genuinely volume-limited sample.
3.8 Newtonian limit
For \(k\gg\mathcal H\), in the plane-parallel weak-field limit,
\[ \mathcal C\longrightarrow \frac{1}{\mathcal H}\partial_\parallel v_\parallel, \qquad \mathcal M\longrightarrow2\kappa. \tag{3.43}\]
The clock and potential terms in ordinary equal-redshift clustering are suppressed by powers of \(\mathcal H/k\) relative to density and the velocity gradient. Neglecting lensing and selection,
\[ \delta_g^{\mathrm{obs}} \longrightarrow b_1\delta_m -\frac{1}{\mathcal H}\partial_\parallel v_\parallel, \tag{3.44}\]
which is the Kaiser field of Chapter 1. The Newtonian formula is the short-distance limit of the same covariant observable, not a different definition of the catalogue field.
The hierarchy of terms can be understood dimensionally. The density and velocity gradient scale as order unity in \(\mathcal H/k\), a velocity or potential-gradient redshift term is typically suppressed by \(\mathcal H/k\), and a potential term by \((\mathcal H/k)^2\), modulo line-of-sight integrals and selection coefficients. These suppressions justify the Newtonian approximation on short scales but do not alter the observable definition.
3.9 Complete linear galaxy-clustering prescription
The preceding sections can now be assembled into a single prescription. For pressureless matter, synchronous-comoving gauge is the constant-matter-proper-time slicing, so
\[ \delta_m^{\mathrm{pt}}=\delta_m^{\mathrm{sc}}. \tag{3.45}\]
For a volume-limited sample, the complete linear observable is therefore
\[ \boxed{ \Delta_g^{\mathrm{obs}}(\widetilde z,\hat{\boldsymbol n}) =b_1\delta_m^{\mathrm{sc}} +(b_e-3)\mathcal T -\mathcal C-\mathcal M }. \tag{3.46}\]
Here the four terms have distinct origins: \(b_1\delta_m^{\mathrm{sc}}\) is the local source response at fixed proper time, \(\mathcal T\) moves that source response to the event selected at fixed observed redshift, and \(\mathcal C\) and \(\mathcal M\) are the radial and transverse Jacobians of the observed-to-physical coordinate map. A general sample adds a selection response \(\Delta_{\mathrm{sel}}\):
\[ \Delta_g^{\mathrm{obs}} =b_1\delta_m^{\mathrm{sc}} +(b_e-3)\mathcal T -\mathcal C-\mathcal M +\Delta_{\mathrm{sel}}. \tag{3.47}\]
For a flux threshold, \(\Delta_{\mathrm{sel}}\) contains the magnification response \(5s\kappa\); the geometric area dilution is already contained in \(-\mathcal M\). Other color, size, or detection cuts require their own response coefficients.
Substituting the definitions of the clock and ruler distortions gives a useful component form valid in any gauge:
\[ \boxed{ \begin{aligned} \Delta_g^{\mathrm{obs}}={}&b_1\delta_m^{\mathrm{sc}} +b_e\Delta\ln a+(b_e-3)H I_A +\frac12 h^i{}_i+v_\parallel +\partial_{\widetilde\chi}\Delta x_\parallel\\ &-2\widehat\kappa +2\frac{\Delta x_\parallel}{\widetilde\chi} +\Delta_{\mathrm{sel}}. \end{aligned} } \tag{3.48}\]
Although the separate terms on the right-hand side depend on the gauge used to perform the calculation, the displayed combination does not. The density is written as \(\delta_m^{\mathrm{sc}}\) because that is the gauge-invariant proper-time density entering the bias relation; all remaining fields may be evaluated in whichever gauge is convenient.
In synchronous-comoving gauge, \(A=B=v=0\) and \(I_A=0\), so \(\mathcal T=\Delta\ln a\). The same result becomes
\[ \boxed{ \begin{aligned} \Delta_g^{\mathrm{obs}}={}&b_1\delta_m^{\mathrm{sc}} +b_e(\Delta\ln a)_{\mathrm{sc}} +\frac12(h^i{}_i)_{\mathrm{sc}} +\partial_{\widetilde\chi}(\Delta x_\parallel)_{\mathrm{sc}}\\ &-2\widehat\kappa_{\mathrm{sc}} +2\frac{(\Delta x_\parallel)_{\mathrm{sc}}}{\widetilde\chi} +\Delta_{\mathrm{sel}}. \end{aligned} } \tag{3.49}\]
The redshift perturbation in this gauge follows directly from the photon-energy equation,
\[ (\Delta\ln a)_{\mathrm{sc}} =\frac12\int_0^{\widetilde\chi}\mathrm d\chi\, (h_\parallel')_{\mathrm{sc}}, \tag{3.50}\]
up to the chosen observer monopole convention. The radial displacement and convergence are obtained by integrating the spatial null-geodesic equations with the observer boundary data of Section 2.3. Thus Eq. 3.49 is a closed algorithm: solve the linear Einstein–matter system in synchronous-comoving gauge, ray trace from the observer, and insert the resulting endpoint and line-of-sight fields.
For a single growing adiabatic mode, every linear quantity in Eq. 3.46 is a linear functional of \(\delta_m^{\mathrm{sc}}\). It is useful to define transfer kernels by
\[ \mathcal T=\mathcal R_{\mathcal T}\,\delta_m^{\mathrm{sc}}, \qquad \mathcal C=\mathcal R_{\mathcal C}\,\delta_m^{\mathrm{sc}}, \qquad \mathcal M=\mathcal R_{\mathcal M}\,\delta_m^{\mathrm{sc}}, \qquad \Delta_{\mathrm{sel}}=\mathcal R_{\mathrm{sel}}\,\delta_m^{\mathrm{sc}}, \tag{3.51}\]
where the kernels include endpoint terms and line-of-sight integrals and therefore depend on \(k\), direction, and redshift. The master equation is then
\[ \boxed{ \Delta_g^{\mathrm{obs}}(\boldsymbol k,\hat{\boldsymbol n},\widetilde z) =\left[ b_1+(b_e-3)\mathcal R_{\mathcal T} -\mathcal R_{\mathcal C} -\mathcal R_{\mathcal M} +\mathcal R_{\mathrm{sel}} \right] \delta_m^{\mathrm{sc}}(\boldsymbol k,\widetilde z) }. \tag{3.52}\]
Eq. 3.52 should be understood as a transfer-function statement, not as a local multiplication in configuration space. In particular, lensing and integrated redshift terms sample the matter perturbation along the entire line of sight. Once the cosmological model fixes the linear transfer functions, however, \(\delta_m^{\mathrm{sc}}\) supplies the single stochastic initial field and the bracket supplies the complete deterministic map to the observed galaxy fluctuation.
The calculation has now closed the chain \[ \delta_m^{\mathrm{sc}} \longrightarrow \text{local tracer response} \longrightarrow \text{clock and ruler map} \longrightarrow \Delta_g^{\mathrm{obs}}(\widetilde z,\hat{\boldsymbol n}). \] Gauge freedom changes the intermediate representation, but not this map.
Summary
Key ideas. A ruler distortion is the ratio of a physical rest-frame separation, defined with the projector \(g_{\mu\nu}+u_\mu u_\nu\) so that the comparison is made at equal proper time, to the separation inferred from observed labels. Because both ends of that ratio are operational, the six distortions are gauge invariant while their metric, velocity, and displacement pieces are not. Only the radial dilation \(\mathcal C\) and the transverse trace \(\mathcal M\) can enter a scalar volume at linear order, since a single transverse vector cannot form a scalar. Finally, the observed count is the product of a source density and a volume ratio, and the convention chosen for the ruler determines which of the two carries the background expansion.
Main results.
- The covariant physical ruler Eq. 3.4 and the master expansion Eq. 3.13 from which all six distortions follow.
- The longitudinal, mixed, and transverse distortions Eq. 3.16, Eq. 3.17, and Eq. 3.18, with the transparent form Eq. 3.19.
- Magnification Eq. 3.22, ruler shear Eq. 3.23, and the distance perturbation Eq. 3.24.
- The volume Jacobian Eq. 3.28, in which \(\mathcal B_i\) is absent at linear order.
- The central result Eq. 3.38, its biased form Eq. 3.40, and the complete prescriptions Eq. 3.46 and Eq. 3.52.
- The flux-selection coefficient Eq. 3.42, separating the geometric dilution \(-2\kappa\) from the selection response \(5s\kappa\).
- The Newtonian limit Eq. 3.44, recovering Chapter 1.
What the next chapter builds on. Part I is now complete: the observed count has been expressed in terms of a proper-time source density, a clock, and two ruler distortions, all of them operationally defined on the past light cone. Chapter 4 turns to the complementary question. Instead of comparing a source with an observer across a finite null geodesic, it asks what a single freely falling observer can measure inside a small laboratory. The answer, curvature rather than connection, supplies the missing justification for the operator basis assumed in Chapter 1 and sets up the tidal physics of Chapter 5.