Comoving Volume and Number Counts

A redshift bin and a patch of sky select a shell on the past light cone, not a Euclidean box. Its comoving volume combines a transverse area $D_M^2d\Omega$ with the radial Jacobian $d\chi/dz$. This page derives that volume element, shows how a comoving population becomes an observed count, and separates geometry from source evolution and survey selection. It treats the homogeneous FLRW background; instrument calibration, current distance-ladder results, and relativistic perturbations of observed counts are handled elsewhere.

Required background. Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields capability used to identify the radial null coordinate and observed sky direction.

Helpful background. The Einstein Equation supplies the Einstein Equations behind $H(z)$; Relativistic Matter and Compact Objects supplies the Relativistic Matter and Compact Objects populations whose densities may be counted; Relativistic Cosmology orients the background distance conventions; and The Wave Equation and Characteristics supplies the Waves, Green Functions, and Geometric Optics interpretation of the past-light-cone rays.

The light-cone volume Jacobian

Normalize $a_0=1$ and write the reference spatial metric as

$$d\ell_c^2 =d\chi^2+S_K^2(\chi)d\Omega^2.$$

The dimensionful constant $K$ has units of inverse length squared, and

$$S_K(\chi)= \begin{cases} \sin(\sqrt K\,\chi)/\sqrt K, & K\gt0,\\ \chi, & K=0,\\ \sinh(\sqrt{-K}\,\chi)/\sqrt{-K}, & K\lt0. \end{cases}$$

Its comoving volume element is

$$dV_c =S_K^2(\chi)\,d\chi\,d\Omega.$$

The transverse comoving distance is $D_M=S_K(\chi)$. The equations below assume an expanding branch with $H\gt0$ on which redshift is monotonic. Along a radial null ray on that branch,

$$\frac{d\chi}{dz}=\frac{1}{H(z)}$$

in units $c=1$. Therefore

$$\boxed{ \frac{dV_c}{dz\,d\Omega} =\frac{D_M^2(z)}{H(z)} } \qquad (c=1).$$

Restoring units multiplies the right-hand side by $c$. Equivalently, because $D_M=(1+z)D_A$,

$$\frac{dV_c}{dz\,d\Omega} =\frac{c(1+z)^2D_A^2(z)}{H(z)}.$$

This is the same result given in Hogg 2000, §9, printed p. 7, eq. (28) (Open PDF). The factors have a transparent origin: $D_M^2$ supplies the comoving transverse area and $c/H$ supplies radial comoving length per unit redshift. Their product has dimensions of volume.

For a contracting monotonic branch, a positive count Jacobian uses $cD_M^2/|H|$. If one observed redshift labels several emission events across a bounce or turnaround, the branches must be integrated and summed separately; a single-valued $dV_c/dz$ is then not a global description.

The corresponding physical volume on the emission-time slice is smaller:

$$dV_{\rm phys}(z) =a^3(z)dV_c =\frac{dV_c}{(1+z)^3}.$$

Mixing physical density with comoving volume introduces exactly this missing factor of $(1+z)^3$.

From density to expected counts

Let $n_c(\vartheta,z)$ be the comoving number density per intrinsic source parameter $d\vartheta$. A survey selection probability $\mathcal S(\vartheta,z,\boldsymbol n)$ may depend on source properties, redshift, and sky direction. The expected differential count is

$$\boxed{ \frac{dN}{dz\,d\Omega} =\frac{cD_M^2(z)}{H(z)} \int d\vartheta\, n_c(\vartheta,z) \mathcal S(\vartheta,z,\boldsymbol n) }.$$

Geometry is only the prefactor. Formation, destruction, mergers, and luminosity or color evolution live in $n_c(\vartheta,z)$; flux limits, quality cuts, masks, and detection efficiency live in $\mathcal S$. A constant comoving density is appropriate only for a conserved population moving with the Hubble flow. Even then, an observed sample generally does not remain constant because its selection can evolve with redshift.

If measurement scatter matters, introduce observed data $y$ and a response density $p(y\mid\vartheta,z,\boldsymbol n)$:

$$\frac{dN}{dz\,d\Omega\,dy} =\frac{cD_M^2}{H} \int d\vartheta\, n_c(\vartheta,z)\, p(y\mid\vartheta,z,\boldsymbol n)\, \mathcal S(y,z,\boldsymbol n).$$

This forward form prevents an important category error: a catalog histogram is not the underlying density multiplied by volume unless measurement noise and selection are negligible.

For a finite bin $B$, the model prediction is the integral of this expression over the bin, not its value at an arbitrarily chosen effective redshift. Calling the observed count Poisson-distributed adds another assumption: independent sources with a fixed intensity. Galaxy clustering, shared calibration errors, and uncertain selection produce covariance beyond Poisson shot noise. The geometric volume element remains unchanged, but the likelihood for the count does not. Keeping the Jacobian, population model, selection, and stochastic model as separate factors makes each assumption testable.

For transient events, let $\mathcal R_c$ be a source-frame rate per comoving volume and source proper time. Observer time is dilated by $dt_o=(1+z)dt_s$, so

$$\frac{d\dot N_o}{dz\,d\Omega} =\frac{cD_M^2}{H} \frac{\mathcal R_c(z)}{1+z} \times\text{selection}.$$

The extra $1/(1+z)$ belongs to rates, not to counts of persistent objects.

Euclidean and curvature checks

At small redshift,

$$D_M(z)=\frac{cz}{H_0}+O(z^2), \qquad H(z)=H_0+O(z).$$

Hence

$$\frac{dV_c}{dz\,d\Omega} =\frac{c^3z^2}{H_0^3}+O(z^3).$$

Integrating over redshift and the full sky gives

$$V_c(<z) =\frac{4\pi}{3} \left(\frac{cz}{H_0}\right)^3 +O(z^4),$$

the ordinary Euclidean sphere. This check catches both a missing radial factor and an incorrect power of distance.

Away from the flat case, the shell uses $S_K^2(\chi)$ rather than $\chi^2$. The curvature dependence is therefore geometric even if two models happened to share the same radial integral. Conversely, differentiating the cumulative volume must return the shell element:

$$\frac{d}{dz} \left[ \int_0^z dz'\int d\Omega\, \frac{cD_M^2(z')}{H(z')} \right] =\int d\Omega\, \frac{cD_M^2(z)}{H(z)}.$$

Boundaries of the background formula

The redshift in the derivation is the exact FLRW background redshift. Real observed redshifts receive peculiar-velocity and gravitational contributions; lensing changes both angular area and selection by magnification. Those terms are developed in the later past-light-cone chapter. Survey masks, calibration, and source classification must also be specified before this geometric Jacobian can support an inference. None of those effects is legitimately absorbed into an “effective volume” without stating its model.

Common pitfalls

Using $\chi^2$ in curved space. The radial coordinate becomes a transverse area through $D_M^2=S_K^2(\chi)$.

Combining mismatched densities and volumes. Use $n_c\,dV_c$ or $n_{\rm phys}\,dV_{\rm phys}$; the two densities differ by $a^3$.

Adding time dilation to static counts. The factor $(1+z)^{-1}$ converts a source-frame event rate to observer time. It is not part of the number of persistent sources in a snapshot.

Exercises

A population has constant comoving density $n_0$, perfect selection, and is observed over sky fraction $f_{\rm sky}$ at sufficiently small redshift. Find the expected count below $z_{\max}$ and its derivative with respect to $z_{\max}$.

Solution

The survey solid angle is $4\pi f_{\rm sky}$. In the Euclidean limit,

$$N(<z_{\max}) =n_0(4\pi f_{\rm sky}) \int_0^{z_{\max}} \frac{c^3z^2}{H_0^3}\,dz =\frac{4\pi}{3}f_{\rm sky}n_0 \left(\frac{cz_{\max}}{H_0}\right)^3.$$

Differentiating gives

$$\frac{dN}{dz_{\max}} =4\pi f_{\rm sky}n_0 \frac{c^3z_{\max}^2}{H_0^3},$$

which is precisely the density times the survey’s shell volume. The $z_{\max}^3$ cumulative scaling and $z_{\max}^2$ differential scaling are the two independent Euclidean checks.

References

• Hogg, David W. “Distance Measures in Cosmology.” arXiv:astro-ph/9905116v4, 16 December 2000. Open PDF.