Conformal Time and Comoving Coordinates

Conformal time and comoving coordinates separate the changing physical scale of an FLRW universe from a fixed spatial coordinate grid. Defining $d\eta=dt/a(t)$ makes the metric conformal to a time-independent spatial geometry and turns radial null paths into 45-degree curves in $(\eta,\chi)$ coordinates. The transformation simplifies causal and wave calculations, but it does not remove expansion: proper clocks, lengths, wavelengths, affine parameters, and curvature still depend on $a$.

Required background. Pseudo-Riemannian Metrics and Signature supplies Lorentzian metrics, causal cones, and conformal rescaling; Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields background; this page uses its null geodesic and affine-parametrization material to distinguish a path from the rate at which it is traversed, not Jacobi-field theory.

Helpful background. The Einstein Equation supplies the Einstein equations determining $a(t)$, while Relativistic Matter and Compact Objects supplies the matter rest frame relative to which distances are called comoving.

Conformal time and the FLRW line element

Start from cosmic time,

$$ds^2=-dt^2+a^2(t)d\Sigma_K^2.$$

Here $a$ is the dimensionless scale factor normalized by $a_0=1$, and the time-independent comoving metric $d\Sigma_K^2$ has constant curvature $K$ with dimensions of inverse length squared. This is the translated convention from the normalized-curvature form on the FLRW-geometry page; $K$ will never denote a Fourier wave number.

The translation includes the coordinates, not just the scale factor. If the normalized-curvature package uses $a_{\rm dim}$, a dimensionless radial coordinate $\chi_{\rm n}$, normalized curvature $\kappa$, and $d\eta_{\rm n}=dt/a_{\rm dim}$, choose $a_{{\rm dim},0}=a_{\rm dim}(t_0)$ and set

$$a=\frac{a_{\rm dim}}{a_{{\rm dim},0}}, \qquad K=\frac{\kappa}{a_{{\rm dim},0}^{2}}, \qquad \chi=a_{{\rm dim},0}\chi_{\rm n}, \qquad \eta=a_{{\rm dim},0}\eta_{\rm n}.$$

Thus $\chi$ and $\eta$ have dimensions of length in the present package, while angular coordinates remain dimensionless. Flat Cartesian coordinates $x^i$ are rescaled in the same way as $\chi$, and their conjugate Fourier wave numbers have dimensions of inverse length. The constant rescaling leaves the physical metric unchanged.

On an interval where $a\gt0$, define

$$d\eta=\frac{dt}{a(t)}, \qquad \mathcal H\equiv\frac{a'}{a}=aH,$$

where a prime denotes $d/d\eta$ and a dot denotes $d/dt$. The metric becomes

$$ds^2=a^2(\eta)\left[-d\eta^2+d\Sigma_K^2\right].$$

The factor $a^2$ multiplies both temporal and spatial terms, so the light cones are those of the conformal metric $d\widetilde s^2=-d\eta^2+d\Sigma_K^2$. The numerical range of $\eta$ depends on the complete expansion history and on its arbitrary additive origin. Baumann introduces the coordinate and its causal use in Baumann, arXiv:0907.5424v2 (2012), §3.2, pp. 16–18 (PDF).

A comoving coordinate separation is fixed along the background fundamental flow. Its instantaneous physical length on a cosmic-time slice is

$$\ell_{\rm phys}(t)=a(t)\ell_{\rm com}.$$

The normalization of $a$ is conventional. If $a_0=1$, present physical and comoving lengths have the same numerical value; at another epoch they do not. A constant rescaling of $a$ accompanied by an inverse rescaling of comoving coordinates changes neither the metric nor an observable distance.

Null paths and causal integrals

For a radial ray in geodesic-polar spatial coordinates, the present $\chi$ is the dimensional coordinate $\chi=a_{{\rm dim},0}\chi_{\rm n}$, and

$$0=ds^2=a^2(\eta)(-d\eta^2+d\chi^2),$$

and therefore

$$\frac{d\chi}{d\eta}=\pm1.$$

The comoving radial interval crossed between emission and observation is

$$\Delta\chi=\eta_o-\eta_e =\int_{t_e}^{t_o}\frac{dt}{a(t)}.$$

This is why conformal time organizes both distance and horizon calculations. An initial endpoint $\eta_i$ yields a particle-horizon candidate $\eta-\eta_i$; a future endpoint $\eta_f$ yields an event-horizon candidate $\eta_f-\eta$. Whether either integral is finite is a global property of the relevant past or future history, not a conclusion from the instantaneous value of $H$.

Conformal transformations preserve unparametrized null geodesics, so the same curve is null in $g_{ab}$ and $\widetilde g_{ab}=a^{-2}g_{ab}$. They do not preserve an affine parameter. If $\widetilde\lambda$ is affine for the conformal metric, then a physical-metric affine parameter satisfies

$$d\lambda=a^2\,d\widetilde\lambda$$

along the ray, up to an overall constant rescaling. This follows by inserting $g_{ab}=a^2\widetilde g_{ab}$ in the connection transformation and demanding $k^b\nabla_bk^a=0$. Frequency and focusing therefore cannot be read from the 45-degree path alone. Tong uses the conformal null-path construction to derive particle- and event-horizon criteria in Tong 2019, §1.1.4, pp. 15–19 (PDF).

Comoving wave numbers and physical scales

On a flat reference slice, expand a perturbation using a comoving wave vector,

$$f(\eta,\mathbf x) =\int\frac{d^3k}{(2\pi)^3} f_{\mathbf k}(\eta)e^{i\mathbf k\cdot\mathbf x}.$$

The coordinate wave number $k=|\mathbf k|$ is constant for a fixed Fourier label and has dimensions of inverse length in this package. The physical wave number and wavelength are

$$k_{\rm phys}=\frac{k}{a}, \qquad \lambda_{\rm phys}=\frac{2\pi a}{k}.$$

This distinction recurs throughout perturbation theory, structure formation, primordial black holes, and gravitational-wave transfer. A comparison written as $k\simeq aH=\mathcal H$ is a comparison with the comoving Hubble scale. It is often a useful dynamical marker, but the Hubble radius is not generally a particle or event horizon.

For curved slices, Fourier plane waves are replaced by eigenfunctions of the spatial Laplacian. The eigenvalue label and its relation to a physical wave number depend on $K$ and on scalar, vector, or tensor type. Writing a flat-space $e^{i\mathbf k\cdot\mathbf x}$ expansion in a curved model without this translation silently changes the spectrum.

What conformal simplification does not erase

Null cones and angles are conformally natural, but massive-particle proper times, physical areas and volumes, geodesic affine parameters, and curvature are not invariant under $g_{ab}\mapsto a^2g_{ab}$. For example,

$$\sqrt{-g}=a^4\sqrt{-\widetilde g}$$

in four dimensions, and an observer’s proper time is $dt=a\,d\eta$. The physical curvature scalar contains $\mathcal H'$ and $\mathcal H^2$ even when the conformal metric is flat. Conformal coordinates therefore reorganize the expansion; they do not transform an FLRW spacetime into Minkowski spacetime.

The method also has a domain. If $a$ reaches zero or diverges, the conformal coordinate may place a physical or merely coordinate boundary at finite or infinite $\eta$. Completeness and extendibility require affine geodesics and curvature, not inspection of the coordinate range alone. Generic exact-spacetime classification remains outside this chapter; here the method is used only for the homogeneous cosmological background.

Exercises

Suppose $a(t)=C t^p$ for $t\gt0$, with $C\gt0$ and $p\neq1$. Find $a(\eta)$ up to constant factors. Check radiation expansion $p=1/2$, matter expansion $p=2/3$, and explain separately what happens for exponential de Sitter expansion.

Solution

Integration gives

$$\eta-\eta_0=\frac{t^{1-p}}{C(1-p)}.$$

Thus $t\propto|\eta-\eta_0|^{1/(1-p)}$ and

$$a(\eta)\propto|\eta-\eta_0|^{p/(1-p)}.$$

Radiation gives $a\propto\eta$ and matter gives $a\propto\eta^2$ after a convenient origin choice. For de Sitter, $a=e^{Ht}$ instead yields $\eta=-e^{-Ht}/H$ on the expanding flat patch, hence $a(\eta)=-1/(H\eta)$ with $\eta\lt0$. Its infinite cosmic future lies at the finite conformal endpoint $\eta=0$.

References

• Baumann, Daniel. “TASI Lectures on Inflation.” arXiv:0907.5424v2, 30 November 2012. Open PDF.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas Term 2019. Official PDF.