Large-Scale Structure Observables in General Relativity
Galaxy Clustering, Cosmic Rulers, and Local Tides
Preface
These lecture notes develop the relation between the matter perturbations of a cosmological model and the numbers that appear in a galaxy catalogue. They are written for graduate students in theoretical physics who are comfortable with actions, symmetries, perturbation theory, and general covariance, but who have had less occasion to ask how a survey turns photons into a statement about spacetime and matter.
That translation is not a secondary data-analysis step. It is part of the theoretical prediction, and it is the subject of the five lectures collected here.
Why relativistic observables matter
A cosmological model is confronted with observation only after it passes through a sequence of maps,
\[ \mathcal S[g_{\mu\nu},\Phi_A,\ldots] \longrightarrow \{g_{\mu\nu},\Phi_A\} \longrightarrow \{n_g,u_g^\mu,S_{ij},\ldots\} \longrightarrow \{\widetilde z,\hat{\boldsymbol{n}},F,\gamma,N,\ldots\} \longrightarrow \text{statistical inference}. \]
The first arrow is dynamics: an action and an initial state determine spacetime and matter fields. The second is source physics: galaxies and other tracers form and respond to their long-wavelength environment. The third is the relativistic observation map: photons propagate from the source to an observer, who records redshifts, directions, fluxes, shapes, and counts. Only the final objects are entries in a catalog. A proposal for new dark-sector physics or modified gravity can affect several arrows at once, and its observational meaning depends on keeping them distinct.
Figure placeholder. The chain from theory to data. A single horizontal schematic showing action, source formation, photon propagation along the past light cone, observer, and catalogue, with the intermediate variables of each stage labelled beneath the corresponding arrow.
Diffeomorphism invariance in operational form
In a field-theory calculation, gauge redundancy is often handled by choosing a convenient gauge and working with the resulting fields. In cosmology, that is not the end of the problem. A telescope does not report the matter-density contrast at a coordinate position. It reports a source at an observed redshift and direction, at a specified observer proper time. Under a diffeomorphism, the coordinate representation of the density, the metric perturbations, and the coordinate position assigned to the emission event all change. The complete observable does not.
The cancellation is not an additional prescription imposed after the calculation; it follows from perturbing a single covariantly defined measurement. This is the technical thread that runs through these notes. In 2 Gauge freedom and covariant number counts it appears as the pullback of a current three-form; in 3 Cosmic rulers, cosmic clocks, and the observed density as the ratio of a physical ruler to an inferred one; in 4 Fermi normal coordinates: the Manasse–Misner construction as the statement that a freely falling observer can remove the connection but not the curvature.
A coordinate-dependent field is not automatically unphysical, and a gauge-invariant combination is not automatically an observable. Define the measurement covariantly, specify the source and observer conditions, and only then perturb the result.
Why the dark universe requires observables
The dark universe is inferred through a logic of consistency. Dark matter, late-time acceleration, and primordial perturbations are not observed as labels attached to individual events. They enter a spacetime model that must account simultaneously for expansion, growth, peculiar velocities, clustering, lensing, and other measured maps. The strength of the standard cosmological picture is that these different observations can be organized into one consistent description.
Its phenomenological success does not by itself identify the microscopic nature of the dark sector. Understanding how the consistent picture is built is therefore a prerequisite for challenging it. A model that reproduces one observable while quietly changing the meaning of another has not been tested; it has been fitted.
Why this concerns theorists outside cosmology
A new degree of freedom, interaction, symmetry, or modification of gravity is not yet a cosmological prediction when its effect has only been computed for a gauge-dependent field. One must also determine how the theory changes tracer formation, local tidal response, photon propagation, and the actual variables used by an observer.
Conversely, the tools used here are familiar theoretical tools placed in an observational setting: functional response and effective field theory for galaxy bias in 1 Newtonian galaxy clustering: bias as a response and redshift-space distortions, differential forms and pullbacks for number counts in 2 Gauge freedom and covariant number counts, gauge symmetry for cosmic clocks and rulers in 3 Cosmic rulers, cosmic clocks, and the observed density, and the exponential map and local curvature for Fermi normal coordinates in 4 Fermi normal coordinates: the Manasse–Misner construction. A reader who has met these objects in another context will find the cosmological application unusual mainly in what it demands of them: they must all remain consistent inside a single measured number.
Large-scale structure as an application of general relativity
In this operational sense, large-scale-structure observables are close to an endgame of general relativity. The difficulty is not only solving the field equations. Covariance, gauge freedom, null propagation, the source and observer split, local freely falling frames, and matter response must all remain consistent in a single prediction.
Three failure modes recur, and each is treated explicitly in what follows. A coordinate artifact can resemble a physical large-scale effect. A genuine physical effect can move between intermediate variables under a gauge transformation. A locally measurable gravitational field begins only at tidal order. The output of the calculation is not a metric perturbation, but a relation among quantities that an observer can measure.
The dark universe requires ideas beyond a phenomenological parameterization. The purpose of these lectures is to provide the translation layer that allows such ideas to become controlled predictions for data.
Course map
These notes develop the relation between matter perturbations, biased tracers, and observables on the observer’s past light cone. The discussion is organized around three constructions: tracer bias as a functional response to long-wavelength gravitational fields, number counts as the pullback of the galaxy-current three-form to observed coordinates, and local gravitational measurements through the Manasse–Misner construction of Fermi normal coordinates. At linear order, cosmic clocks and cosmic rulers convert a density defined on a source proper-time hypersurface into an observed number-count field. The same local-curvature viewpoint then relates the Newtonian tidal operator, intrinsic alignment, and the response to gravitational waves.
The main references are Secs. 2.5 and 9.3 of1,2,3,4, and5.
The central distinction is
\[ \delta_m\big|_{t_F} \xrightarrow{\text{formation response}} \delta_g\big|_{t_F} \xrightarrow{\text{clock, ruler, and pullback}} \delta_g^{\mathrm{obs}}(\widetilde z,\hat{\boldsymbol{n}}). \]
The first map describes how a selected tracer population responds to its long-wavelength environment. The second map describes how source fields and spacetime geometry produce a field on the observed redshift and angle manifold. Local gravitational effects will be expressed in terms of curvature measured in a freely falling orthonormal frame.
The distinction in this map is not only notation. The three fields answer different questions. The matter density contrast describes the state of the matter fluid on a specified spacetime slicing. The galaxy density contrast describes a selected population on a specified source hypersurface. The observed field is the number assigned to a cell in measured redshift and direction at a fixed observer proper time. They coincide only after taking a limit in which the relevant time slicings and coordinate maps can be identified.
A useful rule throughout is to define the geometric or operational object before perturbing it. A gauge transformation then changes the coordinate representation of the source field and the coordinate location of the observed source, but not the full observable. This viewpoint avoids treating gauge invariance as a cancellation to be guessed after a calculation; the cancellation follows from expanding one covariant construction.
The discussion separates two notions of observability. Number counts and ruler distortions are nonlocal observables on the past light cone: they contain source, propagation, and observer contributions. Tidal fields are local observables: after transforming to a freely falling frame, the connection can be removed on the observer’s worldline, while curvature remains. Both notions are needed for large-scale-structure observables, and the division of these notes into two parts follows this split.
How to read these notes
The chapter Notation and conventions collects the assumptions, signature, and symbol conventions used throughout; readers who prefer to start with physics can skip it and return when a symbol is unfamiliar.
Part I builds the observed number count. 1 Newtonian galaxy clustering: bias as a response and redshift-space distortions sets the Newtonian baseline and defines bias as a response. 2 Gauge freedom and covariant number counts replaces Euclidean number conservation by a covariant pullback. 3 Cosmic rulers, cosmic clocks, and the observed density assembles clocks, rulers, and the source density into the central result.
Part II turns to what a freely falling observer can measure locally. 4 Fermi normal coordinates: the Manasse–Misner construction constructs Fermi normal coordinates. 5 Local tides, intrinsic alignment, and gravitational waves applies the resulting local curvature to tidal bias, intrinsic alignment, and gravitational waves.
The appendices are reference material rather than a continuation of the argument. They collect response conventions (Appendix A: Response conventions and functional derivatives), pullback and Lie-derivative identities (Appendix B: Pullback and Lie-derivative identities), a gauge-transformation table (Appendix C: Gauge-transformation table), and the index-order algebra behind the Manasse–Misner metric (Appendix D: Manasse–Misner index order and the quadratic algebra).
Four kinds of highlighted box appear in the text, and each has a fixed meaning:
| Box | Meaning |
|---|---|
| Key idea | A conceptual anchor. The physics that survives when the algebra is forgotten. |
| Historical note | Where a construction came from, in two or three sentences. |
| Tip | An intuition or a mnemonic that is useful but not load bearing. |
| Caution | A convention trap. Ignoring one of these produces a wrong sign or a double count. |
Passages marked as a figure placeholder indicate where an illustration would help. The artwork has not yet been drawn.