Relativistic Cosmology

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Why does distant light arrive redshifted? How did a nearly uniform hot universe form galaxies? What can observations tell us about its earliest moments and its future? Relativistic cosmology connects these questions through spacetime geometry, the evolution of matter and radiation, and the way signals reach observers.

This volume begins with the Friedmann–Lemaître–Robertson–Walker (FLRW) description of a homogeneous and isotropic universe. It then develops thermal history and small departures from uniformity, follows them into the cosmic microwave background (CMB) and large-scale structure, and reaches global evolution, primordial black holes, cosmological gravitational waves, and inference from data. The intended starting point is general relativity; the more specialized thermal, quantum, and statistical preparation is linked where it is needed.

Helpful background. The Einstein Equation supplies Einstein Equations and the site sign convention. Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects stress–energy, fluids, kinetic theory, and microphysical inputs. Individual chapters link the exact geometry, thermodynamics, field theory, probability, global-structure, or numerical preparation they use.

Geometry, matter, and observation form one chain

The starting point is not a fitted value of the Hubble constant but a spacetime model and a family of observers. Homogeneity and isotropy select the FLRW line element; comoving proper time defines $H=\dot a/a$ ; Einstein’s equation and stress–energy conservation determine how a declared matter model can evolve. With dimensionless $a(t_0)=1$ and dimensionful reference curvature $K$ ,

H^2+\frac{K}{a^2} =\frac{8\pi G}{3}\rho+\frac{\Lambda}{3}, \qquad \dot\rho+3H(\rho+p)=0,

and the spatial scalar curvature is ${}^{(3)}R=6K/a^2$ . These equations do not identify the material components by themselves. An equation of state, interaction law, scalar-field model, or kinetic description supplies the closure. Ellis and van Elst develop the observer split and FLRW reduction in Ellis and van Elst 1999, §§2.1 and 4.1–4.3, printed pp. 6–11 and 21–24 (Open PDF).

Observation enters through null propagation. Redshift is the ratio of measured photon frequencies, $1+z=(-U\cdot k)_e/(-U\cdot k)_o$ , not simply a coordinate label. Angular and luminosity distances require a null bundle; number counts require a source current, volume map, selection, and detector response. Perturbations then add a second layer: the event at fixed observed redshift is displaced from the event at fixed background time. The resulting light-cone observable is gauge invariant only after source, path, and observer terms are combined.

The same discipline applies to statistical claims. A parameter is inferred from a named data vector, calibration and selection model, likelihood, nuisance parameters, priors, and model class. A compact schematic is

p(\theta,\phi\mid d,S,M) \propto p(d\mid\theta,\phi,S,M) p(\theta,\phi\mid S,M),

where $\theta$ are cosmological parameters, $\phi$ are nuisance parameters, $S$ is the selection event, and $M$ is the model. Removing any of these conditions can turn a reproducible inference into an apparently universal number.

The thirteen-chapter path

The order follows the scientific dependencies.

Cosmological Spacetimes and Observers derives FLRW geometry, comoving observers, cosmic and conformal time, redshift, and the controlled Bianchi departure from isotropy before choosing a cosmic matter model.
Friedmann Dynamics and Cosmic Components reduces Einstein’s equation and conservation to the constraint, acceleration, continuity, scaling, curvature, vacuum, and multicomponent expansion laws.
Cosmic Distances and Causal Scales propagates null rays and bundles to construct lookback time, distances, reciprocity, volume, horizons, candles, rulers, and cosmographic approximations.
Thermal History and Relic Abundances follows entropy, reaction rates, decoupling, neutrino reheating, nucleosynthesis, recombination, and thermal relic freeze-out through the changing expansion rate.
Cosmological Perturbations and Gauge Invariance builds scalar–vector–tensor decomposition, gauge transformations, invariant potentials and curvature, regular initial modes, and the Einstein–Boltzmann system.
Inflation and Primordial Fluctuations develops accelerated expansion, scalar-field slow roll, canonical quantization, primordial spectra, reheating dependence, and competing early-universe mechanisms.
The Cosmic Microwave Background carries primordial modes through visibility, acoustic oscillations, polarization, lensing, foregrounds, calibration, and versioned likelihoods.
Relativistic Structure Formation evolves matter through linear growth, transfer, shell crossing, nonlinear clustering, halos, bias, redshift-space distortions, and relativistic large-scale limits.
Inhomogeneity and the Observed Light Cone constructs distance, number-count, lensing, velocity, and averaging observables on the actual past light cone rather than a fictitious constant-time survey slice.
Cosmological Horizons, Singularities, and Global Evolution separates causal horizons, geodesic completeness, theorem hypotheses, future fates, and the extra stability assumptions required by bouncing, cyclic, or emergent scenarios.
Primordial Black Holes maps enhanced primordial statistics into collapse, mass functions, evaporation, accretion, clustering, mergers, and conditional observational constraints.
Cosmological Gravitational-Wave Backgrounds compares inflationary, phase-transition, defect, and scalar-induced sources, transfers them across cosmic history, and asks how they can be separated from unresolved astrophysical populations.
Cosmological Inference and Model Appraisal closes the chain with base and derived parameters, normalized likelihoods, correlated data combination, model comparison, tension diagnostics, and reproducible computation.

Tong’s lecture notes provide a compact background-to-perturbations route in Tong 2019, §§1–3 (official PDF). The volume goes further where proof hypotheses, observational release contracts, nonlinear limitations, and current research disagreements must be kept explicit.

One mode, many observables

A useful thread is to follow one comoving Fourier mode. The site uses $e^{i\mathbf k\cdot\mathbf x}$ , so $k$ is a comoving angular wavenumber and $k/a$ is physical. In conformal Newtonian gauge,

ds^2=a^2(\eta) \left[-(1+2\Phi)d\eta^2 +(1-2\Psi)\delta_{ij}dx^i dx^j\right].

Gauge transformations change coordinate perturbations but not a correctly constructed observable. Malik and Wands derive the transformation and gauge-invariant dictionaries in Malik and Wands 2009, §§6–9, printed pp. 31–51 (Open PDF). The pages use that dictionary as a package; importing only a potential sign or velocity definition from another source can reverse an Einstein constraint or Doppler term.

During inflation, quantum mode functions can seed a primordial curvature spectrum. After reheating, the mode remains in the regular superhorizon solution until the relevant dynamical scales catch it. The Einstein–Boltzmann hierarchy transfers it into photon multipoles, matter perturbations, and metric potentials. At recombination those sources project into CMB temperature and polarization. Later, matter growth, bias, redshift-space mapping, lensing, and observer terms determine galaxy and shear observables. An enhanced scalar mode may also source primordial-black-hole collapse and second-order tensor radiation, but both predictions must use the same scalar statistics rather than independently tuned approximations.

This thread exposes several distinctions that recur throughout the volume:

$k=\mathcal H$ or $k=aH$ is a dynamical crossing estimate, not a definition of a particle or event horizon;
the Bardeen potentials, curvature perturbation, matter density contrast, tracer density, and observed number contrast are different variables;
a transfer function maps declared initial data through declared dynamics; it is not a detector response or a likelihood;
a nonlinear fitting formula is tied to its simulation, halo definition, tracer selection, and calibration range;
a theorem about geodesic incompleteness does not by itself prove curvature blow-up, inextendibility, or a unique beginning.

Ma and Bertschinger give a canonical analytic and numerical perturbation system in Ma and Bertschinger 1995, §§2–8 (Open PDF). Here it is paired with explicit gauge translation, hierarchy closures, convergence checks, and observational forward models.

Evidence has a time coordinate too

Background identities and theorem hypotheses can be durable; release-specific calibrations, likelihoods, parameter summaries, tension significances, and exclusion regions are not. The observational comparisons cover results through 2026-08-19. They identify data and software versions, preserve selection and covariance, compare alternatives, and explain what could change the interpretation. The cutoff is not a claim that the subject stopped evolving.

The volume uses several levels of check:

Claim type	Required check
Analytic identity	signs, dimensions, normalization, and an independent limit
Proof-critical result	exact hypotheses, conclusion, locator, and counterexample when a hypothesis is removed
Numerical calculation	pinned inputs, rerun command, convergence, analytic benchmark, and visible failure mode
Observational inference	release, calibration, cuts, covariance, nuisance parameters, priors, and residual or robustness test
Frontier comparison	dated comparison, competing formulations, contrary result or null test, and a test that could change the conclusion

For example, a diagonal-only covariance can make a distance ratio appear more precise; a low-order cosmographic series can change dramatically under a redshift-basis transformation; an abundance exclusion cannot be applied pointwise to an extended primordial-black-hole mass function; and a stochastic gravitational-wave slope cannot identify an early-universe source without transfer and foreground models. The remedy is not to suppress uncertainty but to expose the calculation that carries it.

Reader routes

A first encounter with cosmology. Start with FLRW Geometry and Cosmological Redshift, then work through The Friedmann Equations and Multi-Component Expansion Histories. Follow with Angular-Diameter and Luminosity Distances and Causal Scales. These pages give a first complete account of expansion, light propagation, and observable distances before the thermal and perturbation calculations. Work through the solved exercises before advancing; the solutions expose the reasoning, so they also serve as worked examples.

For a graduate core, read Chapters 1–6 in order, then Chapters 7–8. This route moves from geometry and background evolution to perturbations, the early universe, radiation, and structure. A relativistic-observable route uses Chapters 1–3, 5, and 9–10; it emphasizes observers, null bundles, gauge invariance, light-cone projection, and global causal questions.

For the early universe, take Chapters 2 and 4–7 before Chapters 11–12. The shared questions are when a mode is generated, when it enters the Hubble scale, which species or source stress it couples to, and which later transfer features survive. For data analysis, begin with distances in Chapter 3 and then use Chapters 7–9 and 13. This route keeps measured summaries, calibration, covariance, latent fields, nuisance parameters, and selection distinct.

The computation-facing route connects the multicomponent Friedmann quadrature, freeze-out integration, Einstein–Boltzmann hierarchy, CMB lensing remap, Zel’dovich flow, halo normalization, primordial-black-hole population integral, cosmological-wave transfer, and inference sampler. Each bounded benchmark is a transparent check, not a substitute for a maintained production code.

Boundaries with neighboring volumes

Spacetime Geometry and General Relativity develop Lorentzian geometry, geodesics, curvature, Einstein’s equation, and generic stress–energy. Causality and Global Structure and Mathematical Relativity develop the general causal hierarchy and rigorous theorem framework. This volume applies those results to cosmological symmetry, observers, expansion, null propagation, and global histories.

Relativistic Matter and Compact Objects develops detailed microphysics and transport. Gravitational Collapse and Black-Hole Dynamics develops nonlinear collapse and critical phenomena generally. QFT in Curved Spacetime and Semiclassical Gravity develop Hawking-radiation derivations and renormalized quantum stress–energy. The primordial-black-hole chapter uses those inputs for cosmological abundance and population evolution without rederiving them.

Gravitational Radiation and Waveforms develops wave generation, propagation, and strain conventions; Gravitational-Wave Astronomy develops instrument response, calibrated searches, and observing-run likelihoods. Gravity Beyond General Relativity constructs non-GR theories, while this volume states the GR baseline and treats altered equations as declared model inputs. Maintained executable environments and interactive workflows accompany the relevant scientific topics; the Library pages state the equations, verification targets, and failure modes those workflows must satisfy.

Exercises

Expansion, redshift, and density

Two freely comoving galaxies have fixed comoving separation in an expanding FLRW universe. The scale factor doubles between two times. How do their proper separation, the wavelength of a freely propagating photon, the nonrelativistic matter density, and the radiation energy density change? Assume separately conserved pressureless matter and freely propagating radiation. Which answer contains an effect beyond dilution of particle number?

Solution

Proper separation and photon wavelength both double because each is proportional to $a$ . A comoving volume grows by $2^3=8$ , so its conserved particle number gives $\rho_m\propto a^{-3}$ and the matter density falls to one eighth of its original value. Radiation has the same number dilution, but each photon also loses half its energy through redshift. Thus $\rho_r\propto a^{-4}$ and the radiation energy density falls to one sixteenth. The extra factor of two is the change in energy per photon. Consequently $\rho_r/\rho_m$ halves: expansion can change which component dominates even without converting one component into another.

Check the full chain

Consider a spatially flat universe containing radiation and matter, with

E^2(a)=\frac{H^2(a)}{H_0^2} =\Omega_{r0}a^{-4}+\Omega_{m0}a^{-3}.

First find the equality scale factor. Next consider a scale-invariant primordial tensor mode that enters well inside radiation domination. Find the local present-day transfer slope and convert $k=10^{12}\,\mathrm{Mpc}^{-1}$ to cyclic frequency. Finally explain why a measured excess with that slope would not, by itself, establish an inflationary origin.

Solution

Equality occurs when the two terms are equal:

\Omega_{r0}a_{\rm eq}^{-4} =\Omega_{m0}a_{\rm eq}^{-3}, \qquad a_{\rm eq}=\frac{\Omega_{r0}}{\Omega_{m0}}.

For modes entering during a decelerating constant-equation-of-state era ( $w\gt-1/3$ ), $H\propto a^{-3(1+w)/2}$ . The entry condition $k=a_kH(a_k)$ therefore gives $a_k\propto k^{-2/(1+3w)}$ . After entry a freely propagating tensor amplitude decays as $a^{-1}$ , so its present power acquires a factor $a_k^2$ . For a scale-invariant initial spectrum, the wave energy per logarithmic wavenumber is consequently proportional to $k^2a_k^2$ . Its passive transfer slope is

n_{\rm tr}=\frac{2(3w-1)}{1+3w}.

Radiation has $w=1/3$ , so $n_{\rm tr}=0$ : a scale-invariant primordial tensor spectrum remains locally flat in $\Omega_{{\rm GW},0}$ , apart from changes in relativistic degrees of freedom and free-streaming damping. These assumptions and their exceptions are developed in Gravitational-Wave Transfer through Cosmic History. The frequency is

f_0=1.546\times10^{-15}\,\mathrm{Hz} \left(\frac{k}{1\,\mathrm{Mpc}^{-1}}\right) =1.546\times10^{-3}\,\mathrm{Hz}.

A flat local spectrum is not an origin tag. A string network, a broad phase-transition flank, a scalar-induced spectrum, or an unresolved astrophysical population may approximate the same slope over a finite band. The inference must compare response-folded source and transfer templates, anisotropy or higher-order information where available, foreground and selection models, covariance, and prior sensitivity. The background calculation predicts a shape under stated assumptions; it does not replace the observational likelihood.

References

Ellis, George F. R., and Henk van Elst. 1999. “Cosmological Models.” In Theoretical and Observational Cosmology, 1–116. Dordrecht: Springer. DOI. Open PDF.
Ma, Chung-Pei, and Edmund Bertschinger. 1995. “Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges.” Astrophysical Journal 455, 7. DOI.
Malik, Karim A., and David Wands. 2009. “Cosmological Perturbations.” Physics Reports 475, 1–51. DOI.
Tong, David. 2019. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes. Official PDF.