Relativistic Cosmology

Relativistic cosmology connects spacetime geometry to a chain of increasingly model-dependent inferences: expansion history, thermal relics, perturbations, microwave-background anisotropies, structure, and primordial compact objects. This path is aimed at graduate students and researchers who want that chain to remain mathematically and epistemically visible. It treats homogeneous dynamics as the beginning of cosmology, not as a substitute for perturbation theory or data analysis.

Helpful background. Einstein equations supplies geometric dynamics, Bianchi consistency, and stress–energy coupling. Probability and statistical inference supplies likelihoods, priors, posterior checks, uncertainty propagation, and selection effects. Partial differential equations and Green functions supplies hyperbolic evolution, boundary-value reasoning, distributions, and retarded response needed for perturbations.

Optional orientation. The General Relativity overview can refresh the larger geometric framework, but it is not a core step and does not count toward completion.

Geometry, matter, and cosmological inference

By the end of the path, you should be able to:

• derive and interpret Friedmann dynamics, thermal history, and causal scales;
• develop gauge-aware cosmological perturbations and connect them to microwave-background and structure observables;
• evaluate cosmological inference with model assumptions, likelihoods, degeneracies, and tensions explicit; and
• relate inflation, primordial fluctuations, horizons, and primordial black holes within their separate validity domains.

Keep four layers distinct: a background solution, perturbations about that solution, an observation model, and an inference conditioned on a cosmological model. Agreement at one layer does not validate every assumption at the next.

Check your preparation

1. Starting from $\nabla_\mu T^{\mu\nu}=0$, can you state the continuity equation for a homogeneous perfect fluid?
2. Why can a scalar perturbation of the metric change under a coordinate transformation even when the perturbed spacetime has not changed?
3. In a posterior $p(\theta\mid d)\propto p(d\mid\theta)p(\theta)$, where do calibration, foregrounds, and selection effects enter?

Answers and review

1. For density $\rho$, pressure $p$, and Hubble rate $H=\dot a/a$, conservation gives $\dot\rho+3H(\rho+p)=0$. If deriving this from the covariant equation is difficult, revisit stress-energy conservation in the first graduate general relativity path.
2. A perturbation is defined only after identifying points between the background and perturbed spacetimes; changing that identification changes gauge-dependent components. Physical conclusions must use gauge-invariant combinations or a completely specified gauge. Review the geometry bridge within the same graduate route if this is not yet secure.
3. They belong in the likelihood and its nuisance parameters, with prior information or auxiliary data where justified. Fixing them silently transfers their uncertainty into the cosmological parameters. Review probability and inference.

If the first answer is secure, you can begin immediately. Review the concepts in the second answer before cosmological perturbations and those in the third before the inference phase. If both the first and second are not yet available, complete the graduate-relativity review before proceeding.

Route from expansion to evidence

1. Establish the background. Begin with cosmological spacetimes and observers and Friedmann dynamics and components, then calculate distances from the expansion history and check their low-redshift limits.
2. Connect expansion to causal and thermal history. Study cosmic distances and causal scales followed by thermal history and relics. Keep the particle horizon, event horizon, and Hubble radius conceptually distinct.
3. Develop the perturbative description. Work through cosmological perturbations and inflation and primordial fluctuations. Record gauge choice, initial state, and approximation order in every calculation.
4. Map perturbations to observables. Continue to the cosmic microwave background, calculate the acoustic scales with stated parameter assumptions, and then study relativistic structure formation.
5. Examine global and compact-object consequences. Read cosmological horizons and global evolution, primordial black holes, and cosmological gravitational-wave backgrounds.
6. Appraise the model with data. Finish with cosmological inference and model appraisal. A tension is a comparison among specified analyses, not a theory-independent observable.

Worked calculation: a constant-equation-of-state universe

Consider a spatially flat FLRW spacetime containing one perfect fluid with $p=w\rho$ and constant $w$. Conservation gives

$$\dot\rho+3H(1+w)\rho=0 \quad\Longrightarrow\quad \rho=\rho_0\left(\frac{a}{a_0}\right)^{-3(1+w)}.$$

For vanishing cosmological constant, the Friedmann equation is

$$H^2=\frac{8\pi G}{3}\rho.$$

For $w\gt-1$, the expanding big-bang branch can be written

$$a(t)\propto (t-t_{\mathrm B})^{\frac{2}{3(1+w)}}, \qquad t\gt t_{\mathrm B}.$$

Choosing $t_{\mathrm B}=0$, radiation, $w=1/3$, gives $a\propto t^{1/2}$, while pressureless matter, $w=0$, gives $a\propto t^{2/3}$. For $w=-1$, the density is constant and the expanding solution is exponential, $a\propto e^{Ht}$. For a phantom fluid with $w\lt-1$, the expanding branch is instead $a(t)\propto(t_*-t)^{2/[3(1+w)]}$ for $t\lt t_*$; because the exponent is negative, the scale factor diverges at the finite future time $t_*$. The combination $G\rho$ has dimensions of inverse time squared, as $H^2$ requires.

This calculation also exposes a common overreach. The background relation determines neither the spectrum of perturbations nor the likelihood of a data set. Those require, respectively, initial conditions and perturbation dynamics, followed by transfer functions, astrophysical observation models, foregrounds, and a statistical specification. Even the causal interpretation needs care: the comoving Hubble radius $(aH)^{-1}$ is a local expansion scale, whereas a particle horizon is an integral over the preceding history.

Checkpoints

Checkpoint 1. In a radiation-dominated flat universe, how do $\rho$ and the comoving Hubble radius scale with $a$?

Conceptual hint

Use $\rho\propto a^{-3(1+w)}$ and $H^2\propto\rho$.

Setup hint

Set $w=1/3$, derive the power of $a$ in $\rho(a)$, use Friedmann’s equation for $H(a)$, and only then form $(aH)^{-1}$.

Next-step hint

Keep the comoving Hubble scale distinct from the causal-history integral even if their powers of $a$ happen to agree in this era.

Solution

With $w=1/3$, $\rho\propto a^{-4}$ and $H\propto a^{-2}$. Hence $(aH)^{-1}\propto a$. This growing comoving scale is not automatically identical to the particle horizon, although the two have the same scaling in this idealized power-law era.

Checkpoint 2. Two parameter combinations produce nearly the same CMB angular spectrum over the measured multipoles. Is a narrow one-dimensional marginalized posterior evidence that the degeneracy is absent?

Conceptual hint

Think about priors, projection, and information from other data sets.

Setup hint

Sketch the joint posterior in the two parameter combinations before projecting onto either axis, and mark any prior boundary or auxiliary-data constraint.

Next-step hint

Ask whether the interval stays narrow when the auxiliary data are removed or the prior measure is changed within a justified range.

Solution

No. Marginalization can project a curved or prior-bounded degeneracy into a narrow interval. Inspect the joint posterior, the prior measure, and which auxiliary data break the degeneracy. The physical interpretation must be conditional on those additions.

Transfer task

Choose one claimed early-universe signal. Trace it from a primordial variable through its evolution and transfer function to an observed statistic and likelihood. At every arrow, state the gauge-invariant quantity, approximation, nuisance parameters, and possible degeneracy. Then change one background assumption—for example spatial curvature, an additional relativistic component, or the expansion history—and predict which parts of the chain must be recomputed. The goal is a controlled dependency analysis, not a literature summary.

Capstone: reconstruct a cosmological claim

Produce a self-contained analysis of one cosmological conclusion, using either a reproducible public likelihood or a transparent synthetic data set. Your work should:

• derive the relevant background and perturbation equations at the level needed for the observable;
• state initial conditions, gauge treatment, transfer physics, and domain of validity;
• specify data, likelihood, priors, nuisance model, and selection or foreground treatment;
• include at least one limiting-case or independent numerical check; and
• distinguish what the data favor from what the assumed model makes identifiable.

Assess the capstone on physical derivation, observable mapping, inferential transparency, and robustness. A strong result is correct and reproducible; an exemplary result also shows how its main conclusion moves under one defensible alternative model or nuisance treatment.

When you are ready to move on

You are ready to leave the path when you can derive the worked background result, solve both checkpoints without hints, complete the observable-chain transfer, and meet all four capstone criteria. Continue to Primordial Black Holes for the formation-to-constraint chain, Tests of Gravity for cross-regime inference, or Quantum Fields in Curved Spacetime for quantum fields on cosmological and black-hole backgrounds.