Evidence-Based Cosmology
Cosmology is the application of the field equations of The Einstein Field Equations to the universe as a whole, and it is admitted here on the strength of four independent data streams: the expansion recorded in galaxy redshifts [Hubble:1929], the light-element abundances fixed in the first minutes [Alpher:1948a], the relic microwave background predicted before it was found [Alpher:1948b] [Penzias:1965], and the growth of large-scale structure with its baryon acoustic ruler [Eisenstein:2005]. Each is treated with its primary sources; the CMB measurements get their own experiment chapter, Experiment: The Cosmic Microwave Background.
The chapter's standing claim is that \(\Lambda\)CDM is a parametrization of these data, not yet an explanation of them: six numbers fit everything [Aghanim:2020], while the physical nature of two of the ingredients — the dark matter of The Dark Sector: Evidence Without Explanation and the constant \(\Lambda\) — and the mechanism behind the initial conditions remain unidentified. The chapter states throughout what is measured, what is inferred through the model, and what is not measured at all.
The expanding universe
Nebular redshifts
The observational record begins not with a theory but with a decade of patient spectroscopy. Between 1912 and 1917, working with the 24-inch refractor of the Lowell Observatory, Slipher measured the Doppler displacements of the spectral lines of the brightest spiral nebulae — objects whose nature, star cloud or island universe, was itself still in dispute [Slipher:1917]. Two facts emerged from his radial velocities and survived every later refinement. First, the velocities were enormous by the standards of stellar astronomy: several hundred kilometres per second, reaching \(1100\,\mathrm{km}/\mathrm{s}\) for the faintest objects measured, an order of magnitude beyond any stellar radial velocity then known. Second, the displacements were overwhelmingly toward the red: of the spirals Slipher had measured by 1917, the receding outnumbered the approaching roughly five to one. A sample drawn from bodies in random motion does not vote five to one for recession.
The observable is dimensionless and spectroscopic. If a line identified in the laboratory at wavelength \(\lambda_{\mathrm{em}}\) is received at \(\lambda_{\mathrm{obs}}\), the redshift of the source is \(z=(\lambda_{\mathrm{obs}}-\lambda_{\mathrm{em}})/\lambda_{\mathrm{em}}\), as restated below in Equation (48.1). At small \(z\) the special-relativistic Doppler formula of Lorentz Transformations reads \(z\simeq v/c\) and licenses the word “velocity”; but the derivation of Phenomenon 48.1 will show that a cosmological redshift is not a velocity through space at all — it is a ratio of the sizes of the universe at emission and at reception, and the kinematic reading is legitimate only over distances small enough that the size has not changed. The precise statement is Equation (48.2).
The distance ladder
A redshift is read off a spectrum in an afternoon; a distance must be climbed to. Every extragalactic distance rests on a ladder of overlapping calibrations, and the propagation of error up that ladder is the central systematic of observational cosmology, then and now.
The base of the ladder is trigonometric parallax — pure geometry, the apparent semiannual displacement of a nearby star against the distant sky as the Earth traverses its orbit. It defines the parsec (\(1\,\mathrm{pc}=3.0857\times 10^{16}\,\mathrm{m}\), the distance at which the astronomical unit subtends one second of arc) and it reaches, with modern astrometry, across a usable fraction of the Galaxy — but not remotely to another one.
The rung that first carried distances beyond the Galaxy was found by Leavitt in the Small Magellanic Cloud. Among the periodic variable stars now called Cepheids, she established that the apparent brightness at fixed period was tightly correlated with the period — and since every star in the Cloud sits at essentially one distance, apparent brightness there is absolute brightness up to one common constant [Leavitt:1912]. A Cepheid's period, which no distance error can corrupt, therefore reveals its luminosity; comparing luminosity with observed flux gives the distance. Calibrate the zero point on parallax Cepheids in the Galaxy, and the ladder extends as far as individual Cepheids can be resolved. It was by resolving Cepheids in the spiral nebulae that Hubble settled their nature as galaxies and measured the distances that enter Section 48.1.3 [Hubble:1929]. The physics of the pulsation mechanism itself belongs to stellar astrophysics (Stellar Structure and Nucleosynthesis).
Above the Cepheids sits the brightest standardizable candle known, the type Ia supernova, taken up in Section 48.6: a single event outshines its host galaxy and carries the ladder to redshifts where the expansion history itself becomes measurable. The modern three-rung ladder — parallax, Cepheid, supernova — is the instrument behind the local Hubble-constant measurement of Section 48.7.2 [Riess:2022], and each rung multiplies the error budget of the one below it, which is why so much of Section 48.7.2 turns on calibration.
The Hubble–Lemaître law
The linear law joining Slipher's velocities to the new distances has a double history, and the priority is worth recording precisely. In 1927 Lemaître derived, from the field equations of The Einstein Field Equations applied to an expanding homogeneous universe, the prediction that recession velocity grows in proportion to distance, and extracted from Slipher's velocities and Hubble's early distance estimates an expansion rate of about \(600\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) [Lemaitre:1927]. The paper, in French in the Annales of the Brussels scientific society, went unread; when it was translated into English in 1931, the paragraphs deriving the rate were omitted. Hubble established the relation observationally in 1929, from Cepheid-calibrated distances to two dozen galaxies, with a slope near \(500\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) [Hubble:1929], and the law long carried his name alone. Both rates are some seven times the modern value: the error lay not in the velocities but in the first rung of the distance ladder, whose Cepheid zero point was systematically miscalibrated, and unwinding it took three decades [Freedman:2021].
The spectral lines of galaxies beyond the Local Group are displaced toward longer wavelengths, and for nearby galaxies the displacement grows in proportion to distance,
Slipher's radial velocities of the spiral nebulae established the displacements [Slipher:1917]; Lemaître derived the linear law from the field equations and extracted a rate from those same data [Lemaitre:1927]; Hubble established the relation observationally from Cepheid distances [Hubble:1929]. The present value of \(H_{0}\) is near \(70\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\), with a discrepancy between the two ways of measuring it that is the subject of Section 48.7.2. Rests on Equation (48.3) and Theorem 48.3.
Derivation. Derives Phenomenon 48.1. In the FLRW geometry of Section 48.2 light reaching us radially travels on null curves with \(c\,\dd t=a(t)\,\dd\chi\), where \(\chi\) is the comoving radial coordinate of the source. Two successive wave crests, emitted at \(t_{\mathrm{e}}\) and at \(t_{\mathrm{e}}+\delta t_{\mathrm{e}}\) and received at \(t_{0}\) and \(t_{0}+\delta t_{0}\), traverse the same comoving interval, so
Subtracting the common part of the two integrals leaves \(\delta t_{0}/a(t_{0})=\delta t_{\mathrm{e}}/a(t_{\mathrm{e}})\), and since a wavelength is \(c\) times the crest interval,
The redshift is therefore a ratio of scale factors, not a velocity: the special-relativistic Doppler formula of Lorentz Transformations applies only over distances small enough that \(a\) may be treated as constant.
For a nearby source expand about the present epoch, \(a(t_{\mathrm{e}})\simeq a(t_{0})\left[1-H_{0}\left(t_{0} -t_{\mathrm{e}}\right)\right]\) with \(H_{0}:=\dot{a}\left(t_{0}\right)/a\left(t_{0}\right)\). Then Equation (48.2) gives \(z\simeq H_{0}\left(t_{0}-t_{\mathrm{e}}\right)\), and to the same order the light-travel time is \(t_{0}-t_{\mathrm{e}}\simeq d/c\), whence \(cz\simeq H_{0}d\). Note what the derivation did not use: no field equation, no matter content, no spatial curvature. Linearity at small \(z\) follows from expansion alone, which is why confirming Equation (48.1) establishes that the universe expands without yet constraining what drives the expansion.
∎In SI terms the Hubble constant is an inverse time: \(H_{0}=67.4\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc} =2.18\times 10^{-18}\,/\mathrm{s}\) (computed here from the definition of the parsec), so that \(1/H_{0}\approx14.5\times 10^{9}\,\mathrm{yr}\) — the first crude indication, already available in 1929, that the universe has a history of finite depth comparable to the age of the oldest stars.
The FLRW geometry
Homogeneity, isotropy and the metric
On sufficiently large scales, and at each cosmic epoch, the universe is spatially homogeneous (no point of space is distinguished) and isotropic (no direction is distinguished about any point).
This is a falsifiable idealization, not an axiom of convenience, and the evidence for it is now strong: the microwave background is isotropic to about one part in \(10^{5}\) once the kinematic dipole is removed (Section 48.4.2), and galaxy counts approach uniformity when averaged over scales above roughly \(100\,\mathrm{Mpc}\) [Peebles:1993] [Aghanim:2020]. Below those scales the universe is conspicuously lumpy — that lumpiness is the subject of Section 48.5 — so the principle governs the background on which structure grows.
The mathematics of symmetric spaces was built once, in full generality, in Differentiable Manifolds, Tensors, and Curvature: a metric's continuous symmetries are its Killing vectors (Definition 13.139), a \(D\)-dimensional space admits at most \(D(D+1)/2\) of them, and a space admitting the full complement has curvature of constant-curvature form (Theorem 13.160). Here that machinery is instantiated on the spatial sections of spacetime.
A spacetime obeying Definition 48.2 — foliated by spacelike hypersurfaces each homogeneous and isotropic — admits coordinates \((t,r,\theta,\varphi)\) in which the metric takes the form
with \(r\) dimensionless, \(a(t)>0\) a function of \(t\) alone carrying the dimension of length, and \(k\) the sign of the spatial curvature. The uniqueness of this family was established group-theoretically by Robertson and Walker [Robertson:1935] [Walker:1937]. Rests on Theorem 13.160, Proposition 13.162 and Remark 13.161.
Derivation. Derives Theorem 48.3. Step 1: the slicing and cosmic time. Let the homogeneous slices be labelled by a parameter \(t\) and let \(x^{i}\) be coordinates within each slice, carried from slice to slice along the curves of constant \(x^{i}\). In these coordinates the metric has blocks \(g_{00}\), \(g_{0i}\), \(g_{ij}\). The vector field \(g_{0i}\) restricted to a slice is a spatial vector attached to every point; isotropy about every point forbids any nonvanishing spatial vector field defined by the geometry, so \(g_{0i}=0\): the slices are orthogonal to the curves of constant \(x^{i}\), whose tangents define the comoving observers. Homogeneity makes \(g_{00}\) a function of \(t\) alone, and rescaling \(t\) absorbs it: \(g_{00}=-c^{2}\). The label \(t\) is then the proper time of every comoving observer, synchronized across each slice — cosmic time.
Step 2: each slice has constant curvature. Each slice is a three-dimensional Riemannian space isotropic about every point. By Schur's argument (Remark 13.161), pointwise isotropy in \(D=3>2\) already forces the curvature into the maximally symmetric form Equation (13.318) with a single constant \(K\) over the slice: the slice is a space of constant curvature, of positive, zero or negative sign.
Step 3: the radial form of a constant-curvature 3-metric. For \(K>0\) the slice is (locally) the 3-sphere of radius \(\ell=K^{-1/2}\), realizable by Proposition 13.162 as the quadric \(X^{2}+Y^{2}+Z^{2}+W^{2}=\ell^{2}\) in Euclidean 4-space. Introduce on it the radius \(\rho\) of the 2-sphere orbits, so that \(X^{2}+Y^{2}+Z^{2}=\rho^{2}\) and \(W^{2}=\ell^{2}-\rho^{2}\). Then \(W\,\dd W=-\rho\,\dd\rho\), and the induced line element is
with \(\dd\Omega^{2}=\dd\theta^{2}+\sin^{2}\theta\,\dd\varphi^{2}\). For \(K<0\) the same computation on the quadric of Proposition 13.162 with \(\sigma=-1\) (in signature \((3,1)\)) flips the sign under \(\rho^{2}/\ell^{2}\); for \(K=0\) the slice is flat and the line element is \(\dd\rho^{2}+\rho^{2}\dd\Omega^{2}\). Writing \(\rho=\ell r\) with \(r\) dimensionless unifies the three cases into the bracket of Equation (48.3) with \(k=\sgn K\) and an overall factor \(\ell^{2}\).
Step 4: the scale factor. The one geometric datum distinguishing slice from slice is the curvature radius \(\ell\), which homogeneity makes a single number per slice but which may vary with \(t\): call it \(a(t)\). Isotropy forbids any direction-dependent rescaling between slices (an anisotropic map of one slice onto the next would define preferred directions), and the sign \(k\) cannot change continuously. The metric is therefore Equation (48.3); for \(k=0\) the normalization of \(a\) is conventional, since only \(\dot{a}/a\) and \(\ddot{a}/a\) will appear in observables. That this family is the unique one — that no further functional freedom survives the symmetry — is the content of the group-theoretic classification of Robertson and Walker [Robertson:1935] [Walker:1937], to which we refer for the completeness argument.
∎Comoving coordinates deserve one dynamical remark before use: an object released at rest in them stays at rest (its worldline is a geodesic, by the isotropy of Equation (48.3) about it), and an object moving through them slows down.
Let a free particle move through the flat (\(k=0\)) FLRW geometry with physical momentum \(p\) measured by the comoving observer it passes. Then along its geodesic
Rests on Proposition 13.141 and Equation (48.3).
Derivation. Derives Proposition 48.4. For \(k=0\) the spatial translations \(\pp_{i}\) are Killing vectors of Equation (48.3) (they preserve the flat slice metric, whose overall factor \(a^{2}(t)\) they do not touch), so by Proposition 13.141, for the translation \(\vect{\xi}=\pp_{x}\) along any fixed comoving direction \(x\), the quantity \(\xi_{\mu}u^{\mu}=g_{xx}u^{x}=a^{2}\,\dv{x}{\tau}\) is constant along the geodesic. The physical momentum component measured locally is \(p_{x}=m\,a\,\dv{x}{\tau}\) (the proper distance swept per proper time is \(a\,\dd x\)), so \(p_{x}a=m\,a^{2}\dv{x}{\tau}\) is the conserved quantity, which is Equation (48.4). For a photon the same argument runs with the affine parameter in place of \(\tau\), and \(p\propto1/a\) reproduces the redshift Equation (48.2), since \(p=h/\lambda\). Peculiar velocities of galaxies decay the same way, which is why the comoving frame is not merely a convention but an attractor.
∎The Friedmann equations
Symmetry fixed the metric up to one function \(a(t)\) and one sign \(k\); the field equations of The Einstein Field Equations now fix the dynamics of \(a\). The same symmetry first constrains the source: isotropy about every comoving observer forces the stress–energy tensor to have no energy flux and no anisotropic stress in the comoving frame — the only tensors available are \(g_{\mu\nu}\) and the observer 4-velocity \(u^{\mu}\) — so it must take the perfect-fluid form
with \(\varepsilon(t)\) the energy density (\(\mathrm{J}/\mathrm{m}^{3}\)) and \(p(t)\) the pressure (\(\mathrm{Pa}\)), both spatially constant by homogeneity.
For the metric Equation (48.3) sourced by Equation (48.5), the field equations with cosmological constant \(\Lambda\) reduce to
together with the continuity equation
The first two were obtained from the field equations by Friedmann, for positive and for negative spatial curvature [Friedmann:1922] [Friedmann:1924], and independently by Lemaître [Lemaitre:1927]. Rests on Equation (48.3), Equation (48.5) and Postulate 44.3.
Derivation. Derives Theorem 48.5. Write the metric as \(g_{00}=-c^{2}\), \(g_{ij}=a^{2}(t)\gamma_{ij}\), where \(\gamma_{ij}\) is the time-independent bracket of Equation (48.3): a 3-metric of constant curvature \(k\), whose Ricci tensor is \(R^{(3)}_{ij}=2k\gamma_{ij}\) by Equation (13.318) in \(D=3\). The nonvanishing Christoffel symbols of Theorem 13.150 involving time are, by direct computation from \(\Gamma^{\lambda}{}_{\mu\nu} =\tfrac12 g^{\lambda\sigma}(\pp_{\mu}g_{\sigma\nu} +\pp_{\nu}g_{\sigma\mu}-\pp_{\sigma}g_{\mu\nu})\),
plus the purely spatial symbols of \(\gamma_{ij}\). Assembling the Ricci tensor (Definition 13.153) from these,
and the scalar
The Einstein tensor components follow:
In the comoving frame \(u^{\mu}=(1,0,0,0)\), \(u_{t}=-c^{2}\), so Equation (48.5) gives \(T_{tt}=\varepsilon c^{2}\) and \(T_{ij}=p\,a^{2}\gamma_{ij}\). The \(tt\) component of the field equations \(G_{\mu\nu}+\Lambda g_{\mu\nu}=(8\pi G/c^{4})\,T_{\mu\nu}\) (The Einstein Field Equations) reads
which is Equation (48.6). The \(ij\) components, after dividing out \(a^{2}\gamma_{ij}\) and eliminating \(\dot{a}^{2}/a^{2}+kc^{2}/a^{2}\) with Equation (48.6), collapse to Equation (48.7).
For Equation (48.8), impose \(\nabla_{\mu}T^{\mu t}=0\) — a consequence of the field equations via the Bianchi identities (The Einstein Field Equations), not a separate assumption. With \(T^{tt}=\varepsilon/c^{2}\) and \(T^{ij}=(p/a^{2})\gamma^{ij}\),
using \(\Gamma^{\mu}{}_{\mu t}=3\dot{a}/a\) and Equation (48.9). Only two of the three equations are independent: differentiating Equation (48.6) and using Equation (48.8) reproduces Equation (48.7).
∎The critical density at expansion rate \(H\) is the energy density that makes the \(k=0\), \(\Lambda\)-free universe balance in Equation (48.6):
Each component's density parameter is its present share, \(\Omega_{i}:=\varepsilon_{i}(t_{0})/\varepsilon_{\mathrm{c}}(t_{0})\), with the conventions \(\Omega_{\Lambda}:=\Lambda c^{2}/(3H_{0}^{2})\) and \(\Omega_{k}:=-kc^{2}/(a_{0}^{2}H_{0}^{2})\), in terms of which Equation (48.6) evaluated today reads
For \(H_{0}=67.4\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) the critical density is \(\rho_{\mathrm{c}}=8.5\times 10^{-27}\,\mathrm{kg}/\mathrm{m}^{3}\) (computed here), about five hydrogen atoms per cubic metre — the gulf between that number and any laboratory vacuum is a useful reminder of how empty the universe is on average.
Distances and horizons
In an expanding curved geometry “distance” must be defined operationally, and the two definitions that match what telescopes actually do — comparing fluxes and comparing angles — differ from each other and from the coordinate separation. All are built on the comoving radial coordinate accumulated by a light ray. Define
so that \(r=S_{k}(\chi)\) and the metric reads \(\dd s^{2}=-c^{2}\dd t^{2}+a^{2}[\dd\chi^{2} +S_{k}^{2}(\chi)\,\dd\Omega^{2}]\). A radial null ray obeys \(c\,\dd t=a\,\dd\chi\), so a source seen today at redshift \(z\) sits at
where the dimensionless expansion history
follows from Equation (48.6) once the scaling of each component with \(a\) is known — derived in Section 48.3.1 below.
Let a source of intrinsic luminosity \(L\) (\(\mathrm{W}\)) at redshift \(z\) be observed with energy flux \(F\) (\(\mathrm{W}/\mathrm{m}^{2}\)). Then the luminosity distance \(d_{\mathrm{L}}:=\sqrt{L/(4\pi F)}\) evaluates to
Rests on Equation (48.2), Equation (48.13) and Theorem 48.3.
Derivation. Derives Proposition 48.7. The photons emitted in all directions at comoving radius \(\chi\) are, at reception, spread over a sphere whose proper area is \(4\pi a_{0}^{2}S_{k}^{2}(\chi)\) (set \(\dd t=\dd\chi=0\) in the metric and integrate the angular part at the observer's epoch). Two further factors of \((1+z)\) deplete the flux: each photon arrives with its energy degraded by \(1+z\) (Equation (48.2) applied to \(E=h\nu\)), and the arrival rate of photons is diluted by the same factor (the crest-interval argument in the derivation of Phenomenon 48.1 applies to any clock, photon spacing included). Hence
and solving \(F=L/(4\pi d_{\mathrm{L}}^{2})\) for \(d_{\mathrm{L}}\) gives Equation (48.15).
∎An object of proper transverse size \(\ell\) at redshift \(z\) subtends the angle \(\theta=\ell/d_{\mathrm{A}}\) with
Rests on Equation (48.15) and Proposition 48.7.
Derivation. Derives Corollary 48.8. The proper size \(\ell\) was laid down at emission: setting \(\dd t=\dd\chi=0\) at the emission epoch, the transverse proper length subtending coordinate angle \(\theta\) at comoving radius \(\chi\) is \(\ell=a(t_{\mathrm{e}})S_{k}(\chi)\,\theta\), whence the first equality; the second follows from Equation (48.15) and \(a_{0}/a(t_{\mathrm{e}})=1+z\). The relation \(d_{\mathrm{L}}=(1+z)^{2}d_{\mathrm{A}}\) holds in any metric theory with photon-number conservation, which makes it a test of the framework itself rather than of a parameter set.
∎Two horizons bound what can be seen and what can be reached. The particle horizon is the comoving radius light can have crossed since the initial time,
finite whenever \(a\) grows fast enough near \(t=0\) — in the radiation era \(a\propto t^{1/2}\) and the integral converges at the lower end, so at every epoch there is a finite sphere of causal contact, a fact that becomes the horizon problem of Section 48.8. Today the proper particle horizon is about \(14\,\mathrm{Gpc}\) (computed here from Equation (48.14) with the parameters of Table 48.1). Complementarily, with \(\Lambda>0\) the future integral \(\int_{t}^{\infty}c\,\dd t'/a(t')\) converges (exponential expansion), so an event horizon exists: comoving regions beyond it can never be signalled, and galaxies presently leave our future light cone [Peebles:1993]. The Hubble radius \(c/H_{0}\approx4.5\,\mathrm{Gpc}\) (computed) is not a horizon but a curvature scale; it enters the perturbation theory of Section 48.5.1 as the boundary of the Newtonian regime.
The thermal history
Radiation, matter and $\Lambda$ eras
Let a component have equation of state \(p=w\varepsilon\) with \(w\) constant. Then Equation (48.8) integrates to
so nonrelativistic matter (\(w=0\)) dilutes as \(a^{-3}\), radiation (\(w=1/3\)) as \(a^{-4}\), and the \(\Lambda\) term (\(w=-1\), \(\varepsilon_{\Lambda}=\Lambda c^{4}/8\pi G\) constant) not at all. Rests on Equation (48.8).
Derivation. Derives Proposition 48.9. With \(p=w\varepsilon\), Equation (48.8) reads \(\dd\ln\varepsilon=-3(1+w)\,\dd\ln a\), which is Equation (48.18). The physics of the two nontrivial cases: matter dilutes with the volume, \(a^{-3}\); radiation dilutes with the volume and each quantum loses energy as \(1/a\) by Equation (48.4), giving the fourth power. The constancy of the \(\Lambda\) term is what makes it inevitable that, given enough expansion, it dominates — the ordering of the eras below is not an accident of initial conditions.
∎Because the four terms of Equation (48.14) scale with four different powers, the history divides into eras with sharp crossovers. Radiation–matter equality sits at \(1+z_{\mathrm{eq}}=\Omega_{\mathrm{m}}/\Omega_{\mathrm{r}} \approx3400\) (computed here from the densities of Table 48.1, with the photon density from the measured \(T_{0}\) and the standard three neutrino species); matter–\(\Lambda\) equality at \(1+z=(\Omega_{\Lambda}/\Omega_{\mathrm{m}})^{1/3}\), i.e. \(z\approx0.30\), and the expansion began accelerating (Equation (48.7) changing sign) at \(z\approx0.6\), both computed from the same table. Solving Equation (48.6) in each pure era gives \(a\propto t^{1/2}\) (radiation), \(a\propto t^{2/3}\) (matter), and \(a\propto\exp\left(\sqrt{\Lambda/3}\,ct\right)\) (\(\Lambda\)) — the last being the reason for the event horizon of Section 48.2.3.
Temperature.
The relic radiation cools as \(T\propto1/a\) (Equation (48.24), derived with the CMB in Section 48.4.1); running that backwards makes the early universe arbitrarily hot, and equilibrium thermodynamics (Quantum Statistics) governs the plasma. Two workhorse relations follow. First, for the coupled relativistic plasma the comoving entropy is conserved: the expansion of Equation (48.8) is adiabatic (no heat crosses the boundary of a comoving volume — there is nowhere for it to come from, by homogeneity), so \(s\,a^{3}=\text{const}\) with \(s\propto g_{\ast}T^{3}\) the entropy density of the relativistic species (Quantum Statistics), \(g_{\ast}\) counting their effective degrees of freedom. While \(g_{\ast}\) is constant this reproduces \(T\propto1/a\); when a species annihilates, its entropy is inherited by the rest, and the photon temperature briefly falls more slowly than \(1/a\).
The one clean application is a prediction. Neutrinos decouple from the plasma just before electrons and positrons annihilate, so the \(e^{\pm}\) entropy goes to the photons alone. Before annihilation the electromagnetic sector holds \(g_{\ast}=2+\tfrac{7}{8}\times4=\tfrac{11}{2}\) (photons plus \(e^{\pm}\)); after, \(g_{\ast}=2\). Conservation of \(g_{\ast}(Ta)^{3}\) across the transition gives \((T_{\gamma}a)_{\mathrm{after}}^{3} =\tfrac{11}{4}(Ta)_{\mathrm{before}}^{3}\), while the decoupled neutrinos keep \((T_{\nu}a)\) fixed, so
The relic neutrino sea itself has never been detected directly; its reality is attested indirectly, through the radiation density it contributes at nucleosynthesis (Section 48.3.2) and in the CMB damping tail [Aghanim:2020] — an honest amber, recorded as such.
Second, in the radiation era Equation (48.6) with \(\varepsilon=\tfrac{g_{\ast}}{2}\,a_{\mathrm{B}}T^{4}\) (\(a_{\mathrm{B}}=4\sigma/c\) the radiation constant, Black-Body Radiation and Planck's Hypothesis) and \(a\propto t^{1/2}\) gives the time–temperature relation
so \(T=10^{10}\,\mathrm{K}\) is reached at \(t\approx1\,\mathrm{s}\) (with \(g_{\ast}=10.75\): photons, \(e^{\pm}\), three neutrino species) and \(T=10^{9}\,\mathrm{K}\) at \(t\approx3\,\mathrm{min}\) (with \(g_{\ast}=3.36\) after annihilation); both computed here. The first minutes are nuclear physics territory, taken up next.
Recombination.
The plasma remains ionized long after \(k_{\mathrm{B}}T\) falls below the hydrogen binding energy of \(13.6\,\mathrm{eV}\), because photons outnumber baryons by \(\eta^{-1}\approx1.6\times10^{9}\) (Section 48.3.2) and the Wien tail keeps ionizing. The ionization balance — the Saha equation [Saha:1920], an application of the chemical equilibrium of Quantum Statistics — drops the free-electron fraction steeply only once \(k_{\mathrm{B}}T\approx0.26\,\mathrm{eV}\), i.e. \(T\approx3000\,\mathrm{K}\), at \(z_{\ast}\approx1090\) (the precise value, \(z_{\ast}=1089.9\), is quoted from the fit of Table 48.1). The photons then fly free: that release is the last-scattering surface whose image is the microwave background of Section 48.4.
Big-Bang nucleosynthesis
The proposal that the light elements were assembled in the hot first minutes is the alpha–beta–gamma paper of 1948 [Alpher:1948a]; the governing role of the neutron-to-proton ratio and its weak-interaction freeze-out was worked out by Hayashi [Hayashi:1950]; and the full reaction network in the modern sense — a stiff system of coupled rate equations integrated through the expansion — was first computed by Wagoner, Fowler and Hoyle [Wagoner:1967], whose descendants remain in use [Fields:2020]. The heavier elements are not made here: there is no stable nucleus at mass 5 or 8, and the low density starves the triple-\(\alpha\) process that stars use to jump the gap (Stellar Structure and Nucleosynthesis, [Hoyle:1954]); primordial synthesis stalls at \(^{7}\)Li.
The least processed material accessible to observation — metal-poor extragalactic gas clouds, and the absorption systems seen against high-redshift quasars — carries a common floor of light elements that stellar processing cannot produce: a helium mass fraction close to \(0.25\), a deuterium abundance \(\mathrm{D}/\mathrm{H}\) of a few parts in \(10^{5}\), and traces of \(^{3}\)He and \(^{7}\)Li. In the hot-big-bang picture these abundances form a one-parameter family governed by the ratio of baryons to photons alone [Alpher:1948a] [Fields:2020], and the value of that ratio which fits them agrees with the entirely independent determination from the microwave-background power spectrum [Aghanim:2020]. Deuterium is the sharpest of the four: it is destroyed in stellar interiors and made nowhere else, so any observed deuterium is a lower bound on the primordial value. Rests on Equation (48.20) and Theorem 48.5.
Derivation. Derives Phenomenon 48.10. Step 1: freeze-out of the neutron fraction. Above \(k_{\mathrm{B}}T\approx1\,\mathrm{MeV}\) the weak reactions \(n+\nu_{e}\leftrightarrow p+e^{-}\) and \(n+e^{+}\leftrightarrow p+\bar{\nu}_{e}\) (Nuclear Forces and Nuclear Structure) hold neutrons and protons in chemical equilibrium, so their ratio is the Boltzmann factor of the mass difference \(\Delta mc^{2}=1.293\,\mathrm{MeV}\):
The interconversion rate per nucleon scales as \(\Gamma\sim\left[G_{\mathrm{F}}/(\hbar c)^{3}\right]^{2} \left(k_{\mathrm{B}}T\right)^{5}/\hbar\) (the Fermi coupling squared times the phase space of the five-dimensional final-state integral, Nuclear Forces and Nuclear Structure), while the expansion rate scales as \(H\sim\sqrt{g_{\ast}G/(\hbar^{3}c^{5})}\, \left(k_{\mathrm{B}}T\right)^{2}\) by Equation (48.6); their ratio falls as \(T^{3}\), so equilibrium is abruptly lost. Carrying the numerical factors of the full rates through (we import them from the standard treatment [Hayashi:1950] [Fields:2020]), \(\Gamma=H\) at \(k_{\mathrm{B}}T_{\mathrm{f}}\approx0.8\,\mathrm{MeV}\), where Equation (48.21) gives \(n/p\approx1/5\); residual out-of-equilibrium conversions push the ratio to about \(1/6\) over the following seconds.
Step 2: the deuterium bottleneck. Fusion must proceed through deuterium, \(p+n\to\mathrm{D}+\gamma\), whose binding energy is only \(B_{\mathrm{D}}=2.224\,\mathrm{MeV}\). Although \(k_{\mathrm{B}}T<B_{\mathrm{D}}\) already at freeze-out, each baryon is bathed in \(\eta^{-1}\approx1.6\times10^{9}\) photons, and photodissociation wins as long as the number of photons in the Wien tail above \(B_{\mathrm{D}}\) exceeds the number of baryons — roughly, while \(\eta^{-1}\exp\left(-B_{\mathrm{D}}/k_{\mathrm{B}}T\right)>1\). Solving with \(\eta=6.1\times10^{-10}\) gives \(k_{\mathrm{B}}T\approx B_{\mathrm{D}}/21\approx0.1\,\mathrm{MeV}\); the full rate balance places the breakout slightly lower, near \(k_{\mathrm{B}}T\approx0.08\,\mathrm{MeV}\), i.e. \(T\approx10^{9}\,\mathrm{K}\), reached at \(t\approx3\,\mathrm{min}\) by Equation (48.20). The delay matters because free neutrons decay with mean life \(\tau_{n}=878.4\,\mathrm{s}\) (Nuclear Forces and Nuclear Structure): by breakout the ratio has fallen to \(n/p\approx1/7\).
Step 3: helium. Once deuterium survives, the chain \(\mathrm{D}(p,\gamma)^{3}\mathrm{He}\), \(\mathrm{D}(\mathrm{D},n)^{3}\mathrm{He}\), \(^{3}\mathrm{He}(\mathrm{D},p)^{4}\mathrm{He}\) runs to completion in minutes, because \(^{4}\)He is by far the most bound light nucleus (Nuclear Forces and Nuclear Structure). To first approximation every surviving neutron ends in helium: with \(x:=n/p\) at breakout, the helium mass fraction is
The estimate is robust precisely because it rests on a counting argument, not on cross sections; the full network sharpens it to \(Y_{\mathrm{p}}=0.2469\pm0.0002\) at the CMB baryon density [Fields:2020].
Step 4: the deuterium residue and the baryometer. A trace of deuterium fails to burn, at a level set by the competition between the burning rate (proportional to the baryon density) and the expansion: a higher baryon density burns deuterium more completely, so \(\mathrm{D}/\mathrm{H}\) is a steeply decreasing function of \(\eta\) — the baryometer. The network prediction at the CMB-determined density lies between \(2.5\) and \(2.6\) parts in \(10^{5}\), the spread reflecting nuclear-rate inputs [Fields:2020]; the measurement, from damped Lyman-\(\alpha\) absorbers along quasar sightlines, is \(\mathrm{D}/\mathrm{H}=(2.527\pm0.030)\times10^{-5}\) [Cooke:2018]. The measured helium fraction in metal-poor H II regions, \(Y_{\mathrm{p}}=0.2449\pm0.0040\) [Aver:2015], agrees with Equation (48.22) and its refinement. The full integration of the coupled rate network through the expansion — dozens of reactions, evaluated against measured cross sections — is imported here by reference [Wagoner:1967] [Fields:2020] rather than restated; the analytic skeleton above is what makes its output intelligible.
∎The baryon-to-photon ratio used above, \(\eta=6.1\times10^{-10}\), is computed here from the CMB baryon density \(\Omega_{\mathrm{b}}h^{2}\) of Table 48.1 and the photon density implied by the measured \(T_{0}\) — and the point of Phenomenon 48.10 is that the abundances measured in gas clouds return the same number from entirely different physics at an epoch separated by a factor of \(10^{9}\) in temperature.
The account above is honest only if its one failure is stated: the same network that gets D and \(^{4}\)He right predicts a primordial \(^{7}\)Li abundance about a factor of three above what is measured on the Spite plateau of metal-poor halo stars [Fields:2020]. Whether the fault lies in stellar depletion of lithium, in the astrophysics of the measurement, or in the network itself is not settled; the discrepancy is recorded here as an open defect of the concordance, not explained away.
The prediction of the relic radiation
The hot early phase required by Section 48.3.2 does not switch itself off: if the universe was once a thermal plasma, its radiation must still be here, cooled by the expansion but preserving its thermal form (the proof of that preservation is given with Phenomenon 48.12 below). In 1948 Gamow set out the framework in which nucleosynthesis fixes the conditions at \(T\sim10^{9}\,\mathrm{K}\) [Gamow:1948], and Alpher and Herman, correcting and completing his estimate, drew the explicit conclusion: a residual blackbody radiation should fill the universe today, at a temperature they put at about \(5\,\mathrm{K}\) [Alpher:1948b].
The structure of their inference deserves to be exhibited in modern numbers, because it is a genuine prediction — made seventeen years before the discovery, and not being searched for when the discovery came. The deuterium bottleneck fixes the baryon density at \(T\approx10^{9}\,\mathrm{K}\): capture must win against expansion, which by the cross sections requires \(\rho_{\mathrm{b}}\sim10^{-2}\,\mathrm{kg}/\mathrm{m}^{3}\) there (computed here from the modern network normalization; Alpher and Herman's value was cruder). Astronomy supplies the present mean baryon density, \(\rho_{\mathrm{b},0}\sim 4\times 10^{-28}\,\mathrm{kg}/\mathrm{m}^{3}\). Since \(\rho_{\mathrm{b}}\propto a^{-3}\) while \(T\propto a^{-1}\) (Proposition 48.9 and Equation (48.24)), the expansion factor between the two epochs is \((\rho_{\mathrm{b}}/\rho_{\mathrm{b},0})^{1/3}\approx4\times10^{8}\), and the present radiation temperature must be
With the modern inputs the estimate lands on the measured value; with the 1948 inputs it gave \(5\,\mathrm{K}\) — the right physics and the right order of magnitude, published in Nature, and then forgotten for seventeen years. When Penzias and Wilson found the radiation in 1965 (Section 48.4.1) they were unaware of the prediction, and the interpreting paper [Dicke:1965] cited the hot-big-bang logic afresh. As a case study in how evidence accumulates — prediction, neglect, independent rediscovery — cosmology has no better example (Epistemology and the Scientific Method).
The cosmic microwave background
Discovery
In 1964 Penzias and Wilson, commissioning the \(6\,\mathrm{m}\) horn antenna at Holmdel for radio astronomy, found an excess antenna temperature of \(3.5(1.0)\,\mathrm{K}\) at \(4080\,\mathrm{MHz}\) that no accounting could remove: it was independent of direction, of season, and of every instrumental cause they could test, down to the famous evicted pigeons [Penzias:1965]. The interpretation — that this is the cooled relic of the hot early universe — was supplied in the companion paper by Dicke, Peebles, Roll and Wilkinson [Dicke:1965], whose own purpose-built radiometer was months from completion. The spectrum was finally measured absolutely from orbit by COBE/FIRAS, and it is the most perfect Planck spectrum known (Black-Body Radiation and Planck's Hypothesis); the instruments, data and error budgets belong to the experiment chapter, Experiment: The Cosmic Microwave Background.
The whole sky radiates in the microwave band with a spectrum indistinguishable from a Planck spectrum, at a temperature \(T_{0}=2.72548\,\mathrm{K}\) known to \(0.00057\,\mathrm{K}\) [Fixsen:2009], and isotropic to about one part in \(10^{5}\) once the dipole of Section 48.4.2 is removed. It was found as an irreducible excess antenna temperature at \(4.08\,\mathrm{GHz}\) that no instrumental or local source would account for [Penzias:1965], and identified in the companion paper as the relic of a hot early phase [Dicke:1965] — a state of affairs predicted seventeen years earlier [Alpher:1948b]. Absolute spectrophotometry from orbit found the departures from the Planck form to be below \(10^{-4}\) of the peak intensity [Mather:1990] [Fixsen:1996], making this the closest approach to a blackbody ever measured anywhere, laboratory sources included. Rests on Equation (48.2) and Theorem 48.3.
Derivation. Derives Phenomenon 48.12. What must be shown is that a thermal spectrum survives free expansion with its temperature as the only changing parameter — otherwise the observed Planck form could not be a relic, the universe having expanded by a factor of order \(10^{3}\) since the photons last scattered. Let the radiation be in equilibrium at temperature \(T_{\mathrm{e}}\) when the scale factor is \(a_{\mathrm{e}}\), so that the mean occupation number of a mode of frequency \(\nu\) is the Planck value of Black-Body Radiation and Planck's Hypothesis,
Now expand freely to scale factor \(a\). Two things happen, and only two. Every mode redshifts, \(\nu\to\nu'=\nu\,a_{\mathrm{e}}/a\), by Equation (48.2) applied to the mode itself; and the occupation number of each mode is unchanged, since with no emission, absorption or scattering the photons are neither created nor destroyed while the modes themselves are simply carried along by the expansion. Hence
which is again of Planck form, with
The spectrum is preserved exactly and the single parameter \(T\) scales as \(1/a\). Two consequences follow. The measured \(T_{0}\) fixes the radiation temperature at every earlier epoch, which is what makes the thermal history of Section 48.3 calculable; and any measured departure from the Planck form would be a record of energy injected after thermalization ceased, so the spectral limits quoted above are constraints on the entire history since then.
∎Anisotropies
The first anisotropy seen was the dipole: one side of the sky is hotter than the other by \(3.4\,\mathrm{mK}\), detected from a U-2 aircraft in 1977 [Smoot:1977] and measured by FIRAS at \(3.372(14)\,\mathrm{mK}\) [Fixsen:1996]. It is kinematic: a Doppler modulation of Equation (48.24) by our own motion, \(\Delta T/T_{0}=v/c\), giving \(v\approx370\,\mathrm{km}/\mathrm{s}\) (computed) for the barycentre of the Solar System relative to the frame in which the radiation is isotropic — the comoving frame of Section 48.2.1 made tangible. It is removed before the cosmological analysis begins.
Beneath the dipole, the microwave sky carries temperature fluctuations of about one part in \(10^{5}\): an rms of roughly \(30\,\mu\mathrm{K}\) on ten-degree scales, first detected by the COBE differential radiometers [Smoot:1992]. At angular scales below a degree the fluctuation power is organized into a series of acoustic peaks, the first near multipole \(\ell\approx220\), measured with high precision by the Planck satellite [Aghanim:2020]. The large-angle plateau was predicted as the gravitational imprint of density perturbations on the last-scattering surface [Sachs:1967], and the peak structure as the frozen phases of sound waves in the pre-recombination plasma [Peebles:1970] [Sunyaev:1970] — both before any anisotropy had been seen. The apparatus, maps and likelihoods are the subject of Experiment: The Cosmic Microwave Background. Rests on Equation (48.24) and Phenomenon 48.12.
Derivation. Derives Phenomenon 48.13. Two mechanisms carry the primordial perturbations onto the sky, and each is derivable in outline here.
The Sachs–Wolfe plateau. Consider a large-scale density perturbation at last scattering — large enough that no causal process has acted on it (Equation (48.17)), so only gravity speaks. Let \(\Phi\) be its Newtonian potential (\(\Phi<0\) in an overdensity). Photons climbing out of the well are gravitationally redshifted (The Equivalence Principle and Classical Tests): \(\left.\Delta T/T\right|_{\mathrm{grav}}=\Phi/c^{2}\). But the well also retards the local clocks: the proper time elapsed at coordinate time \(t\) is shifted by \(\delta\tau/\tau=\Phi/c^{2}\), so the plasma in the well reaches its recombination temperature later in cosmic time, by \(\delta t/t=-\Phi/c^{2}\); at that later time the scale factor is larger by \(\delta a/a=\tfrac{2}{3}\,\delta t/t\) (matter era, \(a\propto t^{2/3}\)), and photons released at fixed local temperature from a larger \(a\) suffer less subsequent cooling, arriving hotter by \(\left.\Delta T/T\right|_{\mathrm{delay}}=-\tfrac{2}{3}\Phi/c^{2}\). The sum is
cold spots over overdensities, at one third of the naive gravitational redshift. The bookkeeping above is heuristic in its treatment of time; the gauge-consistent computation, which confirms the factor \(1/3\) for adiabatic perturbations in the matter era, is Sachs and Wolfe's [Sachs:1967] and is imported by reference. With \(\Delta T/T\sim10^{-5}\), Equation (48.25) measures the depth of the primordial potentials, \(\Phi/c^{2}\sim3\times 10^{-5}\) — the normalization every structure-formation calculation in Section 48.5 inherits.
The acoustic peaks. Inside the horizon the pre-recombination plasma is a single tightly coupled fluid, and perturbations in it oscillate as sound waves (the restoring force is photon pressure; the derivation of the sound speed is given with Phenomenon 48.15). All modes of a given wavenumber \(k\) start their oscillation together — at horizon entry, with the same phase, because they share the same super-horizon initial condition — so at the instant of decoupling the modes with \(k\,r_{\mathrm{s}}=n\pi\), where \(r_{\mathrm{s}}\) is the comoving sound horizon (Equation (48.34)), are caught at the extrema of their oscillation. Those wavenumbers are imprinted as excess variance: projected on the sky through the angular-diameter distance of Corollary 48.8, they appear at multipoles
up to an order-unity shift from the projection geometry and from the gravitational driving of the oscillations. Computing \(\theta_{\ast}\) here from Equations (48.13) and (48.34) with the parameters of Table 48.1 gives \(\theta_{\ast}=1.040\times10^{-2}\,\mathrm{rad}\), against the measured \(1.0411\times10^{-2}\) [Aghanim:2020] — agreement at the level of a part in a thousand, which is the concordance of Section 48.7 in miniature. The measured first peak sits at \(\ell_{1}\approx220\). On scales smaller still the peaks die away: photons random-walk out of wavelengths shorter than their diffusion length before decoupling completes, erasing the fluctuations — Silk damping, predicted in 1968 [Silk:1968] and measured in the high-\(\ell\) tail [Aghanim:2020].
∎What the peak pattern buys, quantitatively, is leverage on nearly every parameter of Section 48.7: the sound horizon is calculable from pre-recombination physics, the peak angle measures the geometry through Equation (48.26), the relative peak heights measure the baryon and matter densities (baryons load the oscillating fluid asymmetrically; matter sets the epoch of equality and the driving). The extraction, with the instruments and likelihoods behind it, is Experiment: The Cosmic Microwave Background.
Large-scale structure
Growth of perturbations
The universe of Definition 48.2 is smooth; the universe outside the window is not. The bridge is gravitational instability: the fluctuations of \(10^{-5}\) measured on the last-scattering surface grow under their own gravity into the galaxies and clusters of today. The instability of a self-gravitating medium was analysed by Jeans for a static background [Jeans:1902]; the general-relativistic perturbation theory of an expanding universe is Lifshitz's [Lifshitz:1946]. On scales well inside the Hubble radius and for nonrelativistic matter, Newtonian fluid mechanics (Fluid Dynamics) suffices, and that is the regime derived here; the relativistic treatment is imported by reference where super-horizon scales are concerned [Lifshitz:1946].
Let \(\delta:=\delta\rho/\bar{\rho}\) be the fractional density perturbation of a pressureless matter component in an FLRW background, on scales well inside the Hubble radius. To linear order each Fourier mode \(\delta_{\vect{k}}\) (comoving wavevector \(\vect{k}\)) obeys
with \(c_{s}\) the sound speed of the medium. In the matter era the pressureless growing mode is
power-law growth, not the exponential growth of the static case. Rests on Equation (48.6), Theorem 48.5, Equation (31.10) and Equation (31.19).
Derivation. Derives Theorem 48.14. Start from the continuity and Euler equations of Fluid Dynamics (Equations (31.10) and (31.19)) with the gravitational force \(-\nabla\Phi\), and the field equation of Newtonian gravity in local form, \(\nabla^{2}\Phi=4\pi G\rho\) (Central Forces and Statics). The background solution is the expansion itself: \(\bar{\rho}(t)\propto a^{-3}\), velocity field \(\bar{\vect{v}}=H\vect{r}\), and \(\bar{\Phi}=\tfrac{2}{3}\pi G\bar{\rho}r^{2}\), which satisfies the three equations by virtue of Equations (48.6) and (48.7) restricted to their Newtonian content.
Perturb: \(\rho=\bar{\rho}(1+\delta)\), \(\vect{v}=H\vect{r}+\vect{u}\), \(\Phi=\bar{\Phi}+\phi\), and pass to comoving coordinates \(\vect{x}=\vect{r}/a\), in which \(\nabla_{r}=a^{-1}\nabla_{x}\) and the time derivative at fixed \(\vect{x}\) replaces that at fixed \(\vect{r}\) via \(\pp_{t}|_{r}=\pp_{t}|_{x}-H\vect{x}\cdot\nabla_{x}\). To first order in \((\delta,\vect{u},\phi)\) the three equations become
where the pressure term wrote \(\nabla p=c_{s}^{2}\bar{\rho}\,\nabla\delta\). Take \(\pp_{t}\) of Equation (48.29), insert Equation (48.30), eliminate \(\nabla_{x}\!\cdot\vect{u}\) with Equation (48.29) again and \(\phi\) with Equation (48.31), and Fourier-transform in \(\vect{x}\): the result is Equation (48.27). The Hubble term \(2H\dot{\delta}\) — absent in the static case — acts as friction: expansion carries matter away from the well as fast as gravity gathers it.
Setting \(H=0\), \(\bar\rho\) constant recovers Jeans's dichotomy [Jeans:1902]: modes with \(k<k_{\mathrm{J}}\), where
grow exponentially at rate \(\sqrt{4\pi G\bar{\rho}}\), shorter modes oscillate as sound. In the expanding matter era take \(c_{s}\to0\), \(a\propto t^{2/3}\), so \(H=2/(3t)\) and \(4\pi G\bar{\rho}=\tfrac{3}{2}H^{2}=2/(3t^{2})\) (Einstein–de Sitter values from Equation (48.6)); the ansatz \(\delta\propto t^{n}\) gives \(3n^{2}+n-2=0\), i.e. \(n=2/3\) or \(n=-1\). The growing mode is \(\delta\propto t^{2/3}\propto a\), which is Equation (48.28). During radiation domination the rapid expansion (\(H\) set by the radiation, not by the matter) overwhelms the self-gravity of the matter and growth stalls logarithmically; once \(\Lambda\) dominates, growth stops [Peebles:1993].
∎Theorem 48.14 carries an arithmetic consequence that is one of the strongest arguments in this book. From recombination to today the scale factor grows by \(1+z_{\ast}\approx1100\), so a baryonic fluctuation of the measured amplitude \(\sim10^{-5}\) (Phenomenon 48.13) can have grown by at most that factor — to \(10^{-2}\), and less once the stalls at both ends are counted. Galaxies, with \(\delta\gg1\), could not exist. The resolution is a matter component that does not couple to the photons: its perturbations, unsupported by radiation pressure, began growing at matter–radiation equality (\(z_{\mathrm{eq}}\approx 3400\), Section 48.3.1) and had dug potential wells well before recombination; the baryons, released at \(z_{\ast}\), fell into wells already prepared [Peebles:1982]. This is the structure-formation argument for dark matter cited in Phenomenon 48.16.
The initial spectrum feeding Equation (48.27) was guessed on general grounds before it was measured: Harrison and Zel'dovich argued independently that the only spectrum injecting equal potential fluctuations on every scale — the only one with no built-in preferred length — has spectral index \(n_{s}=1\) [Harrison:1970] [Zeldovich:1972]. The measured value, \(n_{s}=0.9649\pm0.0042\) [Aghanim:2020], is close to but significantly below that: a small tilt, whose possible origin is taken up in Section 48.8.
Baryon acoustic oscillations
The sound waves that produce the CMB peaks (Phenomenon 48.13) do not vanish at recombination: they freeze into the matter distribution too, and their imprint is measurable in galaxy surveys billions of years later — a ruler of known length available at many redshifts.
The two-point correlation function of galaxies carries a single narrow excess at a comoving separation of about \(150\,\mathrm{Mpc}\), riding on an otherwise smooth, nearly power-law clustering. It was first detected in the luminous-red-galaxy sample of the Sloan survey [Eisenstein:2005]. The feature sits at the comoving sound horizon frozen in when the photons decoupled from the baryons — the same length that sets the angular spacing of the acoustic peaks in the microwave background [Aghanim:2020] — so it is a ruler of calculable length laid down at a known redshift. Measuring its apparent size along and across the line of sight at several redshifts maps the expansion history geometrically, with no standard candle and no distance ladder involved. Rests on Equation (48.34), Equation (48.16) and Corollary 48.8.
Derivation. Derives Phenomenon 48.15. Step 1: the sound speed of the coupled fluid. Before recombination, Thomson scattering ties the baryons to the photons into a single fluid whose pressure is supplied almost entirely by the photons, \(p_{\gamma}=\varepsilon_{\gamma}/3\) (Black-Body Radiation and Planck's Hypothesis), and whose inertia is shared. For adiabatic perturbations the photon and baryon densities fluctuate together (equal number-density contrasts, so \(\delta\varepsilon_{\gamma}/\varepsilon_{\gamma} =\tfrac{4}{3}\,\delta\rho_{\mathrm{b}}/\rho_{\mathrm{b}}\)), and the sound speed follows from \(c_{s}^{2}=c^{2}\,\delta p/\delta\varepsilon\):
since only the photons push while both components weigh. Deep in the radiation era \(R\ll1\) and \(c_{s}\to c/\sqrt{3}\); by recombination \(R\) is of order unity and the fluid has audibly slowed.
Step 2: the sound horizon. A pressure wave launched at the initial time has travelled, by decoupling, the comoving distance
the acoustic analogue of Equation (48.17). Evaluating the integral here with Equations (48.14) and (48.33) and the densities of Table 48.1, at the drag epoch \(z_{\mathrm{d}}\approx1060\) (when the baryons, outnumbered \(10^{9}\) to one, are finally released by the photons — slightly after the photons are released by the baryons), gives \(r_{\mathrm{d}}=147\,\mathrm{Mpc}\) comoving, in agreement with the fitted value \(r_{\mathrm{d}}=147.09(26)\,\mathrm{Mpc}\) quoted by Planck [Aghanim:2020].
Step 3: survival as a bump. Picture a single pointlike primordial overdensity: dark matter, baryons, photons and neutrinos superposed. The photon–baryon component launches a spherical sound wave; the dark matter, pressureless, stays and grows by Theorem 48.14. At the drag epoch the wave stalls: the pressure that drove it vanishes with the coupling, and the baryon shell is stranded at radius \(r_{\mathrm{d}}\). Thereafter both the central dark-matter peak and the baryon shell grow by the same linear theory — Equation (48.27) is linear, so the configuration is preserved up to overall amplitude, and gravity gradually shares the shell between the components. Since galaxies form where matter is dense, pairs of galaxies separated by \(r_{\mathrm{d}}\) are slightly more common than the smooth clustering predicts: the correlation function acquires a bump at \(r_{\mathrm{d}}\), of relative amplitude set by the baryon fraction \(\Omega_{\mathrm{b}}/\Omega_{\mathrm{m}}\approx1/6\) — a percent-level feature, which is why finding it took a survey of \(46\,748\) luminous red galaxies over \(0.72\,h^{-3}\,\mathrm{Gpc}^{3}\) [Eisenstein:2005]. Nonlinear evolution at late times broadens the bump slightly but shifts its centre only at the sub-percent level, which is what qualifies it as a standard ruler [Eisenstein:2005] [Aghanim:2020].
∎Used as a ruler, the feature measures \(d_{\mathrm{A}}(z)\) transverse to the line of sight and \(c/H(z)\) along it (Corollary 48.8); anchored to the same \(r_{\mathrm{d}}\) that the CMB calibrates, BAO surveys extend the geometric measurement from \(z_{\ast}\) down through the era of \(\Lambda\) domination, and they are the third leg — with the CMB and the supernovae — of every concordance fit [Aghanim:2020].
Evidence for dark matter
The dark-matter inference is often told as one discovery; it is better told as three independent measurements that agree on a number. Zwicky, applying the virial theorem to the Coma cluster in 1933, found the velocity dispersion of the member galaxies to imply a binding mass vastly exceeding the luminous one [Zwicky:1933]. Rubin and Ford, four decades later, traced the rotation of M31 through the emission lines of its H II regions and found no Keplerian decline [Rubin:1970], a result their survey of field spirals made general [Rubin:1980]. And the microwave background requires, through the peak structure of Phenomenon 48.13 and the growth argument of Section 48.5.1, a gravitating component that does not scatter light, at five times the baryon density [Aghanim:2020]. Gravitational lensing supplies the most direct image: in the merging cluster 1E\,0657\(-\)56 the X-ray–emitting gas — most of the baryonic mass — was stripped and slowed by the collision, while the lensing mass sailed through with the galaxies, mapping a dominant collisionless component spatially separated from the baryons [Markevitch:2004] [Clowe:2006].
The orbital speed of gas and stars in a spiral galaxy does not decline beyond the visible disc: it stays nearly constant out to the largest radius at which any tracer can be found [Rubin:1970]. The same discrepancy appears on the scale of clusters, where the velocity dispersion of the member galaxies implies a binding mass far above the luminous one [Zwicky:1933], and again in the microwave background and in the growth of structure, both of which require a matter component that neither emits nor scatters radiation [Aghanim:2020]. The three arguments are independent, and they agree on the amount. Rests on Equation (48.27) and Theorem 48.14.
Derivation. Derives Phenomenon 48.16. For a test particle on a circular orbit of radius \(r\) inside a spherically symmetric mass distribution, Newtonian dynamics gives
where \(M(r)\) is the mass interior to \(r\). Were the luminous matter all the matter, \(M(r)\) would be essentially constant outside the optical disc and \(v\) would fall off as \(r^{-1/2}\), the Keplerian behaviour familiar from the solar system. The observed constancy of \(v\) instead forces
an enclosed mass still growing linearly at radii where the surface brightness has long since fallen away exponentially. The inference is purely kinematic: it uses only Newtonian gravity in the weak-field, slow-motion regime together with the assumption of a roughly spherical halo, and it fixes how much unseen mass there is without saying anything whatever about what it is made of.
The cluster version is the same argument with the virial theorem in place of the circular orbit, and the theorem is one line from Newton's equations (Newtonian Dynamics). For \(N\) bodies with positions \(\vect{r}_{i}\) and momenta \(\vect{p}_{i}\), let \(\mathcal{G}:=\sum_{i}\vect{p}_{i}\cdot\vect{r}_{i}\); then \(\dot{\mathcal{G}}=2T+\sum_{i}\vect{F}_{i}\cdot\vect{r}_{i}=2T+U\), the last step by Euler's homogeneous-function identity applied to the gravitational potential energy \(U\), homogeneous of degree \(-1\) in the positions. For a bound system \(\mathcal{G}\) stays finite, so the long-time average of \(\dot{\mathcal{G}}\) vanishes: \(2\avg{T}+\avg{U}=0\). A cluster of radius \(R\) whose galaxies move with velocity dispersion \(\sigma\) therefore has mass of order \(M\sim\sigma^{2}R/G\). For Coma, \(\sigma\approx1000\,\mathrm{km}/\mathrm{s}\) and \(R\) of a few megaparsecs give a mass one to two orders of magnitude above what the starlight accounts for — Zwicky's 1933 arithmetic, essentially unchanged [Zwicky:1933].
∎What the particle might be — and the searches that have so far excluded much of the candidate space — is deliberately not discussed here: the evidence above is gravitational and belongs to this chapter; the particle physics is a frontier and belongs to The Dark Sector: Evidence Without Explanation, under the standards of Epistemology and the Scientific Method.
The accelerating universe
Type Ia supernovae
A type Ia supernova is the thermonuclear disruption of a white dwarf driven over its stability limit by accretion (Stellar Structure and Nucleosynthesis); because the igniting mass is nearly universal, so is the luminosity, and what scatter remains is correlated with the light-curve's decline rate — broader is brighter — tightly enough that a measured light curve standardizes the peak magnitude to about \(0.15\) magnitudes [Phillips:1993]. That makes the events standardizable candles visible across half the observable universe, and in the late 1990s two independent teams pushed the Hubble diagram to \(z\approx0.5\)–\(1\) expecting to measure the deceleration of the expansion [Riess:1998] [Perlmutter:1999]. Both found the opposite sign.
Type Ia supernovae near \(z\approx0.5\), standardized through the correlation between light-curve width and peak luminosity, are observed systematically fainter — and so at systematically larger luminosity distance — than a matter-dominated, decelerating universe predicts, by an amount that grows with redshift [Riess:1998] [Perlmutter:1999]. Read through the FLRW distance relations of Section 48.2.3, the observation says that \(\ddot{a}>0\) at the present epoch, that is that the deceleration parameter
is negative. The concordance fit, which now folds in the microwave background and the acoustic ruler of Phenomenon 48.15, attributes about \(69\%\) of the present energy budget to a smooth component whose equation of state is consistent with \(w=-1\) and about \(31\%\) to matter [Aghanim:2020]. What has been measured is an expansion history; no mechanism has been measured. Rests on Equation (48.15), Proposition 48.7, Equation (48.7) and Theorem 48.5.
Derivation. Derives Phenomenon 48.17. Step 1: the kinematic distance–redshift relation. Expand the scale factor about the present,
which defines \(q_{0}\) via Equation (48.36) and uses no field equation. Inverting Equation (48.2) order by order gives the look-back time in terms of \(z\), and inserting it into the null-ray integral \(\chi=\int c\,\dd t/a\) and then into Equation (48.15) (curvature enters only at third order) yields
The first term is Equation (48.1); the second is where the dynamics hides. At \(z=0.1\) the two-term formula agrees with the exact integral for the concordance parameters to a quarter of a percent (computed here), so the expansion is trustworthy where the calibrating supernovae live and begins to bend measurably by \(z\approx0.5\).
Step 2: magnitudes. Optical astronomy measures the distance modulus
so a fractional distance excess \(\epsilon\) registers as \(\Delta\mu=(5/\ln 10)\,\epsilon\approx2.17\,\epsilon\) magnitudes. By Equation (48.37) a change of \(q_{0}\) shifts the modulus at redshift \(z\) by \(\Delta\mu\approx2.17\,z\,\abs{\Delta q_{0}}/2\): distinguishing the matter-only Einstein–de Sitter value \(q_{0}=+\tfrac12\) from an accelerating \(q_{0}\approx-0.5\) requires photometry good to a few tenths of a magnitude at \(z\approx0.5\) — exactly what standardized type Ia photometry provides.
Step 3: what was measured. Both teams found the high-redshift supernovae fainter than any decelerating model allows. Against the exact integrals of Equation (48.13) (computed here at \(z=0.5\)): the concordance universe places a supernova \(0.40\) magnitudes fainter than Einstein–de Sitter, \(0.19\) fainter than an open \(\Omega_{\mathrm{m}}=0.2\) universe, and \(0.12\) fainter than an empty coasting one — and the data preferred distances near the concordance values, \(10\)–\(15\%\) beyond the open model [Riess:1998], with the 42-supernova fit of the second team giving \(\Omega_{\mathrm{m}}\approx0.28\) under flatness [Perlmutter:1999]. Through Equations (48.6) and (48.7), evaluated today,
so the measured \(\Omega_{\mathrm{m}}\approx0.3\), \(\Omega_{\Lambda}\approx0.7\) give \(q_{0}\approx-0.53\) (computed): the expansion accelerates. The systematics that could have mimicked the signal — grey dust along the line of sight, luminosity evolution between the young universe and the old — were the subject of a decade of scrutiny and are bounded by the later data: the Hubble diagram now extends beyond \(z=1\) into the epoch where the matter term of Equation (48.14) dominates and the model predicts the faintness trend must reverse into relative brightening, which is observed [Brout:2022]. Dust and evolution do not reverse sign on cue.
∎The cosmological constant as dark energy
The minimal reading of Phenomenon 48.17 is already in the field equations. The constant \(\Lambda\) was introduced by Einstein in 1917 to allow a static universe [Einstein:1917a] — a motivation the redshifts of Section 48.1 demolished — but the term itself is not optional decoration: it is the one addition the structure of the field equations permits (The Einstein Field Equations), and the data now demand it. With \(\Lambda>0\), Equation (48.7) turns positive once matter dilutes sufficiently, and the fit of Table 48.1 gives (computed here from \(\Omega_{\Lambda}\) and \(H_{0}\))
about \(3.3\,\mathrm{GeV}\) per cubic metre: two protons' worth of rest energy in every cubic metre of vacuum. Interpreted as a fluid, the \(\Lambda\) term has \(w=-1\) exactly (Proposition 48.9); the fitted equation of state, from the combination of CMB, BAO and supernovae, is consistent with \(-1\) at the three-percent level [Aghanim:2020] [Brout:2022]. Nothing in the data yet distinguishes a true constant from a slowly varying impostor.
Two honest difficulties frame the subject. First, the smallness problem: quantum field theory assigns the vacuum an energy density, and every naive estimate of it — cutting the zero-point sum off at any scale from the electroweak to the Planck scale — exceeds Equation (48.39) by some \(55\) to \(120\) orders of magnitude [Weinberg:1989]. Why the measured value is not zero yet absurdly small compared to its natural scale is unexplained; Weinberg's review remains the standard statement of how resistant the problem is to every class of mechanism proposed. Second, the anthropic bound: Weinberg observed, a decade before the acceleration was found, that a \(\Lambda\) much larger than the matter density at the epoch of structure formation would have prevented galaxies from forming at all, so observers can only find themselves in universes with \(\Lambda\) at most of roughly the observed order [Weinberg:1987]. Whether that observation is an explanation is a question about scientific method rather than about data; this treatise records it as an argument, notes that it predicted the right order of magnitude, and does not admit it as evidence (Epistemology and the Scientific Method).
What is measured, in sum: an expansion history \(H(z)\), consistent across three independent probes with a constant vacuum energy of the value Equation (48.39). What is not measured: any property of the component beyond its density and its pressure, or any mechanism setting either. “Dark energy” is a name for the gap between those two sentences.
The $\Lambda$CDM parametrization
Six numbers
The base model that fits every data stream of this chapter is defined by Equation (48.3) with \(k=0\), the Friedmann dynamics of Theorem 48.5 sourced by baryons, cold dark matter, radiation and \(\Lambda\), and a power-law spectrum of adiabatic initial perturbations. Six sampled numbers specify it: two densities, one angle, one optical depth, and the amplitude and tilt of the initial spectrum. Everything else — the Hubble constant included — is derived. Table 48.1 lists the six with the 2018 Planck values, and a selection of the derived quantities this chapter has used; here \(h:=H_{0}/(100\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc})\).
| Parameter | Symbol | Value |
|---|---|---|
| Baryon density | $\Omega_{\mathrm{b}}h^{2}$ | \(0.02237(15)\) |
| Cold-dark-matter density | $\Omega_{\mathrm{c}}h^{2}$ | \(0.1200(12)\) |
| Acoustic angular scale | $100\,\theta_{\mathrm{MC}}$ | \(1.04092(31)\) |
| Reionization optical depth | $\tau$ | \(0.0544(73)\) |
| Primordial amplitude | $\ln\left(10^{10}A_{\mathrm{s}}\right)$ | \(3.044(14)\) |
| Scalar spectral index | $n_{\mathrm{s}}$ | \(0.9649(42)\) |
| Hubble constant | $H_{0}$ | \(67.36(54)\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) |
| Matter fraction | $\Omega_{\mathrm{m}}$ | \(0.3153(73)\) |
| Dark-energy fraction | $\Omega_{\Lambda}$ | \(0.6847(73)\) |
| Fluctuation amplitude | $\sigma_{8}$ | \(0.8111(60)\) |
| Age of the universe | $t_{0}$ | \(13.797(23)\,\mathrm{Gyr}\) |
| Last-scattering redshift | $z_{\ast}$ | \(1089.92(25)\) |
| Sound horizon, drag epoch | $r_{\mathrm{d}}$ | \(147.09(26)\,\mathrm{Mpc}\) |
The model's economy is its strongest and its weakest feature at once. Strongest: six numbers, fitted to the microwave background alone, then predict the BAO scale in galaxy surveys, the supernova Hubble diagram, the light-element abundances through \(\Omega_{\mathrm{b}}h^{2}\), and the broad shape of the matter power spectrum — and the predictions land (Phenomena 48.10, 48.15 and 48.17). Weakest: of the six, only \(\Omega_{\mathrm{b}}h^{2}\) names physics whose microscopic identity is known. The model does not say what the cold component is, why \(\Lambda\) has the value Equation (48.39), or what set \(A_{\mathrm{s}}\) and \(n_{\mathrm{s}}\). It is a parametrization whose parameters are measured superbly and explained not at all; they take their place in the ledger of unexplained constants in The Free Parameters of Physics.
The Hubble tension
The two best determinations of the present expansion rate disagree. Calibrating type Ia supernovae on the Cepheid distance ladder gives \(H_{0}=73.0\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) with an uncertainty near \(1.0\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) [Riess:2022]; fitting the microwave-background power spectra within the same six-parameter model that fits everything else in this chapter gives \(H_{0}=67.4\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) with an uncertainty near \(0.5\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) [Aghanim:2020]. The two differ by about five times their combined uncertainty. The disagreement has survived a decade of scrutiny of both error budgets and of the rungs of the ladder, and it is at present unresolved (What We Observe but Do Not Understand). It is recorded here as a measurement in conflict — one of the two determinations, or the model connecting them, is wrong — and not as evidence for any particular resolution. Rests on Equation (48.26), Phenomenon 48.13, Equation (48.15) and Proposition 48.7.
Derivation. Derives Phenomenon 48.18. What must be exhibited is where each number comes from, since the conflict lives entirely in the inference chains.
The ladder value. The chain of Section 48.1.2: geometric anchors (parallaxes, the megamaser distance to NGC\,4258, detached eclipsing binaries in the LMC) calibrate the Cepheid period–luminosity law; Cepheids in 42 host galaxies calibrate the peak luminosity of type Ia supernovae; the calibrated supernovae, read through Equation (48.38) and Equation (48.37) in the redshift range where the \(q_{0}\) correction is small, give \(H_{0}=73.04(104)\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) [Riess:2022]. The measurement is local (\(z\lesssim0.15\)) and nearly model-independent: no assumption about early-universe physics enters.
The CMB value. The chain of Phenomenon 48.13: the peak spacing measures \(\theta_{\ast}\) to three parts in \(10^{4}\) (Table 48.1); the sound horizon \(r_{\mathrm{s}}(z_{\ast})\) is computed from Equation (48.34), whose only inputs are the densities \(\Omega_{\mathrm{b}}h^{2}\) and \(\Omega_{\mathrm{m}}h^{2}\) — both fixed by the peak heights — and standard pre-recombination physics; the ratio then fixes the comoving distance to last scattering through Equation (48.26), and that distance is the integral Equation (48.13) of \(c/H(z)\) over the entire expansion history. Within \(\Lambda\)CDM — flat, with constant \(\Lambda\) — the integral has no freedom left once the densities are set, and \(H_{0}\) comes out at \(67.36(54)\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) [Aghanim:2020]. The value is thus an inference through the model, anchored at \(z=1090\) and extrapolated across the whole history; the BAO ruler of Phenomenon 48.15, calibrated by the same \(r_{\mathrm{d}}\), confirms the extrapolation at intermediate redshifts.
The comparison. The difference is \(5.7\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) against a combined uncertainty of \(1.2\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\): \(4.8\sigma\) as computed here from the two quoted errors, “about five sigma” as commonly stated [Riess:2022]. For the disagreement to be an accident of errors, one budget must be wrong by five times its stated size. The candidate systematics have been hunted for a decade: Cepheid crowding photometry, metallicity terms and anchor distances on the ladder side — partially checked by the independent tip-of-the-red-giant-branch calibration, which lands between the two values, near \(70\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) with larger uncertainty [Freedman:2021] — and foregrounds and likelihood choices on the CMB side, where independent instruments and analyses reproduce the low value [Aghanim:2020]. The third logical possibility is the model itself: the CMB chain rests on the computed \(r_{\mathrm{s}}\), so any new physics that shrinks the sound horizon by a few percent — extra relativistic energy density before recombination, for instance — would raise the inferred \(H_{0}\). No such extension is currently supported by the data it would have to leave undisturbed. The tension stands, and this treatise records it as standing (What We Observe but Do Not Understand).
∎Inflation: the evidence status
The proposal
The hot big bang of this chapter is spectacularly successful from one second onward and silent about its own initial conditions. Two of those conditions are quantitatively strange, and the arithmetic deserves to be shown.
The horizon problem. The particle horizon at last scattering, Equation (48.17) evaluated through the radiation and matter eras, is a comoving \(280\,\mathrm{Mpc}\) (computed here); the comoving distance to the last-scattering surface is \(13.9\,\mathrm{Gpc}\) (computed, Equation (48.13)). A causally connected patch therefore has an angular radius of about \(1.2^\circ\) on the sky, and the visible last-scattering surface comprises on the order of \(10^{4}\) such regions that, within the hot big bang alone, have never exchanged a signal (computed). Yet Phenomenon 48.12 shows them at a common temperature to a part in \(10^{5}\). Equilibrium without contact is not impossible as an initial condition — initial conditions are free — but it is unexplained structure.
The flatness problem. Write Equation (48.6) as \(\abs{\Omega(t)-1}=\abs{k}c^{2}/(a^{2}H^{2})\): the deviation from critical density scales as the inverse square of \(aH\), which grows throughout radiation and matter domination. Running the measured bound backwards (computed here from Equation (48.14)): today's \(\abs{\Omega-1}<0.002\) (Section 48.8.2) was \(\sim10^{-6}\) at \(z_{\mathrm{eq}}\) and below \(10^{-17}\) at the weak freeze-out that opens nucleosynthesis (\(t\approx1\,\mathrm{s}\), Section 48.3.2). The hot big bang must be handed a universe tuned to critical density to seventeen decimal places for one to be here measuring it.
Guth's 1981 observation is that a single mechanism voids both problems [Guth:1981]: a phase of accelerated expansion before the radiation era.
The comoving Hubble radius \(c/(aH)=c/\dot{a}\) decreases exactly when \(\ddot{a}>0\). Rests on Equation (48.7).
Derivation. Derives Proposition 48.19. \(\dv{}{t}\left(c/\dot{a}\right)=-c\,\ddot{a}/\dot{a}^{2}\), negative iff \(\ddot{a}>0\). During such a phase \(\abs{\Omega-1}=\abs{k}c^{2}/\dot{a}^{2}\) is driven toward zero rather than away — flatness becomes an attractor — and comoving regions vastly larger than the instantaneous Hubble radius were inside it, in causal contact, before the phase began: the uniformity of the last-scattering surface is inherited from a causally connected past. Sixty-odd \(e\)-foldings of such expansion suffice to bring the whole presently observable volume within one pre-inflationary causal patch and to flatten it beyond the measured bounds [Guth:1981]; the counting is standard and we do not repeat it.
∎Guth's own realization stalled in its exit from the accelerated phase; the workable slow-roll versions came a year later [Linde:1982] [Albrecht:1982]. What made the framework more than a cure for two aesthetic complaints is a by-product: the quantum fluctuations of the field driving the phase are stretched to super-horizon scales and become a nearly scale-invariant, slightly tilted spectrum of density perturbations — computed by Mukhanov and Chibisov in 1981 [Mukhanov:1981], eleven years before COBE saw any anisotropy at all (Phenomenon 48.13). A framework that predicted the statistical character of the seeds of all structure, in advance, has earned its place in this chapter.
Measured and unmeasured
The discipline of Epistemology and the Scientific Method requires the balance sheet to be explicit.
Measured, and consistent with the framework. Spatial curvature: the CMB combined with BAO gives \(\Omega_{k}=0.0007\pm0.0019\) — flat to two parts in a thousand [Aghanim:2020], where inflation predicts flatness to experimental precision. The spectral tilt: \(n_{s}=0.9649\pm0.0042\), eight standard deviations below exact scale invariance (computed from the quoted values), with the sign and rough size that generic slow roll predicts [Aghanim:2020] [Mukhanov:1981]. Adiabaticity and Gaussianity: the fluctuations are adiabatic and Gaussian to the limits of measurement [Aghanim:2020], as the simplest single-field realizations require.
Not measured. The distinguishing prediction of the framework — the one signal no alternative shares — is a background of primordial gravitational waves, imprinted as \(B\)-mode polarization in the CMB. None has been detected: the tensor-to-scalar ratio is bounded at \(r<0.036\) at 95\% confidence [Ade:2021]. With no tensor signal there is no measurement of the energy scale of the proposed phase, no identification of the field that drove it, and no test of the mechanism as opposed to its outputs. The consistency successes above are real but not exclusive: they check predictions that some alternative histories could also produce, whereas the tensor background is the framework's own signature.
The verdict, in this treatise's terms: inflation is mainstream cosmology and belongs in this chapter as the leading account of the initial conditions — an account whose generic outputs (flatness, tilt, adiabatic Gaussian perturbations) are measured and agree, and whose distinguishing prediction remains untested. It is reported in exactly those terms, and its speculative extensions — which trade in what cannot be observed — are excluded from this book by the scope rule of Epistemology and the Scientific Method (What We Observe but Do Not Understand records what a future \(B\)-mode detection, or a definitive null, would each settle).