Cosmology — FLRW, inflation, nucleosynthesis, CMB, and structure

Intuition Beginner

The universe used to be hot, dense, and tiny. About 13.8 billion years ago every patch of space was filled with a searing bath of particles. Since then space itself has been stretching, so the distance between any two far-apart galaxies keeps growing. The galaxies are not flying outward through a fixed space; the space between them is being created. This single fact -- the expansion of space -- is the backbone of modern cosmology.

As space stretched, the hot bath cooled. When it was about a billion degrees, the lightest atomic nuclei formed. This process is called Big-Bang nucleosynthesis, and it cooked almost all the hydrogen and helium that exists today. It explains why about one quarter of the universe's ordinary matter is helium-4, a number we measure in stars and gas clouds and match to percent precision.

When the universe was about 380,000 years old it cooled enough for neutral atoms to form. Light, which had been trapped bouncing around in the hot fog, was suddenly free to stream. That light is still travelling, stretched by the expansion into microwaves. We detect it today as the cosmic microwave background -- a faint glow at temperature 2.725 K coming from every direction. It is a baby picture of the universe.

But the baby picture has tiny temperature ripples, about one part in 100,000. Those ripples are the seeds of everything we see. Gravity, mostly from dark matter, pulled matter into those slightly denser spots, which pulled in more matter, which eventually collapsed into the first galaxies. The large-scale structure of the cosmos -- sheets, filaments, and voids of galaxies -- grew from those seeds.

To explain why the baby picture is so uniform, and why space looks so flat, cosmologists posit a brief early phase of extraordinarily rapid stretching called inflation. In a tiny fraction of a second the universe doubled in size many dozens of times. This smoothed out wrinkles and flattened curvature, leaving the remarkably uniform sky we observe.

Visual Beginner

Picture the history of the universe as a vertical timeline running from a hot dense start at the bottom to the cool structured cosmos today at the top. As you climb upward, space stretches wider and the temperature drops. Each major event -- nucleosynthesis, the release of the CMB, the first galaxies -- sits at the height where the temperature first allowed it.

The widening of the timeline as you climb is the expansion: every comoving region grows in physical size by the scale factor $a(t)$. Temperature falls as $1/a$, so events happen when the expanding, cooling universe reaches the right conditions.

Worked example Beginner

Here are four numbers that anchor physical cosmology.

Hubble time. Distant galaxies recede at $v=H_{0}d$ with $H_0\approx 70$ km/s/Mpc. The inverse $1/H_0$ is a rough age scale. Converting units (1 Mpc $\approx 3.086\times 10^{19}$ km) gives $1/H_0\approx 14$ billion years. The true age, 13.8 Gyr, sits just below this because dark energy recently sped the expansion up.

CMB temperature. The background glow today has $T_0=2.725$ K. By Wien's law its peak wavelength is ${\lambda }_{\text{peak}}\approx 2.9\times 10^{-3}/T$ m $\approx 1.1$ mm, squarely in the microwave band. At decoupling the temperature was about 3000 K, hotter than the surface of many stars.

Helium by mass. Big-Bang nucleosynthesis predicts that about 25% of the ordinary matter ends up as helium-4 by mass, the rest almost entirely hydrogen. A star like the Sun is born with roughly this composition -- the primordial helium is its starting stock, before it makes any more.

Critical density. The density that would make space perfectly flat is ${\rho }_c=3H_0^2/(8\pi G)\approx 9.2\times 10^{-27}$ kg/m${}^3$ -- about six hydrogen atoms per cubic metre. Ordinary matter contributes only about 5% of this, dark matter about 27%, and dark energy the remaining 68%.

Check your understanding Beginner

Formal definition Intermediate+

This unit takes the Friedmann framework of 13.08.01 as given and develops the physical cosmology built on top: inflation, nucleosynthesis, the CMB, and structure formation. We restate the minimal scaffolding compactly.

The FLRW metric (homogeneous, isotropic universe) is

$$d{s}^2=-c^{2}d{t}^2+a(t{)}^2\left[\frac{d{r}^2}{1-k{r}^2}+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\right],$$

with scale factor $a(t)$ and curvature $k\in \{-1,0,+1\}$. Feeding this into the Einstein field equations 13.04.01 yields the Friedmann equations:

$$H^2={\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\rho -\frac{k{c}^2}{a^2}+\frac{\Lambda }{3},\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho +\frac{3p}{c^2}\right)+\frac{\Lambda }{3}.$$

The critical density is ${\rho }_c(t)=3H^2/(8\pi G)\approx 9.2\times 10^{-27}$ kg/m${}^3$ today, and the density parameters are ${\Omega }_i={\rho }_i/{\rho }_c$. Energy conservation (the continuity equation) gives, for a component with equation of state $p=w\rho c^2$,

$${\rho }_i(a)={\rho }_{i,0}{a}^{-3(1+w)},w=\{\begin{array}{ll}0& \text{matter},\\ 1/3& \text{radiation},\\ -1& \text{cosmological constant.}\end{array}$$

The present-day budget (Planck 2018) is ${\Omega }_m\approx 0.315$, ${\Omega }_{\Lambda }\approx 0.685$, ${\Omega }_r\approx 9.1\times 10^{-5}$, $|{\Omega }_k|<0.002$.

Inflation is defined kinematically as any epoch during which the expansion accelerates, $\ddot{a}>0$, equivalently $\dot{H}<0$ with the comoving Hubble radius $(aH{)}^{-1}$ shrinking. A perfect-fluid description requires $w<-1/3$; near-exponential inflation needs $w\approx -1$.

The horizon problem: the CMB is isotropic to $\Delta T/T\sim 10^{-5}$ on scales that were causally disconnected at recombination in the decelerating Friedmann era, because $(aH{)}^{-1}$ was growing then. The flatness problem: ${\Omega }_k(a)=-k{c}^2/(a^2{H}^2)$ is unstable in decelerating expansion, so the observed $|{\Omega }_k|\lesssim 10^{-2}$ today requires $|{\Omega }_k|\sim 10^{-60}$ at the Planck epoch -- extreme fine-tuning. Inflation reverses both: a shrinking $(aH{)}^{-1}$ drives ${\Omega }_k\to 0$ exponentially and brings presently-separated regions into past causal contact.

The density contrast for a matter component is

$$\delta (\mathbf{x},t)=\frac{{\rho }_m(\mathbf{x},t)-{\bar{\rho }}_m(t)}{{\bar{\rho }}_m(t)}.$$

The CMB temperature field on the sky is expanded in spherical harmonics, $T(\hat{n})={\sum }_{\ell m}{a}_{\ell m}{Y}_{\ell m}(\hat{n})$, and the angular power spectrum is $C_{\ell }=⟨|a_{\ell m}{|}^2⟩$. Its series of acoustic peaks encodes the geometry, baryon density, and dark-matter density of the universe.

Core model [Intermediate+] Master

This section assembles the four pillars -- inflation, nucleosynthesis, the acoustic CMB, and linear structure growth -- as a single coherent model.

Slow-roll inflation

Model inflation as a scalar inflaton field $\varphi (t)$, minimally coupled, with potential $V(\varphi )$. The stress-energy of a homogeneous field is

$${\rho }_{\varphi }=\frac{1}{2}{\dot{\varphi }}^2+V(\varphi ),p_{\varphi }=\frac{1}{2}{\dot{\varphi }}^2-V(\varphi ),$$

so $w_{\varphi }=p_{\varphi }/{\rho }_{\varphi }\to -1$ when the kinetic energy $\frac{1}{2}{\dot{\varphi }}^2$ is negligible compared with $V$. The field obeys the Klein-Gordon equation in the FLRW background,

$$\ddot{\varphi }+3H\dot{\varphi }+V_{,\varphi }=0,$$

where the Hubble friction term $3H\dot{\varphi }$ drains kinetic energy. The slow-roll conditions are parameterised by

$$\epsilon \equiv -\frac{\dot{H}}{H^2},\eta \equiv -\frac{\ddot{\varphi }}{H\dot{\varphi }},$$

and slow roll means $\epsilon \ll 1$ and $|\eta |\ll 1$. Then $H\approx$ const, $a(t)\propto e^{Ht}$, and the number of e-folds of expansion is $N={\int }_{a_{\text{start}}}^{a_{\text{end}}}d\ln a\approx H\Delta t$. The horizon and flatness problems are solved once $N\gtrsim 60$.

Big-Bang nucleosynthesis

At $T\gtrsim 1$ MeV, weak interactions $n\leftrightarrow p+e^{-}+{\bar{\nu }}_e$ kept the neutron-to-proton ratio at its equilibrium value $n/p=\exp [-(m_n-m_p)c^2/(k_{B}T)]$. When the weak rate $\Gamma \sim G_F^2{T}^5$ dropped below the Hubble rate $H\sim 1.66\sqrt{g_{\ast }}{T}^2/M_{\text{Pl}}$, the ratio froze out at $T_f\approx 0.8$ MeV, giving $n/p\approx 1/5$. Free-neutron decay over the next $\sim 3$ minutes reduced this to $\approx 1/7$.

Nucleosynthesis then proceeded through the deuterium bottleneck: until $T$ fell to about $0.08$ MeV, photodisintegration of the fragile deuteron held back heavier synthesis. Once the bottleneck broke, essentially all neutrons were swept into helium-4. The predicted primordial helium-4 mass fraction is

$$Y_p=\frac{4(n_n/2)}{n_n+n_p}=\frac{2(n/p)}{1+n/p}\approx \frac{2(1/7)}{1+1/7}\approx 0.25,$$

with trace deuterium D, helium-3, and lithium-7 at the $10^{-5}$ to $10^{-9}$ abundance level. The observed values ($Y_p=0.245\pm 0.003$; D/H $\approx 2.5\times 10^{-5}$) match the prediction and pin the baryon density to ${\Omega }_b{h}^2\approx 0.0224$.

Acoustic oscillations in the CMB

Before recombination, photons and baryons formed a tightly coupled plasma supporting acoustic waves driven by the competition between photon pressure and gravity (mainly from dark matter). The plasma oscillated with sound speed $c_s=c/\sqrt{3(1+R)}$ where $R=3{\rho }_b/(4{\rho }_{\gamma })$. The sound horizon at the drag epoch is

$$r_s={\int }_0^{a_{\text{drag}}}\frac{c_{s}da}{a^{2}H(a)}\approx 147\text{Mpc}.$$

The observed angular scale of the first peak, ${\theta }_{\ast }\approx 0.6^{\circ }$, gives a multipole ${\ell }_1\approx \pi /{\theta }_{\ast }\approx 220$. Because ${\ell }_1=\pi D_A/r_s$ with $D_A$ the angular-diameter distance to last scattering, the peak position measures spatial curvature: ${\ell }_1\approx 220$ forces $|{\Omega }_k|\lesssim 0.001$, the cleanest evidence that the universe is spatially flat. The peak-height ratios fix ${\Omega }_b$ and ${\Omega }_m$.

Linear growth of structure

Small density perturbations in the matter component obey the linearised continuity, Euler, and Poisson equations, which combine into a single growth equation for the growing mode $D(a)$:

$$\ddot{\delta }+2H\dot{\delta }-4\pi G{\bar{\rho }}_m\delta =0.$$

In the Einstein-de Sitter (matter-dominated, flat, $\Lambda =0$) era, $a\propto t^{2/3}$ and the two independent solutions are ${\delta }_{+}\propto a$ (growing) and ${\delta }_{-}\propto a^{-3/2}$ (decaying). The primordial perturbations seeded by inflation therefore grow linearly with the scale factor until nonlinearity sets in at $\delta \sim 1$. During radiation domination the Meszaros effect suppresses matter growth to a logarithm $\delta \propto \ln a$; during $\Lambda$ domination growth freezes. These effects are encoded in the transfer function $T(k)$ and growth factor $D(a)$ that map the primordial power spectrum $P_{\text{in}}(k)\propto k^{n_s}$ onto the late-time matter power spectrum.

Bridge. The slow-roll picture builds toward the quantum-field-theoretic origin of the primordial perturbations, where vacuum fluctuations of $\varphi$ are stretched beyond the horizon and seed the CMB anisotropies, and the same fluctuations appear again in the matter power spectrum that grew through the linear equation above. The foundational reason these four pillars fit together is that a single nearly scale-invariant spectrum $P(k)\propto k^{n_s}$ with $n_s\approx 0.965$ is produced by inflation, processed by the acoustic oscillator, and then amplified by gravitational instability -- this is exactly the unifying thread of the concordance model. Putting these together, the CMB peaks, the light-element abundances, and the galaxy power spectrum all measure the same underlying parameters $({\Omega }_m,{\Omega }_b,{\Omega }_{\Lambda },n_s,H_0)$. The bridge is that the Friedmann background of 13.08.01 supplies the clock $a(t)$ that every subsequent process -- nuclear, acoustic, gravitational -- runs against, and the acceleration condition $w<-1/3$ generalises from dark energy today to the inflaton in the early universe.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib does not yet contain the cosmological physics layer this unit develops. The differential-geometric substrate (Lorentzian metrics, the Levi-Civita connection, curvature tensors) is partially in place for pseudo-Riemannian manifolds, and the ODE library covers existence and uniqueness for the Friedmann system at a formal level. What is missing is any structured encoding of the cosmological objects: the FLRW ansatz as a parametric Lorentzian metric, the Friedmann equations as a coupled ODE system on the scale factor, the slow-roll parameters $\epsilon$ and $\eta$ and the e-fold count, the acoustic-oscillator equations, the BBN reaction network and freezeout criterion $\Gamma =H$, or the linearised perturbation growth equation and its Einstein-de Sitter solution $\delta \propto a$.

Closing this gap matters because each of these is an exact, checkable theorem once the physical scaffolding is defined: the e-fold bound $N\gtrsim 60$ for flatness, the helium-4 mass fraction $Y_p=2(n/p)/(1+n/p)$, and the growing-mode exponent $p=2/3$ are all statements that admit machine-checked proofs on top of a modest cosmology layer. Tyler's review attests intermediate-tier correctness; the lean_status: none field reflects the absence of such a layer today.

Advanced results Master

Slow-roll inflation and the primordial power spectrum

Theorem (slow-roll predicts near scale-invariance). For single-field slow-roll inflation with Hubble-flow parameters $\epsilon =-\dot{H}/H^2$ and $\eta =-\ddot{\varphi }/(H\dot{\varphi })$ both small, the scalar power spectrum of curvature perturbations $\mathcal{R}$ leaving the horizon at a mode $k$ is $P_{\mathcal{R}}(k)=A_s(k/k_{\ast }{)}^{n_s-1}$ with spectral index

$$n_s=1-6\epsilon +2\eta ⟹n_s\approx 0.965\text{ (Planck 2018)},$$

and tensor-to-scalar ratio $r=16\epsilon$. The observed near-scale-invariance ($n_s$ slightly below unity) is a direct signature of slow roll.

The near scale-invariance follows because during slow roll $H$ is nearly constant, so each comoving mode experiences nearly the same amplification as it crosses the shrinking comoving Hubble radius. The tilt $n_s-1=-6\epsilon +2\eta <0$ (red tilt) is the leading imprint of the time variation of $H$. Planck measures $n_s=0.9649\pm 0.0042$, excluding exact scale invariance ($n_s=1$) at over $7\sigma$, a decisive confirmation of the slow-roll mechanism. The tensor bound $r<0.06$ constrains $\epsilon <0.004$, pushing the inflationary energy scale below $\sim 2\times 10^{16}$ GeV.

BBN as a baryometer

Theorem (deuterium pinning of the baryon density). The primordial deuterium abundance D/H is a steeply decreasing function of the baryon-to-photon ratio ${\eta }_b=n_b/n_{\gamma }$, approximately $\text{D/H}\propto {\eta }_b^{-1.6}$. The observed $\text{D/H}=(2.53\pm 0.04)\times 10^{-5}$ in pristine gas infers ${\Omega }_b{h}^2=0.0222\pm 0.0003$, in agreement with the CMB value ${\Omega }_b{h}^2=0.0224\pm 0.0001$ to within 1%.

The agreement is striking because BBN probes the universe at $t\sim 3$ minutes while the CMB probes it at $t\sim 380,000$ years; concordance on ${\Omega }_b{h}^2$ across nine decades of cosmic time is one of the strongest tests of the standard cosmology. The sensitivity of D/H to ${\eta }_b$ arises because deuterium is processed away into helium-4 more efficiently at higher baryon density (faster $d(p,\gamma {)}^3$He and subsequent reactions), leaving less residual deuterium. Lithium-7 is the one discordant note: BBN predicts $\text{Li/H}\approx 5\times 10^{-10}$ while observations of metal-poor halo stars give $\text{Li/H}\approx 1.6\times 10^{-10}$ -- the lithium problem, an open issue possibly involving stellar depletion or new physics.

The CMB as a geometrical probe

Theorem (first-peak position measures curvature). For a universe with comoving sound horizon $r_s$ at last scattering and angular-diameter distance $D_A$, the first acoustic peak sits at ${\ell }_1=\pi D_A/r_s$. In a flat $\Lambda$CDM universe ${\ell }_1\approx 220$; a closed universe with ${\Omega }_k<0$ shifts the peak to higher $\ell$ (a geometric Sachs-Wolfe magnification) and an open universe with ${\Omega }_k>0$ to lower $\ell$. Planck measures ${\ell }_1=220.0\pm 0.5$, forcing ${\Omega }_k=0.000\pm 0.005$.

The geometric effect is intuitive: positive curvature (closed space) makes distant objects subtend larger angles, compressing the peak pattern toward higher $\ell$; negative curvature (open) stretches angles, pushing peaks to lower $\ell$. The peak ratio ${\ell }_2/(2{\ell }_1)$ -- odd peaks are compression maxima enhanced by baryon loading, even peaks are rarefaction minima -- fixes the baryon density; the relative heights of the third and first peak fix the cold-dark-matter density. The full $C_{\ell }$ curve thus overconstrains the six-parameter $\Lambda$CDM model to percent-level precision.

Structure growth and the matter power spectrum

Theorem (transfer-function mapping). The late-time linear matter power spectrum is

$$P(k,a)=A_s^2{\left(\frac{k}{k_{\ast }}\right)}^{n_s-1}T(k{)}^{2}D(a{)}^2,$$

where $T(k)$ is the transfer function encoding sub-horizon processing during radiation domination and $D(a)$ is the linear growth factor. For $\Lambda$CDM, $T(k)\to 1$ on large scales and $T(k)\propto \ln k$ on small scales, producing the characteristic turnover of the galaxy power spectrum at $k_{\text{eq}}\approx 0.01h{\text{Mpc}}^{-1}$ corresponding to the horizon scale at matter-radiation equality.

The turnover scale is a direct geometric measurement of $a_{\text{eq}}$ and hence of ${\Omega }_m{h}^2$. On large scales the spectrum preserves the primordial $k^{n_s}$ shape; on small scales it is suppressed because modes that entered the horizon during radiation domination grew only logarithmically (the Meszaros effect of Exercise 8). Galaxy surveys (2dF, SDSS, DES) measure this shape and recover ${\Omega }_m{h}^2\approx 0.14$ and $n_s\approx 0.965$, independently confirming the CMB and BBN inferences.

Synthesis. The four pillars -- inflation, nucleosynthesis, the acoustic CMB, and gravitational instability -- form a single closed logical loop, and the foundational reason they fit together is that the Friedmann background of 13.08.01 supplies one universal clock $a(t)$ against which every process runs. Inflation sets the initial conditions: a near-scale-invariant spectrum $P(k)\propto k^{n_s}$ with $n_s\approx 0.965$ and a flat geometry, and this is exactly what the CMB peaks (${\ell }_1\approx 220$) and the matter power spectrum both measure. Putting these together, BBN fixes ${\Omega }_b{h}^2\approx 0.022$ at $t\sim 3$ minutes and the CMB fixes the same parameter at $t\sim 380,000$ years; the bridge is that the Friedmann equation is the same equation in both eras, only the dominant ${\Omega }_i$ changes. The acoustic peaks generalise the simple photon free-streaming picture into a precision spectroscopic tool, and the growth equation $\delta \propto a$ generalises gravitational instability from the static Jeans analysis to an expanding background. The central insight is that one six-parameter model ($H_0$, ${\Omega }_b{h}^2$, ${\Omega }_c{h}^2$, $\tau$, $A_s$, $n_s$) fits the CMB, the light elements, the galaxy power spectrum, and the Hubble diagram simultaneously, which is why $\Lambda$CDM is called the concordance model.

Full proof set Master

Proposition 1 (e-folds drive flatness). During slow-roll inflation with $\epsilon =-\dot{H}/H^2$ and $H$ approximately constant, the curvature density parameter evolves as ${\Omega }_k(a)={\Omega }_{k,\text{start}}{e}^{-2N(a)}$ where $N(a)=\ln (a/a_{\text{start}})$. Reducing $|{\Omega }_k|$ from unity at the Planck epoch to its observed value today requires $N\gtrsim 60$ e-folds of inflation.

Proof. By definition ${\Omega }_k(a)=-k{c}^2/(a^2{H}^2)$. Differentiate with respect to $\ln a$:

$$\frac{d\ln |{\Omega }_k|}{d\ln a}=-2-2\frac{d\ln H}{d\ln a}=-2+2\epsilon ,$$

because $d\ln H/d\ln a=(\dot{H}/H)/(\dot{a}/a)=\dot{H}/H^2=-\epsilon$. During slow roll $\epsilon \ll 1$, so to leading order $d\ln |{\Omega }_k|/d\ln a=-2$, giving $\ln |{\Omega }_k(a)|=\ln |{\Omega }_{k,\text{start}}|-2N(a)$ and hence ${\Omega }_k(a)={\Omega }_{k,\text{start}}{e}^{-2N(a)}$.

To reach the present $|{\Omega }_{k,0}|\lesssim 10^{-2}$ from a Planck-era value of order unity, accounting for the post-inflationary dilution by radiation and matter domination (which together contribute a factor $\sim 10^{-25}$ of growth in $|{\Omega }_k|$ before today), inflation itself must supply $e^{-2N}\lesssim 10^{-37}$, i.e. $N\gtrsim 43$. Including the requirement that the present horizon scale was inflated beyond the Planck length (a stronger, scale-based bound) gives the canonical $N\gtrsim 60$. $\square$

Proposition 2 (growing mode in Einstein-de Sitter). In a flat, matter-dominated, $\Lambda =0$ universe with $a\propto t^{2/3}$ and ${\bar{\rho }}_m=1/(6\pi G{t}^2)$, the linear perturbation equation $\ddot{\delta }+2H\dot{\delta }-4\pi G{\bar{\rho }}_m\delta =0$ has general solution $\delta (t)=C_{+}{t}^{2/3}+C_{-}{t}^{-1}$, so the growing mode satisfies ${\delta }_{+}\propto a$.

Proof. Substitute $H=2/(3t)$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$ into the growth equation and try $\delta \propto t^p$:

$$p(p-1)t^{p-2}+\frac{4}{3}p{t}^{p-2}-\frac{2}{3}{t}^{p-2}=0⟹p^2+\frac{1}{3}p-\frac{2}{3}=0⟹3p^2+p-2=0.$$

The roots are $p=(-1\pm \sqrt{1+24})/6=(-1\pm 5)/6$, giving $p_{+}=2/3$ and $p_{-}=-1$. The general solution is $\delta (t)=C_{+}{t}^{2/3}+C_{-}{t}^{-1}$. Because $a\propto t^{2/3}$, the growing mode is ${\delta }_{+}\propto a$ and the decaying mode ${\delta }_{-}\propto t^{-1}=a^{-3/2}$. Generic initial conditions excite both, but the decaying component redshifts away, leaving the growing mode to dominate. $\square$

Proposition 3 (helium-4 mass fraction from the $n/p$ ratio). If, at the onset of efficient nucleosynthesis, essentially every free neutron is incorporated into a helium-4 nucleus and the neutron-to-proton ratio is $r\equiv n/p$, then the primordial helium-4 mass fraction is $Y_p=2r/(1+r)$. For $r=1/7$ this gives $Y_p=1/4$.

Proof. Consider a baryon reservoir of $n_p$ protons and $n_n=r{n}_p$ neutrons. Each helium-4 nucleus consumes 2 protons and 2 neutrons, so the number of He-4 nuclei formed (assuming all neutrons are used) is $N_{{}^4\text{He}}=n_n/2=r{n}_p/2$. The mass of each He-4 nucleus is approximately $4m_b$ (neglecting binding-energy corrections at the $\sim 0.1\%$ level). The total helium mass is $M_{\text{He}}=4m_b\cdot r{n}_p/2=2r{m}_b{n}_p$, and the total baryon mass is $M_b=m_b(n_p+n_n)=m_b{n}_p(1+r)$. Therefore

$$Y_p=\frac{M_{\text{He}}}{M_b}=\frac{2r{m}_b{n}_p}{m_b{n}_p(1+r)}=\frac{2r}{1+r}.$$

For the post-freezeout, post-decay value $r=1/7$: $Y_p=2(1/7)/(1+1/7)=(2/7)/(8/7)=2/8=1/4=0.25$, matching observation. $\square$

*Proposition 4 (acceleration requires $w<-1/3$).** *In the FLRW background with a single perfect-fluid component of equation of state $p=w\rho c^2$, the expansion accelerates ($\ddot{a}>0$) if and only if $w<-1/3$.

Proof. The second Friedmann equation (with $k$ and $\Lambda$ absorbed or zero for the single-component case) reads

$$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho +\frac{3p}{c^2}\right)=-\frac{4\pi G\rho }{3}(1+3w).$$

Since $4\pi G\rho /3>0$, the sign of $\ddot{a}$ is the sign of $-(1+3w)$. Acceleration $\ddot{a}>0$ requires $1+3w<0$, i.e. $w<-1/3$. Pressureless matter ($w=0$) and radiation ($w=1/3$) both fail this bound and decelerate; a cosmological constant ($w=-1$) and the slow-roll inflaton ($w\approx -1$) satisfy it and accelerate. $\square$

Connections Master

• To the field equations 13.04.01. Every result in this unit is the Einstein field equation $G_{\mu \nu }+\Lambda g_{\mu \nu }=(8\pi G/c^4)T_{\mu \nu }$ specialised to the FLRW ansatz and then decorated with microphysics. The Friedmann equations are the $tt$ and $ii$ components; the perturbation growth equation is the linearised Einstein equation about FLRW; inflation is just an inflaton stress-energy tensor with $w<-1/3$ fed into those same equations. There is no extra gravitational dynamics in cosmology beyond 13.04.01 -- only a drastic symmetry reduction that turns PDEs into ODEs.

• To curvature and spatial geometry 13.03.01. The acoustic-peak measurement ${\ell }_1\approx 220$ is a direct observation of spatial curvature -- the very Riemann tensor of 13.03.01 evaluated on the largest accessible scale. A nonzero ${\Omega }_k$ would mean the spatial slices carry intrinsic curvature $k\ne 0$, shifting the peak; the observed flatness $|{\Omega }_k|<0.005$ tells us the spatial Riemann tensor nearly vanishes on Gpc scales, a fact inflation explains dynamically via Proposition 1.

• To black holes 13.06.01. The late de-Sitter attractor of 13.08.01 gives the universe an event horizon with a Gibbons-Hawking temperature and Bekenstein-Hawking entropy, structurally identical to a black-hole horizon 13.06.01. The same thermodynamic laws (area theorem, generalised second law) govern both. Cosmological horizons also source the Hawking radiation that sets the ultimate floor on the universe's temperature.

• To the foundational FLRW unit 13.08.01. This unit presupposes the Friedmann equations, $\Lambda$CDM decomposition, de Sitter asymptotics, and conformal-time / horizon machinery developed in 13.08.01. Where that unit derived the background dynamics, this unit overlays the thermal, nuclear, acoustic, and gravitational microphysics that runs on top of that background: inflation sets the initial perturbations, BBN and the CMB are thermal-history events on the clock $a(t)$, and structure growth is gravitational instability against the Friedmann expansion.

Historical & philosophical context Master

Cosmology as an exact science was born when Alexander Friedmann found non-static exact solutions of the Einstein equations in 1922 ^{[Friedmann1922]}, against Einstein's own belief in a static universe (Einstein had inserted $\Lambda$ precisely to keep it static). Georges Lemaitre independently derived the expansion law in 1927 and predicted the velocity-distance relation ^{[Lemaitre1927]}; Edwin Hubble's 1929 observations confirmed it ^[Hubble1929], and Lemaitre further identified an initial "primeval atom" -- the first statement of what would be called the Big Bang.

The hot-Bang picture became quantitative with the alpha-beta-gamma paper of Alpher, Bethe and Gamow in 1948 ^{[AlpherBetheGamow1948]}, which predicted the light-element abundances and, in follow-up work by Alpher and Herman, a relic radiation at a few kelvin. That radiation was discovered serendipitously by Penzias and Wilson in 1965 ^{[PenziasWilson1965]} as an unexplained 3 K antenna temperature, immediately vindicating the hot Big Bang over its steady-state rival. The COBE satellite (1992) detected the $\Delta T/T\sim 10^{-5}$ anisotropies that seed structure.

Inflation was proposed by Alan Guth in 1980-1981 ^[Guth1981] as a resolution of the horizon and flatness problems; its later refinement (Linde, Albrecht and Steinhardt; "new" and "chaotic" inflation) made it a mechanism also for generating the primordial perturbations. The philosophical import is striking: the largest-scale structure of the universe -- its flatness, isotropy, and the seeds of galaxies -- is traced back to microscopic quantum physics in the first $10^{-32}$ s. Open questions remain: the cosmological-constant / coincidence problem (why is ${\Omega }_{\Lambda }\approx {\Omega }_m$ now?), the lithium-7 discrepancy, the nature of dark matter and dark energy, and whether the inflationary perturbations carry a detectable tensor signal. The concordance of BBN, CMB, and large-scale structure across nine decades of cosmic time is the strongest evidence that the Friedmann framework is correct in its domain, even while its dark components remain physically unidentified.

Bibliography Master

@article{Friedmann1922,
  author  = {Friedmann, Alexander},
  title   = {{\"U}ber die Kr{\"u}mmung des Raumes},
  journal = {Zeitschrift f{\"u}r Physik},
  volume  = {10},
  pages   = {377--386},
  year    = {1922}
}

@article{Lemaitre1927,
  author  = {Lema{\^\i}tre, Georges},
  title   = {Un Univers homog{\`e}ne de masse constante et de rayon croissant},
  journal = {Annales de la Soci{\'e}t{\'e} Scientifique de Bruxelles},
  volume  = {A47},
  pages   = {49--59},
  year    = {1927}
}

@article{Hubble1929,
  author  = {Hubble, Edwin},
  title   = {A relation between distance and radial velocity among extra-galactic nebulae},
  journal = {Proceedings of the National Academy of Sciences},
  volume  = {15},
  pages   = {168--173},
  year    = {1929}
}

@article{AlpherBetheGamow1948,
  author  = {Alpher, Ralph A. and Bethe, Hans and Gamow, George},
  title   = {The origin of chemical elements},
  journal = {Physical Review},
  volume  = {73},
  pages   = {803--804},
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@article{PenziasWilson1965,
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  volume  = {142},
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@article{Guth1981,
  author  = {Guth, Alan H.},
  title   = {Inflationary universe: A possible solution to the horizon and flatness problems},
  journal = {Physical Review D},
  volume  = {23},
  pages   = {347--356},
  year    = {1981}
}

@book{Dodelson2003,
  author    = {Dodelson, Scott},
  title     = {Modern Cosmology},
  publisher = {Academic Press},
  year      = {2003}
}

@book{Weinberg2008,
  author    = {Weinberg, Steven},
  title     = {Cosmology},
  publisher = {Oxford University Press},
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@book{Mukhanov2005,
  author    = {Mukhanov, Viatcheslav},
  title     = {Physical Foundations of Cosmology},
  publisher = {Cambridge University Press},
  year      = {2005}
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@book{Ryden2017,
  author    = {Ryden, Barbara},
  title     = {Introduction to Cosmology},
  edition   = {2},
  publisher = {Cambridge University Press},
  year      = {2017}
}