Cosmology

For a homogeneous fluid of energy density $\rho$, pressure $P$, and scale factor $a(t)$, adiabatic work in a comoving volume gives

A constant barotropic equation of state has $P=w\rho$. The cases $w=0$ and $w=1/3$ describe pressureless matter and radiation respectively.

For constant $w\ne -1$, the cosmological continuity equation integrates towhen $a=1$ at the reference time and ${\rho }_0$ is the reference density.

For an expanding spatially flat universe with constant $w>-1$ and $a(t_0)=1$,It vanishes at the finite past time $t_{\ast }=t_0-2/[3(1+w)H_0]$.

At Hubble parameter $H$, the critical energy density is

The density parameter is $\Omega =\rho /{\rho }_{\mathrm{crit}}$. In a Friedmann universe, its departure from one measures the spatial-curvature term relative to the density or expansion term.

In a decelerating expanding universe, $|\Omega -1|$ grows, so its small present value requires extremely fine-tuned early initial data unless an earlier mechanism drives $\Omega$ toward one.

The continuity equation givesDuring expansion with $\rho +3P<0$, the product $\rho a^2$ grows and $|{\Omega }^{-1}-1|\propto (\rho a^2{)}^{-1}$ decreases.

In a decelerating hot Big Bang model, widely separated regions of the observable universe have nearly equal conditions despite having no shared past light cone. Inflation supplies an earlier era with a shrinking comoving Hubble radius, allowing a formerly causal patch to grow beyond the later horizon.

Cosmological radiation has equation of state $P_r={\rho }_r/3$ and density scaling ${\rho }_r\propto a^{-4}$.

A cosmological-constant component has $P_{\Lambda }=-{\rho }_{\Lambda }$, so its energy density is constant during expansion.

A Big Crunch is a future finite-time singularity in which a recollapsing universe reaches vanishing scale factor.

Cosmic inflation is an early period of accelerated expansion, often modeled by a scalar field whose potential energy dominates its kinetic energy.

A canonical homogeneous inflaton $\varphi (t)$ hasand obeys $\ddot{\varphi }+3H\dot{\varphi }+V^{\prime }(\varphi )=0$.

For a canonical scalar field in a spatially flat expanding universe,The field Euler-Lagrange equation is

For nearly constant inflaton potential and constant $H_{\inf }$, the spatial Fourier transform obeys

For $a=-1/(H_{\inf }\tau )$, the rescaled modesatisfies

The Bunch-Davies vacuum selects the mode that approaches the positive-frequency ground-state mode of angular frequency $ck$ in the far past $\tau \to -\infty$.

A Fourier-space variance $P_k$ in three dimensions is scale invariant when the dimensionless power per logarithmic wavenumber interval,is independent of $k$. For a massless inflaton mode at late times,so ${\Delta }^2(k)$ is constant.

For a canonical scalar field with potential $V(\varphi )$, slow roll reduces the field and Friedmann equations to

For $V(\varphi )=m^2{\varphi }^2$ in units where $3H^2\simeq V$ and for positive $\varphi$,and

For $V(\varphi )=\lambda {\varphi }^4/4$ with
$H^2\simeq 8\pi GV/(3c^2)$ and positive $\varphi$,and

The potential slow-roll parameter isUnder the slow-roll equations, ${\dot{\varphi }}^2/(2V)\simeq {ϵ}_V/3$.

The number of e-folds between field values ${\varphi }_i$ and ${\varphi }_f$ iswhen the field rolls monotonically down the potential.

For $V(\varphi )=\lambda {\varphi }^n/n$ under the stated Cambridge Tripos normalization,If slow roll ends by kinetic-potential equality, then

Natural inflation uses a periodic potentialIts potential slow-roll parameter obeyson the branch where the tangent is positive.

For the cosine potential,

For pressureless subhorizon matter perturbations, a Fourier mode in conformal time obeyswhere $\mathcal{H}=a^{\prime }/a$.

The density contrast is the fractional perturbation $\delta =(\rho -\bar{\rho })/\bar{\rho }$.

For a pressureless fluid withthe first-order density contrast $\delta =\delta \rho /\bar{\rho }$ obeys

The Jeans wavenumber separates pressure-supported modes from gravitationally unstable density modes.

During radiation domination in a flat expanding universe, $a(t)\propto t^{1/2}$ and $H=1/(2t)$.

In a flat matter-dominated era with $a\propto {\tau }^2$, one has $\mathcal{H}=2/\tau$ andThe growing mode is proportional to the scale factor.

In cosmic time during matter domination, $a(t)\propto t^{2/3}$ and the pressureless density contrast obeysIts growing and decaying solutions are $t^{2/3}$ and $t^{-1}$.

After neglecting the decaying mode, the linear growth between conformal times ${\tau }_1$ and ${\tau }_2$ is

A mode of comoving wavenumber $k$ and physical wavelength $2\pi a/k$ crosses a conformal horizon of physical size $ac\tau$ when

With $a(t_0)=1$, $k_0=2\pi /(c{t}_0)$, and $1+z_{\mathrm{eq}}=(t_0/t_{\mathrm{eq}}{)}^{2/3}$,

A mode crossing during matter domination acquires the growth factor $({\tau }_0/{\tau }_H{)}^2$. An initial amplitude proportional to ${\tau }_H^2{k}^{1/2}$ therefore becomes proportional to $k^{1/2}$ today.

The density power spectrum records the squared Fourier-mode amplitude, up to the chosen statistical normalization. Scale dependence acquired from primordial amplitudes and subsequent growth determines its spectral shape.

If primordial horizon-crossing perturbations satisfy $V⟨|{\delta }_k{|}^2⟩=C/k^3$, remain frozen during radiation domination, and grow as $a$ during matter domination, then

Cosmological recombination is the epoch in which cooling allowed free electrons and ions to form neutral atoms, sharply reducing the free-electron density.

Chemical equilibrium for $A^{+}+e^{-}\leftrightarrow A^0+\gamma$ givesafter neglecting the ion--atom mass difference in the translational prefactor.

Because equilibrium reactions balance chemical potentials and photons have zero chemical potential,The rest-mass difference in the Maxwell--Boltzmann densities then supplies the Boltzmann factor $e^{-I_A/(k_{B}T)}$.

For $p+e^{-}\leftrightarrow H+\gamma$, the hydrogen binding energy is the rest-energy defect

With singly ionized helium and helium mass fraction $Y_p$, charge neutrality gives

Writing $\mathcal{F}=n_e/n_B$ couples the two equilibrium equations:

When $Y_p=0$, $\mathcal{F}=X_{H^{+}}$, and

$H=\dot{a}/a$, a fluid with $P=w\rho$ has $\rho \propto a^{-3(1+w)}$, and ${\rho }_{\mathrm{crit}}=3c^2{H}^2/(8\pi G)$.

Combining the perfect-fluid continuity equation withshows thatis constant in time. This constant is the spatial-curvature integration constant in the first Friedmann equation.

For energy density $\rho$, pressure $P$, and cosmological constant $\Lambda$,

For a homogeneous pressureless fluid in comoving coordinates, the Euler equation giveswhile the Poisson equation givesTaking the divergence of the first relation and comparing them yieldsThis local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.

For positive spatial curvature, zero cosmological constant, and $\rho ,P\ge 0$, continuity makes $\rho a^3$ nonincreasing during expansion. The negative curvature term then prevents unbounded growth of $a$, while the acceleration equation forces $\dot{a}$ to reach zero in finite time.

With zero cosmological constant, the condition $\rho +3P\ge 0$ implies $\ddot{a}\le 0$ andwherever $H\ne 0$.

If $H>0$ at one time, the inequality $d(H^{-1})/dt\ge 1$ forces $H\to +\infty$ and $a\to 0$ within finite time to the past. If $H<0$, it forces $H\to -\infty$ and $a\to 0$ within finite time to the future.

In a flat radiation-plus-$\Lambda$ universe,The expansion changes from deceleration to acceleration when the radiation and cosmological-constant terms are equal, at

If the present radiation and cosmological-constant densities are respectively $\beta$ and $1$ times the present critical density, while $a_0=1$, positive curvature givesTurning points are therefore the positive roots of $x^2-\beta x+\beta$ with $x=a^2$.

For $\beta >4$, the turning polynomial has two positive roots and its smaller root lies below $2$. A universe expanding from $a=0$ reaches that root with $H=0$ and $\dot{H}<0$, then contracts to a finite-time Big Crunch.

For $\beta =4$ and $a(0)=0$, the expanding branch isIt grows as $2\sqrt{H_{0}t}$ near zero and approaches $\sqrt{2}$ exponentially at late times.

A fluid with $w<-1$ grows in density during expansion and can produce a finite-time big-rip singularity.

The cosmological scale factor $a(t)$ converts comoving separations into physical separations in an FLRW universe.

The Hubble parameter is the fractional expansion rateand satisfies $\dot{H}+H^2=\ddot{a}/a$.

The homogeneous expansion velocity is proportional to physical position:

A comoving coordinate remains fixed for an observer following the homogeneous cosmological expansion.

Peculiar velocity is motion relative to the homogeneous Hubble flow. In comoving coordinates,where $\mathbf{u}$ is physical velocity and $\mathbf{v}$ is peculiar velocity.

Light emitted at $t_e$ and observed at $t_0$ has $1+z=a(t_0)/a(t_e)$.

Observed time intervals from a comoving source are stretched by the same factor $1+z$ as photon wavelengths.

After photon decoupling in an expanding universe, every frequency scales as $\nu \propto a^{-1}$. Therefore a Planck occupation factor retains its thermal form when its temperature is assigned the same scaling,

Luminosity distance is defined by $F=L/(4\pi d_L^2)$ and includes geometric dilution, photon redshift, and arrival-rate dilation.

Expansion reduces bolometric flux by one factor of $1+z$ from photon energy and one from arrival rate, in addition to inverse-area dilution.

The FLRW metric is the homogeneous and isotropic spacetime metric with scale factor $a(t)$ and constant spatial curvature $k$.

The Milne universe is the empty $k=-1$, $\Lambda =0$ FLRW model with scale factor proportional to proper time. It is a hyperbolic-coordinate description of the interior of a future light cone in Minkowski spacetime.

In Cartesian comoving coordinates and signature $(-,+,+,+)$, a spatially flat FLRW universe has

The cases $k>0$, $k=0$, and $k<0$ give spherical, Euclidean, and hyperbolic constant-curvature spatial slices.

Proper distance is the spatial metric length measured along a fixed-cosmic-time slice.

Conformal time satisfies $d\tau =dt/a(t)$ and factors the FLRW metric into $a^2(\tau )$ times a static conformal metric.

For the de Sitter scale factor in flat slicingchoose the additive constant so thatAn eternal expanding patch has $-\infty <\tau <0$.

Comoving radial distance uses the coordinate $\chi =\int dr/\sqrt{1-k{r}^2}$ and removes the radial curvature factor.

In conformal coordinates a radial light ray obeys $d\chi /d\tau =\pm c$, so the cosmological light cone is at forty-five degrees after setting $c=1$.

Energy conservation in FLRW gives $\dot{\rho }=-3H(\rho +P)$ when $\rho$ and $P$ are energy density and pressure.

A cosmological fluid with constant equation-of-state parameter $w$ has $P=w\rho$ and $\rho \propto a^{-3(1+w)}$.

Quintessence is dynamical dark energy with negative pressure, conventionally $-1<w<-1/3$.

For $w=-2/3$, zero cosmological constant, and negative curvature, $a(t)=\sqrt{\beta }t+\gamma t^2/4$ with density and curvature parameters $\gamma ,\beta >0$.

For $w=-2/3$, a quintessence-dominated scale factor grows quadratically and $tH$ approaches two.

The age is the proper cosmic time elapsed from the scale-factor singularity $a=0$ to the present epoch.

The Hubble time $H_0^{-1}$ is the expansion timescale inferred from the present Hubble parameter.

In a curvature-dominated open FLRW regime, the scale factor is approximately linear in cosmic time and $tH$ approaches one.

The deceleration parameter is $q=-a\ddot{a}/{\dot{a}}^2$; negative values describe accelerated expansion.

An accelerating universe has $\ddot{a}>0$, equivalently $q<0$ while expansion continues.

Big Bang nucleosynthesis formed the light nuclei during the first minutes of cosmic expansion after weak freeze-out and the deuterium bottleneck.

With negligible lepton chemical potentials, weak equilibrium gives $n_n/n_p\simeq e^{-Q/(k_{B}T)}$, where $Q=(m_n-m_p)c^2$.

Weak neutron--proton conversion freezes out when its interaction rate falls below the Hubble expansion rate.

If an interaction-to-expansion ratio scales as $(T/\kappa {)}^3$, increasing $\kappa$ raises the temperature at which the ratio drops through one.

Light nuclei cannot accumulate until cooling makes deuterium sufficiently resistant to photodissociation, delaying nucleosynthesis after weak freeze-out.

For chemical equilibrium $D\leftrightarrow n+p$, Maxwell--Boltzmann densities and ${\mu }_D={\mu }_n+{\mu }_p$ give, after taking $m_n\simeq m_p$ and $m_D\simeq 2m_p$,Writing $X_i=n_i/n_B$, $n_B=\eta n_{\gamma }$, and $n_{\gamma }=16\pi \zeta (3)(k_{B}T{)}^3/(hc{)}^3$ givesThe very small baryon-to-photon ratio suppresses deuterium until $T$ is far below the binding-energy scale.

Increasing the baryon-to-photon ratio raises the equilibrium deuterium abundance at fixed temperature, so the deuterium bottleneck ends earlier. Fewer neutrons decay before nuclear burning, and neutron-limited helium synthesis produces a larger primordial helium mass fraction if weak freeze-out is unchanged.

Between weak freeze-out and deuterium formation, beta decay lowers the surviving free-neutron fraction.

If almost all neutrons enter helium-4 and $r=n_n/n_p$, its primordial mass fraction is $Y_p\simeq 2r/(1+r)$.

When protons outnumber neutrons, helium-4 production is neutron-limited and produces about one helium nucleus per two neutrons.

Increasing primordial helium at the expense of hydrogen reduces fuel for long-lived main-sequence hydrogen burning and changes subsequent stellar evolution.

The baryon-to-photon ratio is the number density of baryons divided by the number density of photons. In the present universe it is of order $10^{-9}$.