= Cosmology
{wiki}

= Cosmological perfect-fluid continuity equation
{parent=Cosmology}

For a homogeneous fluid of energy density $\rho$, pressure $P$, and scale factor $a(t)$, adiabatic work in a comoving volume gives
$$
\dot\rho+3\frac{\dot a}{a}(\rho+P)=0.
$$

= Barotropic equation of state
{parent=Cosmological perfect-fluid continuity equation}
{wiki=Equation_of_state_(cosmology)}

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

= Constant-equation-of-state density scaling
{parent=Barotropic equation of state}

For constant $w\ne-1$, the cosmological continuity equation integrates to
$$
\rho(a)=\rho_0a^{-3(1+w)}
$$
when $a=1$ at the reference time and $\rho_0$ is the reference density.

= Flat constant-equation-of-state scale factor
{parent=Constant-equation-of-state density scaling}

For an expanding spatially flat universe with constant $w>-1$ and $a(t_0)=1$,
$$
a(t)=\left[1+\frac32(1+w)H_0(t-t_0)\right]^{2/[3(1+w)]}.
$$
It vanishes at the finite past time $t_*=t_0-2/[3(1+w)H_0]$.

= Critical density
{parent=Cosmology}
{wiki=Critical_density}

At Hubble parameter $H$, the critical energy density is
$$
\rho_{\rm crit}=\frac{3c^2H^2}{8\pi G}.
$$

= Cosmological density parameter
{title2=$\Omega$}
{parent=Critical density}
{wiki=Density_parameter}

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

= Flatness problem
{c}
{parent=Cosmological density parameter}
{wiki=Flatness_problem}

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.

= Inflationary solution of the flatness problem
{parent=Flatness problem}

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

= Horizon problem
{c}
{parent=Cosmology}
{wiki=Horizon_problem}

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.

= Radiation in cosmology
{parent=Cosmology}

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

= Cosmological constant energy
{parent=Cosmology}
{wiki=Cosmological_constant}

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

= Big Crunch
{parent=Cosmology}
{c}
{wiki}

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

= Cosmic inflation
{parent=Cosmology}
{wiki=Inflation_(cosmology)}

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

= Inflaton
{parent=Cosmic inflation}
{wiki}

A canonical homogeneous inflaton $\phi(t)$ has
$$
\rho=\frac12\dot\phi^2+V(\phi),
\qquad
P=\frac12\dot\phi^2-V(\phi),
$$
and obeys $\ddot\phi+3H\dot\phi+V'(\phi)=0$.

= Inflaton action in an expanding universe
{parent=Inflaton}

For a canonical scalar field in a spatially flat expanding universe,
$$
S[\phi]=\int d^3x\,dt\,a^3
\left[
\frac12\dot\phi^2
-\frac{c^2}{2a^2}|\nabla\phi|^2
-V(\phi)
\right].
$$
The field <Euler-Lagrange equation> is
$$
\ddot\phi+3H\dot\phi-\frac{c^2}{a^2}\nabla^2\phi+V'(\phi)=0.
$$

= Inflationary scalar Fourier mode
{parent=Inflaton action in an expanding universe}

For nearly constant inflaton potential and constant $H_{\rm inf}$, the spatial <Fourier transform> obeys
$$
\ddot{\widehat\phi}_{\mathbf k}
+3H_{\rm inf}\dot{\widehat\phi}_{\mathbf k}
+\frac{c^2k^2}{a^2}\widehat\phi_{\mathbf k}=0.
$$

= Canonically rescaled de Sitter scalar mode
{c}
{parent=Inflationary scalar Fourier mode}

For $a=-1/(H_{\rm inf}\tau)$, the rescaled mode
$$
\widetilde\phi_{\mathbf k}=a\widehat\phi_{\mathbf k}
$$
satisfies
$$
\widetilde\phi_{\mathbf k}''
+\left(c^2k^2-\frac2{\tau^2}\right)
\widetilde\phi_{\mathbf k}=0.
$$

= Bunch-Davies vacuum
{c}
{parent=Canonically rescaled de Sitter scalar mode}
{wiki=Bunch%E2%80%93Davies_vacuum}

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

= Scale-invariant inflationary power spectrum
{parent=Bunch-Davies vacuum}

A Fourier-space variance $P_k$ in three dimensions is scale invariant when the dimensionless power per logarithmic wavenumber interval,
$$
\Delta^2(k)=\frac{k^3P_k}{2\pi^2},
$$
is independent of $k$. For a massless inflaton mode at late times,
$$
P_k\longrightarrow\frac{\hbar H_{\rm inf}^2}{2c^3k^3},
$$
so $\Delta^2(k)$ is constant.

= Slow-roll approximation
{parent=Cosmic inflation}
{wiki}

For a canonical scalar field with potential $V(\phi)$, slow roll reduces the field and Friedmann equations to
$$
H^2\simeq\frac{V}{3M_{\rm Pl}^2},
\qquad
3H\dot\phi\simeq-V'.
$$

= Quadratic-potential slow-roll solution
{parent=Slow-roll approximation}

For $V(\phi)=m^2\phi^2$ in units where $3H^2\simeq V$ and for positive $\phi$,
$$
\phi(t)=\phi_i-\frac{2m}{\sqrt3}t,
$$
and
$$
a(t)=a_i\exp\left(
\frac{m\phi_i}{\sqrt3}t-\frac{m^2t^2}{3}
\right)
=a_i\exp\left(\frac{\phi_i^2-\phi(t)^2}{4}\right).
$$

= Quartic-potential slow-roll solution
{parent=Slow-roll approximation}

For $V(\phi)=\lambda\phi^4/4$ with
$H^2\simeq8\pi GV/(3c^2)$ and positive $\phi$,
$$
\phi(t)=\phi_0
\exp\left[-c\sqrt{\frac{\lambda}{6\pi G}}(t-t_0)\right],
$$
and
$$
a(\phi)=a_0
\exp\left[\frac{\pi G}{c^2}(\phi_0^2-\phi^2)\right].
$$

= Potential slow-roll parameter
{parent=Slow-roll approximation}

The potential slow-roll parameter is
$$
\epsilon_V=\frac{M_{\rm Pl}^2}{2}
\left(\frac{V'}V\right)^2.
$$
Under the slow-roll equations, $\dot\phi^2/(2V)\simeq\epsilon_V/3$.

= Slow-roll e-fold count
{parent=Slow-roll approximation}

The number of e-folds between field values $\phi_i$ and $\phi_f$ is
$$
N\simeq\frac1{\sqrt2M_{\rm Pl}}
\int_{\phi_f}^{\phi_i}\frac{d\phi}{\sqrt{\epsilon_V(\phi)}}
$$
when the field rolls monotonically down the potential.

= Monomial slow-roll e-fold count
{parent=Slow-roll e-fold count}

For $V(\phi)=\lambda\phi^n/n$ under the stated Cambridge Tripos normalization,
$$
N_e=\frac{4\pi G}{c^4n}
(\phi_i^2-\phi_f^2).
$$
If slow roll ends by kinetic-potential equality, then
$$
\phi_f^2\simeq\frac{n^2c^4}{48\pi G},
\qquad
N_e\simeq\frac n{12}
\left[\left(\frac{\phi_i}{\phi_f}\right)^2-1\right].
$$

= Natural inflation
{parent=Slow-roll approximation}
{wiki}

Natural inflation uses a periodic potential
$$
V(\phi)=V_0[1+\cos(\phi/f)].
$$
Its potential slow-roll parameter obeys
$$
\sqrt{\epsilon_V}=\frac{M_{\rm Pl}}{\sqrt2f}
\tan\frac{\phi}{2f}
$$
on the branch where the tangent is positive.

= E-fold count for the natural-inflation cosine potential
{parent=Natural inflation}

For the cosine potential,
$$
N=\frac{2f^2}{M_{\rm Pl}^2}
\left[
\log\sin\frac{\phi_i}{2f}
-\log\sin\frac{\phi_f}{2f}
\right].
$$

= Linear cosmological density perturbation
{parent=Cosmology}
{wiki=Structure_formation}

For pressureless subhorizon matter perturbations, a Fourier mode in conformal time obeys
$$\delta''+\mathcal H\delta'-\frac32\Omega_M\mathcal H^2\delta=0,$$
where $\mathcal H=a'/a$.

= Density contrast
{title2=$\delta$}
{parent=Linear cosmological density perturbation}
{wiki=Density_contrast}

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

= Linearized cosmological continuity equation
{parent=Density contrast}

For a pressureless fluid with
$$
\rho=\bar\rho+\epsilon\,\delta\rho,
\qquad
\mathbf v=\epsilon\,\delta\mathbf v,
$$
the first-order <density contrast> $\delta=\delta\rho/\bar\rho$ obeys
$$
\dot\delta=-\frac1a\nabla\cdot\delta\mathbf v.
$$

= Jeans wavenumber
{title2=$k_J$}
{parent=Linear cosmological density perturbation}
{c}
{wiki=Jeans_instability}

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

= Radiation domination
{parent=Linear cosmological density perturbation}
{wiki=Scale_factor_(cosmology)}

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

= Matter-era growing and decaying density modes
{parent=Linear cosmological density perturbation}

In a flat matter-dominated era with $a\propto\tau^2$, one has $\mathcal H=2/\tau$ and
$$\delta=A\tau^2+B\tau^{-3}.$$
The growing mode is proportional to the scale factor.

= Cosmic-time matter density modes
{parent=Matter-era growing and decaying density modes}

In cosmic time during matter domination, $a(t)\propto t^{2/3}$ and the pressureless density contrast obeys
$$
\ddot\delta+\frac4{3t}\dot\delta-\frac2{3t^2}\delta=0.
$$
Its growing and decaying solutions are $t^{2/3}$ and $t^{-1}$.

= Matter-era linear growth factor
{parent=Matter-era growing and decaying density modes}

After neglecting the decaying mode, the linear growth between conformal times $\tau_1$ and $\tau_2$ is
$$D(\tau_2,\tau_1)=\frac{a(\tau_2)}{a(\tau_1)}
=\left(\frac{\tau_2}{\tau_1}\right)^2.$$

= Cosmological horizon crossing
{parent=Linear cosmological density perturbation}

A mode of comoving wavenumber $k$ and physical wavelength $2\pi a/k$ crosses a conformal horizon of physical size $ac\tau$ when
$$\tau_H=\frac{2\pi}{kc}.$$

= Horizon-crossing time across matter-radiation equality
{parent=Cosmological horizon crossing}

With $a(t_0)=1$, $k_0=2\pi/(ct_0)$, and $1+z_{\rm eq}=(t_0/t_{\rm eq})^{2/3}$,
$$
\frac{t_H}{t_0}\simeq
\begin{cases}
(k_0/k)^3,&t_H>t_{\rm eq},\\
(1+z_{\rm eq})^{-1/2}(k_0/k)^2,&t_H<t_{\rm eq}.
\end{cases}
$$

= Matter-era transfer of a horizon-crossing amplitude
{parent=Cosmological horizon crossing}

A mode crossing during matter domination acquires the growth factor $(\tau_0/\tau_H)^2$. An initial amplitude proportional to $\tau_H^2k^{1/2}$ therefore becomes proportional to $k^{1/2}$ today.

= Cosmological density power spectrum
{parent=Linear cosmological density perturbation}
{wiki=Matter_power_spectrum}

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.

= Broken matter power spectrum from horizon entry
{parent=Cosmological density power spectrum}

If primordial horizon-crossing perturbations satisfy $V\langle|\delta_k|^2\rangle=C/k^3$, remain frozen during radiation domination, and grow as $a$ during matter domination, then
$$
P(k)=
\begin{cases}
Ck/k_0^4,&k<k_{\rm eq},\\
Ck_{\rm eq}^4/(k^3k_0^4),&k>k_{\rm eq}.
\end{cases}
$$

= Cosmological recombination
{parent=Cosmology}
{wiki=Recombination_(cosmology)}

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

= Saha ionization equation
{parent=Cosmological recombination}
{c}
{wiki=Saha_ionization_equation}

Chemical equilibrium for $A^++e^-\leftrightarrow A^0+\gamma$ gives
$$
\frac{n_en_{A^+}}{n_{A^0}}
=\frac{g_eg_{A^+}}{g_{A^0}}
\left(\frac{2\pi m_ek_BT}{h^2}\right)^{3/2}
e^{-I_A/(k_BT)},
$$
after neglecting the ion--atom mass difference in the translational prefactor.

= Chemical-potential balance for ionization
{parent=Saha ionization equation}

Because equilibrium reactions balance chemical potentials and photons have zero chemical potential,
$$\mu_{A^+}+\mu_e=\mu_{A^0}.$$
The rest-mass difference in the Maxwell--Boltzmann densities then supplies the Boltzmann factor $e^{-I_A/(k_BT)}$.

= Hydrogen binding energy
{title2=$E_{\mathrm{bind}}$}
{parent=Saha ionization equation}
{c}

For $p+e^-\leftrightarrow H+\gamma$, the hydrogen binding energy is the rest-energy defect
$$
E_{\mathrm{bind}}=(m_p+m_e-m_H)c^2.
$$

= Free-electron fraction during hydrogen--helium recombination
{parent=Cosmological recombination}

With singly ionized helium and helium mass fraction $Y_p$, charge neutrality gives
$$
\frac{n_e}{n_B}
=(1-Y_p)X_{H^+}+\frac{Y_p}{4}X_{He^+}.
$$

= Coupled hydrogen--helium Saha equations
{parent=Free-electron fraction during hydrogen--helium recombination}

Writing $\mathcal F=n_e/n_B$ couples the two equilibrium equations:
$$
\mathcal F\frac{X_{H^+}}{1-X_{H^+}}=K(T)e^{-I_H/(k_BT)},\qquad
\mathcal F\frac{X_{He^+}}{1-X_{He^+}}=2K(T)e^{-I_{He}/(k_BT)}.
$$

= Hydrogen-only Saha equation
{parent=Coupled hydrogen--helium Saha equations}

When $Y_p=0$, $\mathcal F=X_{H^+}$, and
$$
\frac{X_{H^+}^2}{1-X_{H^+}}
=\frac1{n_B}\left(\frac{2\pi m_ek_BT}{h^2}\right)^{3/2}
e^{-I_H/(k_BT)}.
$$

= Friedmann equation
{c}
{parent=Cosmology}
{wiki=Friedmann_equations}

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

= First integral of the Friedmann acceleration equation
{parent=Friedmann equation}
{c}

Combining the perfect-fluid continuity equation with
$$
\frac{\ddot a}{a}=-\frac{4\pi G}{3c^2}(\rho+3P)
$$
shows that
$$
\frac{8\pi G}{3c^2}\rho a^2-\dot a^2
$$
is constant in time. This constant is the spatial-curvature integration constant in the first Friedmann equation.

= Friedmann acceleration equation
{parent=Friedmann equation}
{c}

For energy density $\rho$, pressure $P$, and cosmological constant $\Lambda$,
$$
\frac{\ddot a}{a}
=-\frac{4\pi G}{3c^2}(\rho+3P)
+\frac{\Lambda c^2}{3}.
$$

= Raychaudhuri equation
{c}
{synonym}

= Newtonian fluid derivation of the Raychaudhuri equation
{c}
{parent=Friedmann acceleration equation}

For a homogeneous pressureless fluid in comoving coordinates, the <Euler equations for an inviscid fluid>[Euler equation] gives
$$
\nabla\bar\Phi=-a\ddot a\,\mathbf x,
$$
while the <Poisson equation> gives
$$
\nabla^2\bar\Phi=\frac{4\pi G}{c^2}\bar\rho a^2.
$$
Taking the <divergence> of the first relation and comparing them yields
$$
\frac{\ddot a}{a}=-\frac{4\pi G}{3c^2}\bar\rho.
$$
This local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.

= Recollapse of a closed Friedmann universe with nonnegative pressure
{parent=Friedmann acceleration equation}

For positive spatial curvature, zero cosmological constant, and $\rho,P\geq0$, 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.

= Strong energy condition in a Friedmann universe
{parent=Friedmann acceleration equation}

With zero cosmological constant, the condition $\rho+3P\geq0$ implies $\ddot a\leq0$ and
$$
\dot H\leq-H^2,
\qquad
\frac d{dt}(H^{-1})\geq1
$$
wherever $H\ne0$.

= Finite-time Friedmann singularity under the strong energy condition
{parent=Strong energy condition in a Friedmann universe}

If $H>0$ at one time, the inequality $d(H^{-1})/dt\geq1$ forces $H\to+\infty$ and $a\to0$ within finite time to the past. If $H<0$, it forces $H\to-\infty$ and $a\to0$ within finite time to the future.

= Radiation-to-cosmological-constant transition
{parent=Friedmann acceleration equation}

In a flat radiation-plus-$\Lambda$ universe,
$$
a(t)=\left(\frac{\Omega_{R0}}{1-\Omega_{R0}}\right)^{1/4}
\left[\sinh\!\left(2H_0\sqrt{1-\Omega_{R0}}\,t\right)\right]^{1/2}.
$$
The expansion changes from deceleration to acceleration when the radiation and cosmological-constant terms are equal, at
$$
t_\Lambda=\frac{\operatorname{arsinh}1}
{2H_0\sqrt{1-\Omega_{R0}}}.
$$

= Closed radiation--cosmological-constant turning polynomial
{parent=Friedmann equation}

If the present radiation and cosmological-constant densities are respectively $\beta$ and $1$ times the present critical density, while $a_0=1$, positive curvature gives
$$
H^2=\frac{H_0^2}{a^4}(a^4-\beta a^2+\beta).
$$
Turning points are therefore the positive roots of $x^2-\beta x+\beta$ with $x=a^2$.

= Supercritical closed radiation--cosmological-constant recollapse
{parent=Closed radiation--cosmological-constant turning polynomial}

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.

= Critical closed radiation--cosmological-constant solution
{parent=Closed radiation--cosmological-constant turning polynomial}

For $\beta=4$ and $a(0)=0$, the expanding branch is
$$
a(t)=\sqrt{2(1-e^{-2H_0t})}.
$$
It grows as $2\sqrt{H_0t}$ near zero and approaches $\sqrt2$ exponentially at late times.

= Phantom energy
{parent=Cosmology}
{wiki}

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

= Scale factor
{parent=Cosmology}
{wiki=Scale_factor_(cosmology)}

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

= Hubble parameter
{parent=Scale factor}
{c}
{wiki}

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

= Hubble flow
{c}
{parent=Hubble parameter}
{wiki=Hubble%27s_law}

The homogeneous expansion velocity is proportional to physical position:
$$
\mathbf u=H\mathbf r=aH\mathbf x=\dot a\,\mathbf x.
$$

= Comoving coordinate
{parent=Scale factor}
{wiki=Comoving_and_proper_distances}

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

= Peculiar velocity
{parent=Comoving coordinate}
{wiki}

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

= Cosmological redshift
{parent=Scale factor}
{wiki}

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

= Cosmological time dilation
{parent=Cosmological redshift}

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

= Redshift preservation of a thermal photon spectrum
{parent=Cosmological redshift}

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,
$$
T(t)=\frac{a(t_{\rm dec})}{a(t)}T_{\rm dec}.
$$

= Luminosity distance
{parent=Cosmology}
{wiki}

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

= Cosmological flux dimming
{parent=Luminosity distance}

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

= Friedmann-Lemaitre-Robertson-Walker metric
{parent=Cosmology}
{c}
{wiki=Friedmann–Lemaître–Robertson–Walker_metric}

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

= Milne universe
{c}
{parent=Friedmann-Lemaitre-Robertson-Walker metric}
{wiki=Milne_model}

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>.

= Spatially flat FLRW metric
{parent=Friedmann-Lemaitre-Robertson-Walker metric}
{c}

In Cartesian comoving coordinates and signature $(-,+,+,+)$, a spatially flat FLRW universe has
$$
ds^2=-c^2dt^2+a^2(t)(dx^2+dy^2+dz^2).
$$

= Spatial curvature of an FLRW universe
{parent=Friedmann-Lemaitre-Robertson-Walker metric}

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

= Proper distance in cosmology
{parent=Friedmann-Lemaitre-Robertson-Walker metric}

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

= Conformal time
{parent=Friedmann-Lemaitre-Robertson-Walker metric}
{wiki}

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

= Conformal time during de Sitter expansion
{c}
{parent=Conformal time}

For the <de Sitter scale factor in flat slicing>
$$
a(t)=a_0e^{H_{\rm inf}t},
$$
choose the additive constant so that
$$
\tau=-\frac1{a_0H_{\rm inf}}e^{-H_{\rm inf}t}
=-\frac1{aH_{\rm inf}}.
$$
An eternal expanding patch has $-\infty<\tau<0$.

= Comoving radial distance
{parent=Conformal time}
{wiki=Comoving_and_proper_distances}

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

= Radial null geodesic in FLRW spacetime
{parent=Conformal time}

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 $c=1$.

= Cosmological continuity equation
{parent=Cosmology}
{wiki=Friedmann_equations\#Fluid_equation}

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

= Equation-of-state parameter
{parent=Cosmological continuity equation}
{wiki=Equation_of_state_(cosmology)}

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

= Quintessence
{parent=Equation-of-state parameter}
{wiki=Quintessence_(physics)}

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

= Quintessence scale-factor solution
{parent=Quintessence}

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

= Quintessence-dominated universe
{parent=Quintessence}

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

= Age of an FLRW universe
{parent=Cosmology}
{wiki=Age_of_the_universe}

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

= Hubble time
{parent=Age of an FLRW universe}
{c}
{wiki}

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

= Curvature-dominated universe
{parent=Cosmology}

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

= Deceleration parameter
{parent=Cosmology}
{wiki}

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

= Accelerating universe
{parent=Deceleration parameter}
{wiki=Accelerating_expansion_of_the_universe}

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

= Big Bang nucleosynthesis
{parent=Cosmology}
{c}
{wiki}

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

= Neutron-proton chemical equilibrium
{parent=Big Bang nucleosynthesis}

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

= Cosmological weak freeze-out
{parent=Big Bang nucleosynthesis}

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

= Freeze-out temperature and interaction strength
{parent=Cosmological weak freeze-out}

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

= Deuterium bottleneck
{parent=Big Bang nucleosynthesis}
{wiki}

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

= Deuterium equilibrium abundance
{parent=Deuterium bottleneck}

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 $m_D\simeq2m_p$,
$$
\frac{n_D}{n_pn_n}
\simeq
\left(\frac{\pi m_pk_BT}{h^2}\right)^{-3/2}
 e^{B_D/(k_BT)}.
$$
Writing $X_i=n_i/n_B$, $n_B=\eta n_\gamma$, and $n_\gamma=16\pi\zeta(3)(k_BT)^3/(hc)^3$ gives
$$
\frac{X_D}{X_pX_n}
\simeq\frac{16\zeta(3)}{\sqrt\pi}\eta
\left(\frac{k_BT}{m_pc^2}\right)^{3/2}
 e^{B_D/(k_BT)}.
$$
The very small baryon-to-photon ratio suppresses deuterium until $T$ is far below the binding-energy scale.

= Baryon-density effect on primordial helium
{parent=Deuterium equilibrium abundance}

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.

= Free-neutron decay after freeze-out
{parent=Deuterium bottleneck}

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

= Primordial helium mass fraction
{parent=Big Bang nucleosynthesis}

If almost all neutrons enter helium-4 and $r=n_n/n_p$, its primordial mass fraction is $Y_p\simeq2r/(1+r)$.

= Neutron-limited helium synthesis
{parent=Primordial helium mass fraction}

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

= Primordial composition and stellar lifetime
{parent=Primordial helium mass fraction}

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

= Baryon-to-photon ratio
{title2=$\eta$}
{parent=Cosmology}
{wiki}

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 $10^{-9}$.