Cosmology

Decoupling of a particle abundance from chemical equilibrium when interactions become too slow compared with cosmic expansion. Pair annihilation can continue after equilibrium production is negligible, leaving a nonzero relic abundance.

For $dY/dx=-\lambda Y^2/x^2$ after freeze-out, the exact limit is $Y_{\infty }=[Y_f^{-1}+\lambda /x_f{]}^{-1}$. The frequently used estimate $x_f/\lambda$ additionally neglects the initial inverse abundance.

Relative physical laws and large-scale statistical properties do not depend on the chosen spatial origin. For a continuous deterministic galaxy velocity field, using the same relative-velocity rule from each origin forces additivity and hence a position-independent linear velocity gradient.

The traceless symmetric part of the galaxy velocity-gradient matrix changes shape without isotropic expansion. An antisymmetric part instead represents rigid rotation. A matrix proportional to the identity plus an antisymmetric matrix has no shear.

Nonlinear cosmological perturbations allow finite fluctuations of the spatial scale, density and other fields. Expanding in spatial gradients can remain useful even when a linear expansion in their amplitudes is invalid. The long-wavelength approximation in cosmology is one such organization of the equations.

When physical wavelengths greatly exceed the Hubble radius, spatial derivatives are small compared with temporal expansion scales. A cosmological gradient expansion then approximates neighbouring regions as locally homogeneous universes, while retaining finite perturbation amplitudes. This does not alone discard frozen tensor cosmological perturbations.

With negligible shear, a minimally coupled scalar field serving as a clock, and $\Pi \ne 0$, the Hamiltonian constraint and momentum constraint imply $\dot{H}=-4\pi GN{\Pi }^2$ and ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi$. Differentiating the local Friedmann relation yields $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$ and ${\partial }_i\Pi =-(3H+V_{,\varphi }/\Pi ){\partial }_i\varphi$. Inserting these identities into ${\dot{\zeta }}_i$ cancels all terms, including gradients of the lapse function. The conserved covector becomes the gradient of a uniform-density curvature perturbation at linear adiabatic order. Dropping the shear constraint is essential; this is not a proof of conservation in an arbitrary non-attractor phase.

With vectors and independent tensors neglected, finite scalar cosmological perturbations of the local spatial scale can be encoded by $h_{ij}=a_{\mathrm{bg}}^2{e}^{2\zeta }{\delta }_{ij}$. Scalar lapse and shift potentials complete its 3+1 decomposition of spacetime. The exact shift-square term is $h^{ij}{N}_i{N}_j$, independently of the parametrization of the scalar shift.

The traceless mixed extrinsic curvature of a spatial hypersurface can have leading solution ${\tilde{K}}^i{}_j=C^i{}_j(\mathbf{x})a_{\mathrm{loc}}^{-3}$. Cosmic inflation suppresses this anisotropic expansion. An anisotropic spatial shape can nevertheless freeze to a nonzero constant, so shear damping does not prove the absence of tensor cosmological perturbations.

Baryogenesis is the dynamical production of the cosmic excess of baryons over antibaryons. It requires baryon-number violation, C and CP violation, and departure from thermal equilibrium, or an appropriate generalization of these Sakharov conditions.

The standard conditions for baryogenesis from a symmetric initial ensemble are baryon number violation, both C violation and CP violation, and departure from thermal equilibrium. Conserved baryon number cannot develop a net value from zero; an exact conjugation symmetry pairs production with equal antiproduction. In thermal equilibrium, CPT symmetry makes the equilibrium expectation of net baryon number vanish for zero imposed chemical potentials for charges odd under CPT symmetry. These conditions assume the usual local quantum-field framework and do not alone guarantee a large asymmetry.

In a symmetric initial population of particles and antiparticles, exact CP symmetry equates conjugate decay rates after summing final momenta and helicities. Since parity does not reverse baryon number, this equality prevents a total baryon asymmetry even if charge conjugation alone is violated. Different same-helicity rates can cancel against the opposite-helicity rates.

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

A noninteracting perfect fluid in general relativity has ${\nabla }_{\mu }{T}_i^{\mu \nu }=0$, so ${\dot{\rho }}_i+3H({\rho }_i+P_i)=0$. If energy is transferred between species, source terms $Q_i$ appear and only their sum vanishes. The Friedmann equation still involves the sum of all component stress tensors because all components share one spacetime metric.

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

A coasting fluid has pressure $p=-\rho /3$. The cosmological perfect-fluid continuity equation makes its density scale as $a^{-2}$. If it alone fills a flat expanding universe, the scale factor grows linearly in physical time. Its background density has the same scale-factor dependence as a spatial-curvature term, although the fluid does not itself create spatial curvature.

An isotropic, noninteracting network of domain walls with energy density proportional to $a^{-1}$ has effective equation-of-state parameter $w=-2/3$. This follows from constant-equation-of-state density scaling $\rho \propto a^{-3(1+w)}$.

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 $t_{\ast }=t_0-2/[3(1+w)H_0]$.

Pressureless matter has equation-of-state parameter $w=0$, energy density $\rho \propto a^{-3}$ and negligible stress compared with its rest-mass density.

An isotropic network of long cosmic strings has average equation-of-state parameter $w=-1/3$ and energy density proportional to $a^{-2}$.

An isotropic cosmic string network with $\rho =C/a^2$ has equation-of-state parameter $w=-1/3$. The Friedmann equation gives ${\dot{a}}^2=8\pi GC/3-K$. On the expanding branch with positive right side, $a=B(t-t_B)$, the comoving Hubble radius is constant, and the cosmological density parameter is constant. There is no growing Flatness problem, but flatness is not an attractor. If this exact coasting solution extends all the way to $t_B$, then ${\int }_{t_B}^{t}d{t}^{\prime }/a(t^{\prime })$ diverges and there is no finite particle horizon. A finite starting time changes that global conclusion.

A closed universe containing only a cosmic string network can remain static. Its Friedmann acceleration equation vanishes identically because $P_s=-{\rho }_s/3$, and its Friedmann equation requires ${\rho }_s{a}^2=3K/(8\pi G)$. Once this conserved normalization is tuned, every positive radius gives a static solution. The usual Hubble-normalized cosmological density parameter is undefined at a static epoch.

At fixed string normalization $C={\rho }_s{a}^2$, scale-factor shifts move between static solutions and are neutral, with no restoring force. The acceleration equation gives $\ddot{a}=0$, but the Friedmann constraint also imposes ${\dot{a}}^2=8\pi GC/3-K$. At the static normalization it excludes a nonzero velocity. A slightly larger $C$ admits linearly expanding or contracting branches; a smaller $C$ admits no real closed solution. This distinguishes permitted perturbations from constraint violations.

Consider a spatially flat universe containing radiation and a cosmic string network. If their densities are both ${\rho }_{\mathrm{eq}}$ at scale factor $a_{\mathrm{eq}}$, thenIn conformal time $d\tau =dt/a$, the flat Friedmann equation becomesChoosing the Big Bang at $\tau =0$, its expanding solution is

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.

The cosmic baryon fraction is the ratio of mean baryonic matter density to total matter density. Assigning this fraction to a halo is a simplifying assumption; cooling, ejection and other baryonic processes can change the halo's retained gas and stellar fractions. A baryon conversion efficiency becomes a stellar-to-total-halo mass ratio only after multiplying by the applicable baryon fraction.

For an expanding Friedmann-Lemaitre-Robertson-Walker metric with positive energy density, put $\Omega =8\pi G\rho /(3H^2)$ and $w=P/\rho$, in units $c=1$. The Friedmann equation, Friedmann acceleration equation and cosmological perfect-fluid continuity equation giveFor constant $w$, linearizing at $\Omega =1$ gives $\Omega -1\propto a^{1+3w}$. For variable $w$, replace the power by $\exp \int (1+3w(a))d\log a$. This describes the Flatness problem for decelerating expansion, away from epochs at which $H=0$.

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.

The particle horizon at cosmic time $t$ is the greatest comoving distance from which a light signal could have reached an observer since the initial time:Regions separated by more than twice this distance cannot have exchanged signals by time $t$.

For a radiation-filled expanding universe with $0<{\Omega }_0<1$ and no cosmological constant, set $x=a/a_0$. The particle horizon radial proper distance is $H_0^{-1}{\int }_0^1[{\Omega }_0+(1-{\Omega }_0)x^2{]}^{-1/2}dx$. An inverse-hyperbolic-sine antiderivative gives the formula. The spatially flat limit is $d_{h0}=H_0^{-1}$; radial proper distance is distinct from the area radius of the corresponding horizon sphere.

A horizon-length patch on the matter-radiation equality surface has physical size $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$ in a flat universe with conformal time measured from the Big Bang. Dividing by its angular diameter distance $a_{\mathrm{eq}}({\tau }_0-{\tau }_{\mathrm{eq}})$ gives its small angular size. For the radiation-matter scale factor in conformal time, this is $(\sqrt{2}-1)\sqrt{{\Omega }_{r,0}}/[1-\sqrt{2{\Omega }_{r,0}}]$. A patch diameter is twice the angle associated with one horizon radius. The particle horizon is an integral over past evolution and must not be silently replaced by the instantaneous Hubble radius.

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

For relativistic species in thermal equilibrium at a common temperature $T$, the total energy density can be writtenThe factor $7/8$ is the fermion-to-boson thermal integral ratio for energy density.

For relativistic species at a common temperature, the mass-equivalent energy density is $\rho ={\pi }^2{g}_{\ast }(k_{B}T{)}^4/(30{\hslash }^3{c}^5)$. Using mass rather than energy density accounts for the powers of $c$ in the Friedmann equation.

Dark energy is a smooth component with sufficiently negative pressure to drive accelerated cosmic expansion.

Dynamical dark energy has an energy density or equation-of-state parameter that changes with cosmic time, unlike a cosmological constant.

For $P_{\mathrm{DE}}=w(z){\rho }_{\mathrm{DE}}$, the continuity equation gives

The Chevallier-Polarski-Linder parametrization models dynamical dark energy by $w(a)=w_0+w_a(1-a)$, equivalently $w(z)=w_0+w_{a}z/(1+z)$.

The cosmological constant $\Lambda$ is the constant term proportional to the metric in the Einstein field equations. Interpreted as vacuum energy, it has $P_{\Lambda }=-{\rho }_{\Lambda }$, so its energy density is constant during expansion.

The matter density ${\rho }_m$ is the energy density in nonrelativistic matter. In a homogeneous expanding universe, conservation of particle number and dilution of physical volume give ${\rho }_m\propto a^{-3}$.

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.

Approximately constant positive Hubble parameter gives a nearly de Sitter spacetime expansion in flat slicing. Its scale factor can be approximated by $a=-1/(H\tau )$ with negative conformal time increasing toward zero. Slow variation of the expansion rate limits the accuracy of this approximation.

Reheating transfers the inflaton energy into other particles after cosmic inflation, eventually establishing a hot thermal bath. It requires interactions in addition to a background scalar potential. The duration and averaged equation-of-state parameter of reheating affect the number of e-folds between a chosen scale's horizon exit and the end of inflation.

The reheating temperature characterizes the thermal radiation bath after energy from a nonthermal component has been transferred and thermalized. For sudden decay of a dominant nonrelativistic relic, $T_R\simeq [3M_{\mathrm{Pl}}^2{\Gamma }^2/(c_2{g}_{\star }){]}^{1/4}$. It differs from the residual radiation temperature immediately before decay.