Cosmology

Neighboring freely falling comoving particles in a homogeneous Friedmann-Lemaître-Robertson-Walker metric have separation

r = a (t) χ

and relative acceleration

\ddot{r} = (\overset{a}{¨} / a) r

, where

t

is cosmic time and

χ

is constant. This is a local tidal force effect, of order

H^{2} r

when the dimensionless background parameters are of order one. In a weak-field central-mass model retaining the homogeneous background tide, one writes

\overset{r}{¨} = - GM / r^{2} + ℓ^{2} / r^{3} + (\overset{a}{¨} / a) r

. The detailed residual tide depends on the cosmological embedding; a bound orbit is not required to satisfy

\overset{r}{˙} = Hr

. Comparing

H^{2}

with

GM / r^{3}

estimates whether that background tide can affect its dynamics. A locally vacuum region with a cosmological constant has the corresponding outward term

Λ c^{2} r /3

Physical cosmology

Physical cosmology connects the universe's history to the measurable behaviour of its contents. The Friedmann equations describe cosmic expansion; atomic physics, radiation pressure and gravity determine what that expanding universe looks like. A particularly revealing example is the route from cosmological recombination through the photon-baryon fluid to the Cosmic microwave background acoustic peaks. The universe supplies the initial conditions and the laboratory; the prediction must still pass through a model of the telescope before it can be compared with observations.

The practical reward is that one sky contains several different tests. The Cosmic microwave background power spectrum carries information about the sound horizon, the cosmological baryon density and cosmic geometry. These are related quantities with different observational signatures. Angular sound-horizon degeneracy explains why measuring a characteristic angle alone cannot count the universe's baryons.

Peebles primordial-plasma predictions

P. J. E. Peebles made the connection between microscopic processes and cosmological observables quantitative. His 1968 paper treated cosmological recombination as a kinetic problem. Capturing an electron directly into the hydrogen ground state releases a photon that can ionize another atom. Net neutralization depends on escape of resonance radiation through cosmological redshift and on two-photon spontaneous emission from the

2 s

state. An equilibrium Saha equation alone does not give that history.^[ref]

This matters for an acoustic prediction because the hydrogen ionization fraction fixes the supply of free electrons for Thomson scattering. In physical units the scattering rate per photon is

Γ_{T} = n_{e} σ_{T} c

. The cosmological visibility function describes the distribution of last-scattering times. Thus atomic physics helps determine when the photon-baryon acoustic oscillator is read out, as well as how rapidly tight coupling fails.

In 1970 P. J. E. Peebles and J. T. Yu followed adiabatic cosmological perturbations by numerically integrating the photon collision equation through cosmological recombination. Their calculation connected acoustic evolution and dissipation to residual matter structure and microwave brightness fluctuations. It did not assume that a perfect photon-baryon fluid remained valid throughout last scattering. Their illustrative cosmological models and normalization were historical choices, rather than the present Lambda-CDM model; their predicted brightness variance versus angular resolution should not be relabelled as a modern measured

C_{ℓ}

spectrum.^[ref]

The continuing idea is a forward prediction: specify initial conditions, evolve the coupled matter and radiation, project onto the sky, and compare the result with an instrument's readout. Coherent initial phases are needed for acoustic variance peaks explains why a collection of oscillating modes does not automatically produce a peaked power spectrum. The BOOMERANG acoustic-peak measurement provides a later observational test of this physical programme.

Constant-loading acoustic model ( $y^{''} + k^{2} c_{s}^{2} y = - k^{2} c_{s}^{2} R$ )

A deliberately restricted forced harmonic oscillator shows two effects of the baryon loading parameter without duplicating the full photon Boltzmann hierarchy. Use conformal time

d η = d t / a

in seconds, a comoving wavenumber

k

in inverse metres, constant

R = 3 ρ_{b} / (4 ρ_{γ})

, and equal constant dimensionless Newtonian gauge gravitational potentials

Φ = Ψ \neq = 0

. The energy densities

ρ_{b}

and

ρ_{γ}

use the same units. Neglect expansion drag, Silk damping and evolution of the potentials. The photon-baryon sound speed is

c_{s} = c / 3 (1 + R)

. Writing

Θ_{0}

for the photon monopole temperature perturbation, the photon-baryon acoustic oscillator reduces to

Θ_{0}^{''} + k^{2} c_{s}^{2} Θ_{0} = - \frac{k ^{2} c ^{2}}{3} Ψ.

For

y = (Θ_{0} + Ψ) /Ψ

this gives the displayed forced equation. Constant loading and the neglected expansion drag make this a local teaching model, not a self-consistent expanding cosmology with

R

fixed forever.

Choose

y (0) = 1/3

and

y^{'} (0) = 0

, the static matter-era Sachs-Wolfe combination normalized by

Ψ

. With

q = k c_{s} η

, direct substitution gives

y (q) = (\frac{1}{3} + R) cos q - R, y_{q} (0) = 0.

(2)

R = 0

, successive extrema have equal magnitudes

1/3

. At the stipulated

R = 0.6

y (π) = - 23/15

whereas

y (2 π) = 1/3

. The shifted equilibrium

y = - R

makes alternating extrema unequal. This is an elementary illustration of how the baryon loading parameter changes acoustic amplitudes; these extrema are neither measured peak heights nor a complete Cosmic microwave background power spectrum. Comparing curves at fixed

q

isolates the response; at fixed

k

and

η

, increasing

R

also reduces the photon-baryon sound speed and hence the phase.

For example, holding

c η_{*} = 300 Mpc

gives sound horizons

r_{s} = 173.2 Mpc

R = 0

and

136.9 Mpc

R = 0.6

. Their ratio is

1/ 1.6

. Each term of the forced equation has units of inverse seconds squared because

y

and

R

are dimensionless and

k c_{s}

is an inverse time. In the actual photon-baryon fluid,

R

grows with the scale factor and the sound horizon is the integral of the evolving photon-baryon sound speed.

Figure 1.
Acoustic loading shifts an oscillator; an angular measurement fixes a ratio
. Original analytic examples. The left panel holds the baryon loading parameter and gravitational potential constant and uses the stated initial conditions. The right panel independently stipulates two comoving rulers and distances with the same angular size. Neither panel is an observed cosmic microwave background spectrum. **Details**.

Angular sound-horizon degeneracy ( $θ_{} = r_{s} (z_{}) / D_{M} (z_{}), ℓ_{A} = π / θ_{}$ )

The angular sound horizon compares two lengths using the same distance convention. If

D_{A}

is the angular diameter distance to last scattering at redshift

z_{*}

, define the transverse comoving distance

D_{M} = (1 + z_{*}) D_{A}

. A comoving sound horizon

r_{s}

has proper length

r_{s} / (1 + z_{*})

at emission, so its angular size is

θ_{*} = r_{s} / D_{M}

. Mixing a comoving ruler with

D_{A}

would introduce an erroneous factor

1 + z_{*}

. In a curved FLRW metric,

D_{M}

is generally different from the radial comoving distance.

For a stipulated example,

r_{s} = 144 Mpc

and

D_{M} = 14000 Mpc

give

θ_{*} = 0.010286

radians, or

0.58 9^{\circ}

, and an acoustic scale

ℓ_{A} = 305.4

. Multiplying both lengths by

1.1

gives

158.4 Mpc

and

15400 Mpc

with exactly the same

θ_{*}

and

ℓ_{A}

. This example establishes a ratio degeneracy, without claiming that any arbitrary rescaling corresponds to an allowed cosmological model. Equivalently,

δ ln θ_{*} = δ ln r_{s} - δ ln D_{M} .

Other observables are needed to separate changes that preserve the ratio.

The approximate adiabatic acoustic-peak spacing is

ℓ_{A}

. The first observed temperature peak is shifted by gravitational driving and angular projection of density and Doppler acoustic sources; it need not occur at

ℓ_{A}

. In particular, neither the illustrative

305.4

nor the BOOMERANG acoustic-peak measurement of

197 \pm 6

should be substituted for the other.

The cosmological baryon density enters through the baryon loading parameter and cosmological recombination, changing relative acoustic amplitudes as well as the sound horizon. The baryon-to-photon ratio counts net baryon number per photon, whereas

θ_{*}

is an angle. Constraining cosmological baryon density requires the pattern of relative peak heights within a model, with other parameters fitted jointly. Big Bang nucleosynthesis provides a separate abundance test. Baryon-density consistency between nucleosynthesis and the microwave background is consequently a test of one common cosmic history, rather than two direct measurements of the same angle. A first peak by itself cannot determine every component of the cosmological density parameter or uniquely fix spatial curvature of an FLRW universe.

BOOMERANG acoustic-peak measurement

BOOMERANG's 1998 Antarctic balloon flight scanned a low-dust region with a microwave telescope near

38 km

and bolometers at

0.3 K

. Ground beam measurements and microwave angular dipole calibration connected signals to sky temperature. Frequency-map and time-split differences tested contamination. The 2000 paper fitted the Cosmic microwave background power spectrum, finding

ℓ_{peak} = 197 \pm 6

at one standard deviation. Its near-flat-geometry inference used a chosen model family and parameter priors; secondary peaks were not yet established.^[ref]

Figure 1.
Primordial-plasma predictions meet a calibrated sky measurement
. Original functional diagram. The numerical result is the published peak location; no experimental bandpowers are simulated. **Details**.

Cosmological spectral background ( $I_{ν_{0}}$ )

The cosmological spectral background is received radiant power per detector area, received frequency interval and solid angle, summed over a cosmological source population. Its intensity depends on emitted spectra, their cosmic evolution, the expansion history and absorption. A finite radial comoving distance alone does not guarantee a finite background when high-redshift sources are sampled at increasingly high emitted frequencies.

Cosmological bolometric background from conserved sources ( $F_{0} = c L n_{0} \int a (t) d t$ )

Normalize the scale factor to

a_{0} = 1

. Identical isotropic sources of bolometric luminosity

L

and conserved comoving number density

n_{0} = a^{3} n

produce all-sky radiative flux

F_{0} = c L n_{0} \int a (t) d t

in a transparent FLRW metric. The source counts on an FLRW past light cone and cosmological flux dimming cancel the transverse shell-area factor. For arbitrary proper source density the expression is instead

c L \int n (t) a (t)^{4} d t

. The intensity per solid angle is

F_{0} / (4 π)

. A detector of collecting area

A

receives power

A F_{0}

under an all-sky effective-area convention.

Open dust bolometric background integral ( $F_{0} = \frac{c L n _{0} Ω _{0}^{2}}{4 H _{0} ( 1 - Ω _{0} ) ^{5/2}} J (u_{0})$ )

For an open matter-dominated Friedmann solution, put

u = H_{0} 1 - Ω_{0} τ

a = Ω_{0} (cosh u - 1) / [2 (1 - Ω_{0})]

and

cosh u_{0} = 2/ Ω_{0} - 1

. The cosmological bolometric background from conserved sources has

J (u) = sinh (2 u) /4 - 2 sinh u + 3 u /2

, since

J^{'} = (cosh u - 1)^{2}

and

d t = a d u / [H_{0} 1 - Ω_{0}]

. At fixed

H_{0}

, the limit

Ω_{0} \to 0

c L n_{0} / (2 H_{0})

, whereas the Einstein-de Sitter universe gives

2 c L n_{0} / (5 H_{0})

. Keeping the cosmic age fixed instead reverses this ordering: the two values are respectively

c L n_{0} t_{0} /2

and

3 c L n_{0} t_{0} /5

Cosmological background intensity from comoving emissivity ( $I_{ν_{0}} = \frac{c}{4 π} \int \frac{ϵ _{(1 + z) ν_{0}} ( z )}{( 1 + z ) H ( z )} d z$ )

Let

ϵ_{ν_{e}} (z)

be the comoving spectral emissivity, namely total emitted spectral luminosity per comoving volume. In a homogeneous transparent FRW metric, a radial shell contributes its comoving volume times the cosmological spectral flux-density relation. Its transverse-area factors cancel, leaving the displayed integral. In a flat model this cancellation is explicit:

d V_{c} = r^{2} (c / H) d z d Ω

, whereas

F_{ν_{0}} = (1 + z) L_{ν_{e}} / [4 π (1 + z)^{2} r^{2}]

. Absorption inserts a transmission factor, and evolving sources are represented by the redshift dependence of

ϵ

Einstein-de Sitter spectral-background convergence criterion ( $α < 3/2$ )

For nonevolving comoving number density

n_{0}

, identical spectra

L_{ν} \propto ν^{α}

, and no attenuation, the cosmological background intensity from comoving emissivity in an Einstein-de Sitter universe is proportional to

\int_{0}^{\infty} (1 + z)^{α - 5/2} d z

. The improper integral is finite exactly for

α < 3/2

, with value

1/ (3/2 - α)

. Equality gives logarithmic divergence; larger

α

gives a power-law divergence. A formation epoch or a spectral break can regularize this idealized model.

Cosmological monopole problem

The cosmological monopole problem is the excessive relic density of stable magnetic monopoles produced in some early-universe symmetry-breaking scenarios. Cosmic inflation dilutes their number density as

a^{- 3}

. Successful later reheating must also avoid recreating an excessive abundance.

Horizon-density monopole abundance bound ( $ρ_{M 0} \propto η^{4}$ )

For a radiation-era transition at

T_{i} \sim η

, use

H_{i} = 1.66 g_{*} η^{2} / m_{pl}

and initial number density

n_{i} = C_{n} H_{i}^{3}

. Stable monopoles of mass

C_{M} η

redshift as nonrelativistic matter. With comoving entropy conservation,

ρ_{M 0} \sim C_{n} C_{M} (1.66)^{3} g_{*}^{3/2} \frac{g _{* s, 0}}{g _{* s, i}} \frac{η ^{4} T _{0}^{3}}{m _{pl}^{3}} .

The fourth-power scale dependence makes even a small causal initial abundance incompatible with conventional high-scale grand unification unless inflation, later entropy production or efficient destruction reduces the relic population.

// Target: cosmology.bigb

Structure formation

Structure formation is the growth of cosmological matter inhomogeneities into structures such as galaxies through gravitational instability and gas dynamics. Linear cosmological density perturbations describe the early, small-amplitude regime; nonlinear evolution is required once the density contrast becomes large.

Cosmological principle

The cosmological principle assumes that the universe is statistically homogeneous and isotropic on sufficiently large spatial scales. It applies to a coarse-grained description, not to each galaxy or local void. Combined with the Copernican principle, cosmological isotropy around us supports a homogeneous description rather than a specially centred model. It motivates an FRW metric, whose expansion is governed by a few time-dependent quantities. Observations sample only a finite past light cone, so they support this hypothesis without proving global spatial homogeneity beyond the observed region.

Cosmological isotropy

Cosmological isotropy means that coarse-grained spatial observations or their statistics are unchanged by rotations about an observer. In an FRW metric it holds about every comoving location. cosmological isotropy about a single location is weaker than spatial homogeneity: a radially inhomogeneous model centred on that location can also have an isotropic sky.

Copernican principle

Our observer location is not specially privileged. In cosmology this assumption allows cosmological isotropy observed around us to be interpreted statistically as cosmological isotropy around other typical observers. It is an assumption about typicality, not a deduction from one observer's sky or from the Einstein field equations alone.

Cosmological particle freeze-out

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.

Coannihilation

Number-changing reactions involving a relic candidate and another nearly degenerate species. If reactions interchange the species rapidly enough, the effective annihilation rate depends on their joint abundances and channels. A single-species relic estimate can then fail even for fixed candidate mass.

Exponential freeze-out sensitivity ( $d ln n_{0} / d P = 3/ (2 P) - 1$ )

In a supplied equilibrium-cutoff model with

P = m c^{2} / (k_{B} T_{f})

, present number density can contain

P^{3/2} e^{- P}

. The logarithmic derivative is

3/ (2 P) - 1

, making small changes in a large cosmological particle freeze-out parameter important for the inferred abundance.

Entropy degrees of freedom ( $g_{* s}$ )

The effective count appearing in thermal entropy density

s \propto g_{* s} T^{3}

. Conservation of comoving entropy implies

g_{* s} T^{3} a^{3}

is constant, so stable relic dilution includes

g_{* s, 0} / g_{* s, f}

when expressed using temperatures.

Sharp-freeze-out relic dilution ( $n_{0} = n_{f} (a_{f} / a_{0})^{3}$ )

In a stable-particle approximation with negligible later annihilation, comoving particle number is conserved and physical number density falls as

a^{- 3}

. Replacing scale-factor ratios by photon-temperature ratios additionally assumes no change in the entropy degrees of freedom.

Mass cancellation in sharp-freeze-out number dilution ( $n_{0} = 6 P^{3/2} [k_{B} T_{0} / (ℏ c)]^{3} e^{- P}$ )

For an imposed equilibrium cutoff

m c^{2} = P k_{B} T_{f}

and density

n_{f} = 6 [m c^{2} k_{B} T_{f} / (ℏ c)^{2}]^{3/2} e^{- P}

, dilution by

(T_{0} / T_{f})^{3}

cancels the explicit mass in the final number density. The rest-energy density still scales with particle mass. This result assumes constant entropy degrees of freedom and no post-cutoff annihilation; entropy conservation adds a factor

g_{* s, 0} / g_{* s, f}

when these assumptions are relaxed.

Thermal relic ( $Y = n / s$ )

A thermal relic is a surviving particle population whose earlier abundance was set by chemical thermal equilibrium with the cosmic plasma. After number-changing reactions cease and cosmic entropy is conserved, its comoving abundance is described by the yield

Y = n / s

. A relativistically decoupled species can retain an appreciable yield, whereas a species remaining in thermal equilibrium after becoming nonrelativistic is subject to Boltzmann suppression. Its later mass density, lifetime and momentum distribution determine its cosmological effects.

Thermalization temperature for a massless gauge mediator ( $Γ/ H = α_{Y}^{2} m_{pl} / (1.66 g_{*} T)$ )

A rate proportional to

T

competes with radiation-era expansion proportional to

T^{2}

. The interaction-to-expansion ratio therefore grows as the universe cools, crossing unity near

T_{eq} = α_{Y}^{2} m_{pl} / (1.66 g_{*})

. If a phase transition occurs at

T_{c} ≫ T_{eq}

, this high-temperature interaction never establishes thermal equilibrium before the transition. Subsequent faster number-changing processes may still establish a thermal distribution; the initial state is not automatically Fermi-Dirac just because the particles are relativistic.

Mass-dependent fermion freeze-out ( $m_{nr} = \frac{α _{Y}^{4} m _{pl}}{1.66 g _{*}}, T_{D} = m_{Y} (m_{Y} / m_{nr})^{1/3}$ )

In a radiation-dominated toy model with

Γ = α_{Y}^{4} T^{5} / m_{Y}^{4}

, setting the interaction rate equal to the Hubble parameter gives the displayed decoupling scale. The transition

T_{D} = m_{Y}

defines

m_{nr}

. For smaller masses,

x_{D} = m_{Y} / T_{D} = (m_{nr} / m_{Y})^{1/3} ≫ 1

. An instantaneous chemically equilibrated cosmological particle freeze-out estimate gives

n_{Y} / s = [45 g_{Y} / (2 π^{2} g_{* s}) (2 π)^{- 3/2}] x_{D}^{3/2} e^{- x_{D}}

. This estimate presupposes number-changing thermal equilibrium, radiation domination and the assumed rate law, rather than following from elastic thermalization alone.

Thermal relic density bound for a tunable fermion ( $\frac{ρ _{Y}}{ρ _{B}} = \frac{m _{Y}}{m _{B}} \frac{Y _{Y}}{Y _{B}}, Y_{i} = n_{i} / s$ )

For a stable nonrelativistic thermal relic, compare energy densities rather than number densities. If

m_{Y} = m_{nr} / x^{3}

and

Y_{Y} = C x^{3/2} e^{- x}

, its density relative to baryons is

(C m_{nr} / m_{B} Y_{B}) x^{- 3/2} e^{- x}

. Requiring this ratio below an allowed dark-matter-to-baryon ratio

R

means

x + \frac{3}{2} lo g x \geq lo g [C m_{nr} / (R m_{B} Y_{B})]

. With fixed rate parameters this excludes the weakly interacting large-mass branch, while the sufficiently Boltzmann-suppressed smaller-mass branch can satisfy the density bound. Decay or a nonthermal initial abundance requires a different constraint.

Post-freeze-out relic abundance ( $Y_{\infty}$ )

For

d Y / d x = - λ Y^{2} / x^{2}

after freeze-out, the exact limit is

Y_{\infty} = [Y_{f}^{- 1} + λ / x_{f}]^{- 1}

. The frequently used estimate

x_{f} / λ

additionally neglects the initial inverse abundance.

Spatial homogeneity

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.

Cosmological shear

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.

Cosmological perturbation theory

Cosmological perturbation theory studies departures of the metric tensor and matter fields from a homogeneous and isotropic FRW metric, order by order. Linear cosmological perturbation theory keeps only first-order terms. Higher orders describe nonlinear coupling between perturbations. Nonlinear cosmological perturbation theory also uses expansions that allow finite amplitudes, such as the long-wavelength approximation in cosmology.

Nonlinear cosmological perturbation theory

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.

Long-wavelength approximation in cosmology ( $k / (a H) ≪ 1$ )

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.

Shear-inclusive long-wavelength scalar equations ( $3 H^{2} = 8 π G (Π^{2} /2 + V) + S^{i}_{j} S^{j}_{i} /2$ )

For zero shift vector, negative-expansion extrinsic curvature

K = - 3 H

and

S^{i}_{j} = K^{i}_{j}

, retain the leading spatial-gradient equations of a minimally coupled scalar. They give

\dot{ϕ} = N Π

\dot{Π} = - N (3 H Π + V_{, ϕ})

\dot{H} = - 4 π GN Π^{2} - N S^{i}_{j} S^{j}_{i} /2

, and

\dot{S}^{i}_{j} = - 3 N H S^{i}_{j}

. The first-gradient momentum constraint is

\partial_{i} H = - 4 π G Π \partial_{i} ϕ - \nabla_{j} S^{j}_{i} /2

. Shear decays as

a^{- 3}

during expansion. Its neglect after inflationary damping yields the equations underlying conservation of the nonlinear curvature covector.

Nonlinear curvature covector ( $ζ_{i} = - \partial_{i} lo g a + (H /Π) \partial_{i} ϕ$ )

With negligible shear, a minimally coupled scalar field serving as a clock, and

Π \neq = 0

, the Hamiltonian constraint and momentum constraint imply

\dot{H} = - 4 π GN Π^{2}

and

\partial_{i} H = - 4 π G Π \partial_{i} ϕ

. Differentiating the local Friedmann relation yields

\dot{Π} = - N (3 H Π + V_{, ϕ})

and

\partial_{i} Π = - (3 H + V_{, ϕ} /Π) \partial_{i} ϕ

. Inserting these identities into

\dot{ζ}_{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.

Nonlinear scalar metric in a cosmological gradient expansion ( $h_{ij} = a_{bg}^{2} e^{2 ζ} δ_{ij}$ )

With vectors and independent tensors neglected, finite scalar cosmological perturbations of the local spatial scale can be encoded by

h_{ij} = a_{bg}^{2} e^{2 ζ} δ_{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.

Inflationary shear damping ( $K^{i}_{j} \propto a_{loc}^{- 3}$ )

The traceless mixed extrinsic curvature of a spatial hypersurface can have leading solution

K^{i}_{j} = C^{i}_{j} (x) a_{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

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.

Baryon-asymmetry washout

Inverse decays or number-violating scatterings can erase an asymmetry after its generation. A viable mechanism must generate a surviving charge combination before the relevant washout processes cease to be effective.

Leptogenesis

A mechanism in which an initial lepton asymmetry is generated and then partly converted into a baryon asymmetry, commonly through electroweak anomalous processes. Heavy Majorana neutrino decays offer one possible realization; a seesaw mechanism by itself does not guarantee the required asymmetry.

Baryon-number washout

Baryon-number washout is erasure of a baryon asymmetry by reactions that violate baryon number. Inverse decays and scattering can drive the plasma toward its zero-asymmetry thermal equilibrium. The relevant comparison is their rate against the Hubble parameter, not just the decay width that initially produced the asymmetry.

Heavy-boson decay baryogenesis ( $ϵ = (r - \overset{r}{ˉ}) /2$ )

Consider equal initial abundances of a heavy boson

X

and its antiparticle. Two decay channels with different baryon numbers, for example

X \to uu

and

X \to \overset{ˉ}{d} e^{+}

, have branching fractions

r

and

1 - r

. The conjugate channels have branching fractions

\overset{r}{ˉ}

and

1 - \overset{r}{ˉ}

. Their mean baryon numbers are

r - 1/3

and

- \overset{r}{ˉ} + 1/3

, so each particle-antiparticle pair produces

r - \overset{r}{ˉ}

, or

ϵ

per decaying particle. Unequal branching fractions require CP violation; equal total decay widths are consistent with CPT symmetry. Baryon number cannot be assigned so that both channels conserve it. An initial abundance decaying outside thermal equilibrium, after inverse production becomes ineffective, can therefore generate a nonzero baryon asymmetry. At the microscopic level a decay-rate difference caused by CP violation requires interference of decay amplitudes with both a relative weak phase and an absorptive phase; complex couplings alone do not guarantee it. Subsequent baryon-number washout must be ineffective for the produced asymmetry to survive.

Baryon asymmetry ( $η_{B} = (n_{B} - n_{\overset{ˉ}{B}}) / n_{γ}$ )

The baryon asymmetry is an excess of baryons over antibaryons. Here

n_{B} - n_{\overset{ˉ}{B}}

counts net baryon number per unit volume, including the baryons bound in atomic nuclei. It is dynamically generated by baryogenesis. Once the antibaryon abundance is negligible, its photon-normalized value agrees with the usual baryon-to-photon ratio.

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.

CP rate cancellation in baryogenesis

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.

Cosmological perfect-fluid continuity equation

For a homogeneous fluid of energy density

ρ

, pressure

P

, and scale factor

a (t)

, adiabatic work in a comoving volume gives

\overset{ρ}{˙} + 3 \frac{a ˙}{a} (ρ + P) = 0.

Separately conserved cosmological fluids

A noninteracting perfect fluid in general relativity has

\nabla_{μ} T_{i}^{μν} = 0

, so

\overset{ρ}{˙}_{i} + 3 H (ρ_{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.

Barotropic equation of state

A constant barotropic equation of state has

P = wρ

. The cases

w = 0

and

w = 1/3

describe pressureless matter and radiation respectively.

Coasting fluid ( $w = - \frac{1}{3}, ρ \propto a^{- 2}$ )

A coasting fluid has pressure

p = - ρ /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.

Scalar-field reconstruction of a coasting component ( $V (ϕ) = 2 M^{2} H_{0}^{2} Ω_{X} / a (ϕ)^{2}$ )

For conserved

w = - 1/3

scalar energy,

ρ_{X} = 3 M^{2} H_{0}^{2} Ω_{X} / a^{2}

and

ϕ^{'} = 2 M H_{0} Ω_{X}

on the increasing branch. With dust and radiation, define

b = Ω_{m} / (2 Ω_{X})

and

A = Ω_{m}^{2} /4 - Ω_{r} Ω_{X} / Ω_{X}

. When the discriminant is positive,

a (ϕ) = A cosh [\frac{ϕ - ϕ _{0}}{2 M} + C] - b, C = arcosh \frac{1 + b}{A},

and

V (ϕ) = 2 M^{2} H_{0}^{2} Ω_{X} / a (ϕ)^{2}

. The formulas retain all density parameters without eliminating one by the flatness sum.

Cosmological domain wall network

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

ρ \propto a^{- 3 (1 + w)}

Constant-equation-of-state density scaling

For constant

w \neq = - 1

, the cosmological continuity equation integrates to

ρ (a) = ρ_{0} a^{- 3 (1 + w)}

when

a = 1

at the reference time and

ρ_{0}

is the reference density.

Flat constant-equation-of-state scale factor

For an expanding spatially flat universe with constant

w > - 1

and

a (t_{0}) = 1

a (t) = [1 + \frac{3}{2} (1 + w) H_{0} (t - t_{0})]^{2/ [3 (1 + w)]} .

It vanishes at the finite past time

t_{*} = t_{0} - 2/ [3 (1 + w) H_{0}]

Pressureless matter

Pressureless matter has equation-of-state parameter

w = 0

, energy density

ρ \propto a^{- 3}

and negligible stress compared with its rest-mass density.

Cosmic string network

An isotropic network of long cosmic strings has average equation-of-state parameter

w = - 1/3

and energy density proportional to

a^{- 2}

The equation of state averages the stress of a cosmic string over an isotropic network; it is not the definition of the individual defect.

Coasting cosmic-string universe ( $a (t) = B (t - t_{B})$ )

An isotropic cosmic string network with

ρ = C / a^{2}

has equation-of-state parameter

w = - 1/3

. The Friedmann equation gives

\overset{a}{˙}^{2} = 8 π 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^{'} / a (t^{'})

diverges and there is no finite particle horizon. A finite starting time changes that global conclusion.

Static closed cosmic-string universe ( $ρ_{s} a^{2} = 3 K / (8 π G)$ )

A closed universe containing only a cosmic string network can remain static. Its Friedmann acceleration equation vanishes identically because

P_{s} = - ρ_{s} /3

, and its Friedmann equation requires

ρ_{s} a^{2} = 3 K / (8 π 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.

Homogeneous stability of a static cosmic-string universe

At fixed string normalization

C = ρ_{s} a^{2}

, scale-factor shifts move between static solutions and are neutral, with no restoring force. The acceleration equation gives

\overset{a}{¨} = 0

, but the Friedmann constraint also imposes

\overset{a}{˙}^{2} = 8 π 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.

Radiation--cosmic-string Friedmann solution

Consider a spatially flat universe containing radiation and a cosmic string network. If their densities are both

ρ_{eq}

at scale factor

a_{eq}

, then

ρ_{r} = ρ_{eq} (\frac{a _{eq}}{a})^{4}, ρ_{s} = ρ_{eq} (\frac{a _{eq}}{a})^{2} .

In conformal time

d τ = d t / a

, the flat Friedmann equation becomes

(\frac{d a}{d τ})^{2} = \frac{8 π G a _{eq}^{2}}{3 c ^{2}} ρ_{eq} (a^{2} + a_{eq}^{2}) .

(2)

Choosing the Big Bang at

τ = 0

, its expanding solution is

a (τ) = a_{eq} sinh (a_{eq} \frac{8 π G ρ _{eq}}{3 c ^{2}} τ) .

(3)

Critical density

At Hubble parameter

H

, the critical energy density is

ρ_{crit} = \frac{3 c ^{2} H ^{2}}{8 π G} .

This is the density threshold in the Friedmann equation for spatial flatness; thermodynamic and plasma critical densities are different concepts.

Mass-equivalent cosmological critical density ( $ρ_{crit} = 3 H^{2} / (8 π G)$ )

The mass-density convention for the cosmological critical density. If

u

denotes energy per unit volume,

ρ = u / c^{2}

and

u_{crit} = 3 c^{2} H^{2} / (8 π G)

. Both conventions give the same dimensionless density parameter when numerator and denominator use consistent units. This distinction avoids inserting an extra

c^{2}

into a Friedmann equation written in terms of mass-equivalent density.

Cosmological density parameter ( $Ω$ )

The density parameter is

Ω = ρ / ρ_{crit}

. In a Friedmann universe, its departure from one measures the spatial-curvature term relative to the density or expansion term.

The Friedmann equation relates this ratio to spatial curvature and the expansion rate; the parameter and the evolution equations are distinct concepts.

Density-parameter divergence at a cosmological turnaround ( $Ω = ρ / ρ_{c}, ρ_{c} = 3 H^{2} / (8 π G)$ )

At maximum expansion of a closed matter-dominated Friedmann solution, the Hubble parameter vanishes while the physical mass density stays positive and finite. Its critical density therefore vanishes and the cosmological density parameter diverges. This is a singularity of that ratio rather than a curvature singularity. In the cycloid variable

θ

Ω_{M} = sec^{2} (θ /2)

, diverging at

θ = π

and approaching one at the Big Bang and Big Crunch ends.

Figure 1.
A closed dust universe: the Hubble parameter changes sign while its density parameter diverges at maximum expansion
. **Details**.

Matter density parameter ( $Ω_{m} = ρ_{m} / ρ_{crit}$ )

The matter density parameter is the ratio of total nonrelativistic matter density, including baryonic and dark matter, to the critical density

ρ_{crit} = 3 H^{2} / (8 π G)

at the same epoch. A present-epoch value uses

H_{0}

and

ρ_{m, 0}

. It is not the total cosmological density parameter when radiation or vacuum energy also contributes.

Scale factor in a conformal-time density parameter ( $Ω_{i} = 8 π G a^{2} \overset{ρ}{ˉ}_{i} / (3 H^{2})$ )

For physical density

\overset{ρ}{ˉ}_{i}

and conformal time,

H = a^{'} / a = a H

. Substitution into the definition

Ω_{i} = 8 π G \overset{ρ}{ˉ}_{i} / (3 H^{2})

gives the displayed formula. Omitting

a^{2}

gives

Ω_{i} / a^{2}

, which need not equal one even in a spatially flat matter-only universe. In particular, the conformal growth source

\frac{3}{2} H^{2} Ω_{m}

equals

4 π G a^{2} \overset{ρ}{ˉ}_{m}

with the physical density parameter.

Cosmic baryon fraction ( $f_{b} = Ω_{b} / Ω_{m}$ )

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.

Cluster baryon-fraction estimate of matter density ( $Ω_{m} = Ω_{b} / f_{b, cluster}$ )

If a measured cluster baryon fraction represents the cosmic baryon fraction

Ω_{b} / Ω_{m}

, combining it with the cosmological baryon density inferred from Big Bang nucleosynthesis gives the displayed estimate. Gas and stellar baryons must both be counted, and all distance or Hubble constant dependence must be consistent. Baryon depletion or feedback modifies the identification: with

f_{b, cluster} = b Ω_{b} / Ω_{m}

, the corrected answer is

Ω_{m} = b Ω_{b} / f_{b, cluster}

. The method measures total matter rather than vacuum or radiation density.

Cosmological density parameter flow ( $d Ω/ d lo g a$ )

For an expanding Friedmann-Lemaître-Robertson-Walker metric with positive energy density, put

Ω = 8 π Gρ / (3 H^{2})

and

w = P / ρ

, in units

c = 1

. The Friedmann equation, Friedmann acceleration equation and cosmological perfect-fluid continuity equation give

\frac{H ˙}{H ^{2}} = - 1 - \frac{1}{2} (1 + 3 w) Ω, \frac{d Ω}{d l o g a} = (1 + 3 w) Ω (Ω - 1) .

For constant

w

, linearizing at

Ω = 1

gives

Ω - 1 \propto a^{1 + 3 w}

. For variable

w

, replace the power by

exp \int (1 + 3 w (a)) d lo g a

. This describes the flatness problem for decelerating expansion, away from epochs at which

H = 0

Matter density flow with a cosmological constant ( $Ω_{m}^{'} = H Ω_{m} (Ω_{m} - 1 - 2 Ω_{Λ})$ )

For separately conserved pressureless matter,

ρ_{m} \propto a^{- 3}

. The deceleration parameter identity

\dot{H} / H^{2} = - (1 + q)

and

Ω_{m} \propto ρ_{m} / H^{2}

give

\dot{Ω}_{m} = H Ω_{m} (2 q - 1)

. A dust-plus-vacuum universe has

q = Ω_{m} /2 - Ω_{Λ}

; conversion to conformal time yields the displayed flow. The familiar matter-only equation requires negligible cosmological constant. In a spatially flat dust-plus-vacuum universe,

Ω_{m} + Ω_{Λ} = 1

and

Ω_{m}^{'} = - 3 H Ω_{m} Ω_{Λ}

, so flatness does not require

Ω_{m} = 1

Flatness problem

In a decelerating expanding universe,

∣Ω - 1∣

grows, so its small present value requires extremely fine-tuned early initial data unless an earlier mechanism drives

Ω

toward one.

Inflationary solution of the flatness problem

The continuity equation gives

\frac{d}{d t} (ρ a^{2}) = - H a^{2} (ρ + 3 P) .

During expansion with

ρ + 3 P < 0

, the product

ρ a^{2}

grows and

∣ Ω^{- 1} - 1∣ \propto (ρ a^{2})^{- 1}

decreases.

Flatness bound on the initial field in exponential inflation ( $ϕ_{i} \leq ϕ_{R} - \frac{A m _{pl}}{8 π} \frac{l o g ( a _{R} H _{R} / a _{0} H _{0} )}{1 - A ^{2} / ( 16 π )}$ )

For positive slope parameter

A

, the exponential-potential field increases during cosmic inflation. The quantity

a H

grows by

e^{(1 - ϵ_{H}) N}

with constant

ϵ_{H} = A^{2} / (16 π)

and

N = 8 π (ϕ_{R} - ϕ_{i}) / (A m_{pl})

. Starting from order-one curvature and demanding order-one or smaller present curvature gives the displayed upper bound on the initial field. A specified tighter final curvature tolerance adds one-half its logarithm to the numerator. Thus sufficient cosmic inflation sets an upper, not a lower, initial-field bound for this sign of the potential slope. Instantaneous reheating and subsequent entropy conservation determine

a_{R} H_{R} / a_{0} H_{0}

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.

Particle 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:

χ_{ph} (t) = \int_{t_{initial}}^{t} \frac{c d t ^{'}}{a ( t ^{'} )} .

Regions separated by more than twice this distance cannot have exchanged signals by time

t

Particle horizon in a power-law universe ( $R (t) = A t^{p}, D_{p} (t) = c t / (1 - p), 0 < p < 1$ )

The radial null integral from the Big Bang is

χ_{p} = c \int_{0}^{t} d t^{'} / R (t^{'}) = c t^{1 - p} / [A (1 - p)]

. Multiplication by the scale factor gives the proper radial horizon distance. It is

3 c t

for a flat matter-only model and

2 c t

for a flat radiation-only model. A finite initial time different from zero or an earlier expansion era changes the result.

Angular particle-horizon size in a power-law universe ( $θ_{p} = r^{1 - p} / (1 - r^{1 - p}), r = t_{e} / t_{0}$ )

Divide the particle horizon in a power-law universe at emission by the angular diameter distance in a flat power-law universe. The result is the small angular extent of one radial horizon length. Matter-only and radiation-only histories have powers

1/3

and

1/2

, respectively. These are idealized single-era histories; a real recombination horizon and distance depend on different portions of the expansion history.

Particle-horizon radius versus patch diameter

The proper radial particle horizon is the distance travelled by a null signal since the initial boundary, measured on the current time slice. A spherical patch with that radius has twice that physical diameter. Dividing one or the other by an angular diameter distance therefore differs by a factor of two. A question using horizon size should state which linear extent is intended.

Total comoving visibility radius ( $χ_{max} = \int_{t_{i}}^{t_{f}} c d t / a (t)$ )

In a spatially flat FLRW metric, the supremum of the comoving radial distance from which a signal emitted at the initial time can eventually reach a fixed comoving observer is the total conformal time interval. At an intermediate time

t

, this equals its comoving particle horizon plus the comoving radius of its cosmological event horizon, when both integrals converge. This is a visibility bound for emission arbitrarily close to the initial boundary; it does not assert that a transparent universe or a source existed there.

Total conformal lifetime of a flat matter-Lambda universe ( $χ_{max} = \frac{Ω _{m}^{- 1/3} Ω _{Λ}^{- 1/6}}{3 H _{0}} B (1/3, 1/6)$ )

Use

c = 1

a_{0} = 1

, positive pressureless matter and positive cosmological constant, with

Ω_{m} + Ω_{Λ} = 1

and no radiation. The Friedmann equation gives

χ_{max} = H_{0}^{- 1} \int_{0}^{\infty} d y / (Ω_{Λ} + Ω_{m} y^{3})^{1/2}

. Set

y = (Ω_{Λ} / Ω_{m})^{1/3} u

, then

v = u^{3}

. The remaining integral is

\frac{1}{3} \int_{0}^{\infty} v^{- 2/3} (1 + v)^{- 1/2} d v = \frac{1}{3} B (1/3, 1/6)

, proving the formula by the beta function integral. Its two endpoints converge for positive densities, but the lifetime diverges in either pure-fluid limit.

Comoving particle horizon ( $r_{p} (t) = c \int_{0}^{t} \frac{d t ^{'}}{a ( t ^{'} )}$ )

The comoving particle horizon is the particle horizon expressed in comoving coordinates, assuming a finite initial cosmic-time boundary at zero. In a spatially flat FLRW metric, it is the distance a photon could have travelled since that boundary. The corresponding proper distance in cosmology at time

t

a (t) r_{p} (t)

. It differs from the future-directed cosmological event horizon and from the instantaneous comoving Hubble radius.

Photon arrival from an Einstein-de Sitter particle horizon ( $t_{arrival} = 8 t_{0}$ )

For an Einstein-de Sitter universe normalized by

a (t_{0}) = 1

a = (t / t_{0})^{2/3}

. Its current comoving particle horizon is

3 c t_{0}

. A photon emitted inward at that location at

t_{0}

covers

3 c t_{0} [(t / t_{0})^{1/3} - 1]

in comoving distance by time

t

. Equating these distances gives arrival at

8 t_{0}

, a travel time of

7 t_{0}

Particle horizon through radiation-matter equality ( $r_{p} (a) = \frac{2 c}{H _{0} Ω _{m, 0}} (Ω_{r, 0} + Ω_{m, 0} a - Ω_{r, 0})$ )

Neglecting curvature and dark energy, integration of the Friedmann equation for radiation in cosmology and pressureless matter gives the displayed comoving particle horizon. At matter-radiation equality,

a_{eq} = Ω_{r, 0} / Ω_{m, 0}

, this is

2 (2 - 1)

times the radiation-only extrapolation

c a_{eq} / (H_{0} Ω_{r, 0})

. Its associated inverse length differs by an order-one factor from the matter-radiation equality scale

a_{eq} H_{eq} / c

Direct cosmological communication and a shared causal past ( $d \leq η versus d \leq 2 η$ )

In a spatially flat Friedmann-Lemaître-Robertson-Walker metric with initial conformal time zero, two comoving worldlines at separation

d

can exchange a one-way light signal by time

η

only if

d \leq η

. Their past light cones overlap if

d \leq 2 η

: a common event halfway between them can influence both. These are different statements. For opposite emission events observed at conformal time

η_{0}

, their separation is

2 (η_{0} - η_{e})

, so the respective thresholds are

η_{e} \geq 2 η_{0} /3

and

η_{e} \geq η_{0} /2

Open radiation universe particle horizon ( $d_{h 0} = \frac{arsinh ( 1 - Ω _{0} ) / Ω _{0}}{H _{0} 1 - Ω _{0}}$ )

For a radiation-filled expanding universe with

0 < Ω_{0} < 1

and no cosmological constant, set

x = a / a_{0}

. The particle horizon radial proper distance is

H_{0}^{- 1} \int_{0}^{1} [Ω_{0} + (1 - Ω_{0}) x^{2}]^{- 1/2} d x

. An inverse-hyperbolic-sine antiderivative gives the formula. The spatially flat limit is

d_{h 0} = H_{0}^{- 1}

; radial proper distance is distinct from the area radius of the corresponding horizon sphere.

Horizon angle at matter-radiation equality ( $θ_{eq} ≃ τ_{eq} / (τ_{0} - τ_{eq})$ )

A horizon-length patch on the matter-radiation equality surface has physical size

a_{eq} τ_{eq}

in a flat universe with conformal time measured from the Big Bang. Dividing by its angular diameter distance

a_{eq} (τ_{0} - τ_{eq})

gives its small angular size. For the radiation-matter scale factor in conformal time, this is

(2 - 1) Ω_{r, 0} / [1 - 2 Ω_{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.

Radiation in cosmology

Cosmological radiation has equation of state

P_{r} = ρ_{r} /3

and density scaling

ρ_{r} \propto a^{- 4}

Radiation-dominated universe ( $a (t) \propto t^{1/2}$ )

A radiation-dominated universe has expansion controlled mainly by radiation in cosmology. When spatial curvature and the cosmological constant are negligible, the Friedmann equation and density scaling

ρ \propto a^{- 4}

give the displayed scale factor, with Hubble parameter

H = 1/ (2 t)

Comoving entropy conservation ( $g_{* s} T^{3} a^{3} = constant$ )

Adiabatic expansion conserves entropy in a comoving volume:

s a^{3}

is constant. With

s = (2 π^{2} /45) g_{* s} T^{3}

, it gives

(a_{i} / a_{0})^{3} = (g_{* s, 0} / g_{* s, i}) (T_{0} / T_{i})^{3}

. Changes in entropy degrees of freedom must therefore be retained when transporting an early relic density to today, rather than setting

a T

constant across all particle thresholds.

// Target: cosmology.bigb

Effective number of relativistic energy degrees of freedom ( $g_{*}$ )

For relativistic species in thermal equilibrium at a common temperature

T

, the total energy density can be written

ϵ = \frac{4 π ^{5}}{15 ( h c ) ^{3}} g_{*} (k_{B} T)^{4}, g_{*} = \sum_{bosons} g_{i} + \frac{7}{8} \sum_{fermions} g_{i} .

The factor

7/8

is the fermion-to-boson thermal integral ratio for energy density.

Radiation mass density ( $ρ_{rad}$ )

For relativistic species at a common temperature, the mass-equivalent energy density is

ρ = π^{2} g_{*} (k_{B} T)^{4} / (30 ℏ^{3} c^{5})

. Using mass rather than energy density accounts for the powers of

c

in the Friedmann equation.

Dark energy

Dark energy is a smooth component with sufficiently negative pressure to drive accelerated cosmic expansion. The supernova discovery of accelerating expansion measures an expansion history through luminosity distance; interpreting that history as a component of the stress tensor additionally assumes general relativity. A cosmological constant is one possibility, whereas dynamical dark energy permits a changing equation-of-state parameter.

Dark-energy acceleration criterion ( $q_{0} = \frac{1}{2} Ω_{m, 0} + \frac{1}{2} (1 + 3 w_{0}) Ω_{DE, 0}$ )

Neglect radiation in cosmology and take pressureless matter plus a separately conserved dark energy component with

p_{DE} = w ρ_{DE} c^{2}

, where

ρ

here denotes mass-equivalent energy density. The Friedmann acceleration equation gives

\frac{a ¨}{a} = - \frac{4 π G}{3} [ρ_{m} + ρ_{DE} (1 + 3 w)] .

Thus

w < - 1/3

for a positive-density component is necessary for that component to accelerate expansion, but the total must also overcome pressureless matter. For a cosmological constant,

w = - 1

and

q_{0} = Ω_{m, 0} /2 - Ω_{Λ, 0}

, so acceleration requires

Ω_{Λ, 0} > Ω_{m, 0} /2

. Spatial curvature of an FLRW universe enters the distance relation but has no separate term in this acceleration criterion. A Hubble diagram alone does not identify a microscopic dark energy mechanism.

Negative-w barotropic-fluid gradient instability ( $c_{s}^{2} = w < 0$ )

A propagating adiabatic fluid with the constant closure

p = wρ

has

c_{s}^{2} = d p / d ρ = w

. For

- 1 < w < 0

, a short-wavelength Fourier perturbation obeys

ω^{2} ≃ w k_{phys}^{2} < 0

, producing gradient growth. This does not exclude a negative-pressure perfect-fluid stress tensor with nonadiabatic pressure response, or a canonical scalar with rest-frame sound speed one. A cosmological constant has no independent propagating fluid density mode.

// Target: cosmology.bigb

Dynamical dark energy

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

Dark-energy equation-of-state reconstruction ( $ρ_{de} (z) = ρ_{de, 0} exp [3 \int_{0}^{z} (1 + w) / (1 + z^{'}) d z^{'}]$ )

energy conservation relates a separately conserved dark-energy density to its equation-of-state history. Distance measurements integrate the associated expansion history, making rapid variation difficult to recover uniquely. Growth and potential-evolution probes add information but depend on gravity and clustering assumptions. Consistency with

w = - 1

over a measured interval cannot prove that a field is exactly constant everywhere.

Variable dark-energy equation of state ( $w (z)$ )

For

P_{DE} = w (z) ρ_{DE}

, the continuity equation gives

ρ_{DE} (z) = ρ_{DE, 0} (1 + z)^{3} exp (3 \int_{0}^{z} \frac{w ( z ^{'} )}{1 + z ^{'}} d z^{'}) .