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For $w=-1$, constant-equation-of-state density scaling makes ${\rho }_{\mathrm{DE}}$ constant. In the first Friedmann equation, its contribution combines with the cosmological constant asIn the second equation, $-8\pi G{p}_{\mathrm{DE}}/c^2=8\pi G{\rho }_{\mathrm{DE}}$ produces precisely the same shift of $\Lambda c^2$. Therefore homogeneous expansion measures the sum of these contributions; a constant-density fluid with $p=-\rho c^2$ is dynamically indistinguishable from cosmological constant energy in these equations.

Write $H=\dot{a}/a$ for the Hubble parameter. Subtracting the Friedmann equation from the second equation gives the Friedmann acceleration equation,Since $\dot{H}=\ddot{a}/a-H^2$, eliminating $H^2$ with the first equation yieldsDifferentiating the first equation, with constant cosmological constant and curvature parameter, now givesThus the cosmological perfect-fluid continuity equation is $\dot{\rho }+3H(\rho +p/c^2)=0$. Multiplying by $a^3$ and using $d(a^3)/dt=3H{a}^3$ provesThis is local conservation of energy: expansion changes the energy in a comoving volume through pressure work.

For pressureless matter and zero cosmological constant, the Friedmann acceleration equation gives $\ddot{a}/a=-4\pi G\rho /3$. The deceleration parameter therefore satisfies the dust relation between deceleration and density, $q=4\pi G\rho /(3H^2)$. Evaluating today,Substitute this into the present Friedmann equation to obtain $H_0^2+k{c}^2/a_0^2=2q_0{H}_0^2$, orIn particular, $q_0>1/2$, $q_0=1/2$, and $q_0<1/2$ correspond to positive, zero, and negative spatial curvature.

Well before matter-radiation equality, radiation domination makes the integrand approximately constant:At equality, ${\Omega }_{r,0}{a}_{\mathrm{eq}}^{-4}={\Omega }_{m,0}{a}_{\mathrm{eq}}^{-3}$, so $a_{\mathrm{eq}}\simeq 1/3000$ and ${\Omega }_{r,0}\simeq 10^{-4}$. Extrapolating the radiation-only result gives $r_p(a_{\mathrm{eq}})\simeq 133\mathrm{Mpc}$. This overestimates the horizon because matter is no longer negligible at equality and increases the expansion rate.

Retain both matter and radiation, still neglecting the insignificant early cosmological constant. The particle horizon through radiation-matter equality isThereforeThis is a comoving distance; the proper horizon at equality is approximately $110/3000\mathrm{Mpc}\simeq 37$ kiloparsecs.

For today's horizon, extrapolating matter domination givesThis is an order-of-magnitude estimate: dark energy is significant late, and radiation matters early. Numerical integration of the complete expression with the supplied density parameters and ${\Omega }_{r,0}\simeq 10^{-4}$ givesBoth neglected positive-density terms increase $H$ at fixed $a$ and reduce the horizon relative to the matter-only extrapolation.

Set $c=1$ and let $H=\dot{a}/a$ denote the proper-time Hubble parameter. The Friedmann equation is the expansion constraint: density drives expansion, while spatial curvature and the cosmological constant contribute separately. The Friedmann acceleration equation shows that the active gravitational density is $\rho +3P$; positive pressure gravitates, while positive vacuum energy accelerates expansion.

Subtracting the constraint from the acceleration equation gives $\dot{H}=-4\pi G(\rho +P)+k/a^2$. Differentiating the constraint gives $2H\dot{H}=(8\pi G/3)\dot{\rho }+2kH/a^2$. Eliminate $\dot{H}$ to obtainThis cosmological continuity equation is local energy conservation. For a comoving volume $V\propto a^3$, it is $d(\rho V)=-PdV$, the adiabatic first law: expansion does work against pressure.

Define the deceleration parameter $q=-a\ddot{a}/{\dot{a}}^2$, the critical density ${\rho }_c=3H^2/(8\pi G)$, and the cosmological density parameters ${\Omega }_m={\rho }_m/{\rho }_c$ and ${\Omega }_{\Lambda }=\Lambda /(3H^2)$. The dust-plus-vacuum acceleration equation then gives $q={\Omega }_m/2-{\Omega }_{\Lambda }$. Dust conservation gives ${\rho }_m\propto a^{-3}$, while ${\rho }_{\Lambda }=\Lambda /(8\pi G)$ is constant. At the acceleration transition, ${\rho }_m=2{\rho }_{\Lambda }$, so

For spatial flatness and $a_0=1$, the constraint today sets ${\Omega }_{\Lambda 0}=1-{\Omega }_{m0}$. Thus $H^2=H_0^2[{\Omega }_{m0}{a}^{-3}+1-{\Omega }_{m0}]$. Put $M={\Omega }_{m0}$, $L=1-M>0$ and $y=a^{3/2}$. Then $\dot{y}=(3H_0/2)\sqrt{M+L{y}^2}$. Integrating from $y(0)=0$ gives $\mathrm{arsinh}(y\sqrt{L/M})=(3H_0/2)\sqrt{L}t$, hence the flat dust-and-vacuum expansionFor small $t$, $a\simeq [(3/2)H_0\sqrt{M}t{]}^{2/3}$, the matter-dominated law. At late times, $a\simeq [M/(4L){]}^{1/3}{e}^{H_0\sqrt{L}t}$, approaching de Sitter spacetime. Negative spatial curvature adds a positive $a^{-2}$ contribution to $H^2$. This is subdominant to dust's $a^{-3}$ at small $a$ and to the constant vacuum term at large $a$, so it does not change either leading limit.

The model describes the transition from matter domination to the accelerated late-time expansion indicated by cosmological distance observations, with nearly flat geometry and vacuum-like dark energy. Its formal $t\to 0$ dust limit is not the actual earliest universe: radiation, neglected here, dominates then and gives $a\propto t^{1/2}$.

The Levi-Civita connection has symmetric Christoffel symbols, so antisymmetrization removes its connection terms. Hence the Papapetrou electromagnetic field iswhere the last equality uses the Killing equation. As a differential two-form, $F=d{V}^{♭}$, so the identity $d^2=0$ for the exterior derivative proves ${\nabla }_{[a}{F}_{bc]}=0$. Contracting the preceding second covariant derivative of a Killing vector givesA vacuum spacetime here has $R_{ab}=0$, with no cosmological constant. Therefore both source-free Maxwell equations hold:This is a test electromagnetic field on the given vacuum spacetime; its own stress-energy tensor is not included in that background's Einstein field equations.

Use the mostly-plus Clifford algebra and Riemann curvature tensor conventions of the preceding solutions, with $e=\det (e_{\mu }{}^a)$, ${\kappa }^2=8\pi G$, and a Majorana gravitino. An action for minimal four-dimensional supergravity in the exact transformation normalization given isThe kinetic coefficient is tied to the normalization of the supersymmetry transformations: if ${\Psi }_{\mu }=2{\psi }_{\mu }$ and $\eta =2ϵ$, this becomes the canonical coefficient $-1/2$, with $\delta e_{\mu }{}^a=(\kappa /2)\bar{\eta }{\gamma }^a{\Psi }_{\mu }$ and $\delta {\Psi }_{\mu }={\nabla }_{\mu }\eta /\kappa$. Mixing those two normalizations would spoil the cancellation.

To the retained order, use the torsion-free connection. More generally the 1.5-order formalism allows its variation to be omitted after imposing its own algebraic equation; the resulting contorsion is quadratic in fermions and first affects the displayed action at quartic order. The metric tensor variation induced by the vierbein is $\delta g_{\mu \nu }=4\kappa \overline{ϵ}{\gamma }_{(\mu }{\psi }_{\nu )}$, so the Einstein-Hilbert action varies asFor the fermionic action, varying both Majorana factors gives the same contribution after integration by parts and the Grassmann bilinear interchange rule. ThusThe vector-index Riemann curvature tensor contribution vanishes by the first Bianchi identity. The spin-index contribution is determined byFor the last equality, expand the antisymmetric three-gamma/two-gamma product: the totally antisymmetric Riemann curvature tensor terms vanish by Bianchi, and the surviving terms are the Ricci contraction minus half its trace. Hencewhich cancels the gravitational variation. Variations of vierbeins, gamma matrices and the connection in the fermionic kinetic term carry three dynamical fermions. They lie beyond this linear-fermion cancellation and are cancelled by the omitted four-fermion completion. Thus the action is invariant up to boundary terms at the order specified, rather than claiming that a truncated action is exactly invariant at every fermion order.

For the cosmological extension, add a gravitino mass parameter $m$ and deform the derivative:Indeed the fermion terms combine as $-2{\bar{\psi }}_{\mu }{\gamma }^{\mu \nu \rho }{\mathcal{D}}_{\nu }{\psi }_{\rho }$ because ${\gamma }^{\mu \nu \rho }{\gamma }_{\nu }=-2{\gamma }^{\mu \rho }$. The modified spinor connection curvature isand contraction givesThe same action-variation cancellation therefore works with $G$ replaced by $G+\Lambda g$ precisely forThis is the Anti-de Sitter deformation of minimal supergravity. A cosmological term alone would leave its variation uncancelled; the mass term and the gamma shift of the gravitino transformation are both necessary. A positive cosmological constant cannot arise from this real unbroken minimal deformation.

For electromagnetism, retain $\mathcal{N}=1$ and introduce an Abelian vector multiplet $(A_{\mu },\lambda )$, with Majorana gaugino $\lambda$. A complete quadratic-fermion extension isThe accompanying leading transformations areThey preserve the rigid Maxwell-gaugino action for constant $ϵ$. For a local parameter, the variation containsTo obtain this, vary $F^2$, vary both gaugino factors, and use ${\gamma }^{ab}{\gamma }^{\mu }={\gamma }^{ab\mu }+g^{b\mu }{\gamma }^a-g^{a\mu }{\gamma }^b$. The terms with $\nabla \lambda$ cancel between the two actions; integrating the three-gamma term uses ${\nabla }_{[\mu }{F}_{\nu \rho ]}=0$, leaving exactly the displayed derivative of the parameter. The interaction is $-\kappa e{\bar{\psi }}_{\mu }{j}^{\mu }$, whose variation under $\delta {\bar{\psi }}_{\mu }={\nabla }_{\mu }\overline{ϵ}/\kappa$ cancels that term.

The other linear-fermion variation is also fixed. Put $\mathcal{F}=F_{ab}{\gamma }^{ab}$ and $T_{\mathrm{EM}}^{\mu \nu }=F^{\mu \rho }{F}^{\nu }{}_{\rho }-g^{\mu \nu }{F}^2/4$. Clifford multiplication givesVarying $\lambda$ in the gravitino-current interaction then gives $2\kappa e{T}_{\mathrm{EM}}^{\mu \nu }{\bar{\psi }}_{\mu }{\gamma }_{\nu }ϵ=-2\kappa e{T}_{\mathrm{EM}}^{\mu \nu }\overline{ϵ}{\gamma }_{\nu }{\psi }_{\mu }$, which cancels the Maxwell vierbein variation. This proves the required supergravity coupling of an Abelian vector multiplet through the retained order.

Beyond that order one uses a supercovariant gauge field strength, here ${\hat{F}}_{\mu \nu }=F_{\mu \nu }-2\kappa {\bar{\psi }}_{[\mu }{\gamma }_{\nu ]}\lambda$, and the associated higher-fermion transformation terms and four-fermion interactions. An auxiliary scalar may be retained for off-shell closure, or eliminated in this on-shell formulation. A Maxwell field without a fermionic partner cannot be coupled by merely adding $-F^2/4$ while leaving all transformations unchanged. An alternative is an $\mathcal{N}=2$ gravity multiplet, where the vector is a graviphoton and a second gravitino supplies its extra fermionic states; the explicit construction above is the minimal $\mathcal{N}=1$ matter extension.

Normalize the scale factor by $a(t_0)=1$. Separate conservation for a constant equation-of-state parameter $w$ gives $d\log {\rho }_i=-3(1+w_i)d\log a$. Thus radiation has ${\rho }_R={\rho }_{R0}{a}^{-4}$ while the dark energy component with $w=-1$ has constant density. Flatness gives ${\Omega }_{R0}+{\Omega }_{\Lambda 0}=1$, soHere $\rho$ has mass-density units in the stated Friedmann equation; the corresponding energy density is $\rho c^2$.

Assume both density parameters are positive and choose the Big Bang at $t=0$. The expanding branch satisfies $\dot{a}=H_0\sqrt{{\Omega }_{R0}{a}^{-2}+{\Omega }_{\Lambda 0}{a}^2}$. Set $u=a^2$ and integrate:ConsequentlyThe factor $H_0$ is required because $\beta$ has inverse-time units. As $t\downarrow 0$, $a(t)\sim (2H_0\sqrt{{\Omega }_{R0}}t{)}^{1/2}$, the expected radiation-dominated power law. As $t\to \infty$, $a(t)\sim \alpha e^{H_0\sqrt{{\Omega }_{\Lambda 0}}t}/\sqrt{2}$, the exponential expansion driven by a cosmological constant.

Take the usual smooth late dark energy, in particular a cosmological constant, and hold the primordial perturbations fixed. Increasing its physical density while holding the matter and radiation densities fixed raises $H_0$ in a flat universe. It lowers ${\Omega }_{m0}$, but leaves the physical product ${\Omega }_{m0}{H}_0^2=\kappa {\rho }_{m0}/3$ fixed. It also leaves the early equality ratio $a_{\mathrm{eq}}={\rho }_{r0}/{\rho }_{m0}$ and the equality wavenumber in ${\mathrm{Mpc}}^{-1}$ essentially unchanged, provided dark energy remains negligible at equality. The early cosmological transfer function, including the acoustic scale, is then unchanged.

More dark energy causes accelerated expansion to start earlier, reducing the time for matter growth. The suppression of matter growth by smooth accelerated expansion lowers $D(1)$ and hence the present spectrum. For the subhorizon scales shown,This is approximately a downward amplitude change with the same turnover and wiggle locations in physical inverse-megaparsec units. For matter plus a cosmological constant, an explicit early-normalized growth solution iswith radiation neglected in this late-time growth equation. A numerical factor needs the old and new dark-energy densities. A change to axes expressed using $h$ would additionally relabel wavenumbers and volumes; the printed axes use ${\mathrm{Mpc}}^{-1}$ and ${\mathrm{Mpc}}^3$.

Let $u={\delta }_m/H$. The supplied equation has integrating factor $(aH{)}^3$:Consequently the integral linear growth factor in a matter-Lambda universe gives the two independent solutionsThe first is the conventional matter-era decaying mode. A finite lower limit in the second fixes a decaying admixture; adding a multiple of $H$ can choose a pure growing normalization.

During matter domination, write $H=h{a}^{-3/2}$. The integral solution isRemoving the second term by the independent $H$ solution gives the matter-era linear growth factorFor positive cosmological constant at sufficiently late times, $H\to H_{\Lambda }>0$. The integrand approaches $H_{\Lambda }^{-3}{a}^{-3}$, whose tail is integrable. ThereforeIts remaining approach has leading $a^{-2}$ behavior for the usual matter-plus-Lambda background. Growth of structure freezes rather than continuing as $a$. The basis function $H$ also approaches a constant; subtracting a suitable multiple of it isolates a genuinely decaying late-time solution $\propto a^{-2}$. The name “decaying mode” for $H$ refers to its matter-era behavior.

A four-dimensional vacuum Einstein solution with the fixed cosmological constant $\Lambda$ has $R_{ab}=\Lambda g_{ab}$ and $R=4\Lambda$, constant. Thus every derivative of $f^{\prime }(R)$ vanishes andThe metric tensor is nondegenerate, so this is zero precisely whenThis is the necessary and sufficient Einstein metric condition in f(R) gravity for the specified $\Lambda$, and works for every such Einstein metric, even when its Weyl tensor is nonzero. There is no need to divide by $f^{\prime }(4\Lambda )$; the degenerate case where both $f$ and $f^{\prime }$ vanish at that curvature is included. At $\Lambda =0$, the condition is simply $f(0)=0$.

If the intention is to demand the inclusion for every real $\Lambda$ simultaneously, the stronger functional condition is $R{f}^{\prime }(R)=2f(R)$ for all $R$. On each nonzero half-line it integrates to $f(R)=C{R}^2$; smoothness across zero makes the constants equal. Thus the all-$\Lambda$ version gives $f(R)=C{R}^2$, including $C=0$. This stronger reading is distinct from fixing one cosmological constant.

For a pressureless spherical mass shell with fixed enclosed mass and no shell crossing, the shell theorem gives $\ddot{R}=-GM/R^2$. A cosmological constant is absent in the Einstein-de Sitter universe. Parametric differentiation givesSince $R^2=A^2(1-\cos \theta {)}^2$, the equation of motion holds providedThe expanding branch runs from $\theta =0$ to the turnaround of spherical collapse $\theta =\pi$; the recollapsing branch runs from $\pi$ to $2\pi$. HenceThe common bang-time origin selects the growing overdensity, without an extra arbitrary shift in the time parameter.

In the Einstein-de Sitter universe, $H=2/(3t)$ and $\bar{\rho }=1/(6\pi G{t}^2)$. An unperturbed homogeneous sphere containing the same mass thus has $R_{\mathrm{bg}}^3=(9/2)GM{t}^2$. Comparing its volume with the perturbed sphere givesAt maximum expansion,For the non-dissipative, homologous spherical-collapse model, the potential energy of a uniform sphere is $W=-3G{M}^2/(5R)$. At turnaround the bulk kinetic energy is zero, so $E=W_{\mathrm{ta}}$. At virial equilibrium, $2K+W=0$ gives $E=W_{\mathrm{vir}}/2$. Conservation of energy implies $W_{\mathrm{vir}}=2W_{\mathrm{ta}}$, and therefore $R_{\mathrm{vir}}=R_{\max }/2$. This comparison assumes the same potential-energy structure coefficient; it is the usual top-hat model rather than an exact claim about every halo profile.

Halving the radius multiplies the density by eight. During the interval to $t_{\mathrm{coll}}=2t_{\max }$, the background density falls by four. ThusThe formal pressureless trajectory reaches zero radius; the physical virialized object instead has a finite radius. Its density is evaluated at the same collapse epoch, not at the instantaneous singular solution's density.

For a physical estimated virial radius and enclosed mass, ${\rho }_h=3M/(4\pi r_{\mathrm{vir}}^3)$. Equating this to ${\Delta }_{\mathrm{vir}}{\bar{\rho }}_{m,0}(1+z_f{)}^3$ gives a halo density estimate of formation redshift:In the pure Einstein-de Sitter universe, ${\bar{\rho }}_{m,0}$ equals the present critical density. The inference assumes the measured density retains the collapse-epoch normalization. Later accretion, mergers and a changing virial-radius convention can change it, so it is a model-dependent formation estimate rather than a unique historical date.

Apply the Grassmann Gaussian integral to a regulated finite collection of field components. Up to a field-independent measure normalization and phase,In the continuum this is a formal functional determinant, including spinor and spacetime indices. Its meaningful definition requires a regulator and boundary conditions. A complex Dirac field supplies a determinant, not the inverse square root obtained for a real commuting field.

The comparison is clean after Wick rotation. A free real scalar field with positive Euclidean operator $P_E=-{\partial }_E^2+m^2$ hasTaking a logarithm gives $-\frac{1}{2}\mathrm{Tr}\log P_E$ for the scalar and $+\mathrm{Tr}\log K_E$ for the Dirac field. The opposite statistics sign is the determinant counterpart of the minus sign for a closed fermion loop. The magnitude also differs because a Dirac field has several independent spin and antiparticle degrees of freedom.

The vacuum energy sign of a fermionic oscillator makes the comparison explicit. A real scalar mode contributes $+E/2$; each independent fermionic oscillator contributes $-E/2$. There is one oscillator per scalar momentum, but a massive Dirac field has two particle and two antiparticle oscillators. Thus the vacuum energy densities are formallyEquivalently, the large Euclidean-time vacuum functional behaves as $\log Z_E\sim -\beta V{\rho }_{\mathrm{vac}}$. Both displayed vacuum energies are ultraviolet divergent; a common regularization in quantum field theory and the appropriate renormalization are needed before comparing them. The zero-point vacuum energies of bosons and fermions have opposite signs, but do not cancel without matching masses and degrees of freedom. Normal ordering removes an additive vacuum constant in nongravitating flat-space theory. Coupling to the metric in general relativity makes that constant contribute to the cosmological constant, so it cannot simply be discarded without a renormalization condition.

The canonical Kähler potential gives $e^K=e^{|z{|}^2}$ and inverse Kähler metric one. Substituting the superpotential and its Kähler covariant derivative of a superpotential into the supergravity F-term potential yieldsFor real $m$, the prefactor is simply $m^4$; the modulus form also covers a complex phase. To keep both scalar directions explicit, write $z=x+iy$ and $P=1+x^2+y^2+\beta x$. ThenThe negative term is essential: unlike the global F-term scalar potential, the supergravity F-term potential need not be nonnegative. This is why cancelling the cosmological constant does not force the auxiliary field to vanish.

For the stable branch found above, $t=x+\beta =1$ and $s=1$. Hence the vacuum expectation value isin the stated Planck units. To verify that it is a vacuum rather than merely a zero-energy stationary point, evaluate the Hessian matrix. At either zero-energy stationary branch,Because $H$ and its first derivatives vanish there, the Hessian matrix of $V$ is just $|m{|}^4{e}^{x^2}$ times this Hessian matrix. For $s=1$, both eigenvalues are positive. This proves a strict local minimum in both real scalar directions. For $s=-1$, $H_{xx}<0$, so the alternative $\beta =2+\sqrt{3}$, $⟨z⟩=-\sqrt{3}-1$ is a saddle point and is excluded from the stable zero-energy Polonyi vacuum.

There is also a useful global check. Set $u=x-(\sqrt{3}-1)$ on the stable branch. Directly completing squares givesBoth remaining coefficients are positive. Thus $H\ge 0$ everywhere, with equality only at $u=y=0$. The positive exponential prefactor proves that this is the unique global minimum, not just a metastable vacuum.

Finally, at the stable vacuum $z+\beta =1$ and $D_{z}W=\sqrt{3}{m}^2$. Its supergravity auxiliary field hasThus the Minkowski vacuum breaks supersymmetry, even though its cosmological constant vanishes. If $m=0$, the potential is flat and the displayed tuned parameter and scalar value are not selected.

The cosmological perfect-fluid continuity equation gives ${\rho }_r={\rho }_{r,0}{a}^{-4}$ and ${\rho }_m={\rho }_{m,0}{a}^{-3}$. In conformal time, $\dot{a}=a^{\prime }/a$ and $H=a^{\prime }/a^2$. Thus the flat Friedmann equation isChoose the expanding branch and put the Big Bang at $\tau =0$. Integration givesInverting it supplies the radiation-matter scale factor in conformal time:The early linear term gives radiation domination, while the late quadratic term gives matter domination. The solution presumes the stated radiation-plus-matter model, without a cosmological constant.

At matter-radiation equality, $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$. Since ${\Omega }_{r,0}+{\Omega }_{m,0}=1$,In units $c=1$, the comoving particle horizon at equality is ${\tau }_{\mathrm{eq}}$; its physical size is $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$. The angular diameter distance to the equality surface is $a_{\mathrm{eq}}({\tau }_0-{\tau }_{\mathrm{eq}})$. Therefore the horizon angle at matter-radiation equality corresponding to one horizon length isFor ${\Omega }_{r,0}=10^{-4}$, this is $4.20\times 10^{-3}$ radians, or $0.241^{\circ }$, about $14.4$ arcminutes. If quoting the full diameter of a horizon-sized patch, the result is twice this, $0.481^{\circ }$. The small-angle approximation is excellent. Expanding in $\sqrt{{\Omega }_{r,0}}\ll 1$ would give $4.14\times 10^{-3}$ radians; retaining both cosmic components near equality avoids an inaccurate pure-matter extrapolation there. The scale factor cancels between physical size and angular diameter distance, and $H_0$ cancels from the ratio.

If cosmological recombination occurs only a modest expansion after equality, its causal patch still subtends a small part of the sky. Widely separated regions of the observed cosmic microwave background have nearly the same temperature even though their past particle horizons did not overlap in the ordinary decelerating history. Local thermalization after the Big Bang therefore cannot explain this large-angle uniformity. This is the horizon problem; inflation supplies an earlier connected region that can grow to encompass the observed sky.

In the Lambda-CDM model, nearly Gaussian primordial density contrasts grow under gravity within an expanding universe containing cold dark matter, ordinary baryons and a cosmological constant. The cold dark matter is effectively collisionless and has negligible pressure on galactic scales. Before recombination, baryons are coupled to the photon fluid: radiation pressure and acoustic oscillations prevent their perturbations from behaving like pressureless matter. After recombination they can fall into the gravitational potentials already established by dark matter, subject to gas pressure and the Jeans mass.

For small density contrasts, evolution is linear. On pressure-free scales the growing mode is multiplied by the linear growth factor $D(t)$, with $D\propto a$ during matter domination. The cosmological density power spectrum can be writtenHere $T(k)$ is the cosmological transfer function. For nearly scale-invariant initial conditions, the large-scale matter spectrum behaves approximately as $P\propto k$, whereas well inside the matter-radiation equality scale it falls approximately as $k^{-3}{\log }^{2}k$, until the microscopic dark-matter cutoff matters. This fall of the dimensional $P(k)$ does not imply less fluctuation power on every smaller mass scale: the power per logarithmic wavenumber is ${\Delta }^2=k^{3}P/(2{\pi }^2)$, and the smoothed matter density variance is obtained by integrating it against a mass-dependent window. Over the relevant cold-dark-matter hierarchy, smaller mass windows generally have larger variance.

The hierarchical galaxy formation picture follows: fluctuations on small mass scales typically reach the nonlinear collapse threshold first, while larger objects assemble later through accretion and dark-matter halo mergers. It is a statistical ordering, not a claim that every small object precedes every large rare peak. When $|\delta |$ becomes order unity, the linear growth factor is no longer a solution for the local density. Collisionless dark matter develops multistream regions and bound dark matter halos; phase mixing and violent relaxation redistribute orbital energies, and virialized structures approximately obey the virial theorem.

Baryons have an additional nonlinear route. Infall and shocks convert bulk kinetic energy into thermal energy, with characteristic virial temperatureUnlike collisionless dark matter, the gas can lose this energy through radiative cooling. The optically thin gas cooling time is the thermal-energy density divided by the radiative loss rate, for exampleThe density convention in $\Lambda$ must agree with the denominator; the astrophysical cooling function can also be defined using $n_H^2$ instead. The cooling criterion for galaxy formation compares this time with the collapse or supply time. Rapidly cooling gas loses pressure support, contracts and can form stars. Slowly cooling gas remains in a hot atmosphere. Stable virial shocks are not obligatory in every low-mass system: gas can also arrive in cold streams and cool while being accreted.

Angular momentum prevents indefinite radial contraction. Tidal torque theory supplies an initial halo spin, and later mergers change it. If gas radiates energy while retaining much of its specific angular momentum, it settles into a rotationally supported galactic disk rather than reaching the centre. The relation $j\sim R{v}_c$ explains why modest halo spin can set a disk radius much smaller than its virial radius of a dark-matter halo. Torques, bars and gravitational encounters can transport angular momentum outwards and feed central concentrations; radiative cooling alone does not remove it.

Galaxy mergers alter stellar structure as well as assembling mass. A major galaxy merger can strongly disturb or destroy an existing galactic disk, randomizing stellar orbits and creating a spheroid through violent relaxation. A gas-rich galaxy merger also permits dissipation, inflow and a burst of star formation; gas left over or accreted afterwards can rebuild a galactic disk. Minor galaxy mergers add stars to outer components, thicken disks and grow bulges. Dry galaxy mergers add stellar mass and can increase size without much new star formation. Halo merging therefore does not imply instantaneous merging of its galaxies: satellite orbital decay requires Chandrasekhar dynamical friction and can take a substantial time.

The atomic and molecular cooling thresholds for galaxy formation supply a lower characteristic scale. Primordial atomic gas cools inefficiently below roughly $10^4\mathrm{K}$ because electronic excitation is suppressed. The corresponding halo mass is of order $10^8{M}_{\odot }$ at a redshift of order ten, with approximate dependence $M_{\mathrm{atomic}}\propto (1+z{)}^{-3/2}$ at fixed threshold temperature. Molecular hydrogen can cool gas at hundreds of kelvin and permit smaller early objects, provided it forms and survives dissociating radiation. Metal-line cooling changes these thresholds after enrichment. Thus the atomic threshold is not an absolute minimum mass for all stellar systems.

At the other end, sufficiently massive dark matter halos have high virial temperatures and low-density hot gas. Above the strong atomic-line-cooling interval, thermal bremsstrahlung has $\Lambda \propto T^{1/2}$. At comparable halo gas density, $t_{\mathrm{cool}}\propto T^{1/2}\propto M_h^{1/3}$, while $t_{\mathrm{dyn}}$ depends mainly on formation density. Cooling therefore becomes less able to condense all the gas within the available time. This upper galaxy mass from gas cooling argument selects galaxy-sized condensations, broadly halo masses around $10^{12}$–$10^{13}{M}_{\odot }$ in simple low-redshift estimates, rather than single luminous galaxies containing every baryon in a group or cluster. Its numerical scale depends on epoch, metallicity and gas profile. Subsequent galaxy mergers can build larger stellar systems; cooling is not an absolute upper bound on their final mass.

Finally, a Press-Schechter halo mass function has many low-mass objects and a steep high-mass cutoff, while the luminosity function of galaxies also has a faint component and a bright cutoff, often described by a Schechter function. The two shapes are related through the halo-to-galaxy luminosity mapping, not by identifying luminosity with total halo mass. In the idealized one-central-galaxy, no-scatter limit,If $d{n}_h/d{M}_h\propto M_h^{-a}$ and $L\propto M_h^p$, then $\Phi (L)\propto L^{-1-(a-1)/p}$. A constant conversion gives similar shapes. Actual stellar feedback, inefficient low-temperature cooling and reionization suppress faint galaxies relative to small haloes, while long cooling times and active-galactic-nucleus feedback suppress luminosity at large halo masses. Satellites, scatter, stellar populations and dust further affect the correspondence. The halo hierarchy supplies the gravitational framework; cooling, angular momentum and feedback determine which parts become luminous galaxies.

At fixed mass, decreasing redshift increases the linear growth factor and reduces the halo peak height $\nu$. During matter domination $\dot{D}/D=H$ also falls as the universe expands. As long as $\nu >1$, both effects reduce the positive abundance-growth rate, so $t_d$ grows towards lower redshift. In the very rare-tail limit $t_d\propto D^2/H\propto (1+z{)}^{-7/2}$ provides the approximate trend.

There is a limit to calling this an increase time. At $\nu =1$, the fixed-mass Press-Schechter halo mass function reaches its maximum as a function of time; its logarithmic growth rate vanishes and $t_d$ diverges. For $\nu <1$, the derivative becomes negative, corresponding to net transfer of small objects into more massive systems, so the signed inverse is no longer a positive doubling or increase time. At still later epochs the cosmological constant further suppresses the linear growth factor. These qualifications prevent extrapolating the positive high-redshift growth trend indefinitely.

The contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Combining the proposed equation with stress-energy conservation therefore imposes ${\nabla }_{b}R=0$. Taking its trace gives $R=\kappa T$, with $T=g^{ab}{T}_{ab}$, so for nonzero $\kappa$ the matter trace must also be constant. Ordinary matter need not satisfy this extra condition. For example, if asymptotic flatness forces the constant scalar curvature to vanish, the proposal would require $T=0$ everywhere.