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For an expanding pressureless matter universe with zero cosmological constant, fixed current Hubble parameter $H_0$ and current matter cosmological density parameter $\Omega >0$, the age of a dust universe without a cosmological constant is $H_0{t}_0={\int }_0^1\sqrt{x}/\sqrt{\Omega +(1-\Omega )x}dx$. For every $0<x<1$ the integrand decreases strictly as $\Omega$ increases. Thus open models with $0<\Omega <1$ are older than the flat model, and closed expanding models with $\Omega >1$ are younger. The flat value is $2/3$ and the empty Milne universe limit is $1$. Specifying what is held fixed matters: at fixed present scale factor and energy density, the integral $t_0={\int }_0^{a_0}da/\sqrt{C/a-k}$ instead increases with $k$, provided all three models reach $a_0$ on their expanding branches.

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 $z_{\ast }$, define the transverse comoving distance $D_M=(1+z_{\ast })D_A$. A comoving sound horizon $r_s$ has proper length $r_s/(1+z_{\ast })$ at emission, so its angular size is ${\theta }_{\ast }=r_s/D_M$. Mixing a comoving ruler with $D_A$ would introduce an erroneous factor $1+z_{\ast }$. In a curved FLRW metric, $D_M$ is generally different from the radial comoving distance.

For a stipulated example, $r_s=144\mathrm{Mpc}$ and $D_M=14000\mathrm{Mpc}$ give ${\theta }_{\ast }=0.010286$ radians, or $0.589^{\circ }$, and an acoustic scale ${\ell }_A=305.4$. Multiplying both lengths by $1.1$ gives $158.4\mathrm{Mpc}$ and $15400\mathrm{Mpc}$ with exactly the same ${\theta }_{\ast }$ and ${\ell }_A$. This example establishes a ratio degeneracy, without claiming that any arbitrary rescaling corresponds to an allowed cosmological model. Equivalently,Other observables are needed to separate changes that preserve the ratio.

The approximate adiabatic acoustic-peak spacing is ${\ell }_A$. The first observed temperature peak is shifted by gravitational driving and angular projection of density and Doppler acoustic sources; it need not occur at ${\ell }_A$. 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 ${\theta }_{\ast }$ 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.

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.

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 $\theta$, ${\Omega }_M={\sec }^2(\theta /2)$, diverging at $\theta =\pi$ and approaching one at the Big Bang and Big Crunch ends.

The matter density parameter is the ratio of total nonrelativistic matter density, including baryonic and dark matter, to the critical density ${\rho }_{\mathrm{crit}}=3H^2/(8\pi G)$ at the same epoch. A present-epoch value uses $H_0$ and ${\rho }_{m,0}$. It is not the total cosmological density parameter when radiation or vacuum energy also contributes.

For the expanding empty model, $\dot{a}=H_0$ in the preceding normalization. If $t_B$ denotes the origin of the expanding slicing,The constant-time spatial slices are negatively curved hyperbolic spaces, with curvature radius $R_{\mathrm{curv}}=a/\sqrt{-k}=c(t-t_B)$ and three-dimensional scalar curvature $6k/a^2$. In the usual simply connected description they are copies of hyperbolic three-space. This is the Milne model: its four-dimensional spacetime is actually flat. Setting $T=(t-t_B)\cosh \chi$ and $R=c(t-t_B)\sinh \chi$ converts the metric to the interior of a future light cone in Minkowski spacetime. Its zero-scale-factor boundary is a slicing boundary, not a curvature singularity.

Add separately conserved pressureless matter, with present mass-equivalent density ${\rho }_{M0}$. Define the cosmological density parametersThenCurvature and dust have equal magnitudes at $a_K={\Omega }_M/{\Omega }_K$, givingFor the almost-empty open case $0<{\Omega }_M<1/2$, curvature domination begins at positive redshift. Equality is today for ${\Omega }_M=1/2$ and in the future for $1/2<{\Omega }_M<1$. With ${\Omega }_M=0$, curvature dominates at all times and there is no finite matter–curvature equality; with ${\Omega }_K=0$, curvature never dominates.

Use mass density units for the $\rho$ appearing in the Friedmann equation; an actual energy density is $\rho c^2$. Conservation, $\dot{\rho }+3H(\rho +P/c^2)=0$, gives $\boxed{{\rho }_M={\rho }_{M,0}{a}^{-3},{\rho }_R={\rho }_{R,0}{a}^{-4}}$ for pressureless matter and radiation in cosmology, respectively, with $a_0=1$. The first scaling counts dilution of particle number, while the second includes the cosmological redshift of each photon's energy.

Define the present cosmological density parameters by ${\Omega }_{X,0}=8\pi G{\rho }_{X,0}/(3H_0^2)$. ThenFor $k=0$, conformal time gives $da/d\eta =a^{2}H$, so the expanding branch obeys $(da/d\eta {)}^2=H_0^2({\Omega }_{M,0}a+{\Omega }_{R,0})$. Set $\eta =0$ at the Big Bang. Integrating givesand hence the flat matter-radiation conformal solutionThe limiting pure-radiation and pure-matter cases follow continuously.

For dust alone, write ${\dot{a}}^2=c^2(A/a-k)$, whereSet $\theta =\sqrt{k}c\eta$ and $a=(A/k){\sin }^2(\theta /2)$. Differentiation gives $\dot{a}=c\sqrt{k}\cot (\theta /2)$ once $dt=ad\eta$ is used, verifying the Friedmann equation. Integrating the time relation from the Big Bang yields the closed matter-dominated Friedmann solutionThe Hubble parameter and the cosmological density parameter areThe first decreases from positive infinity through zero at maximum expansion to negative infinity at the Big Crunch. The second approaches one near either singular endpoint and diverges at maximum expansion. This is a density-parameter divergence at a cosmological turnaround: $H=0$ makes the critical density vanish even though the physical dust density there is finite.

For $\Lambda =0$, use the instantaneous cosmological density parameterInsert the Friedmann equation from the preceding part and multiply numerator and denominator by $a^4$:This holds on an expanding or contracting branch with $H\ne 0$; in a recollapsing model the density parameter diverges at turnaround.

During radiation domination, while curvature is small, $|\Omega -1|\simeq |{\Omega }_0-1|a^2/{\Omega }_{R0}$. During matter domination the corresponding behavior is $|\Omega -1|\simeq |{\Omega }_0-1|a/{\Omega }_{M0}$. Thus the exactly flat solution $\Omega =1$ is unstable against relative curvature during decelerating expansion: a small departure grows rather than being dynamically driven to zero. The flatness problem is that a universe with modest present curvature must have had an extremely tiny initial curvature-to-density ratio at an early radiation epoch. The fact that every such solution tends to $\Omega =1$ when extrapolated to $a\to 0$ does not supply a mechanism selecting that small departure at a specified finite initial epoch. An early period of cosmic inflation can instead shrink the comoving Hubble radius and suppress $|\Omega -1|=|k|/(aH{)}^2$.

Normalize the scale factor to $a(t_0)=1$, use $c=1$, and define the instantaneous critical density by ${\rho }_{\mathrm{cr}}(t)=3H(t{)}^2/(8\pi G)$. Separately conserved pressureless matter and radiation in cosmology obey the cosmological continuity equation,The present cosmological density parameters areThe cosmological constant has equivalent energy density ${\rho }_{\Lambda }=\Lambda /(8\pi G)$. Substituting into the Friedmann equation yieldsAt the present epoch this gives $k=H_0^2({\Omega }_0-1)$, or curvature contribution ${\Omega }_{K0}=-k/H_0^2=1-{\Omega }_0$. Dividing each component by $H_0^2$ provesSpatial curvature is not included in the total physical-density parameter ${\Omega }_0$. In particular, positive vacuum density does not have the radiation or matter scaling.

Use natural units and conformal time $dt=ad\tau$. The conformal Hubble parameter is $\mathcal{H}=a^{\prime }/a=aH$. For the barotropic equation of state $P=(\gamma -1)\rho$, the Friedmann acceleration equation givesThe Friedmann equation gives $8\pi G{a}^2\rho /3={\mathcal{H}}^2+k$. Hence the constant-equation-of-state conformal Riccati equation isFor $\alpha \ne 0$, separation and integration of this Riccati equation yield, after choosing the origin of conformal time,The open positive-energy density branch uses the hyperbolic cotangent, rather than the cotangent. Integrating $a^{\prime }/a=\mathcal{H}$ giveswhere the positive constants are fixed by the energy density normalization. These formulas apply on branches where the bases are positive. For $\alpha >0$, take the Big Bang at $\tau =0$. The cosmic time is thenThe integrals also give the general answer after selecting a suitable reference time when a finite Big Bang endpoint does not exist. The exceptional coasting fluid $\gamma =2/3$ has $\alpha =0$, so $\mathcal{H}=h$ is constant: $a=A{e}^{h\tau }$ and $t=(A/h)e^{h\tau }+t_{\ast }$. If $h=0$, instead $a=A$ and $t=A\tau +t_{\ast }$. A flat fluid with $\gamma =0$ is the constant-density de Sitter spacetime case, for which the flat time integral is logarithmic.

During matter domination, $\gamma =1$ and $\alpha =1/2$. Absorbing numerical factors into separate positive constants gives the closed matter-dominated Friedmann solution, the flat Einstein-de Sitter universe and the open matter-dominated Friedmann solution:To compare the age of a dust universe without a cosmological constant, specify the normalization. At fixed present Hubble parameter $H_0$, with $x=a/a_0$ and present matter cosmological density parameter ${\Omega }_m>0$, the Friedmann equation givesThe integrand decreases strictly with ${\Omega }_m$ for $0<x<1$. Therefore age ordering of dust universes at fixed Hubble parameter gives, on the expanding branches,The final bound is approached by the empty Milne universe. This compares models with the same $H_0$; fixing instead the present scale factor and energy density gives a different comparison.

A genuine nondegenerate bounce has ${\dot{a}}_{\ast }=0$ and ${\ddot{a}}_{\ast }>0$; using ${\ddot{a}}_{\ast }\ge 0$ also includes the limiting stationary case. Put $s=a_{\ast }^{-1}=1+z_{\ast }$, and define the present matter density parameter ${\Omega }_m=8\pi G{\rho }_0/(3H_0^2)$, ${\Omega }_{\Lambda }=\Lambda /(3H_0^2)$ and ${\Omega }_k=-k/H_0^2$. The dust-plus-cosmological-constant Friedmann equation and its acceleration equation becomeAt the bounce these giveFor a past bounce with $s>1$, eliminate ${\Omega }_k$ from the first equation:Subtracting ${\Omega }_m{s}^3/2$ and multiplying by the positive denominator givesThus the matter bound from a Friedmann bounce isStrict positive bounce acceleration makes this inequality strict. The zero of the Friedmann equation also requires positive curvature for positive matter and positive $\Lambda$.

An observed source at $z=6$ has $a_e=1/7$. Since the minimum scale factor cannot exceed the scale factor at emission, $z_{\ast }\ge 6$. The denominator of the bound increases with positive $z_{\ast }$, soThe observational lower bound is much larger. Cluster mass-to-light ratios, inferred independently from motions, gas temperatures and gravitational lensing, combined with the observed luminosity density imply substantial cosmic mass density. Measurements of the cosmic baryon fraction from representative clusters, combined with the baryon cosmological density parameter inferred from Big Bang nucleosynthesis, provide another estimate: ${\Omega }_m\simeq {\Omega }_b/f_b$. Large-scale clustering and peculiar velocities provide independent dynamical checks, although their interpretation must account for galaxy bias. These methods support matter densities of order a few tenths, rather than below one percent. The dust-plus-cosmological-constant bouncing model is incompatible with both a quasar at redshift six and the stated observational evidence for ${\Omega }_m\gtrsim 0.2$.

Use units $c=1$ and take the initially expanding branch with ordinary positive matter or radiation density. In the Friedmann-Lemaître-Robertson-Walker metric, $k=+1,0,-1$ respectively describe positive, zero and negative spatial curvature. Separately conserved pressureless matter has ${\rho }_M\propto a^{-3}$ and radiation in cosmology has ${\rho }_R\propto a^{-4}$, so either dominates the curvature and vacuum terms sufficiently far back on a hot Big Bang branch.

With negligible cosmological constant, a spatially flat radiation universe has $a\propto t^{1/2}$ and a flat matter universe has $a\propto t^{2/3}$; both expand forever while decelerating. An open universe also expands forever and eventually becomes approximately curvature dominated, with $a\propto t$. A closed universe containing only ordinary matter/radiation reaches a maximum size and recollapses. For example, dust gives ${\dot{a}}^2=C/a-k$ with $C>0$, so only $k=+1$ admits the positive turning point $a=C$.

Curvature alone does not determine the fate when the cosmological constant is retained. Positive $\Lambda$ can cause late accelerated expansion and prevent even a closed universe from recollapsing; whether a closed expanding branch turns around depends on the relative matter and vacuum terms. Negative $\Lambda$ can cause recollapse even for flat or open spatial sections. These statements follow by inspecting ${\dot{a}}^2=C/a-k+\Lambda a^2/3$ for dust; the curvature sign is a geometrical classification, not by itself a universal destiny criterion.

Define the critical density ${\rho }_c=3H^2/(8\pi G)$ and cosmological density parameters ${\Omega }_M={\rho }_M/{\rho }_c$, ${\Omega }_R={\rho }_R/{\rho }_c$, ${\Omega }_{\Lambda }=\Lambda /(3H^2)$ and ${\Omega }_k=-k/(a^2{H}^2)$. The Friedmann equation gives ${\Omega }_M+{\Omega }_R+{\Omega }_{\Lambda }+{\Omega }_k=1$. Dividing the Friedmann acceleration equation by $-H^2$ givesConsequentlyThe deceleration parameter measures the sign and size of the acceleration relative to $a{H}^2$: $q>0$ means deceleration, $q<0$ accelerated expansion and $q=0$ instantaneously zero acceleration. In particular a falling Hubble parameter alone does not imply deceleration, since $q=-1-\dot{H}/H^2$.

For conformal time $d\eta =dt/a$, the conformal Hubble parameter is $\mathcal{H}=aH$. Using matter conservation and differentiating ${\Omega }_M\propto {\rho }_M/H^2$,This is the matter density flow with a cosmological constant. Neglecting ${\Omega }_{\Lambda }$ during matter domination reduces it to the requested relationThat last equation is not exact with nonzero $\Lambda$. For example, a flat dust-plus-vacuum state with $({\Omega }_M,{\Omega }_{\Lambda })=(0.9,0.1)$ gives ${\Omega }_M^{\prime }/\mathcal{H}=-0.27$, whereas the matter-only expression gives $-0.09$.

For the flatness problem, let ${\Omega }_{\mathrm{tot}}={\Omega }_M+{\Omega }_R+{\Omega }_{\Lambda }$. Curvature satisfies ${\Omega }_{\mathrm{tot}}-1=k/(aH{)}^2$, soIn decelerating near-flat radiation and matter eras, deviations therefore grow as $a^2$ and $a$ respectively. Maintaining small curvature at a late epoch requires very small early deviations unless a prior mechanism suppresses them. Cosmic inflation, with $q\simeq -1$, drives the deviation down approximately as $a^{-2}$. If vacuum energy is present, flatness means ${\Omega }_{\mathrm{tot}}=1$, not ${\Omega }_M=1$.

Define neutral-hydrogen column density along a ray by $N=\int n_{\mathrm{HI}}d\ell$, in atoms per unit area. The neutral-hydrogen column-density distribution per redshift isso $f_{z}dNdz$ is the expected number of absorbers in those intervals along one sightline. A logarithmic column bin instead has distribution $N\ln (10)f_z$; the bin convention must be specified.

Let $n_c$ be a constant comoving number density and ${\sigma }_p$ a constant physical interception area. The corresponding physical number density is $n_p=n_c(1+z{)}^3$. Proper light-path length satisfies $d\ell =c|dt|=cdz/[(1+z)H(z)]$. The expected intersection count, or first-order probability in infinitesimal $dz$, is thereforeThe Friedmann equation, the constant-equation-of-state density scaling for pressureless matter, and the present cosmological density parameters giveCombining them yields the absorber incidence and comoving number density relationHere the constant cross-section is physical, as required for this result; a fixed comoving cross-section would instead decrease in physical area as $(1+z{)}^{-2}$. It is convenient to define absorption distance by $dX/dz=(H_0/H(z))(1+z{)}^2$ and $f_X=f_z/(dX/dz)$, so a population with these non-evolution assumptions has constant incidence per $X$.

Use natural units with $c=1$, write $H=\dot{a}/a$, and measure cosmic time from the Big Bang. The cosmological continuity equation follows by differentiating the Friedmann equation and using the Friedmann acceleration equation; for pressureless matter it gives $\dot{\rho }+3H\rho =0$, hence $\rho ={\rho }_0{a}^{-3}$ when the present scale factor is $a_0=1$. The present critical density and cosmological density parameter giveConsequently, with $C=H_0^2{\Omega }_0$, the Friedmann equation becomes ${\dot{a}}^2=C/a-k$.

Introduce conformal time through $dt=ad\tau$. A prime denotes a derivative with respect to $\tau$, so $a^{\prime }=a\dot{a}$ andSet $u=\sqrt{k}\tau$, with $u=0$ at the Big Bang. On the expanding branch the substitution $a=C(1-\cos u)/(2k)$ satisfies this equation. Integrating $dt=adu/\sqrt{k}$, with $t(0)=0$, gives the closed matter-dominated Friedmann solution:The parametrization continues smoothly through the maximum scale factor, reached at $u=\pi$, onto the contracting branch. Its curve in the $(t,a)$ plane is a rescaled cycloid.

For a presently expanding universe, $H_0>0$ selects $0<u_0<\pi$. The normalization $a(u_0)=1$ givesSubstitution yields the age of a closed dust universe:The first later zero of the scale factor occurs at $u=2\pi$, so the lifetime of a closed dust universe gives the Big Crunch timeThis is the total lifetime measured from the Big Bang; the remaining cosmic time is $t_{\mathrm{BC}}-t_0$. As a check, the limit ${\Omega }_0\downarrow 1$ gives $H_0{t}_0\to 2/3$, while the Big Crunch recedes to infinite time, as expected for a flat matter-dominated universe with zero cosmological constant.

There is a conformal-time normalization error in the printed density parameter. For physical background density, $H=\mathcal{H}/a$, so the cosmological density parameter isThe printed definition lacks the numerator's $a^2$. The scale factor in a conformal-time density parameter is essential for ${\Omega }_c\simeq 1$ in the intended flat matter-dominated approximation.

During matter domination, pressureless matter supplies both the expansion and the gravitational source: $a\propto {\tau }^2$, $\mathcal{H}=2/\tau$ and ${\Omega }_c\simeq 1$ when CDM is taken to constitute that matter. ThusTry ${\delta }_c\propto {\tau }^p$. Its characteristic polynomial is $p(p-1)+2p-6=(p-2)(p+3)$, givingRadiation-era photon pressure and radiation perturbations affect the metric source, and matter alone does not dominate the expansion, so this matter-only closure and its power laws are inadequate before equality. At late acceleration the growth also ceases to follow $a\propto {\tau }^2$ and slows. The question's claim that the differential form itself is valid only in matter domination is too strong: for subhorizon pressureless matter with smooth dark energy it still applies with the correct time-dependent $\mathcal{H}$ and ${\Omega }_m$. Indeed, the pressureless growth equation in proper time is $\ddot{\delta }+2H\dot{\delta }-4\pi G{\bar{\rho }}_m\delta =0$ for smooth dark energy; substituting $d/d{t}_{\mathrm{proper}}=a^{-1}d/d\tau$ gives ${\delta }^{\prime \prime }+\mathcal{H}{\delta }^{\prime }-4\pi G{a}^2{\bar{\rho }}_m\delta =0$, the same conformal differential form. It is the matter-dominated coefficients and simple solutions that are restricted to the stated interval.

Outside the radiation-era horizon, the regular growing adiabatic mode has nearly constant curvature, while its comoving-synchronous density perturbation begins at gradient order $k^2{\tau }^2$. Equivalently its metric trace grows as ${\tau }^2$ and the volume relation in (i) gives ${\delta }_c\propto {\tau }^2$. This is a relativistic superhorizon solution, not instantaneous Newtonian communication between causally disconnected regions. Inside the horizon, pressure makes the dominant radiation oscillate and its gravitational potential decay, while CDM self-gravity is still subdominant. CDM then grows only weakly, logarithmically in the more accurate Mészáros effect; the stated freezing approximation replaces that slow growth by a constant.

Assume all modes being compared start outside the horizon at common ${\tau }_i$, and define ${\tau }_h=2\pi /k$, $k_{\mathrm{eq}}=2\pi /{\tau }_{\mathrm{eq}}$ and $G_0=({\tau }_0/{\tau }_i{)}^2$. Modes with $k\le k_{\mathrm{eq}}$ never undergo the radiation-era frozen interval, so their growth is $G_0$. For $k>k_{\mathrm{eq}}$, multiply growth before horizon entry, freezing until equality, and subsequent matter growth:Thus the frozen-radiation approximation to the matter transfer function isThis $T$ includes total growth because the question defines it directly between two density contrasts. The usual transfer normalized to one on large scales is $T/G_0$.

Power is the variance of a Fourier amplitude, so $P_0=|T{|}^2{P}_i$. The prescribed primordial density spectrum givesIt turns over at the comoving equality-horizon scale $2\pi /k_{\mathrm{eq}}={\tau }_{\mathrm{eq}}$. The turnover and reduced small-scale growth help organize the broad spectrum of galaxy clustering, clusters and superclusters; they do not select one unique object size. The dimensionless power $k^3{P}_0/(2{\pi }^2)$ is approximately flat above equality in this crude approximation, so a decreasing dimensional spectrum does not imply no small-scale structures. Restoring radiation-era logarithmic growth adds logarithmic corrections to the small-scale tail. Baryon acoustic features, a smooth equality transition, late growth suppression and nonlinear halo formation are absent from this simple linear broken-power approximation.

The fluid conservation equation is $\dot{\rho }+3H(\rho +P/c^2)=0$. With the radiation equation of state, it becomes $\dot{\rho }+4H\rho =0$, giving $\rho a^4={\rho }_0$. Define the critical density ${\rho }_{\mathrm{crit}}=3H^2/(8\pi G)$ and cosmological density parameter $\Omega =\rho /{\rho }_{\mathrm{crit}}$. Since $a_0=1$,Evaluating the Friedmann equation today gives $H_0^2+k{c}^2=H_0^2{\Omega }_0$, hence

For pressureless matter, the Friedmann acceleration equation with vanishing cosmological constant isEvaluating at the present cosmic time and using the definition of the deceleration parameter givesThe denominator is the critical density, so the cosmological density parameter satisfies the dust relationThis uses pressureless matter and $\Lambda =0$; spatial flatness has not been assumed.

For fixed cosmological density parameters, the redshift drift has the form $\dot{z}=H_{0}F(z)$. Multiplying the Hubble constant changes the vertical scale of all three sketches but leaves their zeros unchanged. Thus the drift-zero redshift is independent of $H_0$ at fixed density fractions. Every model has the trivial $z=0$ zero; only the specified mixed model has a further positive zero, $z_{\ast }\simeq 2.089$. If instead physical matter and vacuum densities are held fixed while $H_0$ changes, their density fractions change too, so that is a different family of models.

Let $H=\dot{a}/a$ be the Hubble parameter, and normalize the scale factor by $a(t_0)=1$. The cosmological continuity equation gives ${\rho }_M={\rho }_{M0}{a}^{-3}$ for pressureless matter, whereas the cosmological constant has constant energy density ${\rho }_{\Lambda }$. The spatially flat Friedmann equation therefore becomesThe present Hubble constant is $H_0=H(t_0)$. In terms of the present critical density ${\rho }_{\mathrm{crit},0}=3H_0^2/(8\pi G)$, the matter cosmological density parameter is ${\Omega }_M={\rho }_{M0}/{\rho }_{\mathrm{crit},0}$. Spatial flatness gives ${\rho }_{\Lambda }/{\rho }_{\mathrm{crit},0}=1-{\Omega }_M$, so