The Dark-Energy-Dominated Era

Every energy component in an expanding universe dilutes as the space it fills grows, and the rate is fixed entirely by one number, the equation-of-state parameter $w$ in $p=w\rho c^2$ . Solving the continuity equation gives $\rho\propto a^{-3(1+w)}$ : radiation ( $w=1/3$ ) thins as $a^{-4}$ , matter ( $w=0$ ) as $a^{-3}$ , and a component with $w=-1$ does not thin at all. That single fact makes the dark-energy-dominated era inevitable: however subdominant a non-diluting vacuum energy is early on, it cannot be outrun by things falling toward zero, so it must eventually become the whole budget — and the expansion it drives is qualitatively unlike anything before it. The dynamical signature of the era is cosmic acceleration. The second Friedmann equation, $\ddot a/a=-\tfrac{4\pi G}{3}(\rho+3p/c^2)$ , says that in general relativity pressure gravitates alongside density, and the source of cosmic gravity turns repulsive whenever $\rho+3p/c^2<0$ , i.e. $w<-1/3$ — a violation of the strong energy condition that ordinary matter and radiation cannot supply but dark energy with $w\approx-1$ supplies comfortably. The 1998 discovery of this acceleration in the dimming of high-redshift Type Ia supernovae converted the cosmological constant from a discarded fudge factor into the dominant component of the universe and the cornerstone of the $\Lambda$ CDM concordance model. If dark energy is a pure cosmological constant, the far future is a fixed point: matter and radiation dilute to nothing, the Hubble rate freezes at $H_\Lambda=\sqrt{\Lambda c^2/3}$ , the scale factor grows exponentially, and spacetime relaxes toward maximally symmetric de Sitter geometry — carrying a cosmological event horizon at proper distance $c/H_\Lambda$ beyond which the observable universe empties itself rather than growing. It is the deepest structural break in the whole history of cosmic expansion.

Key takeaways

The dark-energy-dominated era is the present and future epoch — roughly $z\lesssim0.3$ — in which a component with negative pressure, $w\approx-1$ , controls the cosmic budget and drives accelerated expansion, $\ddot a>0$ .
Because $\rho\propto a^{-3(1+w)}$ , a cosmological constant ( $w=-1$ ) has strictly constant density while matter and radiation dilute to zero, so it is guaranteed to dominate at late times no matter how small it started.
Acceleration requires $w<-1/3$ , equivalently $\rho+3p/c^2<0$ — a violation of the strong energy condition that no ordinary fluid can achieve.
Established observationally by Type Ia supernovae in 1998 and pinned down by the CMB and baryon acoustic oscillations; the concordance value is $\Omega_{\Lambda,0}\approx0.69$ with $q_0\approx-0.53$ .
The pure- $\Lambda$ future is de Sitter space: $a\propto e^{H_\Lambda t}$ with $H_\Lambda\approx56$ km/s/Mpc, a fixed event horizon at $c/H_\Lambda\approx17$ Gly, and a horizon temperature $T\sim2\times10^{-30}$ K.
Key idea: because a component's density scales as $\rho\propto a^{-3(1+w)}$ , anything with $w<-1/3$ both dilutes slower than matter (so it eventually dominates) and produces $\ddot a>0$ (so it accelerates the expansion); a cosmological constant with $w=-1$ is the limiting case, constant density driving asymptotic exponential de Sitter expansion.

Where this lives today

Measuring the dark-energy equation of state is the central goal of a generation of cosmological surveys, and the stakes are the fate of the universe itself: a pure cosmological constant demands $w=-1$ exactly; quintessence predicts $w>-1$ ; phantom energy, $w<-1$ , would end in a Big Rip. Current data pin $w=-1$ to a few percent for the tightest CMB + BAO + supernova combinations, consistent with a bare $\Lambda$ — but only just. In 2024 the DESI collaboration reported that combining its baryon-acoustic-oscillation measurements with supernovae and CMB data mildly favours a $w$ that evolves with redshift over a constant $\Lambda$ , at the two-to-four-sigma level — a possible first crack in the cosmological constant. The Euclid space telescope, the Vera C. Rubin Observatory's LSST, and the Nancy Grace Roman telescope are all built to map the expansion history precisely enough to measure $w(z)$ at the percent level — and so to decide whether the universe ends as a cold de Sitter emptiness, a Big Rip, or something not yet written down.

The physical picture

Picture the three cosmic ingredients as runners in a race whose track keeps stretching. Radiation tires fastest — its density falls as $a^{-4}$ , one power from volume dilution and one more because expansion redshifts every photon. Matter falls as $a^{-3}$ , pure volume dilution of conserved particles. A cosmological constant never tires: its density is a property of empty space, so doubling the volume gives twice as much vacuum at the same density, $a^0$ . When two runners slow toward a halt and one holds a constant pace, the eventual leader is settled before the gun fires — the dark-energy era is simply "after the tireless runner takes the lead," and the first figure shows where that overtake happens.

Figure

Densities dilute at different rates set by w: radiation ∝ a⁻⁴, matter ∝ a⁻³, dark energy ∝ a⁰. The flat Λ line must eventually top the others — it overtakes matter at scale factor a ≈ 0.77, redshift z ≈ 0.3, and has run the expansion ever since

Negative pressure is the strange but indispensable second ingredient. In general relativity the active gravitational source is $\rho+3p/c^2$ , so pressure gravitates too; push it sufficiently negative and the combination flips sign — gravity becomes a push. A cosmological constant is the extreme case of a uniform vacuum energy whose tension $p_\Lambda=-\rho_\Lambda c^2$ so overwhelms its density in the source term that it drives spacetime apart. Once that repulsion wins, the scale factor stops behaving like a power of time and grows exponentially. The second figure traces that transition through the actual expansion history of our universe.

Figure

Scale factor a(t) for flat ΛCDM: concave-down and decelerating until t ≈ 7.7 Gyr (z ≈ 0.6), where the curve inflects and turns concave-up — acceleration begins, earlier than Λ overtakes matter at z ≈ 0.3. The far future is exponential, a ∝ exp(H_Λ t) with H_Λ ≈ 56 km/s/Mpc and a fixed event horizon at c/H_Λ ≈ 17 Gly

The core equations

The whole era is governed by the two Friedmann equations plus one equation of state. The dynamical statement — the one that defines the era — is the acceleration (second Friedmann) equation, which fixes the sign of $\ddot a$ from the density and pressure.

\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)+\frac{\Lambda c^2}{3} \qquad (1)

where: $a(t)$ — scale factor of the FLRW metric; $\rho$ — mass-energy density of matter plus radiation; $p$ — their total pressure; $\Lambda$ — cosmological constant; $G,\,c$ — Newton's constant and the speed of light. Reading: the density-plus-pressure term always decelerates, a positive $\Lambda$ always accelerates; the era is when the second wins. Each component's density is set by its equation of state through energy conservation. This is the scaling that makes dark-energy dominance inevitable.

\rho_i(a)=\rho_{i,0}\,a^{-3(1+w_i)},\qquad p_i=w_i\,\rho_i c^2 \qquad (2)

where: $\rho_{i,0}$ — present-day density of component $i$ (with $a_0=1$ ); $w_i$ — its equation-of-state parameter. Reading: $w=1/3\to a^{-4}$ (radiation), $w=0\to a^{-3}$ (matter), $w=-1\to a^0$ (constant). A cosmological constant is exactly a perfect fluid with a specific, unchanging equation of state. Absorbing $\Lambda$ into an effective vacuum fluid makes equation (1) a single-source acceleration equation.

\rho_\Lambda=\frac{\Lambda c^2}{8\pi G},\qquad p_\Lambda=-\rho_\Lambda c^2,\qquad w_\Lambda=-1 \qquad (3)

where: $\rho_\Lambda$ — constant vacuum energy density; $p_\Lambda$ — its negative pressure. Reading: because $p_\Lambda=-\rho_\Lambda c^2$ , the source term $\rho_\Lambda+3p_\Lambda/c^2=-2\rho_\Lambda<0$ — the vacuum antigravitates. When matter and radiation have diluted away, the first Friedmann equation leaves only $\rho_\Lambda$ , and the solution is de Sitter space with a constant Hubble rate and exponential growth.

H_\Lambda=\sqrt{\frac{\Lambda c^2}{3}}=\sqrt{\frac{8\pi G\rho_\Lambda}{3}},\qquad a(t)\propto e^{H_\Lambda t} \qquad (4)

where: $H_\Lambda$ — asymptotic constant Hubble parameter. Reading: a single number sets the whole future — the e-folding time $1/H_\Lambda$ , the horizon $c/H_\Lambda$ , and horizon temperature all follow. For the real, mixed universe the expansion history is packaged in the Hubble rate as a function of redshift, the master relation that every distance measurement is compared against.

H^2(z)=H_0^2\left[\Omega_{r,0}(1+z)^4+\Omega_{m,0}(1+z)^3+\Omega_{k,0}(1+z)^2+\Omega_{\Lambda,0}\right] \qquad (5)

where: $\Omega_{i,0}$ — present density parameters for radiation, matter, curvature and $\Lambda$ ; $z$ — redshift, $1+z=1/a$ ; $H_0$ — present Hubble constant. Reading: the four terms carry the four scalings; at low $z$ the constant $\Omega_{\Lambda,0}$ term dominates. Exponential growth converges the future light travel integral, giving the era its most dramatic feature — a fixed cosmological event horizon.

d_{\rm EH}=a(t_0)\int_{t_0}^{\infty}\frac{c\,\mathrm{d}t}{a(t)}\;\xrightarrow{\ \text{de Sitter}\ }\;\frac{c}{H_\Lambda} \qquad (6)

where: $d_{\rm EH}$ — proper distance to the cosmological event horizon; $t_0$ — the present time. Reading: in matter domination this integral diverges (no horizon); in de Sitter it converges to $c/H_\Lambda$ . Finally, the measurable that first announced the era: the deceleration parameter, whose sign defines acceleration.

q_0\equiv-\left.\frac{\ddot a\,a}{\dot a^2}\right|_0=\Omega_{r,0}+\tfrac12\Omega_{m,0}-\Omega_{\Lambda,0}\approx-0.53 \qquad (7)

where: $q_0$ — present deceleration parameter. Reading: $q_0<0$ means the expansion accelerates today — the observational fingerprint of dark energy.

Derivation ADe Sitter expansion from a pure cosmological constant

The goal is to show, from the first Friedmann equation alone, that once matter and radiation have diluted to negligibility the scale factor grows strictly exponentially at a constant Hubble rate — the de Sitter fixed point that every $\Lambda$ -dominated future flows toward. We take the late-time limit in which only the vacuum energy survives, integrate the resulting first-order equation, and check that the exponential solution is genuinely accelerating and consistent with the second Friedmann equation.

Step 1. Write the flat first Friedmann equation $(\dot a/a)^2=\tfrac{8\pi G}{3}\rho_{\rm tot}$ with $\rho_{\rm tot}=\rho_r+\rho_m+\rho_\Lambda$ . As $t\to\infty$ the scale factor grows without bound, so $\rho_r\propto a^{-4}\to0$ and $\rho_m\propto a^{-3}\to0$ , leaving $(\dot a/a)^2\to\tfrac{8\pi G}{3}\rho_\Lambda=\tfrac{\Lambda c^2}{3}$ . Why: the whole point of the era is that the two diluting species vanish and only the non-diluting vacuum term is left standing.

Step 2. The right-hand side is now a constant, so the Hubble rate is constant: $H\equiv\dot a/a=\sqrt{\Lambda c^2/3}\equiv H_\Lambda$ . Why: a constant $H$ is the defining property of de Sitter space and is what makes the future so simple — one number governs everything that follows.

Step 3. Integrate $\dot a/a=H_\Lambda$ , a first-order linear equation with constant coefficient: $\mathrm{d}(\ln a)=H_\Lambda\,\mathrm{d}t$ gives $\ln a=H_\Lambda t+\text{const}$ , hence $a(t)=a_*\,e^{H_\Lambda(t-t_*)}$ . Why: constant fractional growth rate is exactly the statement that the quantity grows exponentially — the signature that separates the dark-energy era from every power-law era before it.

Step 4. Confirm acceleration and self-consistency. Differentiating twice gives $\ddot a=H_\Lambda^2\,a>0$ . Checking against equation (1): with $\rho+3p/c^2=\rho_\Lambda-3\rho_\Lambda=-2\rho_\Lambda$ , the acceleration equation returns $\ddot a/a=-\tfrac{4\pi G}{3}(-2\rho_\Lambda)=\tfrac{8\pi G}{3}\rho_\Lambda=H_\Lambda^2$ , matching. Why: the first and second Friedmann equations agree, so the exponential is a genuine solution, not an artefact of dropping terms.

Go deeper · Derivation B · 1 minThe acceleration threshold and why negative pressure gravitates

Derivation BThe acceleration threshold and why negative pressure gravitates

The goal is to locate the exact equation-of-state value that separates a decelerating universe from an accelerating one, and to show why it is $w=-1/3$ rather than $w=0$ — that is, why acceleration demands not merely low density but genuinely negative pressure. We reduce the acceleration equation to a single fluid, read off the sign condition, and integrate to get the power-law family that connects smoothly to the exponential de Sitter case.

Step 1. Take a flat universe with one fluid $p=w\rho c^2$ and no separate $\Lambda$ term. Substituting into equation (1), $\ddot a/a=-\tfrac{4\pi G}{3}(\rho+3w\rho)=-\tfrac{4\pi G}{3}\rho(1+3w)$ . Since $\rho>0$ , the sign of $\ddot a$ is carried entirely by the factor $(1+3w)$ . Why: this isolates the one algebraic combination that decides braking versus boosting, stripping away every other detail of the model.

Step 2. Acceleration therefore requires $1+3w<0$ , i.e. $w<-\tfrac13$ . This is precisely the condition $\rho+3p/c^2<0$ , a violation of the strong energy condition. Why: it makes explicit that ordinary matter ( $w=0$ ) and radiation ( $w=1/3$ ) can only decelerate — you cannot accelerate the universe by removing gravitating matter, only by adding something that gravitates the wrong way.

Step 3. Solve for the growth. Energy conservation gives $\rho\propto a^{-3(1+w)}$ , so the Friedmann equation reads $\dot a^2\propto a^{-(1+3w)}$ ; integrating yields $a\propto t^{2/[3(1+w)]}$ for $w\neq-1$ . The familiar cases fall out: $w=1/3\Rightarrow a\propto t^{1/2}$ (radiation) and $w=0\Rightarrow a\propto t^{2/3}$ (matter), both decelerating. Why: recovering the standard scalings verifies the machinery and shows the threshold sits between matter and the vacuum.

Step 4. Read the dark-energy window. For $-1<w<-\tfrac13$ the exponent $2/[3(1+w)]$ exceeds one, so $a$ grows faster than linearly — accelerating power law. As $w\to-1$ that exponent diverges, and the power law smoothly hands over to the exponential $a\propto e^{H_\Lambda t}$ of the previous derivation. Why: it stitches quintessence ( $w>-1$ ) continuously onto the pure cosmological constant ( $w=-1$ ), so the whole family of accelerating fates is one connected picture.

Go deeper · Derivation C · 1 minThe finite event horizon of an eternally accelerating universe

Derivation CThe finite event horizon of an eternally accelerating universe

The goal is to show that de Sitter expansion creates a cosmological event horizon — a fixed proper distance beyond which no future signal from us can ever reach a comoving observer — and to contrast this with matter domination, which has no such horizon. We write the comoving distance a photon covers from now to the infinite future, evaluate it for exponential growth, convert to proper distance today, and show why the same integral diverges when the expansion decelerates.

Step 1. A radial photon travels comoving distance $\int_{t_0}^{\infty} c\,\mathrm{d}t/a(t)$ ; the greatest proper distance from which light emitted after $t_0$ can ever reach us is the event horizon $d_{\rm EH}=a(t_0)\int_{t_0}^{\infty} c\,\mathrm{d}t/a(t)$ . Why: an event horizon exists only if this future-directed integral converges — if it diverges, light from every comoving source eventually arrives and there is no horizon.

Step 2. In de Sitter, $a(t)=a(t_0)\,e^{H_\Lambda(t-t_0)}$ , so $\int_{t_0}^{\infty} c\,\mathrm{d}t/a(t)=\tfrac{c}{a(t_0)H_\Lambda}$ . The exponential in the denominator makes the tail integrable — the integral converges even over an infinite future. Why: constant-rate growth outruns light so decisively that the total future reachable comoving distance is finite, unlike any decelerating era.

Step 3. Multiply by $a(t_0)$ to get the proper horizon distance $d_{\rm EH}=c/H_\Lambda$ , a fixed length set entirely by the asymptotic Hubble rate. Why: this is the sharp geometric fact of the era — the observable universe asymptotes to a sphere of fixed proper radius, and comoving galaxies drift permanently outside it.

Step 4. Contrast matter domination, $a\propto t^{2/3}$ : then $\int^{\infty}\mathrm{d}t/t^{2/3}\propto t^{1/3}\big|^{\infty}$ diverges, so there is no event horizon and every source is eventually seen. Why: the event horizon is a specific consequence of acceleration, appearing exactly when — and because — $\ddot a>0$ persists forever.

Limiting cases

The dark-energy era is one band in a spectrum of equation-of-state values, and the boundaries of that band are as physically meaningful as its centre.

Case	Result	Meaning
Pure cosmological constant	$w=-1,\ \rho_\Lambda=\text{const},\ a\propto e^{H_\Lambda t}$	Exact de Sitter limit: density never dilutes, expansion is strictly exponential, and a fixed event horizon $c/H_\Lambda$ forms.
Quintessence (slow-roll scalar)	$-1<w<-\tfrac13,\ a\propto t^{2/[3(1+w)]}$	Still accelerating but the density slowly dilutes; power-law rather than exponential growth, and the event horizon may not be eternal.
Acceleration threshold	$w=-\tfrac13,\ \ddot a=0$	Marginal case $\rho+3p/c^2=0$ : for $w>-\tfrac13$ the expansion decelerates (matter, radiation), for $w<-\tfrac13$ it accelerates.
Phantom energy	$w<-1,\ \rho\to\infty$ at finite $t$	Super-acceleration: density grows with time, ending in a Big Rip where $a\to\infty$ in finite proper time and all bound structures are torn apart.

Worked examples

1 · When the universe started accelerating, and when Λ took over (fully worked)

Using the Planck concordance values $\Omega_{m,0}=0.31$ , $\Omega_{\Lambda,0}=0.69$ (radiation negligible), find the redshift where acceleration began, the redshift where dark energy overtook matter, and the present deceleration parameter. Comment on their ordering.

Step 1. Dark energy overtakes matter when their densities are equal, $\Omega_{m,0}(1+z)^3=\Omega_{\Lambda,0}$ , so $1+z_{\rm eq}=(\Omega_{\Lambda,0}/\Omega_{m,0})^{1/3}=(0.69/0.31)^{1/3}=(2.226)^{1/3}=1.306$ . Thus $z_{\rm eq}\approx0.31$ — dark energy has dominated only since a cosmically recent $z\approx0.3$ .

Step 2. Acceleration begins when $\ddot a=0$ . With matter ( $p_m=0$ ) plus $\Lambda$ , the source term $\rho_m+\rho_\Lambda-3\rho_\Lambda$ vanishes when $\rho_m=2\rho_\Lambda$ . Hence $\Omega_{m,0}(1+z)^3=2\Omega_{\Lambda,0}$ , giving $1+z_{\rm acc}=(2\Omega_{\Lambda,0}/\Omega_{m,0})^{1/3}=(4.452)^{1/3}=1.645$ , so $z_{\rm acc}\approx0.65$ .

Step 3. The present deceleration parameter is $q_0=\Omega_{r,0}+\tfrac12\Omega_{m,0}-\Omega_{\Lambda,0}\approx\tfrac12(0.31)-0.69=\mathbf{-0.535}$ — the negative value is the direct statement that the expansion accelerates today.

Step 4. The ordering $z_{\rm acc}\approx0.65>z_{\rm eq}\approx0.31$ is the key result: acceleration began before dark energy became the majority component, because $\Lambda$ 's pressure contributes to $\ddot a$ with three times the weight of its density. In time, acceleration set in at $t\approx7.7$ Gyr against a present age $t_0\approx13.8$ Gyr — the second half of cosmic history has been accelerating.

2 · The de Sitter future — event horizon and horizon temperature (fully worked)

Take $H_0=67.4$ km/s/Mpc and $\Omega_{\Lambda,0}=0.69$ . Find the asymptotic Hubble rate $H_\Lambda$ , the fixed event-horizon distance $c/H_\Lambda$ , and the de Sitter horizon temperature $T=\hbar H_\Lambda/(2\pi k_B)$ .

Step 1. As matter dilutes away, $H\to H_\Lambda=H_0\sqrt{\Omega_{\Lambda,0}}=67.4\times\sqrt{0.69}=\mathbf{56.0\ km/s/Mpc}$ , with e-folding time $1/H_\Lambda=(1/H_0)/\sqrt{\Omega_{\Lambda,0}}=14.5/0.8307\approx\mathbf{17.5\ Gyr}$ .

Step 2. In SI, $H_\Lambda=1.81\times10^{-18}\ \mathrm{s^{-1}}$ , so the event-horizon distance is $d_{\rm EH}=c/H_\Lambda=(3.00\times10^8)/(1.81\times10^{-18})=1.65\times10^{26}\ \mathrm{m}\approx\mathbf{17.5\ Gly}\ (5.4\ \mathrm{Gpc})$ . Any galaxy that recedes past this radius is lost from causal contact forever.

Step 3. The horizon carries a Gibbons–Hawking temperature $T=\dfrac{\hbar H_\Lambda}{2\pi k_B}=\dfrac{(1.055\times10^{-34})(1.81\times10^{-18})}{2\pi(1.381\times10^{-23})}\approx\mathbf{2.2\times10^{-30}\ K}$ .

Step 4. Interpret the scale. This temperature is colder than the present CMB ( $2.7$ K) by thirty orders of magnitude, so the horizon radiates a negligible whisper today — yet its existence links cosmic acceleration to horizon thermodynamics and a de Sitter entropy $S=k_B c^3 A/(4G\hbar)$ . The far future is a fixed, empty, $10^{-30}$ K de Sitter bubble of radius $17.5$ Gly.

Physics of the portrait

The dark-energy era's mathematics was written down before anyone knew the era was real. In 1917 Einstein added a cosmological constant to his field equations to hold the universe static; in the same year Willem de Sitter found a very different solution to those augmented equations — an empty universe that, far from being static, expands without end. For eighty years that solution looked like a mathematical curiosity. Then the supernovae came in, and de Sitter's empty accelerating cosmos turned out to be the destiny of ours.

Origins · 1 minWillem de Sitter and how this was discovered

Willem de Sitter

1872–1934 · cosmology and the vacuum solutions of general relativity

Willem de Sitter was a Dutch mathematician and astronomer, director of Leiden Observatory, and one of the first to grasp the cosmological reach of Einstein's new gravity. In 1917, weeks after Einstein introduced the cosmological constant $\Lambda$ to force a static universe, de Sitter published a second solution of the same field equations: a spacetime containing no matter at all, only $\Lambda$ , that nonetheless expands exponentially. Contemporaries found it puzzling — it seemed to predict that light from distant test particles would be systematically reddened (the "de Sitter effect"), a foreshadowing of cosmological redshift years before Hubble's 1929 measurements. He also collaborated with Einstein on the 1932 Einstein–de Sitter model, the flat matter-only universe that was the default cosmology for half a century. History inverted the emphasis: the matter-only model he lent his name to describes the past, the empty $\Lambda$ solution he found almost as an aside the future. De Sitter space — maximally symmetric, constant curvature, constant Hubble rate — is now the asymptotic state of the accelerating universe, the fixed point a $\Lambda$ -dominated cosmos flows toward as its matter dilutes to nothing. His empty world, once a paradox, is where everything is headed.

History

1917

— Einstein adds the cosmological constant Λ; de Sitter finds the empty, exponentially expanding vacuum solution

1929

— Hubble establishes cosmic expansion, and Λ falls out of favour as an unnecessary complication

1998

— the Riess and Perlmutter teams find cosmic acceleration in high-redshift Type Ia supernovae

2011

— the Nobel Prize in Physics is awarded for the discovery of the accelerating expansion

Going further

Texts. Carroll, Spacetime and Geometry, ch. 8 (Cosmology), §8.3–8.5 · Wald, General Relativity, ch. 5 and §5.2 on de Sitter space · Misner, Thorne & Wheeler, Gravitation, §27 · Hartle, Gravity, ch. 18–19 · Schutz, A First Course in General Relativity, ch. 12. Primary literature: Riess et al. (1998) AJ 116, 1009; Perlmutter et al. (1999) ApJ 517, 565; Planck 2018 Results VI, A&A 641, A6. The key results — $\rho\propto a^{-3(1+w)}$ from the continuity equation, the $w<-1/3$ acceleration threshold, $a\propto e^{H_\Lambda t}$ for $w=-1$ , and the convergence of the future horizon integral to $c/H_\Lambda$ — are derived in the sections above. The numerical values ( $\Omega_{\Lambda,0}\approx0.69$ , $\Omega_{m,0}\approx0.31$ , $z_{\rm eq}\approx0.3$ , $z_{\rm acc}\approx0.6$ , $q_0\approx-0.53$ , $t_0\approx13.8$ Gyr, $t_{\rm acc}\approx7.7$ Gyr, $H_\Lambda\approx56$ km/s/Mpc, $c/H_\Lambda\approx17.5$ Gly, $T\approx2\times10^{-30}$ K) are Planck-2018 concordance numbers, all recomputable from the quoted density parameters via the $\Lambda$ CDM $H(z)$ relation and the exact flat matter-plus- $\Lambda$ solution $a(t)\propto\sinh^{2/3}(\tfrac32\sqrt{\Omega_\Lambda}H_0t)$ .

Builds on: The Friedmann Equations · The Cosmological Fluid and Equation of State · The Cosmological Constant and Dark Energy · Critical Density and the Density Parameters · The Scale Factor and Hubble Parameter Leads to: Cosmological Horizons · Cosmological Redshift · The Matter-Dominated Era

Ask the tutor: Ask me to work through why acceleration began before dark energy dominated, or to trace what a comoving galaxy looks like as it drifts across the de Sitter event horizon.

Is dark energy the same thing as the cosmological constant?

Not necessarily — the cosmological constant is the simplest candidate, but "dark energy" is the general name for whatever drives cosmic acceleration. A cosmological constant has $w=-1$ exactly and a density that never changes, and it is what $\Lambda$ CDM assumes. Dynamical alternatives such as quintessence (a slowly rolling scalar field) give $w$ slightly greater than $-1$ and possibly time-varying, while phantom models give $w<-1$ . All accelerate the universe because all satisfy $w<-1/3$ ; they differ in the far-future fate they imply. Current data are consistent with a bare cosmological constant to about ten percent, but distinguishing $\Lambda$ from a slowly evolving field is exactly what the next generation of surveys is built to do.

How can pressure cause gravity to push instead of pull?

In Newtonian gravity only mass density sources the field, but general relativity is sourced by the full stress-energy tensor, and for a homogeneous fluid the active gravitational density is $\rho+3p/c^2$ , not $\rho$ alone — pressure gravitates because it carries energy and momentum flux that curve spacetime. For ordinary matter and radiation the pressure is small or positive, so $\rho+3p/c^2$ stays positive and gravity attracts, decelerating the expansion. A cosmological constant has pressure so negative — $p=-\rho c^2$ — that $\rho+3p/c^2=-2\rho<0$ , flipping the sign of the acceleration term. Nothing about the energy is negative; it is the tension of the vacuum that turns the net gravitational effect into a push.

Does a redshift z ≈ 0.3 mean dark energy is a recent arrival?

It was always there, but only recently took over. Because the vacuum density is constant while matter dilutes as $a^{-3}$ , at high redshift matter vastly outweighed dark energy — at $z=1$ matter density was $(1+z)^3=8$ times its present value, dwarfing the unchanged $\Lambda$ . Only when the universe had expanded enough to dilute matter below the fixed vacuum density did dark energy dominate, at $z_{\rm eq}\approx0.3$ , a few billion years ago. So dark energy is not new; what is new is its dominance — and, run forward, that dominance only deepens, since matter keeps diluting and $\Lambda$ never does.

What is the difference between the onset of acceleration and dark-energy dominance?

They are two different thresholds, and acceleration comes first. Dark-energy dominance is about density — it happens when $\rho_\Lambda$ first exceeds $\rho_m$ , at $z\approx0.3$ . Acceleration is about the sign of $\ddot a$ , which flips earlier, when $\rho_m=2\rho_\Lambda$ (matter still twice as dense as dark energy), at $z\approx0.6$ . The reason is the factor of three on pressure in the acceleration equation: the vacuum's negative pressure counts three times over in $\ddot a$ , so it reverses the acceleration before it wins the density contest. The universe therefore spent the interval $z\approx0.6$ to $0.3$ accelerating while matter was still the majority component.

Will the accelerating universe end in a Big Rip?

Only if dark energy is phantom, with $w<-1$ . Then the vacuum density grows with time, the Hubble rate diverges, and the scale factor reaches infinity in a finite time, tearing apart galaxies, stars, planets, and eventually atoms — the Big Rip. For a pure cosmological constant ( $w=-1$ ) there is no Rip: the density is constant, the expansion is eternally exponential, and the fate is a cold, empty de Sitter universe in which everything beyond the Local Group has crossed the event horizon and faded from view. Which fate awaits us depends on measuring whether $w$ sits exactly at $-1$ , just above, or just below.

If galaxies cross the event horizon, does that violate the speed of light?

No. Galaxies are not moving through space faster than light; the space between us and them is expanding, and recession velocity is the sum of many small local expansions along the line of sight. Once a galaxy is far enough that this cumulative recession exceeds $c$ , light it emits toward us is carried away faster than it can approach, and it passes beyond our event horizon. Special relativity forbids local motion faster than light, but places no limit on how fast the distance between two comoving points grows, because that growth is geometry, not a local velocity. This is exactly why an event horizon can form in an accelerating universe.