Equation of State of Dark Energy

Dark energy is the largest single entry in the cosmic ledger — about $70\%$ of everything — and its entire physical character is compressed into one dimensionless number: the equation-of-state parameter $w=p/\rho$ , the ratio of its pressure to its energy density. That ratio is not decoration. It appears in the fluid conservation law, where it fixes how the density dilutes as space expands, and it appears in the Friedmann acceleration equation, where general relativity insists that gravity is sourced by $\rho+3p$ rather than by $\rho$ alone. A component with pressure negative enough — $w<-1/3$ — therefore anti-gravitates, and the universe's expansion speeds up. This is the whole reason dark energy exists as a concept. The prize question of observational cosmology is whether $w$ is frozen at exactly $-1$ , the signature of a true cosmological constant $\Lambda$ (vacuum energy that never dilutes), or whether it drifts with time. A single measured departure — $w\neq-1$ today, or any detected evolution $\mathrm{d}w/\mathrm{d}a\neq0$ — would demolish the simplest picture and force a dynamical mechanism: a slowly rolling scalar field (quintessence), a modification of gravity on the largest scales, or something stranger still. So $w$ is a diagnostic of fundamental physics disguised as a bookkeeping constant. Because dark energy is smooth, we cannot weigh it locally the way rotation curves weigh dark matter. We read $w$ only from how it bends the expansion history $H(z)$ and slows the growth of structure — through standard candles (supernovae), standard rulers (baryon acoustic oscillations), and the high-redshift anchor of the microwave background. This article assembles the machinery that turns those distances into a constraint on $w$ and its evolution.

Key takeaways

The dark-energy equation of state is $w\equiv p/\rho$ (with $c=1$ ); it sets how the density evolves, $\rho\propto a^{-3(1+w)}$ , so $w=-1$ gives a constant density — the cosmological constant $\Lambda$ .
Cosmic acceleration requires $w<-1/3$ ; a smooth component with $w<-1$ ("phantom" energy) grows denser as the universe expands and can drive a future Big Rip.
A rolling scalar field (quintessence) has $w=(\tfrac12\dot\phi^2-V)/(\tfrac12\dot\phi^2+V)$ , confined to $-1\le w\le1$ , and $w$ generically evolves — which is precisely what would distinguish it from $\Lambda$ .
Surveys fit the low-parameter CPL form $w(a)=w_0+w_a(1-a)$ and report the pair $(w_0,w_a)$ ; the point $(-1,0)$ is an exact cosmological constant and is the bullseye every experiment aims at.
Current data (Planck + supernovae + BAO) sit within a few percent of $w=-1$ ; recent DESI BAO hints at mild evolution, but $\Lambda$ CDM remains the concordance baseline.
Key idea: the single number $w=p/\rho$ controls how dark-energy density dilutes with expansion via $\rho\propto a^{-3(1+w)}$ ; $w=-1$ is the fixed point of a true cosmological constant, and any measured evolution of $w$ would betray a dynamical field or modified gravity.

Where this lives today

Constraining $w$ is the organising goal of an entire generation of experiments. The Dark Energy Spectroscopic Instrument (DESI) is mapping tens of millions of galaxies and quasars to trace the baryon acoustic oscillation ruler across $0.1<z<3.5$ ; its 2024–2025 releases produced the field's most talked-about tension — a mild but persistent preference, when BAO is combined with supernovae and the CMB, for a CPL model with $w_0>-1$ and $w_a<0$ over pure $\Lambda$ , at the two-to-four-sigma level depending on the supernova sample. Whether this is the first crack in the cosmological constant or a subtle systematic is the live question. Alongside DESI, the Euclid space telescope and the Vera C. Rubin Observatory's LSST measure weak-lensing shear and the growth rate $f\sigma_8(z)$ , breaking the geometric degeneracies that distances alone cannot, and the Nancy Grace Roman Space Telescope will add a deep space-based supernova and BAO survey. The prize is not just a tighter error bar on $w_0$ but the shape of the $(w_0,w_a)$ contour and its Figure of Merit — the inverse area of that ellipse — because only a resolved evolution would unambiguously rule out $\Lambda$ . Behind the surveys, theorists still ask why the vacuum energy is $10^{120}$ times smaller than naive quantum-field estimates and why it dominates just now — the deep problems that make $w$ far more than a curve-fit.

The physical picture

Ask one question of every substance in the universe: when space doubles in size, does your density fall, stay put, or rise? The answer is dictated entirely by $w$ , because the continuity equation forces $\rho\propto a^{-3(1+w)}$ . Ordinary matter ( $w=0$ ) simply dilutes with volume, $\rho\propto a^{-3}$ . Radiation ( $w=1/3$ ) dilutes one power faster, $\rho\propto a^{-4}$ , because each photon also redshifts. A cosmological constant ( $w=-1$ ) is the strange fixed point where the exponent vanishes and the density never changes — new space arrives already stocked with vacuum energy. And phantom energy ( $w<-1$ ) is stranger yet: stretching space creates density, so dark energy runs away with the expansion. The first figure is nothing but this fan of dilution slopes, each slope being exactly $-3(1+w)$ .

Figure

Each component dilutes as its density is proportional to a to the power −3(1+w), so the slope on this log–log plot is exactly −3(1+w): radiation with w = 1/3 falls as a⁻⁴, matter with w = 0 as a⁻³, the cosmological constant with w = −1 stays perfectly flat, and phantom energy with w < −1 climbs toward the future

The engine behind all of this is negative pressure. In Newtonian gravity only mass attracts; in general relativity the active gravitational density is $\rho+3p$ . Give a substance enough tension — pressure so negative that $\rho+3p<0$ , i.e. $w<-1/3$ — and its gravity reverses sign, pushing galaxies apart faster and faster. A rolling scalar field makes this dynamical and intuitive: kinetic energy $\tfrac12\dot\phi^2$ behaves like stiff positive pressure ( $w\to+1$ ), potential energy $V(\phi)$ behaves like a temporary cosmological constant ( $w\to-1$ ), and a field pinned high on its potential by Hubble friction mimics $\Lambda$ until it begins to roll. The second figure shows the arena where all of this is decided — the $(w_0,w_a)$ plane, where every survey draws its error ellipse and asks whether the cosmological-constant point survives.

Figure

The observational target is the point where w₀ = −1 and wₐ = 0, an exact cosmological constant with no evolution; surveys shrink the joint error ellipse around their best fit and ask whether it excludes that point — a firm detection of w₀ ≠ −1 or wₐ ≠ 0 would rule out Λ and reveal dynamical dark energy

The core equations

The definition is the seed from which everything grows: the equation-of-state parameter is simply the ratio of the isotropic pressure to the energy density of the component. Everything else in this article is a consequence of feeding this ratio into the two governing equations of an expanding universe.

w \equiv \frac{p}{\rho c^2} \qquad (1)

where: $w$ — dimensionless equation-of-state parameter; $p$ — isotropic pressure of the component; $\rho$ — energy density (mass density $\times c^2$ ); $c$ — speed of light, routinely set to $1$ . Reading it: $w$ is negative only when the pressure is negative, i.e. when the substance is under tension rather than pushing outward. Local energy conservation in an expanding, homogeneous universe gives the fluid continuity equation. It is the cosmological first law of thermodynamics: the energy in a comoving volume changes only through the $p\,\mathrm{d}V$ work done as that volume grows.

\dot\rho + 3\frac{\dot a}{a}\,(\rho + p) = 0 \qquad (2)

where: $a$ — cosmic scale factor; $\dot a/a\equiv H$ — Hubble expansion rate; $\dot\rho$ — time derivative of energy density. Reading it: if $p=-\rho$ (that is $w=-1$ ), the bracket vanishes and $\dot\rho=0$ — the density is frozen no matter how fast space expands. Integrating (2) for a constant $w$ gives the master dilution law, the single most useful formula in this subject. It converts the abstract number $w$ into a concrete recipe for how density thins with expansion.

\rho(a) = \rho_0\, a^{-3(1+w)} \qquad (w=\text{const}) \qquad (3)

where: $\rho_0$ — present density (with $a_0=1$ ); the exponent $-3(1+w)$ is the slope drawn in the first figure. Reading it: $w=0\to a^{-3}$ , $w=\tfrac13\to a^{-4}$ , $w=-1\to$ constant, $w<-1\to$ growing. The second Friedmann equation governs acceleration. It is where negative pressure earns its keep: the source term is $\rho+3p$ , so a sufficiently negative $w$ flips the sign of $\ddot a$ and the expansion accelerates.

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

where: $G$ — Newton's constant; $\ddot a$ — second time derivative of the scale factor. Reading it: since $\rho>0$ , the sign of $\ddot a$ is the sign of $-(1+3w)$ , so $\ddot a>0$ demands $w<-1/3$ . When $w$ is allowed to vary, the constant- $w$ solution generalises to an integral over history. This is the form every cosmology code evaluates to build the expansion rate for a dynamical dark-energy model.

\rho_{\rm DE}(a) = \rho_{\rm DE,0}\,\exp\!\left[\,3\int_a^1 \frac{1+w(a')}{a'}\,\mathrm{d}a'\right] \qquad (5)

where: $w(a')$ — the possibly evolving equation of state; $\rho_{\rm DE,0}$ — present dark-energy density. Reading it: at each moment the local exponent is $-3[1+w(a)]$ , and the integral simply accumulates those instantaneous dilution rates. Rather than fit an arbitrary function, surveys adopt the two-parameter Chevallier–Polarski–Linder form. It is linear in the scale factor, well-behaved at high redshift, and captures the leading evolution with just a present value and a slope.

w(a) = w_0 + w_a(1-a) = w_0 + w_a\frac{z}{1+z} \qquad (6)

where: $w_0$ — present value $w(a{=}1)$ ; $w_a=-\,\mathrm{d}w/\mathrm{d}a|_{a=1}$ — evolution parameter; $z$ — redshift with $a=1/(1+z)$ . Reading it: as $z\to\infty$ ( $a\to0$ ), $w\to w_0+w_a$ , so $w_a$ sets the early-time asymptote. A canonical scalar field realises a dynamical $w$ microscopically, as the ratio of a kinetic minus potential term to a kinetic plus potential term. This is quintessence, and it is confined to the physical band $-1\le w_\phi\le1$ .

w_\phi = \frac{\tfrac12\dot\phi^2 - V(\phi)}{\tfrac12\dot\phi^2 + V(\phi)} \qquad (7)

where: $\phi$ — the scalar field; $\tfrac12\dot\phi^2$ — kinetic energy; $V(\phi)$ — potential energy. Reading it: slow roll ( $\dot\phi^2\ll V$ ) drives $w\to-1$ (mimicking $\Lambda$ ), while kinetic domination drives $w\to+1$ (a stiff fluid). Finally, all of this reaches the data through the expansion rate and the distances built on it. The Friedmann equation packages the species into $H(z)$ , and the distance integrals turn $H(z)$ into the observables that supernovae and BAO actually measure.

H^2(a) = H_0^2\!\left[\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_{\rm DE}\,f(a)\right],\qquad D_L(z)=(1+z)\,c\!\int_0^z\!\frac{\mathrm{d}z'}{H(z')} \qquad (8)

where: $H_0$ — present Hubble constant; $\Omega_i$ — present density parameters; $f(a)=\rho_{\rm DE}(a)/\rho_{\rm DE,0}$ carries the $w$ -dependence; $D_L$ — luminosity distance (SNe Ia), with the angular-diameter distance $D_A=D_L/(1+z)^2$ (BAO). Reading it: $w$ never appears alone — it always enters folded into $f(a)$ inside $H(z)$ , which is why it is measured only through distances and growth.

Derivation AFrom continuity to the master dilution law

The goal is to convert the abstract statement "dark energy has equation of state $w$ " into the concrete scaling $\rho\propto a^{-3(1+w)}$ that underlies every plot in this article, and to see why $w=-1$ is special. We start from covariant conservation of the perfect-fluid stress-energy in a homogeneous, isotropic FRW background, insert the equation of state, separate variables, integrate, and read off the limits for each cosmic species — checking that matter, radiation, and $\Lambda$ all fall out correctly.

Step 1. The perfect fluid has $T^{\mu}{}_{\nu}=\mathrm{diag}(-\rho,p,p,p)$ in the comoving frame, and covariant conservation $\nabla_\mu T^{\mu}{}_{0}=0$ with the FRW connection $\Gamma^{i}{}_{0i}=\dot a/a$ yields $\dot\rho=-3H(\rho+p)$ . Why: this is just energy balance — the density falls both because the fluid is diluted and because its pressure does work on the expanding volume.

Step 2. Insert the equation of state $p=w\rho$ to get $\dot\rho=-3H\rho(1+w)$ . Why: the two independent quantities $\rho$ and $p$ collapse to one once $w$ is specified, closing the equation.

Step 3. Use $H=\dot a/a$ to cancel $\mathrm{d}t$ and separate variables: $\mathrm{d}\rho/\rho=-3(1+w)\,\mathrm{d}a/a$ . Why: writing everything as logarithmic differentials turns the problem into a one-line integral when $w$ is constant.

Step 4. Integrate from the present ( $a_0=1,\ \rho=\rho_0$ ): $\ln(\rho/\rho_0)=-3(1+w)\ln a$ , so $\rho=\rho_0\,a^{-3(1+w)}$ . Why: the constant of integration is fixed by anchoring to today's measured density.

Step 5. Read the limits: $w=0\Rightarrow a^{-3}$ (matter dilutes with volume); $w=\tfrac13\Rightarrow a^{-4}$ (radiation redshifts too); $w=-1\Rightarrow$ constant ( $\Lambda$ ); $w<-1\Rightarrow$ density grows. Why: recovering the textbook species scalings is the verification that the derivation is correct, and it exposes $w=-1$ as the knife-edge where the exponent vanishes.

Go deeper · Derivation B · 1 minThe acceleration threshold w below minus one third

Derivation BThe acceleration threshold w below minus one third

The goal is to prove that no ordinary component can make the universe accelerate, and that a dark-energy fluid must have pressure more negative than $-\rho/3$ to do so. We isolate the sign of $\ddot a$ in the second Friedmann equation, substitute the equation of state, and show that the outcome hinges entirely on the sign of a single bracket $(1+3w)$ — turning "acceleration" into a sharp inequality on $w$ and connecting it to the measured deceleration parameter.

Step 1. Write the acceleration equation $\ddot a/a=-\tfrac{4\pi G}{3}(\rho+3p)$ , valid with $c=1$ for a single dominant component. Why: this is the general-relativistic replacement for Newton's $-GM/r^2$ , and crucially the source is $\rho+3p$ , not $\rho$ alone.

Step 2. Insert $p=w\rho$ to obtain $\ddot a/a=-\tfrac{4\pi G}{3}\rho(1+3w)$ , with $\rho>0$ for any physical energy density. Why: pulling out $\rho$ leaves the entire sign information in the dimensionless bracket $1+3w$ .

Step 3. Since $\rho>0$ and $4\pi G/3>0$ , acceleration $\ddot a>0$ occurs if and only if $1+3w<0$ , i.e. $w<-\tfrac13$ . Why: only the bracket can change sign, so the acceleration condition is a pure statement about $w$ .

Step 4. Cross-check against the present deceleration parameter $q_0=\tfrac12\sum_i\Omega_i(1+3w_i)$ ; for $\Omega_m=0.3,\ \Omega_{\rm DE}=0.7,\ w=-1$ this gives $q_0=\tfrac12[0.3-1.4]=-0.55<0$ . Why: a concrete concordance-model number confirms that realistic matter-plus- $\Lambda$ mixtures do accelerate, and that matter alone ( $w=0,\ q_0>0$ ) never can.

Go deeper · Derivation C · 1 minThe closed-form CPL dark-energy density

Derivation CThe closed-form CPL dark-energy density

The goal is to evaluate the general evolving- $w$ integral for the specific CPL ansatz $w(a)=w_0+w_a(1-a)$ , producing the closed form $f(a)=a^{-3(1+w_0+w_a)}e^{-3w_a(1-a)}$ that is fed straight into $H(z)$ inside survey pipelines such as CAMB and CLASS. We substitute the linear ansatz into the integral, split it into a logarithmic piece and a linear piece, combine, and exponentiate — then confirm it collapses to the cosmological constant when $w_0=-1,\ w_a=0$ .

Step 1. Start from the general solution $\ln f=-3\int_1^a\frac{1+w(a')}{a'}\,\mathrm{d}a'$ with $f\equiv\rho_{\rm DE}(a)/\rho_{\rm DE,0}$ . Why: this is equation (5) rewritten as a density ratio, the quantity that actually enters the Friedmann equation.

Step 2. Substitute the ansatz: $1+w(a')=1+w_0+w_a-w_a a'$ . Why: expanding the linear form separates a constant term from a term proportional to $a'$ , which integrate differently.

Step 3. Split the integral: $\ln f=-3(1+w_0+w_a)\int_1^a\frac{\mathrm{d}a'}{a'}+3w_a\int_1^a\mathrm{d}a'$ , where the first integral is $\ln a$ and the second is $(a-1)$ . Why: the $1/a'$ term gives a logarithm (a power law in density) while the constant term gives a linear piece (an exponential in density).

Step 4. Combine: $\ln f=-3(1+w_0+w_a)\ln a+3w_a(a-1)$ . Why: collecting the two contributions readies the expression for exponentiation.

Step 5. Exponentiate to $f(a)=a^{-3(1+w_0+w_a)}\,e^{-3w_a(1-a)}$ , and check that $w_0=-1,\ w_a=0$ gives $f=1$ (a true $\Lambda$ ), while $w_a=0$ gives the constant- $w$ power law $a^{-3(1+w_0)}$ . Why: the correct limits certify the algebra, and the split form makes plain that $w_a$ controls the deviation of $f$ from a simple power law. *The two limits confirm the algebra: $w_0=-1,\,w_a=0$ gives $f=1$ (a true $\Lambda$ ), and $w_a=0$ recovers the constant- $w$ power law $a^{-3(1+w_0)}$ . Numerically, $f(a{=}0.5)=0.976$ for the DESI-like pair $(w_0,w_a)=(-0.9,-0.4)$ .*

Limiting cases

The single scaling $\rho\propto a^{-3(1+w)}$ organises every limiting behaviour of a constant- $w$ component; the acceleration equation supplies the threshold.

Case	Result	Meaning
Cosmological constant	$w=-1\ \Rightarrow\ \rho=\text{const}$	Density independent of $a$ ; exactly vacuum energy $\Lambda$ , the concordance value.
Acceleration threshold	$w=-\tfrac13\ \Rightarrow\ \ddot a=0$	Boundary between decelerating and accelerating expansion for a DE-dominated universe.
Phantom regime	$w<-1\ \Rightarrow\ \rho\ \text{grows with}\ a$	Violates the null energy condition; can end in a finite-time Big Rip singularity.
Quintessence range	$-1\le w\le+1$	Achievable by a canonical scalar field; $w=+1$ is kinetic-dominated (stiff), $w=-1$ potential-dominated.
Matter and radiation	$w=0\ (\rho\propto a^{-3}),\quad w=\tfrac13\ (\rho\propto a^{-4})$	Ordinary components shown for contrast; both dilute, so dark energy inevitably comes to dominate.

Common trapConfusing the equation of state with the sound speed

Do not conflate the equation-of-state parameter $w=p/\rho$ (a background quantity) with the adiabatic sound speed $c_s^2=\delta p/\delta\rho$ (a perturbation quantity). For quintessence $c_s^2\approx1$ regardless of the value of $w$ , and it is that near-unity sound speed — not $w$ — that keeps dark energy from clustering: pressure support erases sub-horizon inhomogeneities, so a smooth $w\neq-1$ fluid perturbs the global $H(z)$ but does not pile up in halos the way dark matter does. A second, separate trap: $w<-1$ does not violate causality or let signals outrun light. It violates the null energy condition $\rho+p\ge0$ , which is a statement about energy positivity, not about speed. Keep the three ideas distinct — background $w$ , perturbation $c_s^2$ , and the energy conditions — and most "paradoxes" of dark energy dissolve.

Worked examples

1 · When did the universe start accelerating? (fully worked)

Take flat $\Lambda$ CDM with $\Omega_m=0.3$ , $\Omega_\Lambda=0.7$ , and $w=-1$ . Find the present deceleration parameter $q_0$ and the redshift $z_{\rm acc}$ at which the expansion switched from decelerating to accelerating.

Step 1. The deceleration parameter is $q_0=\tfrac12\sum_i\Omega_i(1+3w_i)$ . Matter contributes $\Omega_m(1+0)=0.3$ ; the constant contributes $\Omega_\Lambda(1+3(-1))=0.7\times(-2)=-1.4$ . So $q_0=\tfrac12(0.3-1.4)=\mathbf{-0.55}$ . A negative $q_0$ means the expansion is speeding up today.

Step 2. Acceleration turns on when $\ddot a=0$ , i.e. when the source $\sum_i\rho_i(1+3w_i)=0$ . With $w=-1$ for $\Lambda$ and $\rho_\Lambda$ constant this reads $\Omega_m a^{-3}(1)+\Omega_\Lambda(-2)=0$ , so $a^{-3}=2\Omega_\Lambda/\Omega_m=2(0.7)/0.3=4.667$ .

Step 3. Invert: $a_{\rm acc}=4.667^{-1/3}=\mathbf{0.598}$ , hence $z_{\rm acc}=1/a_{\rm acc}-1=\mathbf{0.67}$ . The universe was decelerating under matter's gravity until $z\approx0.67$ (about $6$ Gyr ago), when the ever-diluting matter finally thinned enough for the constant vacuum energy to take over. q₀ = −0.55 today, and acceleration began at z ≈ 0.67 — both in excellent agreement with the supernova Hubble diagram.

2 · Dilution versus growth: reading the sign of 1 + w

A dark-energy component has constant $w$ . Compare how its density at redshift $z=1$ (scale factor $a=0.5$ ) relates to today's for the quintessence-like value $w=-0.9$ and the phantom value $w=-1.1$ .

Step 1. Use the master law $\rho(a)/\rho_0=a^{-3(1+w)}$ . For $w=-0.9$ the exponent is $-3(0.1)=-0.3$ , giving $\rho(0.5)/\rho_0=0.5^{-0.3}=\mathbf{1.23}$ : the density was $23\%$ higher in the past, so it is falling with time — the fluid dilutes, just slowly.

Step 2. For $w=-1.1$ the exponent is $-3(-0.1)=+0.3$ , giving $\rho(0.5)/\rho_0=0.5^{+0.3}=\mathbf{0.81}$ : the density was $19\%$ lower in the past, so it is rising with time — phantom energy grows as space expands.

Step 3. The cosmological constant sits exactly between them: $w=-1$ gives exponent $0$ and $\rho(0.5)/\rho_0=1$ , unchanged. The sign of $1+w$ is the whole story — positive means dilution, zero means frozen, negative means runaway growth toward a possible Big Rip. w = −0.9 dilutes (ratio 1.23), w = −1 is frozen (ratio 1), w = −1.1 grows (ratio 0.81).

3 · A DESI-style evolving model and the phantom crossing

A survey reports CPL parameters $(w_0,w_a)=(-0.9,-0.4)$ . Evaluate $w$ today, at $z=1$ , and in the far past, and find the redshift at which the model crosses the phantom divide $w=-1$ .

Step 1. Today ( $a=1$ ): $w=w_0=\mathbf{-0.9}$ , safely above $-1$ — quintessence-like at present.

Step 2. At $z=1$ ( $a=0.5$ ): $w=w_0+w_a(1-a)=-0.9+(-0.4)(0.5)=\mathbf{-1.1}$ , now below $-1$ — phantom in the past. As $a\to0$ , $w\to w_0+w_a=-1.3$ .

Step 3. The crossing $w=-1$ happens when $w_0+w_a(1-a)=-1$ , i.e. $(1-a)=(-1-w_0)/w_a=(-0.1)/(-0.4)=0.25$ , so $a=0.75$ and $z_{\rm cross}=1/0.75-1=\mathbf{0.33}$ . A model that crosses $-1$ is called "quintom" and cannot be built from a single canonical scalar field — which is exactly why such a signal, if confirmed, would be so consequential. w runs from −1.3 (early) through −1 at z ≈ 0.33 to −0.9 today, crossing the phantom divide.

Physics of the portrait

The equation of state of dark energy is not the discovery of one person but the convergence of a century of cosmology onto a single number. Its portrait is best worn by the experimentalist who forced the number into existence by measuring that the universe accelerates.

Origins · 1 minSaul Perlmutter and how this was discovered

Saul Perlmutter

born 1959 · led the Supernova Cosmology Project; shared the 2011 Nobel Prize in Physics

Saul Perlmutter is an American astrophysicist at the University of California, Berkeley and Lawrence Berkeley National Laboratory who, through the 1990s, built and led the Supernova Cosmology Project — an industrial-scale campaign to find and calibrate distant Type Ia supernovae as standard candles. The gamble was that measuring how the brightness of these explosions falls off with redshift would weigh the universe and reveal its fate. In 1998, his team and the competing High-z Supernova Search Team led by Brian Schmidt and Adam Riess independently found the opposite of what almost everyone expected: the distant supernovae were fainter, and therefore farther, than a decelerating universe allowed. The expansion was speeding up. In the language of this article, the data demanded a component with $w<-1/3$ — a smooth, negative-pressure energy filling space, soon named dark energy. Perlmutter, Schmidt, and Riess shared the 2011 Nobel Prize in Physics for the result. The measurement resurrected Einstein's long-abandoned cosmological constant $\Lambda$ (which he had introduced in 1917 and later called his greatest blunder) and reframed cosmology's central question: is this $w$ exactly $-1$ , the frozen vacuum energy, or does it evolve? Perlmutter has spent the years since pushing that question forward, from the Hubble-diagram precision of surveys to the design of experiments — SNAP, and its descendants Rubin and Roman — built expressly to measure whether $(w_0,w_a)$ ever strays from $(-1,0)$ .

History

1917

— Einstein introduces the cosmological constant Λ into general relativity

1988

— Ratra and Peebles propose quintessence, a slowly rolling scalar field

1998

— Perlmutter, Schmidt and Riess find distant supernovae imply accelerating expansion

2003

— Linder casts evolving dark energy as w(a) = w₀ + wₐ(1 − a)

2024

— DESI baryon acoustic oscillations hint that w may evolve with time

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 29–30 · Ryden, Introduction to Cosmology, 2nd ed., Ch. 4–6 · Dodelson, Modern Cosmology, 2nd ed., Ch. 1, 8 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 3 (cosmological background) · Binney & Tremaine, Galactic Dynamics, 2nd ed. (FRW preliminaries).

Primary sources. Riess et al. (1998), AJ 116, 1009; Perlmutter et al. (1999), ApJ 517, 565 (discovery of acceleration) · Chevallier & Polarski (2001), Int. J. Mod. Phys. D 10, 213; Linder (2003), PRL 90, 091301 (the $w_0$ – $w_a$ parametrisation) · Caldwell (2002), Phys. Lett. B 545, 23 (phantom energy) · DESI Collaboration (2024), BAO cosmological constraints. The core results are: $\rho\propto a^{-3(1+w)}$ from the FRW continuity equation; the acceleration threshold $w<-1/3$ and $q_0=\tfrac12\sum\Omega_i(1+3w_i)=-0.55$ for $\Omega_m=0.3$ flat $\Lambda$ CDM; the CPL density $f(a)=a^{-3(1+w_0+w_a)}e^{-3w_a(1-a)}$ from integrating the evolving- $w$ continuity equation; and the quintessence equation of state $w_\phi=(\tfrac12\dot\phi^2-V)/(\tfrac12\dot\phi^2+V)$ from the scalar-field stress tensor. The worked numbers ( $z_{\rm acc}\approx0.67$ , $\rho$ ratios $1.23/1/0.81$ , phantom crossing $z\approx0.33$ , $f(0.5)=0.976$ ) follow from these with $\Omega_m=0.3$ flat $\Lambda$ CDM and the stated CPL pair; the BAO scale $\sim150$ Mpc and $z_{\rm rec}\approx1100$ are standard Planck values, and the DESI preference is stated qualitatively.

Prerequisites. Dark Energy and the Cosmological Constant · The ΛCDM Concordance Model · Linear Growth of Structure. Leads to. Baryon Acoustic Oscillations · The Sound Horizon and the Standard Ruler · The CMB Anisotropy Power Spectrum.

Ask the tutor: Ask me to walk through why negative pressure anti-gravitates, or to fit a toy $(w_0,w_a)$ to a supernova Hubble diagram step by step.

Is w really a single number, or a whole function of time?

Both, depending on the model. In the simplest and still-favoured picture — a cosmological constant — $w$ is a single number fixed at exactly $-1$ for all time. In any dynamical model (quintessence, modified gravity) $w$ is a function $w(a)$ that can drift with the expansion. Because data cannot yet resolve an arbitrary function, cosmologists compress the possible evolution into two numbers, $w_0$ and $w_a$ , through the CPL form $w(a)=w_0+w_a(1-a)$ . So in practice you will see $w$ quoted either as one number (assuming a constant) or as the pair $(w_0,w_a)$ (allowing linear evolution), and which one is appropriate is itself part of what the data are being asked to decide.

Why does negative pressure cause acceleration rather than collapse?

The intuition from everyday gases — pressure pushes outward and resists compression — is a red herring in cosmology, because a uniform pressure exerts no net force (there is no pressure gradient in a homogeneous universe). What matters gravitationally is the source term in the second Friedmann equation, which general relativity fixes as $\rho+3p$ , not $\rho$ alone. Ordinary matter has $p\approx0$ , so it decelerates the expansion. A substance with $p<-\rho/3$ makes $\rho+3p$ negative, and a negative gravitational source produces repulsion. So acceleration is not pressure "pushing" space apart mechanically; it is the reversal of gravity's sign once the pressure is sufficiently negative.

What is the difference between a cosmological constant and quintessence?

A cosmological constant $\Lambda$ is vacuum energy with a fixed density and exactly $w=-1$ for all time — a property of empty space itself, with no dynamics. Quintessence is a physical scalar field rolling in a potential; its energy density and pressure change as the field evolves, so its $w$ sits in the band $-1\le w\le1$ and generically drifts with time. The observational punchline is that quintessence predicts $w$ slightly above $-1$ and evolving, whereas $\Lambda$ predicts $w=-1$ exactly and unchanging. Detecting any statistically robust deviation of $w$ from $-1$ , or any evolution $w_a\neq0$ , would rule out the cosmological constant and favour a dynamical field.

What is phantom energy and the Big Rip?

Phantom energy is dark energy with $w<-1$ , so negative that its density increases as the universe expands, rather than staying constant or diluting. Because the density climbs without bound, the repulsive acceleration grows ever stronger, and in the simplest models it reaches a singularity in finite time: first galaxy clusters, then galaxies, then solar systems, then atoms are torn apart as the scale factor diverges — the Big Rip. Phantom energy violates the null energy condition ( $\rho+p\ge0$ ) and cannot be built from a stable canonical scalar field without pathologies. It is taken seriously mainly because the current data do not sharply exclude $w<-1$ , and because some CPL fits (including recent DESI-plus-supernova combinations) wander briefly onto the phantom side of $-1$ .

Why is w measured only through distances and growth, never directly?

Because dark energy is smooth and does not clump, there is nothing local to weigh — no rotation curves, no lensing halos, no laboratory sample. Its only handle is the way it changes the global expansion rate $H(z)$ , which feeds two families of observables: geometric distances (the luminosity distance to supernovae, the angular-diameter distance to the BAO ruler), which are integrals of $1/H(z)$ , and the growth rate of structure $f\sigma_8(z)$ , which depends on $H(z)$ through the friction term in the perturbation equation. Every constraint on $w$ is therefore an inference: measure distances and clustering versus redshift, then ask what $w(z)$ inside $H(z)$ reproduces them.

What does the DESI result actually claim, and is Λ dead?

DESI's baryon-acoustic-oscillation data, combined with supernova samples and the CMB, show a mild preference for an evolving equation of state — a CPL fit with $w_0$ a little greater than $-1$ and $w_a$ negative — over a pure cosmological constant. The significance runs from about two to four sigma depending on the supernova compilation, intriguing but short of the five-sigma discovery standard, and it is sensitive to systematics and to the rigidity of CPL itself. The honest summary: $\Lambda$ is not dead, but for the first time in twenty-five years the data are nudging, not merely confirming. The next DESI, Euclid, and Rubin releases will decide whether the nudge sharpens into a detection.