Astronomy & Celestial Mechanics
the sky, computed
Cosmology

The number that would say whether it is a constant

Whether dark energy is a cosmological constant is the question of whether w is exactly −1, and w = −0.9 changes a distance modulus by thirty-four millimagnitudes at redshift a half — a fifth of the scatter of a single supernova, along a degeneracy only the acoustic scale can cut across.

Assumes Dark energy, Baryon acoustic oscillations and Supernovae.

18 min read 8 figures The sky as a clockMeasuring the unreachable

The rung below found an acceleration nobody was looking for: two teams set out to measure how fast the expansion was slowing, and both found a quarter of a magnitude of extra faintness at redshift a half, which is the wrong sign.

A quarter of a magnitude is a large signal and it was measured convincingly within a few years. The question that followed is a very much harder one, and the fact that it is harder by an order of magnitude is the whole content of this rung.

Something is accelerating the expansion. Is it a constant?

w = −0.9 is 34 millimagnitudes, and one supernova scatters by 120. Above: how much the distance modulus moves when the dark energy is not a constant. Each curve is a universe with the same Ωₘ = 0.315 and a different equation of state w, drawn as a difference from w = −1 in magnitudes. At redshift a half, w = −0.9 is worth 34 millimagnitudes — the shaded band is the 0.12-magnitude intrinsic scatter of a single standardised type Ia supernova, and the signal is a fifth of it. Nothing about one object can see this; the measurement is the mean of 1500, whose error on the mean is 3.1 millimagnitudes, and even that only works because the shape of the curve in redshift is different from every systematic anybody has thought of. Below: why the supernovae are not enough on their own. Each locus is the set of (Ωₘ, w) that a measurement cannot tell apart from the fiducial model — computed, not sketched: the supernova curve is the ridge of the same sum of squares a fit would minimise over 0.02–1 in redshift, and the acoustic-scale curve is the exact set of models with the same comoving distance to last scattering, which is what fixes the angle the microwave background's first peak subtends. They cross at 36 degrees. Neither is a measurement of w and the pair is, which is why the constraint on the equation of state is a picture of two loci crossing rather than a number read off a curve — and why −1.03 ± 0.03 is a statement about how well they cross rather than about how well anything was measured.
Fig. 1 The difficulty, in two panels. Above: how much the distance modulus moves when the equation of state is not exactly minus one. At redshift a half, w=−0.9w = -0.9 is worth thirty-four millimagnitudes, and the shaded band is the intrinsic scatter of a single standardised supernova — the signal is a fifth of it. Below: why the supernovae are not enough on their own. Each locus is the set of parameters a measurement cannot distinguish from the fiducial model, computed rather than sketched, and they cross at thirty-six degrees.

What w is

The energy density of any component of the universe evolves as

ρ∝a−3(1+w),\rho \propto a^{-3(1+w)},

where ww is the ratio of its pressure to its energy density. Matter has w=0w = 0 and dilutes as a−3a^{-3}; radiation has w=1/3w = 1/3 and dilutes as a−4a^{-4}, with the extra factor from the redshifting of each photon, and the budget the three make up is five parts ordinary matter in a hundred.

A cosmological constant has w=−1w = -1 exactly, and therefore does not dilute at all: the same energy density in every cubic metre for ever, unchanged as the metres multiply. That is a strange substance and it is also the simplest possible one, because it has no parameters — it is a constant of nature, or a term in the field equations, and there is nothing about it that could vary.

Anything else is a field. A slowly rolling scalar field has ww close to −1-1 but not equal to it, and generically varying with time. There is no shortage of candidates; there is a complete shortage of predictions, since the models can be built to give almost any ww near −1-1.

So the measurement is: is ww exactly −1-1? A confirmed departure of any size would rule out a cosmological constant outright and open the field to physics.

Three densities, two crossings, and which one is in charge. The density of each component in units of today's critical density, against the scale factor, both logarithmic. Nothing is fitted: radiation dilutes as a⁻⁴ because expansion both spreads the photons out and stretches each one, matter as a⁻³ because it is only spread out, and Λ not at all. The three straight lines cross twice, and the crossings are the two dividing lines of cosmic history. Matter overtakes radiation at a = 2.92e-4, which is z = 3419; Λ overtakes matter at a = 0.772, z = 0.29, when the universe was 10.3 Gyr old — only 3.5 Gyr ago. The second crossing is the reason the composition today is an unrepresentative snapshot: matter ran the expansion from the age of 50,474 years until 10.3 Gyr, which is three quarters of the history so far, and before that radiation did.
Fig. 2 Why the question is answerable at all. The densities of the three components against time: matter and radiation fall steeply and the dark energy does not, so the epoch at which each dominates is set by where the curves cross. A different ww tilts the dark-energy line, which moves the crossing — and moving the crossing changes the whole expansion history, which is what a distance measurement sees.

Why the signal is small

Changing ww from −1-1 to −0.9-0.9 changes the dark energy density’s behaviour by a thirty-per-cent tilt over the observable range of redshift. That sounds large. What it does to a distance is not.

Distance is an integral of 1/H(z)1/H(z), and H(z)H(z) contains the dark energy only through a term that is a third of the total at redshift one and less beyond. Integrating over redshift smears the difference further. The net effect at redshift a half is 0.034 magnitudes.

A type Ia supernova, after standardisation by its light-curve shape and colour, has an intrinsic scatter of about 0.12 magnitudes. The signal is a fifth of the noise on one object.

The measurement therefore does not exist for any individual supernova. It exists only as a statistical statement about a sample, and the error on the mean of NN objects is 0.12/N0.12/\sqrt{N} — three millimagnitudes for fifteen hundred, which is a ten-sigma detection of the difference between w=−0.9w = -0.9 and w=−1w = -1.

The systematic floor

The square root does not go on for ever, and where it stops is the actual limit of the field.

Averaging reduces the random scatter. It does nothing at all to an error common to the whole sample, and a distance modulus is a difference of magnitudes measured through filters, calibrated against standard stars, corrected for dust, and compared between high- and low-redshift objects observed with different instruments in different bands.

Every one of those steps has a systematic at the ten-to-thirty-millimagnitude level, which is the same size as the entire signal.

Photometric calibration. The zero points of the photometric system have to be transferred between telescopes and between bandpasses, and the current best-effort calibration of a supernova sample is uncertain at about 0.01 magnitudes.

Dust. Extinction reddens and dims, and separating it from a supernova’s own intrinsic colour variation requires assuming a reddening law. Whether that law is the same in every host galaxy is not known, and it is the leading systematic in most modern analyses.

Evolution. High-redshift supernovae explode in younger, more metal-poor galaxies than nearby ones. If the standardisation depends on host properties — and there is evidence that it does, at the 0.05-magnitude level — then the comparison across redshift carries a drift that mimics a change in ww.

The other reason the sample is not just numbers

Standardisation is a two-parameter correction and both parameters have to be fitted from the sample itself, which introduces a subtlety worth naming because it costs precision that a naive count does not show.

The observed peak magnitude is corrected as

mcorr=m+αx−βc,m_{\text{corr}} = m + \alpha x - \beta c,

with xx the light-curve stretch and cc the colour, and α\alpha and β\beta fitted simultaneously with the cosmology. That means the cosmological parameters and the standardisation parameters are correlated: a change in β\beta can be partly absorbed by a change in ww, and the error on ww includes the error on β\beta.

Worse, β\beta appears to depend on the host galaxy — split the sample by host stellar mass and the two halves prefer different values, by an amount that is significant and that nobody can derive. The standard treatment is to fit the two halves separately and quote the difference as a systematic, which is honest and is not an explanation.

Three densities, two crossings, and which one is in charge. The density of each component in units of today's critical density, against the scale factor, both logarithmic. Nothing is fitted: radiation dilutes as a⁻⁴ because expansion both spreads the photons out and stretches each one, matter as a⁻³ because it is only spread out, and Λ not at all. The three straight lines cross twice, and the crossings are the two dividing lines of cosmic history. Matter overtakes radiation at a = 3.69e-4, which is z = 2711; Λ overtakes matter at a = 0.715, z = 0.40, when the universe was 9.9 Gyr old — only 4.5 Gyr ago. The second crossing is the reason the composition today is an unrepresentative snapshot: matter ran the expansion from the age of 80,285 years until 9.9 Gyr, which is three quarters of the history so far, and before that radiation did.
Fig. 3 The three densities with the matter density lowered to 0.25 from 0.315. The crossings move — matter–radiation equality later, matter–Λ equality earlier — and the shapes of the three curves are unchanged, because each one’s slope is fixed by its own equation of state and nothing else. The whole content of a cosmology is where the lines cross, and ww is the parameter that decides whether the Λ line is horizontal at all.

The degeneracy

Even with a perfect sample there is a second problem, and it is geometric rather than statistical.

The distance to a given redshift depends on ww and on Ωm\Omega_m together, and over the redshift range supernovae reach, a change in one can be nearly cancelled by a change in the other. Increasing Ωm\Omega_m and making ww more negative both slow the recent expansion, and the combination that leaves the distances unchanged is a curve in the plane rather than a point.

The curve is not a sketch: for each ww there is a specific Ωm\Omega_m minimising the difference in distance modulus over the sample’s redshift range, and that locus is what the lower panel of the first figure draws.

What cuts across it is a measurement that depends on the same parameters differently. The microwave background fixes the angle the sound horizon subtends at last scattering, and therefore the comoving distance to redshift 1090 — an integral over the whole history rather than the recent part. And it fixes the physical matter density Ωmh2\Omega_m h^2 very well from the ratios of the acoustic peak heights.

Holding Ωmh2\Omega_m h^2 fixed and requiring the same distance to last scattering gives a second locus, running at a large angle to the first. Their intersection is the measurement.

The third measurement, which is the same ruler at low redshift

There is a third constraint and it is the cleanest of the three, because it uses the same physical ruler at a completely different epoch.

The sound horizon imprinted on the microwave background is also imprinted on the distribution of galaxies, as a slight excess of pairs separated by about 150 megaparsecs. Measuring that scale in a galaxy survey at redshift 0.5 gives a distance; measuring it along the line of sight gives H(z)H(z) directly rather than an integral of it.

That last point matters. Supernovae measure integrated distance and therefore respond to ww weakly and with a lag; a direct measurement of H(z)H(z) at a redshift where dark energy matters responds to ww immediately.

Four expansion histories that agree exactly today. The scale factor against time, with the present at zero and every model normalised to a = 1 and an expansion rate of 67.36 km/s/Mpc there. That normalisation is the figure: four universes that are indistinguishable from a measurement made now, separated entirely by what is behind and ahead of them. Each curve is integrated from da/dt = aH₀E(a) rather than from its own closed form, so the four are compared through one routine. The age each implies is where its curve meets zero: empty (Ω = 0) 14.52 Gyr, matter only (Ω = 1) 9.68 Gyr, ΛCDM (Planck 2018) 13.80 Gyr, closed (Ω = 2) 8.29 Gyr. The empty universe's 14.52 Gyr is the Hubble time 1/H₀ exactly, which is what makes it the natural thing to measure an acceleration against. The closed model turns over and is stopped at its turnaround.
Fig. 4 What a different equation of state does to the history rather than to the observables. Four expansion histories at the same present rate: the accelerating one is 13.8 billion years old against 9.7 for a matter-only universe, and the difference is not merely a scaling — the accelerating case decelerates for nine billion years and then does not. The number this essay is about decides where that inflection falls, and moving it by a tenth moves the age of the universe by half a billion years.
The same universe, four times, as a fraction of itself. Each bar is the fractional contribution of the four components to the total density at one epoch, computed from the Planck 2018 parameters by scaling each component from today: radiation as (1+z)⁴, both kinds of matter as (1+z)³, and Λ as a constant. The familiar figure — five per cent baryons, twenty-six dark matter, sixty-nine dark energy — is the top bar and only the top bar. At recombination the same universe is three-quarters dark matter and Λ is one part in ten million; before matter–radiation equality it is mostly radiation. A pie chart of the contents of the universe is therefore a statement about a moment, and the moment is the one it happens to be drawn in.
Fig. 5 The composition at four epochs chosen to bracket where the constraint comes from: today, redshift a half where the supernovae are, redshift ten where nothing is yet measured, and equality. Dark energy is a rounding error at every epoch except the most recent, which is why every constraint on ww comes from the last few billion years and why extending a survey to higher redshift buys so much less than extending it to more objects.

Why the three probes are combined rather than compared

It is tempting to treat three measurements of the same quantity as three independent checks, and the way they are actually used is different and worth setting out.

They are not three measurements of ww. None of them measures ww at all on its own. Supernovae measure relative distances over 0<z<20 < z < 2; the microwave background measures one distance to z=1090z = 1090 and two densities; galaxy surveys measure a distance and an expansion rate at a few redshifts in between. Each constrains a combination of Ωm\Omega_m, ww and H0H_0, and each combination is different.

The value of the combination is therefore geometric rather than statistical. Two loci crossing at thirty-six degrees give a small intersection; two crossing at five degrees give a long thin one, however precise each is. That is why the quoted improvement from adding the acoustic scale to the supernovae is a factor of several rather than the modest gain that adding an independent measurement of equal precision would give.

It also means a systematic in one probe propagates into the joint answer in a direction set by the geometry rather than by its own size — which is the reason every joint analysis publishes the result with each probe removed in turn.

What the answer currently is

Combining supernovae, the acoustic scale in the microwave background, and the acoustic scale in galaxy surveys gives

w=−1.03±0.03,w = -1.03 \pm 0.03,

consistent with a cosmological constant.

Three things about that number deserve saying.

It assumes ww is constant. Fitting a time-varying ww instead — usually parameterised as w(a)=w0+wa(1−a)w(a) = w_0 + w_a(1-a) — gives constraints that are far weaker and that have shown mild departures from a constant in some recent combinations of data, which is currently the most interesting unresolved question in the field.

It assumes flatness. Allowing curvature as a free parameter widens the constraint on ww substantially, because curvature and dark energy affect distances in similar ways.

And 0.03 is close to the systematic floor. Doubling the supernova sample would improve the statistical error by 30 per cent and the total by very little, which is why the effort has moved to the geometric probes and to controlling calibration rather than to collecting more objects.

Why a pressure accelerates anything

The ratio of pressure to energy density is an odd thing for a cosmological measurement to be about, and the reason it is the right variable is worth a section, because it is the one step of the argument that has no Newtonian analogue at all.

In Newtonian gravity, pressure does not gravitate. A gas cloud’s pull on a distant body depends on its mass and not on how hard it is pushing outward, and pressure enters only through the forces it exerts across surfaces inside the cloud.

In general relativity the source of gravity is not mass but the whole stress–energy tensor, pressure included. The expansion obeys

a¨a=−4πG3(ρ+3pc2),\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right),

and the second term inside the bracket is the departure. Every ordinary substance has p≥0p \geq 0, so it contributes to the deceleration on top of its own density — a hot gas pulls harder than a cold one of the same mass, which is a genuine and measurable prediction and is one of the ways pressure’s gravitation has been tested.

Setting p=wρc2p = w\rho c^2, the bracket is ρ(1+3w)\rho(1 + 3w), which changes sign at w=−1/3w = -1/3. Anything with ww more negative than a third accelerates the expansion, and that is the whole condition — no substance is required to push, and nothing is being blown apart. A cosmological constant at w=−1w = -1 clears it comfortably; a network of cosmic strings at w=−1/3w = -1/3 sits exactly on the boundary and does nothing.

The negative pressure itself is not exotic when written the other way round. A constant energy density means that expanding a volume creates energy in proportion to the volume, and the work done in that expansion is p dVp\,dV — so the pressure must be negative and equal in magnitude to the energy density, which is w=−1w = -1 arrived at from thermodynamics rather than from the field equations. The strange thing is the constant density, and the negative pressure is its bookkeeping.

The same recession law from two different galaxies. Twenty galaxies at fixed comoving positions after the whole picture has been multiplied by 1.34, with each galaxy's displacement drawn from where it was to where it is. Every arrow's tail sits at the old separation and its tip at the new one, so the arrow is exactly 0.34 times its tail's distance from the highlighted galaxy — proportional to separation for one reason and no other: a uniform scaling moves everything in proportion to its distance from whatever point the scaling is measured about. The right-hand panel measures from a different galaxy and gets the identical law with the identical constant. That is the content of a linear velocity–distance relation. It is the signature of an expansion with no centre, and the observation that every galaxy recedes is therefore not evidence that this one is the centre — it is evidence that none of them is.
Fig. 6 The expansion drawn with no dynamics at all: galaxies at fixed comoving positions, the picture scaled up, and every displacement proportional to the separation it starts from. This is what “the universe expands” means and it contains no ww whatever — the equation of state enters only through how the scale factor depends on time, which this figure deliberately does not show. Everything in this essay is a second derivative of this drawing.

The side of minus one that should not be reachable

The measured value is −1.03±0.03-1.03 \pm 0.03, which is centred on the wrong side of the boundary, and the region below −1-1 has a name and a set of problems.

Dark energy with w<−1w < -1 is called phantom, and its density grows as the universe expands. Everything else in the budget dilutes; a phantom component does the opposite, so it comes to dominate more and more completely, and the domination runs away. The expansion rate diverges in finite time, and the divergence tears apart structures in order of decreasing binding energy — clusters, then galaxies, then the solar system, then atoms. That is the big rip, and for w=−1.03w = -1.03 it is some hundreds of billions of years off.

The theoretical objection is stronger than the eschatological one. A component with w<−1w < -1 violates the null energy condition, and the simplest field theories that produce it have kinetic terms of the wrong sign, which makes the vacuum unstable to producing arbitrary amounts of the field and its negative-energy partner. Nothing that behaves this way is easy to write down as a well-behaved theory.

So the honest reading of −1.03±0.03-1.03 \pm 0.03 is that a value one standard deviation into a region nobody can construct a model for is exactly what a measurement centred on −1-1 looks like half the time. The result’s interest is in its width and not in its centre, and a determination at ±0.005\pm 0.005 that stayed where it is would be a very different statement — which is the specification the next generation of surveys was written to.

What a departure would mean

It is worth stating why three hundredths of a magnitude is worth a generation of instrument-building, because the quantity is abstract and the effort is not.

A cosmological constant is a term that can be written into the field equations with no dynamics attached, and its measured value is about 10−12010^{-120} in the natural units the underlying physics is written in. Nobody has an account of that number. It is the largest unexplained ratio in physics, and one standard response is that a constant requires an explanation of a kind that a field would not, because a field can roll to wherever it is now.

So a confirmed w≠−1w \neq -1 would not merely add a parameter. It would say that the thing driving the expansion has dynamics, that it has a history, and that its present value is a consequence rather than a coincidence.

A confirmed w=−1w = -1 to high precision says the opposite, and is the harder result to live with.

Three densities, two crossings, and which one is in charge. The density of each component in units of today's critical density, against the scale factor, both logarithmic. Nothing is fitted: radiation dilutes as a⁻⁴ because expansion both spreads the photons out and stretches each one, matter as a⁻³ because it is only spread out, and Λ not at all. The three straight lines cross twice, and the crossings are the two dividing lines of cosmic history. Matter overtakes radiation at a = 2.92e-4, which is z = 3420; Λ overtakes matter at a = 0.772, z = 0.30, when the universe was 10.3 Gyr old — only 3.5 Gyr ago. The second crossing is the reason the composition today is an unrepresentative snapshot: matter ran the expansion from the age of 50,451 years until 10.3 Gyr, which is three quarters of the history so far, and before that radiation did.
Fig. 7 And why the constant is only measurable now. Dark energy’s density does not change with the scale factor while matter’s falls as the cube, so the two cross once — at z=0.29z = 0.29 — and everything before that crossing is a universe in which the component this essay is about contributes almost nothing. A survey at z=3z = 3 is measuring a term at a per cent of the total. That is the whole difficulty: the quantity is dominant today, negligible when the universe was young, and the leverage is all in a narrow band of redshift.

What would settle it

Ten thousand supernovae with a calibration good to five millimagnitudes. That is the specification of the surveys now running, and its binding constraint is the calibration rather than the count.

A percent-level acoustic-scale measurement over a wide range of redshift. Spectroscopic surveys of tens of millions of galaxies give the ruler at many epochs, which turns a single distance into an expansion history.

And a measurement of H(z)H(z) that does not go through a ruler at all. Standard sirens do exactly that, and enough of them with identified hosts would give the expansion rate at several redshifts with systematics that share nothing with any of the above. One more matter density shows how weakly the present composition constrains the past one.

Three densities, two crossings, and which one is in charge. The density of each component in units of today's critical density, against the scale factor, both logarithmic. Nothing is fitted: radiation dilutes as a⁻⁴ because expansion both spreads the photons out and stretches each one, matter as a⁻³ because it is only spread out, and Λ not at all. The three straight lines cross twice, and the crossings are the two dividing lines of cosmic history. Matter overtakes radiation at a = 2.30e-4, which is z = 4338; Λ overtakes matter at a = 0.836, z = 0.20, when the universe was 10.7 Gyr old — only 2.5 Gyr ago. The second crossing is the reason the composition today is an unrepresentative snapshot: matter ran the expansion from the age of 31,362 years until 10.7 Gyr, which is three quarters of the history so far, and before that radiation did.
Fig. 8 The three densities in a universe with a matter fraction of 0.4. The two crossings move by a few hundred million years and the shape of the history is unchanged, which is why the equation-of-state parameter is measured from the derivative of the expansion rather than from the composition.

Where this ladder goes next

This rung has taken the question that followed the discovery and shown why it is harder by an order of magnitude: the signal is a fifth of the scatter of one object, it lies along a degeneracy, and the error is now dominated by terms that averaging does not touch.

The rung above is the time dependence. A constant ww is already a restrictive assumption, and the parameterisation used to relax it has two parameters and a figure of merit built around the area of the ellipse they occupy — which is the quantity every survey since has been designed to shrink.

Beside it lies the alternative that removes dark energy entirely: modifying gravity on large scales rather than adding a substance. The distinguishing observable is not the expansion history, which can be matched, but the growth of structure within it — which is why redshift-space distortions, measuring how fast galaxies fall into overdensities, are the other half of every survey design.

And below it, the habit: a discovery and a measurement of the same thing can differ by an order of magnitude in difficulty. Detecting the acceleration needed a quarter of a magnitude and a hundred objects. Characterising it needs three hundredths of a magnitude and a calibration nobody had, and twenty-five years later the answer is still consistent with the simplest possibility.

Where this goes next

Distance modulus against redshift, for three universes. The distance modulus μ = 5 log₁₀(D_L/10 pc) against redshift for three universes, all with H₀ = 67.36 km/s/Mpc, with 60 model supernovae drawn from the ΛCDM curve with 0.15 magnitudes of scatter. The point of the figure is how little difference there is: across two decades of redshift the three curves stay within a few tenths of a magnitude, and at z = 0.5 the accelerating and decelerating cases differ by 0.387 mag. A cosmology is not read off this plot. It is read off the residual, which is the next figure. An expansion that was supposed to be slowing Gravity is attractive, so an expanding universe full of matter must be decelerating, and the only question was by how much. Two teams set out to measure the deceleration and both found a quarter of a magnitude of extra faintness at redshift half — which is the wrong sign. A bump at a hundred megaparsecs. The two-point correlation function of galaxies, multiplied by the square of the separation so that the interesting part is not buried under the power law. The smooth dashed curve is the broad-band clustering — the ordinary fact that galaxies are near other galaxies, which carries no cosmological information and is treated as a nuisance term in the real analysis. The bump on top of it is the whole measurement, and its position here is not fitted: it is the comoving sound horizon at the drag epoch, integrated in this file from ∫c_s da/a²H with c_s = c/√(3(1+R)), which comes out at 146.9 Mpc — 99.0 in the h⁻¹ Mpc a survey works in. The excess says that a galaxy is very slightly more likely to have a companion at that separation than at 90 or 115, by about one part in two hundred, and the reason is that a pressure wave in the photon–baryon fluid ran outward from every overdensity for four hundred thousand years and stopped where it was when the photons let go. The points are drawn with the error a survey of a million galaxies achieves; a single pair of galaxies at 100 Mpc means nothing, which is why the measurement waited for the surveys. The same ruler measured twice, ten billion years apart The sound wave that produced the acoustic peaks in the microwave background also left a faint excess in how galaxies are spaced, at a separation of about a hundred megaparsecs. It is the only cosmological distance indicator whose length is set by physics rather than by a chain of calibrations. A light curve that is a decay chain. Bolometric luminosity against days since explosion, for three Type Ia models synthesising 0.3 M☉, 0.6 M☉, 0.9 M☉ of ⁵⁶Ni, from Arnett's one-zone diffusion model. Nothing in the shape is fitted: the two timescales are the laboratory half-lives of ⁵⁶Ni and ⁵⁶Co, 8.8 and 111.3 days, and the rise is those decays seen through an envelope that takes 11.3, 15.5, 18.6 days to leak. Each peaks at 14, 18, 21 days, and at maximum the luminosity equals the instantaneous deposited power to within 0.6% — Arnett's rule, and the only reason a peak brightness can be read as a mass of nickel. After maximum the curves settle to nearly straight lines on this logarithmic axis, declining at 0.0360 magnitudes a day against ⁵⁶Co's own 0.0098. The difference is gamma-ray escape: the ejecta become transparent to the very photons that are supposed to be heating them, and the tail is therefore steeper than the isotope. The one thing the tail is not is a property of the star — it is a half-life, plus a column density that is falling as t⁻². The candle that has to be standardised A Type Ia's light is the decay of half a solar mass of nickel-56 seen through an expanding envelope. Their peak brightnesses span six-tenths of a magnitude — and it is the width of the light curve, not anything else, that says which.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Acoustic scaleCosmological constantDark energy densityDeceleration parameterDegeneracy directionEquation of stateFigure of meritJoint constraintPhotometric calibrationQuintessenceStandardisationSystematic floor