Unsolved Problems in Physics: 95% of the Cosmos Still Has Only a Name

0. Key points

• Of the energy content of the universe, the matter we know about — atoms — accounts for only about 5%. Another 27% or so is dark matter and about 68% is dark energy, and both have a name and nothing else.
• The case for dark matter follows from one simple observation: the rotation speed of a galaxy does not fall off in its outskirts. In this article we compute the mass of the Milky Way by hand and confirm numerically that more than four times the visible stellar mass is required.
• Dark energy was discovered in 1998 from supernova observations. If it is the energy of the vacuum, the prediction of particle theory and the observed value disagree by 120 orders of magnitude. This is probably the worst discrepancy in the history of physics.
• Inflation in the early universe resolves the horizon problem and the flatness problem beautifully, but what actually drove inflation is unsettled. The primordial gravitational waves that would settle it have not been found.
• General relativity and quantum mechanics collide head-on at the centre of a black hole and at the beginning of the universe. After more than ninety years there is still no theory uniting them.
• Every one of these is a problem to which nobody yet knows the answer. Past the last page of the textbook, blank pages remain.

1. Motivation: the era when physics was declared finished

At the end of the nineteenth century a certain sense of completion hung over physics. Newtonian mechanics accounted for the motion of the heavens, Maxwell’s equations for electricity, magnetism and light, thermodynamics for heat engines — everything, apparently, had been explained. The story is told of young people who asked whether to take up physics and were advised against it: nothing was left, they were told, but filling in decimal places.

Two small stains, however, remained in those decimal places. The spectrum of blackbody radiation did not match classical theory, and the speed of light appeared to be the same measured from any inertial frame（Definition 2.1）［Einstein and Feynman］. From those two stains grew quantum mechanics and relativity, the twin pillars of twentieth-century physics. What had looked like a blemish turned out to be the entrance to a new continent.

Physics today explains far more, and far more precisely, than it did then. The Standard Model of particle physics predicts the magnetic moment of the electron to twelve decimal places (the most accurate prediction humanity has ever made（Example 6.6）［Einstein and Feynman］), and general relativity is used routinely to correct the clocks aboard GPS satellites (the calculation of GPS clock drift（Example 4.3）［Physics in Everyday Life］; the practical side is treated in Physics in Everyday Life). And yet — or rather, because of this — several stains have come into sharp focus. This time they are not small ones: 95% of the energy of the universe is unidentified.

This article takes up four of the large mysteries that remain in modern physics: three belonging to cosmology (dark matter, dark energy, inflation) and one belonging to the foundations of theory (quantum gravity). If you finish it thinking “so much is still unknown”, the article has done its job. That is where science currently stands.

2. Preliminaries: three senses of “unsolved”, and the cosmic budget

2.1. Unsolved problems come in kinds

“Unsolved problem” covers rather different situations. It is worth sorting them out.

Kind	Content	Examples
Observation without explanation	The phenomenon is certainly occurring, but the entity responsible has not been identified	Dark matter, dark energy
Theory without verification	A plausible theory exists, but the decisive experiment or observation is missing	Inflation, supersymmetry
Theories that contradict each other	Two successful theories break down when used together	Quantum gravity, the black hole information problem
Principle understood, equations unsolved	The fundamental equations are known, but the behaviour of their solutions is intractable	Turbulence, high-temperature superconductivity

The last row may be the surprising one. The Navier–Stokes equations governing the motion of water have been available since the nineteenth century, yet the statistical properties of turbulence still cannot be derived from first principles. Knowing an equation and knowing its consequences are two different things. This distinction connects with the discussion in Laplace’s Demon (the exact solution of the logistic map and its Lyapunov exponent（Proposition 4.2）［Laplace's Demon and Determinism］).

2.2. Tools for drawing up the cosmic budget

To argue about how much of what the universe contains, we need a yardstick for comparison. Cosmology uses the following quantities.

Definition 2.1（Critical density and density parameter）

Suppose the universe expands homogeneously and isotropically, and let its expansion rate be the Hubble constant $H$ (the fractional stretching per unit time). With $G$ the Newtonian constant,

$$\rho_c = \frac{3H^2}{8\pi G}$$

is called the critical density. For a component $i$ of the universe with mean density $\rho_i$,

$$\Omega_i = \frac{\rho_i}{\rho_c}$$

is called its density parameter. When the sum over all components $\Omega_{\text{tot}} = \sum_i \Omega_i$ is exactly $1$, the space of the universe is flat (Euclidean geometry holds).

The critical density is “the density that makes space exactly flat”. Denser than this and space closes up like a sphere; thinner and it opens out like a saddle. The density parameter writes the cosmic budget in the form “fractions of a total income set equal to 1”.

Example 2.2（Putting numbers on the cosmic budget）

Observations of the cosmic microwave background (CMB) by the Planck satellite give $H_0 = 67.4\ \mathrm{km/s/Mpc}$. We first compute the critical density. Since $1\ \mathrm{Mpc} = 3.086 \times 10^{22}\ \mathrm{m}$,

$$H_0 = \frac{6.74 \times 10^{4}\ \mathrm{m/s}}{3.086 \times 10^{22}\ \mathrm{m}} = 2.18 \times 10^{-18}\ \mathrm{s^{-1}}$$

Substituting into Definition 2.1,

$$\rho_c = \frac{3 \times (2.18 \times 10^{-18})^2}{8\pi \times 6.674 \times 10^{-11}} = \frac{1.43 \times 10^{-35}}{1.68 \times 10^{-9}} = 8.5 \times 10^{-27}\ \mathrm{kg/m^3}$$

The mass of a single hydrogen atom is $1.67 \times 10^{-27}\ \mathrm{kg}$, so this amounts to five hydrogen atoms per cubic metre. Averaged over the whole universe, space is far emptier than a laboratory vacuum.

The same observations break the total down as $\Omega_b \simeq 0.049$ (ordinary matter made of atoms), $\Omega_c \simeq 0.265$ (dark matter) and $\Omega_\Lambda \simeq 0.685$ (dark energy). The density of ordinary matter is therefore

$$0.049 \times 8.5 \times 10^{-27} = 4.2 \times 10^{-28}\ \mathrm{kg/m^3}$$

that is, 0.25 protons per cubic metre. What we have been calling “matter” is a rounding error in the cosmic budget.

flowchart LR
A["10^-36 s<br/>Inflation<br/>Mystery: what drove it?"] --> B["3 minutes<br/>Synthesis of light elements<br/>Established"]
B --> C["380,000 years<br/>Recombination<br/>Established"]
C --> D["Several hundred Myr<br/>First stars and galaxies<br/>Mystery: what is dark matter?"]
D --> E["About 5 Gyr ago<br/>Expansion turns to acceleration<br/>Mystery: dark energy"]
E --> F["Today<br/>13.8 Gyr"]

The history of the universe and the mysteries left inside it: established stretches and unsolved ones alternate

3. Invisible matter — dark matter

3.1. Zwicky’s indignation and Rubin’s persistence

In 1933 the Swiss astronomer Fritz Zwicky noticed that the galaxies in the Coma cluster were moving about far too quickly. In a system bound by gravity, speeds that are too high mean the system flies apart. His estimate was that holding the cluster together required several hundred times the mass of the luminous matter actually visible. Zwicky called this dunkle Materie — dark matter — and the astronomical community of the day paid it almost no attention. That he was the sort of man who called his colleagues “spherical bastards” (bastards whichever direction you look at them from) probably did not help.

The decisive result came forty years later. In 1970 Vera Rubin and Kent Ford measured the rotation speed of the Andromeda galaxy precisely and showed that the rotation speed does not fall off even in the outer parts of the galaxy. This is entirely unlike the behaviour of planets in the solar system. Neptune orbits far more slowly than Mercury, yet stars in the outskirts of a galaxy were circulating about as fast as those near the centre.

Definition 3.1（Dark matter）

Matter that interacts almost not at all with electromagnetic radiation (light, radio waves, X-rays and so on), so that it cannot be observed directly, but whose existence is confirmed through gravity, is called dark matter. It is thought to consist of cold (non-relativistic) particles carrying no electric charge and interacting only very weakly with ordinary matter.

3.2. What a rotation curve tells us

Why does “the rotation speed does not fall off” amount to “there is invisible mass”? The argument can be followed completely with high-school mechanics.

Proposition 3.2（Flat rotation curves and the mass distribution）

Consider a star moving on a circular orbit of radius $r$ with speed $v(r)$ inside a galaxy with a spherically symmetric mass distribution. Writing $M(r)$ for the total mass inside radius $r$,

$$v(r) = \sqrt{\frac{G M(r)}{r}}$$

holds. Consequently, if $v(r)$ equals a constant $v_0$ over some range, then over that range

$$M(r) = \frac{v_0^2}{G}\, r, \qquad \rho(r) = \frac{v_0^2}{4\pi G}\cdot \frac{1}{r^2}$$

That is, the enclosed mass keeps growing in proportion to the radius, while the density falls off as $r^{-2}$.

Proof（Proposition 3.2）

For a spherically symmetric distribution, the gravitational pull of the matter outside radius $r$ cancels (Newton’s shell theorem), and the mass $M(r)$ inside exerts the same force as if it were concentrated at the centre. The centripetal force on a star of mass $m$ is therefore $GM(r)m/r^2$, and the equation of motion for circular motion is

$$\frac{G M(r) m}{r^2} = \frac{m v^2}{r}$$

Dividing both sides by $m$ and multiplying by $r$ gives $v^2 = GM(r)/r$, that is, $v = \sqrt{GM(r)/r}$.

Now set $v(r) = v_0$ (constant). Solving the relation above for $M(r)$ gives $M(r) = v_0^2 r / G$. For the density, the mass contained in the shell $[r, r+dr]$ is $dM = 4\pi r^2 \rho(r)\, dr$, so

$$\rho(r) = \frac{1}{4\pi r^2}\frac{dM}{dr} = \frac{1}{4\pi r^2}\cdot \frac{v_0^2}{G} = \frac{v_0^2}{4\pi G r^2}$$

Starlight is concentrated near the centre of a galaxy, and there is almost no luminous matter in the outskirts. If $M(r)$ nevertheless keeps growing in proportion to $r$, then something that does not shine must be spread out well beyond the visible edge.

∎

A galactic rotation curve. The speed predicted from visible matter alone (dashed) falls off outwards, while the actual observation (solid) does not. The gap is the evidence for dark matter

Example 3.3（Estimating the mass of the Milky Way by hand）

The Sun orbits at a distance $r = 8.2\ \mathrm{kpc}$ from the galactic centre with speed $v = 235\ \mathrm{km/s}$. We use Proposition 3.2 to find the mass inside the Sun’s orbit. Since $1\ \mathrm{kpc} = 3.086 \times 10^{19}\ \mathrm{m}$, we have $r = 2.53 \times 10^{20}\ \mathrm{m}$ and $v = 2.35 \times 10^{5}\ \mathrm{m/s}$.

$$M = \frac{v^2 r}{G} = \frac{(2.35\times 10^{5})^2 \times 2.53 \times 10^{20}}{6.674\times 10^{-11}} = 2.1 \times 10^{41}\ \mathrm{kg}$$

Dividing by the solar mass $M_\odot = 1.99 \times 10^{30}\ \mathrm{kg}$ gives $1.1 \times 10^{11}\, M_\odot$, roughly a hundred billion solar masses. So far this is consistent with the statement that the Milky Way contains about a hundred billion stars.

The trouble lies further out. Even at $r = 50\ \mathrm{kpc}$ the rotation speed remains close to $200\ \mathrm{km/s}$. Computing the enclosed mass with the same formula, with $r = 1.54\times 10^{21}\ \mathrm{m}$ and $v = 2.0\times 10^{5}\ \mathrm{m/s}$,

$$M = \frac{(2.0\times 10^{5})^2 \times 1.54\times 10^{21}}{6.674\times 10^{-11}} = 9.2\times 10^{41}\ \mathrm{kg} = 4.6\times 10^{11}\, M_\odot$$

But beyond $8.2\ \mathrm{kpc}$ there are hardly any luminous stars compared with the interior. If the luminous matter stopped at $1.1\times 10^{11}\,M_\odot$, the rotation speed at $50\ \mathrm{kpc}$ would have to be

$$v = \sqrt{\frac{6.674\times10^{-11} \times 2.1\times 10^{41}}{1.54\times 10^{21}}} = 9.5\times 10^{4}\ \mathrm{m/s} = 95\ \mathrm{km/s}$$

less than half the observed value. A factor of two in speed is a factor of four in mass. The conclusion is that the invisible mass exceeds the visible mass by more than a factor of three.

3.3. What about “the law of gravity is wrong”?

A natural objection arises here. “Rather than postulating invisible matter, would it not be more economical to suppose that gravity deviates from Newton’s law at large distances?” The question is a reasonable one, and in fact Mordehai Milgrom proposed such a theory in 1983: MOND (Modified Newtonian Dynamics), which changes the gravitational law in regions where the acceleration is smaller than $a_0 \simeq 1.2 \times 10^{-10}\ \mathrm{m/s^2}$.

Example 3.4（The Bullet Cluster: matter and gravity pulled apart）

In 2006 Clowe and collaborators studied the “Bullet Cluster” (1E 0657-56), a pair of galaxy clusters caught in the act of colliding. There, two things can be measured separately:

1. The distribution of ordinary matter (the hot gas that makes up most of a cluster’s mass) — visible in X-rays.
2. The distribution of the total mass — reconstructed from the gravitational lensing that distorts the images of background galaxies.

The result was dramatic. The hot gas had been slowed by friction in the collision and left behind in the middle, while the centres of gravity sat on either side, in two lumps, as if they had passed straight through. The source of gravity is not where the ordinary matter is. Explaining this by modifying the law of gravity is extremely awkward; explaining it as “there is invisible matter that passes through even in a collision” is natural.

Remark 3.5

MOND reproduces rotation curves remarkably well on the scale of individual galaxies. On the scale of clusters, in the ratio of the heights of the acoustic peaks of the CMB, and in cases such as Example 3.4, it runs into difficulty. The mainstream view today is dark matter, but “why MOND works so well at galactic scales” remains an open question in its own right. Unsolved problems nest inside one another.

Candidates for the identity of dark matter include as-yet-undiscovered particles: WIMPs (weakly interacting massive particles), axions and primordial black holes. Detectors buried deep underground (XENONnT, LUX-ZEPLIN and others) have been waiting for decades, but as of the mid-2020s no definitive signal has been captured. WIMPs, long regarded as the leading candidate, are having their allowed parameter space squeezed steadily narrower.

4. Accelerating expansion — dark energy

4.1. A discovery nobody expected

In the 1990s two international teams were pursuing the same goal: to use distant Type Ia supernovae as “standard candles” and measure how much the expansion of the universe was decelerating. Gravity attracts, so the expansion had to be slowing down. The only issue, supposedly, was by how much.

In 1998 both teams produced the same answer, and it was one nobody had expected. Distant supernovae were fainter than predicted under decelerating expansion — by about 0.2 magnitudes at redshift $z \simeq 0.5$. Since magnitude $\Delta m$ and brightness (flux) $F$ are related by $F \propto 10^{-0.4\Delta m}$, a value $\Delta m = 0.2$ means a brightness factor of $10^{-0.08} = 0.83$, that is, 17% fainter. Brightness falls off as the inverse square of distance, so the distance is larger by a factor $1/\sqrt{0.83} = 1.10$, about ten per cent. The supernovae had travelled further than predicted: the expansion was accelerating.

Definition 4.1（Dark energy and the cosmological constant）

The energy component with negative pressure that accelerates the expansion of the universe is called dark energy. When the ratio $w = p/\varepsilon$ of its pressure $p$ to its energy density $\varepsilon$ equals $w = -1$ and does not vary in time, it is equivalent to the cosmological constant $\Lambda$ of the Einstein equations, and is interpreted as a constant energy density carried by space itself (the energy of the vacuum).

That acceleration requires negative pressure is counter-intuitive. Roughly speaking, in general relativity pressure is also a source of gravity, and the expansion decelerates when $\varepsilon + 3p$ is positive and accelerates when it is negative. For $w = -1$ we get $\varepsilon + 3p = -2\varepsilon < 0$, so the expansion does indeed accelerate. Ordinary matter ($p \simeq 0$) and ordinary radiation ($p = \varepsilon/3$) do not manage this.

For this discovery Perlmutter, Schmidt and Riess received the 2011 Nobel Prize in Physics. The irony is that the cosmological constant — introduced by Einstein in 1917 to produce a static universe, and withdrawn after the expansion was discovered as what he is said to have called the greatest blunder of his life — came back after eighty years. A physicist’s “blunder” is often a correct answer arrived at too early.

4.2. A discrepancy of 120 orders of magnitude

What exactly is the problem with dark energy? Let us put a number on it.

Example 4.2（The density of dark energy is three protons' worth）

Take $\rho_c = 8.5\times 10^{-27}\ \mathrm{kg/m^3}$ from Example 2.2, multiply by $\Omega_\Lambda = 0.685$, and multiply by $c^2$ to convert to an energy density.

$$\varepsilon_\Lambda = 0.685 \times 8.5\times 10^{-27} \times (3.00\times 10^{8})^2 = 5.2\times 10^{-10}\ \mathrm{J/m^3}$$

The rest energy of a proton is $938\ \mathrm{MeV} = 1.50\times 10^{-10}\ \mathrm{J}$, so this is the energy of 3.5 protons per cubic metre. The agent driving the accelerated expansion of the universe is, as a density, absurdly dilute.

Remark 4.3（The vacuum energy catastrophe）

In quantum field theory the vacuum too has energy, because each mode of oscillation carries a zero-point energy $\hbar\omega/2$. Summing over all modes diverges, so one cuts the sum off at the highest energy for which the theory is trusted. Taking the cutoff at the Planck energy, where gravity begins to matter, the vacuum energy density comes out at roughly $10^{113}\ \mathrm{J/m^3}$.

The observed value is the $10^{-10}\ \mathrm{J/m^3}$ of Example 4.2. The ratio is $10^{123}$. Even lowering the cutoff to the electroweak scale leaves a mismatch of some $10^{55}$. This discrepancy is known as the cosmological constant problem, and since Weinberg’s 1989 review it has been regarded as one of the most serious open problems in theoretical physics. The disagreement is not two or three orders of magnitude. It is 120.

On the observational side, effort continues to determine whether $w$ is really $-1$ or varies in time (the possibility of a dynamical field, called quintessence). At present $w$ agrees with $-1$ to within a few per cent, but the precision is not yet sufficient. Whether dark energy is the energy of the vacuum or something else entirely rests with future observations.

5. The first instant — inflation

5.1. Two embarrassments for Big Bang cosmology

Big Bang cosmology gave a splendid account of the abundances of the light elements and of the existence of the CMB. But it left two uncomfortable features behind.

Example 5.1（The horizon problem: places that cannot have met are at the same temperature）

The CMB is light from the moment, about 380,000 years after the birth of the universe, when electrons and protons combined and the universe became transparent. Its temperature is $2.725\ \mathrm{K}$ in every direction on the sky, with directional differences of only about $10^{-5}$.

Now, at that moment, how far could information have travelled (the particle horizon)? In a matter-dominated expanding universe, including the effect of the scale factor growing as $a \propto t^{2/3}$, the horizon size is $3ct$. With $t = 3.8\times 10^{5}\ \mathrm{yr} = 1.2\times 10^{13}\ \mathrm{s}$,

$$d_{\text{hor}} = 3 \times 3.00\times 10^{8} \times 1.2\times 10^{13} = 1.1\times 10^{22}\ \mathrm{m}$$

The universe has since expanded by a factor of about 1090, so in today’s ruler this is $1.2\times 10^{25}\ \mathrm{m} \simeq 380\ \mathrm{Mpc}$. The distance to the surface that emitted the CMB, on the other hand, is about $13900\ \mathrm{Mpc}$ in today’s ruler. The angle this horizon subtends on the sky is therefore only

$$\theta \simeq \frac{380}{13900} = 0.027\ \mathrm{rad} = 1.6^\circ$$

Covering the whole sky, $4\pi\ \mathrm{sr}$, with circles of diameter $1.6^\circ$ (solid angle $\pi(0.8^\circ)^2 = 6.1\times 10^{-4}\ \mathrm{sr}$) takes about twenty thousand of them. In other words the CMB sky is made of roughly twenty thousand regions that have never once been in contact with one another, and all of them are at the same temperature. It is as if twenty thousand candidates who had never exchanged a word before the exam handed in papers agreeing to five decimal places.

The other is the flatness problem.

Proposition 5.2（The flatness problem）

In the Friedmann equation for a homogeneous and isotropic universe,

$$H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2}$$

($a$ the scale factor, $k$ the spatial curvature), writing the total density parameter as $\Omega = \rho/\rho_c$ in the notation of Definition 2.1, we have

$$\Omega - 1 = \frac{kc^2}{\dot{a}^2}$$

Consequently $|\Omega - 1| \propto a^2$ during radiation domination ($a \propto t^{1/2}$) and $|\Omega - 1| \propto a$ during matter domination ($a \propto t^{2/3}$), so that in either case $|\Omega-1|$ grows with time. That is, $\Omega = 1$ is an unstable state.

Proof（Proposition 5.2）

Divide both sides of the Friedmann equation by $H^2$. Using $H = \dot{a}/a$,

$$1 = \frac{8\pi G \rho}{3H^2} - \frac{kc^2}{a^2 H^2} = \Omega - \frac{kc^2}{\dot{a}^2}$$

and rearranging gives $\Omega - 1 = kc^2/\dot{a}^2$. Since $k$ and $c$ are constants, the time dependence of $|\Omega - 1|$ is governed by $\dot{a}^{-2}$ alone.

During radiation domination $a \propto t^{1/2}$, so $\dot{a} \propto t^{-1/2}$ and hence $\dot{a}^{-2} \propto t \propto a^2$. During matter domination $a \propto t^{2/3}$, so $\dot{a} \propto t^{-1/3}$ and hence $\dot{a}^{-2} \propto t^{2/3} \propto a$. In both cases $|\Omega-1|$ grows as $a$ grows.

Here is why that is a problem. Present observations give $|\Omega_0 - 1| < 0.01$. Trace this value back into the past. Back to matter–radiation equality ($a$ equal to $1/3400$ of today’s value) the quantity scales as $a$, so $|\Omega-1| < 0.01/3400 = 3\times 10^{-6}$; from there back to the epoch of Big Bang nucleosynthesis ($a$ equal to $4\times 10^{-10}$ of today’s value, that is $1.4\times 10^{-6}$ times the value at equality) it scales as $a^2$, so

$$|\Omega - 1| < 3\times 10^{-6} \times (1.4\times 10^{-6})^2 \simeq 6\times 10^{-18}$$

was required. At three minutes of age the universe had to differ from flatness by less than one part in $10^{17}$. It is like standing a pencil on its point and having it stay up for 13.8 billion years. Dismissing this with “it just happened to be so” is unsatisfying — that is the flatness problem.

∎

5.2. Inflation as a prescription

In 1981 Alan Guth and Katsuhiko Sato independently proposed the picture that the universe underwent a brief episode of exponential expansion immediately after its birth. Refinements by Linde, Albrecht and Steinhardt brought it to its present form.

Definition 5.3（Inflation）

The hypothesis that in the early universe (roughly between $10^{-36}$ and $10^{-32}$ seconds after the beginning) there was a period in which the scale factor grew exponentially as $a \propto e^{Ht}$ (with $H$ nearly constant) is called inflation. The expansion factor is thought to have been at least $e^{60} \simeq 10^{26}$.

This single stroke solves both problems.

• The horizon problem. The entire universe we now observe was contained, before inflation, in a single region far smaller than the horizon. Of course its temperature is uniform: everybody was originally in the same classroom.
• The flatness problem. During inflation $H$ is constant, so $\dot{a} = Ha \propto e^{Ht}$. From the relation in Proposition 5.2, $|\Omega-1| \propto e^{-2Ht}$, and the universe is driven exponentially towards flatness. With 60 e-folds the factor is $e^{-120} \simeq 10^{-52}$. It is exactly the metaphor of a balloon: blow it up enough and every part of the surface looks flat.

There was a welcome bonus as well: quantum fluctuations during inflation are stretched out and become the density fluctuations that later seed galaxies. Moreover the prediction was quantitative — the spectrum of those fluctuations should be nearly scale-invariant (comparable at all sizes) but weighted very slightly towards large scales, so that the index $n_s$ comes out a little below $1$.

The Planck satellite measured $n_s = 0.965 \pm 0.004$. Not $1$, but close to $1$, exactly as predicted.

5.3. Why it is nevertheless unsolved

With so much going right, why is inflation still an “unsolved problem”?

First, we do not know what drove inflation. A scalar field called the inflaton is presumed responsible, but its identity, the shape of its potential and its relation to any particle of the Standard Model are all undetermined. Hundreds of models have been proposed. When hundreds of models make the prediction that came true, that is not what we call an established theory.

Second, decisive evidence is still missing. Inflation should produce not only density fluctuations but primordial gravitational waves, which would leave a swirling “B-mode” pattern in the polarisation of the CMB. The tensor-to-scalar ratio $r$ measures its strength. In 2014 the BICEP2 team announced a detection at $r \simeq 0.2$, to great excitement; a year later most of the signal turned out to come from dust within our own galaxy. The current upper limit is $r < 0.036$ (BICEP/Keck, 2021), and many models are being ruled out.

Note

The BICEP2 episode tends to be told as a cautionary tale, but less than a year passed between announcement and refutation. It is equally an instance of independent verification working and an error being corrected in public. Science is trustworthy not because it does not make mistakes, but because it has mechanisms for finding them built in.

Third, in many models inflation once started never stops, spawning innumerable “bubble universes” (eternal inflation). If that is right, our universe is one of a multiverse — but the other bubbles are unobservable in principle. From this arises a dispute that touches the methodology of physics itself: can a prediction that cannot be falsified count as a scientific prediction?

6. Two theories in collision — quantum gravity

6.1. Where the quarrel breaks out

The two pillars of twentieth-century physics are each perfect on their own ground. General relativity deals with heavy objects, large scales and smooth spacetime. Quantum mechanics deals with light particles, small scales and probabilistic superposition（Definition 2.1）［Schrödinger's Cat］. As long as their territories are far apart, the two never quarrel.

The trouble comes with things that are heavy and small: specifically, the centre of a black hole and the instant of the universe’s birth. There one must use both theories at once, and using both at once makes the calculation collapse.

Where that boundary lies can be found by dimensional analysis alone.

Definition 6.1（Planck units）

The units of length, time and energy that can be built from the Newtonian constant $G$, the reduced Planck constant $\hbar$ and the speed of light $c$ alone are called the Planck length, Planck time and Planck energy. They mark the scale at which gravity ($G$), quantum mechanics ($\hbar$) and relativity ($c$) all matter simultaneously.

Proposition 6.2（Dimensional analysis of the Planck length）

The only quantity with the dimension of length that can be formed from $G$, $\hbar$ and $c$ is, up to a dimensionless multiple,

$$\ell_P = \sqrt{\frac{\hbar G}{c^3}}$$

Its value is $\ell_P = 1.62 \times 10^{-35}\ \mathrm{m}$.

Proof（Proposition 6.2）

Set $\ell = G^{\alpha}\hbar^{\beta}c^{\gamma}$. The dimensions of the three quantities are

$$[G] = \mathrm{m^3\,kg^{-1}\,s^{-2}}, \quad [\hbar] = \mathrm{kg\,m^2\,s^{-1}}, \quad [c] = \mathrm{m\,s^{-1}}$$

(the second because $\hbar$ is an energy times a time, $= \mathrm{kg\,m^2 s^{-2}} \times \mathrm{s}$). Matching the exponents dimension by dimension so that $[\ell] = \mathrm{m}$,

$$\begin{aligned} \mathrm{m}\ &: \quad 3\alpha + 2\beta + \gamma = 1 \\ \mathrm{kg}\ &: \quad -\alpha + \beta = 0 \\ \mathrm{s}\ &: \quad -2\alpha - \beta - \gamma = 0 \end{aligned}$$

The second equation gives $\beta = \alpha$. Substituting into the third gives $\gamma = -2\alpha - \beta = -3\alpha$. Putting these into the first gives $3\alpha + 2\alpha - 3\alpha = 1$, that is $2\alpha = 1$, so $\alpha = 1/2$. Hence $\beta = 1/2$ and $\gamma = -3/2$, giving $\ell_P = \sqrt{\hbar G/c^3}$. The solution of the linear system is unique, so no other form is possible.

Now the numbers. With $\hbar G = 1.055\times 10^{-34} \times 6.674\times 10^{-11} = 7.04\times 10^{-45}$ and $c^3 = 2.69\times 10^{25}$,

$$\ell_P = \sqrt{\frac{7.04\times 10^{-45}}{2.69\times 10^{25}}} = \sqrt{2.61\times 10^{-70}} = 1.62\times 10^{-35}\ \mathrm{m}$$

This is twenty orders of magnitude below the size of a proton (about $10^{-15}\ \mathrm{m}$) — an outrageously small length.

∎

Corollary 6.3（The Planck energy and the size of an accelerator）

The energy corresponding to the Planck length is $E_P = \sqrt{\hbar c^5/G} \simeq 1.22\times 10^{19}\ \mathrm{GeV} = 1.22\times 10^{16}\ \mathrm{TeV}$. A single proton beam at the LHC has an energy of $6.8\ \mathrm{TeV}$, so the ratio is about $1.8\times 10^{15}$.

Proof（Corollary 6.3）

Energy has dimension $\mathrm{kg\,m^2\,s^{-2}}$. Following the same procedure as in the proof of Proposition 6.2, setting $E = G^{\alpha}\hbar^{\beta}c^{\gamma}$ gives $3\alpha + 2\beta + \gamma = 2$ for $\mathrm{m}$, $-\alpha + \beta = 1$ for $\mathrm{kg}$, and $-2\alpha - \beta - \gamma = -2$ for $\mathrm{s}$. The second gives $\beta = \alpha + 1$. Adding the first and third gives $\alpha + \beta = 0$, and combining the two yields $\alpha = -1/2$, $\beta = 1/2$, $\gamma = 5/2$. Hence $E_P = \sqrt{\hbar c^5/G}$. Numerically, from $\hbar c^5 = 1.055\times 10^{-34} \times 2.42\times 10^{42} = 2.55\times 10^{8}$,

$$E_P = \sqrt{\frac{2.55\times 10^{8}}{6.674\times 10^{-11}}} = \sqrt{3.83\times 10^{18}} = 1.96\times 10^{9}\ \mathrm{J}$$

Dividing by $1\ \mathrm{GeV} = 1.602\times 10^{-10}\ \mathrm{J}$ gives $1.22\times 10^{19}\ \mathrm{GeV}$. The LHC beam is $6.8\ \mathrm{TeV} = 6.8\times 10^{3}\ \mathrm{GeV}$, so the ratio is $1.22\times 10^{19}/6.8\times 10^{3} = 1.8\times 10^{15}$.

∎

Example 6.4（If we built an accelerator reaching the Planck energy）

For a circular accelerator with a given magnet technology, the energy attained is roughly proportional to the circumference. The LHC is $27\ \mathrm{km}$ around, so multiplying by the ratio from Corollary 6.3,

$$27\ \mathrm{km} \times 1.8\times 10^{15} = 4.8\times 10^{16}\ \mathrm{km}$$

Dividing by $1$ light year $= 9.46\times 10^{12}\ \mathrm{km}$ gives about 5000 light years. The Milky Way is 100,000 light years across, so a good fraction of the galaxy would be filled with accelerator. We shall not ask about the budget.

This is why quantum gravity cannot be probed experimentally. That theorists are forced into speculation is not laziness but a physical constraint.

6.2. What breaks

If one tries to treat general relativity naively as a quantum field theory, infinities appear in the calculations. Infinities as such are nothing unusual in particle physics: as long as the procedure of “renormalisation” can push them into a finite number of parameters, the theory remains usable. The Standard Model succeeded on exactly those terms.

For gravity this does not work. In 1986 Goroff and Sagnotti showed that pure Einstein gravity in vacuum has a divergence at two loops that cannot be removed. Absorbing the infinities would require infinitely many parameters, and the theory loses its predictive power. The current understanding is that general relativity is an excellent effective theory at low energies, but at the Planck scale the theory itself must be replaced.

Another serious symptom is the black hole information problem. Between 1974 and 1976 Hawking showed that a black hole emits thermal radiation and evaporates. If the radiation is perfectly thermal, the information about whatever fell in is lost forever. But time evolution in quantum mechanics is unitary — it preserves information. Either general relativity or quantum mechanics must be wrong. The problem is still under active study as an examination question telling us what conditions a theory of quantum gravity has to satisfy.

6.3. The candidates

Theory	View of spacetime	Strength	Weakness
Superstring theory	Particles are strings, not points; spacetime is ten-dimensional	Perturbatively consistent and contains gravity; derived the entropy of certain black holes microscopically	Experimental test effectively impossible; some $10^{500}$ choices of vacuum
Loop quantum gravity	Space itself is a discrete network	Requires no background spacetime; areas and volumes are quantised	Whether it reproduces general relativity in the low-energy limit is unsettled
Asymptotic safety	Ordinary field theory throughout, reaching a fixed point at high energy	Stays within the existing framework	Existence of the fixed point relies on approximate calculations
Causal sets	Spacetime is a discrete partially ordered set of events	Treats causal structure as the most fundamental structure	The dynamics is not yet constructed

None is decisive. Despite more than ninety years of effort, there is still no theory one can call the right one. The problem Feynman and Einstein worked on is still sitting on the desk (their personalities are the subject of Famous Physicists).

Quantum mechanics, for its part, has an unresolved problem at its own foundation: the measurement problem — what physically happens when “observation collapses the wave function”. For details see the measurement problem（Remark 5.1）［Schrödinger's Cat］ and Schrödinger’s Cat.

7. Plenty more unsolved problems

For reasons of space this will be brisk, but here are problems of comparable importance. The “big four” are not the only open ones.

Problem	Content	Status
Matter–antimatter asymmetry	If the Big Bang made matter and antimatter in equal amounts, they should have annihilated leaving only light. In fact about one proton of matter survived per $10^{9}$ photons	CP violation in the Standard Model is not large enough
Neutrino masses	Oscillations established that they have mass, but neither the absolute values nor the reason they are more than $10^{6}$ times lighter than other particles is known	Cosmology bounds the sum of the masses at about $0.12\ \mathrm{eV}$
The strong CP problem	The strong interaction is allowed to violate CP symmetry, yet experiment bounds the violation below $10^{-10}$	A proposed solution invokes the axion; searches are under way
The hierarchy problem	Why is the Higgs mass $125\ \mathrm{GeV}$ rather than the Planck scale?	Supersymmetry was the leading candidate, but nothing has been found at the LHC
The Hubble tension	The expansion rate from the CMB, $67.4$, disagrees at the $5\sigma$ level with $73.0\ \mathrm{km/s/Mpc}$ from nearby supernovae	Debated as systematics versus new physics
High-temperature superconductivity	The mechanism by which cuprates superconduct above $100\ \mathrm{K}$ is not understood	Conventional BCS theory cannot explain it
Turbulence	The Navier–Stokes equations are known, yet turbulence statistics cannot be derived from first principles	Even mathematically, existence and smoothness of solutions is open

The last one, turbulence, is continuous with the difficulty of the statistical physics treated in Maxwell’s Demon (Boltzmann's entropy（Definition 2.2）［Maxwell's Demon］). Between obtaining the fundamental equations and understanding the world there is still a wide gap.

The day physics is finished does not look like arriving any time soon. Had the advisers of the late nineteenth century been right, the physics of the twentieth century would not exist. If someone tells you today that all the important things are done, they are probably wrong again.

8. Exercises

Exercise 8.1Easy

Suppose that in the outskirts of the Milky Way, at $r = 30\ \mathrm{kpc}$ from the centre, the rotation speed is still $v = 200\ \mathrm{km/s}$. Find the total mass inside this radius in units of the solar mass. You may use $1\ \mathrm{kpc} = 3.086\times10^{19}\ \mathrm{m}$, $G = 6.674\times 10^{-11}\ \mathrm{m^3 kg^{-1} s^{-2}}$ and $M_\odot = 1.99\times 10^{30}\ \mathrm{kg}$.

Solution

Use $M(r) = v^2 r/G$ from Proposition 3.2. In SI units, $r = 30 \times 3.086\times 10^{19} = 9.26\times 10^{20}\ \mathrm{m}$ and $v = 2.00\times 10^{5}\ \mathrm{m/s}$.

$$M = \frac{(2.00\times 10^{5})^2 \times 9.26\times 10^{20}}{6.674\times 10^{-11}} = \frac{4.00\times 10^{10} \times 9.26\times 10^{20}}{6.674\times 10^{-11}} = 5.55\times 10^{41}\ \mathrm{kg}$$

Dividing by the solar mass,

$$\frac{5.55\times 10^{41}}{1.99\times 10^{30}} = 2.8\times 10^{11}\, M_\odot$$

about 280 billion solar masses. That is more than 2.5 times the $1.1\times 10^{11}\,M_\odot$ inside the Sun’s orbit found in Example 3.3, and there are almost no luminous stars between $8.2\ \mathrm{kpc}$ and $30\ \mathrm{kpc}$. Dark matter is what fills the gap.

Exercise 8.2Standard

Using the numbers in Example 2.2 and Example 4.2, find by what factor the energy density of dark energy exceeds the rest-energy density of ordinary (baryonic) matter.

Solution

From Example 2.2, the mass density of ordinary matter is $\rho_b = 4.2\times 10^{-28}\ \mathrm{kg/m^3}$. Converting to a rest-energy density,

$$\varepsilon_b = \rho_b c^2 = 4.2\times 10^{-28} \times 9.00\times 10^{16} = 3.8\times 10^{-11}\ \mathrm{J/m^3}$$

Dark energy is $\varepsilon_\Lambda = 5.2\times 10^{-10}\ \mathrm{J/m^3}$, so the ratio is

$$\frac{5.2\times 10^{-10}}{3.8\times 10^{-11}} = 14$$

about 14. This agrees with the ratio of the density parameters, $0.685/0.049 = 14$ (as it must, which makes it a useful check). Stars, planets and human beings are a supporting cast worth less than a fourteenth of the cosmic energy budget.

Exercise 8.3Standard

By the same method as in Proposition 6.2, find the quantity with the dimension of time (the Planck time) $t_P$ that can be built from $G$, $\hbar$ and $c$, and compute its value.

Solution

Set $t = G^{\alpha}\hbar^{\beta}c^{\gamma}$ and write down the conditions for $[t] = \mathrm{s}$. Using the same table of dimensions as in the proof of Proposition 6.2,

$$\begin{aligned} \mathrm{m}\ &: \quad 3\alpha + 2\beta + \gamma = 0 \\ \mathrm{kg}\ &: \quad -\alpha + \beta = 0 \\ \mathrm{s}\ &: \quad -2\alpha - \beta - \gamma = 1 \end{aligned}$$

The second gives $\beta = \alpha$. The first gives $\gamma = -3\alpha - 2\beta = -5\alpha$. Substituting into the third gives $-2\alpha - \alpha + 5\alpha = 1$, that is $2\alpha = 1$, so $\alpha = 1/2$. Hence $\beta = 1/2$, $\gamma = -5/2$ and

$$t_P = \sqrt{\frac{\hbar G}{c^5}}$$

Numerically, from $\hbar G = 7.04\times 10^{-45}$ and $c^5 = (3.00\times 10^{8})^5 = 2.42\times 10^{42}$,

$$t_P = \sqrt{\frac{7.04\times 10^{-45}}{2.42\times 10^{42}}} = \sqrt{2.91\times 10^{-87}} = 5.4\times 10^{-44}\ \mathrm{s}$$

This agrees with $\ell_P / c = 1.62\times 10^{-35}/3.00\times 10^{8} = 5.4\times 10^{-44}\ \mathrm{s}$: it is the time light takes to cross a Planck length. About the universe before this much time had elapsed since the Big Bang, present-day physics has nothing to say.

Exercise 8.4Hard

Some claim that “dark matter is just ordinary matter that happens to be too dim to see — faint stars, planets, cold gas”. Give two reasons, based on different observations, why this cannot be so.

Solution

Reason 1: Big Bang nucleosynthesis. In the first few minutes after the birth of the universe, deuterium, helium-4 and lithium-7 were built from protons and neutrons. The amounts produced depend sensitively on the density of protons and neutrons (baryons) at that time; deuterium in particular is burned away more thoroughly the higher the baryon density. Working backwards from the observed abundances of deuterium and helium gives a baryon density of $\Omega_b \simeq 0.05$, only one sixth of the total matter density $\Omega_m \simeq 0.315$. Faint stars, planets and cold gas are all made of protons and neutrons, so they fall under this ceiling. It is not enough.

Reason 2: the acoustic peaks of the CMB. Before recombination, baryons coupled to light oscillated in a tug of war between gravity and radiation pressure. Those oscillations are imprinted on the temperature fluctuations of the CMB as a series of peaks, and the ratio of the heights of the first and second peaks depends on the ratio of the amount of baryons to the amount of dark matter. The Planck measurements show with high precision that matter not coupled to light is about five times as abundant as baryons. This is independent evidence for matter other than baryons.

(As a supplement, a third reason can be given. In the Bullet Cluster of Example 3.4, the ordinary matter was decelerated by the collision while the centres of gravity passed straight through; faint stars or gas would be affected by the collision to some degree. Furthermore, if large numbers of faint stars or planet-sized bodies existed, they would produce gravitational microlensing brightenings as they crossed in front of background stars — and the searches for this, such as the MACHO survey, did not find the required numbers.)

References

• Planck Collaboration, “Planck 2018 results. VI. Cosmological parameters”, Astronomy & Astrophysics 641 (2020), A6. arXiv:1807.06209 — every cosmological parameter used in this article is taken from here.
• V. C. Rubin and W. K. Ford Jr., “Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions”, The Astrophysical Journal 159 (1970), 379 — the classic paper on flat rotation curves.
• D. Clowe et al., “A Direct Empirical Proof of the Existence of Dark Matter”, The Astrophysical Journal 648 (2006), L109. arXiv:astro-ph/0608407 — the Bullet Cluster.
• S. Weinberg, “The cosmological constant problem”, Reviews of Modern Physics 61 (1989), 1 — the classic review of the cosmological constant problem.
• A. H. Guth, “Inflationary universe: A possible solution to the horizon and flatness problems”, Physical Review D 23 (1981), 347 — the original paper on inflation.
• Takahiko Matsubara, Gendai Uchūron: Jikū to Busshitsu no Kyōshinka (Modern Cosmology: The Coevolution of Spacetime and Matter), University of Tokyo Press, 2010 (in Japanese) — a textbook covering everything from the Friedmann equations to inflation.
• Hirosi Ooguri, Jūryoku towa Nanika: Einstein kara Chōgenri Riron e, Uchū no Nazo ni Semaru (What Is Gravity? From Einstein to Superstring Theory), Gentosha Shinsho, 2012 (in Japanese) — a popular account surveying quantum gravity without equations.

Appendix: On the numbers in this article

Sources. The cosmological parameters ($H_0 = 67.4\ \mathrm{km/s/Mpc}$, $\Omega_b = 0.049$, $\Omega_c = 0.265$, $\Omega_\Lambda = 0.685$, $n_s = 0.965$) are rounded values based on the Planck 2018 TT,TE,EE+lowE+lensing analysis. They are used to about two significant figures; the finer digits vary with the analysis method.

Galactic numbers. The Sun’s distance from the galactic centre, $8.2\ \mathrm{kpc}$, and its rotation speed, $235\ \mathrm{km/s}$, are representative values from recent estimates including those from the Gaia satellite. The rotation speed of $200\ \mathrm{km/s}$ at $50\ \mathrm{kpc}$ is an approximate figure inferred from the motions of outer satellite galaxies and globular clusters, with an uncertainty of about $\pm 20\ \mathrm{km/s}$. The conclusion of Example 3.3 — that the invisible mass is more than three times the visible mass — is unaffected by errors of this size.

Treatment of approximations. The calculation of the angle subtended by the horizon in Example 5.1 used $d_{\text{hor}} = 3ct$, valid under matter domination, together with a comoving distance to the CMB of $13900\ \mathrm{Mpc}$. Treating the contribution of radiation and the change in expansion history due to dark energy exactly would shift the result by tens of per cent, but the conclusion “somewhere between $1^\circ$ and $2^\circ$” stands. The ratio $10^{123}$ in Remark 4.3 is likewise an estimate depending on the choice of cutoff; following convention, we speak of ”$120$ orders of magnitude”.

Numbers still in motion. The Hubble tension ($67.4$ against $73.0\ \mathrm{km/s/Mpc}$) and the upper limit $r < 0.036$ on the tensor-to-scalar ratio are figures likely to change as observations advance. This article describes the situation in the mid-2020s; that an article about unsolved problems goes out of date in a few years should be taken as evidence that research is moving.