Princeton Frontiers in Physics

Peter Fisher

• Publisher: Princeton University Press
• Year: 2022
• ISBN:
  9780691185910
• Topic: Physical Sciences
• Subtopic: Condensed Matter

What we know about dark matter and what we have yet to discover Astronomical observations have confirmed dark matter's existence, but what exactly is dark matter? In What Is Dark Matter?, particle physicist Peter Fisher introduces readers to one of the most intriguing frontiers of physics. We cannot actually see dark matter, a mysterious, nonluminous form of matter that is believed to account for about 27 percent of the mass-energy balance in the universe. But we know dark matter is present by observing its ghostly gravitational effects on the behavior and evolution of galaxies. Fisher brings readers quickly up to speed regarding the current state of the dark matter problem, offering relevant historical context as well as a close look at the cutting-edge research focused on revealing dark matter's true nature.Could dark matter be a new type of particle—an axion or a Weakly Interacting Massive Particle (WIMP)—or something else? What have physicists ruled out so far—and why? What experimental searches are now underway and planned for the near future, in hopes of detecting dark matter on Earth or in space? Fisher explores these questions and more, illuminating what is known and unknown, and what a triumph it will be when scientists discover dark matter's identity at last.

1

SOME BACKGROUND

Gravity plays a central role in everything that happens in the universe, and we will need to know a little about gravity to understand the dark matter story. The original ideas of gravity came from Galileo Galilei and Johannes Kepler in the early 1600s. Isaac Newton developed the full theory in 1687, explaining both the motion of objects acted on by a force and how massive bodies produce the forces that act on one another. In the nineteenth century, experiments began to show that Newton’s laws of motion were not strictly obeyed, leading to Albert Einstein’s Special Theory of Relativity in 1905, which gave universal laws of motion, with Newton’s laws as approximations for objects moving at much less than the speed of light. In 1916, Einstein’s General Theory of Relativity showed that gravity produced its effects through changes in the structure of space and time, with Newton’s law of gravitation giving an approximation for weak gravitational forces. Einstein’s theory has remained unmodified ever since.

We begin with essential background: Section 1.1 describes the relationship among mass, weight, and energy; Newton’s law of gravitation; and distance scales in the universe. Mass, weight, and energy have precise meanings that are necessary for understanding how gravity works. Next, we will look at how Newton’s law of gravitation exerts forces on distant objects, which will be essential for understanding how we know about dark matter. Since gravity works between distant bodies, Section 1.2 describes the typical sizes and separations of planets, stars, and galaxies in the expanding universe. Sections 1.3 and 1.4 describe the redshift phenomenon—crucial for measuring velocities in the cosmos—and dark energy.

1.1 MASS, WEIGHT, AND ENERGY

In 1687, Newton published Principia, in which he laid down three laws of motion. Newton’s first law defines inertia and why objects in motion will remain in motion. The second law defines force as a change in momentum. The second law says the force on an object is the product of the object’s mass and its acceleration:

$$\begin{array}{lll}\hfill \mathrm{force}=\mathrm{mass}\times \mathrm{acceleration}.& & \hfill \text{(1.1)}\end{array}$$

We can then measure the mass of something by computing the force acting on it, measuring its acceleration, and forming the ratio of force to acceleration:

$$\begin{array}{lll}\hfill \mathrm{mass}=\frac{\mathrm{force}}{\mathrm{acceleration}}.& & \hfill \end{array}$$

The second law is a kinematic law, meaning it relates quantities of motion like velocity, force, mass, acceleration, and so on. A kinematic law tells us how objects respond to a force, while a dynamical law tells us what the forces are.

Figure 1.1. Two equal-mass bodies exert equal gravitational force on each other.

Figure 1.2. Doubling the mass of one body doubles the gravitational force felt by both bodies.

Newton’s third law explains how a force acting between two massive bodies causes an equal and opposite force on each body.

Principia also set forth Newton’s theory of gravity—a dynamic law of gravitational force between two bodies. While the same law applies whether the bodies are at rest or moving, for the purposes of example, if we label the bodies 1 and 2, and assume they are at rest, Newton’s dynamic law of gravitation says:

• The force acting on each body has the same strength for each body and acts along the line connecting the two bodies (Fig. 1.1).

• The size of the force is proportional to the mass of each body separately—double the mass of either body, and the force acting on both bodies doubles (Fig. 1.2).

• The force diminishes as the square of the distance between the bodies—double the distance, and the force on each body goes down by a quarter (Fig. 1.3).

Figure 1.3. Doubling the distance between the bodies decreases by a quarter the gravitational force between them.

Figure 1.4. Doubling one body’s mass doubles the acceleration of the other body.

When astronomers observe distant bodies moving under the force of gravity, they can measure their positions, velocities, and accelerations. We can combine the kinematic second law (Eq. 1.1) and the dynamic law shown in Fig. 1.1 to get the acceleration of the two bodies from their gravitational interaction. If we focus on body 1, the gravitational law says that the force on body 1 is proportional to its mass m₁. The second law says the acceleration, which we can measure, is equal to the force on body 1 divided by its mass m₁, which means the acceleration on body 1 from gravity does not depend on its mass. Thus if we double m₁, the mass of the first body, the acceleration of body 1 does not change, and the acceleration of body 2 doubles (Fig. 1.4).

Figure 1.5. Doubling one body’s mass doubles the acceleration of the other body.

Figure 1.6. Doubling the distance between two bodies quarters the acceleration of each body.

• If the mass of the second body, m₂, doubles, the acceleration of body 1 doubles, and acceleration on body 2 remains the same (Fig. 1.5).

• If the distance between the objects doubles, the acceleration on each decreases by a quarter (Fig. 1.6).

Since the acceleration of an object from gravity does not depend on the object’s mass, two objects starting at the same point in space and moving in the same direction will follow the same trajectory, even if one object is a bowling ball and the other is a feather. In the universe, this means structures like our solar system, galaxies, and clusters of galaxies stay together more than they would if they were subject to some other force for which acceleration depends on mass.² Think of two solar systems that are far apart in comparison with the largest planetary orbit in each, and that are stationary relative to each other. In solar system 1, the planet farthest away from solar system 2 will have a smaller acceleration than the planet closest to solar system 2 because of the diminution of its acceleration with distance. Both planets will accelerate away from the star at the center of the solar system. This is an example of a force, called a tidal force, acting to pull the solar system apart.

Think about using a rocket whose engine produces a force to deflect an asteroid on course to hit Earth. If the asteroid is the size of a softball, the rocket will easily push the asteroid away. If the asteroid is huge—say, the size of a ship—the rocket will push the asteroid more slowly. The mass of the ship-sized asteroid is much larger than the mass of a softball-sized asteroid; so, given the same force, the smaller asteroid will move out of the way faster.

People sometimes say “weight” when they mean “mass.” Weight expresses the force of gravity on an object near the Earth’s surface. We measure weight in pounds and convert pounds to kilograms by dividing the weight by 2.2 pounds per kilogram, which is fine on Earth. If we measured the weight of The Physics of Energy by Jaffe and Taylor (an excellent book, and the second-heaviest book I own) using a spring scale, we would find it weighs 5 lb. 13 oz., or 5.8 lb. A spring scale measures the amount of gravitational force acting on a body by balancing the gravitational force against the force a spring must apply to keep the object from falling. The book’s weight, 5.8 lb., is a measure of the force that the Earth’s mass exerts on the book and is equal to the book’s mass times the gravitational acceleration near the Earth’s surface, 9.8 m/s². On Earth, the conversion factor from the force measurement in pounds to a mass in kilograms works out to 0.455 kilograms/pound, so the book has mass of 2.6 kg.

If we took the book to the Moon, its mass would still be 2.6 kg. However, near its surface, the Moon has a gravitational acceleration of 1.63 m/s², or 1/6 of Earth’s. If we used the same spring scale as we used on Earth, it would measure 1/6 as much force on the Moon as on Earth, and the book would appear to weigh just under 1 lb. Weight depends on the force of gravity at the object’s location, while mass remains the same everywhere.

If you have some knowledge of special relativity, you will notice the expressions above are classical, not relativistic. I mostly use the classical expressions here, because they are accurate enough just about anywhere we will need them in our exploration of dark matter.

A rich array of particles and radi...

Table of contents

• Cover Page
• Series Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Introduction: The Dark Matter Problem
• 1. Some Background
• 2. Evidence for Dark Matter from Astronomy
• 3. Normal Matter: The Standard Model
• 4. What Dark Matter Is Not
• 5. Searching for WIMPs on Earth
• 6. Searching for Dark Matter in Space
• 7. Searching for Axions
• 8. Epilogue
• Glossary
• Suggested Readings
• Index