Introduction to Cosmology and the Cosmic Microwave Background — Part 1

Introduction to Cosmology and the Cosmic Microwave Background: Lesson 1

1. Introduction and On Understanding the Universe

Why do human beings exist? Perhaps this is one of the hardest questions to answer. Because we do not merely live. We look around, try to understand what we see, and, in time, begin to ask questions. Who are we? Where did we come from? Who came before us? How did our ancestors live? What are the stars we see in the sky? How did the world we inhabit form? And beyond all of these lies the most fundamental question.

What is the universe?

With the limited knowledge at our disposal, we began to seek answers to these questions. Humanity was made for physics. First, we observed our surroundings. We followed the stars. Over time, we learned to measure more precisely, observe more closely, and express the patterns in nature through mathematics. Yet the questions grew along with our knowledge. What does the universe contain? How do stars form? Why does matter exist? What is time? What kind of thing is space?

And... what was there at the beginning of this universe?

Human curiosity took us farther and farther back. Physics seemed to be about asking the most complicated question, but it was not. It was about asking the simplest one: What was there in the beginning?

To understand the present, we began to look into the past. From the motions of galaxies, we learned that the universe is expanding. We saw that stars and galaxies have not always existed in their present forms, and that the universe has changed over time.

Now, suppose we could pick up a camera and rewind into the past. What was happening in the universe's earliest times?

We cannot observe the answer to this question directly :D In other words, taking a camera back to the universe's first moments and seeing what happened there does not seem particularly feasible. But nature leaves traces of its past.

We can look for those traces in light, matter, particles, and the structure of the universe we observe today.

The purpose of physics is not merely to write down formulas. We try to uncover the order behind the phenomena we see in nature. The ideas we use to understand how a particle behaves can sometimes take us all the way to stars and galaxies. And when we try to understand the universe on its largest scales, we may have to return to the physics of the smallest particles.

That is exactly what we will be doing throughout this series.

We will begin with the expansion of the universe, examine the relationship between temperature and energy, and try to understand what kind of environment the early universe was. We will then look at how light and matter interacted in that environment, how photons became free, and how the Cosmic Microwave Background radiation that we still observe today was formed.

We will follow these questions as well. On this journey, from the smallest particles to structures spanning billions of light-years, our aim will not simply be to learn what the universe is, but to try to understand it.

And perhaps the whole story will return to that simple question at the very beginning: How did the universe begin?

The Expansion of the Universe and the Concept of Physical Distance

Our main aim in this section is to understand what the expansion of the universe means and to examine how the physical distance between two points is defined in an expanding space.

We all know how to measure the distance between two points in everyday life. We can use a ruler to measure it directly. But we cannot use the same method for celestial objects. Suppose, for example, that we want to measure the distance between Earth and a distant celestial body. Our first thought might be to use a very long ruler, but it is clear that this is not physically possible :D

Thinking more like physicists, we might instead decide to use light. After all, its speed is constant. We send a light signal from Earth toward the celestial body and start a stopwatch at the moment we send it. When the light signal reaches the object and returns to Earth, we stop the watch. Since we know that the speed of light is constant, we can calculate the object's distance from the light's round-trip travel time.

There is, however, an important point to notice. When making this measurement, we make certain assumptions about the geometry of the space through which the light travels. We assume that space does not change during the measurement and that the distance we wish to measure remains fixed.

But what happens if, while we are making the measurement, not only the celestial objects but space itself changes over time?

In that case, the distance between two celestial objects may change even if the objects do not move through space. In cosmology, therefore, we must study not only the motion of objects, but also how the space containing them changes with time.

Here it is more appropriate to think of galaxies very far apart rather than planets. Gravitationally bound structures such as planetary systems and galaxies do not directly follow the cosmological expansion. The expansion of the universe is observed primarily in the distances between galaxies that are sufficiently far apart and move together on large scales.

Now imagine two galaxies very far from one another. Suppose we neglect their local motions through space. Even so, if we measure the physical distance between them at different times, we may find that it has increased.

This naturally raises the following question:

Are the galaxies moving through space, or is the space between them expanding?

To make this idea easier to understand, imagine two glasses resting on a tablecloth. There are two different ways to increase the distance between them. In the first, we can move one or both glasses across the tablecloth. In the second, we can keep the glasses fixed relative to the cloth and stretch the cloth itself.

In the first case, the glasses move relative to the surface beneath them. In the second, they do not move relative to the surface; yet the distance between them increases because the surface itself changes.

The expanding-universe picture used in cosmology resembles the second case. Galaxies can remain fixed at particular coordinates in expanding space while the physical distance between them increases over time. What changes need not be the coordinate positions of the galaxies. What changes is the geometry of space that determines the physical distance between those coordinates.

We must therefore distinguish between two kinds of motion in cosmology:

1. The local motion of objects through space
2. Recession caused by the expansion of space

We will later call an object's local motion through space its peculiar velocity. Recession due to cosmological expansion, by contrast, arises because the scale of space changes with time.

To understand the expansion of the universe more clearly, let us now use a simple balloon analogy. We will examine how the distance between two points on the surface of an inflating balloon changes.

Imagine a balloon and choose two points on its surface.

Let us call these points $A$ and $B$. Before the balloon is inflated, let the physical distance between $A$ and $B$ be

$$d_1$$

Now imagine that we inflate the balloon. As its surface expands, the distance between $A$ and $B$ will increase:

Let us denote the new physical distance between the two points after the balloon has been inflated by

$$d_2$$

In that case,

$$d_2 > d_1$$

The most important point here is that $A$ and $B$ do not move across the balloon's surface on their own. They remain at the same positions relative to the surface. Nevertheless, because the balloon's surface expands, the physical distance between them increases.

The points therefore do not have to move over the surface for the distance between them to increase. An expansion of the surface itself can also increase that distance.

Similarly, in an expanding universe, the physical distance between two galaxies can change over time even if their coordinate positions in space do not. The distance between the galaxies may increase not because they are moving through a fixed space, but because the scale of the space between them grows with time.

At this point, the fundamental question we need to answer is:

How is the physical distance between two galaxies defined in an expanding universe?

To answer this question, let us first consider the coordinate distance that describes the positions of galaxies in expanding space. Before doing so, however, we need to understand why any definition of distance necessarily requires an observer.

1.1 Observer Dependence and Comoving Observers

When we speak of an object's position or motion in physics, we must specify the reference frame relative to which that position or motion is defined. Whether an object is moving or at rest depends on the observer we choose. The same object may appear stationary to one observer and moving to another.

A similar situation arises in cosmology. To describe the positions of galaxies in an expanding universe, we choose special observers who follow the expansion of space. These observers do not move through space on their own; they are carried along with expanding space.

Let us think about this using the balloon analogy. Place two observers at points $A$ and $B$ on the balloon's surface. Suppose they are stuck to the surface and cannot walk across it.

As the balloon inflates, the observers do not move across its surface under their own power. Each remains at the point where they began. Yet because the balloon's surface expands, the physical distance between the observers increases over time.

Definition: Such observers are called comoving observers.

Here, “comoving” does not mean that the observers move toward or away from one another under their own velocity. On the contrary, they move with the flow of expanding space. In other words, they are at rest with respect to the expansion of space.

Now imagine that we draw coordinate lines on the balloon. Let the line on which observer $A$ is located be $L_A$, and the line on which observer $B$ is located be $L_B$. As the balloon inflates, the physical distance between these lines increases. Nevertheless, observer $A$ remains on $L_A$ and observer $B$ remains on $L_B$. In other words, their coordinates do not change.

Therefore, the coordinate difference between $L_A$ and $L_B$ does not change with time either. The observers have fixed coordinates on the expanding surface.

Definition: The distance between these fixed coordinates is called the comoving distance.

Comoving distance is usually denoted by $r$.

Thus, even as the balloon expands, the comoving distance between observers $A$ and $B$ remains constant:

$$r = \text{constant}$$

The quantity that remains constant here is not the actual physical distance between the observers. What stays fixed is the coordinate distance that specifies their positions within the expanding coordinate system.

As the balloon inflates, the physical distance between the observers increases, while the distance between their comoving coordinates remains unchanged. We therefore arrive at two different notions of distance for the same pair of observers: comoving distance, the coordinate distance that does not change with the expansion; and physical distance, the actual distance that changes over time because of the expansion.

This distinction forms the basis for defining distance in an expanding universe. We can now establish the relationship between the constant comoving distance and the time-dependent physical distance.

2. Definition of Physical Distance

Let us return once more to points $A$ and $B$ on the balloon. We previously said that these points were attached to the balloon's surface and did not move across it on their own as the balloon inflated. Their comoving coordinates therefore remain unchanged.

As the balloon inflates, however, the geometry of its surface changes and the actual distance between the two points increases. Thus, although the comoving distance remains fixed, the actual distance that could be measured with a ruler depends on time.

Physical distance. The actual, measurable distance between two points at a given instant is called the physical distance.

Let us denote physical distance by $d_{\text{phys}}(t)$. Here, $t$ indicates that the distance may change with time.

If we measure the arc length between $A$ and $B$ while the balloon is small, we obtain a certain value. If we measure the arc length between the same two points again after inflating the balloon, we obtain a larger value.

The comoving coordinates of the points have not changed. What has changed is the scale of the balloon's surface, which determines the physical length between those coordinates.

We must therefore distinguish between the following two concepts:

$$d_{\text{com}} = \text{the fixed distance in the expanding coordinate system}$$

and

$$d_{\text{phys}}(t) = \text{the actual physical distance measured at a given instant}$$

To establish the relationship between these two distances, we now need a quantity that describes how much the balloon has grown.

Let the balloon's initial radius be $R_0$, and let its radius at any time $t$ after inflation be $R(t)$.

We can express how much the balloon's radius has grown relative to its initial value through the ratio

$$a(t) = \frac{R(t)}{R_0}$$

Scale factor. The time-dependent, dimensionless quantity that indicates how much all physical lengths in space have grown or shrunk relative to a chosen initial time is called the scale factor and is denoted by $a(t)$.

At the initial time,

$$R(t_0) = R_0$$

and therefore

$$a(t_0) = \frac{R_0}{R_0} = 1$$

Thus, we have normalized the scale factor to equal one at the initial time.

If the balloon is expanding,

$$R(t) > R_0$$

and consequently

$$a(t) > 1$$

If the balloon is shrinking, on the other hand,

$$a(t) < 1$$

The scale factor tells us how the constant comoving distance is converted into the physical distance at a particular time. The physical distance between two comoving observers is given by

$$d_{\text{phys}}(t) = a(t)\, d_{\text{com}}$$

The physical meaning of this relation is very important. The comoving distance $d_{\text{com}}$ does not change with time. But because the scale factor $a(t)$ multiplying it does change, the physical distance changes with time as well.

Since $a(t_0) = 1$ at the initial time,

$$d_{\text{phys}}(t_0) = d_{\text{com}}$$

At a later time, if the universe has expanded and the scale factor has increased, then

$$d_{\text{phys}}(t) > d_{\text{com}}$$

Thus, even if two galaxies remain fixed in comoving coordinates, the physical distance between them can increase. This increase does not arise because the galaxies are moving within the coordinate system, but because the physical scale between the coordinates grows with time.

This relation is the foundation for defining distance in an expanding universe. In this lesson and those that follow, we will spend quite some time explaining what $a(t)$ is and what effects it has on the universe. Before that, however, let us continue by explaining one of the most important constants in physics.

3. Deriving the Hubble Relation

We obtained the relationship between physical distance and comoving distance as

$$d_{\text{phys}}(t) = a(t)\, d_{\text{com}}$$

Our next step is to determine how quickly this physical distance changes with time. Before taking the derivative, however, we need to spend a little time on what we mean by a “constant” in physics.

What Does “Constant” Mean in Physics?

We encounter many constants in physics. The speed of light $c$, Planck's constant $h$, the gravitational constant $G$, and the Hubble constant $H_0$ are some examples. Yet these quantities are not all “constant” in the same sense.

It is important here to distinguish among three different cases: fundamental constants of nature, constants determined through measurement, and physical parameters that express a value at a particular time.

Planck's constant, for example, has an exactly fixed value and is a constant by definition. The gravitational constant $G$, on the other hand, is determined experimentally. Whether these fundamental constants vary in nature, and how they may be related to one another, are subjects of detailed scientific research.

The situation is somewhat different when it comes to the Hubble constant. At first glance, the name Hubble “constant” may suggest that this quantity remains unchanged throughout the entire history of the universe. After all, it is a “constant”—why would it change?

To avoid conceptual confusion, we should remember the following. While $H(t)$ is a time-dependent quantity, we call the present-day value of this parameter the Hubble constant.

Hubble parameter. The time-dependent quantity that gives the fractional expansion rate of the universe at a particular instant is called the Hubble parameter and is denoted by $H(t)$.

Hubble constant. The present-day value of the Hubble parameter is called the Hubble constant and is denoted by

$$H_0 = H(t_0)$$

Here, $t_0$ represents the present cosmic time. The Hubble constant is determined from observations. Its measured value therefore carries experimental and observational uncertainties. Results obtained through different observational methods may also differ.

Let us now derive the physical origin of the Hubble parameter step by step.

The Change in Physical Distance with Time

Consider two comoving galaxies that are sufficiently far apart. Because they are carried along with the expansion of space, their comoving coordinates do not change.

For the comoving distance, we can therefore write

$$\frac{d}{dt} d_{\text{com}} = \dot{d}_{\text{com}} = 0$$

Here, the dot notation denotes a derivative with respect to time:

$$\dot{X} = \frac{dX}{dt}$$

We previously defined the relationship between physical distance and comoving distance as

$$d_{\text{phys}}(t) = a(t)\, d_{\text{com}}$$

Now let us differentiate both sides of this equation with respect to time:

$$\dot{d}_{\text{phys}}(t) = \frac{d}{dt}\left[a(t)\, d_{\text{com}}\right]$$

Applying the product rule gives

$$\dot{d}_{\text{phys}}(t) = \dot{a}(t)\, d_{\text{com}} + a(t)\,\dot{d}_{\text{com}}$$

Because the galaxies' comoving coordinates do not change, $\dot{d}_{\text{com}} = 0$. The second term therefore vanishes:

$$\dot{d}_{\text{phys}}(t) = \dot{a}(t)\, d_{\text{com}}$$

We can write the comoving distance in terms of the physical distance. From our original relation,

$$d_{\text{com}} = \frac{d_{\text{phys}}(t)}{a(t)}$$

Substituting this into the derivative equation gives

$$\dot{d}_{\text{phys}}(t) = \dot{a}(t)\,\frac{d_{\text{phys}}(t)}{a(t)}$$

Rearranging, we arrive at

$$\dot{d}_{\text{phys}}(t) = \frac{\dot{a}(t)}{a(t)}\, d_{\text{phys}}(t)$$

The change in physical distance with time gives the cosmological recession velocity between the two galaxies. Let us denote this velocity by

$$v_{\text{rec}}(t) = \dot{d}_{\text{phys}}(t)$$

Here, $v_{\text{rec}}$ differs from the galaxies' local velocity through space. It is the rate at which their physical separation increases because of the expansion of space.

Recession velocity. The rate of change, due to cosmic expansion, of the physical distance between two comoving galaxies is called the recession velocity. Thus,

$$v_{\text{rec}}(t) = \frac{\dot{a}(t)}{a(t)}\, d_{\text{phys}}(t)$$

The ratio that repeatedly appears in this equation,

$$\frac{\dot{a}(t)}{a(t)}$$

is called the Hubble parameter:

$$H(t) = \frac{\dot{a}(t)}{a(t)}$$

Substituting this definition into the previous equation, we obtain

$$v_{\text{rec}}(t) = H(t)\, d_{\text{phys}}(t)$$

This relation is called the Hubble relation or Hubble's law.

The Physical Meaning of the Hubble Parameter

The Hubble parameter is defined by

$$H(t) = \frac{\dot{a}(t)}{a(t)}$$

Here, $\dot{a}(t)$ is the rate of change of the scale factor with time. But using $\dot{a}(t)$ alone is not enough to describe the universe's expansion rate, because the same amount of change can have different physical meanings for scale factors of different magnitudes.

We therefore divide $\dot{a}(t)$ by the scale factor itself. In this way, we obtain the fractional change in the scale factor per unit time.

In other words, the Hubble parameter tells us not how much the universe grows in absolute terms, but how rapidly it expands relative to its current size.

Interpreting the Hubble Relation

The relation we obtained,

$$v_{\text{rec}}(t) = H(t)\, d_{\text{phys}}(t)$$

tells us that recession velocity is directly proportional to physical distance.

Since $H(t)$ is the same for all comoving observers at the same cosmic time, a more distant galaxy has a greater recession velocity.

For example, if one of two galaxies is twice as far from the observer as the other, then, when we consider cosmological expansion alone, its recession velocity is also twice as large.

This result can be summarized as

$$\text{as } d_{\text{phys}}(t) \text{ increases, } v_{\text{rec}}(t) \text{ increases}$$

This does not, however, mean that distant galaxies are being hurled outward from a center through a fixed and unchanging space. The physical distance increases because all the space between them is expanding.

Returning to the balloon analogy, two points that are farther apart have more balloon surface between them. When each small region of the balloon expands by the same proportion, the total increase in distance is greater for points with more surface between them.

Similarly, the greater the physical distance between two galaxies, the more expanding space there is between them. This is why more distant galaxies appear to have greater cosmological recession velocities.

Peculiar Velocity and Cosmological Recession Velocity

Real galaxies do not merely participate in the expansion of the universe. They can also move locally through space because of the gravitational fields around them.

When considering a galaxy's total physical velocity, we can separate it into two contributions: the recession velocity due to cosmological expansion and the galaxy's local peculiar velocity. Symbolically, the galaxy's total observed radial velocity can be written as

$$v_{\text{total}} = v_{\text{rec}} + v_{\text{pec}}$$

For nearby galaxies, peculiar velocities can be significant compared with the cosmological expansion velocity. Consequently, not every galaxy in our local neighborhood follows the Hubble relation perfectly. Some nearby galaxies may be approaching us because of their local gravitational motion.

On sufficiently large cosmological scales, however, the effect of peculiar velocities becomes relatively small, and the overall expansion behavior emerges more clearly. But have we ever observed this?

Yes. So far, under the assumption of an expanding universe, we have examined how physical distance and recession velocity change. But how do we know that this assumption is correct? In the following installments, we will seek an answer to this question.