Scaling Factor, Hubble's Parameter, and the Age of the Universe

In this lesson, we'll introduce three concepts which are essential to cosmology: the scaling factor, Hubble's parameter, and Hubble's law. The scaling factor allows us to define the distance between galaxies as a function of time. Hubble's law, which was deduced from measurements made by the astronomer Edwin Hubble, demonstrates that the distance $$D$$ between galaxies is proportional to their recessional speed $$V$$ away from us; in other words, the farther away from us a galaxy is, the faster it is moving away from us. Hubble deduced that $$D∝V$$ by taking numerous measurement of the distances $$D$$ of speeds $$V$$ of many other galaxies, plotting the measurements on a graph, and then realizing that the line of regression passing through all of the points should be a straight line. The slope of this line, today, is known as Hubble's constant; but the ratio $$D/V$$ is not constant with time. The slope of $$D(V)$$ vs. $$V$$ as a function of time is known as Hubble's parameter. We'll also show in this lesson that we can use Hubble's parameter and Hubble's law to estimate the age of the universe to be ~14 billion years old. This estimate, however, is a little off. The correct answer is actually 13.8 billion years old.

Spectroscopy

According to Planck's relation each element in the periodic table emits only discrete frequencies of light, not a continuous spectrum. Also, each element has its own unique signature and emits particular frequencies and wavelengths of light. An instrument called a spectroscope allows us to measure and record the particular frequencies of light emitted by a substance which, in turn, allows us to determine what particular kinds of elements that substance is comprised of. Using this technique, astronomers were able to determine the compositions of the atmospheres of other planets, of the Sun and other stars, and of entire galaxies.

Friedman-Robertson-Walker (FRW) Equation

Using Newton's law of gravity and shell theorem, we can derive the FRW equation. The FRW equation is a differential equation whose solution is the scaling factor $$a(t)$$. The FRW equation essentially give us the relationship between the distribution of mass/energy and the scaling factor; we can use the FRW equation to find out what $$a(t)$$ for various different kinds of mass/energy distributions including a universe dominated by matter, one dominated by radiation, or one dominated by vacuum energy. We'll see that the FRW equation predicts scenarios where the universe could be expanding. One might point out, and correctly so, that this contradicts Newton's law of gravity since, according to this law, gravity cannot act as a repulsive force. To reconcile this (and, we'll do this in a separate lesson), we could derive the FRW equation from Einstein's general theory of relativity by plugging the FRW metric into the Einstein Field Equations; plugging in the FRW metric, one could simplify the Einstein Field Equations to the FRW equation. This essentially would resolve the contradiction of the FRW equation predicting that, under the right circumstances, the universe could be expanding because general relativity (unlike Newton's law of gravity) allows for things like expanding universes.

Solving the FRW Equation for the Scaling Factor in different scenarios

In this lesson we'll solve the FRW equation (one of the EFE's) for the scaling factor $$a(t)$$ to determine the expansion rate of the universe in two different idealized scenarios: a universe filled with only radiation and a universe filled with only matter. These two different scenarios are called a radiation dominated universe and a matter dominated universe, respectively.

Dark Energy

In this lesson we'll use the FRW equation to solve for the scaling factor $$a(t)$$ in a universe dominated by vacuum energy or dark energy. We'll see that the scaling factor grows exponentially with time; this means that not only is the size of the universe getting bigger and bigger with time, but the rate at which it is doing so is increasing. In other words, the expansion of the universe is accelerating with time.

Surface of Last Scattering

According to the Big Bang theory, for hundreds of thousands of years the entire universe was hotter than a star. The universe was so hot that electrons could not bind with atomic nuclei; this meant that the universe was opaque to radiation. If you were living in such a universe, you wouldn't be able to see very far; in fact, you would only be able to "see things" that are microscopic distances away. But as the universe expanded, everything cooled down. What we'll do in this lesson is calculate that after about 300,000 years since the origin of the universe, the universe cooled enough (due to expansion) for electrons to bind with atomic nuclei. After this happened, light was no longer constantly scattering off of nucleons in a plasma soup; once the universe cooled enough for all of the plasma (which was everywhere in the universe) in the universe to become a gas, light could travel through space freely without constantly "bumping into things."

Genesis of the Elements

During the 20th century, scientists discovered the genesis of the elements in the periodic table. Einstein's theory of gravity precipitated a revolution and renascence period in cosmology; it transformed our picture of the large-scale universe. We learned that, contrary to the ideas which has prevailed for centuries since Newton's time, the universe is actually expanding and that in some distant epoch—the earliest moments of the young universe—all of the matter and energy in the universe must have been on top of each other and concentrated into a single, very small amount of space no bigger than the size of a grape fruit. The temperatures and pressures were so extreme in this early universe that hydrogen and helium could be formed. Later, the universe cooled and vast aggregates of atoms condensed into galaxies and stars. In the latter-half of the 20th century, we learned that the heavier elements in the periodic table were created in the nuclear furnaces and death-roes of the stars. We really are made of star stuff!

Cosmic Microwave Background Radiation

The mapping of the Cosmic Microwave Background Radiation (CMBR) was hailed by Stephen Hawking as one of the greatest achievements in 20th-century science because it gave us an image of our "baby universe" when it was very young. The CMBR proved, beyond a shadow of a doubt, that the Big Bang theory—a prediction of Einstein's general theory of relativity—is correct. Before the CMBR, many did not think of cosmology as a serious science since it lacked high-precision experiments and measurements like astronomy; many viewed cosmology as merely theoretical speculation that could never be confirmed by experiment. Many people view the mapping of the CMBR as the turning point when cosmology transitioned from being purely theoretical to a truly experimentally rigorous science.

Bentley's and Olber's Paradoxes

Shortly after Isaac Newton published his law of gravitation, the philosopher Richard Bentley and the astronomer Heinrich Olbers pointed out two paradoxes that arise from this law. The first, which is called Bentley's paradox, points out that if the universe is finite in size then, since the force of gravity is always attractive, all of the stars and galaxies in the universe should collapse in on themselves. The second, called Olbers' paradox, states that if the universe is infinite and if the distribution of stars in the universe is uniform then the night sky should be filled with infinitely many stars and the night sky should therefore be blindingly bright.

Dark Matter

Since most of the mass in our home Milky Way galaxy is at its center we would expect, from Newton's law of gravity and second law, that the tangential velocity of each star in our galaxy would increase with increasing distance away from the Milky Way's center. But observations showed that, actually, the tangential velocity of each star is roughly constant. There were many hypotheses put forward around the time of this discovery which attempted to reconcile Newton's laws with experiment. But today, the prevailing explanation of why the tangential velocities of the stars in our galaxy is roughly constant is the theory that an invisible substance known as dark matter pervades the galaxy. Researchers speculate that this substance is comprised of particles which do not interact with light or radiation and are therefore invisible.

Origin of Structure and "Clumpyness"

Fundamental questions which have been asked since the beginning of humanity are: How did structure arise in the cosmos? Why is there something rather than nothing? Why is there “clumpyness” in the cosmos (in other words, why is there more stuff here than there)? For a long time, answering such questions would have been impossible. But in the past few decades cosmologists, by using the scientific method and advanced technology, have unraveled these deep and profound mysteries which humans have long pondered since their origin. Let’s start out by answering the second question: Why is there something rather than nothing? The fundamental principles of quantum mechanics predict a myriad of mathematical and physical consequences. One of them is the time-energy uncertainty principle which predicts that over any given time interval, the observed energy of a particle is always uncertain and will be within a range of values because you must give it a “kick” to observe it. The energy of something can never be exactly zero (because then $$ΔE=0$$ and the time-energy uncertainty principle would be violated). Therefore, not even the vacuum of empty space can have no energy. This energy of empty space can be transformed into mass (a property of matter) according to Einstein’s mass-energy equivalence principle and the rules of particles physics. Since the possible energy values of any region of space are random within a certain range of values, it follows that the distribution of energy, mass, and matter also must be random throughout space. But if the fundamental principles of quantum mechanics and the time-energy uncertainty principle in particular explain the origin of matter and energy in the universe, then how did the cosmos attain structure and clympyness? Over the next few paragraphs, we’ll begin a discussion which starts with analyzing the predicament (namely, applying quantum mechanics alone predicts that the universe should have been too uniform for galaxies and other structures to arise); then, after that, I’ll take you through a very brief account of how the universe developed structure and clumpyness.

In the earliest epochs of our universe, all of the matter and energy existed in the form of virtual particles popping in and out of existence. If the distribution of this matter was random then, on average, there would be just as many virtual particles over here as over there. But in order for structure to arise in the cosmos, the distribution of matter must start out clumpy and not almost completely uniform as the uncertainty principle predicts. The uncertainty principle predicts that the distribution of matter and virtual particles throughout space starts out more or less uniform. This seems to contradict today’s observations since if all of the matter density is initially uniform, it’ll more or less stay uniform. The Big Bang theory is widely regarded as one of the greatest triumphs in science of the 20th century. It successfully predicted an enormous range of things: the CMBR; where the light elements came from as well as the correct amounts and proportion of each of them; and the observed redshifts and recessional velocities of distant galaxies. But there were many things which the Big Bang theory could not account for such as how the universe got to its present size, how the recorded temperatures of the CMBR are so uniform, and how the CMBR has slight fluctuations to one part in $$10^5$$. Cosmologists generally agree that we must postulate the existence of a physical mechanism which correctly predicts all of these observed features. This is precisely what inflationary theory accomplishes. Many have criticized the physical mechanisms postulated by inflationary theory as ad hoc despite the success in inflationary theory of predicting what has been experimentally observed.

The universe is presently 13.8 billion years old and we are causally connected with all of the matter and energy within a sphere whose radius is 13.8 billion light years. When the universe was only one second old, its temperature was $$10^{10}K$$; the photons buzzing around back then were 3.7 billion times hotter than their present temperature today of $$2.725K$$. By Boltzman’s relationship $$E∝T$$, Plank’s relationship $$E∝λ$$, and the relationship between wavelength and the scaling factor ($$λ∝a^{-1}$$), it follows that the entire observable universe must have been 3.7 billion times smaller than it is today—or about only 3.7 light years in radius. Back in that distant epoch, since the universe was only one second old, photons emitted by charged particles could, at most, have travelled a distance of one light second. The gravitational field generated by energetic particles propagates at the speed of light and, therefore, would have had enough time to only spread out a distance of one light second. Any signal emitted by an energetic particle therefore could not travel farther than one light second and could not communicate with other energetic particles more than one light second away. Therefore, any pair of energetic particles separated by more than one light second from one another were causally disconnected.

The causal disconnection between matter and energy in a region of the universe 3.7 light years in radius when it was only one second old is significant because it implies that the energy density of regions separated by more than one light-second should be radically different. Therefore, the CMBR should not be so uniform. However, it successfully explains the origin of the observed fluctuations in matter and energy density: all of the particles within very small regions move towards one another due to gravity without effecting distant regions. This is how the first fluctuation in matter density originated. The clumpyness of matter was initially very slight; but over enormous time intervals (billions of years), gravity amplified this clumpyness to form the cosmic web, filaments, galaxy superclusters and clusters and groups, starts, planets, moons, comets, asteroids, and all of the large-scale structure and clumpyness we see in the universe today. On a medium size scale, the force of electromagnetism pulls and subsequently binds together atoms and molecules to form complex chemistry—the basis of all living organisms. This is the origin of structure and clumpyness on a medium size scale. The strong nuclear force binds together protons and neutrons in order to keeps atoms held together and form structure and clumpyness on an even smaller scale. The slight non-uniformities in matter density imparted by quantum fluctuations eons ago and three fundamental forces (gravity, electromagnetism, and the strong force), together, seeded and eventually developed all of the structure and clumpyness in the cosmos we see today.