The Evolution & Fate of the Universe

Posted 2004-02-09 (月) 午後6:04 by terra

The ultimate fate of the universe holds a perpetual fascination with mankind. Ever since Hubble discovered in 1929 that the universe is expanding, people have wondered whether the universe would continue expanding forever, or whether its expansion would one day come to a halt and be reversed, leading eventually to a “big crunch”. In this essay, you should trace through the pioneering work of Hubble, Einstein and others, and review modern cosmological theories as well as current evidence for the fate of the universe.

A Static Universe
A New Era
Towards a New Theory of Cosmology
The Expanding Universe
New Cosmological Models
The Cosmic Microwave Background
The Inflationary Universe
The Age of the Universe
The Omega Point
Conclusion
Bibliography

A Static Universe

When Issac Newton (1642-1727) published his masterpiece Philosophiae Naturalis Principia Mathematica in 1687, it brought him international approbation. For not only did his laws of motion and gravity seem to apply to all things, it suggested a new age of philosophy – that the world was really a complex cosmic machine, and that all things in the Universe were components of it. Unlike teleological views of centuries past, Nature was deterministic; the future state of anything that is in motion can be predicted from knowledge of its existing state.

In the Principia, Newton postulated a law of Universal Gravitation, which states that the force of the attraction between two given particles is inversely proportional to the square of their distance apart. This law, together with his second law of mechanics, correlated to the three laws of Johannes Kepler, recorded more than half a century before.

Newton believed in the concept of a static, unchanging universe, as was the general thought of the time. It appeared logical then (and even to now), to assume that time and space were absolute and independent of each other. Newton’s Law of Universal Gravitation did however create a quandary. Since all celestial bodies attract each other, they should eventually be pulled together – it could not explain why stars had not yet collapsed in on themselves after all this time.

Nevertheless, Newton’s model of an infinite static universe became adopted as the standard model of the Universe.

A New Era

New theories and discoveries seek to supersede old ones. Just as the Copernican-Newtonian revolution toppled the centuries-old Aristotle system, the dawn of the Twentieth century proved to be the start of a new wave of physics that would challenge the two-hundred-year supremacy of Newtonian physics.

In 1905, Albert Einstein, while working as a third class Technical Expert in the Swiss Patent Office, released his first scientific paper on special relativity – "Zur Elektrodynamik bewegter Körper" ("On the Electrodynamics of Moving Bodies"). It was to become a revelation in the scientific world for it abandoned the Newtonian concept of absolute space and absolute time.

The paper held that "light is always propagated in empty space with a definite velocity c which is independent on the state of motion of the emitting body" and that the laws of physics are the same in all inertial reference frames. Mass and Energy are equivalent; no matter can travel faster than light.

However, Einstein noticed a flaw in his theory of special relativity, which assumes the absence of gravitational fields. In Newtonian physics, gravitation is a force. Since the laws of physics are the same in all inertial reference frames, therefore according to special relativity, gravitational effects observed should be the same for observers in different reference frames. However, this is not the case.

For example, an astronaut accelerating towards Earth by Earth’s gravity would feel weightless. To another observer standing on Earth, in the same gravitational field however, he would feel the force of gravity. Therefore the effects of acceleration are relative.

In 1907, Einstein formulated the equivalence principle where he established that that the effects of acceleration are not differentiable from the effects of a uniform gravitational field. This was a consequence of the equivalence of gravitational mass and inertial mass. The free-fall acceleration of an object in a uniform gravitational field is proportional to the gravitational field strength. Since gravitational mass is equivalent to inertial mass, therefore gravity and acceleration due to it are equal.

	a = (m_g/m_i) * g		where acceleration a is proportional to gravitational field strength g.
	Since m_g= m_i —> m_g/m_i= 1
	Therefore a = g

The equivalence principle led Einstein to embark on a new theory of gravity, which he called the theory of general relativity. It was published in 1915.

In general relativity, gravity is depicted as a distortion in the geometry of spacetime. Four-dimensional spacetime is non-Euclidean – its curvature is dependant on the presence of matter in the Universe. According to Einstein, freely moving bodies move along trajectories of spacetime. These trajectories are called geodesics, the shortest distance between two points in curved spacetime. The ‘phenomena’ of the bending of light in a gravitational field is attributed to the fact that photons always travel along geodesics.

General relativity predicted, among many things, the degree of deflection of starlight as a result of the sun’s gravity, gravitational time dilation, etc. It was also to be the catalyst for the development of new cosmological models of the Universe.

Towards a New Theory of Cosmology

Since Newtonian times, the Universe was believed to be unchanging and eternal. The resilience of Newtonian physics carried that belief to the Twentieth century.

When Einstein used general relativity to explain the Universe however, he discovered to his surprise that his equations eliminated the possibility of a static universe. If general relativity was right, the Universe could only be expanding or contracting. This implied that the Universe had a beginning and possibly an end. Einstein was reluctant to accept this per se, and he considered the possibility that there was something wrong with his equations. Reluctantly, Einstein added into his equations a cosmological constant, which would accommodate his vision of a static cosmological model.

Another supporter of the static universe model was a Dutch cosmologist named Willem de Sitter. In 1917, he published a several papers that discussed the static model using Einstein’s equations. Unlike Einstein however, his model did not require a cosmological constant.

Ironically, it was discovered later that de Sitter’s model did in fact imply an expanding universe. His model assumed that the Universe was empty. But if tiny particles were scattered in it, the result would be an exponentially expanding universe.

The Expanding Universe

In 1920s, an American astronomer Edwin Powell Hubble began his investigations on the structures of nearby nebulae. With the new 100-inch Hooker telescope that was installed on Mt. Wilson, he was able to pick out some Cepheid variables – stars that emit light at predictable varying intensities. With these Cepheids, he was able to estimate the distances of the stars from Earth from their absolute and apparent luminosities.

Hubble discovered that there were some stars that appeared to be much further away than the rest of the stars. He concluded that the stars could not belong to the Milky Way but were part of other galaxies. With this discovery, Hubble began work on classifying observable galaxies, studying their gravitational redshifts and measuring their distances from Earth.

Redshifts were first described by an Austrian physicist Christian Andreas Doppler in his 1842 paper On the Coloured Light of the Double Stars and Certain Other Stars of the Heavens. Doppler observed that the relative motion of a source and a receiver would result in a change in frequency of any signal from the source. Waves from a moving source would be observed to have a higher frequency as it approaches than when it departs. This is due to the fact that successive wave crests take a shorter time to reach the receiver as it approaches. The light spectrum of an object moving away would thus be shifted towards the red.

Hubble found that with the galaxies he was studying, with the exception of some nearby galaxies that showed gravitational blueshifts, most of the galaxies exhibited gravitational redshifts. This meant that the distant galaxies were in fact moving away from the Earth at very fast velocities. He also discovered that the gravitational redshifts were directly proportional to the distance between the galaxies, which implies that distant galaxies recede at greater rates than galaxies that are closer by. This became known as Hubble’s Law.

The implications were enormous. Contrary to Einstein’s view of a static universe, observational evidence demonstrated that the Universe was expanding. In 1929, Hubble announced his discovery to the world.

New Cosmological Models

Seven years before Hubble concluded that the Universe was expanding, a Russian mathematician Aleksandr Friedmann put to print a paper that discussed various possible universal models based on Einstein’s general relativity equations. He, like Einstein, realised that the equations of general relativity did not allow for a static universe model. But unlike Einstein, he was not biased towards the static universe belief. Friedmann recognised that without Einstein’s cosmological constant, the equations of general relativity allowed for more than one possible model of the Universe. Right from the beginning, Friedmann built in the concept of an expanding universe into his models.

According to Friedmann, the Universe is homogeneous and isotropic. Apart from some slight deviations, it is uniform and looks the same in all directions. The Universe is boundless and there is no centre from which it expands. Its rate of expansion is the same everywhere, i.e. Hubble’s law can be applied to any galaxy in it.

The expanding Friedmann models suggest that it is possible to extrapolate to a point in the past whereby the distances between the galaxies were zero. Moving further into the past, the density of the universe would be infinite. In accordance to the theory of general relativity, the curvature of spacetime would also be infinite. This hints at the possibility that the Universe could have emerged from a singularity. This singularity is known as the Big Bang singularity.

According to general relativity, gravity would cause two objects in the Universe to eventually fall into each other. An expanding universe whereby all galaxies in the universe are moving further away from each other could offset this attraction, preventing the galaxies from collapsing into each other.

The eventual fate of the universe is hence determined by the rate of its expansion and the amount of matter in the universe. If the average mass density of the Universe exceeds a certain critical figure, the rate of expansion of the universe will not be sufficient to overcome the attracting force of gravity between the galaxies.

There are three different types of Friedmann models based on this theory. The structures of these models are dependant on Ω (Omega), the symbol scientists use to represent the ratio of the average mass density and critical mass density.

The first type of Friedmann model describes a closed universe where Ω > 1. The Universe starts off with the Big Bang singularity, expands to a maximum and eventually collapses onto itself. The geometry of the closed universe non-Euclidean and is analogous to that of a sphere where the galaxies are objects located on its surface.

The second type of universe is an open universe where Ω < 1. It has a beginning, is spatially infinite and it expands forever. Its geometry is the reverse of that of the closed-universe models, and can be likened to the surface of a saddle.

The third type of universe is a flat universe where Ω = 1. It is the boundary between the closed and open universe models. Like the open universe, it begins with the Big Bang and it is spatially infinite. However, unlike the open universe, it does not expand forever. Gravitation causes the rate of expansion to eventually decrease to a near-zero value but it is not quite enough to trigger a contraction.

When they were first published, the Friedmann cosmological models were largely ignored because of the inherent belief in a static universe. They are now however, widely accepted as the foundations of modern cosmology.

Cosmological Models The figure depicts the 3 main possible cosmological models. In the open universe model (a), the universe expands eternally. In the case of the flat universe (b), expansion halts when the universe reaches a certain size. In the closed universe model (c), the universe expands and then recollapses.

The Cosmic Microwave Background

According to the Big Bang theory, the Universe stemmed from a singularity and has been expanding ever since.

In the late 1940s, an American physicist George Gamow quantitatively described the conditions of the early universe. He predicted that the present Universe would be filled with a residue sea of radiation, leftover from the primordial Big Bang. Since the Universe expands without a centre, at the same rate everywhere, according to the Friedman universal models, this radiation would be evenly distributed throughout the Universe. Gamow’s coworkers, Ralph Alpher and Robert Herman estimated this temperature to be approximately 5ºK

In 1965, two American physicists Arno Penzias and Robert Wilson, while working at the Bell Laboratories in New Jersey, discovered by accident a low temperature microwave radiation that was propagating from beyond the Earth’s atmosphere. This cosmic microwave radiation was homogeneous and constant, and though they did not know it at the time of discovery, similar to Gamow’s prediction over a decade before.

In the very early universe, at about the first one-hundredth second after its formation, the universe is in a state of near perfect thermal equilibrium with a temperature of approximately 10¹¹ºK. The Universe at this time is an extremely dense mixture of matter and radiation. Neutrons and protons are constantly being bombarded with the more abundant particles such as electrons and positrons resulting in the interconversion of these protons and neutrons.

As time passes and the Universe expands and cools, conditions become suitable for the formation of stable nuclei like helium and other elements.

The detection of the cosmic background radiation provided strong evidence for the Big Bang theory.

The Inflationary Universe

Some years after the discovery of the uniform background radiation, two worrisome problems became evident. The first problem was this: if the cosmic radiation were perfectly homogeneous, galaxy formation and the irregularities of their distribution would be hard to account for.

This problem was resolved in 1992, when the Cosmic Background Explorer (COBE) discovered minute fluctuations in spectrum of the microwave background.

The second problem was more difficult to answer. Fluctuations in the background radiation meant that different parts of the universe must have had some form of thermal contact at some time during the formation of the universe in order to achieve the present equilibrium temperature of 2.7ºK. Although the expanding Friedmann cosmological models describe a universe that was smaller in the past, there is still not enough time for the radiation to travel from one region of the universe to a remote region to attain thermal equilibrium.

American scientist Alan Guth put forward a solution to this problem. He suggested that the "uniformity of the observed universe is built into the theory by postulating that the universe began in a state of uniformity", and that "the uniformity in temperature throughout the observed universe is a natural consequence of inflation."[1]

In the inflationary theory, the observed universe starts out much smaller than that assumed in the Big Bang theory, enabling the observed universe to attain thermal equilibrium. As about 10-35s, the universe undergoes an interim stage exponential expansion, attaining a size that is predicted by the standard theory. After this initial state of inflation, the universe assumes a state of evolution not unlike that of the standard theory.

The Age of the Universe

The discovery of a ubiquitous background radiation in the Universe substantiated the Big Bang Theory and provided direct evidence of the birth of the Universe from a primeval fireball. This implied that the Universe is of a finite age.

To obtain an approximation of the age of the Universe, one can use the reciprocal of Hubble’s constant. This reciprocal is known as Hubble time.

Hubble’s constant, which gives the value for the rate of expansion of the Universe, can be derived from Hubble’s Law. Hubble’s Law states that the velocity of recession of a remote galaxy is directly proportional to the distance between the galaxy and the observer.

	i.e. v +prop. d		velocity of recession v is directly proportional to distance d
	v = H_od		where H_o is the Hubble Constant

The age of the universe, obtained from the reciprocal, is based on the assumption that the Universe has been undergoing a uniform expansion since it’s birth. Present estimates of Hubble’s constant put it to about 55 km per second per Megaparsec, which makes the age of the Universe to be about 18 billion years old. In reality, Hubble’s constant is not static, but through the influence of gravity, decreases through time, making the Universe younger than that of the Hubble time.

The Omega Point

The average mass density of the universe is not constant – it decreases as the universe expands. As the Universe expands, the critical mass density required to prevent its collapse also decreases. Therefore, Ω can change with time if the critical mass density and average mass density decrease at different rates.

In the open and closed universes, Ω decreases continuously in the former and increases continually in the latter. However, if Ω was precisely at 1 since the beginning, it continues to remain at 1 as the universe expands.

Physicists have reason to believe that our Universe is a flat one. The argument for this is that the mass density of the universe is still approximately equal to the critical mass, billions of years after the Big Bang. The inflationary model suggested by Alan Guth predicts this situation.

Still, the estimated amount of mass in the universe is subject to discrepancies. There is the possibility of the existence of dark matter, which cannot be seen, but can only be detected by its gravitational pull on visible matter.

If there is a significant amount of dark matter in the Universe, it could tip the balance of a unitary Ω to a value larger than 1. Our Universe would then be that of a closed universe: Billions of years after the last flames of our sun extinguishes, the expanding universe will begin to contract. The temperature of the cosmic microwave background will increase, leading to its eventual Omega Point, the Final Crunch where the universe dies in a contracting region of fire not unlike the situation that was first seen in the Big Bang.

Conclusion

"When you finally understand the universe, it will not only be stranger than you imagine, it will be stranger than you can imagine."
- Arthur C. Clarke

Although the Big Bang and inflationary theories are predominantly the most accepted views of the universe today, theoretical studies of the universe have also led to the development of many other interesting propositions of its origins. One example would be oscillating universe model, suggested by Robert Dicke, where the universe undergoes an eternal series of expansion and contraction, and that our universe is merely one epoch of an infinite time-span.

Scientists believe that will one day be a Grand Unified Theory (GUT) that describes the fundamental interactions of all forces[2] in one set of mathematical equations. They believe that a GUT will aid our understanding of the very early moments of the Big Bang.

The possibility that the universe may have a finite lifespan, and that it may end in a Final Crunch is terrifying. A universe without a future is arguably a meaningless one as humanity’s contributions to the universe are ultimately rendered futile.

Nevertheless, while we may be ineffective in influencing the final outcome of the universe, our innate curiosity propels us to learn more about the reasons for our existence.

The search for answers to the origins and evolution of the universe will continue.