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HOW BIG IS OUR UNIVERSE?
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Cosmic Concerns - Theories of Cosmology Cosmic Concerns - Theories of Cosmology

HOW BIG IS OUR UNIVERSE?

The question is perplexing. In a cosmology where the universe is considered to be infinitely big now from a starting point that was infinitely small, it's hard to put your finger on it. In mythology, I do not recall from where, there is a similar conundrum and an enlightening tale.

Once upon a time there was a great Sultan who had a lovely young daughter with whom he was much pleased and to whom he lavished every fine thing in his realm. However, she became seriously depressed and appeared in need. When he asked the child the nature of her distress, the princess replied that she did not like it when the moon failed to be present in the evening sky, and how she wished to have the moon in her possession to rectify this inconsistency. The Sultan knew this was a very serious request to consider and sought advice from his council who explained to him how big it was and cautiously hedged on how difficult it might be to obtain. But as sultans do, he offered a great reward to anyone in his kingdom who could satisfy his daughter's request, and added the usual grievous penalties for failure to discourage trivial attempts. One brave young and handsome lad accepted the challenge with the caveat of a simple request; that he be able to visit the young princess before he set upon his task. To make a long story short, he was, he did and he asked the sad child what she perceived the moon to be. After she described the moon to be a lustrous ivory sphere about the size of her thumbnail, the task was quickly accomplished, the young man ultimately gained the young woman's hand, and all lived happily ever after.

The Thumbnail

Can we do the same thing? Is the universe any bigger than we can perceive even in our enlightened way, and any more constrained than a child's fantasy? In addition to it's vast size a principle problem is that everything is changing and there is little that may be used as a touchstone. Two assertions that are accepted as fundamental and stable: time flows uniformly, and the velocity of light is a constant.

La Grande Poof

The "Big Bang" cosmology, or if you prefer a more elegant foreign title; "La Grande Poof", has great merit as a paradigm. Aside from the stark name, many including myself have moved away from the more serene Steady State concepts of years ago, and it will be used as the basis of our inquiry.

The receding galaxies and the Doppler shift of their radiation yield the now famous Hubble Constant which defines their proportional rate with distance. Conversely in reverse time, and because it is linear, the galaxies converge to a single point at a single time if their courses are reversed. Therefore this beginning point in time is also determined by the value of the Hubble constant. Other age criteria such as the age of meteorites, the apparent visual depth of the universe, and stellar or even galactic evolutionary schemes yield supportive but not fully complementary numbers and at present, the range in age of the universe varies from as old as 20 billion-years (Byrs) to as young as 10 Byrs. A lingering question but fifteen billion-years old appears quite realistic and 16 Byrs is just 'elegant' for this discussion and will be used. The value of Hubble's constant for a universe 16 Byrs old is about 20 km/sec per million-light-years (Mlyrs) distance.

Interestingly and while not commonly expressed, Hubble's Constant is a doubling constant similar to other rates such as biological growth, radio-active decay and interest-bearing accounts: a factor applicable to the amount present and changing with time. If it had been measured 8 Byrs ago, it would have been twice as big. Variables are often hard to deal with but there is another clever way to look at it this problem. Consequently, to apply this rate to an expanding universe we will consider doublings of time, or what can be equivalently expressed in the "power of 2's", eg. 2,4,8 etc. As it turns out, this is a convenient sum (in your head and on your fingers) with the fact there is 32x106 sec/yr and the power relationship that 103 is approximately equal to 210. Therefore, 1 yr = 25+20sec and 16 Byrs = 24+30 yrs or summing together, 25+34 = 59, we can say there have been 59 doublings of time since the universe was just one second old.

Earlier than one second old, modern physicists feel quite comfortable reaching back to 10-35sec and then some, a number of reverse doublings from 1 sec much greater Than the mere 59 forward doublings. However the domain from 1 second forward and what might be perceived within it, is both familiar and quite adequate for our purposes.

Our numerical notation will use the standard Ba or Ma for billions and millions of years backward in time (Giga- could be substituted for Billion). We will also use Bu or Mu for the same number of years forward from the beginning of the universe. Therefore 1 Bu = 15 Ba is the same point in time when the universe was 1 Billion years old. Other examples are 8 Bu = 8 Ba, the half way point and 1 Mu is approximately 16 Ba or more precisely 15.999. The light-year yardsticks, Blyrs and Mlyrs are distance measurements and Byrs and Myrs are time measurements! All doubling numbers of time, "D" will be in reverse time from the present.

Looking out and back

We will establish as an important reference the almost uniform (since the COBE spacecraft determined slight variations) 2.7o Kelvin background radiation, a thermal remnant of a cooling and expanding universe, conveniently at a point in time when the radiation was about 3000oK and only 1 Myrs old. This radiation can be crudely felt with the palm of the hand held up to a clear dry evening sky or seen as the cause of frost on a sleeping bag on a high mountain summer night. It is a very cold sky!

If our time and velocity of light constants are consistent, we can say our background radiation has traveled for 16 Byrs and has covered an accumulated distance of 16 Blyrs wherever it came. If we segment the duration into reverse doubling packages, in effect freezing the expansion, it can be numerically summed as follows Le Euler elements of our doubling periods:

The sum of 8 Blyrs + 4 Blyrs + 2 Blyrs . . . 2 Mlyrs + 1 Mlyrs = 15.999 Blyrs, with 14 terms and similarly the time to travel the distance: The sum of 8 Byrs + 4 Byrs + 2 Byrs . . . 2 Myrs + 1 Myrs = 15.999 Byrs.

While this exercise presents a consistent picture for the constants we have selected, it implies our universe was very big when it was very young, contrary to the thesis of it being very small. Is there a more fruitful way to look at it? In fact there is and possibly more realistic and comprehensive.

If we are to perceive a smaller universe we must consider the location of the source of the radiation when it started its journey. It will be analogous to a series of messengers, each originally starting in a different environment and all arriving at our door simultaneously. So our first example: consider the place the background radiation originated has moved and the universe has expanded 14 times (1 Mu to 16 Bu = 24x 210) during the trip in reaching us. If we use an analogy similar to lightning and thunder, and the difference in arrival times for sight and sound, a concept we readily recognize, how far away were we when that bolt struck? By we, I mean our place in the universe, as we were still 16 Byrs shy of being born. It can be easily calculated by using the number of doublings and the following principle. If we consider the relationship that within each doubling period, light has twice the time to cover twice the distance, it therefore travels an equal fractional segment of the original distance during each doubling period over the course of the trip. Read that again slowly! For our reference background radiation at l Mu and a doubled distance at 2 Mu, it was 1/14th the distance from us in the first doubling period, covering another 1/14th of the original distance in the second doubling period, etc., for 14 segments of 1 Mlys each or 14 Mlyrs total. Therefore we can say our visible universe at 1 Mu was 14 Mlyrs in one direction or 28 Mlyrs for both directions, a very big sphere. Yet in comparison, this is simply 7 times farther away than we see the Andromeda galaxy today which is a naked eye object.

However as a second thought, keep in mind, what we actually see today is radiation from a spherical shell from the outside surface of that sphere at that earlier time. If we all survive to be centenarians, that shell would have been about 1/40th of a light-year thick for us and has now expanded to 100 light-years. All the other radiation within the sphere has since expanded, cooled and propagated past us. The shell itself still appears around us and at a virtual distance of 16 Blyrs, but it is not an ocean away, it is here and we're immersed in it, like not-so-warm bath water.

The Event Horizon

Since we live but an astronomical moment, our visible horizon is also an event horizon, like the lightning and thunder analogy. We are only going to hear (see) one bolt at any one distance. Earlier ones have long passed by, while others further away will not reach us in our short span, a humble relationship to keep in mind. This horizon has a linear relationship with distance until we get to distances in which the expanding universe becomes a factor to consider. An example in the linear range is Andromeda Galaxy which is 2 Mlyrs away and the image we receive is 2 Myrs old.

Recently one of the most distant objects observed, 4C41.17 - a quasar (Scientific American, Jun 93), is estimated to be about 12 Blyrs away and possibly a universe age of only a few Byrs. The type of object this represents may be the most distant objects there are. There is also reason to believe that these objects existed only about that time during a so called Quasar Era, and have disappeared or changed to something else as yet unrecognized. Therefore quasars exist only at a specific distance in our event horizon. We will return to this idea.

Proceeding as we did with the background radiation, when and where are we observing this object? If it is indeed, only 2 Bu old, there have been only 3 doublings of time since then. The object light, propagating to us has covered the distance in 3 parts and at 2 Bu would then calculate to have been about 6 Blyrs away. Calculating one more distance example, if there were some hypothetical object on our horizon at 1 Bu. it would have been only 4 Blyrs away, the light from the object covering the distance in 4 parts during 4 doublings of time and distance.

Compared to the linear time-distance relationship near by. we begin to see the visible universe is becoming progressively smaller at earlier times but it is also bigger in a secondary respect. Our universe was some number of times larger than the light-year yardstick that is linear with time to start. So another way to describe the size of our visible universe is to compare it to the duration of time it would take light to traverse it, if the universe were not expanding at that specific time.

We will tabulate the distance of the examples in Fig. 1.



Starting from the present, we see back in time 1 light-sec per 1 second, ---1 Blyrs per 1 Ba, or a one for one linear scale of time and distance, but further out we see some of the contraction of the universe with our quasar at 2 Bu being 6 Blyrs, our hypothetical object at 1 Bu at 4 Blyrs, and our background radiation at 1 Mu at 14 Mlyrs. These more distant examples are then 3 times, 4 times and 14 times larger than our velocity of light yardstick. This progression would continue as we approach Bu = 0, but in an ever smaller universe. However this relationship terminates at a special point in time, a little before our 1 Mu background radiation reference. This is known as the time when matter and radiant energy "decoupled". Prior to this, the universe was opaque to radiation and not transparent in which mass and energy were confined together in a local place, interacting and colliding over very short distances (In effect, no relic propagation). So in this very early segment, in reverse time, the universe was again linearly smaller with each halving (reverse doubling) to our 1 second starting point and maintaining the 14 times velocity of light yardstick size.

Therefore in Fig. 1 we can also delineate our universe into 3 significant segments, moving forward in time these are: a) expanding in linear doublings when space was opaque, b) a curved expanding then decreasing event horizon during the middle expansion and c) a linear with reverse time, event horizon cone closing in on our point in time and space. The divisions are placed schematically with different doublings "D" as a base, reflecting the size of the universe at that time.

Consequently the maximum size of what we may call our visible universe is about 8 Blyrs at 4 Bu, the quarter way point, a region of very distant radio galaxies and super galactic clusters. The minimum size occurs at both ends. First at the present end we may pick any arbitrary unit of duration and size, such as 1 sec = 1 light-sec, about the distance we see the moon. And at the other extreme 1 sec. our starting point, is only 14 light seconds large, condensing and containing, in those first few moments, the full complement of the basic precursor particles of hydrogen and helium that will build the universe in which we live. Everything we know and see is contained in that moment within a size much less than the distance to the sun and about 12 times farther away than the moon. So all dimensions of our universe are much less than infinite and some well within our sense of touch and feel as with the Sultan's daughter's perception of the moon.

Lustrous ivory

Having guidelines as to the size of our universe we can now look in broad terms at what is contained within it and more specifically, how we infer it. The first distinction of importance is between an object and radiation. It is not trivial and includes the question if an object has duration, if radiation has a source and if we know these things. For simplicity, we will consider an object being in a fixed place (compared to the velocity of light) and continuously emitting radiation like an endless series of lightning bolts. This emission is then ordinary radiation from that point on, perpetually on the move and what we ultimately sense. While invariably radiation needs a source object, we may identify a unique case with our background radiation that is somewhat analogous to the in-depth blue of the sky.

Background radiation

The critical "decouple" point is not precise but occurs when the background radiation had a time and place in space when it was free to propagate essentially without interference. Radiation progressively earlier had higher probabilities of being absorbed and re-emitted. Incidentally, the radiation temperature and it's degradation is attributable to both the expansion of space and photon interaction over time. Fig. 2 shows the portion of our background radiation for a number of doubling steps around the critical "opaque to transparent" time, compared to a hypothetical object or equivalent point in space.



Some portion of the radiation may have had a more distant and earlier source, making the transition point soft, half object-like and half sourceless. At this time there were no visible objects, the total sky was brilliant like the surface of the sun, but thereafter we can proceed to classify three types of objects that have developed since, and a number of examples with which we are familiar.

Relic objects

Essentially, a relic object includes what we can hold in the palm of our hand or more poignantly, what a young bride may elect to wear as something old and cherished. Within our model, the oldest examples are hydrogen and helium. While invisibly small, they are gases with which we may fill colorful balloons, being born and relics from the first few seconds and minutes of our universe, very familiar segments of time. Everywhere this matter condensed from more exotic particles, but as far as place, the boundary of an area as vast as our galaxy, was at most only a few meters away from our position in space at that time.

Most of the other heavier elements about us are derived after that time, created and scattered during a variety of stellar events until about 11 Bu or 5 Ba. However radioactive elements with long-term half-lives are possibly centered about 5 Ba and expelled in supernova explosions also related to our own solar system condensation. And while all this diverse elemental material pervades us in the "relic" sense, it is not individually distinct and it is even harder to "cozy up" to nuclear materials, such as uranium and thorium which have better dated origins.

A more dramatic object and final example, might be something like a polished pallasite meteorite, a beautiful olivine cummulate in a iron-nickel metal matrix and worn as a pendant, a very precise 4.5 Byrs old and literally unchanged since. In summary these are all things we can touch. As far as age, whether we go on hearsay that grandma made the object or use more involved dating techniques, they are in our immediate environment with no period of time separating them from us. The other remaining types we can only see and hence represent a different type of information gap from knowledge about them.

Quasarian objects

Quasarian objects are a class of unique objects that existed within a specific age of the universe and at no other time. Therefore, we may only observe them on our event horizon at a certain distance. Quasars, the name-sake, appear to be the classic example and are found within a small span of time and at a universe age of about 2 Bu. We may find other such objects between then and now particularly in the young universe, but no others are specifically identified that I'm aware of. Star clusters may be very old but also seem to have a variety of ages. One possible example may be ourselves at 16 Bu. here on earth. Another may be our solar system in spite of the fact we see solar systems presently condensing a few thousand light-years away in Orion's belt, as one school of thought expresses what a unique event the condensation and development of our solar system might have been (Taylor, 1992). In a comparative and summary definition, a quasarian is very different than say: an octogenarian. An octogenarian is a person (object) that passes through his 8th decade, like people on a subway, always in existence as a class but are in actuality some individuals getting on and others getting off, which brings us to the last classification and it's most subtle significance.

Hertzsprung-Russell objects

Hertzsprung-Russell objects are objects which exist over a period of time, with many examples along our event horizon, at different sizes, distances, magnitudes and ages, all representing objects with duration but with different stages of development. If these objects are statistically abundant and gradational in appearance in various respects, they may be plotted into an evolutionary scheme which is the classic namesake diagram. This is a seemingly trivial idea, until asked by any perceptive student,"How do we know?", implying we being ephemeral and confidently stating the age of comparative immortals. We know because we deduce clever ways to eliminate the apparent variables of our event horizon and establish certain absolutes independent of the observer. This is of course the basis of the determination and interpretation of stellar evolution. So instead of just seeing uncorrelated objects on an evening celestial sphere, like poorly differentiated light-specks on a planetarium ceiling, we know we see back in time and space and ultimately back to square one, the size of our universe.

Tying loose ends

Like putting a string on our balloons let's try to tie together a few loose ends. Now that we have a broad conceptual image of what is in our universe we can take a momentary look at what is not. Fig. 3 is such a summary.



Like a moon crescent, our event horizon is a sliver of the whole. What was always beyond and outside will always be. While events and objects may exist within our universe sphere, our limited existence again reduces them to only our event horizon. While objects with duration within our sphere may pass through that horizon, events that occur earlier or later will not. So we really only know them for a moment. For some, they will never be in our ken and for others, they are no longer in our universe, and if they still exist they are long removed by distance from our observational cone. However for any object we do see, we can think of three locations for it: Its position when its radiation started the journey toward us, that position the object appears to us now, and the position it would be today. For example: the Quasar which appears to be 14 Blyrs distance now, at an age 2 Byrs old was at a distance of 6 Blyrs away when its radiation left and can now be imagined to be 32 Blyrs away and far beyond 'our' universe.

This is a good point to distinguish between 'our' universe which is the subject here and the 'infinite' universe, that we only imagine and cannot know and considered to have no preferred position or reference frame. In the first case we are truly at the center but we do need the latter case to frame our belief in the invariance of time and place in physical laws: the center is anywhere or nowhere.

In addition to our' vs. the 'infinite' universe, probably the next most difficult aspect of the 'Big Bang' universe to understand is where we fit in, from something that starts from infinitely small, and extends out to a very dim, endlessly far horizon today. Look again at Fig. 3, was not our position in space also at the location of the Big Bang, when it occurred? In Fig. 4, we bring these two places together, hopefully to clarify the concept.



Study the diagram carefully. It is a projection of our event horizon, and what we see simultaneously at all depths of the night sky. It is plotted on a radial diagram with an approximate log scale for both time and distance. Back in time spirals around while distance is directed radially outward. It is bizarre in that it demonstrates you and the Big Bang were at the same place albeit a few years apart and that your event horizon has two very distinctive ends. One is a hairline that has expanded to a universe in size and the other narrow end flairs to the width of your lifetime. Enjoy!

Even if we acknowledge our centrist viewpoint and also our belief in the physical invariance of other reference frames, how much is really changing and how valid is what is considered constant? How much have we erred in an assumption of an absolute place over time? The expansion of space remains an enigma but we have avoided complex quantum and relativistic effects by the choice of our time domain starting at 1 second. At the other end and taking a closer look at home, the expanding universe should change the distance to the sun by some 10 meters/yr and almost by definition of the Hubble yardstick, Andromeda galaxy at 2 Mlyr should be receding at two times our reference or 40 km/sec, but in both cases local forces and motions put the magnitude of these measurements in the noise(2). Many of our physical constants can be measured with great precision in the laboratory of the "here and now" but it becomes difficult to deduce ways to know if they are valid over vast distances and earlier times. Discovering new events and objects and equally important, new ideas(3) as to where they fit in, has made the total image less obscure with time. So maybe it's just as well that everything is not close enough to touch or possess, as with our example of the Sultan's daughter and as commented by Henry David Thoreau, "If men could pluck a star from the midnight sky, there would be none left to see and few to buy.".

Reference for Big Bang background radiation;

Michael Zeilik, 1991, Astronomy, the Evolving Universe: John Wiley and Sons, p568.

Reference for Quasar features:

Miley, G.K. and Chambers, K.C.,1993, Most distant Radio galaxies: Scientific American, June 1993, p54-61.

Reference for Solar system uniqueness:

Taylor, R.S., 1992, Solar System Evolution: Cambridge Univ. Press, p307

(1) A "decouple point" for the background radiation at 300-500 Kyrs would be a little more accurate or one doubling period earlier, leaving to the student to calculate the distance for that D number. See Fig. 2 to help with the answer.

(2) Per the Hubble expansion, Andromeda is misbehaving and is actually being attracted towards us.

(3) Even the "D" number has merit in providing perspective, particularly when the Hubble Constant, the real basis of the expanding universe, is really only known to be valid to a value of about Z = 3, the recessional ratio and comparable to D = 2, but "D" a time unit has larger values as used herein.

Technical Addendum - Redshift and 'Z' and 'D' Numbers

Redshift is the observed effect of the light spectrum in distant galaxies as determined by the offset of numerous atomic element absorption frequencies toward the red end. While there may be a number of causes, two accepted are: Light climbing out of a gravitational gradient or a recessional velocity known as Doppler effect.

'Z' Number or the interpreted recessional velocity of distant galaxies - the second cause, is a ratio of light wavelengths, specifically (lambda minus lambda-reference) divided by lambda- reference. This equation normalizes 'Z' to equal zero locally and interpreted to reveal an expanding Universe of distant objects.

'D' Numbers as defined in the text are the count of halvings (reverse doublings) in the age of the Universe, an assumption that also interprets an expanding Universe. As the Hubble Constant implies the expansion is linear, namely, all objects in reverse time are together in one place - Big Bang - both 'Z' and 'D' have correspondence at certain values. A trivial example, a local object with light in the present has zero redshift and zero age with respect to the past, hence Z =D= 0. Also at an age of half the Universe age or 8 Bu, L or wavelength is twice the reference or Z = (2-1)/1 = 1 and one half in age, D = 1 reverse doubling.

However, for larger values they differ significantly. 'Z' is empirical with limitations on measurement while 'D' is mathematical and geometric. Comparing values and not considering Relativistic effects:

When 'D' equals 0, 1, 2, 3, 4,---10, 'Z' equals 0, 1, 3, 7, 15,---1023 = 1024-1

 
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