Cosmic Concerns - Theories of CosmologyCosmic Concerns - Theories of Cosmology
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Cosmic Concerns - Theories of CosmologyCosmic Concerns - Theories of Cosmology
'D' NUMBERS AND COSMOLOGY
Cosmology is an esoteric subject, straining all elements of our perception with vast magnitudes, a strange dynamism and a grand evolutionary tale. Ironically and at the same time, the source of this intellectual grandeur is nothing more than the interpretation of the more or less still and serene view of the night sky. Add to this the good fortune of even having this view and it not being obscured by countless other factors such as the daytime sun boldly there half the time as a pertinent example and bias to be reckoned with. So too, this ageless challenge to understanding is portrayed in an ancient lexicon of star names with a colored mythology and now modern tools of observation and space technology. Here we consider one more insight or tool to possibly aid in this mystical and profound epiphany - 'D' numbers and 'D' number calculations.
In a summary definition, 'D' numbers are mathematically rigorous associated with two accepted physical invarients; the flow of time and the velocity of light, compatible with the Big Bang paradigm as established from the Hubble constant, comparable with 'Z' numbers at low values - the Doppler red-shift value for distant objects and graphically comparative to other potential cosmologies for an early Universe. In a technical definition, 'D' numbers are a count of reverse doublings of time for any assumed age of the Universe as implied by the Hubble constant. All of these elements need explanation as does the practicality of the concept.
As suggested, the difference between day and night is profound and our perception is fundamentally biased by this contrast. The day is brilliant and an instant active interplay of our earthly scene. The night is quiet, still and only moves with ponderously slow determinations or startling cometary or meteoritic indeterminates. While historically more mystical, modern interpretations must still surmount these almost reflexive conditionings. The essence of this comparison is two-fold. During the day, a visual time delay appears trivial or nonexistent in our mundane active scene. And while the night scene is extra-terrestrial and has apparent depth and even a modest recognition of time delay, space and time are inseparably welded together and the only window through which we may view the whole, both a constraint and separate insight.
As a mathematical entity 'D' numbers may be the most secure reference among the many variables of Cosmology. If any parameter might be considered inviolate, sacred or any other gut feeling, it is likely the flow of time. While 'D' numbers are a geometric interpretation of time as doubling numbers, it is mathematically rigorous and relates well to the present Big bang paradigm at both ends of our interpretation: From physical expansion features within the first '3 Minutes' mostly of a theoretical nature to observational features looking back from the present into deep space.
While we are dealing with both time and distance, such units from seconds to parsecs do not easily transform or are not tied down with many space parameters subject to observation or interpretation as in 'Z' numbers. Granted these things must be figured out but as in Quantum theory, perturbations from the simplest model maybe the way to go as 'D' numbers may be used. Nothing can change if you say half the age of the Universe or equivalently 'D' = 1.
What can we see and how can we interpret it? The 'Size of our Universe' (Theme 2) introduces the concept of 'D' numbers and assumes a universe age of 16 billion years old as convenient with a simple computation of 59 doublings of time to the present from a universe age of just one second old and a number of observational examples of when and where in such a Hubble expansion, Big bang scenario. Our knowable sliver of space-time is graphically shown to close in upon itself - our self - in the concluding view, a radial display of cosmic time and distance.
Slightly modified and more fully explained, Figure 1 is a display of 'D' numbers in relation to time and distance with you at the center.
While any total age of the Universe may be used, time is rotational with 'now' at the apex, winding counter-clockwise to an early Universe of 16 billion years ago at the base. D is zero 'now', equal to one at 8 Byrs ago and goes to infinity at the base 16 Byrs ago, and overall a linear depiction of time. Distance is radial outward and geometric in simple units of 1 light-second, 1 light-year, etc, to millions and billions of light-years away. This is the playing field for the subsequent figures and discussion.
Looking out into deep space at any particular image, we can imagine what and where we are seeing something three different ways: Where it appears to be, where it was, and where it is now.
a) Where it appears to be has several elements: All this constitutes the interpretation of what the object is and its consequent distance, with the further acceptance of the actual visual radiation information as here and now - eg. Whatever, it took some time getting here.
b) Where it was has also multi-elements of interpretation: That it is an event, then and there, possibly exciting like a super-nova or just plain existence issuing its daily radiation toward us at that time.
c) Where it is now is a natural extension of our relationship with it but also a question or an interpretation not only of position but possible fate for the object. Is it still out there?
In Figure 2, Case a) appears off in the far distance and distant past, but nothing greater than 16 Blyrs, our extreme horizon. Case b), is that distance where it was in our expanding universe paradigm when its radiation started its journey and its path taken. Case c) continues the paradigm path to the present but possibly much farther than our 16 Blyr horizon away. Two examples, near and far are given including some basis of determination:
Andromeda Galaxy
a) 2 Mlyrs by Cepheid varaibles
b) 2 Mlyrs - 20Km/sec x 2Myrs
c) 2 Mlyrs + 20Km/sec x 2Myrs
a Quasar viewed at 12 Blyrs away
a)12 Blyrs by 'Z' = 3, recessional vel.
b) 8 Blyrs 'D' = 2, time doublings
c) 32 Blyrs 'D' = 2, space doublings
In the relatively near case of Andromeda Galaxy, these three distances are straight forward. Not that the distance hasn't had its determination trauma this last century, but the apparent distance, Case a is given at 2 Mlyrs or what is recognized as the present look-back distance, while Case b, where it was when light started its travel, is minus the Hubble recession velocity and Case c, where we might conjecture it to be today, is plus that distance, about 2 parts in 16,000 (2 Myr/16 Byr), essentially in the noise.
However in the Quasar example, the chain of technique to establish where and what requires complex, fastidious and often brilliant astronomic observation and theory. The methods expressed here are interpretive extensions of that data to cosmological considerations with 'D' number calculations. Here Fig 2 dramatically show the relevant illusions. Illusion in the sense that all that really exists is the incipient radiation. Case a, where 'Z' the recessional velocity and equivalent Hubble interpretation of 12 Blyrs away or 12 Byrs ago, lead to Case b of 4 Byrs (16-12 Byrs) Universe age and two time doublings ago hence 'D' is equal to 2, and Case c, from that position, extrapolated out two space doublings to the present, now far out of our visible Universe. Where it was is 'The distance light can travel in its original doubling period times the number of doublings to the present' (4 Blyrs x 2 = 8 Blyrs). This is based on the principle that within each successive 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. Where it is now is "Where it was and doubling the distance for each doubling period forward' (8 Blyrs x 2 x 2 = 32 Blyrs).
What else may be done within the framework of this diagram and 'D' numbers? Interpreting the Big Bang and other potential cosmologies such as a Big Crunch or a version of a Steady State, here called a Big Appearance is shown in Fig 3 as BB, BC and BA respectively and related to what we might observe or conjecture.
It is busy, mostly with open ideas of what the quadrants represent, the type of views we are presented with and the possible deviations in different cosmological models. The Universe is evolving but how? The Big Bang is not the only possibility nor necessarily the most rational.
Consider the conceptual views we behold. View A is the basis of our evolutionary cosmological data, the present back to Z=1, the whole panoply of the cosmic heavens over the last 8 Blyrs and reasonable viewing depths to see the profound complexity. View B is reaching the far observational depths usually strained to instrument physical limits, Hubble deep field and Z to values of 3 to 5 with space doubling back at an earlier time, in a possible Riemannian space with an ambiguous Cosmological constant. View C is mostly theoretical for a Big Bang with physics clues on Background radiation, H/He ratios etc; profound in theory but with anisotropies and possible alternatives still in the offing.
Correspondingly with both views and theory, Quadrant 1 is the foundation of our cosmological concepts, discovery of variety and evolutionary trends of space, mass and energy from stars to nucleons. Quadrant 2 shows a bifurcation of possibility between a Big Bang from nothing and an Infinite Universe of time or space. At this stage, many searched-for proofs have been evading us: Universe age vs Hubble constant, galaxy counts vs size or age, Inflation scenarios and missing mass. Further, there are knowledge gaps outside Relativity or particle/gravity associations or within space dimensional concepts. Quadrant 3 exists on paper as an extension beyond the Big Bang evolutionary picture. Is the Universe possibly older or even rebounding eg. was there time before Time zero? Incidentally, the Big Crunch is seldom thought as a precursor to our universe but if so might - would have to - dovetail into the one we see in the rebound. In this case, there would be meaningful negative 'D' numbers of the compression with possible correspondence to the positive states in the expansion. There would also be some maximal non-singularity compression state of space at the critical point.
The Big Appearance or Steady State Universe suggests a less traumatic start but just extending back with other causes for the appearances we see and termed Big Apperance because it has to overcome the apparent Hubble expansion observations. More specifically, distant red-shifts might have other interpretations than just Doppler recession velocities.
The Epilog or An Arabian Tale greater than 1000 and 24 Nights
One of the legends of great numbers from antiquity concerned the poor beggar that inadvertently did the Sultan a great deed. What the deed was evades me, but that isn't relevant anyway. As the yarn goes the Sultan offered the beggar anything he wished in gratitude. The foxy beggar pulled out a chess board and said 'Give me one grain of rice for the first square, two for the second and double again for each of the 64 squares.'
The Sultan thought this novel and had a slave bring forth a bushel of rice and proceeded to play the game. By the time they got to the tenth square the board was getting messy, and in both frustration and boredom, the Sultan proceeded to pour the whole bushel on the chess board and was quoted to say 'Enough! Begone with you.' The beggar was miffed for he knew the agreement was for an astronomical amount of rice and while he had an Eastern abacus, he was in no position to argue his technical claim and possibly chance maybe losing his head in the matter. Even if he had a modern calculator with an y^{x} function key which would compute in a moment that the 64th square was 1.8 x 10^{19} grains of rice and the total due was twice that number with the sum of the other squares from 63 down to the first (remember Gauss' similar instant sum of the digits 1 to 100?). The Sultan failed to comprehend the magnitude of his commitment and laughed the beggar off as merely stupid to waste his largess on this silly game. And while the beggar never had another such opportunity, he decided he had to work on his communication skills. He pondered that if he had countered that 10 doublings were 1024 or about a 1000, or ten to the 3rd power, that 60 such doublings would be ten to the 18th power and only 4 more doublings to go.
We now know vast numbers are not strange, such as Avogadro's at about ten with 23 zeros trailing. But there is another number almost as big and as interesting: How old or how big is the Universe in a consistent unit of measure, say seconds or light-seconds? A practical and useful way to deal with this is the beggars doubling technique, translated, how many doublings of time or space from when the Universe was one second old? The only complication hangup is the number of seconds in a year, given here as about 32 million and the Universe 16 billion years old, the 'powers of two' sum is therefore 5 + 20 + 4 + 30 respectively or a total of 59. So what can be done with this number? Comprehension is the game.
Therefore in a general conclusion, while the hand-grenade example in Theme 3, 'Life in a Flat Universe' has no inherent limits, what are the perceptual limits of the Hubble phenomenon? For sure, it is a deep space perception with limited observational range quite stretched out and further stretched theoretically herein. Yet its dimensional base is arbitrary whether per million light-years or per million parsec. Can we get more meaning out of it? What is its relationship with the velocity of light (Voc) itself? Two ideas have already been mentioned: The Background radiation at an apparent distance of 16 Blyrs is expanding at the Voc 'now' or 'our' Universe at one second large, expanding at that velocity per light-second of space or one on one. But from that starting point, with each doubling, the expansion is half per unit of space as previously. From 59 doublings ago, that expansion is reduced to 1 over 2^{59} or one part in 2 Billion-billion per second. Not much to see today. In the distance of one light-second, about the distance to the moon, space is expanding only a fraction of a nanometer per second, an amount vastly below perception or observation. So only a rate of change as in the distance to Andromeda Galaxy does it start to take on comprehension even in the equally dim view of what a million light-years might really be to poor beggars of time or space.
Personally, I may be more like the Sultan, failing to realize the larger meaning of modern Cosmology. I have eliminated gravity in calculations as a cause of cosmologic effects, yet I can argue: This linear interpretation provides a first order understanding compatible with the observational data of a Flat Universe. The crux of the issue is that we traditionally hear that 'Gravity slows the expansion' and now there might be 'Acceleration of the expansion'. Until we have a better understanding can't we leave gravity out of it. To me, the open possibility of an Infinite Universe would render gravity to a null in all directions and distance.
Dimming light and Accelerating expansion and a Paradox
Stars in the night have a wide range of brightness. Their apparent brightness is the combination of the objects surface brightness and its size - either actually big, like a red giant or how far it is away. Makes sense, but tree trunks in a forest all appear equally bright and we see only difference in their size with distance. Stellar objects in effect behave the same, except we can not distinguish their size hence see brightness as the variable. Going deeper, galaxies have an inherent brightness (quite dim if you observe Andromeda) but are made visible to us by multiplying their image size and light-gathering power with a telescope or in cases of a photo by multiplying the light collection time. But like the tree trunks they are similarly bright.
However in much deeper space there are other reasons why their surface brightness may appear brighter or dimmer (neglecting any obscuration). Principally, if the Z number is due to cosmic-space-expansion, it has two opposite effects. First, light is a spectrum of energy frequencies and if it is shifted to the red it is diminished in intensity, but secondly, the same spreading of the wavelength along the transmission direction, which is the cause, also increases the apparent size of the object to the same extent on the viewing surface (note this does not refer to the galaxy but the light in transit). An object at Z=3 should have about the same apparent size as one at Z=1, hence for that reason appear not as far and be confusingly bright.
The latest determination from 1A Supernovas, seen at great distances suggest they are dimmer than their Z distance would imply. The tentative conclusion is that the Universe expansion rate was slower then, when their light started its journey than that observed for objects closer to home. I cannot judge all the factors, but concerning the above discussion, the 1A Supernovas as standard candles are more like stars having no size, than galaxies where we do discern size. At this stage I'd leave the conclusion open until separating the myriad potential causes. But there is a possible paradoxal effect of which I haven't heard mentioned and discussed here in both 'Z' and 'D' numbers:
If looking out on an endless savanna with a number of elephants at various distances and assuming them all to be the same size, they would be visually smaller based on their relative distance away. We are at a stage now that this distance relationship may not be true when viewing very distant galactic objects. While absolute galactic size, compared to elephants, might be a poor assumption because great distance also means a much earlier time in the Universe with an unknown effect on galaxy size, the following hypothetical image below of four similar sized galaxies but partially obscured by being one behind another, might actually exist in a yet to be observed Hubble deep field image. But if it doesn't, there should be a big question: Why not?
In addition to their apparent size, from left to right, each galaxy might be measured to have a 'Z' number of 0.75, 1.0, 3.0, and 7.0 and 'D' numbers of .75, 1, 2 and 3, respectively. In an expanding Universe and based on Hubble recession concepts, this would mean:
Up front, Galaxy #1 has Z = .75 or 'D' = less than an interpreted distance of 6 Billion-light-years
Behind it, Galaxy 2 has Z = 1.0 and 'D' = an interpreted distance of 8 Billion light years
Behind it, Galaxy 3 has Z = 3.0 or 'D' = 2 an interpreted distance of 12 Billion light years
Behind it, Galaxy 4 has Z = 7.0 or 'D' = 3 an interpreted distance of 14 Billion light years
Galaxy # 2 and 3 appear the same size yet one is 4 Blyrs farther distant. Galaxy # 1 and 4 appear the same size and # 4 looks larger than # 2 and 3 yet is farther from them all. How can this be? The explanation is simple, the basis of the red-shift or expanded wavelength in the radial direction (toward us) is attributed to expanding space over the duration of the light travel. So be it, but that also occurs in the surface direction normal to us also, hence a larger size in the same proportion.
Again, based on an expanding Universe, a separate intuitive interpretation is that Galaxy # 1 and 4 were about the same distance from us when their light started its journey, albeit #4 left (14 - 6) billion years earlier. Galaxy #2 and 3 are about the same and the maximum distance of any objects in 'our' universe based on the time their radiation left. This is no more than a consequence of all being together at the Big bang and the expansion of space regardless of acceleration rates. While pushing the observational window, if this observation is not valid then serious questions would have to be raised as to the basic interpretation of what wavelength variation, as measure in 'Z' number, really means.
The 20th Century closes with many questions yet what it has uncovered has been truly amazing. 'D' numbers might help with these revelations. But the greater value of the 'D' number axiom is the potential to associate the great magnitudes of the Cosmos, in terms of the Big Bang paradigm, into more comprehensive space-time-object relationships to a interested public. It's quite simple. I personally have taught these concepts to such an average public, including the math, with no more visual aids than a campfire and starry sky overhead deep within the walls of Grand Canyon.
Cosmic Concerns - Theories of CosmologyCosmic Concerns - Theories of Cosmology