The Universe Marches to Its Own Drummer

Blink your eyes. Are you done yet? Okay, that took about one-third of a second. For us humans, that seems like a very short time, occurring in "the blink of an eye." But by the time the early Universe was about an eye-blink old, a lot of the interesting stuff had already happened. For example, by about a millionth of a second, the sub-microscopic particles making up all ordinary matter- electrons, protons and neutrons- had been produced. But it would be a very long time- 380,000 years in fact- before the Universe contained any stable atoms. Only then would the blazingly hot Universe have cooled down enough to let those tiny building blocks stick together long enough to form atoms. And perhaps prophetically, most of those early atoms were hydrogen, the most abundant atom in our bodies.

But we need a far less anthropomorphic view of Time to think about the seminal events in the early Universe. During the first fraction of a second after its birth, the Universe was much more tumultuous and rapidly changing than at any later time.

First came the Big Bang that started it all about 13.7 billion years ago. This was of course really "big", since it was the beginning of our Universe, when space, mass-energy, and even time itself somehow appeared, apparently out of nothing. But scientists now believe that, for a time extremely shorter than a millionth of a second, the nascent Universe expanded in a rather leisurely fashion- not at all like a Big Bang. Then, according to a 1979 theory of Alan Guth, the true "Bang" began an exceedingly short time later. Guth termed his theory Inflation, a mild term indeed for the posited tremendous exponentially increasing explosion in the size of the Universe.

This proposed Inflation "explosion", occurring very early in the life of the Universe, could answer a number of otherwise very puzzling questions about our present-day Universe, 13.7 billion years later:  1. Why is the Universe (on a very large scale) the same, and in fact so smooth, in all directions?  2. Why is the Universe so "flat"? Yes, it seems strange to describe the three-dimensional Universe using a word conventionally applied to two-dimensional objects like a perfect plain. But for the Universe, "flat" simply means that the angles in any gigantic triangle add up to 180 degrees,  just like those triangles featured in our high school trigonometry classes.  3. Any magnet contains both a North and South magnetic pole, and so do the pieces we get by cutting the magnet in half again and again. But any electrical charge has only a single "pole": it's  either positive or negative. So why hasn't anyone ever found an analogous magnet (a "magnetic monopole"),  that also has only a single pole? Inflation provides the answer to that question too.

Because Inflation yields answers to all three of these profound questions about today's Universe, it is a widely accepted theory. But there are two fundamental unanswered questions about this proposed enormous explosion:

1.  We don't know know why or when Inflation might have either started or stopped. We do know that the beginning and end of Inflation (at least in our local part of the Universe) must have occurred at times unimaginably soon after the birth of the Universe, something like a trillionth of a trillionth of a trillionth second later. And during this extremely short time, the Universe doubled in size at least 90 times, growing from inconceivably small to the size of something we can actually imagine, for example a golf ball or a basketball. During Inflation, parts of the Universe expanded far faster than the speed of light- surely the greatest explosion that ever has or ever will occur. A "big bang" indeed! [For anyone concerned about a possible contradiction here with Einstein's Special Theory of Relativity: although Einstein's theory states that no information or (real) matter can travel within the Universe faster than light, there is no limit on how fast the Universe itself can expand.] When Inflation ended (if indeed it ever started), the Universe returned to a more laid-back expansion rate. 
    But curiously, this rate is being accelerated by a very mysterious form of energy discovered in the 1990s, Dark Energy . The expansion caused by Dark Energy is fairly mild today, but its rate increases as the Universe itself expands. So scientists hypothesize that, many billions of years from now, the effect of Dark Energy will become strong enough to cause a Big Rip, ultimately tearing apart the Universe and even its constituent atoms. The Universe would then turn into a vast thin dark and cold gruel, with the sad probable fate of forever becoming only ever bigger, darker, and colder.

2.  Unfortunately, we don't yet have any direct experimental evidence that Inflation ever happened. A part of the problem is that we can't detect the light of the Universe at any age earlier than 380,000 years after its birth. This is because, as noted above, only then could stable atoms form from sub-atomic particles. So light particles could finally cease incessantly bumping off of the charged sub-atomic particles; and so voyage freely through space for 13.7 billion years, finally to reach our telescopes today. What we can see today of the 380,000 year-old Universe is termed the Cosmic Microwave Background (CMB), whose discovery in 1964 provided convincing evidence for the Big Bang theory. It does seem poignant that we will very probably never pierce this barrier that prevents any glimpse of our Universe during its frenzied early infancy. 

Quite recently, there was optimism that direct evidence for Inflation had finally been obtained. Early in 2014, observations by astronomers using the BICEP2 telescope at the South Pole, of particular light patterns in the CMB, indicated that these patterns arose from the effects of Inflation. But data produced by the Planck spacecraft later that year showed that the BICEP2 data could have arisen entirely from effects of dust in our own galaxy, instead of from Inflation.

So stay tuned. It seems likely that current intense and ever more precise investigations of the CMB will determine whether Inflation- potentially able to explain profoundly puzzling aspects of our current present Universe- ever actually occurred.

(sciencequandaries.blogspot.com)

Infinities Galore

What do you see when you picture Infinity? Perhaps you imagine a very, very large number. If so, please don't. Infinity is not a number at all, but a concept. Infinity is actually a seeming oxymoron, an unreachable limit of a never-ending series of numbers. A simple example is a list of all the positive integers: 1,2,3....., where the dots indicate that this list never stops. And, to further complicate matters, there is not just one infinity, but instead multiple ones, each larger than the one before.

A warning is in order here: The concept of Infinity is initially quite mysterious and unintuitive, because in the mathematics of our daily life we deal with actual numbers such as the cost, weight, and speed of a new cell phone. Thus thinking about the enigma of Infinity might cause headaches, hallucinations, or other adverse reactions. So proceed at your own risk!

An often-used example, The Infinite Hotel, illustrates the properties of the smallest infinity. Here is my version: 

Picture a hypothetical hotel with an infinite number of rooms, named imaginately Room 1, Room 2, ... This is the hotel's high season, and so the hotel is completely full with, of course, an infinite number of guests. Then, very late one night, a young couple shows up. These two are clearly in love (or at least lust), and just as clearly in desperate need of a private place with a bed, and so ask the hotel manager for a room. The aged night manager patiently explains to the young lovers that the hotel is full. But the couple entreats him to use all of his powers to try to find them a room. Well, who among us could not open his/her heart (and hotel) to such a plea? The old manager, remembering well the urgency of youthful desires, is deeply moved by the duo's plight. And he also knows that he could actually provide the lovers a room in the hotel. But doing that would require a lot (actually an infinite amount) of distasteful interactions with an infinite number of sleepy, irate guests. So with a heavy sigh (but a light heart!), the manager moves the guests in Room 1 to Room 2, those in Room 2 to Room 3, and so on right up the line. The lovers of course then get to move into Room 1. And since the hotel is infinite, none of the hotel's current guests will have to sleep out in the cold.

Clearly there's a problem here, since no real-world hotel could work like this. Though the room list of this hypothetical hotel will eventually reach any whole number we can think of, the list will never reach the non-number Infinity. And because Infinity is not a number, it is meaningless to use this concept in standard numerical equations; e.g., Infinity - Infinity = ?  

But surprisingly, there are many different kinds of infinity. The brilliant mathematician Georg Cantor showed in 1878 that infinities come in different sizes. Cantor gave the name Aleph-Null to the smallest Infinity: the set of all "rational numbers", which include the whole numbers in the hotel example above, plus all fractions like 3/4, 1/137, 20/21 that can be written as a ratio of two integers. Cantor showed that it is possible to count all of the Aleph-Null numbers, including the fractions, in a diagonal manner that would eventually arrive at any specified number. So he termed Aleph-Null the countable ("denumerable") Infinity. It turns out that even if you multiply Aleph-Null by itself, the result s still Aleph-Null. So what manner of infinity could possibly be larger than Aleph-Null?

Cantor showed that there is indeed an infinity larger than Aleph-Null, termed Aleph-One, that contains all of the "real" numbers. The real numbers include all the Aleph-Null rational numbers. But these real numbers 
additionally contain all the "irrational" numbers (like the square root of 2 and pi), that can't be expressed as a simple ratio of integers. These irrational numbers  can instead be represented only by a never-ending series of digits. Cantor demonstrated that Aleph-One is a non-countable infinity, since between any two rational numbers lie infinitely many irrational numbers. 
So Aleph-One is a larger infinity than Aleph-Null. Cantor's proofs clearly showed that Infinity could no longer be viewed as a single concept. In fact, Cantor showed that there are infinitely many infinities!

Could there be any infinities lurking between Aleph-Null and Aleph-One? Cantor believed the answer is no (and
long attempted to prove this), a conjecture now termed the "continuum hypothesis". It is not known whether Cantor was correct. Even worse, this question is an undecidable problem, of the type shown in 1931 by Kurt Godel's incompleteness theorems to plague the basis of arithmetic. So Cantor's conjecture can't be either proven or disproven within any standard (e.g., ZFC) arithmetic system. Sadly, we will thus never know for sure whether there is a long-lost middle sibling lying between the two smallest known infinities. 

So not only is Infinity a concept or limit rather than a number, there is actually an infinite number of infinities, each larger than the one before. When it comes to infinities, size does indeed matter.

(sciencequandaries.blogspot.com)