In previous essays, we discussed the multiverse's sole "prediction" that we are typical observers in a typical universe. However, this ignored the major problem that it doesn't seem possible for an infinite varied multiverse to use probabilities to make any prediction whatsoever because there are an infinite number of copies of every possible universe!
To get around this problem, multiverse scientists use something called measures to compute probabilities in an infinite multiverse and then determine what a typical universe looks like. They thereby attempt to rescue multiverse theory from the fatal flaws of a naive multiverse, but end up running directly into the measure problem we'll discuss next time.
Highlights of this essay:
From a Naive Multiverse to Measures
Now that we’re finally up measures, let’s review the line of reasoning from the past three essays. In essay 6, we discussed a multiverse theory that only contains the first two premises - that there exists an infinite multiverse, and that the constants vary from universe to universe. We illustrated how a theory that explains fine tuning by simply positing that in an infinite varied multiverse everything must occur somewhere, is a naive multiverse that falls prey to three serious problems. Make sure to read that essay to get the complete picture.
Then, in essay 7, we showed that a complete multiverse theory that adds the third premise - the Typical Universe Premise - could theoretically explain fine tuning and avoid these problems by making a prediction - that we are typical observers in a typical universe.
Of course, while making a true prediction is good and would save multiverse theory from the problems of a naive multiverse, making a false prediction is bad and would doom a complete multiverse for the reason of its being just plain wrong. In essay 8, we discussed two problems multiverse scientists faced in verifying their prediction: the Boltzmann Brain and the Grand Universe problems. These problems indicate that our universe isn’t typical, an apparent falsification of multiverse theory.
Despite the significance of these problems, in truth, essay 8 was somewhat out of order. To simplify the presentation, we skipped over a major conceptual problem that multiverse scientists face in making any prediction whatsoever in an infinite multiverse.
If you recall, we referenced the problem of taking probabilities for infinite sets as soon as we brought up the multiverse’s prediction that our universe is typical. But, we decided to wait until this essay to take up this important but complicated point. We admit, however, that conceptually this essay's problem belonged earlier.
In this essay, we’ll begin by presenting the problem of evaluating probabilities in an infinite multiverse. This will naturally lead us to measures, the only way multiverse scientists can make any prediction in an infinite multiverse. In the next essay, we’ll elucidate the serious problems with this approach and slowly develop the multi-layered measure problem.
Some of the science in this essay will be difficult to follow. But don’t worry. We’ll give another helpful marble analogy that will help you get the basic idea. As long as you understand the analogy, you’ll get the essential point of this essay.
The Problem of Infinities and Probabilities
Before we present what measures are, we need to understand why they’re needed. As we discussed in prior essays, to avoid the problems of a naive multiverse, the multiverse must predict that our universe is typical.
But this presents the obvious problem. If there are an infinite number of universes, then there are an infinite number of duplicate copies of every universe. As physicist Alan Guth said: “In an eternally inflating universe, anything that can happen will happen; in fact, it will happen an infinite number of times.”
But if there really are an infinite number of copies of each and every universe, what sense is there in claiming that one universe is typical or more common than any other? Every single universe exists an infinite number of times!
As opposed to finite quantities, infinities present unique challenges in math and physics. This is especially the case when trying to compute probabilities in an infinite multiverse. By positing an infinite number of each type of universe, an infinite multiverse makes it impossible to make any straightforward calculation of which type of universe is more likely than others.
Let's give an analogy to help illustrate the problem of computing probabilities in infinite sets. Imagine a row of many silver and gold marbles. If the row has a million silver marbles and only one gold marble, a randomly selected marble is a million times more likely to be a typical silver marble than the one atypical gold marble. However, if the row contains an infinite number of silver marbles and an infinite number of gold marbles, it’s impossible to say which color is typical or more likely to be randomly chosen. There are an infinite number of both colors!
The same is the case with an infinite multiverse. If there are infinitely many identical copies of each type of universe, how can it be more probable for us to exist in one type of universe rather than some other type of universe? It seems impossible for an infinite multiverse to make any predictions at all!
To bring the point home, what sense is there in saying that our universe is a typical universe with intelligent observers when there is also an infinite number of universes with intelligent observers and fire-breathing dragons, an infinite number of universes with intelligent observers and unicorns, and an infinite number of every single type of universe with intelligent observers?
Infinities and Measures
This problem has been a major obstacle for eternal inflation since its inception. Remember, eternal inflation is the primary method multiverse scientists use to establish the existence of an infinite number of universes. Andrei Linde, one of the founders of eternal inflation, discusses this point in his 2010 joint paper with Mahdiyar Noorbala, Measure Problem for Eternal and Non-Eternal Inflation:
While you may not follow this, or some of the following quotes, it’s important to hear for yourself that measures and the associated measure problem aren’t something we just came up with on our own. Rather, it’s something that multiverse scientists acknowledge and explicitly attempt to deal with. And sometimes, you need to hear a complicated quote for that to come across.
Before we get to the quote, keep in mind that when they say the phrase “different types of vacua”, they mean universes with different values for their constants of nature.
Here’s the quote:
The main problem here is that in an eternally inflating universe the total volume occupied by all, even absolutely rare types of the “universes,” is indefinitely large. Therefore comparison of different types of vacua involves comparison of infinities. As emphasized already in the first papers on the probability measure in eternal inflation, such a comparison is inherently ambiguous and depends on the choice of the cutoff, which is required to regularize the infinities.
To help understand its meaning, let’s explain the terms “measure” and “choice of cutoff” one at a time. Let’s start with a measure.
A measure is a rule introduced to deal with the problem of taking probabilities for infinite sets by assigning or weighting probabilities. The externally imposed qualitative rule, the measure, tells you to order the infinite set in a specific quantitative way that allows probabilities to be meaningful.
Let’s try to understand this by returning to the analogy of our infinite row of marbles. When there are an infinite number of silver marbles and an infinite number of gold marbles, the infinities prevent us from naturally computing the probability of randomly picking a silver marble from this row. To get around this problem, let’s assert that the row should be arranged in a specific arrangement that weights the silver marbles more heavily than the gold marbles: that is, we repeatedly alternate a million silver marbles for every one gold marble. The qualitative nature of this arrangement is that it’s a repeating pattern - as opposed to a random assortment - and the quantitative element is the ratio of a million silver to one gold - as opposed to any other ratio.
While this arrangement still contains an infinite number of both colored marbles, they are now arranged in a specific manner that’s more conducive to a probabilistic analysis. There’s now a sense in saying that picking a silver marble is a million times as likely as picking a gold marble.
Of course, even with this artificial arrangement, there are still an infinite number of both color marbles in the infinite row. That’s why there’s one more element, the cutoff, that’s needed to calculate a probability by making the set finite. The cutoff refers to the consequential point at which you choose to limit the infinite set and thereby calculate probabilities from the remaining finite set.
In the example, if we cut off the infinite row of marbles after a million and one marbles, we would have a million silver marbles and 1 gold marble, thereby making the ratio a million to 1. However, if we cut off the row after 1.5 million marbles, we would have 1.5 million silver marbles and still only one gold marble, making the ratio 1.5 million to 1. As you can see, the probabilities are affected by both the way you choose to order the row and by where you choose to cut it off.
In a similar vein, multiverse scientists use measures and cutoffs to attempt to make predictions in an infinite multiverse. Even though an infinite multiverse contains an infinite number of every type of universe, multiverse scientists choose a measure that functions like a metalaw by asserting that some types of universes are weighted more heavily than others. And then they choose a cutoff for that measure that allows them to calculate which universe is typical.
These weights and their associated cutoffs can then be used to compute probabilities for different types of universes. They use this to determine what the typical universe with intelligent observers looks like and thereby check the measure’s predictions. The hope is that their measure will predict that our universe is a typical universe with typical observers.
True and False Predictions
We’re now in a position to understand how multiverse scientists attempt to establish the Typical Universe Premise. In other words, what does it mean that the multiverse either predicts or fails to predict that we’re typical observers in a typical universe?
Once again recall that since it’s impossible to truly take probabilities given an infinite number of universes, multiverse scientists must choose a particular measure and cutoff that allows them to compute finite probabilities and thereby determine what a typical universe would look like.
For any particular measure that multiverse scientists choose, a true prediction would mean that the measure predicts that typical intelligent observers are something like us - real humans in our grand universe. A false prediction, on the other hand, can occur in many ways. One example would be if the measure predicts that the typical observer is a Boltzmann Brain surrounded by chaos. Once again, it’s important to realize that whether or not the multiverse’s prediction turns out to be true depends upon the specific measure chosen.
To illustrate this point, let’s return to our analogy of the infinite row of silver and gold marbles. Let’s assume we randomly choose a marble and find that it’s gold. To assess whether chance is a good explanation for this golden selection, we must consider whether we would have predicted getting a gold marble in advance. To do so, we’d need to compute the probability of getting a gold marble and thereby determine whether a gold marble is typical.
Because there’s no straightforward way to compute probabilities in the infinite row of marbles, we’d have to consider their order. If they were ordered as above - a repeating pattern of a million silver marbles followed by one gold marble - then the typical marble would be silver. If so, getting the gold marble would prove our prediction false and thereby refute the hypothesis that the marble was chosen at random. We’d need to look for a different explanation - maybe the selection was rigged.
If, on the other hand, the marbles were ordered differently - for example, a repeated pattern of a million gold marbles followed by one silver marble - then we would have predicted picking a typical golden marble. If so, our golden selection would be perfectly consistent with the hypothesis that the marble was selected randomly.
Just like the gold marble being typical depends on the ordering of the infinite row of marbles, so too our universe being typical depends on the specific qualitative and quantitative features of the measure that’s posited as a metalaw that weights the different universes in the infinite multiverse.
While the marble analogy is easy to follow, it’s much more difficult to understand exactly what’s going on regarding the multiverse and its measures. Nevertheless, you can keep the marble analogy in mind when we get a bit more specific about how multiverse scientists use measures to determine if our universe is typical.
The Boltzmann Brain Measure
Multiverse scientists have spent many years trying out different possible measures and cutoffs and checking to see whether each one’s predicted typical universe corresponds to our own observed universe. While multiverse scientists have tried out many potential measures, we want to briefly mention one measure that Andrei Linde referred us to in response to our email question about Boltzmann brains.
While this point may be a bit complicated, it’s extremely valuable to see a concrete example of how measures are used to solve the Boltzmann brain problem. And, it’s also valuable to hear top scientists use terms like “freak observers” and see them perform technical calculations of “Boltzmann brain nucleation rates” so you can realize just how far multiverse scientists have gone in their attempt to explain away fine tuning.
Don't worry if you don’t follow the details - just try to follow the main gist of the quote. Linde’s paper, coauthored with Guth, Vilenkin, and others, is called, Boltzmann brains and the scale-factor cutoff measure of the multiverse. This paper attempts to use one particular measure, called the scale-factor cutoff measure, to find general conditions under which Boltzmann brain domination is avoided.
“Boltzmann brain domination” is the unwanted condition of a multiverse that is dominated by Boltzmann brains such that the typical observer is a disembodied brain. Go ahead, Aaron. Here’s the abstract of the paper:
To make predictions for an eternally inflating “multiverse,” one must adopt a procedure for regulating its divergent spacetime volume. Recently, a new test of such spacetime measures has emerged: normal observers — who evolve in pocket universes cooling from hot big bang conditions — must not be vastly outnumbered by “Boltzmann brains” — freak observers that pop in and out of existence as a result of rare quantum fluctuations. If the Boltzmann brains prevail, then a randomly chosen observer would be overwhelmingly likely to be surrounded by an empty world, where all but vacuum energy has redshifted away, rather than the rich structure that we observe. Using the scale-factor cutoff measure, we calculate the ratio of Boltzmann brains to normal observers. We find the ratio to be finite, and give an expression for it in terms of Boltzmann brain nucleation rates and vacuum decay rates. We discuss the conditions that these rates must obey for the ratio to be acceptable, and we discuss estimates of the rates under a variety of assumptions.
It's hard to believe that the top physicists in the world are taking freak observers so seriously. But they have no choice. They’re forced to make these calculations for multiverse to even potentially work. This is the consequence of avoiding an intelligent cause of one fine tuned universe and instead positing an infinite number of universes.
While the details of the scale-factor cutoff measure are too technical to get into, the abstract of the paper illustrates the general approach that multiverse scientists take in order to complete multiverse theory and establish the Typical Universe premise. By choosing different measures that enable them to calculate probabilities, they are hoping to find one measure that will predict that typical observers will be normal observers like us and not freak observers like Boltzmann brains. If they can do so, they can successfully verify the prediction of the multiverse and thereby rescue it from the lethal problems of a naive multiverse.
Now that we understand what measures are and why they’re needed, we’re finally ready to address the legitimacy of establishing the typical universe premise using measures.
Next time we’ll culminate our entire line of reasoning by discussing the multi-layered measure problem. You’ll clearly see why the multiverse isn’t a viable scientific solution to fine tuning, or to anything at all. So stay tuned!
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