Viraj Patel

The idea of a multiverse is easily one of the most ambitious ideas that has come from physics, but it has been heavily exaggerated in popular culture. Despite its origins lying in the Many-Worlds interpretation of quantum mechanics [1], some people have approached the idea from a statistical point of view: what is the probability that a universe just like ours could exist? This article is going to outline an approach that has been used many times in popular science and explain why it would or wouldn’t work.

**Our Unique Universe**

Much of physics can be explained in terms of the fundamental constants of the universe such as the gravitational constant, the speed of light, the Planck constant, and many more. These constants are so finely tuned that changing them even by a small amount can lead to significant consequences [2]. The gravitational constant, according to General Relativity, describes how much mass and energy distort space-time, so a different value of this constant would affect the entirety of astrophysics [3].

Due to the anthropic principle [4], we can form a single idea that there are only a small number of designs of a universe that can support any form of life (not limited to carbon-based) and several universes were made with different designs each with an equal probability of being chosen [5]. This would mean that the fundamental constants of nature have different values in each universe, leading to differing behaviours in each universe (one universe could have no light while another could have high gravity) [4].

**The Power of Monkeys**

So, what do monkeys have to do with anything? In 1913, Émile Borel developed the Infinite Monkey Theorem (IMT) [6]. In the paper, he claims that if a monkey was left in front of a typewriter for an infinite amount of time, there is a chance (albeit small) that the monkey would type any given text like the complete works of Shakespeare. This conclusion can be reached using simple probability theory. Moreover, if we modify the problem so that there are *n* monkeys typing, then it can be shown that the probability of a single monkey typing a given text approaches 1 as *n* approaches infinity (this is left as an exercise to the reader, but the answer is referenced) [7].

Strangely enough, in 2002, an experiment was conducted at the University of Plymouth where 6 monkeys were left in an enclosure with a computer keyboard for a month and the typed text was stored and sent to a website. The results were far from desired as not a single word was typed and the document consisted of five pages of only the letter “S”, the document can be found here. From the experiment, it became clear that more monkeys were needed, more time was required, and the thought experiment did not consider the behaviour of the monkeys [8]. Due to the final point, the theorem was amended to describe a monkey as an object that types a random letter on the keyboard with equal probability of typing a given letter.

The IMT has had many applications across maths, computer science [9] and physics [10] [11]. For this article, we will be focusing its application to popular science and the theory of the multiverse.

**Monkey Cosmology**

Let us define the multiverse as an infinite set of universes, and we exist in just one iteration of the pool. In the search for a universe exactly like ours, the problem isn’t too different from the IMT, in this case our universe is the monkey typing actual text.

Firstly, the IMT assumes the monkeys are left for an infinite amount of time. This is unphysical as it has already been proved using the cosmic microwave background [12] that the age of the universe is finite. If we assume other universes also have a finite age, we find that this constrains the IMT to a finite amount of time. If the IMT was used for a finite amount of time, the probability of the monkeys typing a given text is incredibly small. So small, in fact, that we assume it’s impossible [13]. By applying this to the idea of a multiverse, we can conclude that it is virtually impossible to find another universe exactly like ours.

The only other possibility is that other universes have been in existence for an infinite amount of time. This would agree with the conditions of the IMT but would those universes be able to recreate the finely tuned conditions of our universe? To consider this case, we can apply the IMT to the number pi. Let the first monkey type the first N_{1} digits of pi after the decimal point, the second type the digits of pi until it reaches the N_{2}^{th }digit, and so on. As the number of monkeys tends to infinity, it can be shown that, with probability one, no set of monkeys (finite or infinite) will ever reproduce the number pi. This can be shown using the Borel-Cantelli lemma or otherwise [14]. The fundamental constants of nature are like pi, most of them involve an infinite decimal expansion. So, they canot be perfectly reproduced using the IMT. This means, using the IMT, you can show that, if other universes are created, things could happen (e.g. you could materialise in another universe), but you cannot reproduce the fine tuning of the universe [15].

**Conclusion**

In this article, we discussed the idea of a finely tuned universe with just the right conditions for intelligent carbon-based life and the problem with a multiverse containing a similar universe to ours. It is possible that there are other universes out there, some of which may support life, but it is not probable that there is a universe exactly like ours with the same fundamental constants of nature.

**References**

[1] D. Wallace, The Emergent Multiverse: Quantum Theory According to the Everett Interpretation, Oxford University Press, 2012, pp. 37-38.

[2] C. J. Hogan, “Why the universe is just so,” *Rev. Mod. Phys., *vol. 72, no. 4, pp. 1149-1161, 2000.

[3] G. F. Lewis and L. A. Barnes, “The Fundamental Constants of Nature,” in *A Fortunate Universe: Life in a Finely Tuned Cosmos*, Cambridge University Press, 2016, pp. 31-32.

[4] J. Schombert, “Anthropic Principle,” [Online]. Available: http://abyss.uoregon.edu/~js/cosmo/lectures/lec24.html. [Accessed 1 January 2021].

[5] B. Harrub and B. Thompson, “Our Finely Tuned Universe,” Apologetics Press, Montgomery, 2003.

[6] É. Borel, “La mécanique statique et l’irréversibilité.,” 1913.

[7] M. McKubre-Jordans and P. L. Wilson, “Infinity in Computable Probability,” *Journal of Applied Logics, *2019.

[8] BBC News, “No words to describe monkeys’ play,” BBC, 9 May 2003. [Online]. Available: http://news.bbc.co.uk/1/hi/3013959.stm. [Accessed 2 January 2021].

[9] X.-S. Yang, S. F. Chien and T. O. Ting, “Computational Intelligence and Metaheuristic Algorithms with Applications,” *The Scientific World Journal , *vol. 2014, pp. 2356-6140, 2014.

[10] P. Réfrégier, Noise theory and application to physics: from fluctuations to information., Springer Science & Business Media, 2004.

[11] Christopher R. S. Banerji et al., “A notion of graph likelihood and an infinite monkey theorem,” *Journal of Physics A: Mathematical and Theoretical, *vol. 47, no. 3, p. 035101, 2014.

[12] L. Knox, N. Christensen and C. Skordis, “The age of the universe and the cosmological constant determined from cosmic microwave background anisotropy measurements,” *The Astrophysical Journal Letters, *vol. 563, no. 2, p. L95, 2001.

[13] BK, “Infinite Monkeys versus Infinite Universes,” 12 August 2017. [Online]. Available: http://christiancadre.blogspot.com/2017/08/infinite-monkeys-versus-infinite.html. [Accessed 2 January 2021].

[14] A. P. Godbole, “The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets (or, Can a Monkey Really Type Hamlet?),” in *Number Theory and Its Applications*, IntechOpen, 2020.

[15] A. D. Aczel, “The Infinite Multiverse and Monkeys Typing Pi,” Science 2.0, 6 June 2014. [Online]. Available: https://www.science20.com/the_greatest_science_mysteries/the_infinite_multiverse_and_monkeys_typing_pi-138060. [Accessed 2 January 2021].

I’m doing an MSci in Physics with Theoretical Physics. I’m interested in quantum field theory, quantum computing, general relativity and complexity & network science. When I’m not studying or rushing that assignment due tomorrow, I’m sketching or playing cricket.