The Degree of Fine-Tuning in Our Universe — and Others
Fred C. Adams, Physics Reports 807 (2019) 1–111; doi:10.1016/j.physrep.2019.02.001; arXiv:1902.03928.
Excerpt (pp. 7, 51–52, 139, 146): §1 the sensitive-vs-hierarchical taxonomy; §4.1–4.2 the cosmological-constant problem and structure-formation bounds; Table 2; §10.4 “Is our Universe Maximally Habitable?”. Ellipses mark omitted material; full text at the source.
A second type of tuning arises when a parameter of interest has a vastly different value from that expected (usually on theoretical grounds). The cosmological constant provides an example of this issue: The observed value of the cosmological constant is smaller than some expectations of its value by ∼ 120 orders of magnitude. This second type of tuning is thus hierarchical. In the first example, the strong nuclear force can apparently vary by only ∼ 10 percent without rendering deuterium unstable or diprotons stable. Nuclear structure thus represents a possible instance of Sensitive Fine-Tuning. In the second example, the value of the cosmological constant could be a million times smaller or larger (if the fluctuation amplitude Q is also allowed to vary) and nothing catastrophic would happen, but the values would still be much smaller than the Planck scale (by ∼ 100 orders of magnitude or more). The cosmological constant is thus an example of Hierarchical Fine-Tuning.
In addition, when an unexpected hierarchy arises due to some quantity being much smaller than its natural scale, one way to get such an ordering is for two large numbers to almost-but-not-quite cancel. This near cancellation of large quantities can be extremely sensitive to their exact values and could thus require some type of tuning. This state of affairs arises, for example, in the cosmological constant problem [110, 546]
(see Section 4). This general concept is known as Naturalness.
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Given the large number of possible mechanisms by which the universe could determine the magnitude of its vacuum energy density, it is likely that other regions of space-time (other universes in the multiverse) would have different values of the cosmological constant [545]. The value of ρΛ (equivalently Λ or λ) is thus expected to vary from universe to universe. In addition, unless the size of the cosmological constant is determined by as-yet-unknown physics, we expect many other universes to have much larger values of ρΛ than that observed locally. As a result, the most common universes, with large ρΛ , will not be able to produce structure (see the following section) and will quickly evolve to become almost empty de Sitter space [126]. In this context, our universe is unusual in its profligate complexity.
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4.2. Bounds on the Vacuum Energy Density from Structure Formation Although the energy density ρΛ of the vacuum is not predicted by any known theory, its value in our universe is nonetheless constrained by cosmological considerations. A relatively weak upper bound follows from the requirement that the universe must have a matter dominated era (see equation [46]). A stronger bound results from the requirement that galaxy formation occurs before the the expansion of the universe starts to accelerate due to the vacuum energy. This section reviews bounds of this type. The original constraints of this type [43, 167, 457, 545] require that quasars form by redshift z = 4.5 and can be written in the form ρΛ < (500/729) ρeq Q3 , (71) where ρeq is the density at the epoch of matter/radiation equality and where the numerical coefficient is derived using a simple spherical infall model for galaxy formation [424]. For the the amplitude of density fluctuations observed in our universe, Q = 10−5 , one obtains the upper limit ρΛ ∼ < 200ρ (obs). Λ
This bound on ρΛ was subsequently generalized to estimate the likely values of the cosmological constant [372]. This updated treatment finds the probability of an observer measuring a given value of ρΛ , conditioned by the requirement that a given universe can produce observers. In this context, the formation of collapsed cosmological structures — essentially dark matter halos — serves as a proxy for the production of observers. Because values of ρΛ much larger than the observed value (in our universe) tend to shut down structure formation, observers in any universe are likely to measure values of ρΛ relatively close to the observed value. This conclusion continues to hold with alternate assumptions, for example using the amount of material in stars as a proxy for observers and using more detailed models of structure formation [423, 499].
The conclusion that typical observers will measure a value of ρΛ roughly comparable to our own follows from assuming that the other cosmological parameters are held constant. Bounds of this type become substantially weaker
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Ranges of Parameter Values for Viable Universes
quantity symbol observed value range (decades)
Up quark mass mu 2.3 MeV >3
Down quark mass md 4.8 MeV 0.85
Electron-proton mass ratio β 1/1836 5
Up-down quark mass difference δm 2.5 MeV 1
Gravitational constant αG 6 × 10−39 > 10
Weak coupling constant αw 10−5 6+
Fine structure constant α 1/137 4
Strong coupling constant αs 15 3
Fluctuation amplitude Q 10−5 4
Baryon to photon ratio η 6 × 10−10 6
Dark matter abundance δ 3 × 10−9 6
Vacuum energy scale λ 0.003 eV 10+
AM Radio fAM 535 – 1605 kHz 0.0043
FM Radio fF M 88 – 108 MHz 0.00087
Table 2: Table of the fundamental and cosmological parameters and their allowed ranges. The ranges are expressed in terms of decades of allowed variation, as defined by equation (226). The ranges expressed for the mass of the up quark and the gravitational constant are lower limits on the full range. The ranges listed for the weak force and the energy scale of the vacuum correspond only to variations to greater values (any lower values are allowed). For comparison, the final two lines specify the amount of tuning required for radio reception (see text).
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10.4. Is our Universe Maximally Habitable? Discussions of fine-tuning often implicitly assume that our universe is optimized for the development of observers [341]. However, it is interesting to revisit this panglossian assumption [38] and ask if different choices for the fundamental constants or the cosmological parameters could lead to universes that are even more favorable to the development of life, or at least the production of complex cosmic structure (see also [393]). In the realm of extra-solar planetary systems, researchers are now considering whether or not Earth is the best prototype for a habitable planet [275, 428]. Asking the same question on a cosmic scale, we find that several parameter choices could lead to possible improvements of the universe: • Smaller vacuum energy: The cosmological constant could vanish (equivalently, ρΛ = 0). In this case, the universe would never enter into a late-time phase of accelerated expansion. This scenario would favor structure formation, as the matter dominated era would never end. In addition, ever larger cosmological structures would continue to enter the horizon, so that large scale structures of ever larger masses could be constructed [12, 143, 190, 328, 390]. Even if the energy density of the vacuum is nonzero, it could be much smaller than the value realized in our universe, so that structure formation could continue over a longer span of time.