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Cosmic Fine Tuning and the Hypothesis

Colin S. Coleman Defence and Technology Organisation Edinburgh, Australia 5111

ABSTRACT

The observable is necessarily hospitable for life. There are indications, however, that the laws of physics and cosmological parameters need not take the form and values observed, and if they were slightly different life could not exist. A common approach to this fine tuning problem is to propose a with an ensemble of domains, mostly inhospitable for life. A Bayesian method is used to show that this hypothesis is more credible than a homogeneous fine tuned universe. This conclusion is straightforward for a finite ensemble, but can be extended to an infinite ensemble by applying a formulation of the Principle of Mediocrity.

A common approach to the fine tuning 1. Introduction problem is to propose a cosmos that is far larger than that observed, much of it being involves several challenges unsuitable for observers. In this so called that distinguish it from other disciplines. It multiverse, the local region has properties concerns domains that are inaccessible to that permit observers and so appears fine observation, it confronts the notion of tuned. The entire multiverse, however, is infinity as an expression of reality, and it deemed not to be fine tuned. is strongly influenced by observer selection effects. Observer selection means The multiverse concept continues an the apparent properties of the universe are enduring trend in cosmology. Previous selected in that they must be such as to anthropocentric views (geocentric, permit the emergence of observers. What heliocentric, galactocenteric) all proved to is observed, therefore, may be very be false. Multiverse cosmology takes this a unrepresentative of what exists. step further by removing any central status from the entire . The Despite strong agreement between theory properties of domains beyond the horizon and observation in modern cosmology, may differ, but all observers must be critical gaps in knowledge remain. These located in regions where the properties include the of the , the permit observers to exist. of dark and , and whether the universe is finite or infinite. Multiverse cosmology has been criticized Also, if the properties of the universe were for being unfalsifiable, and hence slightly different, no life could exist. unscientific. It is tempting to dismiss the concept on these grounds, but a lack of This last issue is known as the fine tuning falsifiability does not mean an idea is problem, which was first noted in the false. To dismiss unfalsifiable hypotheses coincidences of stellar nucleosynthesis is to impose another anthropocentric (Hoyle, 1965) and subsequently in a condition on the universe. Falsifiable broader cosmological context Carr & theories are naturally desirable because Rees, 1979). It concerns the fact that the their credence increases in response to physical laws and cosmological ever more stringent attempts to reject parameters are not constrained to fall them. If this is not possible, however, an within the range that permits observers to alternative approach is required. emerge. If these laws apply everywhere, this circumstance requires explanation. Multiverse are addressed here A four level taxonomy of multiverse by using Bayes' theorem to determine their cosmologies has been proposed (Tegmark, credence relative to a homogeneous fine 2007). The first level includes the infinite tuned universe. Key to this approach is the ergodic universe, which is a consequence treatment of observer selection effects by of chaotic . This comprises many the Self Sampling Assumption (SSA) Hubble volumes with all possible initial (Bostrom, 2002). This requires that the conditions but homogeneous laws. observer be regarded as a random member of the set of all observers. Multiverse The level 2 multiverse envisages many cosmology is shown to be more credible replications of level 1, viewed as ‘bubbles' than a homogeneous fine tuned universe, in an inflating medium where inflation has and the strength of this inference is given ceased. Spontaneous symmetry breaking by the degree of fine tuning. results in different laws and cosmological parameters in each such bubble. This result is direct for a finite multiverse, but less so for an infinite ensemble. One Level 3 is motivated by the many-worlds problem is that observers may occur due to interpretation of quantum mechanics. random fluctuations of matter, albeit with Every quantum state of the universe exists extremely low probability. Such 'freak' simultaneously, rather than colapsing to a observers are not confined to domains that single state upon making an observation. permit observers to emerge naturally, and This adds no content beyond level 2, the may be more numerous than ntural main distinction being that the ensemble observers if observer-supporting exists in an infinite dimensional Hilbert are sufficiently rare. This makes it space rather than real physical space. impossible to treat observer selection effects by applying SSA. Finally, the level 4 multiverse is motivated by a principle of mathematical democracy, To address this problem a Principle of in which all possible mathematical Mediocrity is proposed whereby a section structures exist in reality. This derives of the the multiverse containing the from philosophical considerations, and observer is unexceptional if viewed on a serves to close the multiverse . sufficiently large scale. A weak form of this principle is used to show that The key feature of all multiverse models is observer-supporting members of the the variation of physical laws. If the laws ensemble comprise an infinite subset with were homogeneous, with variation only in non-zero measure. This allows freak initial conditions, the concept offers no observers to be neglected, and ensures the resolution to the fine tuning problem. This ratio of prior probabilities is non-singular, is the case for the infinite ergodic universe thereby extending the result to the case of at level 1. In general, however, the laws an infinite multiverse. may vary over distances larger than the Hubble scale to provide the necessary inhomogeneity. The level 2 multiverse 2. Multiple Universe envisages disjoint domains with different Cosmologies physical laws. For the purpose of resolving the fine tuning problem, it is of no concern The multiverse concept is not founded whether the variation is continuous or solely upon the fine tuning problem. discrete. Inflation cosmology and quantum theory provide a strong basis for a heterogenous ensemble of universes (Linde, 2007). String theory, for example, suggests the existence of a vast number of domains with different physical laws depending on the geometry of the compact dimensions.

3. Bayesian Inference and 4. A Thought Experiment the Self Sampling Assumption To illustrate the rationale, consider an Science progresses by enhancing the observer with no knowledge of the outside credence of a hypothesis in response to world. She finds herself in a room with evidence. When an observer's existence one feature, a four digit number N equal depends on the hypothesis in question, the year of her birth. What can be deduced observer selection effects must be taken about the nature of such rooms? into account. In cosmology this is expressed by the Anthropic Principle (AP) Two alternate hypotheses may be (Carter, 1974; Barrow & Tipler 1986) considered. In the first hypothesis (H1) originally stated by Carter as ...what we there are many rooms, each with the same can expect to observe must be restricted by four digit number. An alternate hypothesis the conditions necessary for our presence (H2) has many rooms with an ensemble of as observers. different numbers. The evidence E is that this room has a number equal to the Various forms of AP have been proposed. observer’s birth year. Some have been criticized for being tautological or offering no prescription for Assuming no causal relationship between application. These concerns have been room numbers and observers, it would be addressed by re-casting AP as the Self a coincidence for all rooms to have a Sampling Assumption (SSA) stated as feature specific to the observer. The ratio -4 follows: One should reason as if one were of prior probabilities P(H1) / P(H2) = 10 a random sample from the set of all if in H2 all four digit numbers are equally observers in one's reference class probable. The conditional probability of E (Bostrom, 2002). under each hypothesis is P(E/H1) = 1 and -4 P(E/H2) = 10 . Then by Bayes’ theorem: This approach avoids tautology and indicates a methodology for expressing the −4 P(H1 / E) P(E / H1 )P()H1 10 probabilistic connection between theory = = = 1 P H / E P E / H P H 10−4 and observation. It provides a means to ()2 ()()2 2 determine observational consequences given theories about the distribution of Thus the evidence E offers no reason to observers. The intent of SSA is to treat an favor either hypothesis. observation as a random element of the set, or reference class, of all comparable With no causal relationship between observations. This has been formalized by rooms and observers, no observer Bostram in an observation equation giving selection effect is involved. Such an effect the probability of a hypothesis h as a may be introduced by specifying that an consequence of evidence e. occupant can survive only in a room with the property that its number is equal to the No attempt is made here to apply the year of her birth. All observers must then observation equation directly to specific find themselves in such a room. multiverse models. Current prescriptions lack the detail required to calculate the In this case it remains a coincidence that distribution of observers and construct the all rooms have a feature specific to the -4 observer reference class. Instead, Bayes' observer, so P(H1) / P(H2) = 10 as before. theorem and SSA are used to determine Applying SSA with a reference class of all the credence of a composite multiverse observers in all rooms, the conditional hypothesis relative to a fine tuned universe probability of E under H1 or H2 is unity, with homogeneous properties that permit and hence: observers. P(H / E) P(E / H )P()H 1 = 1 1 = 10−4

P()H 2 / E P()()E / H 2 P H 2 When observers are constrained to occupy hypothesis H2, and the greater the fine rooms that are amenable to their existence, tuning the greater the inference favoring therefore, the ensemble hypothesis is H2 over H1. This argument holds for a favored over one in which all rooms are finite ensemble because the existence of identical. the means that S is not empty, and the ratio of prior probabilities is not singular. 5. The Multiverse Hypotheses 6. The Infinite Multiverse The observable universe appears to be fine tuned. There is dispute over the degree to There is no reason to expect the multiverse which this is the case, but it is widely ensemble to be finite. Indeed a countably accepted that at least some cosmological infinite ensemble is a more natural model. parameters are not constrained to take The previous analysis does not hold for an their observed values, and the allowed infinite ensemble for two reasons. Firstly, range of values is much greater than that if the set of observer-supporting elements which permits the emergence of observers. is finite, or infinite with zero measure, the Fine tuning may be quantified by a ratio of prior probabilities is singular and parameter F, the probability of selecting the relative credence of the two by chance properties that permit the hypotheses is indeterminate. Secondly, emergence of observers. freak observers may occur in any element of M that contains matter, which may be Now consider cosmological hypotheses more numerous than those that permit H1, H2 and evidence E as follows: observers to emerge by normal processes. Consequently, the assumption that all H1: Physical laws and cosmological observers occur in observer-permitting parameters are homogeneous and elements of the ensemble is false. permit observers. To deal with the infinite ensemble it is H2: Physical laws and cosmological necessary to show that the measure of S is parameters occur in a finite non-zero. With no established theory of ensemble M with a non-empty universe formation, however, it is not subset S that permit observers. possible to determine the probability that a given element of M has properties that E: The observed universe permits permit observers, and hence no way to observers determine the measure of S.

Following the method of the thought The Principle of Mediocrity is commonly experiment, cosmology H1 is fine tuned accepted in cosmology. This holds broadly but H2 is not, and the ratio of prior that a region containing the observer is not probabilities P(H1) / P(H2) = F. Applying special if viewed on a sufficiently large SSA with the reference class of all scale. Stated differently, the local region is observers in the ensemble, the conditional statistically unexceptional within the set of probability of E under H1 or H2 is unity, all similar regions at large scales. This since in H2 all observers are in an element principle may also be applied to the of S. Then by Bayes’ theorem: multiverse. The observable universe is special in that it has properties that permit P()H / E P(E / H )P()H observers, but a large finite section of the 1 = 1 1 = F multiverse, containing the observable P()H 2 / E P()()E / H 2 P H 2 universe, should be unexceptional within the set of all similar sections. If the observable universe is fine tuned with F<<1, the evidence that it permits observers strongly favors the ensemble Consider a partition of the multiverse into H1: Physical laws and cosmological a series of disjoint finite domains . Each parameters are homogeneous and Pi contains mi elements of M and si ≤ mi permit observers. elements of S. The proportion of observer- supporting elements ρi = si/mi satisfies a H2: Physical laws and cosmological distribution function F(ρi) = P(ρj≤ρi); j≠i. parameters occur in an infinite The local universe may be assumed to be ensemble M with subset S of non- in P1 and the Principle of Mediocrity zero measure that permit defined as follows: observers.

For partition Pi and variable ρi with E: The observed universe permits distribution F(ρi), F(ρ1) ≠ 0 or 1. observers.

This definition requires that the partition Again cosmology H1 is fine tuned and element containing the observable cosmology H2 is not, and the ratio of prior universe is unexceptional in the probabilities P(H1) / P(H2) = F where F is distribution F(ρi). In fact it requires only the fine tuning parameter. Applying SSA that ρ1 not be so exceptional that the set of with the reference class of all observers in more extreme members has measure zero. the ensemble, the conditional probability of E under H1 or H2 is unity as all If the multiverse satisfies this Principle of observers in H2 are in elements of S. Freak Mediocrity, S must be infinite. To see this observers may be neglected due to their assume S is finite and choose P1 large extreme improbability, and the fact that enough so that it contains all elements of the measure of S is non-zero. Then by S. Then ρ1 ≠ 0 and ρi = 0 for i≠1, hence Bayes' theorem: F(ρ1) = 1. This contradicts the definition, and the assumption that S is finite is false. P(H1 / E) P(E / H1 )P()H1 = = F P H / E P E / H P H The Principle of Mediocrity also implies ()2 ()()2 2 that the S has non-zero measure. Since P1 is finite and contains at least one observer- Thus fine tuning favors a heterogenous multiverse over a fine tuned homogeneous supporting member, ρ1≠0. By definition universe, and the strength of this inference F(ρ1) ≠ 0 or 1, hence P(ρj≤ρ1) = A with A ≠ 0 or 1. A lower bound L on the measure of is given by the fine tuning parameter.

S is given by L = A.0 + (1-A).ρ1 ≠ 0.

The simplest assumption concerning the 6. Conclusions and composition of M is a steady state infinite Implications multiverse with no natural dimension or timescale. Individual elements of the The result here rests on three assumptions. ensemble may exist for different periods, Firstly, that probability theory may be but the overall composition does not vary. applied to the credence of hypotheses, and In a steady state multiverse a suitable the Self Sampling Assumption is a valid observer reference class is the set of all way to treat observer selection effects. observers at any epoch. For a non-steady Secondly, that a homogeneous or hetero- multiverse it is necessary to include geneous cosmos are a priori equally observers at all epochs, but this distinction likely, but fine tuning biases the prior is not critical for the Bayesian argument. probability to a heterogeneous multiverse. Finally, the multiverse obeys a Principle of Returning to the calculation of the Mediocrity such that a section containing credence of multiverse and non-multiverse the observer is unexceptional if sampled cosmologies, consider hypotheses H1, H2 on a sufficiently large scale. and evidence E as follows:

The first assumption is widely accepted, observers is small, indicating strong fine the second is strongly supported by theory tuning, this constitutes evidence favoring a and observations, and the third is plausible heterogenous multiverse. but cannot be experimentally validated. The multiverse hypothesis may never be The approach followed is to define two subjected to experimental falsification. classes of cosmologies; one in which the While the idea has substantial explanatory physical laws are homogeneous and permit power, it may yield no direct observational observers, and another in which an consequences. This suggests it may be ensemble of domains spans the possible regarded as a discipline of philosophy laws. Two hypotheses are then defined rather than science. Nevertheless, there are according to whether the cosmos conforms sound reasons to suggest it may be true. to one class or the other. Bayes' theorem is applied to determine the relative credence Perhaps the most significant implication of of the two hypotheses. the multiverse concept is that it highlights a limitation of science itself. If the quest Note that these hypotheses do not include for a full theory of physics succeeds, and all possible cosmologies. Models in which explains all phenomena in the observable the entire multiverse is fine tuned are not universe, it may be only a special case addressed as they offer no resolution of the from a vast ensemble of such theories. The fine tuning problem. Also, cosmologies existence and properties of such other involving the agency of intelligence are realms may remain of speculation, excluded by the tacit assumption that forever beyond the reach of experiment or elements of the multiverse ensemble are observation. generated by natural processes, and thus without planning. Any conscious design mechanism makes the fine tuning problem moot, but at the cost of introducing the larger problem of explaining the existence and mechanism of primordial intelligence.

Note that the intercession of intelligence does not imply supernatural phenomena. It has been proposed that life may advance to the extent that it achieves the capability to create universes and select their properties to favor the subsequent emergence of life, indicating a process akin to universal natural selection (Harrison, 1995).

The present analysis does not address these speculations. It indicates only the relative likelihood of two hypotheses. The conclusion may be stated as follows: If the observable universe is fine tuned, a heterogenous multiverse is more likely than a homogeneous universe.

The strength of the inference favoring the multiverse is given by the degree of fine tuning. It may prove difficult to determine this parameter until a full theory of physics is available, which allows calculation of the distribution of cosmological properties. If the chance that these properties support Barrow, J. D. and Tipler, F. J., 1986. The Anthropic (Oxford University Press).

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Carter, B., 1974. Large Number Coincidences and the Anthropic Principle in Cosmology, in Confrontations of Cosmological Theories with Observational Data, ed. M. Longair (Dordrecht: Reidel).

Harrison, E. R., 1995. The Natural Selection of Universes Containing Intelligent Life, Quarterly Journal of the Royal Astronomical Society 36, 193.

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