arXiv:1302.1293v2 [gr-qc] 20 Feb 2013 eto’ h lmnso h yohssae(slightly are hypothesis the of [2]): from elements se- paraphrased The natural ‘cosmological of sin- lection’. process gravitational a of through number laws gularities, by greatest branch- dominated the be entire produce to the which comes universes and of with nature, ensemble form ing of to laws grav- universes which different more in new slightly nature, process cause a of of singularities laws result itational the the be of may and — — generally Model Standard the mln[]hspooe httefe aaeesof parameters free the that proposed has [1] Smolin .Bakhl iglrte oneadeov oinitial to evolve and bounce singularities universes. hole of Black ensemble 2. an of consists world The 1. .Udrvr idasmtosfrtefins func- fitness the for assumptions mild very Under 5. change small a is there event creation such each At 4. aver- the to equal function fitness a is there Hence 3. .SOI’ OMLGCLNATURAL COSMOLOGICAL SMOLIN’S I. ttso xadn universes. expanding of states e sna oa aiu ntrso universe of terms in maximum mem- local every a production. almost near which is in universe ber one many to after steps converges creation ensemble the tion, pro- holes black random of small number average a duced. the to to leading change properties universe in universe transition. a bounce by a such produced in holes initiated black of number age n o oa piat aiieuies rdcin nwhi m inheritance in Keywords universe production, the universe in involvement maximize their to to point f optima fa evolved may local that and possibility fitness not the both and open of presence faithfulness maximization the and in fitness objects: fidel ensem selection distinct high the natural two for has that dominates account reveals to law it mechanism inheritance Moreover, a faithful provides crea thus most universe Principle pr The of reduction model The process laws. bey CNS. branching principle in multitype this operate general generalize would to that laws one allows inheritance (1982) gr Karlin favors m selection of vary that of themselves Principle variation u Reduction theory of evolutionary mechanisms offspring evolutionary When strategy pa to the here. the the adopted through is rise of This transmission, give inheritance occurs. genetic and holes laws variation transmission black for rep in variation which universe laws of The in process parameters. — Darwinian a (CNS) by Selection for accounted be might ASnumbers multiverse. PACS 5.2, Theorem Karlin’s mixing, EETO HYPOTHESIS SELECTION mln(92 rpsdta h n-uigpolmfrpara for problem fine-tuning the that proposed (1992) Smolin mlctoso h euto rnil for Principle Reduction the of Implications eovn oiieoeao,sprrtcl Galton-Wat supercritical, operator, positive Resolvent : soit dtrof Editor Associate 88.s 61.i 62.r 72.n 72.g 97.60 87.23.Kg, 87.23.-n, 06.20.Jr, 96.10.+i, 98.80.Es, : omlgclNtrlSelection Natural Cosmological BioSystems e Altenberg Lee oi Institute; Ronin ; raineet.Sm icsini rsne n[10]: in presented is discussion Some universe produce from events. might parameters creation Model that Standard mechanisms the to the changes — ‘genetics’ verse themselves. mechanisms for genetic account to the theory of was evolution area the ear- the this Among in developments [12]. synthesis” enough liest “extended extensive an theory called of be body to a into ge- grown Syn- Mendelian have with netics “Modern selection century natural Darwinian 20th of the thesis” of diversifica- Augmentations and adaptation tion. — Darwin orig- by phenomena identified the inally beyond considerably broadened has evolutionary the address I Rather, dynamics. da- universe. the our of do over of light also in probabilities tum interpreted I how be of should [2–11]. issue spaces in multiverse re- unsettled discussed and the are address Critiques regard not physics. this the in to sponses respect with elements mlnspooa evsmsl pntemte funi- of matter the open mostly leaves proposal Smolin’s dynamics evolutionary of theory the years, recent In these of any of plausibility the here address not do I .Teeoe o admycoe ebro the of member chosen randomly a for Therefore, 6. ae atfleso elcto.Atheorem A replication. of faithfulness eater naetlprmtr r compromises, are parameters undamental rmtepeetvle ilices h numbers the produced. Model increase holes Standard will black values of the no present of almost the parameters true, from the are be above that in can made conclude change universe we hypotheses chosen our the randomly Since if be decrease. to a assumed unchanged to holes leads black of or production ei- value the present leads its from ther Model Standard the parame- of the ters in change small every almost ensemble, nrdcdb e 16)t understand to (1967) Nei by introduced n ilgclgntc oteunknown the to genetics biological ond fvraini neiac mechanisms inheritance in variation of incnann optn inheritance competing containing tion echanisms. hyaesbett eda’ (1972) Feldman’s to subject are they , tflihrtne rdosbetween Tradeoffs inheritance. ithful dfirgns hs ehdlg is methodology whose genes, odifier t fihrtnebtenuniverses. between inheritance of ity aeesaeas ujc oCSif CNS to subject also are rameters nil o N silsrtdwt a with illustrated is CNS for inciple outo omlgclNatural Cosmological — roduction hcs hi oa non-optimality local their case ch l fuiess h Reduction The universes. of ble altenber@hawaii.edu eeso h tnadModel Standard the of meters son, iesswt lgtyaltered slightly with niverses .Lf C 0 eiru,got and growth semigroup, 2
The hypothesis that the parameters p change mation needed to specify both an effective law and an by small random amounts should be ulti- effective state.” The meta-state took the form of a gene mately grounded in fundamental physics. We posited to control the recombination rate between two note that this is compatible with string the- other genes under natural selection. ory, in the sense that a great many string The first rigorous stability analysis of this model by vacua likely populate the space of low-energy Feldman [16] showed that new forms of the modifier gene parameters. It is plausible that when a re- could grow in number if and only if they reduced the gion of the universe is squeezed to the Planck recombination rate when they occurred in populations density and heated to the Planck tempera- near a stationary state. The same Reduction Principle ture, phase transitions may occur, leading to [17] was observed for modifiers of mutation rates and jumps from one string vacua to another. But dispersal rates as well as recombination (see [18–24] for so far there have been no detailed studies of a sample of the early studies). these processes which would have checked the The Reduction Principle proposes that populations at hypothesis that the change in each generation a balance between transmission and natural selection will is small. One study of a bouncing cosmol- evolve to reduce the rate of error in transmission. The ogy in quantum gravity also lends support to fundamental implication is that evolution has two differ- the hypothesis that the parameters change in ent properties that it operates to maximize: organismal each bounce [13]. fitness, and the faithfulness of transmission. One need not wait for these issues to be resolved before The mathematical basis of the Reduction Principle was investigating how universe inheritance laws would enter discovered by Karlin [25] (although he did not realize into the evolutionary dynamics. Moreover, one can ask: it) in a fundamental theorem on the interaction between What if universe inheritance laws are themselves subject growth and mixing: in a system of objects that (1) are to evolution? Here, another point made by Smolin [14] changed from one state to another by some transforma- is relevant: tion processes, and (2) grow or decay in number depend- ing on their state, then greater mixing produces slower [T]he evolution of laws implies a breakdown growth or faster decay. To be precise, Karlin’s theorem of the distinction between law and state. An- states: other way to say this is that there is an en- larged notion of state—a meta-state which Theorem 1 (Karlin [25, Theorem 5.2]). Let M be an codes information needed to specify both an irreducible stochastic matrix, and D a diagonal matrix effective law and an effective state, that the with positive diagonal elements. If D 6= c I for any c ∈ effective law acts on. R, then the spectral radius r([(1−α)I+αM)D) is strictly The ‘meta-state’ to account for the evolution of universe decreasing in α ∈ [0, 1]. inheritance laws would be information coding for the in- heritance laws, which allows them to vary between uni- This phenomenon is not restricted to finite state or verses and thus be subject to CNS. discrete time models. Karlin’s theorem has now been ex- tended to general resolvent positive operators on Banach spaces [26]. This relationship between mixing and growth II. MODIFIER THEORY FOR THE is therefore an extremely general mathematical property. EVOLUTION OF INHERITANCE MECHANISMS The implication for the theory of cosmological natural selection is that the parameters of the Standard Model Evolutionary theory in biology took just this approach may be optimized not simply for universe replication in trying to understand the evolution of biological inheri- rate, but also for faithfulness of universe replication, and tance mechanisms when Nei [15] posited the first modifier these two objectives may not coincide, but could poten- gene model. Mendel’s laws of genetic transmission had tially involve tradeoffs. Therefore, any finding that pa- hitherto been handled as exogenous parameters in the rameters of the Standard Model are not optimal for the mathematical models of evolution of the Modern Syn- rate of universe replication, via the production of black thesis. There was no reason to believe, however, that the holes in the case of Smolin’s proposed mechanism, cannot material basis for Mendel’s laws was any different from be taken on face value as falsification of the Cosmological the material basis for any other character of organisms. Natural Selection hypothesis. A further examination of The development of molecular biology confirmed that the how such parameters may be constrained by the mecha- mechanisms of inheritance — recombination, mutation, nism of universe inheritance, or how they may impact the DNA repair, etc. — utilized the same objects as organ- faithfulness of universe replication, must be considered. ismal traits involved with survival and fitness: proteins, nucleotides, regulatory sequences, gene interaction net- Main Implication. Parameters in the Standard Model works, and self-organizing structures and activities in the that are not local optima for the rate of production of cell and organism. black holes may be involved in the inheritance mecha- Nei’s modifier gene concept [15] explicitly “enlarged nisms of universe creation and subject to contravening notion of state” with “a meta-state which codes infor- selection for the preservation of universe properties. 3
III. ILLUSTRATION WITH A BRANCHING The following theorem gives the asymptotic behavior PROCESS MODEL of a super-critical branching process.
The modifier gene methodology may be applied to in- Theorem 2 (Kesten and Stigum [27] from [28, Theorem vestigate the evolution of hypothesized universe inheri- 1, p. 192]). Nn N tance mechanisms through addition of another degree of Let x(t) ∈ 0 , t ∈ 0 be vector of the number of indi- freedom to the universe state to allow for variation in the viduals xi of type i in an n-type Galton-Watson branching inheritance laws. process. Assume the following: Variation in the transmission laws can be drawn from 1. The process is nonsingular, i.e. it is not true that a very large choice of alternatives. Variation in the mag- for every j there is an i such that p(e , j)=1. nitude of changes in the free parameters of the Standard i Model is a natural choice. However, to model the con- 2. The process is supercritical, i.e. r(N) > 1. sequences of variation in magnitude one has to include a model of the map from parameters to reproductive prop- 3. N is irreducible and aperiodic. erties, a speculative and complicating exercise. The other principle alternative is to vary the probabilities among a Let v(N) be the Perron vector of N (the eigenvector fixed set of changes in state. The simplest variation in associated with the spectral bound of N) normalized so probabilities is to scale equally all of the transitions be- e⊤v(N)=1. tween states. Such variation is employed here. Then (almost surely, a.s.) The process of universe creation through black hole 1 creation is naturally modeled as a continuous time lim x1(t)= c v(N), branching process with an infinite number of types and t→∞ r(N)t with aging to account for individual universe develop- ment. The purpose here, however, is merely to illustrate where c ≥ 0 is a random variable such that Pr[c> 0] > 0 how universe inheritance mechanisms can evolve, and for if and only if for each i, j ∈{1,...,n} this it is sufficient to analyze a simplified branching pro- cess with discrete time, finite number of types, and no Lij := X xi log(xi) p(x, j) < ∞. (1) n aging. The multitype Galton-Watson process is defined x∈N0 as follows. First, some notation is introduced. The whole num- Let us now frame a multitype branching process for bers are represented as N0 := {0, 1, 2,...}. Vectors are cosmological natural selection. Let U represent the set of represented as x, v, e, etc. and matrices as N, P , D, I, possible universe types, with a finite number |U| of possi- etc.. Matrix elements are Nij ≡ [N]ij . The vector of ble types. Time is censused in epochs t =0, 1,.... From ⊤ ones is e, its transpose e , and ei is the vector with 1 at one epoch to the next, each universe produces offspring position i and 0 elsewhere. Let Dx := diag[xi] represent universes according to the multitype Galton-Watson pro- a diagonal matrix with diagonal elements xi. A column cess. The persistence of a universe between time epochs stochastic matrix P satisfies P ≥ 0 and e⊤P = e⊤. An is accounted for by considering it one of its own offspring. irreducible nonnegative matrix A has for every pair (i, j) Some structure is now introduced for the expectation t some t ≥ 1 such that [A ]ij > 0, and if it is aperiodic, matrix N. We decompose Nij in terms of the total ex- t A > 0 for some t ≥ 1. Letting λi(A) be the eigenvalues pected number of offspring and the distribution among of A, the spectral radius is r(A) = maxi |λi(A)|, and the that total. Let spectral bound is s(A) = maxi Re(λi(A)), where Re(x) ⊤ is the real part of x. βj = X Nij = X e x p(x, j) n Nn Let x ∈ N0 be the n-vector of the number of indi- i∈U x∈ 0 viduals of each of n types. The probability that a sin- gle individual of type j produces a vector of offspring be the expected total number of offspring from type j. numbers x is p(x, j). Starting with an initial popula- This value βj represents the fitness of universe type j. tion x(0) at time t = 0, each individual produces an Let Tij ≥ 0 be the expected fraction type i among the N offspring vector x according to p(x, j), and these vectors offspring of j, so for each j, Pi∈U Tij = 1. Then is are summed to create the population at the next time decomposed into a fitness component and an offspring step, x(1). The reproduction process then repeats. This distribution component: Nij = Tij βj . Define the matrix defines the branching process {x(t): t =0, 1, 2,...}. T := [Tij ]i,j∈U . Then in matrix notation: The expected number of offspring of type i from parent j is N = TDβ. The offspring distribution T is further decomposed. The Nij = X xi p(x, j). n expected proportion of offspring that differ from parent j x∈N0 is given as µj . Thus 1−µj is the expected fraction of type It is assumed that Nij exists for all i, j = {1,...,n}. The j’s offspring that are also type j, i.e. Tjj =1 − µj . The expectation for the process is E[x(t)|x(0)] = N tx(0). expected fraction of offspring of type j that are type i 4 among the non-type-j offspring is Pij , hence Tij = µj Pij Now let us examine the evolution of the multiverse for i 6= j. This decomposition in matrix form is ensemble
ν×|U| T = (I − Dµ)+ PDµ = I + (P − I)Dµ. (2) x := (x1, x2,..., xν ) ∈ N0 where T and P are stochastic matrices. It should be where there are ν different transmission laws in the en- noted that many different probability measures p(x, j) semble, where each transmission law a ∈ T is carried by |U| can produce the same matrix of expected proportions T . sub-population xa ∈ N0 . The matrix of expected num- Now let us include variation in the branching pro- bers for the entire ensemble is a direct sum of diagonal cesses. We enlarge notion of state i ∈ U to a meta- blocks Na: state (a,i) ∈ U ′ := T ×U, where T is the set of pos- sible transmission laws for U. A transmission law for N := N1 ⊕ N2 ⊕···⊕ Nν . the branching process is the probability measure pa(x, j), |U| The diagonal block structure is the result of the trans- pa : N0 × U → [0, 1], that defines the branching process. An important simplifying assumption made here is that mission laws themselves being faithfully transmitted. the transmission law states a ∈ T are themselves trans- With framework defined, we have the following result. mitted faithfully in offspring universes. A model in which Theorem 3 (Domination of the Most Conservative the transmission laws themselves transform in universe Transmission Law). creation is necessary to a more complete theory, but is deferred to further study. By making the transmission 1. Consider an ensemble of universes reproducing ac- law faithfully replicated, we know that its evolution is cording to a branching process, where transmission due solely to its effects on the universe replication be- law a ∈ T determines the transmission law parame- havior. ter ma ∈ (0, 1] which scales the expected proportion Further, variation in transmission law states a ∈ T is of universes i ∈U that differ from their parent uni- assumed for now to be neutral, in that it leaves unchanged verse types j ∈U. the vector of the mean number of offspring, β. Variation in inheritance laws will be encapsulated 2. Let the matrix of expected numbers of offspring uni- solely through a parameter ma ∈ [0, 1] (a ∈ T ) that verses of type i from a universe of type j be scales equally the expected proportion of offspring uni- verses that differ from their parents’ type. In the popu- Na = [I + ma(P − I)Dµ]Dβ, for each a ∈ T , lation genetics literature this is referred to as linear vari- where µ is the vector of the fractions of offspring ation [29]. In matrix form, the scaling by ma enters into that are not identical to their parents, ma scales µ, the expected proportion matrix Ta, a ∈ T , from (2) as β > 0 is the vector of expected numbers of offspring universes for each type in U, and P is the matrix of Ta = (I − maDµ)+ maPDµ = I + ma(P − I)Dµ. expected distributions of offspring type i from parent Thus the matrix of the expected numbers of offspring type j 6= i, Pii =0. for universes within the transmission class a ∈ T is 3. In addition make the technical assumption that for each a ∈ T using p (x, j) in (1) that L < ∞ for N = [I + m (P − I)D ]D . (3) a ij a a µ β all i, j ∈U.
Remark 1. A model in which there is variation in the 4. Assume that the ensemble branching process is magnitude of changes between parent and offspring uni- supercritical, so the spectral radius is r(N) = verses would also be manifest as changes in N, but they maxa∈T r(Na) > 1. would not be linear variation. For example, let changes in a universe parameter γj follow a Gaussian distribu- 5. Let stochastic matrix P be irreducible and aperi- tion (a case considered by Vilenkin [11]), parameterized odic, and let some βi 6= βj . Let the transmission by ma, to give expected distribution of offspring i having parameters be ordered so that m1 In linear variation, ma scales all Tij (ma) equally for i 6= j, where c1 is as in Theorem 2. and disappears from the derivative of Tij (ma), but clearly Therefore, the ensemble of universes becomes dom- this is not the case in (4). The analysis of r(T (ma)Dβ) inated by the transmission law with the highest ex- under such non-linear variation is very much an open pected preservation of universe types in universe creation problem. events. 5 Proof. For the branching process for each transmission reproduction rates, we may give type a its own vector of type a ∈ T , Theorem 2 gives fitnesses βa, and then the matrix of expected offspring numbers becomes 1 lim xa(t)= ca v(Na) a.s.. t→∞ t r(Na) Na = [I + ma(P − I)Dµ]Dβa . (6) Suppose that r(Nb) < r(Na). Then It is clear that if ma < mb then there is some region of t βa values that maintains r(Na) > r(Nb), and this range 1 r(Nb) 1 includes values for which βai <βbi for each universe type lim t xb(t) = lim t t xb(t) t→∞ r(Na) t→∞ r(Na) r(Nb) i ∈U. t r(Nb) Hence, universe type (a,i) can become dominant over = cb v(Nb) lim t = 0 a.s.. t→∞ r(Na) another type (b,i) even though βai < βbi, due to ma < mb. Thus we need to know when r(Nb) < r(Na). This occurs Therefore, to find a universe type (b,i) ∈T ×U with when mb >ma, as shown by Theorem 1. An equivalent fitness greater than our own universe type (a,i) does not form from [26] is that necessarily falsify Cosmological Natural Selection, but rather points to the involvement of property a in uni- d s(D + mA) III.1. When Transmission Fidelity is in Conflict V. DISCUSSION with Universe Production Theorem 3 shows that the ensemble of branching uni- The variation in universe transmission laws is assumed verses comes to be dominated by universe transmission to be neutral as framed Theorem 3: ma varies without laws which are the most faithful in preserving the fun- varying the expected offspring production βi. However, damental constants among offspring universes. This is a nature need not produce such a clean separation of ef- manifestation of the Reduction Principle from population fects. If the transmission type a ∈ T also affects the biology applied here in a new setting. 6 The ubiquitous manifestation of this phenomenon is tradeoff between these two properties, then the ob- due to the fundamental mathematics of how mixing in- servation of non-optimal values of Standard Model teracts with heterogeneous growth. The most general parameters with respect to black hole production characterization of this interaction is for operators on may point to the involvement of those parameters Banach spaces in [26], that in the universe transmission law. d The model analyzed here, while more general than a s(mA + V ) ≤ s(A), dm toy model, also leaves out of consideration how trans- mission laws may be involved in their own transmission. where A is resolvent positive operator on a Banach space, There are also many other higher-order phenomena in V is a real-valued operator of multiplication, s() is the evolutionary dynamics that may prove valuable to in- spectral bound, and m> 0. Resolvent positive operators clude in models of cosmological natural selection, includ- A include Schr¨odinger operators, second order elliptic op- ing the evolution of mutational robustness, the evolution erators, diffusions, integral operators, and in general, any of evolvability, and the evolution of modularity in the generator of a positive strongly continuous semigroup. genotype-phenotype map. This theorem derives from Kato’s [1982] generalization While the Cosmological Natural Selection hypothe- to Banach spaces of Cohen’s [1981] theorem that for fi- sis remains largely speculative, inclusion of higher-order nite matrices A, s(A+D) is convex in diagonal matrices phenomena of evolutionary dynamics may provide ad- D. ditional guidance in deriving falsifiable predictions from This dynamic has two principle implications for the the theory. Cosmological Natural Selection hypothesis: 1. It gives a causal mechanism to incorporate universe ACKNOWLEDGMENTS transmission laws into the framework of cosmolog- ical natural selection, and predicts that transmis- I thank Megan Loomis Powers for introducing me to sion of the laws of physics between parent and off- A. Garrett Lisi, whose collaboration with Lee Smolin spring universes should be very conservative. brought to my attention Smolin’s Cosmological Natural Selection hypothesis. I thank Laura Marie Herrmann 2. It demonstrates that in addition to selection for the for assistance with the literature search. I thank Martin maximal number of offspring produced, selection is Rees, George Ellis, Alexander Vilenkin, and Cl´ement Vi- also indirectly minimizing the probability of change dal for their comments, and Cl´ement Vidal and Brian D. between parent and offspring. Should there be a Josephson for noting typos in an earlier draft. [1] Lee Smolin, “Did the universe evolve?” Classical and [12] Massimo Pigliucci and Gerd B. 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