Speculative String-Cosmologies as Geodesically Incomplete.

Penrose diagrams of de Sitter space in the flat (left) and static (right) slicings that each cover only part of the whole de Sitter space, and that are both geodesically incomplete.

Alan Guth, Alvin Borde and Alexander Vilenkin have argued that within the framework of a future-eternal inflationary multiverse, as well as some more speculative string-cosmologies, all worldlines are geodesically incomplete and, thus, the multiverse has to have a beginning. Unfortunately, if future-eternal inflation is true, all “hypotheses about the ultimate beginning of the universe would become totally divorced from any observable consequences. Since our own pocket universe would be equally likely to lie anywhere on the infinite tree of universes produced by eternal inflation, we would expect to find ourselves arbitrarily far from the beginning. The infinite inflating network would presumably approach some kind of steady state, losing all memory of how it started […] Thus, there would be no way of relating the properties of the ultimate origin to anything that we might observe in today’s universe.” (Guth).

On the other hand, Andrei Linde has argued that the multiverse could be past-eternal, because either all single world lines might have to start somewhere, but not the whole bundle of them (Linde), or there could even exist some (albeit strange) space-times with single past-eternal world lines.

This issue is not settled, and even in those scenarios a global arrow of time may not necessarily exist. However, there are other frameworks possible – and they have even already been developed to some extent, where a future-eternal inflationary multiverse is both not past-eternal and beginningless but arise from some primordial vacuum which is macroscopically time-less. Thus, again, the beginning of some classical space-times is not equivalent with the beginning of everything.

We can even imagine that there is no multiverse, but that the whole (perhaps finite) universe – our universe – once was in a steady state without any macroscopic arrows of time but, due to a statistical fluctuation above a certain threshold value, started to expand  – or to contract, bounce and expand – as a whole and acquired an arrow of time. In such a case the above-mentioned reply, which was based on the spatial distinction of a beginning of some parts of the world and the eternity of the world as a whole, would collapse.

Nevertheless it is necessary to distinguish between the different notions and extensions of the term “universe”. In the simplest case, Kant’s antinomy might be based on an ambiguity of the term “world” (i.e. the difference between “universe” and “multiverse”), but it does not need to; and it was not assumed here that it necessarily does. The temporal part of Kant’s first antinomy was purely about the question whether the macroscopic arrow of time is past-eternal or not. And if it is not past-eternal this does not mean that time and hence the world has an absolute beginning in every respect – it is still possible that there was or is a world with some underlying microscopic time. (By the way, one can also imagine that, even if our arrow of time is past- and/or future-eternal, there might exist “timeless islands” someday: for instance isolated black holes if they would not ultimately radiate away due to quantum effects, or empty static universes if they could split off of our space-time.)

Of course it is possible that firstly a natural principle of plentitude is realized and different multiverses (sets of universes) exist totally independent from each other, and secondly that some of them are truly past-eternal while others have an absolute beginning and others have only local starting points of local arrows of time as it was suggested here. If so, we might not be able to tell in what kind we live in. And this would be irrelevant in the end, because then every possible world is actual and probably exists infinitely often. But we do not know whether such an extreme principle of plentitude does apply or if cosmology is ultimately just and only a matter of pure logical consistency, allowing us finally to calculate the complete architecture of the world by armchair-reasoning.

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Spreading Dynamics Over Trading Prices in the Market

Market time series can be seen as a composite of the set of M interacting dynamical sub-system. Investors put their trading decisions due to their portfolio and market strategies, shaping the prices of the traded stocks. Over time, the prices are depicted the dynamical processes within the collective behavior of the investors. The vicissitudes of a price could affect the dynamic of other prices due to their portfolios. Capturing the dynamics of spreading ups and downs within the market is observing the information flow from one price to one another. For instance we have a source system 𝒴(𝑡) as the source of information affecting other sub-system 𝒳(𝑡), collecting the remaining sub-systems in the vector of 𝒵(𝑡). From the information theoretic studies, we know that the differential entropy of a random vector 𝒳 is defined,

h(𝒳(𝑡)) = −∫ 𝑑 𝑝(𝒙)ln𝑝(𝒙)𝑑𝒙

as the random vector takes value in 𝔑𝑑 with probability density function 𝑝(𝒙). When the random variable 𝒳(𝑡) is multivariate discrete of all possible values of 𝑥 ∈ {𝑥₁, 𝑥₂, … , 𝑥_𝑛}, the entropy is

𝐻(𝒳(𝑡)) = − ∑𝑛𝑖=1 𝑝(𝑥) ln 𝑝(𝑥𝑖)

where now, 𝑝 is the probability mass function of 𝒳. Thus, the transfer entropy,

𝒯_{𝑌(𝑡)→𝑋(𝑡)|𝑍(𝑡)}

of the previous 𝒳(𝑡), 𝒴(𝑡), and 𝒵(𝑡) is written as,

𝒯_{𝑌(𝑡)→𝑋(𝑡)|𝑍(𝑡)} = 𝐻(𝑋(𝑡)|⟦𝑋⁻(𝑡), 𝑍⁻(𝑡)⟧) − 𝐻(𝑋(𝑡)|⟦𝑋⁻(𝑡), 𝑌⁻(𝑡), 𝑍⁻(𝑡)⟧)

where 𝐻(𝐴) denotes the entropy of the variable 𝐴, 𝐻(𝐴|𝐵) the conditional entropy,

𝐻 ( 𝑋 ( 𝑡 ) | 𝑌 ( 𝑡 ) ) = − ∑^𝑛_{𝑖 = 1} ∑^𝑚_{𝑗 = 1} 𝑝 (𝑥_𝑖,𝑦_𝑖) l n 𝑝 (𝑥_𝑖 | 𝑥_𝑖)

for 𝑚 can be different with 𝑛, and 𝑝(𝑥_𝑖|𝑥_𝑖) as the conditional probability, as to

𝐻 ( 𝑋 ( 𝑡 ) | 𝑌 ( 𝑡 ) ) = − ∑^𝑛_{𝑖 = 1} ∑^𝑚_{𝑗 = 1} 𝑝 (𝑥_𝑖,𝑦_𝑖) l n 𝑝 (𝑥_𝑖 | 𝑥_𝑖)

with 𝑝(𝑥_𝑖,𝑥_𝑖) as the joint probability. The past of vectors 𝒳(𝑡), 𝒴(𝑡), and 𝒵(𝑡) are respectively 𝑋⁻(𝑡) = {𝑋(𝑡 − 1), 𝑋(𝑡 − 2), … , 𝑋(𝑡 − 𝑝)}, 𝑌⁻(𝑡) = {𝑌(𝑡 − 1), 𝑌(𝑡 − 2), … , 𝑌(𝑡 − 𝑝)}, and 𝑍⁻(𝑡) = {𝑍(𝑡 − 1), 𝑍(𝑡 − 2), … , 𝑍(𝑡 − 𝑝)} with the length vector 𝑝, and the vectors in the bracket ⟦𝐴, 𝐵⟧ are concatenated.

From there we have,

𝒯_{𝑌(𝑡)→𝑋(𝑡)|𝑍(𝑡)} ≡ ∑ 𝑝(𝑋(𝑡), 𝑋⁻(𝑡), 𝑌⁻(𝑡), 𝑍⁻(𝑡))𝑙𝑛 𝑝(𝑋(𝑡)|𝑋⁻(𝑡),𝑌⁻(𝑡),𝑍⁻(𝑡))/(p(𝑋(𝑡)|(𝑋(𝑡),𝑍⁻(𝑡))

𝑌(𝑡)→𝑋(𝑡)|𝑍(𝑡) 𝑝(𝑋(𝑡)|𝑋(𝑡),𝑍−(𝑡))

where 𝑝(𝐴) is the probability associated with the vector variable 𝐴, and 𝑝(𝐴|𝐵) = 𝑝(𝐴,𝐵)/𝑝(𝐵) probability of observing 𝐴 with knowledge about the values of 𝐵.

The notion of the entropy is an information theoretic terminology that can be regarded as the measure of the disorder level within the random variable of the time series data. Transfer entropy from 𝒴(𝑡) to 𝒳(𝑡) is reflecting the amount of disorderliness reduced in future values of 𝒳(𝑡) by knowing the past values of 𝒳(𝑡) and the given past values of 𝒴(𝑡). Time “moves” as entropy is transferred and observed in flowing information from series to series.

We have two regressions toward 𝑋(𝑡), the first is the moving series without putting the 𝑌(𝑡) into account,

𝑋(𝑡) = 𝐴⟦𝑋⁻(𝑡), 𝑍⁻(𝑡)⟧ + ∈₁(𝑡) and the other one which regard to the information transfer from 𝑌(𝑡) to 𝑋(𝑡),

𝑋(𝑡) = 𝐴⟦𝑋⁻(𝑡), 𝑌⁻(𝑡), 𝑍⁻(𝑡)⟧ + ∈₂ (𝑡)

where A is the vector of linear regression coefficient, and the ∈₁ and ∈₂ are the residuals of the regression. The residuals have respective variances of 𝜎(∈₁) and 𝜎(∈₂), and under Gaussian assumption, the entropy of 𝑋(𝑡) is,

𝐻(𝑋(𝑡)| 𝑋⁻(𝑡), 𝑍⁻(𝑡)) = 1/2 (ln 𝜎(∈₁) + 2𝜋𝑒))

and

𝐻(𝑋(𝑡)| 𝑋⁻(𝑡), 𝑍⁻(𝑡)) = 1₂ (ln 𝜎(∈₂) + 2𝜋𝑒))

Thus, we can get the estimated transfer entropy

𝒯_{𝑌(𝑡)→𝑋(𝑡)|𝑍(𝑡)} = 1/2 ln 𝜎(∈₁)

This information theoretic notion opens the bridging discussions to the statistics of the autoregressive methods of Granger-causality. The idea of Granger-causality came from understanding that 𝒴(𝑡) is said to cause 𝒳(𝑡) for 𝒴(𝑡) helps predict the future of 𝒳(𝑡). This is a statistical concept equivalent with the transfer entropy, of which in our case, the Granger-causality is estimated as,

𝒢_{𝑌(𝑡)→𝑋(𝑡)|𝑍(𝑡)} = ln 𝜎(∈₁)/ 𝜎(∈₂)= 2 𝒯_{𝑌(𝑡)→𝑋(𝑡)|𝑍(𝑡)}

Thus, the entropy transferred can be seen as causal relations among random variables, with which we can learn the spreading dynamics over trading prices in the market represented by the multivariate data.