Relative Structure: A New Measure for Cosmology

We propose a new concept of relative structure (RS) as a way to quantify the degree of order and complexity in a universe. RS is defined as the ratio of the actual information content of a universe to the maximum possible information content given its size and energy. RS ranges from 0 to 1, where 0 corresponds to a state of maximum entropy and 1 corresponds to a state of maximum order. We show that RS can be used to compare different universes and to study the evolution of structure formation in our own universe.

The study of cosmology aims to understand the origin, structure, and fate of the universe as a whole. One of the fundamental questions in cosmology is how the universe evolved from a simple and homogeneous state to a complex and heterogeneous one, where stars, galaxies, planets, and life can exist. This question is related to the concept of entropy, which measures the degree of disorder or randomness in a system. The second law of thermodynamics states that the entropy of an isolated system can only increase or remain constant over time, implying that the universe tends to become more disordered as it expands and cools.

However, this does not mean that order and complexity cannot emerge in some regions of the universe, as long as they are compensated by an increase in entropy elsewhere. In fact, many physical processes, such as gravitational collapse, nuclear fusion, chemical reactions, and biological evolution, can generate local structures that decrease the entropy of their subsystems. These structures can be seen as islands of order in a sea of chaos, or as pockets of negative entropy that resist the overall trend of increasing entropy.

The question then arises: how can we measure and compare the amount of order and complexity in different regions of the universe, or in different universes altogether? Is there a universal metric that can capture the essence of structure formation in cosmology? In this paper, we propose such a metric, which we call relative structure (RS). RS is defined as the ratio of the actual information content of a universe to the maximum possible information content given its size and energy. RS ranges from 0 to 1, where 0 corresponds to a state of maximum entropy and 1 corresponds to a state of maximum order. RS is analogous to Kelvin, where the value 0 is meaningful and represents absolute zero.

We show that RS can be used to compare different universes and to study the evolution of structure formation in our own universe. We also discuss some implications and applications of RS for physics, philosophy, and fiction.

We define RS as follows:

RS = I / Imax

where I is the actual information content of a universe and Imax is the maximum possible information content given its size and energy.

The information content of a universe can be measured by its Kolmogorov complexity, which is the minimum length of a program that can generate the description of the universe on a universal Turing machine. The Kolmogorov complexity captures both the randomness and the regularity of a system, and it is invariant under any computable transformation. The Kolmogorov complexity is also related to the Shannon entropy, which measures the average amount of information per symbol in a message. The Shannon entropy can be seen as an upper bound on the Kolmogorov complexity, since any message can be compressed to its Kolmogorov complexity but not further.

The maximum possible information content given the size and energy of a universe can be estimated by using the Bekenstein bound, which states that the entropy (and thus the information) of a physical system cannot exceed a certain limit proportional to its surface area and inversely proportional to its energy. The Bekenstein bound implies that there is a finite amount of information that can be stored in any region of space-time. The Bekenstein bound can be generalized to apply to any universe with arbitrary geometry and topology by using the holographic principle, which states that any universe can be described by a lower-dimensional theory on its boundary.

Using these concepts, we can calculate RS for any universe by dividing its Kolmogorov complexity by its Bekenstein bound. This gives us a dimensionless number between 0 and 1 that reflects how much structure exists in a universe relative to its size and energy.

Using RS as a metric, we can compare different universes and see how they rank in terms of order and complexity. For example, we can compare our own observable universe with other hypothetical universes with different physical laws or initial conditions. For instance, we can imagine a universe with the same size and energy as ours, but with different values of the fundamental constants, such as the speed of light, the gravitational constant, or the fine-structure constant. These changes would affect the formation and stability of atoms, molecules, stars, and galaxies, and thus the amount of information that can be encoded in matter and radiation. We can also imagine a universe with a different initial state, such as a higher or lower entropy, a different inflationary scenario, or a different symmetry breaking pattern. These changes would affect the initial fluctuations and the subsequent evolution of structure in the universe.

We can also compare our own universe at different stages of its history and see how RS changes over time. According to the standard cosmological model, our universe began in a state of high entropy and low RS, where it was nearly homogeneous and isotropic. As the universe expanded and cooled, small fluctuations in density and temperature grew due to gravitational instability and formed the seeds of structure formation. As a result, RS increased as matter and radiation became more organized and complex. This process reached its peak at the epoch of recombination, when atoms formed and photons decoupled from matter. At this point, RS was maximal, as the universe contained a rich variety of structures at different scales, from galaxies and clusters to stars and planets.

However, after recombination, RS started to decrease again, as the universe entered a phase of accelerated expansion driven by dark energy. This expansion diluted the matter density and stretched the wavelengths of photons, reducing the information content of both components. Moreover, this expansion also increased the size and energy of the universe, raising the Bekenstein bound and lowering RS. As a consequence, RS has been decreasing ever since recombination and will continue to do so in the future. In the far future, RS will approach zero again, as the universe will become dominated by dark energy and will approach a state of maximum entropy and minimum structure.

We have introduced a new concept of relative structure (RS) as a way to quantify the degree of order and complexity in a universe. RS is defined as the ratio of the actual information content of a universe to the maximum possible information content given its size and energy. RS ranges from 0 to 1, where 0 corresponds to a state of maximum entropy and 1 corresponds to a state of maximum order. We have shown that RS can be used to compare different universes and to study the evolution of structure formation in our own universe.