Advances in Astronomy

Volume 2015 (2015), Article ID 135025, 11 pages

http://dx.doi.org/10.1155/2015/135025

## Cosmic Rays Report from the Structure of Space

^{1}Department of Physics, University of Helsinki, 00014 Helsinki, Finland^{2}Department of Biosciences, University of Helsinki, 00014 Helsinki, Finland

Received 3 June 2015; Revised 14 August 2015; Accepted 24 August 2015

Academic Editor: Alberto J. Castro-Tirado

Copyright © 2015 A. Annila. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Spectrum of cosmic rays follows a broken power law over twelve orders of magnitude. Since ubiquitous power laws are manifestations of the principle of least action, we interpret the spectrum accordingly. Our analysis complies with understanding that low-energy particles originate mostly from rapidly receding sources throughout the cosmos. The flux peaks about proton rest energy whereafter it decreases because fewer and fewer receding sources are energetic enough to provide particles with high enough velocities to compensate for the recessional velocities. Above 10^{15.6} eV the flux from the expanding Universe diminishes below the flux from the nearby nonexpanding part of the Universe. In this spectral feature, known as the “knee,” we relate to a distance of about 1.3 Mpc where the gravitational potential tallies the energy density of free space. At higher energies particles decelerate in a dissipative manner to attain thermodynamic balance with the vacuum. At about 10^{17.2} eV a distinct dissipative mechanism opens up for protons to slow down by electron-positron pair production. At about 10^{19.6} eV a more effective mechanism opens up via pion production. All in all, the universal principle discloses that the broad spectrum of cosmic rays probes the structure of space from cosmic distances down to microscopic details.

#### 1. Introduction

Cosmic rays [1, 2] impinge on Earth’s atmosphere with energies that have been measured over twelve orders of magnitude. The particle flux versus energy, when displayed on the log-log plot, compiles mostly from straight lines; that is, the data follows a broken power law [3–9]. Since the power law is a manifestation of the principle of least action in its original form by De Maupertuis [10–13], we are motivated to use this universal imperative to account for particle propagation from cosmic origins to observatories on Earth.

Not only does the spectrum of cosmic rays display power laws, but also other astronomical observations display scale-free patterns [14]. Furthermore, we are motivated to employ the ubiquitous law of nature because it has helped to interpret propagation of light from Type Ia supernovae [15] as well as to reevaluate some other astronomical observations [16–18]. Since production of particles often couples with emission of light, we find the thermodynamic tenet, where everything depends on everything else, as justifiable approach to analyze also the cosmic ray spectrum.

Today, one century since the discovery of cosmic rays, wealth of data has been acquired about these perplexing particles and understanding has emerged, yet it seems hard to put all pieces of the puzzle together [19–21]. Undoubtedly the principle of least action also fails to exhaust all open questions, but the holistic perspective may provide some new thoughts how to interpret data.

#### 2. The Least-Time Principle

The principle of least action in its original form generalizes Fermat’s least-time principle for propagation of energy in any form, that is, not only in the form of light. Thus the universal law of nature simply says that evolution, that is, changes of state, will consume difference in energy of any kind in least time [12, 13]. For instance, when a particle with mass propagates with velocity along its least-time path, that is, along geodesic, the conservation of quanta requires that a change in kinetic energy equates changes in scalar and vector potentials; that is, where the change in kinetic energy is invariably coupled with changes both in the scalar potential and in energy which is dissipated from the system to its surroundings or* vice versa*. The change in mass is for many changes of state, for example, for chemical reactions, small and even minute; nevertheless, if it is omitted, the account is incomplete and conceptual conundrums will follow. Hence ought to be acknowledged and transcribed to the dissipative change in energy by the familiar mass-energy equivalence in the vacuum.

The equation of evolution (1) from one state to another can be derived from the notion of probability in statistical mechanics of open system [13, 22]. It is also easy to recognize, just as De Maupertuis did, that (1) is Newton’s 2nd law of motion for a change in momentum in its original complete form when multiplying with velocity ; that is, where the familiar term of acceleration is identified with the gradient of , since , as well as the change in mass with dissipation using the mass-energy equivalence for the conversion of bound energy in mass to energy of freely propagating photons in the vacuum which is characterized by the squared speed of light . So, the least-time imperative is equivalent to Newton’s notion of net force as it should. Namely, the least-time path points along the resultant force.

The use of mass-energy equivalence (see (2)) in the context of a seemingly nonrelativistic equation may though raise the eyebrows. Note that here for the vacuum is regarded merely as the special form of the general form , known as* vis viva* [23] in the 18th century, most notably, by du Châtelet, s’ Gravesande, Leibniz, and de Maupertuis. Later, when kinetic energy was reduced to , the notion narrowed to quantify only changes in the body’s motion, thereby ignoring concomitant changes in the surrounding energy landscape. For example, when a particle of cosmic origins slows down by losing energy via dissipative processes, the surrounding vacuum will absorb the emitted energy, and hence its state will change too. In other words, every change in the state of system is coupled with corresponding change in its surrounding energy density [12, 13]. Thus, any equation of motion is incomplete if it ignores motional changes in the surroundings that are concurrent with changes in the state of a particle. Therefore, we conclude that the general principle of nature applies also to the cosmic rays that propagate along least-time paths through the evolving Universe.

The principle of least action is, of course, widely used in physics; however, it is primarily used in the nondissipative and deterministic form as was devised by Lagrange. In contrast, according to (1), the dissipative evolution from one state to another is a nondeterministic process. An outcome of this nonholonomic process depends not only on the initial state but also on the taken path. History emerges because the motion consumes its driving forces which, in turn, affect the motion and so on. In other words, when variables cannot be separated, the equation of evolution (see (1) and (2)) cannot be solved exactly. Although Maupertuis’ form did not meet the onetime expectations of a computable law, there is nothing wrong with it. On the contrary, it accurately accounts for evolving nature.

Undoubtedly our reappraisal of the least-time principle in its old original form prompts many a physicist to query for physical assumptions that underlie the tenet. These grounds are minimal and elementary. Namely, quanta of actions are regarded to embody everything, including the vacuum [24]. Specifically the photon is the quantum of action. Its invariance is quantified by Planck’s constant as the product of photon’s energy and period . The photon, as the most elementary action, is a force carrier that moves energy from the system to its surroundings. For instance, X-rays are emitted when electrons are forced to curve on a path. Additionally, the photon itself will lose energy by shifting to red in quest of attaining energetic balance with its surroundings when propagating from the energy-dense nascent Universe to the energy-sparse contemporary surroundings. In other words, no system can change from one state to another without either absorbing or emitting energy.

Maupertuis’ principle does not divide the system to an invariant body of mass propagating along some path. Instead integration of momentum (of a body) on its path yields the entity known as the action . It sums the system in question from quantum of actions up to a certain integer . This 18th century perception of nature in geometrical terms of geodesic actions was consistent and comprehensive. For instance, the dissipative change in mass (see (2)) did not puzzle Euler who quantified the change in mass as a change in the geodesic curvature of an action’s path relative to the straight paths of universal reference, that is, the vacuum. Thus the concept of mass as Euler’s characteristic merely quantifies how much rays of light will deviate in the vicinity of a body away from their otherwise straight paths in the free space [18, 25]. This quality of mass is, of course, contained in general relativity too.

In general, the least-time consumption of free energy results in a sigmoid curve, whose central region dominates on the log-log scale as a power law [13]. Similarly the least-time free energy consumption by several mechanisms over a range of energies yields a series of sigmoid curves that follow each other as consecutive lines on a log-log plot; that is, the data follows a broken power law as the spectrum of cosmic rays.

Since the broken power law characterizes the cosmic ray spectrum, we will employ the universal least-time principle to examine the flux of particles over the entire range of measured energies with the aim of relating spectral features to certain mechanisms of free energy consumption. So, we will not provide some specific new results on any particular aspect of cosmic rays; instead we will compile a holistic account on the propagation of particles across cosmic distances down to microscopic details of dissipation.

Of course, many an expert may not be in need of comprehensive interpretation by the general principle of physics but prefers a particular mechanistic explanation for each spectral characteristic. We do not discard mechanisms either but see them only in the service of least-time free energy consumption. By the same token we do not regard the power-law characteristic merely as a phenomenological model but a fundamental consequence.

It is perhaps worth emphasizing that the general principle of least time does not unveil specific mechanisms of energy transfer. It only relates a change in the spectral index with the change from one mechanism to another. Therefore we are only able to associate a distinct change with a particular mechanism based on what is known by previous studies about the onset energy. Of course, it is conceivable that in a given spectral band there are two or more main mechanisms. For example, it has deduced that along with particle production processes toward the end of the spectrum, and also the composition of flux changes [26–28]. We cannot exclude these model-dependent conclusions but prefer to account for the changes in spectral index by the least number of parameters that are needed in our model.

#### 3. Spectral Characteristics

For the principle to be universal also in the context of cosmic rays, it ought to make sense of the particle propagation over the entire broad band of energies spanned by the particles of cosmic origins. Moreover, each change in the flux ought to be associated with an onset or ending of free energy consumption by a particular mechanism. So, in the following, we will proceed to relate, step by step, each spectral band to a particular process.

##### 3.1. Stationary Flux

Fine features of the cosmic ray spectrum are customarily exposed so that the flux is presented when multiplied by kinetic energy raised to a power of the spectral index . The conventional choice is motivated, as we will show below, because it corresponds to the flux produced by a steady-state partition of sources. Namely, the evolutionary equation of motion (see (1)) integrates to the familiar virial theorem [29], , in the particular case when the system has already attained a free energy minimum state. At the steady state there is no net dissipation; that is, , and hence at thermodynamic balance the least-time paths are stationary.

We find from the equipartition theorem that the steady-state flux must fall as one over the cube of kinetic energy, that is, , to comply with a constant total flux of particles integrated over all energies in unit time per unit area originating from a maximum entropy partition of sources in a constant volume, here simply a sphere with radius , over the density of states with average energy ; that is,is a constant.

So, when the observed flux is multiplied with , the published data of several measurements [30–45] are roughly constant in the range from some million gigaelectron volts (eV) to about billion gigaelectron volts (Figure 1).