Cosmological expansion on a local scale is usually neglected in part due to its smallness, and in part due to components of bound systems (especially those bound by non-gravitational forces such as atoms and nuclei) not following the geodesics of the
cosmological metric. However, it is interesting to ask whether or not experimental tests of cosmological expansion on a local scale (well within our own galaxy) might be experimentally accessible in some manner. We point out, using the Pioneer satellites as an example, that current satellite technology allows for this possibility within time scales of less than one human lifetime.
In a recent letter, the AMS collaboration reported the detailed and extensive data concerning the distribution in energy of electron and positron cosmic rays. A central result of the experimental work resides in the energy regime $30 {rm GeV}< E < 1
{rm TeV}$ wherein the power law exponent of the energy distribution is measured to be $alpha ({rm experiment})=3.17$. In virtue of the Fermi statistics obeyed by electrons and positrons, a theoretical value was predicted as $alpha ({rm theory})=3.151374$ in very good agreement with experimental data. The consequences of this agreement between theory and experiment concerning the sources of cosmic ray electrons and positrons are briefly explored.
We here argue that the knee of the cosmic ray energy distribution at $E_c sim 1$ PeV represents a second order phase transition of cosmic proportions. The discontinuity of the heat capacity per cosmic ray particle is given by $Delta c=0.450196 k_B$.
However the idea of a deeper critical point singularity cannot be ruled out by present accuracy in neither theory nor experiment. The quantum phase transition consists of cosmic rays dominated by bosons for the low temperature phase E<E_c and dominated by fermions for high temperature phase $E > E_c$. The low temperature phase arises from those nuclei described by the usual and conventional collective boson models of nuclear physics. The high temperature phase is dominated by protons. The transition energy $E_c$ may be estimated in terms of the photo-disintegration of nuclei.
We have recently shown that the cosmic ray energy distributions as detected on earthbound, low flying balloon or high flying satellite detectors can be computed by employing the heats of evaporation of high energy particles from astrophysical sources
. In this manner, the experimentally well known power law exponents of the cosmic ray energy distribution have been theoretically computed as 2.701178 for the case of ideal Bose statistics, 3.000000 for the case of ideal Boltzmann statistics and 3.151374 for the case of ideal Fermi statistics. By ideal we mean virtually zero mass (i.e. ultra-relativistic) and noninteracting. These results are in excellent agreement with the experimental indices of 2.7 with a shift to 3.1 at the high energy ~ PeV knee in the energy distribution. Our purpose here is to discuss the nature of cosmic ray power law exponents obtained by employing conventional thermal quantum field theoretical models such as quantum chromodynamics to the cosmic ray sources in a thermodynamic scheme wherein gamma and zeta function regulation is employed. The key reason for the surprising accuracy of the ideal boson and ideal fermion cases resides in the asymptotic freedom or equivalently the Feynman parton structure of the ultra-high energy tails of spectral functions.
