Cosmic Time Transformations in Cosmological Relativity

The relativity of cosmic time is developed within the framework of Cosmological Relativity in five dimensions of space, time and velocity. A general linearized metric element is defined to have the form $ds^2 = (1+\phi) c^2 dt^2 - dr^2 + (1+\psi) \tau^2 dv^2$,wherethecoordinatesaretime$t$, radial distance $r=\sqrt{x^2+y^2+z^2}$ for spatials $x$, $y$ and $z$, and velocity $v$, with $c$ the speed of light in vacuum and $\tau$ the Hubble-Carmeli time constant. The metric is accurate to first order in $t\mathrm{/}\tau$ and $v\mathrm{/}c$. The fields $\varphi$ and $\psi$ are general functions of the coordinates. By showing that $\varphi =\psi$, a metric of the form $ds^2 = c^2 dt^2 - dr^2 + \tau^2 dv^2$ is obtained from the general metric, implying that the universe is flat. For cosmological redshift $z$, the luminosity distance relation $D_L(z,t)=r(1+z)\mathrm{/}\sqrt{1-t^2\mathrm{/}{\tau }^2}$ is used to fit combined distance moduli from Type Ia Supernovae up to $z<1.5$ and Gamma-Ray Bursts up to $z<7$, from which a value of ${\mathrm{\Omega }}_M=0.800\pm 0.080$ is obtained for the matter density parameter at the present epoch. Assuming a baryon density of ${\mathrm{\Omega }}_B=0.038\pm 0.004$, a rest mass energy of $( 9.79 \pm 0.47 ) , {\rm GeV}$ispredictedfortheanti-baryonic$\bar{Y}$andthe$\Phi^{*}$particleswhichdecayfromahypothetical$\bar{X}1$ particle. The cosmic aging function $g_1(z,t)=(1+z)(1-t^2\mathrm{/}{\tau }^2)$ makes good fits to light curve data from two reports of Type 1a supernovae and in fitting to simulated quasar like light curve power spectra separated by redshift $\mathrm{\Delta }z\approx 1$. We determine the multipole of the first acoustic peak of the Cosmic Microwave Background radiation anisotropy to be $l \approx 224 \pm 5$andasoundhorizonof$\theta{sh0} \approx (0.805 \pm 0.020 ) {}^{\circ}$ on today’s sky.