The Nonlocal Temperature of Nonlocal Energy

1. Introduction

The scale factor of the standard $\mathrm{\Lambda }\mathrm{CDM}$ model has a hyperbolic solution of the form $\widehat{a}\left(t\right)\propto \sinh {(t/{\widehat{R}}_0)}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$, which can be mapped exactly on the hyperbolic solution of the “empty” de Sitter spacetime. The question is what it means. To answer this, we shall deviate in two ways from standard cosmology. One is introduction of the conformal metric

$$\overline{g}=a{\left(t\right)}^{-2}g,$$

(1)

where g is the FLRW metric. In standard coordinates, the line element of $\overline{g}$ is

$$d{\overline{s}}^2=a^{-2}d{t}^2-{\displaystyle \frac{d{r}^2}{1-k{r}^2/r_0^2}}-r^{2}d{\mathrm{\Omega }}^2,k=-1,0,1.$$

(2)

While the light cone is preserved, the conformal transformation trades spatial expansion for gravitational time dilation. The significance of this is that it establishes maximal symmetry in a static cosmic space, which singles out constant curvature spacetimes, i.e., de Sitter, Anti-de Sitter and Minkowski (incl. Milne). The constant ordinary matter density ${\overline{\rho }}_{\mathrm{m}}$ in $\overline{g}$ induces positive constant curvature,

$${\overline{\rho }}_{\mathrm{m}}={\rho }_{\mathrm{m}0}=R_0^{-2},$$

(3)

which leaves de Sitter (including reparametrizations like the $\mathrm{\Lambda }\mathrm{CDM}$ model) as single solution in $\overline{g}$. The question is what this implies to cosmology in g.

The second deviation from concordance cosmology is substitution of the total $\mathit{local}$ energy density of ordinary matter and dark components by the total $\mathit{nonlocal}$ gravitational field energy of ordinary matter only. Like with photon radiation, the kinetic energy of the isotropic gravitational field can be decomposed into 6 independent components associated with 6 cardinal directions, which exert a hexadic pressure in every point. The nonlocality of gravitational field energy thus offers a multiplier that is not recognized in the restricted notion of local energy [1]. This suggests association of a total nonlocal temperature with the total nonlocal gravitational energy.

Part of standard theory, nonetheless delicate, is the prominent role of Weyl density in FLRW context, i.e., the peculiar kinetic energy density of baryon acoustic oscillation (BAO) associated with the inhomogeneous matter. Differentiation of the null equation $ar=R_0$ at the de Sitter horizon returns $d{a}^2/a^2=d{r}^2/r^2$, which shows equilibrium of (what seems to represent) Ricci and Weyl densities. Of course, Weyl density of BAO can only arise in de Sitter spacetime in the presence of inhomogeneous matter, which is conceivable in the maximally symmetric static space of the conformal frame $\overline{g}$ (where, instead of dark energy, the constant matter density in fact maintains a constant positive curvature, $R_0^{-2}={\overline{\rho }}_{\mathrm{m}}$).

The ultimate simplicity of de Sitter spacetime eliminates arbitrariness at the cosmic scale. It suggests there must be a reason for every cosmological feature, i.e., all cosmological parameters must be related, perhaps up to a single common scale, like $R_0^{-2}={\overline{\rho }}_{\mathrm{m}}={\rho }_{\mathrm{m}0}$. We show this for the Hubble constant, CMB temperature and a few exact dimensionless ratios. This involves gravitational time dilation at the event horizon in $\overline{g}$, which can be obtained from differentiation of the constant radius $r_0=ar$, i.e., $dr=d(r_0/a)=r_{0}d(1+z)=r_{0}dz$, showing a linear distance-redshift relation $z\left(r\right)=r\left(z\right)/r_0$. This produces a redshift $z^{\prime }=1$ at the horizon and a corresponding value of the scale factor $a^{\prime }={\displaystyle \frac{1}{2}}$, so that proper time at the horizon in $\overline{g}$ appears to run twice as fast as the proper time of the comoving observer, i.e.,

$$d{t}^{\prime }/dt=1/a^{\prime }=1+z^{\prime }=2.$$

(4)

Some interesting properties of the conformal metric $\overline{g}$ have been studied by Deruelle and Sasaki [2]. The authors show that mass m in the FLRW frame transforms in the conformal frame into

$$\overline{m}=am.$$

(5)

This means that presumed constant particle mass m in expanding space is in conflict with the notion of a conserved particle mass $\overline{m}$ in the time-translation symmetry of the conformal frame. Turning the argument around: constant $\overline{m}$ in $\overline{g}$ implies $m\propto a^{-1}$ in the expanding frame g (in fact a realization of Mach’s principle, $m\left(R\right)\propto GM/R$, due to Schr$\ddot{\mathrm{o}}$dinger [1,3]), so that m evolves exactly like cosmic temperature $T_{\mathrm{CMB}}\propto a^{-1}$. Then, baryon density would evolve in g like

$${\rho }_{\mathrm{b}}\left(t\right)={\rho }_{\mathrm{b}0}a{\left(t\right)}^{-4},$$

(6)

consistent with radiation density. This suggests that the local energy density of the ordinary matter contents of baryons, photons and neutrinos evolves uniformly as

$${\rho }_{\mathrm{m}}={\rho }_{\mathrm{b}}+{\rho }_{\gamma }+{\rho }_{\nu }={\rho }_{\mathrm{m}0}{a}^{-4}={\overline{\rho }}_{\mathrm{m}}{(1+z)}^4.$$

(7)

Maximal symmetry of the conformal frame $\overline{g}$ thus imposes constraints on what can possibly exist in the expanding space of g. It underlines the thermal nature of baryonic matter, which further on leads to prediction of CMB temperature and baryon/photon density ratio from their combined density.

2. Friedmann Equation from Null Propagation at the Horizon

For ingoing/outgoing null geodesics at the de Sitter event horizon, one has, in terms of $t^{\prime }$, $d{R}^{\prime }/d{t}^{\prime }=R^{\prime }/R_0\pm 1$. This shows for the ensemble of ingoing and outgoing null modes in some radial direction a total density

$${\displaystyle \frac{d{R}^{\prime 2}}{R^{\prime 2}d{t}^{\prime 2}}}={\displaystyle \frac{2}{R_0^2}}\left(1+{\displaystyle \frac{R_0^2}{R^{\prime 2}}}\right)=2{\overline{\rho }}_{\mathrm{m}}\left(1+a^{\prime }{\left(t^{\prime }\right)}^{-2}\right),$$

(8)

with solution $a^{\prime }\left(t^{\prime }\right)=R^{\prime }\left(t^{\prime }\right)/R_0=\sinh (\sqrt{2}{t}^{\prime }/R_0)$. Invariant transformation to the frame of the observer, $a\left(t\right)=\sinh {(2\sqrt{2}t/R_0)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=\sqrt{a^{\prime }\left(t^{\prime }\right)}$, returns

$${\displaystyle \frac{{\dot{R}}^2}{R^2}}={\displaystyle \frac{2}{R_0^2}}\left(1+{\displaystyle \frac{R_0^4}{R^4}}\right)=2{\overline{\rho }}_{\mathrm{m}}\left(1+a{\left(t\right)}^{-4}\right).$$

(9)

Proper time $d{t}^{\prime }=2dt$ at the horizon in $\overline{g}$ runs twice as fast as the time of the observer, which makes curvature density $\propto a^{\prime }{\left(t^{\prime }\right)}^{-2}$ at the horizon appear as radiation density $\propto a{\left(t\right)}^{-4}$ in the frame of the observer, consistent with Equation (7). It gives meaning to the somewhat abstract notion of ’curvature density’. Moreover, it suggests reinterpretation of CMB radiation in the stationary universe in $\overline{g}$, as shown hereafter.

With independent components of radiation arriving from 6 cardinal directions, one obtains a total density, i.e., the Friedmann equation of a stationary hyperbolic de Sitter spacetime [1],

$${\overline{H}}^2\equiv 6{\displaystyle \frac{{\dot{R}}^2}{R^2}}={\displaystyle \frac{12}{R_0^2}}\left(1+{\displaystyle \frac{R_0^4}{R^4}}\right)=6\left({\displaystyle \frac{{\dot{a}}^2}{a^2}}+{\displaystyle \frac{{\dot{r}}^2}{r^2}}\right).$$

(10)

The stationary density ${\overline{H}}_0^2=24{\overline{\rho }}_{\mathrm{m}}=12R_0^{-2}+12R_0^{-2}$ matches the sum of the Ricci scalar density and an equal Weyl density in the equilibrium of a nonempty de Sitter universe. It is not difficult to show that this equilibrium solution is consistent with concordance cosmology.

The total nonlocal energy density $24{\overline{\rho }}_{\mathrm{m}}$ can be validated using estimates of baryon density ${\overline{\rho }}_{\mathrm{b}}\approx {\overline{\rho }}_{\mathrm{m}}={\rho }_{\mathrm{m}0}$, like the accurate BBN estimate ${\left({\mathrm{\Omega }}_{\mathrm{b}}{h}^2\right)}_{\mathrm{BBN}}=0.02166\pm 0.00026$ by Cooke et al. [4]. Total density $24{\overline{\rho }}_{\mathrm{m}}$ implies ${\mathrm{\Omega }}_{\mathrm{m}}={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$24$}\right.}$, therefore predicts a Hubble constant

$${\overline{H}}_0\approx \sqrt{24{\left({\mathrm{\Omega }}_{\mathrm{b}}{h}^2\right)}_{\mathrm{BBN}}}=72.1\pm 0.4\mathrm{km}/\mathrm{s}/\mathrm{Mpc}.$$

(11)

The concordance model represents BAO energy density, derived from CMB anisotropy across the sky. This is the Weyl part, which matches the Ricci part, $6{\dot{r}}^2/r^2=6{\dot{a}}^2/a^2$. Therefore, it only accounts for half the total density. However, transformation of $6{\dot{a}}^2/a^2$ into $\mathrm{\Lambda }$CDM form by substitution of the scale factor, $a={\widehat{a}}^{3/4}$, gives ${\dot{\widehat{a}}}^2/{\widehat{a}}^2={\displaystyle \frac{16}{9}}{\dot{a}}^2/a^2$, which nearly compensates for the omitted half of total density. So one arrives at the concordance model parameters $\widehat{H_0}={\displaystyle \frac{4}{3}}\sqrt{{\displaystyle \frac{1}{2}}{\overline{H}}_0^2}=67.98\pm 0.40\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ and (with little algebra) ${\widehat{\mathrm{\Omega }}}_{\mathrm{m}}=1-{\widehat{\mathrm{\Omega }}}_{\mathrm{\Lambda }}={[\sinh {\left({\displaystyle \frac{4}{3}}\mathrm{asinh}\left(1\right)\right)}^2+1]}^{-1}=0.3179..,$ which closely matches Planck 2018 concordance model estimates.

3. Cosmic Temperature

The Stefan-Boltzmann law for the energy density of a photon gas in thermal equilibrium

$${\rho }_{\gamma }{c}^2={\displaystyle \frac{4\sigma }{c}}{T}_{\gamma }^4$$

(12)

offers a direct and accurate way to obtain the photon energy density ${\rho }_{\gamma }$ of the CMB from its observed temperature $T_{\mathrm{CMB}}$ [5]. The following shows that, in the present nonlocal energy framework, this same law also relates the (thermalized) ordinary matter energy density to $T_{\mathrm{CMB}}$, which subsequentially enables prediction of the baryon/photon density ratio of the Universe. To this end, the Stefan-Boltzmann equation (12) is to be interpreted in the cosmological context of nonlocal energy density, where indeed photons arrive from 6 cardinal directions, instead of the single direction from a hot surface. Evidently, these 6 components add equally to the thermal energy density in every point of FLRW space. However, temperature is measured in a single direction, like the CMB temperature by FIRAS. A second factor is that (like everything else in de Sitter) temperature is associated with the horizon, but is only measured at the location of the observer, which, according to Equation (4), means it is redshifted by a factor $1+z^{\prime }=2$ as it reaches us. Together, these specifically cosmological conditions suggest that the universe must have a total nonlocal energy density ${\overline{\rho }}_{\mathrm{u}}$ on the one side of the Stefan-Boltzmann equation, which corresponds to a total nonlocal temperature ${\overline{T}}_{\mathrm{u}}=(2·6){\overline{T}}_{\mathrm{CMB}}=12{\overline{T}}_{\mathrm{CMB}}$ on the other side. Adding temperatures may feel uncomfortable, but the argument can be reversed: the total nonlocal temperature ${\overline{T}}_{\mathrm{u}}$ is associated with total (thermalized) nonlocal density ${\overline{\rho }}_{\mathrm{u}}$ of the Universe; what we measure as CMB temperature is a single (redshifted) component of the 6 independent radiation components that reach us.

Planck 2018 results give a total (thermalized) local energy density of baryons and photons, of $({\overline{\rho }}_{\mathrm{b}}+{\overline{\rho }}_{\gamma })=4.185\pm 0.056·{10}^{-28}\mathrm{kg}/{\mathrm{m}}^3$ [6]. Total nonlocal energy density of baryons and photons amounts to 12 Ricci + 12 Weyl units of density $({\overline{\rho }}_{\mathrm{b}}+{\overline{\rho }}_{\gamma })$. Yet, the one Ricci unit that is aligned with the timelike worldline of the freely falling observer (i.e., the FIRAS instrument on the COBE satellite) is unobservable, which leaves 11 observable Ricci units and 12 Weyl units (the missing Ricci unit may still appear as Unruh radiation to accelerated observers). So we assume an effective total density ${\overline{\rho }}_{\mathrm{u}}=23({\overline{\rho }}_{\mathrm{b}}+{\overline{\rho }}_{\gamma })$. Then, the Stefan-Boltzmann law

$${\overline{\rho }}_{\mathrm{u}}{c}^2=23({\overline{\rho }}_{\mathrm{b}}+{\overline{\rho }}_{\gamma })c^2={\displaystyle \frac{4\sigma }{c}}{\left({\overline{T}}_{\mathrm{u}}\right)}^4$$

(13)

returns a total nonlocal temperature ${\overline{T}}_{\mathrm{u}}=32.70\pm 0.12\mathrm{K}$, which according to the above reduces to a predicted observed temperature

$${\overline{T}}_{\mathrm{CMB}}={\overline{T}}_{\mathrm{u}}/12=2.725\pm 0.009\mathrm{K},$$

(14)

which is within the tight confidence limits of the observed $T_{\mathrm{CMB}}=2.7255\pm 0.0006\mathrm{K}$ [5].

4. Baryon/Photon Density Ratio

Since the standard form of the Stefan-Boltzmann law in Equation (12) relates CMB temperature directly to photon density, we can extend Equation (13) on the right hand side, i.e.,

$$23({\overline{\rho }}_{\mathrm{b}}+{\overline{\rho }}_{\gamma })c^2={\displaystyle \frac{4\sigma }{c}}{\left(12{\overline{T}}_{\mathrm{CMB}}\right)}^4={12}^4·{\overline{\rho }}_{\gamma }{c}^2,$$

(15)

from which derives a predicted baryon/photon ratio of ${\overline{\rho }}_{\mathrm{b}}/{\overline{\rho }}_{\gamma }={\displaystyle \raisebox{1ex}{${12}^4$}\!\left/ \!\raisebox{-1ex}{$23$}\right.}-1=900.56..$ which is an exact result, within confidence limits of the observed ratio (Planck 2018) ${\rho }_{\mathrm{b}0}/{\rho }_{\gamma 0}=4.180\pm 0.056\mathrm{E}-28/4.640\pm 0.004\mathrm{E}-31=900.8\pm 12$.

5. Conclusions

The nonlocal energy density of the hyperbolic de Sitter model in the maximal symmetry of the conformal frame $\overline{g}=g/a^2$ explains the success of the concordance model, but it does so without dark matter or dark energy. The symmetry and equilibrium of the gravitational field at the de Sitter horizon is reflected in equal Ricci and Weyl densities, and repeated in a nonlocal cosmic temperature, of which we only measure a single, redshifted, component in one direction. The existence of the maximally symmetric conformal frame $\overline{g}$ by itself makes a strong case for a stationary universe, where indeed one expects to find mature galaxies at high redshifts.