Observing evolution from steady state

TL;DR

This paper reinterprets cosmological observations within a stationary universe framework, deriving matter density and Hubble constant predictions that align with current empirical measurements.

Contribution

It introduces a novel stationary universe model based on conformal transformations, predicting cosmological parameters consistent with observations.

Findings

Predicts matter density Ω_m ≈ 1/24

Derives Hubble constant h ≈ 0.72

Aligns with Planck 2018 results

Abstract

The time-translation symmetry of the conformal FLRW frame $\bar{g}=a^{-2}g$ allows reinterpretation of cosmological observation in the static space of a stationary universe, where constant matter density $\bar{\rho}_{\textrm{m}}=\rho_{\textrm{m0}}$ induces constant curvature $R_{0}^{-2}$. A hyperbolic de Sitter solution arises from equipartition of the kinetic energy of recessional and peculiar components of the gravitational field, corresponding to a total density $24R_{0}^{-2}$ of twice the scalar curvature. This predicts a matter density $\Omega_{\textrm{m}}=\frac{1}{24}$, or a Hubble constant $h=\sqrt{24\rho_{\textrm{m0}}}\approx0.72$, in agreement with distance-ladder estimates. Projecting the equilibrium state onto the $\Lambda\textrm{CDM}$ model returns $\hat{h}=\frac{4}{3}\sqrt{12\rho_{\textrm{m0}}}\approx0.68$ and exact densities $\hat{\Omega}_{\textrm{m}}=1-\hat{\Omega}_{\Lambda}=[\textrm{sinh}(\frac{4}{3}\textrm{asinh}(1))^{2}+1]^{-1}=0.3179...$, within confidence limits of Planck 2018 results.

Equations

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

$$d{\bar{s}}^2=d{\eta }^2-\frac{d{r}^2}{1-k{r}^2/r_0^2}-r^{2}d{\Omega }^2.$$

$$R_0^{-2}={\bar{\rho }}_{\text{m}}={\rho }_{\text{m}0}.$$

$$\bar{m}=am,$$

$${\rho }_{\text{b}}(a)={\rho }_{\text{b0}}{a}^{-4}={\rho }_{\text{b0}}(1+z{)}^4={\bar{\rho }}_{\text{b}}(z),$$

$${\rho }_{\text{m}}={\rho }_{\text{b}}+{\rho }_{\gamma }+{\rho }_{\nu }={\rho }_{\text{m0}}{a}^{-4}.$$

$$d{\tilde{s}}^2=a^{-2}(1+k{r}^2/r_0^2{)}^{2}d{t}^2-d{r}^2-r^{2}d{\Omega }^2.$$

$$\frac{{\dot{a}}^2}{a^2}=\frac{1}{R_0^2}(1+k{r}^2/r_0^2{)}^2=\frac{1}{R_0^2}(1+k{a}^{-2}{)}^2.$$

$$\frac{{\dot{r}}^2}{r^2}=\frac{1}{R_0^2}(1+k{r}^2/r_0^2{)}^2=\frac{1}{R_0^2}(1+k{a}^{-2}{)}^2=\frac{{\dot{a}}^2}{a^2}.$$

$$\frac{{\dot{a}}^2}{a^2}=\frac{2}{R_0^2}(1+k^2{a}^{-4})=\frac{{\dot{r}}^2}{r^2}.$$

$$H^2\equiv \frac{{\dot{a}}^2}{a^2}=\frac{6}{R_0^2}(1+a^{-4})=\frac{{\dot{r}}^2}{r^2}.$$

$${\tilde{H}}^2\equiv \frac{{\dot{R}}^2}{R^2}=\frac{{\dot{a}}^2}{a^2}+\frac{{\dot{r}}^2}{r^2}=\frac{12}{R_0^2}(1+a^{-4})=2H^2.$$

$${\Omega }_{\text{m}}=\frac{1}{24}=\mathrm{0.04166...}.$$

$$h=\sqrt{24{\Omega }_{\text{b}}{h}^2}=0.7332\pm 0.0016,$$

$$a(t)=A^{\text{nicefrac}14}\text{sinh}(2\sqrt{6}t/R_0{)}^{\text{nicefrac}12},$$

$$t_0=\frac{1}{2\sqrt{6}}{R}_0\text{asinh}(1)$$

$$\hat{a}(\hat{t})={\hat{A}}^{\text{nicefrac}14\beta }\text{sinh}(2\sqrt{6}\hat{t}/R_0{)}^{\text{nicefrac}12\beta }.$$

$${\hat{\Omega }}_{\text{m}}=1-{\hat{\Omega }}_{\Lambda }=\hat{A}(1+\hat{A}{)}^{-1}=\mathrm{0.3179...}.$$

$$\hat{H}(t)^{2}\equiv\frac{\dot{\hat{a}}^{2}}{\hat{a}^{2}}=\frac{1}{\beta^{2}}\frac{12}{R_{0}^{2}}\bigl{(}\hat{\Omega}_{\Lambda}+\hat{\Omega}_{\textrm{m}}\hat{a}(t)^{-3}\bigr{)}.$$

$${\hat{H}}_0^2=\frac{1}{{\beta }^2}\cdot 12R_0^{-2}=\frac{16}{9}\cdot 12R_0^{-2}=21\frac{1}{3}{\rho }_{\text{m}0}.$$

Full text

Invariance of de Sitter universe with respect to expansion of the scale factor is physically attributed to unknown vacuum energy. However, in an essay of 1906 on the relativity of space, Poincarï¿œ imagined invariance of an expanding universe as arising from uniform spatial expansion; that is, expansion without exception of gravitationally bound objects, so that everything expands, including photon wavelength and ruler, but nothing seems to change [1]. The uniformly expanding universe thus shows the time-translation symmetry of de Sitter universe, while constant matter density can be related to constant curvature $R_{0}^{-2}$. The expansion of physical units of length makes space in the frame of the comoving observer static and intrinsically stable. In this frame the universe must be infinitely old and in thermodynamic equilibrium, meeting the perfect cosmological principle. Repeated observation of cosmological parameters, like density or temperature, will show no change, hence zero redshift drift, as evidence of a stationary state in a static space. However, in terms of the present unit of length, the universe continues to expand from past into future, as revealed by cosmological redshift on the past light cone. The same universe then appears evolving and of finite age, like in Big Bang cosmology. This presents the duality of Poincarï¿œ’s uniformly expanding universe.

Uniform expansion can be represented by the conformal metric $\bar{\boldsymbol{g}}=a^{-2}\boldsymbol{g}$, where $\boldsymbol{g}$ is the FLRW metric. In standard coordinates, the line element of $\bar{\boldsymbol{g}}$ is

[TABLE]

Properties of $\bar{\boldsymbol{g}}$ have been studied before, e.g., by Deruelle and Sasaki [2], implications of which we discuss. Since conformal frames have the light cone in common, cosmological observation in the FLRW frame can be reinterpreted in terms of gravitational time dilation in the static space of the conformal frame $\bar{\boldsymbol{g}}$, which is our main subject.

One expects a cosmology of gravitational time dilation in the thermostatic equilibrium of the conformal frame to deviate considerably from Big Bang cosmology in the FLRW frame. Still, the two frames remain each other’s conformal dual, so the two cosmologies must be consistent. The conformal frame has the distinct property of time-translation symmetry, i.e., is maximally symmetric, and clearly this must have distinct implications to cosmology in the FLRW frame. This becomes transparent when expressing the line element in conformal time $\textrm{d}\eta=a^{-1}\textrm{d}t$, so that the metric assumes a static form,

[TABLE]

Florides showed that the existence of a static representation constrains solution space in the FLRW frame to spacetimes of constant curvature [3]. In particular, if the constant density is positive, this regards de Sitter spacetime. Constant energy density of the total particle matter fluid in the conformal frame, $\bar{\rho}_{\textrm{m}}=\rho_{\textrm{m}0}$, can then be considered to act as cosmological constant, i.e., (in units where $\frac{8}{3}\pi G=c=1$),

[TABLE]

From this assumption we derive properties and observational viability of a nonempty de Sitter cosmology in the conformal frame. The Friedmann equation is obtained as null solution of the gravitational field at the de Sitter horizon, that is, from the FLRW null metric in isotropic coordinates. Conformal invariance of the null solution means that the Friedmann equation is identical in both frames, yet is to be interpreted differently. For example, in the expanding frame an energy density evolves with time, $\rho(t)=\rho_{0}a(t)^{-n}$, and is homogeneous, i.e., constant in space, while in the conformal frame the same energy density evolves spatially, i.e., $\bar{\rho}(r)=\rho_{0}(1+z(r))^{n}$, yet is constant in time. The two meet on the light cone (’then=there’). The present state in the expanding frame therefore coincides with the local stationary state in the conformal frame, hence is constant, $\rho(t_{0})=\rho_{0}=\bar{\rho}(0)$. Evolution in the uniformly expanding universe thus seems to regard past and future only. This means that in the dual universe density parameters (e.g., of the $\Lambda\textrm{CDM}$ model) in fact are fixed and must somehow relate to steady state parameters, as shown. For simplicity we will refer to ’recessional’ and ’peculiar’ energy densities in both frames, even while recessional energy density is transformed into curvature of time in the conformal frame.

*Properties of $\mathit{\mathit{\bar{\boldsymbol{g}}}}$.—*In the static frame of Eq.(2), particle rest mass $\bar{m}$ is constant, i.e., $d\bar{m}/d\eta=0$. Since $ad\bar{m}/dt=d\bar{m}/d\eta=0$, mass $\bar{m}$ must be constant in the conformal frame too. Indeed, the conformal factor $a^{-1}$ restores the time-translation symmetry that is lost in the expanding universe. Deruelle and Sasaki [2] show that mass $m$ in the FLRW frame transforms in the conformal frame into

[TABLE]

and point out that CMB temperature $\bar{T}=aT=\textrm{const}$. This suggests that presumed constant particle mass $m$ in the expanding frame (as implied by $\rho_{\textrm{b}}a^{3}=\textrm{const}$) would conflict with constant particle mass $\bar{m}=am$ in the time-translation symmetry of the conformal frame. Turning the argument around: $\bar{m}=\textrm{const}$ in the conformal frame implies $m\propto a^{-1}$ in the expanding frame, like cosmic temperature $T\propto a^{-1}$. Then, on the light cone of the uniformly expanding universe, baryon density evolves as

[TABLE]

that is, as pressureless matter density in terms of evolving rest mass $m\propto a^{-1}$. On the other hand, Eq.(5) seems to recognize the inherent relativistic aspect of baryons, both at the subatomic level and in the interaction with the cosmic gravitational field. For our purposes, the main thing is consistency of the two interpretations, so that in either view the total matter content of baryons, photons and neutrinos appears as a single radiation density

[TABLE]

A uniform equation of state decouples evolution of the scale factor from particle fluid composition (as one expects in de Sitter universe). Hereafter, we will not assume $\rho_{\textrm{m}}\propto a^{-4}$, instead independently retrieve this relation further on, as part of the Friedmann equation.

Curvature of time in flat space.—Spatial flatness of de Sitter universe follows directly from the well known differential equation of the event horizon radius, $\dot{R}_{\textrm{e}}=HR_{\textrm{e}}-1$ [4], where constant $R_{\textrm{e}}=R_{0}$ implies $H=R_{0}^{-1}$, i.e., flat de Sitter expansion at deceleration $q=-1$. Intrinsic spatial flatness of de Sitter universe is appealing since it avoids the unstable equilibrium of flat space in standard cosmology. Also, it may explain the remarkable flat volume of Misner-Sharp mass, regardless of $k$ [5]. These properties suggest that, physically, curvature energy density in de Sitter universe can only be associated with curvature of time. Since $q\equiv-\ddot{a}a/\dot{a}^{2}$ is dimensionless, curvature energy density indeed is free to curve time in spatially flat de Sitter universe. The FLRW metric in isotropic coordinates allows relocation of all curvature to the time dimension by the conformal factor $a^{-1}(1+kr^{2}/r_{0}^{2})$, so that space is intrinsically flat, static and stable, and the line element of the conformal metric $\tilde{\boldsymbol{g}}$ reads

[TABLE]

In the following we relate $k=-1,+1$ to, respectively, the in- and outgoing peculiar component of the gravitational field at the horizon. Then the evolving curvature density represents the peculiar kinetic energy density of the field, while the constant ’vacuum’ density $\rho_{\textrm{m0}}=R_{0}^{-2}$ represents the recessional kinetic energy density of the field. One expects the evolving curvature density to cause evolving gravitational time dilation. This, in turn, affects both vacuum and curvature density, and so on, until some point of equilibrium. To account for this interaction of peculiar and recessional kinetic energy we shall rely on the metric.

Metric analysis.—The Friedmann equation of de Sitter universe can be derived from evaluation of the in- and outgoing null solutions at the event horizon, where the inward directed stationary null front satisfies $ar=R_{0}$. Substitution and differentiation of $r=R_{0}/a$ in the radial null metric ($d\Omega=0$) returns the Friedmann equation of the recessional kinetic energy density, i.e.,

[TABLE]

This is mirrored by substitution of $a=R_{0}/r$ in the metric, returning an identical peculiar kinetic energy density

[TABLE]

This equipartition of kinetic energy shows the duality of permanent equilibrium, $\dot{a}^{2}/a^{2}=\dot{r}^{2}/r^{2}$, across evolution into past and future, relative to a constant present state. As suggested, we associate $k=-1,1$ with, respectively, the ingoing and outgoing gravitational field at the horizon. Since these fields coexist, the corresponding kinetic energy densities in Eq.(9) coexist too. Hence, the energy density of the ensemble is given by the sum over $k=\pm 1$, where the alternating cross term vanishes, i.e.,

[TABLE]

This is where we see curvature energy density appearing in the form of gravitational radiation density, i.e., as a uniform total matter density, in agreement with $\rho_{\textrm{m}}\propto a^{-4}$ in Eq.(6). Accounting for equipartition of kinetic energy in 3-dimensional space, the recessional and peculiar energy densities amount to

[TABLE]

where $H\equiv\dot{a}/a$ is the Hubble parameter, expressing expansion rate. Since $(dR/R)^{2}=(da/a+dr/r)^{2}$, the recessional and peculiar kinetic energy densities combine into a total kinetic energy density (recessional and peculiar motion can be seen to occupy independent degrees of freedom [6]). Like before, the cross term vanishes in the ensemble, hence the Friedmann equation of total density in a nonempty de Sitter universe is

[TABLE]

The constant $\tilde{H}_{0}^{2}=24R_{0}^{-2}=24\rho_{\textrm{m0}}=2H_{0}^{2}$ equals twice the scalar curvature of de Sitter spacetime. This predicts a density of local energy of the matter

[TABLE]

Since $\Omega_{\textrm{m}}\approx\Omega_{\textrm{b}}$, this can be validated using the baryon density estimate $\Omega_{\textrm{b}}h^{2}=0.0224\pm 0.0001$ (Planck 2018 [7]). Without introducing dark components, this predicts a Hubble constant,

[TABLE]

within the range of some of the most accurate distance ladder estimates of the Hubble constant, e.g., $0.732\pm 0.013$ by Riess et al. [8], or $0.733\pm 0.018$ by Wong et al. [9]. The slightly lower, but nearly as accurate BBN estimate of baryon density $\Omega_{\textrm{b}}h^{2}=0.02166\pm 0.00026$ by Cooke et al. [10] predicts $h=0.7210\pm 0.0043$, still within range of the above distance ladder estimates.

The Friedmann equation in Eq.(12) shows some interesting properties. The constant radius of curvature $R_{0}$ implies constant present densities, while the total matter density is seen to evolve into past and future, i.e., past divergence of $\rho_{\textrm{m}}\propto a^{-4}$ still refers to a hot past. At the same time, curvature of time prevents a past singularity to actually occur in the local frame of the observer. The solution thus unifies favorable aspects of Big Bang and steady state scenarios [11, 12].

A curious implication of constant densities in the conformal frame is that even the ’age’ parameter $t_{0}$ is a constant. The Friedmann equation (11) has solution

[TABLE]

where the density ratio $A\equiv\Omega_{\textrm{m}}/\Omega_{\Lambda}=1$. Solving for $a(t_{0})=1$ gives a constant ’age’

[TABLE]

as ultimate consequence of time-translation symmetry in the conformal frame. The constant age parameter obviously does not represent elapsed clock time. This seemingly problematic notion of age is still sensible, provided that clock rate slows down, so that age becomes a constant $t_{0}$ in terms of the continuously stretching units of present time (as Eq.(16) shows, in fact representing the constant curvature of de Sitter universe). Deceleration of clock rate is seen indeed in the conformal metrics in Eqs.(1) and (7), i.e., locally ($r=0$) proper time of the observer dilates as conformal time $d\eta=a^{-1}dt$, therefore diverges into the past of an infinitely old stationary universe, while $t_{0}$ is fixed. So in the past, at arbitrary redshifts, the age of the universe was $t_{0}$, like today. This would make the recently reported existence of very massive galaxies at high redshifts conceivable [13].

Transformation to $\varLambda$CDM.—Temperature anisotropy of CMB radiation is due to baryon density fluctuations, which in stationary state are characterized by harmonic acoustic oscillations. The angular power spectrum represents the peculiar kinetic energy densities of the harmonic components, the sum of which is taken as a measure of cosmic energy density $\rho_{0}$, therefore of the recessional kinetic energy density represented by the Hubble constant $H_{0}^{2}=\dot{a}(t_{0})^{2}/a(t_{0})^{2}$. This same relationship is seen in the vacuum/radiation model of Eq.(11). In fact, this model can be transformed into the basic $\Lambda\textrm{CDM}$ model of vacuum and pressureless matter densities, so that the calculated $\Lambda\textrm{CDM}$ parameters can be compared with concordance model estimates. This is rather straightforward since the two models are related by transformation of the scale factor $\hat{a}=a^{\nicefrac{{1}}{{\beta}}}$ along with scaling of the time coordinate $t\rightarrow\hat{t}=t/\beta$. Choosing $\beta=\nicefrac{{3}}{{4}}$ converts radiation density $\propto a^{-4}$ into the form of pressureless matter density $\propto\hat{a}^{-3}$. The transformation brings the solution of the scale factor in Eq.(15) into the form

[TABLE]

For $\hat{a}(\hat{t}_{0})=1$ and $\hat{t}_{0}=t_{0}/\beta=\frac{4}{3}t_{0}$, the density ratio is \hat{A}\equiv\hat{\Omega}_{\textrm{m}}/\hat{\Omega}_{\Lambda}=\textrm{sinh}\bigl{(}\frac{4}{3}\textrm{asinh}(1)\bigr{)}^{-2}=0.4659...\;. Thus,

[TABLE]

This matches the Planck 2018 estimate $\Omega_{\textrm{m}}^{\textrm{Planck}}=0.315\pm 0.007$ [7]. The Friedmann equation in terms of $\hat{a}(\hat{t})$ follows from differentiation of the solution in Eq.(17) and some algebra. Finally, relabeling $\hat{t}\rightarrow t$, the equation of the basic $\Lambda\textrm{CDM}$ model in the frame of the observer is

[TABLE]

By our initial assumption, $R_{0}^{-2}=\rho_{\textrm{m}0}$, the Hubble constant of the $\Lambda\textrm{CDM}$ model can be related to the total matter density, i.e.,

[TABLE]

With $\Omega_{\textrm{b}}\approx\Omega_{\textrm{m}}=\frac{1}{24}$, the BBN estimate of baryon density $\Omega_{\textrm{b}}h^{2}=0.02166\pm 0.00026$ by Cooke et al. [10] gives an estimate of the Hubble constant $\hat{h}=\sqrt{\frac{8}{9}\cdot 24\Omega_{\textrm{b}}h^{2}}=0.6798\pm 0.0041$ for the $\Lambda\textrm{CDM}$ model. This is in agreement with the Planck 2018 estimate $\hat{h}^{\textrm{Planck}}=0.674\pm 0.005$. Yet, the higher Planck 2018 estimate of the baryon density $\Omega_{\textrm{b}}h^{2}=0.0224\pm 0.0001$ [7]) predicts $\hat{h}=0.691\pm 0.002$, at some distance from $\hat{h}^{\textrm{Planck}}$.

The $\Lambda\textrm{CDM}$ expression $\hat{H}_{0}^{2}=\frac{8}{9}\cdot 24R_{0}^{-2}$ shows a substantially lower value of the Hubble constant than the total density expression $\tilde{H}_{0}^{2}=24R_{0}^{-2}$ of the vacuum/radiation model in Eq.(12), i.e., $\tilde{H}_{0}/\hat{H}_{0}=\sqrt{9/8}\approx 1.061$. This may seem to explain most of the distance between CMB and distance ladder estimates of the Hubble constant. It is not evident, however, that distance ladder estimates exactly express total density, as represented by $\tilde{H}_{0}$, even though one expects both recessional and peculiar kinetic energy density to contribute to cosmological redshift. Nonetheless, the present analysis indicates the possibility of alternative interpretations of the Hubble constant and confirms the strong model dependence of parameter estimates, as noted in [7].