Cosmology from conservation of global energy

TL;DR

This paper proposes that conserving the quasi-local Misner-Sharp energy within the apparent horizon offers a new perspective on cosmology, resolving issues related to energy conservation and explaining spatial flatness without the problems of standard models.

Contribution

It introduces a conservation-based approach using Misner-Sharp energy, leading to a relational, Machian cosmological model that aligns with observations and avoids standard paradigm issues.

Findings

Misner-Sharp energy conservation yields de Sitter as the unique solution.

Total matter energy scales as curvature energy, inducing gravitational time dilation.

Open de Sitter appears empty but contains matter effects explaining spatial flatness.

Abstract

It is argued that many of the problems and ambiguities of standard cosmology derive from a single one: violation of conservation of energy in the standard paradigm. Standard cosmology satisfies conservation of local energy, however disregards the inherent global aspect of energy. We therefore explore conservation of the quasi-local Misner-Sharp energy within the causal horizon, which, as we argue, is necessarily an apparent horizon. Misner-Sharp energy assumes the presence of arbitrary mass-energy. Its conservation, however, yields "empty" de Sitter (open, flat, closed) as single cosmological solution, where Misner-Sharp total energy acts as cosmological constant and where the source of curvature energy is unidentified. It is argued that de Sitter is only apparently empty of matter. That is, total matter energy scales as curvature energy in open de Sitter, which causes evolution of the cosmic potential and induces gravitational time dilation. Curvature of time accounts completely for the extrinsic curvature, i.e., renders open de Sitter spatially flat. This explains the well known, surprising, spatial flatness of Misner-Sharp energy, even if extrinsic curvature is non-zero. The general relativistic derivation from Misner-Sharp energy is confirmed by a Machian equation of recessional and peculiar energy, which explicitly assumes the presence of matter. This relational model enhances interpretation. Time-dilated open de Sitter is spatially flat, dynamically close to $\Lambda$CDM, and is shown to be without the conceptual problems of concordance cosmology.

Tables

Table 1. Table 1: Model parameter estimates

	$H^2/H_0^2$		$\Omega ,q$		$H_0$
ΛCDM	${\Omega }_d{a}^{-3}+{\Omega }_{\Lambda }$		${\Omega }_d=$0.278,${\Omega }_{\Lambda }=1-{\Omega }_d$		70.	0
open dS’	${\Omega }_m{a}^{-2}+{\Omega }_{\Lambda }$		${\Omega }_m={\Omega }_{\Lambda }=\frac{1}{2}$		69.	9
power law	$a^{-2-2q}$		$q=-0.36$ $\to a\left(t^{\prime }\right)=t^{\prime 1.56}$		69.	3
Milne	${\Omega }_k{a}^{-2}$		${\Omega }_k=1$		66.	6
flat dS	${\Omega }_{\Lambda }$		${\Omega }_{\Lambda }=1$		74.	5

Table 2. Table 2: Model performance

		${\chi }^2/n$		$n_p$		$\Delta$BIC		wi(BIC)			$\Delta$AIC		wi(AIC)
ΛCDM		0.	9694	2		3.	22	0.	16		0		0.	62
open dS’		0.	9746	1		0		0.	82		1.	14	0.	35
power law		0.	9798	2		9.	46	7.	3E-3		6.	24	0.	027
Milne		1.	0404	1		37.	9	4.	9E-9		39.	0	2.	1E-9
flat dS		1.	6481	1		304.	7	5.	7E-67		305.	9	2.	4E-67

Equations

$${\dot{a}}^2+k+K{a}^2=0,$$

$$R_g^2=\frac{1}{H^2+\kappa a^{-2}}=\frac{1}{H^2-\frac{8}{3}\pi G{\rho }_k},$$

$$E_{MS}(R)=\frac{R}{2G}(1-{\nabla }^{a}R{\nabla }_{a}R).$$

$$E_{MS}(R)=\frac{4}{3}\pi R^3(\rho -{\rho }_k),$$

$$E_g=\frac{R_g}{2G}>0.$$

$$R_g=const,$$

$$E_g=\frac{4}{3}\pi R_g^3(\rho -{\rho }_k),$$

$$E_g=\frac{4}{3}\pi R_g^3{\rho }_E,$$

$$\rho ={\rho }_E+{\rho }_k.$$

$${\rho }_E=const>0,$$

$${\rho }_E={\rho }_{\Lambda },$$

$$E_g=\frac{R_g^3}{2G}(H^2-\frac{8}{3}\pi G{\rho }_k)=\frac{4}{3}\pi R_g^3{\rho }_E.$$

$$E=1=R_g^2(H^2-\frac{8}{3}\pi G{\rho }_k)=\frac{8}{3}\pi G{R}_g^2{\rho }_E.$$

$$T=H^2{R}_g^2=\frac{8}{3}\pi G({\rho }_E+{\rho }_k)R_g^2=1-k{a}^{-2}=E-V.$$

$$\rho ={\rho }_E+{\rho }_k=(1-k{a}^{-2}){\rho }_E.$$

$${\rho }_m={\rho }_k.$$

$$-\phi \equiv E-V=\frac{8}{3}\pi G({\rho }_E+{\rho }_k)R_g^2=1-k{a}^{-2}.$$

$${\phi }_N=-2\pi G\rho R_g^2,$$

$$V_{ij}=-G{m}_i{m}_j{R}_{ij}^{-1}$$

$$T_{ij}=\frac{1}{2}\frac{V_{ij}}{{\phi }_p}{\dot{R}}_{ij}^2.$$

$${\phi }_p=\frac{1}{3}{\phi }_N.$$

$${\phi }_r={\phi }_N.$$

$$T_r=\underset{{\mathcal{V}}_g}{\int }\frac{1}{2}\frac{\text{d}{\phi }_r(r,\theta ,\varphi )}{\frac{1}{3}{\phi }_r}{r}^2{\dot{a}}^2=\frac{3}{4}{r}_g^2{\dot{a}}^2=\frac{3}{4}{H}^2{R}_g^2,$$

$${\phi }_r=-2\pi G({\rho }_E+{\rho }_m)R_g^2,$$

$$T_r=\frac{3}{4}{H}^2{R}_g^2=2\pi G({\rho }_E+{\rho }_m)R_g^2=\frac{3}{4}(1-k{a}^{-2})=E_r-V_r.$$

$$T_p=\frac{1}{4}{H}^2{R}_g^2=\frac{2}{3}\pi G({\rho }_E+{\rho }_m)R_g^2=\frac{1}{4}(1-k{a}^{-2})=E_p-V_p.$$

$$T=H^2{R}_g^2=\frac{8}{3}\pi G({\rho }_E+{\rho }_m)R_g^2=1-k{a}^{-2}=E-V.$$

$$\rho ={\rho }_E+{\rho }_k={\rho }_E+{\rho }_m,$$

$$d{t}^2=g_{t^{\prime }{t}^{\prime }}d{t}^{\prime 2},$$

$$\frac{1}{a^2}\frac{d{a}^2}{d{t}^{\prime 2}}\propto {\rho }_E+{\rho }_k=(1-k{a}^{-2}){\rho }_E.$$

Full text

I Introduction

Conservation of energy in the expanding universe is controversial. It is incompatible with basic elements of standard cosmology, in particular the non-uniform scaling of matter densities and the non-stationarity of FLRW spacetimes. In addition, there is no unanimously agreed upon notion of gravitational energy within general relativity, which of course affects the subject of energy conservation too. Fundamentally this is due to the non-local nature of gravitational energy, which has no coordinate independent local representation like the stress-energy tensor. Standard cosmology is derived from geometry (the Friedmann equation) and conservation of local energy (the continuity equation). The global aspect of energy is in fact ignored. This in itself may be totally justified, considering the homogeneity of the Universe on the large scale. The issue, though, is that conservation of global energy likely constrains the evolution of density parameters, while the continuity equation permits different forms and mixtures of matter, potentially giving rise to odd behavior, like, e.g., violation of energy conservation, the coexistence of multiple “causal” horizons and inconsistency of the different forms of redshift. It is quite curious that the continuity equation ensures ongoing conservation of local energy of radiation, while within an infinitesimal time interval the same radiation fails conservation of energy due to redshift, a specifically non-local feature. This indeed suggests that matter in global perspective cannot have arbitrary equation of state, and that the scaling must relate to global properties of spacetime.

The other issue is non-stationarity. Since FLRW spacetimes are non-stationary, it is often argued that energy can not be conserved in the expanding Universe. While this indeed applies to most universe models, it is not generally the case. As Florides showed (Florides, 1980), there are exactly six FLRW spacetimes (the ones with constant curvature) which have a static representation, namely: Minkowski, Milne, de Sitter (with $k=-1,0,1$, for open, flat and closed geometry, respectively), and anti-de Sitter. The common Friedmann form of these spacetimes is given by Florides as

[TABLE]

where $K=-\Lambda/3$ represents the cosmological constant. Of these six spacetimes, the flat de Sitter universe is cosmologically highly relevant in the early (inflating) and late universe, and total vacuum energy within the de Sitter event horizon is indeed conserved. So conservation of global energy is not totally alien to cosmology. The issue of course is that all six spacetimes of Eq.(1) are “empty” of matter.

So there are certainly arguments within standard cosmology to give up on conservation of energy in the Universe. Violation of conservation of energy is however not the single fundamental problem of standard cosmology. As we argue, multiple ambiguities and inconsistencies arise from violation of conservation of global energy. Indeed it turns out that imposing conservation of global energy resolves other well known issues too. Moreover, this dramatically constrains the solution space of FLRW spacetimes, which in turn gives rise to reconsideration of standard notions of energy.

The purpose of this note is to explore the cosmological implications of conservation of global energy within the context of general relativity. This requires a notion of non-local energy within the causal horizon. We approach this in two parallel ways: through the Misner-Sharp quasi-local energy (Misner and Sharp, 1964) and through a Machian expression of total energy (Telkamp, 2016). The former becomes global if applied to the causal sphere of the Universe, while the latter considers the causal sphere by principle. The two lead to consistent energy equations of total energy. However, the meaning of energy terms in the Misner-Sharp equation is rather implicit, while the relational Machian expression explicitly models recessional and peculiar energy of matter within the causal horizon. The Misner Sharp equation, on the other hand, enables full treatment of the subject within general relativity. The Machian notions mainly serve interpretation. In addition to conservation of energy, we make as little as possible assumptions, e.g., just the presence ($\rho_{m}>0$) of otherwise unspecified matter. Initially, we will be concerned exclusively with physical consistency of the model. Then, we will evaluate the results for observational and theoretical relevance.

In terms of local energy, the causal horizon may seem a rather irrelevant feature of the Universe; the horizon does not appear in the standard formulation of the Friedmann equation. But evidently, the distance to this horizon is a key parameter of global energy. Identification of the causal horizon is therefore an essential step. This is not quite a trivial matter, since the notion of causal horizon is ambiguous in standard cosmology. This ambiguity however dissolves in the de Sitter case. Not surprisingly, conservation of total energy indeed implies a de Sitter universe (open, flat, or closed).

The most significant aspect of this result, however, is that we assume the presence of matter, yet obtain “empty” de Sitter solutions. This may seem a departure from general relativity, but it follows immediately from application of the apparent horizon to the Misner-Sharp energy, which holds in any spherically symmetric spacetime.

Another main result is that the curved solutions, open and closed de Sitter, show evolution of the gravitational potential, therefore necessarily induce gravitational time dilation. This prompts adjustment of the (constant) metric coefficient $g_{tt}\rightarrow g_{t^{\prime}t^{\prime}}(a)$. As it turns out, curvature of time accounts completely for extrinsic curvature in open and closed de Sitter. This means that, taking gravitational time dilation into account, all three de Sitter solutions are necessarily spatially flat. Cosmologically, flat de Sitter, $k=0$, can be identified with the spacetime of the freely falling comoving observer, while open de Sitter, as we show, matches accelerated observational coordinates on the past light cone of the observer. In this way “time dilated open de Sitter” can match CMB observations (due to flat space) as well as SNIa observations (due to gravitational time dilation). Open de Sitter has a present deceleration $q_{0}=-\frac{1}{2}$, which makes it quite close in expansion dynamics to ΛCDM in the redshift range of SN1a observations. Open de Sitter actually outperforms ΛCDM in BIC score on the Union2.1 compilation.

Open de Sitter has many favorable properties and leads to natural explanations of both unidentified “dark” matter and vacuum energy. It may thus provide a viable cosmological model, without the conceptual problems of ΛCDM.

II Concordance cosmology

The ΛCDM model, even while observationally favorable, is physically not understood for multiple reasons, i.e., the horizon problem, coincidence problem, flatness problem, the age problem and the unidentified dark matter and vacuum energy. Inflation theory successfully addresses some of these problems, but does not simplify cosmology and relies on the hypothetical inflaton field, which is not physically understood either. One may add that there are several ambiguities and fundamental issues in the theoretical concepts and assumptions underlying standard cosmology. This regards of course in particular violation of the conservation of energy in the standard paradigm, but also a number of less recognized issues, addressed as follows.

In standard cosmology, matter densities scale differently due to different equation of state, i.e., $\rho_{dust}\propto a^{-3}$ and $\rho_{radiation}\propto a^{-4}$. The extra factor $a$ of radiation accounts for redshift due to expansion of space. Any difference in dilution rates, however, not only sacrifices conservation of energy, it also violates the equivalence of the different forms of mass-energy, at least in the strict interpretation due to Eddington, where the different forms are just different appearances of one and the same property, i.e., energy (Eddington, 1920; Fernflores, 2012). One would therefore expect dust and radiation to scale identically to maintain mass-energy equivalence. Then, there is the question of coordinates being physical or not. Cosmological time is commonly assumed to represent physical (clock) time of a comoving observer. Not only at present, but through all era of the universe. The implicit assumption here is constancy of the potential, thus the absence of gravitational time dilation (the lapse function in the FLRW-metric being set to $g_{tt}=1$). But is this assumption justified? Then there are a number of ambiguous concepts involved, like the unclear relationship between sources and curvature, the coexistence of different horizons with “causal properties” (Ellis and Uzan, 2015), and seemingly incompatible forms of redshift (Chodorowski, 2011; Faraoni, 2010; Melia, 2012). At the bottom of this, there is the question of the representation of energy, i.e., local versus non-local (Szabados, 2009). Within general relativity the non-localness of gravitational energy is recognized, however not satisfactorily resolved.

One may not subscribe to all of these concerns, but the list is simply too long to not question certain fundamentals of standard cosmology. On the other hand, there may well be only few underlying causes of the issues jointly. Violation of conservation of energy is likely one, since it is connected with all other issues. Thus there is reason to reconsider notions of energy, in particular the scaling of the energy densities of matter. We will however reverse order by considering conservation of global energy, and make inferences therefrom. This involves identification of the causal horizon.

III Causal horizon

An observer’s causal horizon is the spherical boundary between particles which are connected with the observer, and the particles which are not (Ellis and Uzan, 2015). The question is when exactly a particle is connected? For our purposes, we will narrow this to the question which masses contribute momentarily to the gravitational potential at the origin. Rather than trying to answer this immediately, we will note the ambiguities surrounding the subject and point out three consistent cases.

The cosmological event, particle and apparent horizon all three have certain “causal properties”. The ambiguity arises from the limited causal relevance of these horizons individually, as well as from their coexistence at different distances (for instance, the apparent horizon and the event horizon in accelerating universes). The issue centers around the temporal aspect of connection. One could argue that the particle horizon, the farthest distance we receive particles from today, is a clear demarcation of causality (Ellis and Uzan, 2015). However, in a presently accelerating universe, the massive source of these particles has long disappeared behind the event horizon, so is disconnected forever and does not contribute anymore to the potential. By definition of the apparent horizon (Rindler, 2006), an inward directed photon emitted at present time from a galaxy crossing the observer’s apparent horizon has momentarily zero physical speed, as measured in the frame of the observer; nothing from this galaxy is moving into our direction now. Thus at this point the mass-energy of this galaxy, or from anything beyond, does not contribute to the gravitational potential in the frame of the observer, so is disconnected. This is actually what one expects to happen at the causal boundary now. But in case of the apparent horizon this is only momentarily so; it is not a null surface in general. An inward directed photon, emitted from a galaxy crossing this horizon, will still move towards the observer in terms of the comoving speed, thus will eventually reach the observer nevertheless. In the case of accelerating universes, it is only beyond the farther located event horizon that photons, emitted now, will not reach the observer ever. The event horizon therefore unambiguously demarcates the forever disconnected region. Even so, a photon emitted exactly from the event horizon now will only reach the observer in infinite time. So one may question the immediate relevance of this photon (Ellis and Uzan, 2015), and in fact of all particles located in between apparent and event horizon, as these do not contribute at present to the potential in the observer’s frame. So in fact, all three horizons have causal characteristics, but none is unequivocally a boundary of causal connection on its own.

The interpretative differences between the horizons however disappear in a few particular cases (Rindler, 2006). In de Sitter the apparent horizon coincides with the event horizon (the particle horizon diverges). In this case, disconnected now unambiguously means disconnected forever. Similarly, in a radiation universe the apparent horizon coincides with the particle horizon (the event horizon diverges). In that case, connected now means connected forever. This universe however does not match solution set Eq.(1), which is surprising in a way. It is worthwhile mentioning that there is one other consistent case: the static Minkowski universe, where all three horizons diverge. Minkowski is part of the solution set of Eq.(1).

Note that the apparent horizon, least of all considered causal, is the single horizon being part of all consistent cases. It is always nearest and always exists (Faraoni, 2011). Most importantly, it is the only horizon with an instant character, which is an indispensable feature to meaningfully act as a causal boundary now. It therefore seems reasonable to assume that the causal horizon necessarily is an apparent horizon. The reverse of course need not be the case, since we expect only specific solutions of the apparent horizon to meet conservation of global energy.

IV Global energy equation

IV.1 Misner-Sharp quasi-local energy

Representation of the non-local gravitational energy is an issue in general relativity, which primarily is a local theory. It is well known that the gravitational field vanishes in the inertial frame, along with local spatial curvature. Indeed, gravitational energy is intimately connected with curvature, but a single general definition of gravitational energy is lacking. To overcome this, various notions of quasi-local mass have been introduced for different spacetimes. The Misner-Sharp mass $E_{MS}(R)$ applies to all spherically symmetric spacetimes and expresses the total internal energy contained in a sphere of radius $R$ (Misner and Sharp, 1964). It is the sum of internal kinetic energy and internal potential energy. Note that in the cosmological context both recessional motion and peculiar motion contribute to total internal energy $E_{MS}$. The requirement of spherical symmetry fits FLRW universes, which are homogeneous and isotropic, but still strictly spherical by the presence of a gravitational causal horizon at proper distance $R_{g}$. In this way, the Misner-Sharp energy $E_{MS}(R_{g})$ provides a notion of global energy within the radius of the Universe.

As motivated above, we tentatively consider the causal horizon to be an apparent horizon. We make no other assumptions, just that the Universe contains matter $\rho_{m}>0$ (massive particles, radiation,…), which implies non-zero energy of both recessional and peculiar motion. The areal radius $R_{g}\equiv ar_{g}$ of the apparent horizon is (in units where $c=1$)

[TABLE]

where $H=\sqrt{\frac{8}{3}\pi G\rho}$ is the Hubble parameter, with total density $\rho$, and where $\kappa=k/R_{g0}^{2}$ is the Gaussian curvature, with index $k=-1,0,1$ for spatially open, flat and closed geometries, respectively (Rindler, 2006). Note that curvature energy density $\rho_{k}$ is always removed from the denominator of Eq.(2). Misner-Sharp energy includes all internal energy and is defined (Misner and Sharp, 1964)

[TABLE]

A well known, surprising property of Misner-Sharp energy is (Faraoni, 2011; Binetruy and Helou, 2015)

[TABLE]

i.e., the r.h.s. shows a flat volume, even if $\rho_{k}\neq 0$. In case of the apparent horizon, $\nabla^{a}R_{g}\nabla_{a}R_{g}=0$ (Faraoni, 2011). Misner-Sharp energy within the apparent horizon $E_{g}\equiv E_{MS}(R_{g})$ therefore reduces to (the Schwarzschild mass)

[TABLE]

From this simple result alone it follows that if total energy $E_{g}$ is conserved, then

[TABLE]

which unambiguously points at the de Sitter universe, where the constant Misner-Sharp total energy density acts as cosmological constant. This solution resolves the ambiguity between apparent and event horizon in accelerating universes, as the two coincide. De Sitter ($k=-1,0,1$) is in agreement with the solution set of Eq.(1). Anti-de Sitter ($\Lambda<0$) and Milne ($R_{g}\neq const$) drop out as solutions (since we have $E_{g}>0$ and $R_{g}=const$). Minkowski is a bit more complicated, as it is a limiting case of de Sitter for $R_{g}\rightarrow\infty$, so we defer a conclusion. Technically though, it is de Sitter, therefore can be treated as such.

The intriguing part of this result is that we assumed the presence of matter, $\rho_{m}>0$, nevertheless obtained “empty” de Sitter as the unique solution. This suggests that de Sitter space is actually non-empty. Moreover, from a relational perspective, inertia, energy, space and time all emerge from the interaction of matter (Mach, 1893; Huggett and Hoefer, 2017). So in this view empty spacetime cannot even exist (this was actually subject of a long debate between Einstein and de Sitter, where Einstein held this Machian view for a long time, but finally accepted emptiness of de Sitter space (Ann M. Hentschel (1998), translation)). In section IV.6 we infer that gravitational time dilation makes de Sitter apparently empty. Misner-Sharp energy holds for arbitrary spherically symmetric spacetimes, so the assumption of the presence of matter is justified, even though this assumption is rather implicit in Eq.(3). Non-emptiness of de Sitter also follows, quite explicitly, from the Machian derivation in section IV.4.

From Eqs.(2,5) we obtain,

[TABLE]

which is Eq.(4) in the specific case of $R=R_{g}$ (in section V.1 we find that curvature of time renders curved de Sitter spatially flat, which explains the remarkable flat volume of Misner-Sharp energy). Total energy $E_{g}$ may be represented by the density $\rho_{E}$,

[TABLE]

hence, total density

[TABLE]

Given constancy of both $E_{g}>0$ and $R_{g}$, also

[TABLE]

which therefore acts as constant vacuum energy density, i.e.,

[TABLE]

and so provides a natural interpretation of the cosmological constant. Using Eqs. (2,5,8) one obtains the following expression of Misner-Sharp global energy,

[TABLE]

With Eq.(5) this can be simplified to global energy per unit mass $E\>(\equiv E_{g}/M_{g}=c^{2}=1)$, i.e.,

[TABLE]

This is the Friedmann equation multiplied on both sides by $R_{g}^{2}$, which expresses the Friedmann equation as a global energy equation. Note that the l.h.s. of Eq.(13) adds an energy constraint to the Friedmann equation, which, as we have seen, forces $R_{g}$ and subsequently $\rho_{E}$ to be constant. For simplicity, all energies hereafter are given per unit mass, so can be regarded as potentials.

Recalling that Misner-Sharp energy represents the sum of internal kinetic energy and potential energy, one may recast equation (13) into the classical form $T=E-V$, where $T=R_{g}^{2}H^{2}$ is kinetic energy and $V=-\frac{8}{3}\pi G\rho_{k}R_{g}^{2}=ka^{-2}$ is potential energy due to curvature, i.e.,

[TABLE]

This equation contains only global energies, $T$, $E$ and $V$. Note that total density can be written

[TABLE]

Clearly, total density consists of only two components: total energy $\rho_{E}=const$ and curvature energy $\rho_{k}=\rho_{E}a^{-2}$. Where then has matter $\rho_{m}$ gone? According to Eq.(14), $R_{g}^{2}H^{2}=1-ka^{-2}$, no matter what one assumes $\rho_{m}$ to be. However, turning this around: if the apparent horizon is to meet conservation of total energy $E$, then this highly constrains the equation of state of the density components to either $w=-1$ ($\rho_{E}$), or $w=-\frac{1}{3}$ ($\rho_{k}$). Hence, since total energy is represented by $\rho_{E}$, the conservation of Misner-Sharp energy within the apparent horizon suggests that the *energy *density of total matter behaves as curvature energy, i.e.,

[TABLE]

This result seems at odds with the equation of state of individual matter components, which agrees with the local notion of energy where an isolated particle represents a certain energy. Eq.(16) however regards the energy of total matter on the cosmological scale. In section IV.3 we connect non-local Misner-Sharp energy with a notion of relational energy between particles, where an isolated point particle fundamentally does not represent any energy at all. Instead, the interaction of the particle with its cosmic background determines the energy associated with each particle. Specifically, we find that the energy associated with matter is quite different in an expanding universe than in peculiar motion in a fixed background, where most of our physical notions and intuition come from (see also section V.2 on dark matter).

IV.2 Cosmic potential

Both $E$ and $V$ are considered to contribute to the cosmic potential, i.e.,

[TABLE]

We see that in the curved case the potential evolves, which necessarily induces gravitational time dilation. This feature is not recognized in the standard FLRW metric, but has profound implications (see section V.1). Note that in the flat (free fall) case the potential is constant, $\varphi=-c^{2}$, which agrees with Sciama’s cosmic potential (Sciama, 1953). The cosmic potential reminds of the Newtonian potential at the center of the causal sphere,

[TABLE]

but the latter is off by a factor $\frac{3}{4}$, i.e., $\varphi_{N}=\frac{3}{4}\varphi$. This points at an interesting relationship. Kinetic energy $T=H^{2}R_{g}^{2}$ is, for the appearance of $H$, naturally associated with recessional motion of matter. However, Misner-Sharp energy $E$ also includes energy of peculiar motion, which one may not immediately relate to the Hubble parameter. That is unless the two, peculiar and recessional energy, maintain a fixed ratio. This is what actually derives from the Machian equation (section IV.4), which expresses recessional and peculiar energy separately.

IV.3 Non-locality of energy

In general relativity, notions of local and non-local energy coexist. The non-local character of quasi-local Misner-Sharp energy fits the relational view, where energy is exclusively a mutual property between causally connected particles. Any relational notion of energy is therefore intimately connected with the density and size of the causal sphere, i.e., with the presence of other matter. This view has become known through Einstein as Mach’s principle, but actually dates back to the relationalism of Descartes, Ockham or even Aristotle, and has, apart from Mach and Einstein, been represented in particular by Leibniz, Berkeley, Poincaré, Schrödinger, Sciama, Brans and Dicke (Huggett and Hoefer, 2017; Berkeley, 1992; Brans and Dicke, 1961; Einstein, 1916, 1918; Mach, 1893; Schroedinger, 1925; Sciama, 1953; Barbour and Pfister, 1995). Mach’s principle (usually taken as: inertial properties arise from the presence of cosmic mass), fits into the relational view, where inertia, energy, space and time emerge from the interaction of particles (the terms relational and Machian have a different historical context, but are largely exchangeable). In the relational view, energy is a shared property, therefore not an intrinsic property of a particle. This means that local energy in fact does not exist in the relational universe; energy density is merely global energy per unit volume. This may be understood realizing that the potential energy of a particle of mass $m$ equals $m\varphi$, where $\varphi$ is the cosmic potential. Without the cosmic mass-energy present, the potential energy of the particle would vanish. A similar argument applies to photon energy $h\nu$, where $\nu$ is the photon frequency. A vanishing potential would redshift the photon frequency to zero. Indeed, according to Eq.(5), Misner-Sharp energy within the apparent horizon equals the Schwarzschild mass. Hence the idea that particle energy disappears in absence of other matter is not uncommon. It can however be largely disregarded in a Universe of constant potential, which is nothing but a flat empty Minkowski background to local physics, where only local inhomogeneities of matter are of interest. As Sciama noted, Newton’s laws hold without referencing any cosmic parameters (Sciama, 1953).

Depending on $k$, the cosmic potential $-\varphi=1-ka^{-2}$ may actually evolve, which changes the picture completely. Then the energy associated with a particle appears to be explicitly related to the presence of surrounding mass. In de Sitter, the comoving distance to the apparent horizon evolves , $r_{g}(a)\propto a^{-1}$. The number of particles contributing to the cosmic potential $\varphi$ depends on the comoving distance to the horizon, therefore evolves too. In light of this, it is conceivable that energy density not necessarily dilutes as $a^{-3}$ in order for energy to be conserved. Moreover, all different forms of mass-energy contribute to the cosmic potential, and therefore to the energy associated with each particle, regardless of species. Hence, the whole notion of separate energy densities of distinct fluids is lost in the relational view. Instead, the cosmic matter seems to behave as a single fluid with a uniform dilution rate. This is a departure from standard cosmology, but not from general relativity: uniform scaling is actually required to maintain (strict) mass-energy equivalence, thus is in this sense a requirement within general relativity. To the contrary, different scaling of radiation and dust does violate both the mass-energy equivalence and the conservation of energy. Indeed, as we have seen, uniform scaling follows from conservation of Misner-Sharp energy within the apparent horizon.

Note that uniform scaling of matter density $\rho_{m}$ does not mean that the actual *number *densities of the different species of particles maintain a fixed ratio. Particles may transform freely from one form into another, so one still expects the number density of photons to decay faster than the number density of massive particles, due to the cool down of the Universe. For energy is conserved in such conversions, these do not affect the total energy density of matter. A single, uniformly scaling “matter fluid” does therefore not conflict with the evolutionary picture of the Universe.

IV.4 Machian energy

Berkeley, an early critic of Newton, noted that one can not meaningfully attribute a position or velocity to a single (point) particle in empty space. Consequentially, this applies to kinetic energy too. He continues noting that of two particles in otherwise empty space, only their radial distance is observable (Berkeley, 1992). Motion in any perpendicular direction, like with these two particles in circular orbit of each other, is unobservable in an empty background. Therefore (and this is crucial), motion in non-radial direction, does not represent energy between two point particles. This means that both the kinetic energy $T_{ij}$ and potential energy $V_{ij}$ between point particles $i$ and $j$ depend only on their separation $R_{ij}$ (or time derivative thereof) (Schroedinger, 1925; Barbour and Pfister, 1995). Actually, Newtonian potential energy

[TABLE]

is perfectly Machian; it is indeed a mutual property between two connected particles and depends geometrically only on their separation. Newtonian kinetic energy, on the contrary, is defined relative to a frame of reference, so is clearly not relational. Schrödinger (Schroedinger, 1925; Barbour and Pfister, 1995) reproduced Einstein’s expression of the anomalous perihelion precession from the following definition of Machian kinetic energy,

[TABLE]

The potential $\varphi_{p}$ is a scaling constant to match Newtonian kinetic energy in peculiar motion (Telkamp, 2016). Definition Eq.(20) meets the Machian requirements: kinetic energy $T_{ij}$ is mutual between two particles, is frame independent, depends only on the radial component of motion, and vanishes if $V_{ij}\rightarrow 0$. As a consequence of the exclusively radial relationship, the mass $m_{j}$ only contributes to the kinetic energy $T_{ij}$ if $\dot{R}_{ij}\neq 0$. This implies that only a part ($\varphi_{p}$) of the total Newtonian potential $\varphi_{N}=-2\pi G\rho R_{g}^{2}$ adds to the Newtonian inertia $m_{i}$ of a particle $i$ in peculiar motion. In a homogeneous, isotropic sphere this part is \bigl{\langle}\dot{R}_{ij}^{2}\bigr{\rangle}/\bigl{\langle}v_{ij}^{2}\bigr{\rangle}=\frac{1}{3}, where $v_{ij}$ is the relative speed, so that the effective potential in peculiar motion is (cf. (Telkamp, 2016))

[TABLE]

Note that what we normally regard as inertial mass relates to peculiar motion. Different from peculiar motion, recession is purely radial motion between all particles, therefore the kinetic energy of recession balances with the full potential, i.e., the potential in recessional motion is

[TABLE]

This however means that a particle in recessional motion effectively has three times the inertial mass of the same particle in peculiar motion. This is an intriguing consequence of Berkeley’s conjectures, which actually hints at the origin of unidentified “dark” matter, as discussed in more detail in section V.2.

Due to the frame independent formulation of $T_{ij}$, even a particle at rest has kinetic energy associated with it. In this way, one can express the recessional energies associated with a particle at rest in the Hubble flow. Initially we shall ignore peculiar motion. Adopting the elementary definition Eq.(20), integration over the cosmic comoving volume $\mathcal{V}_{g}=\mathcal{V}(r_{g})$ yields the recessional Machian kinetic energy $T_{r}$, between a unit mass test particle at rest in the Hubble flow and all receding masses within the horizon (Telkamp, 2016),

[TABLE]

where $\textrm{d}\varphi_{r}(r,\theta,\phi)$ represents the potential at the origin due to the receding infinitesimal cosmic volume element at spherical coordinates ($r,\theta,\phi$). According to Eqs.(18,22), the potential in recessional motion is the Newtonian potential

[TABLE]

where we explicitly assume the presence of matter $\rho_{m}$ next to total energy $\rho_{E}$. Eqs.(14,23) show that recessional kinetic energy is $\frac{3}{4}$ of total kinetic energy, thus accordingly $\varphi_{r}=\frac{3}{4}\varphi=-\frac{3}{4}(1-ka^{-2})$. With Eqs.(23,24) this gives the Machian equation of recessional energy per unit mass

[TABLE]

Recalling that the effective potential in peculiar motion is only $\frac{1}{3}$ of the potential in recessional motion, $\varphi_{p}=\frac{1}{3}\varphi_{r}$, we expect the balancing kinetic energies to maintain the same ratio, i.e., the kinetic energy of peculiar motion per unit mass is $T_{p}=\frac{1}{3}T_{r}$, therefore

[TABLE]

The Machian recessional and peculiar energy combined add to

[TABLE]

IV.5 Cross-interpretation

The Machian equation Eq.(27) is the same as Eq.(14) for Misner-Sharp energy, except that $\rho_{m}$ in the Machian case takes the place of $\rho_{k}$ in the Misner-Sharp case. In both equations total energy and kinetic energy have the same meaning and the same value, i.e., $E=T+V=1$, and $T=H^{2}R_{g}^{2}$. So the potential energy $V$ in both equations must be identical too. Stated otherwise, total density in both expressions is equal,

[TABLE]

hence $\rho_{k}=\rho_{m}$, in agreement with the earlier conclusion of Eq.(16). That is, subject to conservation of global energy, matter behaves as curvature energy, which identifies matter as source of curvature energy. For we demand the presence of matter, i.e., $\rho_{m}>0$, this seems to single out open de Sitter ($\rho_{k}>0$) as the unique solution. Since $\rho=\rho_{E}+\rho_{m}>0$, Minkowski is excluded as a possible limiting case of open de Sitter.

The particularity of curvature energy is that it represents extrinsic curvature; it appears in accelerated frames and vanishes in the free fall frame, revealing flat de Sitter. Even so, we do not expect matter energy $\rho_{m}$ (nor $\rho_{k}$) to actually nullify, for it represents energy. Indeed, nullification of $\rho_{m}$ implies $\rho_{E}=\rho_{m0}=0$, which contravenes Eq.(10). As will follow, the vanishing of matter/curvature in free fall is only apparent, as an effect of gravitational time dilation.

IV.6 Gravitational time dilation

Having established $\rho_{k}=\rho_{m}>0$, we assume hereafter that the Universe is negatively curved and can be represented by the open de Sitter spacetime. According to Eq.(17), curvature $k=-1$ implies evolution of the potential, which induces gravitational time dilation, i.e.,

[TABLE]

where $t$ is the unaccelerated cosmic time. The prime denotes the accelerated time coordinate $t^{\prime}$ in open de Sitter, which we shall identify below as observational time coordinate on the past light cone. Unaccelerated cosmic time is associated with flat de Sitter, where $\rho=\rho_{E}$ and $\varphi=1$. Hence, in the freely falling comoving frame, the curved open de Sitter universe is transformed into flat de Sitter. Note that free fall in this context specifically concerns the time dimension; the observer in ($r$,$t^{\prime}$) coordinates is accelerated in the time dimension and is therefore not in free fall, even though spatially at rest in the Hubble flow like the unaccelerated freely falling comoving observer in ($r$,$t$) coordinates. This is analog to (Parikh, 2002), where Parikh shows the transformation of coordinate time in static de Sitter to the Painlevé-de Sitter time coordinate of the free fall observer. Here, the mechanism of transformation from open de Sitter to flat de Sitter is through transformation of total density by gravitational time dilation, as follows.

In terms of the curved time coordinate $t^{\prime}$, the Friedmann equation of open de Sitter ($k=-1$) is

[TABLE]

Referencing definition Eq.(29), in free fall Eq.(30) transforms to flat de Sitter ($k=0$),

[TABLE]

so that indeed curvature/matter energy seems to vanish due to the effect of time dilation,

[TABLE]

Hence, total density in free fall equals $\rho_{E}$, as expected, but still consists underlying of both transformed components, as is evident from Eq.(31). It is this property of spacetime which turns the evolving potential $\varphi=1-ka^{-2}$ in free fall to the constant $\varphi=1$, thereby hides matter/curvature energy. Thus also time dilation explains why de Sitter is only apparently empty. Furthermore, the constant density $\rho=\rho_{E}$ falsely suggests non-evolution. Inspection of Eq.(31) shows that the contribution of matter/curvature to constant density equals

[TABLE]

which scales down to zero as $a\rightarrow\infty$. The contribution of total energy, on the other hand, ramps up asymptotically to $\rho_{E}$ according to

[TABLE]

Flat de Sitter of course reminds of the non-evolutionary steady state theory (Bondi and Gold, 1948; Hoyle, 1948) by Bondi, Gold and Hoyle, which however features continuous creation of matter to maintain constant density, instead of gravitational time dilation in the present cosmology.

The identification of matter with curvature energy is reminiscent of K-matter (Kolb, 1989) and Milne cosmology. It fits the idea of coasting galaxies in the homogeneous, zero-field universe, as pointed out, e.g., by Layzer (Layzer, 1954). Here, however, matter/curvature density is always in the company of constant total energy density, i.e., $\rho_{E}=a^{2}\rho_{m}$, yielding exclusively open/flat de Sitter.

IV.7 Cosmological solutions

From the foregoing we conclude that conservation of global energy in a non-empty universe is consistent with open de Sitter in terms of the accelerated time coordinate $t^{\prime}$,

[TABLE]

where $H_{\varLambda}^{2}\equiv c^{2}/R_{g}^{2}$. This solution transforms in free fall to flat de Sitter, in terms of unaccelerated cosmic time $t$,

[TABLE]

V Observational validity?

Open/flat de Sitter is the unique solution of conservation of global energy. Nevertheless, observation seems to rule out both representations. CMB data indicate that our Universe is spatially flat, which seems to leave flat de Sitter as the single option. Supernova observations, however, clearly disfavor flat de Sitter relative to ΛCDM. On the other hand, the open de Sitter model Eq.(35), is close to ΛCDM in expansion dynamics. It has a deceleration $q_{0}=-\frac{1}{2}$ at present epoch, quite close to ΛCDM with $q_{0}\sim-0,55$ ($\Omega_{m}\sim 0.3$, $\Omega_{\Lambda}\sim 0.7$). But obviously, open de Sitter, as derived from the standard metric, is spatially curved, therefore is not supported by CMB data. The two de Sitter models thus seem to match different sets of observations quite well, but neither fits both. The situation however changes once gravitational time dilation is taken into account.

V.1 Curvature of time vs. curvature of space

Gravitational time dilation is not accommodated for in the standard FLRW metric,

[TABLE]

where $g_{tt}=1=const$. Evolution of the potential thus prompts adjustment of the metric as follows.

The metric to the free fall flat de Sitter model is the flat standard metric,

[TABLE]

As pointed out in section IV.6, open de Sitter and its free fall representation, flat de Sitter, only differ by their time coordinates due to gravitational time dilation, according to Eq.(32). The metric Eq.(38) in terms of the accelerated time coordinate $t^{\prime}$ is

[TABLE]

where time is curved, but space is flat. Thus index $k$ represents the extrinsic curvature of time instead of space. An insightful way to retrieve open de Sitter is to evaluate the metric Eq.(39) on the light cone ($ds=0$) at the horizon ($r=r_{g}\propto a^{-1}$) and setting $k=-1$, while using $r_{g}da+adr_{g}=0$. This returns the Friedmann equation of open de Sitter, though derived from a spatially flat metric, Eq.(39). This means that curvature of the time dimension accounts completely for the extrinsic curvature of open de Sitter. As a result, open de Sitter is necessarily spatially flat, which makes this model observationally viable. Furthermore, the derivation of open de Sitter from the metric on the light cone identifies the accelerated coordinate $t^{\prime}$ as observational time coordinate on the past light cone.

Note that gravitational time dilation gives an explanation to why the Misner-Sharp mass in Eq.(8) coincides with a flat volume, regardless of curvature $k$.

V.2 Relational physics of missing matter

The relative density of baryonic matter is currently estimated at $\widetilde{\Omega}_{b}\sim 0.05$ (Ade et al., 2016). In the ΛCDM model the relative density of dust is $\widetilde{\Omega}_{d}\sim 0.30$, so that the deficit of matter is $\widetilde{\Omega}_{c}\sim 0.25$, which is attributed to dark matter. The relative density of matter in the open de Sitter model of Eq.(35) is $\Omega_{m}=\frac{1}{2}$, meaning that an even larger fraction $\Omega_{c}\sim 0.45$ of matter energy is to be identified. Recall that density $\Omega_{m}$ regards the global potential energy associated with matter, not the local energy density of particles. As pointed out in section IV.4, Berkeley’s ontological notions on radial motion entail a three times larger inertial mass of a particle in (the purely radial) recessional motion as compared with the same particle in peculiar motion. Our unit of inertial mass relates to peculiar motion, therefore inertial mass gets boosted in recessional motion. An additional effect, due to time dilation, is that at present time the densities in the accelerated time coordinate $t^{\prime}$ are twice the densities in the free fall cosmic time coordinate. This doubles the present energy density of matter in the observational coordinates of the open de Sitter model. The ratio of inertial mass in the different cases is reflected in the ratio of the corresponding kinetic energies as follows. According to Eq.(26), in peculiar motion the kinetic energy per unit mass equals $T_{p}={\textstyle\frac{1}{4}}(1-ka^{-2})$. Thus in the free fall frame ($k=0$) of flat de Sitter this is $T_{p}={\textstyle\frac{1}{4}}$. This is the frame in which we define our standard unit of inertial mass in peculiar motion. The kinetic energy in recessional motion equals according to Eq.(25) $T_{r}={\textstyle\frac{3}{4}}(1-ka^{-2})$, therefore $T_{r}={\textstyle\frac{3}{4}}$ in free fall. The combined peculiar and recessional motion in the same frame amounts to $T=T_{p}+T_{r}={\textstyle\frac{1}{4}}+\frac{3}{4}=1$. Finally, transformation from cosmic time to the accelerated observational time coordinate $t^{\prime}$ in open de Sitter doubles the kinetic energy at present time to $T^{\prime}_{0}=2$. Thus the standard unit of mass in peculiar motion in flat de Sitter gets multiplied by a factor $T^{\prime}_{0}/T_{p}=8$ in the observational open de Sitter model. The present cosmology therefore predicts a relative density of baryonic matter (in standard units related to peculiar motion) of exactly $\Omega_{b}=\Omega_{m}/8=1/16=0.0625$, which is quite close to current estimates of $\sim 0.05$. Though, the global energy density associated with this baryonic matter in open de Sitter is $\frac{1}{2}.$ This is relational physics.

V.3 Model comparison using SNIa data

The physics involved in the present cosmology (in particular gravitational time dilation instead of spatial curvature) must be accounted for accordingly in the physical models underlying cosmological probes such as CMB, BAO and SNIa. Hence, comparison with models of standard cosmology is not necessarily on the same premises. Open de Sitter in standard cosmology implies spatial curvature, whereas time-dilated open de Sitter (open dS’) is spatially flat. This difference impacts cosmological calculations, e.g., of the transverse comoving distance, even while the two models have the exact same Friedmann equation. In the case of SNIa, this difference can be accounted for straightforwardly. The physics of CMB and BAO is more involved and requires deeper analysis, beyond present scope. Thus the following model comparison is only indicative, since it is limited to SNIa.

The Union2.1 compilation of $n=580$ SNIa samples from the Supernova Cosmology Project (SCP) (Suzuki et al., 2012) gives observed distance modulus $\widetilde{\mu}(z)$ versus redshift $z$. We use this data set without any processing. Following Suzuki et al. (Suzuki et al., 2012), we minimize

[TABLE]

where $\mu(z)$ is the model prediction of $\widetilde{\mu}$, and $C$ is a diagonal weighing matrix of sample error variances without systematics, as provided by SCP. The prediction is the theoretical value of the distance modulus, given (in $Mpc$) by $\mu(z)=m(z)-M=5log_{10}(D_{L})+25$. The luminosity distance $D_{L}$ relates according to $D_{L}=(z+1)D_{M}$ to the transverse comoving distance $D_{M}$, while $D_{M}$ in standard cosmology relates to the comoving distance $D_{c}$, depending on spatial curvature index $k$, according to

[TABLE]

where

[TABLE]

In the present cosmology, extrinsic curvature only regards the time dimension; space is manifestly flat. That is, $D_{M}=D_{c}$, regardless of $k$. Finally, the comoving distance $D_{c}$ follows from

[TABLE]

The models considered are: flat ΛCDM, open dS’, power law, Milne and flat de Sitter. The results of minimizing $\chi^{2}$ are summarized in Tables 1 and 2, which show model structure, the number of free parameters $n_{p}$, best fit parameter values, and the values of different model performance criteria (the best score per column in boldface). The models are sorted by $\chi^{2}$ score. As usual, ΛCDM performs best on $\chi^{2}$. Open dS’, however, fits the data nearly as well, while it has only one degree of freedom $n_{p}=1$, against $n_{p}=2$ for ΛCDM. This is rewarded by the significantly better BIC scores of open dS’, while ΛCDM remains preferable according to AIC scores. The $\textrm{w}_{i}(\textrm{BIC)}$ and $\textrm{w}_{i}(\textrm{AIC)}$ are estimated model probabilities, which allow for somewhat easier interpretation than $\varDelta$BIC and $\varDelta$AIC scores (Wagenmakers and Farrell, 2004). From BIC and AIC together one may slightly favor open dS’, but given the noise levels of the data, the difference is too small to be conclusive. Moreover, the purpose here is not to decide on best model, but to point at open dS’ as a viable alternative of ΛCDM.

The power law model does rather well, but is purely empirical; it lacks physical motivation. Milne and flat de Sitter are cosmologically relevant (being part of Florides’ model set), but perform relatively poorly. These models however support the conclusion that open dS’ and ΛCDM are quite close.

VI Theoretical evaluation

Theoretical issues of concordance cosmology, or standard cosmology in general, are revisited for the present cosmology as follows.

1. a)

Horizon problem. The smoothness of the CMB is difficult to explain without divergence of the particle horizon, by which all matter is causally connected at initial time. ΛCDM lacks this property due to deceleration $q>0$ at early times. A short flat de Sitter inflation phase at initial time is hypothesized to provide the causal connection of all matter at initial time. The present de Sitter solution has accelerated expansion from initial time onward, i.e., $q\leq 0$, therefore has a diverging particle horizon at initial time, which provides smoothness of the CMB, without inflation. 2. b)

Flatness problem. Flat space is an unstable equilibrium of the ΛCDM model; any small deviation from a perfectly flat ΛCDM universe causes density to run away from critical density. Inflation creates an extremely flat initial state of the ΛCDM universe, as a way to delay notable curvature. The flatness problem does not apply to the present cosmology; the spatial part of the metric Eq.(39) is manifestly flat, even if $k\neq 0$, since all extrinsic curvature is (necessarily) in the time dimension. 3. c)

Coincidence problem. The densities $\rho_{E}$ and $\rho_{m}$ $(=\rho_{E}a^{-2})$ in open de Sitter are obtained analytically and are exactly equal at present time $t^{\prime}_{0}$. Thus coincidence of nearly equal densities at present time, as in ΛCDM, is absent. Flat de Sitter obviously has no coincidence problem either. 4. d)

Unidentified dark matter. According to section V.2, a density of baryonic matter $\Omega_{b}$ in the standard definition accounts for an effective energy density of baryonic matter $\Omega_{m}=8\Omega_{b}$ in open de Sitter. Since $\Omega_{m}=\frac{1}{2}$ in open de Sitter, this predicts a baryonic density of $\Omega_{b}=0.0625$, which is close to present estimates of around $\widetilde{\Omega}_{b}=0.05$ (Ade et al., 2016). This result follows specifically from relational considerations. 5. e)

Unidentified dark energy. The density of Misner-Sharp total internal energy within the de Sitter horizon is identified as a cosmological constant, i.e., $\rho_{E}=\rho_{\Lambda}=const$. The global Misner-Sharp energy per unit mass amounts to $E=1$ and equals the sum of total energy of recessional ($E_{r}=\frac{3}{4}$) and peculiar motion ($E_{p}=\frac{1}{4}$), as derived from the Machian energy equations, Eqs.(25,26). 6. f)

Age problem. Since $a(t^{\prime})=\sinh(H_{\Lambda}t^{\prime})$, the age of the open de Sitter Universe is $t^{\prime}_{0}=H_{\Lambda}^{-1}\textrm{asinh}(1)=\sqrt{2}H_{0}^{-1}\textrm{asinh}(1)\sim 1.25H_{0}^{-1}\sim 17.9$ billion years (at $H_{0}=68$ km/s/Mpc), which is a comfortable high age, given constraints near $H_{0}^{-1}\sim 14.4$ billion years. Estimated ages of the oldest known objects are around the predicted age of 13.8 billion years of the ΛCDM universe (Ade et al., 2016). 7. g)

Violation of the conservation of energy. The present cosmology is derived from conservation of quasi-local Misner-Sharp energy within the causal sphere, thus necessarily satisfies conservation of this global energy. The result agrees with the Machian expression of total energy. 8. h)

Violation of the (strict) mass-energy equivalence principle. In standard cosmology this violation is due to different scaling of the energy densities of different forms of matter. In Machian context, energy is a mutual property between connected particles of mixed sorts, the total effect of which is aggregated into the cosmic potential. Hence all forms of matter contribute to the energy associated with each particle, thus total matter behaves as a single fluid. The present analysis shows that, subject to conservation of Misner-Sharp global energy, the total energy density of (an arbitrary mixture of) matter behaves as curvature energy density. How uniform scaling of potential energy of total matter relates to the different equation of state of particles in local physics remains to be answered. The present cosmology suggests that the different equation of state of different forms of matter rule the evolution of number densities of these forms, but not the potential energy of total matter. 9. i)

Unclear relationship of sources and curvature energy. The present cosmology points at identity of the matter energy and curvature energy, i.e., $\rho_{m}=\rho_{k}$. 10. j)

Coexistence of different “causal” horizons at different distances. In the de Sitter universe the event horizon and apparent horizon coincide, so the ambiguous causal status of particles in between these horizons disappears. 11. k)

Incompatible forms of redshift. There are compelling reasons to assume that cosmological, gravitational and Doppler redshift are all three relevant to cosmology. Even so, a long history of controversy about the subject has not resolved the apparent inconsistency of these different forms of redshift in Friedmann universes (Chodorowski, 2011; Faraoni, 2010; Harrison, 1981). Without going into this debate, one may note that the three only need to be consistent in the unique spacetime of our Universe. Melia (Melia, 2012) showed that the three forms of redshift are consistent in all six constant curvature spacetimes of Eq.(1), which includes de Sitter. 12. l)

Representation of physical time? Conservation of Misner-Sharp energy yields two consistent cosmological pictures of the Universe: the evolutionary open dS’ universe, as observable on the past light cone, and the (apparent) steady state flat de Sitter universe of the unaccelerated free fall observer, each representation with its own time parameter, which is the only difference between the two. Which time is our clock time? In standard cosmology, cosmic time in flat de Sitter represents comoving clock time, which presumes constancy of the potential. This indeed holds in free fall coordinates. The comoving observer would decide that in flat de Sitter nothing changes and that the Universe must be infinitely old. On the other hand, the presence of matter in expanding flat de Sitter points at evolution, which is confirmed by observation. In the present cosmology, evolution goes together with a declining potential $-\varphi=1+a^{-2}$, causing acceleration of clock rate relative to cosmic time, which suggests that a comoving clock indicates accelerated time $t^{\prime}$, rather than cosmic time $t$.

VII Conclusion

Conservation of energy underlies many, if not all, physical laws and relationships. One therefore does not expect a cosmology in violation of energy conservation to be consistent, unambiguous and explicable. We therefore considered the theoretical aspects of a cosmology derived from conservation of Misner-Sharp global energy, leading to a single cosmological solution without the conceptual problems of concordance cosmology, as summarized in the previous section. Observational validity of the model is only indicative, since limited to SNIa, where open dS’ and ΛCDM perform equally well.

Standard cosmology assumes a local notion of energy, admits different equation of state of matter, and considers de Sitter to be empty. In contrast, the relational view of physics assumes an exclusively global notion of energy, considers a single total matter fluid of uniform equation of state, and precludes the existence of empty space. While this seems at variance with general relativity, conservation of Misner-Sharp energy within the apparent horizon of a non-empty universe yields de Sitter as the unique solution, with energy density of total matter $\rho_{m}=\rho_{k}$, while total energy density $\rho_{E}$ acts as cosmological constant. The same results follow from a Machian model. This shows, quite in the original spirit of Einstein (Einstein, 1916), that global energy in general relativity is consistent with relational notions of energy, in particular Berkeley’s ontology of the exclusively radial relationship between particles. The relational definitions allow for a more explicit treatment, leading to additional results, like the quantification of peculiar and recessional energy, and a predicted baryon density of exactly $\Omega_{b}=\frac{1}{8}\Omega_{m}=0.0625$, which accounts for an effective global energy density of total matter of $\Omega_{m}=\frac{1}{2}$ in open dS’. This emphasizes the relational aspect of energy.