Autor: A. Jaster | Datum: 2026/05/25 | DOI: 10.5281/zenodo.20376169
Building on the causal-Machian emergent time theory [Jaster2026a,Jaster2026b], we examine the cosmological implications of the scalar time field Θ(x) ∈ [0,1]. In exactly homogeneous FLRW spacetime Θ=1 identically and standard cosmology is fully reproduced. In the inhomogeneous universe, Θ deviates from unity only at the level imposed by the causal horizon averaging; because a typical cosmic void occupies a volume fraction ∼4 × 10−7 of the causal horizon, the resulting deviation 1−Θrm is exponentially suppressed by the flat C∞ saturation function S∞ and is quantitatively negligible in the present epoch. The phenomenological consequences of Θ < 1 in voids – including possible contributions to the σ8 tension, the modified ISW effect, void clocks, and Θ-shell lensing – are therefore of unmeasurably small amplitude and are discussed as conceptual predictions only. The CMB hemispheric asymmetry has a possible structural source in the directional properties of UHμ, independently of Θ < 1. The model does not resolve the Hubble tension; in the heuristic background-level treatment any correction would have the wrong sign, but its amplitude is negligible. The cosmological constant remains necessary. Two conceptual solution classes are identified: timeless solutions and Θ-shells. Spacetimes with closed timelike curves are structurally excluded from the regular sector. Planck-mass black hole remnants are possible dark matter candidates if Planck-scale smearing halts evaporation.
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\begin{document}
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\begin{center}
{\LARGE\bfseries Cosmological Implications of the\\[4pt]
Causal-Machian Time Field}\\[1.2em]
{\large A.~Jaster}\\[0.4em]
{\normalsize \today}
\end{center}
\vspace{0.8em}
\begin{abstract}
Building on the causal-Machian emergent time theory
\citep{Jaster2026a,Jaster2026b}, we examine the cosmological
implications of the scalar time field $\Theta(x)\in[0,1]$.
In exactly homogeneous FLRW spacetime $\Theta=1$ identically and
standard cosmology is fully reproduced. In the inhomogeneous
universe, $\Theta$ deviates from unity only at the level imposed by
the causal horizon averaging; because a typical cosmic void occupies
a volume fraction $\sim4\times10^{-7}$ of the causal horizon, the
resulting deviation $1-\Theta_{\rm void}$ is exponentially suppressed
by the flat $C^\infty$ saturation function $S_\infty$ and is
quantitatively negligible in the present epoch. The phenomenological
consequences of $\Theta<1$ in voids -- including possible contributions
to the $\sigma_8$ tension, the modified ISW effect, void clocks, and
$\Theta$-shell lensing -- are therefore of unmeasurably small amplitude
and are discussed as conceptual predictions only. The CMB hemispheric
asymmetry has a possible structural source in the directional
properties of $U_H^\mu$, independently of $\Theta<1$. The model does not resolve the Hubble
tension; in the heuristic background-level treatment any correction
would have the wrong sign, but its amplitude is negligible.
The cosmological constant remains necessary. Two conceptual solution
classes are identified: timeless solutions and $\Theta$-shells. Spacetimes
with closed timelike curves are structurally excluded from the regular
sector. Planck-mass black hole remnants are possible dark matter
candidates if Planck-scale smearing halts evaporation.
\end{abstract}
\vspace{1em}
% =============================================================================
\section{Introduction and Model}
\label{sec:intro}
% =============================================================================
General Relativity~(GR) describes spacetime and gravitation with
extraordinary precision. Open questions remain: the cosmological
constant requires extreme fine-tuning, the Hubble tension and the
$\sigma_8$ tension are statistically significant discrepancies, and
the nature of dark energy is unknown.
The causal-Machian emergent time theory \citep{Jaster2026a} proposes
a conservative extension of GR: physical proper time is defined by
matter and energy in the causal past of each event. The scalar field
$\Theta(x)=S_\infty(\Omega_{H,\lP}/\Omega_{\rm ref})\in[0,1]$ measures
the strength of this coupling via the disformal clock metric:
\begin{equation}
\hat{g}_{\mu\nu} = q_{\mu\nu}
+ \frac{1-\Theta}{c^2}\,U_{H\mu}\,U_{H\nu},
\label{eq:ghat}
\end{equation}
where $q_{\mu\nu}$ is the FLRW-normalised reference geometry and
$U_H^\mu$ the preferred horizon rest-frame vector. In exactly
homogeneous FLRW $\Theta=1$ identically and the theory reduces to GR.
The scale-separation extension of \citet{Jaster2026c} is cosmologically
identical here since $\langle\Theta_{\rm loc}\rangle=1$ at all
cosmological scales.
\paragraph{Magnitude of $1-\Theta$ in voids.}
The field $\Theta=S_\infty(r)$ is determined by the horizon-averaged
ratio $r=\Omega_H/\Omega_{\rm ref}$. As a conservative heuristic estimate on a constant-$T_q$ slice
(the full four-dimensional causal average requires integrating
$W_{H,\lP}$ over the retarded causal history and is not computed here),
a void of radius $\sim30\,h^{-1}$\,Mpc occupies a fraction
$f_{\rm void}^{\rm 3D}\approx4\times10^{-7}$ of the causal horizon
volume ($\sim(c/H_0)^3$). With $\delta\rho/\rho\approx-0.3$, the
resulting deviation is
$\delta r\approx f_{\rm void}^{\rm 3D}\,\delta\rho/\rho\approx-1.3\times10^{-7}$.
The four-dimensional weight $f_{\rm void}^{\rm 4D}$ may differ
depending on the kernel weighting and causal history; the estimate
strongly suggests exponential suppression, but a definitive value
requires the full four-dimensional causal average.
Since $S_\infty$ is a $C^\infty$ saturating function with \emph{all
derivatives vanishing} at $r=1$ -- in particular $S_\infty'(1)=0$ --
the deviation of $\Theta$ from unity is exponentially suppressed:
\begin{equation}
1-\Theta_{\rm void}
\;\sim\; \exp\!\left(-\frac{1}{1-r_{\rm void}}\right)
\;\sim\; \exp(-7.9\times10^6)
\;\approx\; 0.
\label{eq:Thetavoid}
\end{equation}
A linear estimate $1-\Theta\approx S_\infty'(1)\,\delta r$ would give
zero identically, since $S_\infty'(1)=0$. The actual value is
non-zero but transcendentally small. This result has fundamental
consequences: all void-based phenomenological predictions of $\Theta<1$
are quantitatively negligible in the present epoch.
% =============================================================================
\section{Modified Friedmann Equations}
\label{sec:friedmann}
% =============================================================================
From the $(00)$ component of the field equation
$\hat{G}_{\mu\nu}+\Lambda\hat{g}_{\mu\nu}=(8\pi G/c^4)
(T_{\mu\nu}+T^C_{\mu\nu})$
with lapse $N=\sqrt\Theta$ one obtains at the background (homogeneous)
level:
\begin{equation}
\left(\frac{\dot{a}}{a}\right)^2
= \Theta(t)\!\left[
\frac{8\pi G}{3}\rho + \frac{\Lambda c^2}{3}
\right],
\qquad
\frac{\ddot{a}}{a}
= \Theta\!\left[-\frac{4\pi G}{3}
\!\left(\rho+\frac{3p}{c^2}\right)
+\frac{\Lambda c^2}{3}\right]
+ \frac{\dot\Theta H}{2\Theta}.
\label{eq:Friedmann}
\end{equation}
\begin{remark}[Scope of equations \eqref{eq:Friedmann}]
Equations \eqref{eq:Friedmann} are valid at the background level in
homogeneous FLRW, where $\Theta=1$ and $\dot\Theta=0$ identically.
The use of a spatially varying $\Theta(t,\vec x)$ in an
inhomogeneous context goes beyond what can be derived from a single
Friedmann equation; a consistent treatment of inhomogeneities requires
a perturbation theory of the $\Theta$ field, which is not yet available.
Furthermore, the constraint tensor $T^C_{\mu\nu}$, which has not yet
been constructed \citep[Open Problems~1 and~5]{Jaster2026a}, would
enter the right-hand side. All inhomogeneous applications below
should be understood as qualitative structural statements, not as
derived quantitative predictions.
\end{remark}
In homogeneous FLRW, $\Theta=1$ and $\dot\Theta=0$; equations
\eqref{eq:Friedmann} reduce exactly to the standard Friedmann
equations.~\checkmark In the present epoch, $1-\Theta_{\rm void}$
is exponentially suppressed (equation \eqref{eq:Thetavoid}), so no
deviation from $\Lambda$CDM is expected in the background dynamics.
% =============================================================================
\section{Solutions of the Field Equations and Compact Objects}
\label{sec:solutions}
% =============================================================================
\subsection{Known Solutions of the Einstein Field Equations}
In all regions where the local causal matter connection satisfies
$r\geq1$ -- that is, regions that are at least as dense as the
reference -- $\Theta=1$ and $\hat{g}_{\mu\nu}=q_{\mu\nu}$. All GR
solutions in such regions, and in the limit $T^C_{\mu\nu}\to0$, are
exact solutions of the modified theory. More precisely: all solutions
of the Einstein field equations remain solutions wherever
$\Theta=1$ holds and the constraint sector closes consistently.
\paragraph{Vacuum exteriors.}
For the exterior Schwarzschild or Kerr solution, $T_{\mu\nu}=0$
locally; the argument for $\Theta=1$ cannot use local density.
Instead, the causal past $J_q^-(x)$ of an event in the exterior
vacuum contains the entire history of the central mass $M$, which
contributes to $\Omega_H(x)$ through the retarded integral. The
concentrated mass in the causal past gives $\Omega_H>0$ even in the
vacuum exterior; whether this yields $r\geq1$ and hence $\Theta=1$
requires an explicit computation that is identified as an open
problem. It is plausible that $\Theta=1$ in the exterior vacuum,
but this is not derived here.
\paragraph{Pure vacuum solutions.}
For Minkowski, de~Sitter, and anti-de~Sitter spacetimes,
$T_{\mu\nu}=0$ everywhere at all times, giving $\Omega_H=\Omega_{\rm ref}=0$
and $r=0/0$. These spacetimes do not belong to the regular sector
$\mathcal{M}_{\rm reg}$ in a strict sense. In the cosmological
context, where a pure vacuum phase is approached as a limit
($\rho_m\to0$ with both integrands vanishing uniformly), $r\to1$ and
$\Theta\to1$, formally recovering GR. This limiting procedure is
not rigorously established and is identified as an open problem.
\subsection{New Solution Classes}
\label{sec:newclasses}
\paragraph{Timeless solutions.}
Two distinct cases must be separated. (i)~\emph{Matter-free event
within a universe that contains matter elsewhere}: the causal past of
event $x$ is devoid of matter, but the reference density
$\Omega_{\rm ref}>0$ because the FLRW reference is populated. Then
$\Omega_H(x)=0$, $r=0$, and $\Theta(x)=S_\infty(0)=0$. This is the
well-defined timeless case: $\hat{g}_{\mu\nu}$ degenerates and no
physical proper time exists at $x$. (ii)~\emph{Completely empty
universe with no matter at any time}: then also $\Omega_{\rm ref}=0$
and $r=0/0$, which is the pure vacuum case discussed above.
Case~(i) is the intended timeless solution; it realises the Machian
principle precisely: without matter in the causal past, no time. The
transition $\Theta\to0$ is smooth via $S_\infty$. In the observable
universe, case~(i) is never realised since cosmological matter is
always causally present. As a conceptual limiting case it is
important.
\paragraph{$\Theta$-shells.}
At boundaries between cosmic voids and filaments or clusters, a
spatial gradient $\nabla\Theta\neq0$ arises in principle. These
\emph{$\Theta$-shells} are a conceptual solution class -- transition
regions of modified spacetime that exist in principle but whose
amplitude is exponentially suppressed (equation~\eqref{eq:Thetavoid})
and quantitatively negligible in the present epoch. Physical
consequences are discussed in Section~\ref{sec:thetashells} as
structural predictions only.
\subsection{Singularities and Planck Density}
The model does not alter the singularity structure of GR in regions
where $\Theta=1$. Near Planck density a complete treatment requires
\citet{Jaster2026b}; whether singularities are regularised is open.
\subsection{Closed Timelike Curves: Kerr Interior and G\"odel Universe}
\label{sec:ctc}
\paragraph{Kerr metric.}
Outside the outer horizon ($r>r_+$) one plausibly has $\Theta=1$
(see vacuum exterior discussion above); GR is recovered.
Inside the inner Cauchy horizon ($r0$ the universe enters de~Sitter expansion. Since
$\Omega_H$ and $\Omega_{\rm ref}$ decrease together, $r\to1$ and
$\Theta\to1$ at all future times. The de~Sitter future is identical
to $\Lambda$CDM.
% =============================================================================
\section{Outlook}
\label{sec:outlook}
% =============================================================================
\paragraph{Planck era.}
Near $t\sim t_P$, quantum gravitational fluctuations $\delta\rho/\rho\sim1$
may cause $\Theta$ to deviate substantially from unity. A complete
treatment requires \citet{Jaster2026b} and the constraint action.
\paragraph{Cosmological constant and normalisation.}
A fundamental change to the normalisation structure -- for example a
time-fixed reference density, a coupling to vacuum energy, or a
Hubble-volume causal domain rather than the full particle horizon --
could yield $\dot\Theta\neq0$ in the background and potentially
render $\Lambda$ superfluous. Any such modification would also
change the magnitude of void-based effects. This shows that the
underlying principle (time couples to matter) admits different
technical realisations with qualitatively different cosmological
predictions; further exploration is a direction for future research.
% =============================================================================
\section{Open Problems}
\label{sec:open}
% =============================================================================
\begin{enumerate}[label=(\arabic*)]
\item \textbf{Vacuum exterior of Schwarzschild/Kerr.}
Whether the retarded causal integral over the concentrated
mass history gives $r\geq1$ and hence $\Theta=1$ in the vacuum
exterior requires explicit computation.
\item \textbf{Pure vacuum solutions.}
The limit $\Omega_H=\Omega_{\rm ref}\to0$ for Minkowski,
de~Sitter, and AdS needs rigorous treatment.
\item \textbf{CTC exclusion as dynamical statement.}
Whether the structural exclusion of CTC spacetimes can be
promoted to a consequence of the field equations.
\item \textbf{Planck-remnant stabilisation and dark matter.}
Complete derivation of the UV cutoff at $M\sim m_P$; quantitative
density estimate for primordial black hole remnants as dark matter.
\item \textbf{Perturbation theory of $\Theta$.}
A consistent first-order perturbation theory for $\delta\Theta$
in perturbed FLRW, including metric perturbations, gauge choice,
$\delta U_H^\mu$, and $T^C_{\mu\nu}$. This is the prerequisite
for all quantitative void-based predictions.
\item \textbf{Realistic $\Theta(x)$ profiles.}
Computing $\Theta$ from actual density distributions to establish
whether $1-\Theta_{\rm void}$ can exceed the exponential suppression
through non-perturbative or modified-normalisation effects.
\item \textbf{Hubble tension.}
The model does not resolve the Hubble tension. In the current
heuristic background-level treatment, any non-negligible $\Theta<1$
correction would have the wrong sign; its amplitude is negligible
in the present formulation. No resolution exists in the current
framework.
\item \textbf{$U_H^\mu$ from realistic matter distribution.}
Quantitative computation of the preferred frame $U_H^\mu$ and
its directional structure for the CMB hemispheric asymmetry.
\item \textbf{CMB transfer function.}
A perturbation-theory derivation of corrections to $C_\ell$
at large angular scales.
\item \textbf{Constraint tensor $T^C_{\mu\nu}$}
(inherited from \citealt{Jaster2026a}).
\item \textbf{Planck era and early causal structure}
(inherited from \citealt{Jaster2026a,Jaster2026b}).
\end{enumerate}
% =============================================================================
\section{Summary}
\label{sec:summary}
% =============================================================================
\begin{table}[h]
\centering
\caption{Cosmological predictions of the model.}
\label{tab:summary}
\renewcommand{\arraystretch}{1.45}
\begin{tabular}{>{\bfseries}p{5.1cm} p{3.3cm} p{4.3cm}}
\toprule
Phenomenon & Model prediction & Agreement \\
\midrule
Homogeneous FLRW & GR exactly & Yes \\
GR solutions ($\Theta=1$, $T^C\to0$) & Identical & Yes (conditional) \\
Vacuum exteriors (Schwarzschild) & $\Theta=1$ plausible & Open (computation needed) \\
Pure vacuum solutions & Outside $\mathcal{M}_{\rm reg}$ & Open \\
Singularities & As GR & Open (Planck era) \\
Kerr (outside $r_+$) & $\Theta=1$; GR & Yes (conditional) \\
Kerr CTC region / G\"odel & Outside $\mathcal{M}_{\rm reg}$ & Structural exclusion \\
Black hole mergers / GW & Identical to GR & Yes \\
Neutron stars / pulsars & $\Theta=1$; GR & Yes \\
Type Ia supernovae & Identical to GR & Yes \\
Nucleosynthesis / CMB & Identical to GR & Yes \\
Inflation / horizon problem & Not addressed & --- \\
Timelike geodesics & As GR ($\Theta\approx1$) & Yes \\
Future of the universe & As $\Lambda$CDM & --- \\
Conceptual solution classes & Timeless, $\Theta$-shells & Conceptual \\
Dark matter (Planck remnants) & Possible if stable & Speculative \\
$1-\Theta_{\rm void}$ amplitude & $\sim e^{-7.9\times10^6}\approx0$ & --- \\
$\sigma_8$ tension & Correct direction; negligible amplitude & Not quantitatively reproduced \\
CMB hemispheric asymmetry & $U_H^\mu$ least suppressed & Possible structural source \\
CMB large angle ($\ell<5$) & IR damping; structural & Structural direction only \\
ISW / lensing / void clocks & Correct direction; negligible & Not quantitatively reproduced \\
Hubble tension & Wrong sign; amplitude negligible & Not resolved \\
Cosmological constant & $\Lambda$ required & --- \\
\bottomrule
\end{tabular}
\end{table}
The causal-Machian time field reproduces GR exactly in the homogeneous
limit. All known GR solutions remain valid in regions where $\Theta=1$
and the constraint sector closes; the vacuum exterior of compact
objects and pure vacuum solutions require separate treatment. Two
new solution classes arise conceptually: timeless solutions and
$\Theta$-shells. Spacetimes with closed timelike curves are
structurally excluded from the regular sector, providing a chronology-protection selection rule at the
level of the model definition.
The central heuristic estimate of this analysis is
$1-\Theta_{\rm void}\sim e^{-7.9\times10^6}\approx0$: the horizon
averaging and the $C^\infty$ saturation together suppress void-based
deviations from GR to unmeasurable levels. A linear estimate using
$S_\infty'(1)=0$ would give zero identically. As a consequence,
all void-based phenomenological predictions -- $\sigma_8$, ISW,
CMB lensing, $\Theta$-shell lensing, void clocks -- have qualitatively
correct directions but quantitatively negligible amplitudes in the
current model. The CMB hemispheric asymmetry is the least suppressed structural
candidate effect, as it depends on the directional structure of
$U_H^\mu$ independently of $\Theta<1$. The model does not resolve
the Hubble tension; in the heuristic background-level treatment any
non-negligible correction would have the wrong sign, but the amplitude
is negligible in the present formulation. The cosmological constant
remains necessary.
% =============================================================================
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\end{document}