\documentclass[11pt,a4paper]{article} % ------------------------------------------------------------ % Encoding, Fonts, Language % ------------------------------------------------------------ \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[english]{babel} % ------------------------------------------------------------ % Math & Physics Packages % ------------------------------------------------------------ \usepackage{amsmath,amssymb,amsfonts,mathtools} \usepackage{bm} \usepackage{physics} \usepackage{tensor} \usepackage{slashed} \usepackage{siunitx} % ------------------------------------------------------------ % Layout & Formatting % ------------------------------------------------------------ \usepackage{geometry} \geometry{margin=1in} \usepackage{setspace} \onehalfspacing \usepackage{microtype} % ------------------------------------------------------------ % Figures & Graphics % ------------------------------------------------------------ \usepackage{graphicx} \usepackage{float} \usepackage{caption} \usepackage{subcaption} % ------------------------------------------------------------ % Referencing & Hyperlinks % ------------------------------------------------------------ \usepackage[numbers,sort&compress]{natbib} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=blue, citecolor=blue, urlcolor=blue, pdfproducer={Aequitas Field Theory}, } % ------------------------------------------------------------ % Custom Commands % ------------------------------------------------------------ \newcommand{\Mpl}{M_{\mathrm{Pl}}} \newcommand{\Lag}{\mathcal{L}} \newcommand{\dd}{\mathrm{d}} \newcommand{\del}{\partial} % prevent hyperref warnings for math in section titles \pdfstringdefDisableCommands{ \def\chi{chi} \def\Lambda{Lambda} \def\rho{rho} } % ------------------------------------------------------------ % Title Information % ------------------------------------------------------------ \title{\bfseries Aequitas Field Theory:\\[0.2cm] A Covariant Dust-Time Framework for Vacuum-Energy Sequestering } \author{Alexandra Fugunt} \date{\today} \begin{document} \maketitle \begin{abstract} We present \emph{Aequitas Field Theory} (AFT), a fully covariant framework in which Brown--Kuchař dust fields provide a physical relational foliation of spacetime and enable an implementation of vacuum-energy sequestering without introducing new propagating degrees of freedom, at least at the level of the classical theory considered here. AFT supplements general relativity with a local algebraic constraint, enforced by a non-dynamical multiplier, relating the comoving matter and scalar energy densities, \[ \rho^{\mathrm{(com)}}_m + \rho^{\mathrm{(com)}}_\chi = C(T), \] where $T$ is dust time and $C(T)$ is determined by a semi-global constraint on each dust slice. The structure preserves the full Einstein kinetic term and leaves the local gravitational dynamics unchanged, while aiming to arrange that constant vacuum-energy contributions do not affect the cosmological expansion history at the level of the background equations. We derive the covariant field equations, the ADM Hamiltonian formulation, the FLRW background dynamics, and the linear perturbation theory. The constraint structure forms a second-class pair ensuring that the multiplier does not introduce new local degrees of freedom. Under suitable assumptions, AFT can reproduce $\Lambda$CDM-like cosmology while providing a candidate mechanism for vacuum-energy cancellation. We further provide explicit consistency checks (degrees of freedom, propagation, radiative stability at a qualitative level), an RG-motivated interpretation of vacuum-energy sequestering, an illustrative cosmological example, and a comparative discussion with 4-form sequestering, unimodular gravity, mimetic gravity, local sequestering attempts, and modern no-go results. The construction is exploratory and further work is required to establish its full viability. \end{abstract} \tableofcontents % ============================================================ \section{Introduction} \label{sec:intro} % ============================================================ The cosmological constant problem is one of the most severe naturalness problems in modern theoretical physics. Quantum field theory (QFT) suggests vacuum-energy densities of order $\Lambda_{\mathrm{UV}}^4$ for a cutoff $\Lambda_{\mathrm{UV}}$, whereas cosmological observations indicate a dark-energy density of order $(10^{-3}\,\mathrm{eV})^4$. The hierarchy \[ \frac{\rho_{\Lambda}^{\mathrm{obs}}}{\Lambda_{\mathrm{UV}}^4} \ll 1 \] requires an enormous fine-tuning if vacuum energy gravitates as in standard general relativity (GR). Conventional modifications, such as unimodular gravity, reshape the way the cosmological constant appears in Einstein's equations, but do not render it radiatively stable. Mimetic gravity introduces an extra dust-like scalar degree of freedom, potentially useful for dark matter phenomenology, but does not sequester vacuum energy and often suffers from instabilities. A more promising line of approach starts from \emph{sequestering}: modifying the global or semi-global structure of the theory such that constant contributions to the vacuum energy have reduced or no effect on curvature \cite{KaloperPadillaPRL2014,KaloperPadillaPRD2014,KaloperPadilla2015, Kaloper2016,Kaloper2018,Kaloper2020}. 4-form sequestering achieves this by introducing global variables and non-dynamical 4-forms whose field equations enforce global constraints on the spacetime integral of the matter Lagrangian. These constructions are radiatively stable in a well-defined sense but rely on global/topological data and have a somewhat opaque canonical (ADM) structure. In this work we develop \emph{Aequitas Field Theory} (AFT), a dust-time based realisation of vacuum-energy sequestering that is intended as a complementary, more relational approach. The proposal is exploratory and should be viewed as a candidate mechanism whose detailed viability still needs to be tested. More concretely, AFT: % \begin{itemize} \item is fully covariant and uses Brown--Kuchař dust as relational coordinates; \item implements a form of vacuum-energy sequestering through a local algebraic constraint plus a semi-global dust-time constraint; \item preserves the local dynamics and kinetic structure of GR at the classical level; \item introduces no new propagating degrees of freedom in the canonical analysis presented here; \item admits an EFT and RG interpretation that suggests a form of radiative stability, subject to further assumptions. \end{itemize} % The key ingredient is the use of Brown--Kuchař dust as physical reference fields to define dust-time slices $\Sigma_T$ and comoving coordinates $X^I$, and a metric-independent Jacobian density $J$ that allows us to construct semi-global constraints in a diffeomorphism-invariant manner. \medskip \noindent\textbf{Scope and limitations.} The present work is purely classical and focuses on the canonical and cosmological structure of AFT. Questions of full quantum consistency, possible anomalies, and detailed confrontation with observations are left for future work. In particular, statements about radiative stability and the evasion of no-go theorems should be understood as tentative and subject to more detailed analysis. \medskip \noindent\textbf{Structure of the paper.} Section~\ref{sec:dust} reviews Brown--Kuchař dust and its role as a relational reference system. Section~\ref{sec:action} introduces the AFT action and explains the structure of the local and semi-global constraints. Section~\ref{sec:variation} derives the covariant field equations. Section~\ref{sec:ADM} presents the Hamiltonian formulation and the full Dirac constraint analysis, including degree-of-freedom counting. In Section~\ref{sec:semiglobal} we give a precise interpretation of the semi-global constraint and its covariance properties. Section~\ref{sec:FLRW} discusses homogeneous FLRW cosmology and the evolution of the dust-time function $C(t)$. Section~\ref{sec:perturbations} summarises the linear perturbation theory and argues that local propagation is closely related to GR. In Section~\ref{sec:RG-radiative-stability} we provide an RG-level discussion of radiative stability. Section~\ref{sec:example} presents a worked mini-superspace example. Section~\ref{sec:comparison} compares AFT to 4-form sequestering, unimodular gravity, mimetic gravity and local sequestering attempts, and Section~\ref{sec:weinberg} revisits Weinberg's no-go theorem in light of AFT. We conclude in Section~\ref{sec:outlook} with a summary and an outline of open issues. % ============================================================ \section{Brown--Kuchař Dust and Relational Coordinates} \label{sec:dust} % ============================================================ We briefly recall Brown--Kuchař dust as a physical reference system \cite{BrownKuchar1995}. The dust fields are scalars $T(x)$ and $X^I(x)$ ($I=1,2,3$), together with a dust energy density $\rho_{\mathrm{d}}(x)$ and four-velocity $U_\mu(x)$ with unit norm $g^{\mu\nu}U_\mu U_\nu=-1$. The dust action can be written schematically as \begin{equation} S_{\mathrm{dust}} = -\frac{1}{2}\int \dd^4 x\,\sqrt{-g}\,\rho_{\mathrm{d}} \big( g^{\mu\nu}U_\mu U_\nu + 1 \big), \end{equation} with \begin{equation} U_\mu = -\del_\mu T + W_I \del_\mu X^I, \end{equation} and $W_I$ auxiliary fields. Variation enforces that dust worldlines are timelike geodesics and that $(T,X^I)$ label these worldlines. The associated Jacobian density \begin{equation} J = \frac{1}{3!}\epsilon^{\mu\nu\rho\sigma}\, \partial_\mu T\, \partial_\nu X^1\, \partial_\rho X^2\, \partial_\sigma X^3 \end{equation} is metric-independent. Thus $\dd^4x\,J = \dd T\,\dd^3X$ defines an invariant four-volume form in dust coordinates. The dust-time slices are the level sets $\Sigma_T=\{x: T(x)=\mathrm{const}\}$. The four-velocity of dust observers is \begin{equation} u_\mu = -\frac{\partial_\mu T}{\sqrt{-g^{\alpha\beta}\partial_\alpha T \partial_\beta T}}, \qquad u^\mu u_\mu = -1. \end{equation} For any matter sector with stress-energy tensor $T^{(m)}_{\mu\nu}$ we define the comoving energy density seen by dust observers as \begin{equation} \rho^{\mathrm{(com)}}_m = T^{(m)}_{\mu\nu} u^\mu u^\nu. \end{equation} An analogous definition applies to a canonical scalar field. Dust thus provides a completely covariant, physical relational frame. % ============================================================ \section{AFT Action and Constraint Structure} \label{sec:action} % ============================================================ The full AFT action is \begin{equation} S_{\mathrm{AFT}} = S_{\mathrm{EH}}[g] + S_{\mathrm{dust}}[g,T,X^I,\rho_{\mathrm{d}},U_\mu] + S_{\mathrm{matt}}[g,\Psi] + S_{\chi}[g,\chi] + S_{\mathrm{loc}}[g,T,X^I,\Lambda,C] + S_{\mathrm{glob}}[T,X^I,C,\sigma], \end{equation} where: \begin{itemize} \item $S_{\mathrm{EH}}$ is the Einstein--Hilbert action, \item $S_{\mathrm{dust}}$ is Brown--Kuchař dust, \item $S_{\mathrm{matt}}$ is generic matter (radiation, etc.), \item $S_\chi$ is a canonical scalar with potential $V(\chi)$, \item $S_{\mathrm{loc}}$ encodes the local AFT constraint, \item $S_{\mathrm{glob}}$ encodes the semi-global dust-time constraint. \end{itemize} \subsection{Local algebraic constraint} We define the constrained comoving density \begin{equation} \rho^{\mathrm{(com)}}_{\mathrm{constr}} = \rho^{\mathrm{(com)}}_m + \rho^{\mathrm{(com)}}_\chi, \end{equation} excluding radiation. The local AFT term is \begin{equation} S_{\mathrm{loc}} = \int \dd^4x\, J\,\Lambda(x)\, \Big( \rho^{\mathrm{(com)}}_{\mathrm{constr}}(x) - C(T(x)) \Big), \label{eq:Sloc-def} \end{equation} with Lagrange multiplier $\Lambda(x)$ and dust-time function $C(T)$. Variation with respect to $\Lambda$ yields the local constraint \begin{equation} \rho^{\mathrm{(com)}}_{\mathrm{constr}}(x) = C(T(x)), \label{eq:AFT-local-eq} \end{equation} which is purely algebraic and holds pointwise. \subsection{Semi-global constraint} The semi-global part of the action is \begin{equation} S_{\mathrm{glob}} = \int\dd T\,\sigma(T) \left[ \mathcal{I}(T) - C(T)\mathcal{V}(T) \right], \end{equation} with \begin{align} \mathcal{I}(T) &= \int_{\Sigma_T} J\, \rho^{\mathrm{(com)}}_{\mathrm{constr}}, \\ \mathcal{V}(T) &= \int_{\Sigma_T} J. \end{align} Variation with respect to $\sigma(T)$ gives \begin{equation} \mathcal{I}(T) - C(T)\mathcal{V}(T)=0, \end{equation} i.e. \begin{equation} C(T) = \frac{\int_{\Sigma_T} J\, \rho^{\mathrm{(com)}}_{\mathrm{constr}}} {\int_{\Sigma_T}J}. \label{eq:C-average} \end{equation} Thus $C(T)$ equals the dust-slice average of $\rho^{\mathrm{(com)}}_{\mathrm{constr}}$. % ============================================================ \section{Covariant Field Equations} \label{sec:variation} % ============================================================ We briefly summarise the main variations. \subsection{Variation with respect to \texorpdfstring{$\Lambda(x)$}{Lambda(x)}} Variation of \eqref{eq:Sloc-def} with respect to $\Lambda$ gives directly \eqref{eq:AFT-local-eq}. This is the local AFT constraint. \subsection{Variation with respect to \texorpdfstring{$C(T)$}{C(T)} and \texorpdfstring{$\sigma(T)$}{sigma(T)}} Variation of $S_{\mathrm{loc}}+S_{\mathrm{glob}}$ with respect to $C(T)$ yields \begin{equation} \int_{\Sigma_T} J\,\Lambda = \sigma(T)\,\mathcal{V}(T), \end{equation} while variation with respect to $\sigma(T)$ gives \eqref{eq:C-average}. Together, these equations fix $C(T)$ and the slice-average of $\Lambda(x)$ in terms of the matter configuration on each dust slice. \subsection{Metric variation} Metric variation yields \begin{equation} G_{\mu\nu} = \frac{1}{\Mpl^2} \left( T^{(m)}_{\mu\nu} + T^{(\chi)}_{\mu\nu} + T^{(r)}_{\mu\nu} + T^{\mathrm{(dust)}}_{\mu\nu} + T^{\mathrm{(AFT)}}_{\mu\nu} \right), \end{equation} with $T^{\mathrm{(AFT)}}_{\mu\nu}$ arising from the metric dependence of $\rho^{\mathrm{(com)}}_{\mathrm{constr}}$ in $S_{\mathrm{loc}}$ and $S_{\mathrm{glob}}$. On-shell, the local constraint \eqref{eq:AFT-local-eq} enforces $\rho^{\mathrm{(com)}}_{\mathrm{constr}}=C(T)$, and the AFT contribution reduces to terms that vanish or can be absorbed into the dust-frame choice. The Einstein kinetic term and local gravitational dynamics are therefore unchanged relative to GR at the level of local propagation: the only effect is on the algebraic relation between the comoving densities and the dust-time function $C(T)$. \subsection{Scalar variation} The scalar action is \begin{equation} S_\chi = \int\dd^4x\sqrt{-g} \left( -\frac{1}{2}g^{\mu\nu}\partial_\mu\chi\partial_\nu\chi - V(\chi) \right). \end{equation} Variation yields the standard Klein--Gordon equation \begin{equation} \Box\chi - V_{,\chi} = 0, \end{equation} because the AFT sector couples to the scalar only through the algebraic constraint; it does not introduce kinetic or gradient terms for $\chi$. % ============================================================ \section{Hamiltonian Formulation and Constraint Analysis} \label{sec:ADM} % ============================================================ We now outline the ADM formulation and show explicitly that AFT introduces no additional propagating degrees of freedom beyond those of GR plus the scalar field. \subsection{ADM decomposition} Using ADM variables, the metric is \begin{equation} \dd s^2 = -N^2\dd t^2 + h_{ij}(\dd x^i + N^i\dd t)(\dd x^j + N^j\dd t), \end{equation} with lapse $N$, shift $N^i$, and spatial metric $h_{ij}$. The canonical pairs are $(h_{ij},\pi^{ij})$, $(\chi,\pi_\chi)$, dust variables, and the AFT multiplier $(\Lambda,\pi_\Lambda)$. $N$ and $N^i$ are Lagrange multipliers enforcing the Hamiltonian and momentum constraints. \subsection{Primary and secondary AFT constraints} Because $\Lambda$ has no kinetic term, its canonical momentum vanishes: \begin{equation} \pi_\Lambda(x) = \frac{\partial \Lag}{\partial \dot{\Lambda}(x)} = 0. \end{equation} This is a primary constraint $\phi_1(x)\approx 0$. The AFT contribution to the Hamiltonian constraint in dust gauge $T=t$ is \begin{equation} \mathcal{H}_{\mathrm{AFT}}(x) = -\sqrt{h}\,\Lambda(x) \left( \rho^{\mathrm{(com)}}_m +\rho^{\mathrm{(com)}}_\chi - C(t) \right). \end{equation} Requiring preservation of $\phi_1$ gives the secondary constraint \begin{equation} \phi_2(x) := \sqrt{h}\Big( \rho^{\mathrm{(com)}}_m +\rho^{\mathrm{(com)}}_\chi - C(t) \Big) \approx0, \end{equation} i.e.\ the local AFT constraint. One finds \begin{equation} \{\phi_1(x),\phi_2(y)\}=-\sqrt{h(x)}\delta^{(3)}(x-y)\neq 0, \end{equation} so $(\phi_1,\phi_2)$ form a second-class pair and remove $(\Lambda,\pi_\Lambda)$ from the reduced phase space. \subsection{DoF counting} Per space point, the canonical variables are \begin{align} (h_{ij},\pi^{ij}) &\Rightarrow 12,\\ (\chi,\pi_\chi) &\Rightarrow 2, \\ (\Lambda,\pi_\Lambda)&\Rightarrow 2, \end{align} totalling $16$ phase-space dimensions. The $4$ first-class constraints (Hamiltonian and momentum) remove $2\times 4=8$ dimensions. The second-class pair $(\phi_1,\phi_2)$ removes $2$ more. Thus \begin{equation} N_{\mathrm{phys,phase}} = 16 - 8 - 2 = 6, \end{equation} corresponding to $3$ configuration-space degrees of freedom: the two graviton polarizations and the scalar field $\chi$. In particular, there are no additional propagating modes from the AFT sector in this analysis. % ============================================================ \section{Semi-Global Constraint: Precise Interpretation} \label{sec:semiglobal} % ============================================================ The semi-global constraint \begin{equation} C(T) = \frac{\int_{\Sigma_T}J\,\rho^{\mathrm{(com)}}_{\mathrm{constr}}} {\int_{\Sigma_T}J} \end{equation} is global in space but local in dust time. It fixes the homogeneous mode of $\rho^{\mathrm{(com)}}_{\mathrm{constr}}$ on each dust slice. Because $J\,\dd^4x=\dd T\,\dd^3X$ is metric-independent and $T$, $X^I$ are scalars, these integrals are covariant under diffeomorphisms. The constraint does not introduce nonlocal propagation: the local equations remain second-order hyperbolic and the light cone is unchanged. The semi-global constraint acts only on a global degree of freedom (the slice-average of the constrained density) and is enforced by the dust-time multiplier $\sigma(T)$, which carries no local dynamics. In the constraint algebra, the semi-global condition fixes the zero mode of $C(T)$ and ensures compatibility of the local constraint. It affects only the global part of the AFT sector and leaves the local first-class constraints of GR intact. Compared with global 4-form sequestering, which integrates over all of spacetime, the AFT constraint is weaker (slice-wise) but, if consistent, can be sufficient to sequester constant contributions to the vacuum energy. % ============================================================ \section{Homogeneous Cosmology} \label{sec:FLRW} % ============================================================ We now specialise to spatially flat FLRW cosmology, adopting dust gauge $T=t$, $X^I=x^I$. \subsection{Background equations} The metric is \begin{equation} \dd s^2 = -\dd t^2 + a^2(t)\delta_{ij}\dd x^i\dd x^j, \end{equation} and we take homogeneous $\chi(t)$, $\rho_m(t)$, and $\rho_r(t)$. The comoving densities coincide with physical densities. The local constraint \eqref{eq:AFT-local-eq} becomes \begin{equation} \rho_m(t)+\rho_\chi(t) = C(t), \label{eq:local-FLRW} \end{equation} and the Friedmann equation reads \begin{equation} 3\Mpl^2 H^2 = C(t)+\rho_r(t), \label{eq:Friedmann-AFT-FLRW} \end{equation} where $H=\dot{a}/a$. Dust contributes negligibly as a physical fluid; its role is to provide the relational frame. The scalar field has energy density and pressure \begin{equation} \rho_\chi = \frac{1}{2}\dot{\chi}^2 + V(\chi), \qquad p_\chi = \frac{1}{2}\dot{\chi}^2 - V(\chi), \end{equation} and obeys the standard Klein--Gordon equation \begin{equation} \ddot{\chi} + 3H\dot{\chi} + V_{,\chi} = 0. \label{eq:KG-FLRW} \end{equation} Matter and radiation obey \begin{align} \dot{\rho}_m &= -3H\rho_m,\\ \dot{\rho}_r &= -4H\rho_r. \end{align} \subsection{Evolution of \texorpdfstring{$C(t)$}{C(t)}} Differentiating \eqref{eq:local-FLRW} gives \begin{equation} \dot{C} = \dot{\rho}_m + \dot{\rho}_\chi. \end{equation} Using \begin{align} \dot{\rho}_m &= -3H\rho_m,\\ \dot{\rho}_\chi &= -3H(\rho_\chi + p_\chi) = -3H(\dot{\chi}^2), \end{align} we obtain \begin{equation} \dot{C}(t) = -3H(t)\big[\rho_m(t)+\dot{\chi}^2(t)\big]. \label{eq:C-evolution} \end{equation} At late times, when matter redshifts and $\dot{\chi}^2\ll V(\chi)$ in a slow-roll regime, $\dot{C}\approx 0$ and $C(t)$ approaches a constant $C_\infty$. Then \eqref{eq:Friedmann-AFT-FLRW} becomes \begin{equation} 3\Mpl^2 H^2 \to C_\infty, \end{equation} yielding de Sitter-like expansion without the need for explicit fine-tuning of $V(\chi)$ or vacuum energy in the background equations. Whether this behaviour is fully robust in more general settings and under quantum corrections remains an open question. \subsection{Summary of homogeneous dynamics} The homogeneous cosmology is governed by: \begin{itemize} \item Friedmann equation \eqref{eq:Friedmann-AFT-FLRW}, \item local constraint \eqref{eq:local-FLRW}, \item matter and radiation continuity equations, \item Klein--Gordon equation \eqref{eq:KG-FLRW}, \item evolution equation \eqref{eq:C-evolution}. \end{itemize} The only modification relative to GR is the replacement $\rho_m+\rho_\chi\to C(t)$ in the Friedmann equation. $C(t)$ acts as an effective dark-energy density that can dynamically approach a constant in simple scenarios. % ============================================================ \section{Linear Cosmological Perturbations} \label{sec:perturbations} % ============================================================ We briefly summarise the perturbation structure and argue that local propagation remains GR-like. \subsection{Tensor modes} Tensor perturbations $h_{ij}^{\mathrm{(T)}}$ satisfy \begin{equation} \dd s^2 = -\dd t^2 + a^2(t)\left(\delta_{ij}+h_{ij}^{\mathrm{(T)}}\right)\dd x^i\dd x^j, \qquad \partial_i h^{\mathrm{(T)}\,ij}=0, \quad h^{\mathrm{(T)}\,i}{}_i = 0. \end{equation} On an FLRW background, the AFT sector contributes only algebraic constraints and no kinetic terms. The tensor quadratic action is identical to GR: \begin{equation} S^{(2)}_{\mathrm{tensor}} = \frac{\Mpl^2}{8}\int\dd t\,\dd^3x\,a^3 \left[ \dot{h}_{ij}^{\mathrm{(T)}}\dot{h}^{\mathrm{(T)}\,ij} - a^{-2}\partial_k h_{ij}^{\mathrm{(T)}} \partial^k h^{\mathrm{(T)}\,ij} \right]. \end{equation} The equation of motion is \begin{equation} \ddot{h}_{ij}^{\mathrm{(T)}} + 3H\dot{h}_{ij}^{\mathrm{(T)}} - a^{-2}\nabla^2 h_{ij}^{\mathrm{(T)}}=0, \end{equation} so gravitational waves propagate at light speed with GR dispersion, at least within the approximations used here. \subsection{Scalar modes} In Newtonian gauge, \begin{equation} \dd s^2 = -(1+2\Phi)\dd t^2 + a^2(t)(1-2\Psi)\delta_{ij}\dd x^i\dd x^j. \end{equation} The linearised AFT constraint implies that the comoving perturbations of matter and scalar obey an algebraic relation (in dust gauge the perturbation of $C$ is homogeneous). This modifies the constraint sector but not the kinetic term. The local scalar propagation equations (Poisson equation, wave equation for $\delta\chi$) therefore retain their GR form in simple setups; the AFT constraint correlates matter and scalar perturbations but does not introduce new propagating modes or change their leading-order propagation speeds. A more complete, mode-by-mode analysis in Fourier space, including possible subtle effects from the semi-global constraint, is left for future work. % ============================================================ \section{Explicit Consistency Checks} \label{sec:checks} % ============================================================ Here we collect explicit checks of: \begin{enumerate} \item absence of new propagating degrees of freedom; \item no modification of local propagation at the level of the equations of motion; \item radiative stability of vacuum energy in an EFT-inspired sense. \end{enumerate} A full ADM analysis in Section~\ref{sec:ADM} showed that the AFT sector forms a second-class pair and removes no additional local symmetries, leaving the physical DoF count unchanged. Tensor and scalar perturbation analyses in Section~\ref{sec:perturbations} indicate that local dispersion relations match those of GR for the modes considered. In Appendices (or Section~\ref{sec:example}) we provide a mini-superspace example demonstrating explicitly that constant vacuum-energy shifts do not affect the Hubble rate $H(t)$ in AFT at the background level. A more exhaustive analysis, including higher-order perturbations, non-linear effects and quantum corrections, remains to be carried out. % ============================================================ \section{RG-Based Radiative Stability} \label{sec:RG-radiative-stability} % ============================================================ We now recast radiative stability in RG language at a qualitative level. Consider a Wilsonian effective action \begin{equation} S_{\mathrm{eff}}[\Phi,g;\mu] = \int\dd^4x\sqrt{-g} \left[ \frac{\Mpl^2}{2}R - \Lambda_{\mathrm{vac}}(\mu) + \mathcal{L}_{\mathrm{matt}}(\Phi,g;\mu) \right] + S_{\mathrm{AFT}}[g,T,X^I,\Lambda,C,\sigma]. \label{eq:RG-Seff-full} \end{equation} The vacuum-energy parameter runs as \begin{equation} \mu\frac{\dd\Lambda_{\mathrm{vac}}}{\dd\mu} = \beta_\Lambda(\mu)\sim\sum_i\frac{M_i^4}{16\pi^2}+\cdots. \end{equation} In AFT, a constant shift of $\mathcal{L}_{\mathrm{matt}}$, \begin{equation} \mathcal{L}_{\mathrm{matt}}\to\mathcal{L}_{\mathrm{matt}}-\Delta\rho_{\mathrm{vac}}, \end{equation} shifts $\rho^{\mathrm{(com)}}_{\mathrm{constr}}\to \rho^{\mathrm{(com)}}_{\mathrm{constr}}+\Delta\rho_{\mathrm{vac}}$. This can be compensated by \begin{equation} C(T)\to C(T)+\Delta\rho_{\mathrm{vac}}, \qquad \Lambda_{\mathrm{vac}}(\mu)\to\Lambda_{\mathrm{vac}}(\mu)+\Delta\rho_{\mathrm{vac}}, \end{equation} without changing the local equations or curvature at the classical level. The AFT sector is invariant under this spurionic transformation. Thus curvature depends only on the combination \begin{equation} I_{\mathrm{vac}}(T;\mu) := \rho^{\mathrm{(com)}}_{\mathrm{constr}} - C(T) - \Lambda_{\mathrm{vac}}(\mu), \end{equation} which vanishes by the constraint and Einstein equations within the classical framework described here. RG running of $\Lambda_{\mathrm{vac}}(\mu)$ can in this sense be absorbed into $C(T)$, and curvature remains insensitive to the absolute value of the vacuum energy parameter. We therefore argue that AFT may realise a form of radiative stability in the EFT sense: vacuum-energy loops need not require order-by-order fine-tuning of a bare parameter in the gravitational sector. A more complete quantum treatment, including an explicit loop-level calculation in a specific model, is an important open problem. % ============================================================ \section{Publication-Level Example: Scalar-Field Cosmology} \label{sec:example} % ============================================================ We summarise the main steps of a mini-superspace example (details can be extended as needed). The metric is FLRW with lapse $N(t)$ and scale factor $a(t)$, scalar $\chi(t)$, radiation $\rho_r(t)$, and AFT multiplier $\Lambda(t)$. The GR+scalar+radiation action is \begin{equation} S_{\mathrm{GR}+\chi+r} = \int\dd t \left[ - \frac{3\Mpl^2 a\dot{a}^2}{N} + \frac{a^3}{2N}\dot{\chi}^2 - Na^3 V(\chi) - Na^3\rho_r \right]. \end{equation} The AFT local term adds \begin{equation} S_{\mathrm{loc}} = \int\dd t\,a^3\Lambda(t)\left(\rho_\chi(t)-C(t)\right), \end{equation} where $\rho_\chi=\frac{1}{2}\dot{\chi}^2/N^2+V(\chi)$. The canonical momenta are \begin{align} p_a &= -\frac{6\Mpl^2 a\dot{a}}{N},\\ p_\chi &= \frac{a^3}{N}\dot{\chi} + a^3\Lambda\frac{\partial\rho_\chi}{\partial\dot{\chi}},\\ p_\Lambda &= 0, \end{align} so $p_\Lambda\approx0$ is primary. Preservation of $p_\Lambda$ yields the secondary constraint $a^3(\rho_\chi-C(t))\approx0$. The pair is second-class, eliminating $(\Lambda,p_\Lambda)$ exactly as in the full ADM analysis. One first-class Hamiltonian constraint remains. Counting DoF confirms that the only physical configuration variable is $\chi(t)$ (up to gauge choices for $a$). A constant shift $V(\chi)\to V(\chi)+\Delta V$ yields $\rho_\chi\to\rho_\chi+\Delta V$, which can be compensated by $C(t)\to C(t)+\Delta V$, leaving $H(t)$ invariant at the background level. Thus in this simple example constant vacuum-energy shifts are sequestered from the Hubble evolution, illustrating the intended mechanism in a controlled setting. % ============================================================ \section{Comparison with Other Frameworks} \label{sec:comparison} % ============================================================ We briefly compare AFT with other approaches. \subsection{4-form sequestering} 4-form sequestering uses non-dynamical 4-form field strengths and global variables to impose global constraints on the matter vacuum energy \cite{KaloperPadillaPRL2014,KaloperPadillaPRD2014,KaloperPadilla2015, Kaloper2016,Kaloper2018,Kaloper2020}. There are no new propagating modes and radiative stability is achieved in a precise sense. However, the mechanism relies on global topological sectors and nontrivial boundary conditions. AFT aims for a similar decoupling of vacuum energy but uses Brown--Kuchař dust and semi-global slice constraints instead of 4-forms. It is comparatively straightforward to formulate in canonical language and has a clear relational interpretation, while sharing some of the conceptual subtleties associated with semi-global constraints. \subsection{Unimodular gravity} Unimodular gravity fixes $\sqrt{-g}$ and yields Einstein equations with a cosmological constant as an integration constant \cite{Weinberg1989,Unruh1989,HenneauxTeitelboim1989}. It does not sequester vacuum energy: loop corrections still shift the integration constant. No new DoF are introduced, and local dynamics match GR, but radiative stability is not achieved. AFT differs precisely by attempting to make vacuum energy unphysical for curvature via the AFT constraints. \subsection{Mimetic gravity} Mimetic gravity constrains the metric via a non-invertible transformation, introducing a new scalar DoF that behaves like dust \cite{ChamseddineMukhanov2013,MimeticReview}. It does not sequester vacuum energy and often suffers from instabilities. AFT also uses something ``dust-like'', but Brown--Kuchař dust is a relational reference system and does not add propagating DoF; the AFT constraint is second-class and purely algebraic. \subsection{Local sequestering attempts} Local or manifestly local sequestering models attempt to avoid global variables altogether, e.g.\ with divergence-free currents or compensator fields \cite{CopelandPadillaSaffin2015,Padilla2015Notes}. Many such models introduce additional scalar modes and have subtle or problematic constraint algebras. AFT, in contrast, is local in its dynamics but semi-global in its constraints, and has a clean Dirac constraint structure in the analysis presented here: no new propagating modes, GR-like local dynamics, and a candidate for radiative stability. \subsection{Summary table} \begin{center} \begin{tabular}{l|c|c|c|c} \hline Framework & Radiatively stable? & New DoF? & Locality & Mechanism \\ \hline 4-form sequestering & yes & no & global+local & topological 4-forms \\ Unimodular gravity & no & no & local & fixed $\sqrt{-g}$, integration constant \\ Mimetic gravity & no & yes & local & kinematic scalar constraint \\ Local sequester attempts & unclear & often yes & local & divergence-free currents \\ AFT & potentially & no & local + semi-global & dust-time constraints, second-class pair \\ \hline \end{tabular} \end{center} % ============================================================ \section{Weinberg No-Go Theorem and AFT} \label{sec:weinberg} % ============================================================ Weinberg's classic no-go theorem \cite{Weinberg1989} states that under assumptions of locality, general covariance, and fields sitting at stationary points of a local potential, there is no adjustment mechanism avoiding fine-tuning of the cosmological constant. Modern reviews \cite{Padilla2015Notes,PadillaNoGoReview} emphasize that evasion requires relaxing at least one assumption, for example via global variables or non-locality. AFT appears to evade the theorem through: \begin{itemize} \item the presence of Brown--Kuchař dust, which changes the structure of the vacuum state (dust provides a preferred relational slicing); \item a second-class constraint pair $(\Lambda,\rho^{\mathrm{(com)}}_{\mathrm{constr}}-C(T))$ that removes sensitivity of curvature to constant vacuum-energy shifts at the classical level; \item semi-global constraints on dust slices rather than purely local potentials. \end{itemize} The theory remains covariant and local in its dynamics, but is not strictly of the form assumed in Weinberg's original proof. In particular, the vacuum energy is absorbed into a constrained variable $C(T)$ and does not correspond to a dynamical degree of freedom directly sourcing curvature in the same way as in GR. A full, rigorous analysis of the compatibility between AFT and the various modern forms of no-go theorems is beyond the scope of this work and constitutes an interesting direction for future research. % ============================================================ \section{Limitations, Phenomenology, and Outlook} \label{sec:outlook} % ============================================================ AFT preserves the local dynamics of GR and modifies only the constraint structure in a way that aims to eliminate constant vacuum-energy contributions from the curvature sector. While the resulting cosmology can closely resemble $\Lambda$CDM in simple scenarios, several aspects merit further study: \begin{itemize} \item Detailed constraints on deviations from $\Lambda$CDM encoded in the evolution of $C(t)$, e.g.\ via precision measurements of $H(z)$ and growth of structure. \item Nonlinear structure formation and possible signatures in large-scale structure or lensing. \item Extensions including additional scalar sectors or coupling of AFT to modified gravity scenarios. \item Quantum aspects and the role of AFT in semiclassical and quantum gravity, including possible anomalies in the constraint algebra. \item A careful treatment of the semi-global constraint in non-compact spacetimes and its implications for boundary conditions and cosmic initial data. \end{itemize} \medskip\noindent \textbf{Conclusion.} AFT combines a covariant, relational dust framework with a second-class constraint structure and semi-global dust-time constraints to realise a candidate mechanism for vacuum-energy sequestering. In the classical analysis presented here, it retains the local dynamics of GR, introduces no new propagating degrees of freedom, and offers an EFT-inspired interpretation of vacuum-energy decoupling. The construction is exploratory and should be viewed as a starting point rather than a final solution. 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