1 Introduction

The Diósi–Penrose proposal  assigns a spatial superposition of a mass the decoherence rate \(\EG/\hbar\), where \(\EG=\tfrac{G}{2}\iint\Delta\rho(\mathbf{x})\Delta\rho(\mathbf{y})/|\mathbf{x}-\mathbf{y}|\) is the self-energy of the difference \(\Delta\rho=\rho_L-\rho_R\) of the branch mass densities. Perturbative graviton environments give rates of second order in \(G\) . A companion analysis shows why . For a static, inertial source with branches of equal mass, quantized linearized gravity has no zero-frequency noise that couples to \(\Delta\rho\) at any temperature, and relational clock readouts depend on mass and time only through \(GMt\). The first-order rate is reproduced exactly by a source-independent white potential noise with correlation \(\hbar G/|\mathbf{x}-\mathbf{y}|\). That noise has weight at zero frequency and nonzero wavenumber, which no free relativistic field supplies.

This paper asks whether such noise can arise from gravity’s thermodynamic side. If the Einstein equation is an equation of state  or an entanglement-equilibrium condition , the coarse-grained geometry is an equilibrium variable, and equilibrium variables fluctuate. We compute two things:

• the stiffness of those fluctuations from generalized entropy (Section 2);

• their relaxation rate, using modular flow (Section 3).

We then derive the consequences for decoherence, spontaneous radiation and heating (Section 4).

The outcome is a model rather than a derivation from first principles. The stiffness follows at linear order from three assumptions. The relaxation requires a fourth, because general relativity by itself propagates the relevant fluctuations. Section 5 lists all four. Within the model, the rate coefficient is fixed, \(C=1/\pi\), and the noise is thermal. Both features distinguish it from the standard Diósi–Penrose model.

2 Stiffness from generalized entropy

Area deficit.

For a geodesic ball of fixed proper volume \(V=\tfrac{4\pi}{3}\ell^3\) in a three-geometry with scalar curvature \(R\) at its centre, the boundary area differs from its flat value by \[\begin{equation} A|_V-A_{\rm flat}(V)=-\frac{4\pi\ell^4}{30}\,R+\order{\ell^5}, \label{eq:deficit} \end{equation}\] the relation underlying the entanglement-equilibrium derivation of the Einstein equation . Consider a static, conformally flat fluctuation \(h_{ij}=e^{2\omega}\delta_{ij}\) with \(\omega\simeq-\Phi/c^2\). Its curvature is \(R=-e^{-2\omega}(4\nabla^2\omega+2|\nabla\omega|^2)\), and integration by parts gives the exact identity \[\begin{equation} \int\!\sqrt{h}\,R\,d^3x=2\int e^{\omega}|\nabla\omega|^2\,d^3x . \label{eq:identity} \end{equation}\] Tile space with such balls, counted by proper volume. The geometric part of generalized entropy then changes by \[\begin{equation} \Delta S_{\rm geom}=-\frac{\ell c^3}{20G\hbar}\int|\nabla\omega|^2\,d^3x+\order{\omega^3}, \label{eq:dS} \end{equation}\] which is negative definite and carries \(1/G\). Counting balls by coordinate volume instead reverses the sign, so proper-volume counting is an assumption with consequences.

Fluctuation weight.

When geometry fluctuates at a fixed matter state, the matter entropy does not change; its redshifted energy is exchanged with the reservoir. The fluctuation weight is therefore \(\exp(\Delta S_{\rm geom}-\beta\,\Delta E_{\rm mat})\), where \(E_{\rm mat}=\int\mu N\), \(\mu\) are fixed mass elements and \(N\) is the lapse. The static quadratic Einstein–Hilbert form in the lapse and the spatial potential is indefinite, so we restrict fluctuations to the slice on which the two potentials agree. On that slice, at linear order, \[\begin{equation} P[\Phi]\propto\exp\!\big(-\beta E_N[\Phi]\big),\qquad E_N[\Phi]=\int\Big[\frac{|\nabla\Phi|^2}{8\pi G}+\mu\,\Phi\Big]d^3x . \label{eq:gibbs} \end{equation}\] The maximum of \(\eqref{eq:gibbs}\) is the attractive Poisson equation, and its quadratic part is the stiffness \((1/8\pi G)\int|\nabla\delta\Phi|^2\). The inverse temperature in both terms is the same: the volume average of the modular inverse temperature \(2\pi(\ell^2-r^2)/(2\ell c)\) of the ball , \[\begin{equation} k_BT_{\rm eff}=\frac{5\hbar c}{2\pi\ell}. \label{eq:Teff} \end{equation}\] Adding the matter entropy instead of the reservoir term produces a functional whose maximum is either repulsive or has three times Newton’s constant. Using the reservoir term is therefore essential.

Kernel.

Equation \(\eqref{eq:gibbs}\) implies \(\langle\delta\Phi(\mathbf{x})\delta\Phi(\mathbf{y})\rangle= Gk_BT_{\rm eff}/|\mathbf{x}-\mathbf{y}|\). This is Diósi’s spatial kernel, with \(k_BT_{\rm eff}\) in place of \(\hbar\) times a correlation time.

3 Relaxation

Relaxation is necessary and not supplied by general relativity.

With the static statistics of Section 2, the dephasing exponent grows linearly in time only if the fluctuations relax. Propagating fluctuations with the same equal-time statistics give a saturating exponent, as in the linearized analysis . A fluctuation on the slice of Section 2 violates the Hamiltonian constraint. At linear order, general relativity carries such violations through the subsystem \(\partial_t\mathcal{H}=-\nabla\!\cdot\!\mathcal{M}\), \(\partial_t\mathcal{M}=-\nabla\mathcal{H}\), whose modes propagate at \(c\). Relaxation must therefore be an additional ingredient. We write it as damping at a rate \(\kappa\) independent of wavenumber: \(\partial_t\mathcal{H}=-\nabla\!\cdot\!\mathcal{M}-\kappa\mathcal{H}\) and \(\partial_t\mathcal{M}=-\nabla\mathcal{H}-\kappa\mathcal{M}\). Each mode then decays as \(e^{-\kappa t}e^{\pm ickt}\), and the long-time dephasing slope is \[\begin{equation} \frac{F}{t}\to\frac{2k_BT_{\rm eff}}{\hbar\kappa}\,\frac{E_G^{(\kappa)}}{\hbar},\qquad E_G^{(\kappa)}=\frac{G}{2}\iint\Delta\rho(\mathbf{x})\Delta\rho(\mathbf{y})\, \frac{1-e^{-\kappa|\mathbf{x}-\mathbf{y}|/c}}{|\mathbf{x}-\mathbf{y}|}. \label{eq:rate} \end{equation}\] This is the Diósi–Penrose form, with the Newtonian kernel regularized at \(R_0=c/\kappa\) and coefficient \(C=2k_BT_{\rm eff}/(\hbar\kappa)\).

A candidate rate: modular mixing.

In the modular state of a Rindler wedge  or a ball , the modular flow has inverse temperature \(2\pi\). The correlation of a dimension-\(\Delta\) primary along the flow decays as \((4\sinh^2(s/2))^{-\Delta}\sim e^{-\Delta s}\) in the modular parameter \(s\). The wedge case follows from the interval between points on one boost orbit, \(\sigma^2=-4\rho^2\sinh^2(s/2)\); the ball case from the flow \(t=\ell\tanh(s/2)\) at the centre with its conformal factors. The decay rate per unit modular parameter is therefore \(\Delta\).

We posit that the coarse geometry relaxes at the modular mixing rate of the field that carries the noise (assumption A4 in Section 5). That field’s statistics and dynamics are two faces of one KMS state. For coarse, ball-uniform observables the modular generator is \(\int\beta(\mathbf{x})T_{00}=\langle\beta\rangle E\), with the same average that fixed \(\eqref{eq:Teff}\). In modular units the temperature is \(1/(2\pi)\) and the decay rate is \(\Delta\), so \[\begin{equation} C=\frac{2k_BT_{\rm eff}}{\hbar\kappa}=\frac{1}{\pi\Delta}, \label{eq:C} \end{equation}\] independently of how modular and physical time are matched. Evaluating the temperature and the rate at different points of the same state would instead introduce a spurious factor \(5/2\), and would violate detailed balance.

The field carrying the noise.

The noise couples to matter through \(\Phi\). A dimension-one field in a KMS state has the equal-time correlator \((T/4\pi r)\coth(\pi rT)\), in units \(\hbar=c=k_B=1\). For \(rT\gg1\) this becomes the classical \(T/(4\pi r)\) kernel of Section 2. We therefore take \(\Delta_\Phi=1\). This is motivated, not derived: in general relativity \(\Phi\) is a constrained variable, not a conformal primary. With it, \[\begin{equation} C=\frac1\pi\approx0.318,\qquad \kappa=\frac{5c}{\ell},\qquad R_0=\frac{\ell}{5}. \label{eq:values} \end{equation}\] Since \(\Delta\geq1\) for unitary scalars in four dimensions, \(C\leq1/\pi\) for every operator in this mechanism. The Diósi–Penrose value \(C=1\) is not reached.

4 Consequences

Decoherence.

Equation \(\eqref{eq:rate}\) with \(\eqref{eq:values}\) gives \(\Gdec=C\,E_G^{(\kappa)}/\hbar\). For separated bodies larger than \(R_0\) this reduces to \(\EG/(\pi\hbar)\). At the cross-term benchmark of a \(1~\mu\)g mass separated by \(1\) mm, the decoherence time is \(\pi\hbar d/(GM^2)\approx4.96\) ns, compared with \(1.58\) ns for \(C=1\). At times shorter than \(R_0/c\) the decay is Gaussian rather than exponential.

The noise is thermal.

Combining \(\eqref{eq:Teff}\) and \(\eqref{eq:values}\), \[\begin{equation} k_BT_{\rm eff}\,R_0=\frac{C\hbar c}{2}=\frac{\hbar c}{2\pi}, \label{eq:link} \end{equation}\] so \(T_{\rm eff}R_0\approx3.64\times10^{-4}\) K m, independently of \(\ell\). The noise that decoheres superpositions is therefore the fluctuation of a bath at finite temperature. It obeys detailed balance, \(S(-\omega)=e^{-\hbar\omega/k_BT_{\rm eff}}S(\omega)\). The decoherence rate uses \(S(\omega\to0)\), where the quantum and classical spectra agree, so it is unaffected. Standard white Diósi–Penrose noise corresponds instead to infinite temperature.

Spontaneous radiation.

Absorbing energy \(\hbar\omega\) from the bath carries the Bose factor \[\begin{equation} e^{-\hbar\omega/k_BT_{\rm eff}}=e^{-2\pi\omega R_0/c}. \end{equation}\] At \(R_0=4\times10^{-10}\) m, \(k_BT_{\rm eff}\approx79\) eV, and emission at \(10\) keV is suppressed by a factor below \(10^{-55}\). Searches for hard spontaneous radiation, which exclude the parameter-free Diósi–Penrose model , therefore constrain this model far more weakly; searches at soft X-ray energies, where \(E\lesssim k_BT_{\rm eff}\), do not (Section 5). Treating the same noise classically would put on-shell weight at \(k=\omega/c\) and wrongly predict more hard emission than Gaussian-smeared white noise. That is a Rayleigh–Jeans artefact.

Heating.

Low-frequency momentum diffusion is not Bose-suppressed. A free constituent of mass \(m\) gains energy at the rate \[\begin{equation} \frac{dE}{dt}=\frac{C\hbar G m}{2}\int\!\frac{d^3k}{(2\pi)^3}\,k^2\tilde K(k), \label{eq:heating} \end{equation}\] where \(\tilde K\) is the Fourier transform of the spatial kernel. The coarse-grained description has no modes below \(\ell\), so \(\tilde K=4\pi/(k^2(1+k^2R_0^2))\) for \(k<1/\ell\) and zero above. The heating rate is then \((C\hbar Gm/\pi)(x-\arctan x)/R_0^3\) with \(x=R_0/\ell=1/5\). Relative to Gaussian-smeared white noise, \(C\hbar Gm/(4\sqrt\pi R_0^3)\), the ratio is \(r_k\equiv(4/\sqrt\pi)(x-\arctan x)\approx5.9\times10^{-3}\) at the same \(C\) and nominal \(R_0\). Heating-based lower bounds on \(R_0\) therefore weaken by the cube root of that ratio, a factor \(\approx0.18\) at equal \(C\), or \((Cr_k)^{1/3}\approx0.12\) against bounds quoted for \(C=1\) (Section 5). Matter would eventually equilibrate at \(T_{\rm eff}\). For \(R_0\geq4\times10^{-10}\) m this takes much longer than the age of the universe, so saturation has no practical consequence. If the kernel’s Yukawa tail were instead to extend to nuclear scales, heating would exceed the Gaussian value by \(\approx2R_0/r_n\). The cutoff at \(\ell\) is therefore essential to viability.

5 Assumptions and outlook

Table 1 collects the assumptions. A4 is where the model departs from general relativity, which propagates rather than relaxes the relevant fluctuations. The model does not derive the Diósi–Penrose rate from quantized gravity; the companion analysis shows that no such derivation exists at linear order . It replaces a free rate postulate with a thermodynamic mechanism and a fixed coefficient.

Distinguishing predictions.

Relative to Gaussian-smeared Diósi–Penrose noise, the model predicts:

1. a rate coefficient \(C=1/\pi\), a factor \(\pi\) below \(C=1\);

2. a Yukawa rather than Gaussian regularization of the kernel, with \(R_0=\ell/5\);

3. the parameter link \(\eqref{eq:link}\);

4. exponential suppression of spontaneous emission above \(k_BT_{\rm eff}\);

5. heating reduced by \(\approx6\times10^{-3}\) at equal \(R_0\).

Items (4) and (5) change which experiments constrain the model. We compare with prior work and then recompute the published bounds.

Relation to prior work.

Dissipative extensions of the Diósi–Penrose model exist , as do dissipative and coloured extensions of continuous spontaneous localization ; see the reviews . In all of them the noise temperature is a further free parameter, fixed by fiat (typically to about \(1\) K); none ties it to the smearing length, and all remain white in time, so their temperature does not cut off the emission. Coloured models cut off emission with a power law , not exponentially. The obstruction that motivates A4—the Newtonian potential is a constraint, and graviton decoherence vanishes in the vacuum—is due to Anastopoulos and Hu . Newtonian-potential fluctuations at a causal-diamond temperature \(\propto\hbar c/\ell\) appear in the interferometer proposals of Verlinde and Zurek , without a spatial kernel or matter decoherence; Kay  obtains an \(\EG\)-like exponent from matter–gravity entanglement, without a rate. Steady decoherence rates fixed by KMS consistency together with an Ohmic channel are known for Killing horizons ; those authors find no such rate for an inertial laboratory. Assumption A4 supplies the missing channel. We found no prior Yukawa regularization of the kernel, no prior \(C=1/\pi\), and no prior relation \(k_BT_{\rm eff}R_0\sim\hbar c\). Bahrami et al.  showed that a cold dissipative bath forces dissociating momentum kicks; the present bath is hot (\(k_BT_{\rm eff}\sim\) keV at allowed \(R_0\)) and so weakly coupled that equilibration takes far longer than the age of the universe.

Recomputed bounds.

Radiation searches quote bounds for white Gaussian-smeared noise. The Gran Sasso  and MAJORANA  analyses use the Penrose normalization, whose rate is \(8\pi\) times Diósi’s ; XENONnT  and the heating bounds use Diósi’s. Relative to a reference model with coefficient \(C_{\rm ref}\) at the same \(R_0\), the present model multiplies the emission at photon energy \(E\) by \((C/C_{\rm ref})\,r_k\,F(E/k_BT_{\rm eff})\), with \(F(u)=u/(e^{u}-1)\) the detailed-balance factor relative to classical white noise and \(r_k=(4/\sqrt\pi)(x-\arctan x)=5.9\times10^{-3}\) the ratio of the momentum-diffusion constants. Holding each experiment’s signal budget fixed and averaging over its window with the reference \(1/E\) weight gives Table 2. For XENONnT the true low-energy spectrum is orbital-resolved ; the \(1/E\) weighting is an approximation of the window average only. Heating and force-noise bounds  probe phonon and millihertz frequencies, where the bath is classical; they rescale by \((Cr_k)^{1/3}=0.12\).

The Bose factor protects only above \(k_BT_{\rm eff}=\hbar c/(2\pi R_0)\), so the constraint migrates to the softest window. The hard-photon search that excludes parameter-free Diósi–Penrose noise weakens by more than two orders of magnitude in \(R_0\), and heating is subdominant; the binding bound is the soft-X-ray one, \(R_0\gtrsim2.7\times10^{-11}\) m, hence \(\ell\gtrsim1.3\times10^{-10}\) m, a factor \(17\) below the Gaussian bound, with a shape-robust floor of \(4\times10^{-12}\) m from MAJORANA. No upper bound follows from these data. Inside the window the model predicts an emission spectrum \(\propto F(E/k_BT_{\rm eff})/E\) with \(k_BT_{\rm eff}\leq1.2\) keV, tied to the decoherence rate by \(\eqref{eq:link}\). A sub-keV analysis of existing xenon or germanium data with this template is the decisive test; hard-photon searches and low-frequency force-noise measurements are not.

Limitations.

The analysis is linear in \(G\) and Newtonian. It assumes static, inertial sources and a single coarse-graining scale. The coarse-graining scale \(\ell\) is not fixed by the model; only the ratio \(R_0/\ell\) is. The four assumptions are not derived.

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