Phantom Rate in Theoretical Physics

• Phantom Rate is a fundamental parameter that quantifies the evolution (decay, accretion, or relaxation) of physical quantities in systems exhibiting phantom energy (w < -1).
• It arises in diverse contexts—from vacuum decay in phantom cosmology and black hole mass loss to the evolution of scalar fields and anomalous relaxation in non-Hermitian circuits—with specific dependencies on cutoff scales and coupling constants.
• Observational constraints and theoretical analyses rigorously limit the phantom rate, impacting models of dark energy, perturbation growth, and cosmic evolution across multiple physical domains.

The term "phantom rate" arises in several domains of theoretical and mathematical physics, with its most technical manifestations occurring in cosmology, black-hole physics, and the theory of non-Hermitian random circuits. The unifying theme is that the "phantom rate" quantifies the evolution—decay, accretion, or relaxation—of a physical quantity in systems dominated by phantom matter or effective phantom behavior (i.e. equations of state with $w<-1$), or in systems exhibiting emergent dynamics disconnected from naïve spectral expectations. The details depend crucially on context: the physics of vacuum decay in phantom cosmology, accretion rates in black-hole environments, the time evolution of density perturbations in modified gravity, and anomalous relaxation in non-Hermitian Markovian settings.

1. Phantom Rate in Vacuum Decay and Phantom Fluid Cosmology

In quantum phantom cosmologies, phantom fields with negative energy can cause the vacuum to decay into Standard Model or hidden-sector degrees of freedom. The rate of this decay, denoted $\Gamma$, is called the phantom vacuum-decay rate. Even a "sterile" phantom ghost $\phi$ coupled only by gravity yields finite $\Gamma$, with explicit dependence on the phase-space cutoff $\Lambda$ and possible portal mediator scale $M_i$. The microphysical calculation yields

$$\Gamma_g \simeq 4.4 \times 10^{-9} \frac{\Lambda^8}{m_P^4} \left[ 1-\left(\frac{m_{\nu_s}}{\Lambda}\right)^2 \right] \exp\left[ -5.3\left( \frac{m_{\nu_s}}{\Lambda} \right)^{4.2} \right],$$

for decay into hidden-sector neutrinos, with analogous expressions for scalar or vector portal couplings: $$\Gamma_n \simeq 1.1 \times 10^{-5} \Lambda^8 \begin{cases} M_s^{-4} \exp\left[-6.7\left( \tfrac{m_{\nu_s}}{\Lambda} \right)^{2.1} \right] \ (\rm scalar) \ M_v^{-4} \exp\left[-5.7\left( \tfrac{m_{\nu_s}}{\Lambda} \right)^{4.2} \right] \ (\rm vector) \end{cases}$$ This vacuum decay rate sources the continuity equations for the emergent phantom and hidden-sector fluids, yielding an effective equation of state $w_{\rm eff}(z=0)\approx -1.3$ to $-1.5$, manifestly in the phantom regime $w<-1$. The late-time cosmological impact of a nonzero phantom rate $\Gamma$ includes a modest upward shift in $H_0$ and reduction in $S_8$, potentially ameliorating current cosmological tensions. Observational non-detection of decay products constrains $\Lambda \lesssim 19$ MeV and portal scales $M_i \gg 10^8$ GeV for MeV-scale $\Lambda$ (Cline et al., 2023).

2. Phantom Rate in Black Hole Accretion

In black-hole thermodynamics, the phantom rate refers to the rate at which negative-energy phantom fluids are accreted onto a black hole, consequently reducing its mass. In $(2+1)$-dimensional BTZ black hole backgrounds, the mass loss rate due to phantom energy accretion is

$\dot{M} = 4\pi r_+ u^2 (\rho + p)$

where $r_+ = \ell\sqrt{M}$ is the horizon radius, $u<0$ the radial inflow velocity, and the defining feature of phantom fluids, $\rho + p < 0$ for $w < -1$, guarantees $\dot{M} < 0$. This rate is independent of $M$ aside from the geometric factor. The generalized second law further imposes a lower bound on the phantom pressure to ensure $dS_{\rm total} \geq 0$ (Jamil et al., 2010).

Similarly, in the cosmological context of Brans-Dicke theory, the phantom rate of accretion onto primordial black holes is given by

$$\dot{M}_{\rm ph} = 16\pi G(t)^2 M^2 (1+w_{\rm ph}) \rho_{\rm ph}(t)$$

where $G(t)$ is the time-varying gravitational "constant." Phantom accretion eventually dominates over radiation accretion at late times, drastically reducing black-hole lifetimes (from $\tau \sim 10^{121}$ s to $\sim 4.4\times10^{43}$ s for $w_{\rm ph} \approx -1.1$ and realistic densities) (Nayak et al., 2011).

3. Phantom Rate in Quintessence and Effective Dark Energy

In scalar-field cosmology, particularly for minimally coupled quintessence fields $\phi$ with interaction to matter, a time-dependent effective dark energy equation of state $w_{\rm DE}(z)$ can cross and remain below $-1$. The "phantom rate" in this context refers to the redshift derivative $\frac{d w_{\rm DE}}{dz}$, particularly at the point where $w_{\rm DE}(z) = -1$. The instantaneous "phantom-rate"

$$\frac{d w_{\rm DE}}{d N} = \frac{ (1+\epsilon) \frac{d w_\phi}{dN} - w_\phi \frac{d\epsilon}{dN} }{ (1+\epsilon)^2 }$$

where $N = \ln a$ and $\epsilon = (A_m-1)\bar{\rho}_m/\rho_\phi$, is determined by the coupling of the quintessence field to matter. Numeric solutions in string-inspired models exhibit $|dw/dz| \lesssim 1$ for $z \lesssim 1$, consistent with current constraints ($w_a \approx -1$), while the qualitative feature $w_{\rm DE}(z)<-1$ is generically sourced by the appropriate evolution of the coupling function $A_m(\phi)$ and sufficiently steep exponential potentials (Andriot, 15 May 2025).

4. Phantom Rate in Perturbation Growth (Phantom Brane)

On the normal (ghost-free) branch of the Dvali–Gabadadze–Porrati (DGP) braneworld, the background expansion is effectively phantom-like ($w_{\rm eff} < -1$) without ghost instabilities or future singularities. Here, the "phantom rate" refers to the growth rate $f$ of linear matter perturbations,

$f = \frac{d\ln\delta_m}{d\ln a}$

In contrast to the standard parametrization $f = \Omega_m^\gamma$, in the phantom brane $f$ is most accurately described by

$$f(\Omega_m, H) = \Omega_m^{\,6/11 + 0.00729 (1-\Omega_m) + 0.025/(\ell H)} \left(1 + \frac{3.383}{\ell H} \right)^{0.084}$$

where $\ell$ is the brane crossover scale. This "phantom rate" tracks the perturbation growth with sub--$0.1\%$ error for all observationally allowed parameters, whereas standard GR-inspired ansätze fail due to the nonmonotonic evolution of $\Omega_m$ in the phantom-brane background (Viznyuk et al., 2018).

5. Phantom Relaxation Rate in Non-Hermitian Random Circuit Dynamics

In the context of non-Hermitian evolution—specifically, in random circuit theory—the "phantom relaxation rate" describes an emergent asymptotic decay rate of observables (e.g., average purity) that does not correspond to any finite spectral gap. For a Markovian evolution $I(t+1) = M I(t)$, the standard relaxation rate is set by $|\lambda_1|$, the subleading eigenvalue. Phantom relaxation arises when, due to large Jordan blocks or non-Hermitian skin effects, the long-time decay

$$\|I(t) - I(\infty)\| \sim \beta^t, \quad \text{with}\quad |\lambda_1| < \beta < 1$$

has $\beta$ not among the eigenvalues of $M$. For the staircase Haar circuit, $\beta = \alpha/(1-\alpha)$ with $\alpha = d/(d^2+1)$. The underlying mechanism involves the localization of generalized eigenvectors and exponential growth of spectral expansion coefficients, so that the spectral gap becomes a poor predictor of relaxation. Instead, the pseudospectral radius of $M$ governs the relaxation envelope. This is especially relevant in many-body open dynamics with non-Hermitian structure, as canonical eigenmode analysis underestimates actual relaxation times (Znidaric, 2023).

6. Cross-Contextual Overview and Parameter Dependence

The following table summarizes the technical meaning of "phantom rate" by physical context:

Context	Core Quantity	Principal Dependence
Phantom fluid cosmology	Vacuum decay rate $\Gamma$	$\Lambda^8$, $M_i^{-4}, m_{\nu_s}$
Black hole accretion (BTZ/PBH/BD Theory)	Mass loss/accretion rate $\dot{M}$	$\rho+p$, $G(t)$, $M^2$, $w_{\rm ph}$
Quintessence/DE effective EOS	Slope $d w_{\rm DE}/dz$ ("phantom-rate")	$A_m'(\phi)$, $V'(\phi)$, $\rho_\phi$
Braneworld perturbation growth	Growth rate $f$ of density perturbations	$\Omega_m$, $H$, $\ell$, $\gamma$
Non-Hermitian Markov dynamics	Relaxation rate $\beta$ (phantom relaxation)	$\alpha$ (circuit param.), Jordan block

The magnitude, sign, and phenomenological implications of the phantom rate depend distinctly on the details: cutoff scales, mediator couplings, background expansion, couplings of scalar fields, or matrix spectral structure.

7. Phenomenological and Observational Constraints

Stringent bounds on the phantom rate are obtained from various sources:

• Gamma-ray non-observations (e.g., COMPTEL) constrain the phase-space cutoff for vacuum decay ($\Lambda \lesssim 19$ MeV).
• Type Ia supernovae and full cosmological datasets (CMB, BAO, DES, Pantheon) constrain the energy injection rate $\Gamma_\rho$ and associated parameters (e.g., $\Gamma_\rho/\Lambda^4 \lesssim 10^{-38}$ MeV).
• For perturbation growth in braneworlds, distance data restricts $\Omega_\ell \lesssim 0.1$ to ensure subpercent-level consistency with cosmological observables.
• In black-hole environments, the effect of the phantom rate on mass loss becomes dominant only in the late universe, and is tightly limited by the requirement to preserve the generalized second law (Cline et al., 2023, Jamil et al., 2010, Andriot, 15 May 2025, Nayak et al., 2011, Viznyuk et al., 2018, Znidaric, 2023).

In summary, the "phantom rate" designates a fundamental rate parameter—whether it describes vacuum instability, mass accretion or loss, equation-of-state slope, perturbation growth, or non-Hermitian relaxation—distinctive for systems with $w < -1$ (phantom energy) or analogous emergent dynamics, and is tightly circumscribed by both theory and observation.