Power Vacuum Law in Cosmology

• Power Vacuum Law is a framework where vacuum energy decays over time as a power law, influencing cosmic expansion and revealing quantum and gravitational effects.
• The law integrates quantum decay, modified gravity models, and attractor solutions to yield unique, testable predictions on large-scale cosmological observables.
• Its implications extend to resolving the cosmological constant problem by dynamically suppressing vacuum energy from Planck-scale to current values.

The Power Vacuum Law describes a class of cosmological dynamics in which vacuum energy, or effective contributions from modified gravity, decays as a power law in cosmic time or related variables. This framework arises across multiple domains—including quantum cosmology, modified gravity models, and fundamental bounds on vacuum energy dictated by the hierarchy of forces—and is characterized by distinctive, testable predictions for large-scale evolution. The origins and mathematical structures of power-vacuum laws reflect both deep quantum mechanical principles and the constraints imposed by symmetry or dynamical attractors.

1. Quantum Decay and the Running Vacuum Law

Quantum mechanics predicts that the survival probability of an unstable “false vacuum” state develops a late-time power-law tail, as determined by the Paley–Wiener theorem. The survival amplitude for such a state $|M\rangle$ takes the form

$$A(t) = \int_{E_{\min}}^{\infty} \omega(E) e^{-i E t / \hbar} dE$$

where $\omega(E)$ is a positive spectral function. For $t \gg T$ (switching time), the instantaneous energy of the false vacuum approaches that of the true vacuum, modified by a leading inverse power correction: $$E_{\text{false}}(t) \simeq E_{\text{true}} \pm \alpha / t^{2}$$ This leads, at the semiclassical level, to a vacuum energy density law: $$\Lambda(t) = \Lambda_{\text{bare}} + \alpha / t^{2}$$ with $\Lambda_{\text{bare}}$ the true vacuum component and $\alpha > 0$ set by the power-law decay coefficient in the late-time regime (Szydlowski, 2015). As a result, the cosmological “constant” dynamically evolves, decaying as $t^{-2}$, with observational consequences for expansion history and cosmological observables.

2. Power Vacuum Law in Modified and Extended Gravity

Power-law decaying vacuum dynamics appear not only in quantum corrections to vacuum energy but also as attractor solutions in alternative gravity theories:

• Teleparallel gravity with nonminimal coupling: In models with Lagrangian density $$\mathcal{L} = T(1/K + \xi\phi^n) + (\partial_\mu\phi)^2 - 2V_0 \phi^m$$, the vacuum regime yields power-law solutions for the scalar and Hubble parameter:

$$H(t) \propto t^{-2n/(n+2)}, \qquad \phi(t) \propto t^{2/(n+2)}$$

when $n > 2$ and $m \leq -n$ (Skugoreva, 2017). For specific exponents (e.g., $n=2$), the solution reduces precisely to $H(t)\sim 1/t$.

• Multidimensional Gauss–Bonnet gravity: Anisotropic cosmological solutions with power-law scale factors $a_i(t) \propto t^{p_i}$ exist if generalized Kasner-like conditions are satisfied:

$$\sum_{i=1}^N p_i = 3, \qquad \sum_{i<j<k<l} p_i p_j p_k p_l = 0$$

Stability of these solutions, even with homogeneous magnetic fields, depends on subsets $p_{2n-1}+p_{2n} < 2$ for every independent 2-plane, ensuring that power-law “vacuum” is an attractor in the approach to the initial singularity (Chirkov et al., 2012).

3. Noether Symmetry and Unique Power-Law Solutions in $F(R)$ Gravity

Imposing Noether symmetry on vacuum $F(R)$ gravity models in Robertson–Walker minisuperspace uniquely selects the curvature invariant $F(R)\propto R^{3/2}$. This leads to a first fundamental form that is cyclic, granting a conserved charge $\mathcal{Q} = a \sqrt{R}$. The resulting cosmological dynamics admit power-law solutions for the scale factor: $$a(t) = \sqrt{a_4 t^4 + a_3 t^3 + (\frac{3a_3^2}{8 a_4} - k) t^2 + a_2 t + a_1}$$ with pure $a(t)\propto t^{3/2}$ behavior emerging for specific coefficients. The selection of $F(R)\propto R^{3/2}$ is both mathematically unique under Noether symmetry and leads to scale factor evolution not generally capable of reproducing the full standard cosmological history unless supplemented by additional terms (Sarkar et al., 2012).

4. Hierarchical Power Law Bounds on Vacuum Energy

A distinct but related power-law emerges in anthropic or structural bounds on vacuum energy. By demanding that vacuum domination occurs after galaxy formation and that proto-galactic clouds are sub-horizon at matter-radiation equality, one obtains

$$\frac{\rho_{\Lambda}}{M_{\text{pl}}^{4}} \leq C \left( \frac{\alpha_{G}}{\alpha} \right)^{9/2}$$

where $\alpha_G = Gm_p^2 / \hbar c$ (gravitational fine structure constant), $\alpha$ is the electromagnetic fine structure constant, and the exponent $9/2$ results from a combination of galaxy formation and causal horizon constraints (Alexander et al., 2017). This expresses an upper bound on vacuum energy parametrically tied to the extreme smallness of the gravitational-to-electromagnetic force ratio, offering a microphysical selection mechanism for the observed cosmological constant and deepening the vacuum energy–hierarchy relation.

5. Observational Signatures and Phenomenology

The decaying power-vacuum law, specifically $$\Lambda(t) = \Lambda_{\text{bare}} + \alpha / t^{2}$$, modifies cosmological observables such as the Hubble parameter, deceleration parameter $q(t)$, and jerk $j(t)$: $q(t) = -\frac{\ddot{a}/a}{H^{2}}, \qquad j(t) = \frac{\dddot{a}/a}{H^{3}}$ In this context, the jerk parameter departs from the constant $j=1$ prediction of $\Lambda$CDM unless the sum $\Omega_{m,0} + \Omega_{\Lambda,0} = 1$. Thus, any robust late-time detection of $j \neq 1$ constitutes a critical test (“experimentum crucis”) for vacuum decay models (Szydlowski, 2015).

6. Cosmological Constant Problem and Naturalness

A central motivation for the power vacuum law is its capacity to resolve the cosmological constant problem. With $\Lambda(t) \propto 1/t^{2}$, the vacuum energy density transitions from a Planck-scale value at early times to the observed minuscule value today, spanning $120$ orders of magnitude simply by cosmic expansion: $$\frac{\rho_{\Lambda}(t_{\text{Pl}})}{\rho_{\Lambda}(t_{0})} \simeq \frac{\alpha t_{\text{Pl}}^{-2}}{3 H_{0}^{2} \Omega_{\Lambda,0}} \sim 10^{120}$$ This natural, dynamical suppression aligns with observational constraints, although statistical analyses still mildly favor the standard $\Lambda$CDM scenario (Szydlowski, 2015).

7. Attractor Dynamics, Stability, and Limitations

Power-law vacuum regimes are commonly found as attractors in both teleparallel and Gauss–Bonnet gravity frameworks under specific parameter choices and in the absence of matter. Stability analysis is typically performed via linear perturbations, pinpointing conditions for decay of deviations. Physical implications include the possibility of late-time “loitering” regimes, but these solutions may be destabilized or dominated by even small matter densities at late times, constraining their phenomenological relevance and demanding further extension for complete cosmological viability (Skugoreva, 2017, Chirkov et al., 2012).

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References:

• "Cosmological model with decaying vacuum energy law from principles of quantum mechanics" (Szydlowski, 2015)
• "Late-time power-law stages of cosmological evolution in teleparallel gravity with nonminimal coupling" (Skugoreva, 2017)
• "On a Relation of Vacuum Energy to the Hierarchy of Forces" (Alexander et al., 2017)
• "On stability of power-law solution in multidimensional Gauss-Bonnet cosmology" (Chirkov et al., 2012)
• "Why Noether symmetry of F(R) theory yields three-half power law?" (Sarkar et al., 2012)