Bohmian Quantum Cosmology

• Bohmian quantum cosmology is a deterministic framework that uses pilot‐wave guidance laws and quantum potential corrections to evolve cosmological variables.
• The methodology employs the Wheeler–DeWitt equation, Madelung decomposition, and Hamilton–Jacobi formulation to derive quantum‐corrected Friedmann dynamics and avoid singularities.
• This model offers insights into early universe structure formation, the quantum-to-classical transition, and observational signatures such as modified primordial power spectra.

A Bohmian quantum cosmological model is a canonical quantization of cosmology in which the universal wave functional or minisuperspace wave function is supplemented by actual geometric and matter “configurations” (beables) governed by first-order pilot-wave guidance laws. This approach—variously called de Broglie–Bohm, pilot-wave, or causal quantum cosmology—yields a deterministic, observer-independent evolution of the universe’s fundamental degrees of freedom, resolving conceptual and physical issues in quantum cosmology such as the quantum-to-classical transition, singularity avoidance, the problem of time, and the fate of quantum fluctuations. The Bohmian model is systematically constructed from the Wheeler–DeWitt equation via the Madelung (hydrodynamic) or Hamilton–Jacobi–Bohm decomposition, leading to quantum corrected Friedmann equations and guidance relations for the minisuperspace variables. Applications range from early-universe structure formation and the origin of cosmic inhomogeneities, to late-time features such as the absence of Boltzmann brains.

1. Fundamental Construction and Methodology

In Bohmian quantum cosmology, the basic variables are the three-metric $h_{ij}(\mathbf{x})$ and matter fields $\phi(\mathbf{x})$, or in minisuperspace truncations, degrees of freedom such as the FLRW scale factor $a(t)$, scalar field $\phi(t)$, and possibly anisotropic/shear parameters. The quantum state $\Psi$ is a solution of the Wheeler–DeWitt (WDW) equation,

$\widehat{\mathcal{H}}\,\Psi = 0,$

where $\widehat{\mathcal{H}}$ is the Hamiltonian (super-)constraint operator with appropriate operator-ordering and minisuperspace metric. Canonical quantization is performed by promoting canonical momenta (e.g., $p_a, p_\phi$) to derivatives and imposing the constraint as a functional differential equation. In many physically relevant cases, such as the flat FRW + scalar field model, the system is exactly or nearly exactly solvable by separation of variables and canonical transformation, e.g., to hyperbolic oscillator variables (Basilakos et al., 21 Dec 2025).

The Bohmian approach then takes the polar decomposition

$\Psi = R\,e^{iS/\hbar},$

and postulates deterministic evolution of the variables via guidance laws, e.g.,

$$\dot a \propto \frac{\partial S}{\partial a}, \quad \dot\phi \propto \frac{\partial S}{\partial \phi},$$

with coefficients determined by the precise kinetic structure of the minisuperspace action (Pinto-Neto et al., 2018).

The modified Hamilton–Jacobi equation is derived from the real part of the WDW equation: $\mathcal{H}_{\text{cl}}[\partial S] + Q = 0,$ where the quantum potential $Q$ collects all terms involving second (or fractional) derivatives of $R$ (Basilakos et al., 21 Dec 2025, Torres et al., 2018, Pinto-Neto et al., 2018). In models using the Madelung fluid language, the WDW equation is alternatively mapped to continuity and Euler–type equations for a hypothetical cosmological fluid, with the quantum Bohm potential appearing as an explicit “pressure” or force term (Kuzmichev et al., 2024).

2. Quantum Potential, Guidance Equations, and Physical Effects

The quantum potential $Q$ in Bohmian cosmology is responsible for non-classical dynamical effects that are not accessible in conventional semiclassical (WKB) cosmology. Its general form in minisuperspace is

$$Q = -\frac{\hbar^2}{2R}\,G^{AB}\,\partial_A \partial_B R$$

for variables $q^A$ and minisuperspace metric $G^{AB}$ (Passman et al., 2011, Pinto-Neto et al., 2018).

The key guidance equations for the actual configuration variables (beables) take the Hamilton–Jacobi/pilot-wave form: $\dot q^A = G^{AB} \frac{\partial S}{\partial q^B}$ (Pinto-Neto et al., 2018). With the quantum potential included, the Q-corrected Friedmann–like or Raychaudhuri dynamics generically regularizes cosmological singularities, produces non-singular bounces (Frion, 2020, Torres et al., 2018), and imprints dynamical effects on background and perturbation spectra (Maniccia et al., 2024, 0707.1088).

3. Applications: Singularity Resolution, Bouncing Solutions, Quantum Origins of Structure

Bohmian quantum cosmology systematically yields non-singular early-universe dynamics across diverse models:

• For FLRW plus perfect fluid or scalar field, the quantum potential acts as a repulsive stiff-fluid–like component ($\rho_B \sim a^{-4}$), inducing a nonzero minimal scale factor and non-singular bounce (Kuzmichev et al., 2024, Frion, 2020, 0707.1088).
• In cosmologies with nonminimal couplings, higher-curvature, or teleparallel terms, Bohmian quantization with appropriate operator ordering (e.g., via conformable fractional derivatives (Torres et al., 2018)) or affine quantization (Małkiewicz et al., 2022) consistently yields robust avoidance of singularities.
• Bouncing and cyclic Bohmian trajectories are widely observed for appropriate wave packets, and phase space sector analysis can identify acceleration regions compatible with observational constraints, including gravitational wave speed bounds (Torres et al., 2024).

Cosmological perturbation theory is naturally formulated on quantum-corrected Bohmian backgrounds. The effective Schrödinger equation for Mukhanov–Sasaki variables $v_k$ is solved on the pilot-wave–corrected geometry, and the resulting power spectrum incorporates quantum gravity ($\hbar^2$) corrections (Maniccia et al., 2024, Goldstein et al., 2015). In the formalism developed by Pinto-Neto, Struyve, Goldstein, and others, guidance equations for perturbations take the form: $v_k' = \partial_{v^*_k} S_k + \frac{z'}{z} v_k,$ where $S_k$ is the phase of the Fourier mode's conditional pilot wave (Pinto-Neto et al., 2018, Goldstein et al., 2015).

4. Quantum-to-Classical Transition, Symmetry Breaking, and Absence of Boltzmann Brains

A central conceptual achievement of the Bohmian formalism is the natural, observer-independent explanation of the quantum-to-classical transition for cosmological fluctuations. Even if the wave functional is perfectly homogeneous and isotropic (e.g., Bunch–Davies vacuum), sampled Bohmian configurations are invariably spatially inhomogeneous and anisotropic (Goldstein et al., 2015). The deterministic pilot-wave dynamics transports these initial seeds into the super-Hubble regime, generating effective stochastic yet observer-independent realizations with the correct power spectra and correlation properties. There is no recourse to wave-function collapse or an external measuring apparatus.

Conversely, in the asymptotic future (e.g., de Sitter expansion), the Bohmian pilot-wave guidance equations halt further dynamical field rearrangement: actual field modes freeze, preventing the formation of "Boltzmann brain" configurations (Goldstein et al., 2015). This sharply distinguishes Bohmian cosmology from standard many-worlds or Copenhagen approaches, where vacuum fluctuations suffice for a diabatic proliferation of Boltzmann brains over cosmological timescales.

5. Generalizations: Alternative Gravities, Internal Time, and Nonlocality

Bohmian quantum cosmological models extend to modified gravity scenarios, including:

• Nonminimal derivative coupling and Fab Four models, where self-tuning of the cosmological constant and quantum avoidance of singularities are achieved via fractional-derivative Hamiltonians (Torres et al., 2024, Torres et al., 2018).
• Teleparallel gravity with boundary terms, where the quantum Bohm potential is structurally incorporated and deparameterization with respect to a scalar field recasts the WDW constraint as a genuine Schrödinger equation in internal time $\chi$ (Amiri et al., 2024).
• Unimodular or f(R)–like gravity and anisotropic (Bianchi I) geometries, where pilot-wave guidance equations on an extended minisuperspace with an explicit clock variable produce rich bouncing phenomenology and clarify the issue of time reparameterization invariance (Małkiewicz et al., 2022).

The inherently nonlocal structure of the Bohmian guidance equations, where the pilot-wave phase $S$ is a functional on the total configuration space, is key to resolving quantum measurement and ontology in cosmology. The deterministic cosmic trajectory—specified by initial data in quantum equilibrium—is completely fixed once $\Psi$ is chosen, with all observable predictions emerging as properties of the actual path through the multidimensional configuration space (Passman et al., 2011).

6. Quantum Gravity Corrections, Primordial Power Spectrum, and Observational Signatures

Explicit perturbative analysis demonstrates that the Bohmian quantum cosmological model predicts corrections to the inflationary background at order $\hbar^2$, modifying the spectral properties of primordial cosmological perturbations (Maniccia et al., 2024). The modified time-dependent oscillator frequency takes the form: $$\omega_k^2(\eta) = k^2 - \frac{2}{\eta^2} + \mu \hbar^2 \eta^4,$$ where the quantum correction term induces small but distinctive features, including a nontrivial running of the spectral index and its derivatives. Although subleading in amplitude, the structure and hierarchy of running parameters (e.g. $\beta_s$ exceeding $\alpha_s$ above a threshold $k^*$) distinguish the Bohmian scenario from standard QFT-in-curved-space inflation (Maniccia et al., 2024).

Additionally, the quantum potential’s effective energy density can act as an early-universe “stiff-fluid”, providing observational imprints such as a modified pattern in the cosmic microwave background or primordial gravitational wave spectrum, as well as a natural mechanism for singularity avoidance and cosmological constant self-tuning (Kuzmichev et al., 2024, Ali et al., 2014).

7. Summary Table: Key Bohmian Cosmological Models

Model Class	Wave Equation Structure	Bohmian Phenomena/Effects
Flat FRW + scalar	Wheeler–DeWitt, separable to 2D oscillator	Lambda-CDM era interpolation, analytic pilot-wave Hubble law
Madelung–Bohm fluid	Schrödinger–type with quantum pressure	Bounce, quantum “radiation” density, CMB imprints
Nonminimal coupling/Fab Four	Fractionally quantized WDW	Self-tuned $\Lambda$, quantum-induced cycles/bounces
Teleparallel (boundary term)	WDW with internal scalar “clock”	Classical limit via vanishing Q, Schrödinger time evolution
Bianchi/Anisotropic	Affine quantized, effective repulsive terms	Symmetric/clock-dependent bounce in q(t), gauge-invariance issue
FLRW + perturbations	Background + mode Schrödinger equation	Observer-independent realization of structure, no Boltzmann brains

All models realize singularity avoidance, deterministic evolution of geometry and fields, and natural breaking of cosmological symmetries independent of observation or measurement. Further, quantum gravity corrections are reflected in testable features of the primordial power spectrum (Basilakos et al., 21 Dec 2025, Goldstein et al., 2015, Ali et al., 2014, Maniccia et al., 2024).