Friedmann-RG Flow Equations in Cosmology

• Friedmann-RG Flow Equations are a synthesis of classical FLRW evolution and renormalization group improvements, integrating quantum and thermal corrections.
• The framework introduces running couplings for Newton's constant and the cosmological constant, enabling a scale-dependent analysis of early-universe dynamics.
• RG-improved cosmological dynamics allow for the study of phase transitions, fixed point structures, and quantum gravity phenomenology in an evolving universe.

The Friedmann-RG flow equations constitute the synthesis of classical cosmological evolution—encoded by the Friedmann equations on a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) background—and quantum/thermal renormalization group (RG) techniques that capture scale-dependent modifications of Newton's constant, the cosmological constant, and matter couplings. The system establishes a coupled dynamical framework in which cosmological observables evolve alongside the running of effective couplings, governed by functional or Wilsonian RG flows, potentially including both quantum and thermal effects. This framework is foundational for investigating quantum gravity phenomenology and the influence of high-energy physics on early-universe dynamics.

1. Classical Framework and Renormalization Group Improvement

The classical starting point is the action of a scalar-tensor theory in the Jordan frame, typically:

$$S = \int d^4x\,\sqrt{-g}\left[\frac{1}{2}(m_{p}^2 + \xi\phi^2)R + \frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi - V(\phi)\right]$$

with $m_p \equiv (8\pi G)^{-1/2}$ and $\xi$ denoting a possible nonminimal coupling.

The corresponding Friedmann equation for cosmic scale factor $a(t)$ is: $$H^2(t) \equiv \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3} \rho_{\text{matter}} + \frac{\Lambda_{\text{cosmo}}}{3}$$ where $V(0) \equiv m_p^2 \Lambda_{\text{cosmo}}$ sets the classical cosmological constant.

Renormalization group improvement is achieved by promoting $G$ and $\Lambda_{\text{cosmo}}$ (as well as scalar self-couplings $g_i$) to running couplings $G(k)$, $\Lambda(k)$, $g_i(k)$, satisfying scale-dependent RG flow equations. The RG scale $k$ parametrizes the typical energy/momentum cutoff for fluctuations integrated out. The RG-improved Friedmann equation reads: $$H^2(k) = \frac{8\pi G(k)}{3} \rho(k) + \frac{\Lambda(k)}{3}$$ with the couplings running along the RG trajectory as $k$ evolves.

2. Quantum and Thermal RG Scales: Introduction of Dimensionless Temperature

The framework distinguishes between two RG cutoffs:

• $k$, the Wilsonian momentum cutoff for quantum (zero-temperature) fluctuations.
• $k_T$, the cutoff for thermal (Matsubara) modes.

A crucial identification is made: $T \equiv k_T = \tau k$ where $\tau$ is a dimensionless temperature parameter that quantifies the relative scale of the thermal cutoff to the quantum cutoff. For $\tau \ll 1$, quantum fluctuations dominate; for $\tau \gtrsim 1$, thermal fluctuations contribute comparably. In this approach, $\tau$ tracks the cosmic thermal history, and observables are organized as functions of $\tau$ rather than directly in terms of $T$ or $k_T$. This strategy allows for dimensionless RG flow equations with no explicit $k$-dependence on the right-hand side, simplifying the search for fixed points and enabling a clean thermal extension of cosmological RG analyses (Marian et al., 13 May 2024).

3. Structure and Properties of the RG Flow Equations

The matter and gravitational sectors are governed by coupled dimensionless RG flow ("beta-function") equations. Within the Local Potential Approximation, and specializing to $d=4$ and a regulator $R_k(p) = (k^2 - p^2) \theta(k^2-p^2)$ (Litim’s choice), the finite-temperature Wetterich equation for the effective potential reads:

$$k\partial_k V_k(\phi) = 2\alpha_3 \cdot 3\, k^5 \left[ \frac{1}{2\sqrt{k^2+V_k''(\phi)}} \coth\left(\frac{\sqrt{k^2+V_k''(\phi)}}{2\tau k}\right) \right]$$

Passing to dimensionless field $\tilde \phi$ and potential $\tilde V_k$: $$\phi = k^{(d-2)/2}\tilde \phi,\qquad V_k(\phi) = k^d \tilde V_k(\tilde\phi)$$ the RG flow for the potential becomes: $$\Big[ d - \frac{d-2}{2} \tilde\phi \partial_{\tilde\phi} + k\partial_k \Big] \tilde V_k(\tilde\phi) = 2\alpha_{d-1}(d-1) \frac{1}{2\sqrt{1 + \tilde V_k''(\tilde\phi)}} \coth\left(\frac{\sqrt{1 + \tilde V_k''(\tilde\phi)}}{2\tau}\right)$$ with the right-hand side being independent of $k$. This property enables identification of nontrivial fixed points and mapping of the phase diagram.

Expanding $$\tilde V_k(\tilde\phi) = \tilde g_{0,k} + \frac{1}{2}\tilde g_{2,k} \tilde \phi^2 + \frac{1}{4!} \tilde g_{4,k} \tilde \phi^4 + ...$$, the $\beta$-functions for the dimensionless mass ($\tilde g_{2,k}$) and quartic coupling ($\tilde g_{4,k}$) are (for $d=4$):

$$(2 + k\partial_k)\tilde g_{2,k} = \frac{1}{6\pi^2} \left[ -\tilde g_{4,k} \frac{\coth(\frac{\sqrt{1+\tilde g_{2,k}}}{2\tau})}{4 (1+\tilde g_{2,k})^{3/2}} -\tilde g_{4,k} \frac{\mathrm{csch}^2(\frac{\sqrt{1+\tilde g_{2,k}}}{2\tau})}{8\tau (1+\tilde g_{2,k})} \right]$$

$$k\partial_k \tilde g_{4,k} = \frac{1}{6\pi^2} \left[ \frac{9\tilde g_{4,k}^2}{8(1+\tilde g_{2,k})^{5/2}} \coth\left(\frac{\sqrt{1+\tilde g_{2,k}}}{2\tau}\right) + \frac{9\tilde g_{4,k}^2}{16\tau(1+\tilde g_{2,k})^2} \mathrm{csch}^2\left(\frac{\sqrt{1+\tilde g_{2,k}}}{2\tau}\right) + ...\right]$$

The flow for the cosmological constant and Newton's constant is given by: $$\lambda(k) \equiv \Lambda_{\text{cosmo}}(k) k^{-2},\qquad g(k) \equiv G(k) k^2$$ with $k\partial_k g(k) = 2g(k)$ in the absence of gravity loops, and: $$k\partial_k \lambda(k) = 8\pi [g(k) k\partial_k \tilde g_{0,k} + \tilde g_{0,k} k\partial_k g(k)]$$

All $\beta$-functions are expressed in terms of $\tau$ and the dimensionless couplings, and have no explicit $k$-dependence.

4. RG-Improved Cosmological Dynamics and Scale Identification

The evolution of the Universe is determined by integrating the RG-improved Friedmann system:

$$H^2(k) = \frac{8\pi G(k)}{3} \rho(k) + \frac{\Lambda(k)}{3}$$

$k\partial_k g(k) = 2g(k)$

$$k\partial_k \lambda(k) = 8\pi [g(k) k\partial_k \tilde g_{0,k} + \tilde g_{0,k} k\partial_k g(k)]$$

An identification $k=k(t)$ (e.g., $k\propto 1/t$ or $k\propto H$) is required to turn RG flow evolution into time evolution, ensuring consistency with the Bianchi identities. As the universe expands, the RG scale flows downward and couplings interpolate from UV to IR regimes. As $\tau$ evolves (slowly, following cosmic cooling), this formalism interpolates between high-temperature (thermal) and low-temperature (quantum-dominated) epochs.

The resulting RG-improved cosmologies demonstrate scale-dependent departures from standard FLRW dynamics, with the matter, vacuum, and gravitational sectors directly coupled to the RG flow of effective parameters. In the ultraviolet, the system enables studies of fixed-point inflation or RG-driven quasi–de Sitter phases; in the infrared, standard $\Lambda$CDM cosmology is recovered as the flow freezes.

5. Ultraviolet Divergences, Vacuum Energy, and Triviality

The approach incorporates explicit mechanisms for handling vacuum energy divergences and for addressing the triviality problem of $\phi^4$ theory in four dimensions (Marian et al., 13 May 2024):

• UV divergences in the vacuum energy (dimensionless $g_{0,k}$) manifest as $k^4$ and $k^2$ terms in the bare $\beta$-function. Subtracting these terms restores the Gaussian fixed point for $(\tilde g_{0,k}, \tilde g_{2,k}, \tilde g_{4,k})$.
• For $\tau=0$, the Wilson–Fisher and Gaussian fixed points coincide in $d=4$, leading to “trivial” flow; no phase transition connects symmetric and broken phases. For $\tau>0$, the RG separatrix tilts, and a genuine finite-temperature phase transition emerges at a nonzero critical $\tau_c$, with a thermal critical exponent $\nu$ defined by the linearized stability matrix.
• A plausible implication is that finite-$\tau$ effects "solve" the triviality of $\phi^4$ theory by replacing the would-be crossover with a true critical point determined by the temperature parameter.

6. Assumptions, Approximations, and Boundary Conditions

Central approximations underlying the Friedmann–RG flow system include:

• The Local Potential Approximation (LPA), neglecting wavefunction renormalization and restricting the effective action to momentum-independent terms.
• Use of the Litim regulator for computational efficiency.
• Truncation of the effective potential to quartic ($\phi^4$) order, with the subtraction of leading vacuum divergences, and neglect of gravity loops (so $k\partial_k g(k) = 2g(k)$).
• Initial boundary conditions are specified by the choice of initial couplings $$(\tilde g_{0,\Lambda}, \tilde g_{2,\Lambda}, \tilde g_{4,\Lambda}, \tau)$$ at some UV scale $k=\Lambda$. The system is then integrated downward in $k$, holding $\tau$ fixed to generate the effective action at given cosmic temperature. Iterating this for different $\tau$ reconstructs the full thermal history through cosmological epochs.

These assumptions render the system suitable for exploring early-universe quantum corrections, inflationary scenarios, and the RG structure of cosmological phase transitions. Limitations stem from the neglect of higher-derivative or nonlocal operators, the scheme-dependence of $k$-identification, and the potential backreaction from gravity or matter sectors not captured at the level of truncation.

7. Extensions, Generalizations, and Related Paradigms

Alternative RG-improved cosmological frameworks—such as those developed within the ADM functional RG or with inclusion of higher-derivative gravitational operators (Platania et al., 2017, Alwis, 2018)—extend the Friedmann-RG flow approach. Notable distinctions include:

• Direct inclusion of gravity-sector beta functions, e.g., the full Einstein–Hilbert truncation and treatment of gravitational fixed points and their critical exponents.
• Systematic treatment of higher-derivative and nonlocal interactions (e.g., $R\square^n R$ operators) within the Wilsonian RG, which can modify the flows of the cosmological and Newton constants and induce additional running.
• Analysis of the spurious nature of curvature-squared ghosts, eliminated via cutoff-dependent field redefinitions and shown not to propagate at physical scales, thus refining the reliability of RG-improved cosmologies (Alwis, 2018).

Within all such frameworks, the integration of RG flow equations with cosmological evolution equations enables systematic paper of quantum-gravity corrections, scale-dependent dynamics, inflationary fixed points, and high-energy cosmological phenomena.

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The Friedmann-RG flow paradigm provides a mathematically complete system for tracking quantum and thermal corrections to cosmological evolution, grounded in functional RG techniques and equipped to address both vacuum and finite-temperature effects. Its implementation bridges microphysical RG flows and macroscopic cosmological signatures, offering a controlled and extensible approach for probing the quantum structure of the early Universe.