Ginzburg-Landau Theory of Dark Energy: A Framework to Study Both Temporal and Spatial Cosmological Tensions Simultaneously

TL;DR

GLTofDE presents a GLT-inspired dark-energy framework where DE undergoes a phase transition, producing a redshift-dependent Λ_eff and a dynamical Ω_k^{like} that jointly address both temporal tensions (e.g., H0, high-z reconstructions) and spatial CMB anomalies (hemispherical asymmetry and quadrupole–octopole alignment). The approach combines a mean-field homogeneous background with beyond-mean-field anisotropies, yielding a late-time Goldstone mode that induces dipole modulation and patch-like CC variations, explained via TDGL dynamics and Alan–Cahn evolution. Best-fit analyses show a transition around z_t ≈ 0.74, with H0 ≈ 71–72 and negative Ω_k^{like}, and reveal distinctive fingerprints in H(z) and D_V(z) in the 0.5–1.5 redshift range, offering testable predictions for future surveys. Overall, GLTofDE provides a coherent, testable framework to interpret both temporal and spatial cosmological tensions within a single physical mechanism grounded in critical phenomena.

Abstract

A dark energy model (DE) is proposed based on Ginzburg-Landau theory of phase transition (GLT). This model, GLTofDE, surprisingly provides a framework to study not only temporal tensions in cosmology e.g. $H_0$ tension but also spatial anomalies of CMB e.g. the hemispherical power asymmetry and quadrupole-octopole alignment. In the mean field (or Landau) approximation of GLTofDE, there is a spontaneously symmetry breaking exactly like the Higgs potential. We modeled this transition, phenomenologically, and showed that GLTofDE can resolve both the $H_0$ tension and Lyman-$α$ anomaly in a non-trivial way. According to $χ^2$-analysis the transition happens at $z_t=0.738\pm0.028$ while $H_0=71.89\pm0.93$ km/s/Mpc and $Ω^{like}_{k}=-0.225\pm0.049$ which are consistent with the latest $H(z)$ reconstructions. In addition, the GLTofDE proposes a framework to address the CMB anomalies when it is considered beyond the mean field approximation. In this regime existence of a long wavelength mode is a typical consequence which is named the Goldstone mode in the case of continuous symmetries. This mode, which is an automatic byproduct in GLTofDE, makes cosmological constant, direction dependent. This means one side of the sky should be colder than the other side in agreement with what has been already observed in CMB. In addition between initial stochastic pattern and the final state with one long wavelength mode, we can observe smaller patches or protrusions of the biggest remaining patch in the simulation. Our simulations show these protrusions are few in numbers and will be evolved according to Alan-Cahn mechanism. These protrusions can give an additional effect on CMB which is the existence of aligned quadrupole-octopole mode and its direction should be orthogonal to the dipole direction. We conclude that GLTofDE is a fertile framework both theoretically and phenomenologically.

Figures (14)

• Figure 1: In this figure we have plotted GL-action for $T>T_c$ by dotted line and for $T<T_c$ by solid line. Before phase transition we do have $\Lambda_1$ as the value of potential which is bigger than its value after phase transition i.e. $\Lambda_2$. The existence of terms like $\Phi^3$ or $H.\Phi$ can break the $\mathbf{Z}_2$ symmetry and results in difference in the value of potential's minimum on the right and left. This difference will be crucial for us when we study the anisotropic part of our model.
• Figure 2:
• Figure 3: We have plotted normalized $D_V(z)$ and $D_M(z)$ for GLTofDE in solid and dashed lines respectively. Note that we have not used some of these data points in our $\chi^2$ analysis but GLTofDE is very consistent with distance data points. Interestingly, GLTofDE predicted a very non-trivial behavior of $D_V(z)$ in $z\sim 0.4-0.7$ which follows the trend of BOSS-DR12 and DR14-LRG data points. Obviously GLTofDE (almost) solve Lyman-$\alpha$ tension by predicting less $D_M(z)$ around $z\sim 2.4$ while it is compatible with DES data point around $z\sim 0.7$.
• Figure 4: A cartoon sketched based on our simulations for $T<T_c$ (see Appendix \ref{['TDGL']}). In very early times i.e. figure (a), the system is very stochastic but while time is going then we see the appearance of structures i.e. patches in figure (b). The final state will be as half red/blue (figures (c) and (d)) but then one color will be dominant which has not been in this figure. In cosmological scenario red and blue colors represent $\Lambda_2-\delta\Lambda_L$ and $\Lambda_2+\delta\Lambda_R$ (c.f. FIG. \ref{['fig:potential']}) which means different effective CC. Hence for redshifts before DM-DE equality i.e. $z>z_{eq}$, CC has no effect so we do not see any effects of patches in our model. But near $z_{eq}$ these patches will affect the cosmology and for our purposes we expect $z_{eq}$ be a little bit before (almost) final state i.e. figures (c) and (d). Obviously there is a dipole in (d) which can address the hemispherical asymmetry in CMB via (integrated) Sachs-Wolfe effect of different CC's. In figure (e) we removed the dipole structure and we can see the remaining gives a structure with higher multi-poles. In this simulation we can see three cold/hot patches which gives aligned quadrupole and octopole very interestingly. In addition in bottom-left of (e) we could get a cold/hot spot. We emphasize that for quantitative arguments we need more simulations which will be remained for the future works. However GLTofDE is a rich framework to address CMB anomalies as well as $H_0$ tension.
• Figure 5: $5$ random $2d$ cross-sections from $3d$ cubic lattice in simulation (\ref{['equTDGLSim']}) for simulation steps a) $15$, b) $39$, c) $158$, d) $630$, e) $1584$, f) $9999$, g) $25118$, h) $39810$. Obviously for early times we have stochastic pattern in the system. As time goes on the patterns start to be formed and finally one state becomes dominant. However before the final state it is obvious to see the appearance of a long wavelength mode in (f).