A Natural Dark Energy Scale from the Entropy of a Quantum Gravity Foam

1. Introduction

Imagine the cosmological horizon not as a continuous membrane but as a mosaic of Planck-sized quantum pixels – the spin-network punctures of loop quantum gravity. In the earliest moments after inflation, a far-from-equilibrium phase transition – the Coherent–Decoherent Spacetime Transition (CDST) – scrambled these pixels, converting the pristine coherent condensate of the quantum-gravity field into a hot, decohered gas of gauge-singlet gravitational quanta: a virtual foam. Today, the entropy stored in this foam still permeates the horizon, and it is this entropy, we argue, that drives the observed cosmic acceleration.

The late-time acceleration of the Universe [1,2] remains a profound challenge. The cosmological constant $\Lambda$ fits the data [3], but its tiny value lacks a natural explanation in quantum field theory. Dynamical dark-energy models, such as quintessence, often require ad-hoc potentials and initial conditions. Quantum gravity, where the Planck scale $M_{\mathrm{P}}$ provides the only fundamental reference, should offer a deeper answer.

Group Field Theory (GFT) [4] is a background-independent approach to quantum gravity in which spacetime and matter emerge from a condensate of pre-geometric tetrahedra. In the coherent condensate phase, a macroscopic occupation of a single quantum state yields an effective scalar–tensor dynamics that can drive inflation [5,6]. The same mean field has also been shown to produce a phantom dark-energy equation of state ($w<-1$) [6,7], where late-time acceleration emerges from quantum corrections to the condensate potential.

Here we explore a different, physically distinct sector of the theory. After inflation, the homogeneous condensate decays according to the infrared scaling of the GFT renormalisation group. Its amplitude today is suppressed by a factor ${(H_0/M_{\mathrm{P}})}^{2d_{\sigma }}\sim {10}^{-146}$ and cannot account for the observed dark energy (Section 3). The remnant is a decohered gas of gauge-singlet gravitational fluctuations – the virtual foam. We propose that this foam, not the coherent mean field, constitutes the dark energy.

By linking the foam to the thermodynamics of the apparent cosmological horizon [8], we obtain a natural holographic scaling ${\rho }_{\mathrm{DE}}\sim M_{\mathrm{P}}^2{H}^2$ and, ultimately, the observed meV energy density. The central quantity is the active foam fraction

$$\begin{array}{|c|}\hline {\alpha }_{\mathrm{foam}}\equiv {\displaystyle \frac{S_{\mathrm{foam}}^{\mathrm{active}}}{S_{\mathrm{hor}}}}\\ \hline\end{array},$$

(1)

which measures the share of horizon microstates actually occupied by the foam. In the simplest spin-$1/2$ puncture picture, ${\alpha }_{\mathrm{foam}}=N_{\mathrm{act}}/N_{\mathrm{hor}}$ is the fraction of punctures that are decohered (Section 5). The exact horizon-thermodynamic relation involves a dynamical temperature correction, absorbed into an effective active fraction ${\alpha }_{\mathrm{foam}}^{\mathrm{eff}}$. At the present epoch,

$$\begin{array}{|c|}\hline {\mathsf{\Omega }}_{\mathrm{DE},0}={\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}\\ \hline\end{array}.$$

(2)

This is the master formula of the paper. The observed ${\mathsf{\Omega }}_{\mathrm{DE},0}\simeq 0.69$ [3] fixes ${\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}\simeq 0.69$; for the benchmark late-time background adopted here, the raw puncture activation fraction is ${\alpha }_{\mathrm{foam},0}\approx 0.95$ – a perfectly natural order-one number.

To obtain a consistent expansion history, this holographic scale is embedded into an effective thawing quintessence model with an exponential potential $V\left(\overline{\varphi }\right)=V_0{e}^{-{\lambda }_{\mathrm{DE}}\overline{\varphi }/M_{\mathrm{P}}}$ (Section 6). The slope ${\lambda }_{\mathrm{DE}}$ is semi-analytically estimated from the scaling dimension of the dominant foam operator at the GFT fixed point, yielding a benchmark value of $0.65$. Numerical integration produces specific, testable predictions for the equation of state $w\left(z\right)$ and the growth of structure, which lie within the projected sensitivity of upcoming Stage-IV surveys. A Gaussian entropy formula for ${\alpha }_{\mathrm{foam}}$ (Section 7) and an illustrative constraint $f_{\mathrm{act}}\overline{s}\approx 5.97$ (Appendix C) provide precise microscopic targets for future GFT decoherence calculations. Inflation, driven by the same coherent condensate, is relegated to Appendix D.

The paper is organized as follows. Section 2 recalls the kinematical data of a GFT quantum. Section 3 demonstrates the decay of the homogeneous condensate. Section 4 presents the horizon entropy counting. Section 5 defines the active foam fraction. Section 6 derives the dark-energy scale and the scalar field dynamics, including numerical predictions. Section 7 provides the Gaussian target formula and the semi-analytic RG estimate of ${\lambda }_{\mathrm{DE}}$. Section 8 summarizes the model’s ingredients, and Section 9 discusses fifth-force sequestering and quantum stability. We conclude in Section 10.

2. the GFT Quantum

The GFT field is a complex function on four copies of the Lorentz group and an internal gauge group [5,9]:

$$\mathsf{\Phi }:G^{\times 4}\times H_{\mathrm{int}}^{\times 4}\to \mathbb{C},\mathsf{\Phi }(g_1,\dots ,g_4;h_1,\dots ,h_4),$$

with $G=\mathrm{SL}(2,\mathbb{C})$ and an internal group $H_{\mathrm{int}}\supset \mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$. After imposing the simplicity constraints, the gravitational data of a single tetrahedron are four $\mathrm{SU}\left(2\right)$ spins $j_a\in {\displaystyle \frac{1}{2}}{\mathbb{N}}_0$ ($a=1,\dots ,4$) and an intertwiner i that couples them to a gauge-invariant singlet. Internal gauge data are labelled by representations $R_a$ of $H_{\mathrm{int}}$ and an internal intertwiner $\lambda$. A single quantum is thus completely characterised by the set

$$\mathcal{Q}=\{j_1,\dots ,j_4;i;R_1,\dots ,R_4;\lambda \}.$$

(3)

This object carries both geometric and matter degrees of freedom.

2.1. Area Spectrum and Additivity

Each triangular face of the tetrahedron has a physical area given by the LQG area operator. We use the reduced Planck length ${\ell }_{\mathrm{P}}^2=8\pi G$, $M_{\mathrm{P}}=1/{\ell }_{\mathrm{P}}$. In these units the standard LQG area spectrum $A\left(j\right)=8\pi \gamma G\sqrt{j(j+1)}$ becomes [10]

$$A\left(j_a\right)=\gamma {\ell }_{\mathrm{P}}^2\sqrt{j_a(j_a+1)},$$

(4)

with the Barbero–Immirzi parameter $\gamma$. A macroscopic horizon is a surface in the emergent geometry that is punctured by many tetrahedra. Because the quanta are independent, the total horizon area is the sum of the individual puncture areas:

$$A_{\mathrm{hor}}=\sum _{\mathrm{punctures}}A\left(j_a\right).$$

(5)

This additivity is a direct kinematical consequence of the discrete geometry.

2.2. Microstates from Intertwiner Configurations

For a fixed set of spins $\left\{j_a^{\left(i\right)}\right\}$ on the $i=1,\dots ,N$ punctures, the microscopic degrees of freedom are the intertwiners. Schematically, the local degeneracy at each puncture is $dim{\mathcal{I}}_i·dim{\Lambda }_i$ (with ${\mathcal{I}}_i$ the gravitational intertwiner space and ${\Lambda }_i$ the internal singlet space). The full horizon Hilbert space must also respect the global horizon gauge and projection constraints. Before imposing those, one may write

$$\mathsf{\Omega }\left(\left\{j_a\right\}\right)={\displaystyle \prod _{i=1}^N}\left(dim{\mathcal{I}}_i·dim{\Lambda }_i\right).$$

(6)

In the entropy formula below, the coefficient ${\gamma }_0$ absorbs the details of the full constrained counting [10,11].

3. Homogeneous Background Condensate and Its Infrared Decay

In the condensate phase, the GFT field acquires a non-vanishing expectation value in a homogeneous, isotropic, gauge-singlet state ${\mathsf{\Psi }}_0$, $\langle \mathsf{\Phi }\rangle =\sigma \left(t\right){\mathsf{\Psi }}_0$. The effective dynamics of the modulus $\sigma \left(t\right)$ is obtained by expanding the GFT action around this mean field and coarse-graining to an FLRW universe:

$$S_{\mathrm{hom}}=\int {\mathrm{d}}^{4}x\sqrt{-g}\left[{\displaystyle \frac{1}{2}}{M}_{\mathrm{P}}^{2}R-{\displaystyle \frac{1}{2}}{g}^{\mu \nu }{\partial }_{\mu }\sigma {\partial }_{\nu }\sigma -V\left(\sigma \right)\right].$$

(7)

The precise form of $V\left(\sigma \right)$ is not critical; what matters is that the theory possesses a non-Gaussian ultraviolet fixed point, as suggested by functional renormalisation group studies of tensor models and GFT [12,13].

At such a fixed point, the scalar field acquires an anomalous dimension ${\eta }_{\sigma }$, giving a scaling dimension

$$d_{\sigma }=1+{\displaystyle \frac{{\eta }_{\sigma }}{2}}.$$

(8)

For a large class of models ${\eta }_{\sigma }>0$, so $d_{\sigma }>1$. In the infrared, the Hubble rate H provides the natural cutoff, leading to

$$\sigma \left(t\right)\sim {\left(H\left(t\right)\right)}^{d_{\sigma }}{M}_{\mathrm{P}}^{1-d_{\sigma }}.$$

(9)

At late times the field is Hubble-frozen. To estimate the energy density we adopt the physically motivated hypothesis that the effective mass of the scalar is tied to the cosmological scale, $m\sim H$, and that the potential is dominated by its quadratic part, so that both kinetic and potential terms scale as $H^2{\sigma }^2$:

$${\rho }_{\sigma }\left(t\right)\sim H{\left(t\right)}^2\sigma {\left(t\right)}^2.$$

(10)

(This is a natural expectation in an infrared GFT phase; if the quartic term or a different scaling prevailed the homogeneous component would be even more suppressed.) Substituting (9) yields

$${\rho }_{\sigma }\left(t\right)\sim H{\left(t\right)}^{2+2d_{\sigma }}{M}_{\mathrm{P}}^{2-2d_{\sigma }}.$$

(11)

Today,

$${\displaystyle \frac{{\rho }_{\sigma ,0}}{{\rho }_{\mathrm{crit},0}}}\sim {\left({\displaystyle \frac{H_0}{M_{\mathrm{P}}}}\right)}^{2d_{\sigma }}\sim {10}^{-146}(\mathrm{for}{d}_{\sigma }\approx 1.2),$$

(12)

completely negligible. Hence the homogeneous condensate cannot explain dark energy; the decohered foam must take over.

4. Horizon Entropy from GFT Boundary States

A macroscopic horizon of area A is pierced by many tetrahedral faces with quantised areas (4). Summing over all spin configurations and intertwiner states, the microcanonical entropy $S_{\mathrm{hor}}\left(A\right)=ln\mathsf{\Omega }\left(A\right)$ grows as [10]

$$S_{\mathrm{hor}}\left(A\right)={\displaystyle \frac{{\gamma }_0}{\gamma }}{\displaystyle \frac{A}{4G}}+\mathcal{O}(lnA),$$

(13)

with ${\gamma }_0$ a model-dependent constant (e.g. ${\gamma }_0=ln2/\left(\pi \sqrt{3}\right)$ for the simplest $j=1/2$ dominance). Fixing the Immirzi parameter to $\gamma ={\gamma }_0$ recovers the Bekenstein–Hawking formula

$$\begin{array}{|c|}\hline S_{\mathrm{hor}}={\displaystyle \frac{A}{4G}}+\mathcal{O}(lnA)\\ \hline\end{array}.$$

(14)

We stress that this is not a first-principles derivation of the Immirzi parameter from GFT; it is the usual entropy-matching condition inherited from the LQG horizon counting, which the GFT boundary states faithfully reproduce. Thus the GFT boundary states provide a microscopic realisation of the leading horizon entropy area law.

5. Active Foam Fraction

5.1. Definition and Microscopic Interpretation

The full horizon Hilbert space is not entirely activated by the foam; only a subset of microstates participates in the dark-energy sector. We therefore define the active foam fraction

$$\begin{array}{|c|}\hline {\alpha }_{\mathrm{foam}}\left(t\right)\equiv {\displaystyle \frac{S_{\mathrm{foam}}^{\mathrm{active}}\left(t\right)}{S_{\mathrm{hor}}\left(t\right)}}\\ \hline\end{array},0\le {\alpha }_{\mathrm{foam}}\left(t\right)\le 1.$$

(15)

Its value is dynamical and must be computed from the CDST.

In the spin-$1/2$ dominance picture, the horizon is pierced by $N_{\mathrm{hor}}$ punctures, each contributing $ln2$ of entropy: $S_{\mathrm{hor}}=N_{\mathrm{hor}}ln2$. After the CDST, $N_{\mathrm{act}}$ of them are decohered and behave as classical records; if each active puncture is maximally mixed, $S_{\mathrm{foam}}^{\mathrm{active}}=N_{\mathrm{act}}ln2$. Hence

$$\begin{array}{|c|}\hline {\alpha }_{\mathrm{foam}}={\displaystyle \frac{N_{\mathrm{act}}}{N_{\mathrm{hor}}}}\\ \hline\end{array}.$$

(16)

Thus ${\alpha }_{\mathrm{foam}}$ is the fraction of punctures that are actively decohered into the foam – the central microscopic identification of this work. Figure 1 gives a schematic illustration.

5.2. Naturalness

The CDST is a far-from-equilibrium process, expected to scramble an appreciable fraction of the available microstates. Hence ${\alpha }_{\mathrm{foam}}$ should be order-unity. Its precise value must come from a first-principles decoherence calculation; we provide a Gaussian target formula in Section 7 and an illustrative constraint in Appendix C.

6. Dark Energy: Holographic Scale and Scalar Field Dynamics

6.1. Holographic Origin of the Dark-Energy Scale

For a flat FLRW universe, the apparent horizon has radius $R_{\mathrm{ah}}=1/H$ and area $A_{\mathrm{ah}}=4\pi /H^2$. In the quasi-de Sitter regime, the Gibbons–Hawking temperature is $T_{\mathrm{ah}}\simeq H/\left(2\pi \right)$. Assigning the thermodynamic energy $E_{\mathrm{foam}}=T_{\mathrm{ah}}{S}_{\mathrm{foam}}^{\mathrm{active}}$ and dividing by the horizon volume $V_{\mathrm{ah}}=4\pi /\left(3H^3\right)$ gives

$${\rho }_{\mathrm{foam}}=3{\alpha }_{\mathrm{foam}}{M}_{\mathrm{P}}^2{H}^2.$$

(17)

The exact dynamical surface gravity of the apparent horizon introduces a correction (see Appendix A):

$${\kappa }_{\mathrm{ah}}=-H\left(1+{\displaystyle \frac{\dot{H}}{2H^2}}\right),T_{\mathrm{ah}}={\displaystyle \frac{|{\kappa }_{\mathrm{ah}}|}{2\pi }}={\displaystyle \frac{H}{2\pi }}\left|1-{\displaystyle \frac{3}{4}}(1+w_{\mathrm{tot}})\right|.$$

To retain the simple form (17) we define an effective active foam fraction

$$\begin{array}{|c|}\hline {\alpha }_{\mathrm{foam}}^{\mathrm{eff}}\left(t\right)\equiv {\alpha }_{\mathrm{foam}}\left(t\right)\left|1-{\displaystyle \frac{3}{4}}\left(1+w_{\mathrm{tot}}\left(t\right)\right)\right|\\ \hline\end{array},$$

(18)

so that the exact relation is

$${\rho }_{\mathrm{DE}}=3{\alpha }_{\mathrm{foam}}^{\mathrm{eff}}{M}_{\mathrm{P}}^2{H}^2.$$

(19)

Today ${\rho }_{\mathrm{crit},0}=3M_{\mathrm{P}}^2{H}_0^2$; consequently

$$\begin{array}{|c|}\hline {\mathsf{\Omega }}_{\mathrm{DE},0}={\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}\\ \hline\end{array}.$$

(20)

With ${\mathsf{\Omega }}_{\mathrm{DE},0}\simeq 0.69$ we obtain ${\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}\simeq 0.69$. For the benchmark late-time background adopted here ($w_{\mathrm{tot},0}\approx -0.59$), the raw puncture activation fraction (16) is then ${\alpha }_{\mathrm{foam},0}={\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}/|1-{\displaystyle \frac{3}{4}}(1+w_{\mathrm{tot},0})|\approx 0.95$, an order-one value requiring no fine-tuning.

6.2. Need for a Dynamical Dark-Energy Model

If ${\rho }_{\mathrm{DE}}\propto H^2$ with constant ${\alpha }_{\mathrm{foam}}$ were used at all times, the Friedmann equation would force $w=0$ and no acceleration. Therefore Eq. (20) only fixes the present-day density. The time evolution is supplied by an effective scalar field that captures the collective foam dynamics.

6.3. Effective Scalar Field Description

The long-wavelength foam is modelled by a canonically normalised scalar $\overline{\varphi }$, minimally coupled to gravity:

$$S_{\mathrm{DE}}=\int d^{4}x\sqrt{-g}\left[{\displaystyle \frac{1}{2}}{M}_{\mathrm{P}}^{2}R-{\displaystyle \frac{1}{2}}{g}^{\mu \nu }{\partial }_{\mu }\overline{\varphi }{\partial }_{\nu }\overline{\varphi }-V\left(\overline{\varphi }\right)\right]+S_m[g_{\mu \nu },{\psi }_m].$$

(21)

The potential is chosen to be exponential,

$$V\left(\overline{\varphi }\right)=V_0{e}^{-{\lambda }_{\mathrm{DE}}\overline{\varphi }/M_{\mathrm{P}}},$$

(22)

motivated by (i) non-perturbative GFT instanton sums, (ii) the scaling symmetry of the GFT fixed point, and (iii) the RG estimate Section 6.4. It is adopted as an effective ansatz.

6.4. Semi-Analytic Estimate of ${\lambda }_{\mathrm{DE}}$

Let the dominant foam operator ${\mathcal{O}}_{\mathrm{foam}}$ have scaling dimension $d_{\mathcal{O}}$ at the GFT fixed point. Its coupling runs as $g_{\mathcal{O}}\left(k\right)\propto k^{4-d_{\mathcal{O}}}$. Identifying the collective field with the RG scale, $k\left(\overline{\varphi }\right)=k_0{e}^{-\overline{\varphi }/M_{\mathrm{P}}}$, yields $V\left(\overline{\varphi }\right)\propto e^{-(4-d_{\mathcal{O}})\overline{\varphi }/M_{\mathrm{P}}}$. Hence

$$\begin{array}{|c|}\hline {\lambda }_{\mathrm{DE}}=4-d_{\mathcal{O}}\\ \hline\end{array}.$$

(23)

If ${\mathcal{O}}_{\mathrm{foam}}$ is a composite scalar with canonical dimension 2 and an anomalous dimension ${\gamma }_{\mathcal{O}}\simeq 1.35$ (typical of melonic tensor models [9]), then $d_{\mathcal{O}}\simeq 3.35$ and

$${\lambda }_{\mathrm{DE}}\simeq 0.65.$$

(24)

This benchmark is used throughout; a first-principles GFT RG calculation must eventually determine it.

6.5. Initial Condition and Parameter Fixing

The CDST supplies a natural matching at the onset of matter domination:

$$\overline{\varphi }\left(t_{\mathrm{onset}}\right)={\alpha }_i{M}_{\mathrm{P}},\dot{\overline{\varphi }}\left(t_{\mathrm{onset}}\right)=0,{\alpha }_i=\mathcal{O}\left(1\right),$$

(25)

with reference value ${\alpha }_i=1$. The normalisation $V_0$ is fixed by requiring ${\mathsf{\Omega }}_{\mathrm{DE},0}=0.6889$ [3]. Using ${\mathsf{\Omega }}_{\mathrm{DE},0}={\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}$,

$$V_0=3{\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}{e}^{{\lambda }_{\mathrm{DE}}{\overline{\varphi }}_0/M_{\mathrm{P}}}{M}_{\mathrm{P}}^2{H}_0^2.$$

(26)

With ${\overline{\varphi }}_0\simeq M_{\mathrm{P}}$, ${\lambda }_{\mathrm{DE}}=0.65$, ${\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}=0.69$, one finds $V_0\simeq 3.96M_{\mathrm{P}}^2{H}_0^2$.

6.6. Numerical Integration and Thawing Predictions

The background equations are given explicitly in Appendix B. For the benchmark slope ${\lambda }_{\mathrm{DE}}=0.65$ the solution yields a thawing equation of state: the field is frozen at $w\simeq -1$ at high redshift and has begun to roll only recently. The exact evolution of $w_{\overline{\varphi }}\left(a\right)$ is shown as the solid red curve in Figure 2.

A low-redshift CPL parametrisation, $w\left(a\right)=w_0+w_a(1-a)$, fitted over $0\le z\le 2$ (i.e. $1/3\le a\le 1$), gives

$$w_0\simeq -0.86,w_a\simeq -0.15.$$

(27)

These values come from the explicit numerical integration of the autonomous system; they differ modestly from the slow-roll estimate $1+w_0\simeq {\lambda }_{\mathrm{DE}}^2{\mathsf{\Omega }}_{\mathrm{DE},0}/3$ (which would give $w_0\simeq -0.90$ for ${\lambda }_{\mathrm{DE}}=0.65$) because the field is not in an exact slow-roll regime at $z\lesssim 1$.

The growth factor $D\left(a\right)$ is obtained from $D^{\prime \prime }+\left(2+{\displaystyle \frac{H^{\prime }}{H}}\right)D^{\prime }-{\displaystyle \frac{3}{2}}{\mathsf{\Omega }}_m\left(a\right)D=0$ and shows a suppression of a few percent relative to $\Lambda$CDM for the same ${\mathsf{\Omega }}_{m,0}$. At recombination, the dark-energy density parameter is $\lesssim {10}^{-5}$, well within observational bounds.

6.7. Why Now?

The effective mass today is $m_{\mathrm{eff},0}^2=V^{\prime \prime }\left({\overline{\varphi }}_0\right)\simeq ({\lambda }_{\mathrm{DE}}^2/M_{\mathrm{P}}^2){\rho }_{\mathrm{DE},0}$, giving $m_{\mathrm{eff},0}\simeq 0.94H_0$. The field becomes light only recently, so thawing occurs naturally at the present epoch once the holographic scale $M_{\mathrm{P}}^2{H}_0^2$ is fixed.

7. Microscopic Target Formulas and Semi-Analytic Derivations

7.1. Gaussian Entropy Formula for ${\alpha }_{\mathrm{foam}}$

In the Gaussian approximation, each foam mode $(I,k)$ carries an occupation number $n_{I,k}$ from the CDST Bogoliubov coefficients. The von Neumann entropy of a single mode is

$$S_{I,k}=(n_{I,k}+1)ln(n_{I,k}+1)-n_{I,k}ln{n}_{I,k}.$$

(28)

Summing over all gauge-singlet foam modes, weighted by a horizon-crossing window $W_{I,k}^{\mathrm{hor}}\left(t\right)=exp[-{\displaystyle \frac{1}{2{\sigma }_H^2}}{(k/\left(aH\right)-1)}^2]$, and using the comoving volume $V_{\mathrm{com}}=a^{-3}{V}_{\mathrm{ah}}$ and a Planckian UV cutoff $k_{\max }\sim a{M}_{\mathrm{P}}$, we obtain

$$S_{\mathrm{foam}}^{\mathrm{active}}\left(t\right)=V_{\mathrm{com}}\left(t\right)\sum _{I\in \mathrm{foam}}{g}_I{\int }_0^{k_{\max }}{\displaystyle \frac{{\mathrm{d}}^{3}k}{{\left(2\pi \right)}^3}}{W}_{I,k}^{\mathrm{hor}}\left(t\right)S_{I,k}\left(t\right).$$

(29)

The horizon entropy is

$$S_{\mathrm{ah}}\left(t\right)={\displaystyle \frac{A_{\mathrm{ah}}}{4G}}=8{\pi }^2{\displaystyle \frac{M_{\mathrm{P}}^2}{H^2}}.$$

(30)

Thus the instantaneous active foam fraction is

$$\begin{array}{|c|}\hline {\alpha }_{\mathrm{foam}}\left(t\right)={\displaystyle \frac{V_{\mathrm{com}}\left(t\right){\sum }_{I\in \mathrm{foam}}{g}_I{\int }_0^{k_{\max }}\frac{{\mathrm{d}}^{3}k}{{\left(2\pi \right)}^3}{W}_{I,k}^{\mathrm{hor}}\left(t\right)\left[(n_{I,k}+1)ln(n_{I,k}+1)-n_{I,k}ln{n}_{I,k}\right]}{8{\pi }^2{M}_{\mathrm{P}}^2/H^2}}\\ \hline\end{array}.$$

(31)

A necessary consistency requirement is that the integral scales as $H^{-2}$, i.e. the active foam entropy respects the area law. This is a non-trivial condition to be verified by a full CDST calculation. Equation (31) is the precise microscopic target formula of this framework.

7.2. Status and Rigorous Bounds

The only rigorous bounds are $0\le {\alpha }_{\mathrm{foam}}\left(t\right)\le 1$. Qualitatively, the integral is dominated by modes in a shell $\Delta k\sim aH$ around horizon crossing. If occupation numbers there are order unity, each mode contributes a few units of entropy, and the total scales with the number of horizon punctures. Hence ${\alpha }_{\mathrm{foam}}$ is naturally order-unity. An illustrative translation into a horizon-cell picture and a quantitative constraint $f_{\mathrm{act}}\overline{s}\approx 5.97$ are given in Appendix C.

7.3. Rg Estimate of ${\lambda }_{\mathrm{DE}}$

The same RG reasoning used earlier is detailed here. The coupling of the dominant foam operator runs as

$${\displaystyle \frac{\mathrm{d}ln{g}_{\mathcal{O}}}{\mathrm{d}lnk}}=4-d_{\mathcal{O}}+\mathcal{O}\left(g_{\mathcal{O}}^2\right),$$

so $g_{\mathcal{O}}\left(k\right)\propto k^{4-d_{\mathcal{O}}}$. With $k\left(\overline{\varphi }\right)=k_0{e}^{-\overline{\varphi }/M_{\mathrm{P}}}$, the effective potential inherits the exponential form $V\left(\overline{\varphi }\right)\propto e^{-(4-d_{\mathcal{O}})\overline{\varphi }/M_{\mathrm{P}}}$, leading again to ${\lambda }_{\mathrm{DE}}=4-d_{\mathcal{O}}$ and the benchmark $0.65$.

7.4. Fifth-Force Sequestering

Linear mixing between the foam collective mode and non-singlet matter modes vanishes by internal-sector orthogonality. Couplings to gauge-invariant composites (e.g. $\overline{\varphi }{T^{\mu }}_{\mu }/M_{\mathrm{P}}$) are assumed to be suppressed in the infrared GFT projection. This provides a microscopic sequestering argument for the absence of leading fifth forces, and it is technically natural if the foam coordinate is orthogonal to the matter sector.

8. Status of the Model

Table 1. Summary of the model’s main ingredients and their epistemic status.

Quantity	Type	Derivation / Value
$S_{\mathrm{hor}}=A/\left(4G\right)$	Derived (LQG)	GFT/LQG boundary counting, Immirzi matching
${\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}$	From data	$\simeq 0.69$ (observed ${\mathsf{\Omega }}_{\mathrm{DE},0}$)
${\alpha }_{\mathrm{foam},0}$ (raw)	Inferred	$N_{\mathrm{act}}/N_{\mathrm{hor}}\simeq 0.95$ for adopted background
${\lambda }_{\mathrm{DE}}$	Semi-derived (RG)	$4-d_{\mathcal{O}}$, benchmark $0.65$
$V_0$	Normalised	$3{\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}{e}^{{\lambda }_{\mathrm{DE}}{\overline{\varphi }}_0/M_{\mathrm{P}}}{M}_{\mathrm{P}}^2{H}_0^2$
$w_0,w_a$	Predicted (numerical)	$-0.86,-0.15$
Microscopic target	Target for CDST	$f_{\mathrm{act}}\overline{s}\approx 5.97$ (Appendix C)
Fifth force	Sequestering	Orthogonality + EFT sequestering

Table 1. Summary of the model’s main ingredients and their epistemic status.

Quantity	Type	Derivation / Value
$S_{\mathrm{hor}}=A/\left(4G\right)$	Derived (LQG)	GFT/LQG boundary counting, Immirzi matching
${\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}$	From data	$\simeq 0.69$ (observed ${\mathsf{\Omega }}_{\mathrm{DE},0}$)
${\alpha }_{\mathrm{foam},0}$ (raw)	Inferred	$N_{\mathrm{act}}/N_{\mathrm{hor}}\simeq 0.95$ for adopted background
${\lambda }_{\mathrm{DE}}$	Semi-derived (RG)	$4-d_{\mathcal{O}}$, benchmark $0.65$
$V_0$	Normalised	$3{\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}{e}^{{\lambda }_{\mathrm{DE}}{\overline{\varphi }}_0/M_{\mathrm{P}}}{M}_{\mathrm{P}}^2{H}_0^2$
$w_0,w_a$	Predicted (numerical)	$-0.86,-0.15$
Microscopic target	Target for CDST	$f_{\mathrm{act}}\overline{s}\approx 5.97$ (Appendix C)
Fifth force	Sequestering	Orthogonality + EFT sequestering

9. Theoretical Consistency and Discussion

9.1. Absence of Fifth Forces

The dark-energy scalar is minimally coupled in the physical frame; no tree-level Brans–Dicke force arises. Residual direct operators are suppressed by the orthogonality of foam and matter sectors.

9.2. Quantum Stability

The exponential form is technically natural for a dilaton-like collective coordinate; one-loop and graviton-loop corrections are strongly suppressed.

9.3. Comparison with Other GFT Dark-Energy Models

Earlier GFT condensate models obtained phantom dark energy from the coherent background [6,7]. In contrast, our model identifies the incoherent virtual foam as the source, leading to a thawing scalar with $w>-1$.

9.4. Inflation

The same coherent condensate can drive inflation when equipped with a large dynamical non-minimal coupling, as outlined in Appendix D.

10. Conclusions

The central result of this paper is the identification

$$\begin{array}{|c|}\hline {\mathsf{\Omega }}_{\mathrm{DE},0}={\alpha }_{\mathrm{foam},0}^{\mathrm{eff}}\\ \hline\end{array}.$$

It unifies the microscopic origin of horizon entropy with the late-time cosmic acceleration: the same GFT boundary states that yield $S=A/\left(4G\right)$ also supply the active foam whose entropy drives the acceleration. The observed ${\mathsf{\Omega }}_{\mathrm{DE},0}\approx 0.69$ fixes the effective active fraction to $0.69$; for the benchmark late-time background used here, the raw fraction of activated horizon punctures is $\approx 0.95$, a perfectly natural order-one efficiency.

We embedded this holographic scale into a thawing quintessence model with an exponential potential, whose slope ${\lambda }_{\mathrm{DE}}\approx 0.65$ is semi-analytically estimated from GFT fixed-point scaling. The model predicts $w_0\approx -0.86$, $w_a\approx -0.15$, and a mild suppression of structure growth – all lying within the projected sensitivity of DESI and Euclid.

The theoretical challenge now is to compute the CDST occupation numbers from first-principles GFT. The Gaussian entropy formula (31) and the illustrative constraint $f_{\mathrm{act}}\overline{s}\approx 5.97$ provide precise targets. Solving this problem will turn the dark-energy puzzle into a quantitatively well-posed target for future GFT decoherence calculations.

Appendix A.1. Kms Thermality in de Sitter Space

In exact de Sitter space, the Bunch–Davies vacuum restricted to the static patch is a KMS (thermal) state at the Gibbons–Hawking temperature $T_{\mathrm{GH}}=H/\left(2\pi \right)$ [14,15]. The reduced density matrix is

$${\rho }_{\mathrm{static}}={\displaystyle \frac{e^{-\beta H_{\mathrm{static}}}}{e^{-\beta H_{\mathrm{static}}}}},\beta ={\displaystyle \frac{2\pi }{H}}.$$

Appendix A.2. Extension to the Apparent Horizon

In a flat FLRW universe, the apparent horizon has radius $R_{\mathrm{ah}}=1/H$. Its surface gravity is

$${\kappa }_{\mathrm{ah}}=-H\left(1+{\displaystyle \frac{\dot{H}}{2H^2}}\right),$$

giving the dynamical Unruh temperature

$$T_{\mathrm{ah}}={\displaystyle \frac{|{\kappa }_{\mathrm{ah}}|}{2\pi }}={\displaystyle \frac{H}{2\pi }}\left|1-{\displaystyle \frac{3}{4}}(1+w_{\mathrm{tot}})\right|.$$

In the quasi-de Sitter regime ($w_{\mathrm{tot}}\approx -1$), this reduces to $H/\left(2\pi \right)$ up to an order-unity factor, which is absorbed into ${\alpha }_{\mathrm{foam}}^{\mathrm{eff}}$ as defined in Eq. (18).

Appendix C.1. Horizon-Cell Variables

The apparent-horizon entropy is $S_{\mathrm{ah}}=8{\pi }^2{M}_{\mathrm{P}}^2/H^2$. Defining the number of effective Planck-sized horizon cells $N_{\mathrm{cell}}=A_{\mathrm{ah}}/{\ell }_{\mathrm{P}}^2=4\pi M_{\mathrm{P}}^2/H^2$, we have $S_{\mathrm{ah}}=2\pi N_{\mathrm{cell}}$. If $N_{\mathrm{act}}$ cells are active, each with average entropy $\overline{s}$,

$$S_{\mathrm{foam}}^{\mathrm{active}}=N_{\mathrm{act}}\overline{s},{\alpha }_{\mathrm{foam}}={\displaystyle \frac{N_{\mathrm{act}}}{N_{\mathrm{cell}}}}{\displaystyle \frac{\overline{s}}{2\pi }}=f_{\mathrm{act}}{\displaystyle \frac{\overline{s}}{2\pi }},$$

with $f_{\mathrm{act}}=N_{\mathrm{act}}/N_{\mathrm{cell}}$.

Appendix C.2. Average Entropy from Occupation Numbers

For a bosonic mode with occupation number n, the von Neumann entropy is $s\left(n\right)=(n+1)ln(n+1)-nlnn$. Representative values are given compactly as:

For large n the asymptotic form $s\left(n\right)\approx lnn+1+O(1/n)$ is accurate to better than 5% already for $n\gtrsim 10$.

Appendix C.3. the Constraint from the Observed Dark-Energy Density

The raw active foam fraction today is ${\alpha }_{\mathrm{foam},0}\approx 0.95$ for our benchmark background. This imposes

$$\begin{array}{|c|}\hline f_{\mathrm{act}}\overline{s}\simeq 2\pi {\alpha }_{\mathrm{foam},0}\simeq 5.97\\ \hline\end{array}.$$

(A4)

Any complete GFT calculation of the CDST must produce a foam state satisfying this relation.

Appendix C.4. What the Constraint Implies

Because $f_{\mathrm{act}}\le 1$, the average entropy per active horizon mode must obey $\overline{s}\ge 5.97$. Using the asymptotic relation $\overline{s}\approx ln{\overline{n}}_{\mathrm{hor}}+1$, this translates into a lower bound on the typical occupation number in the horizon band:

$$\begin{array}{|c|}\hline {\overline{n}}_{\mathrm{hor}}\gtrsim 150\\ \hline\end{array}.$$

Thus a foam with occupation numbers of order unity is ruled out; the CDST must produce significant particle creation and squeezing.

The physically most natural scenario is $f_{\mathrm{act}}\approx 1$ (almost all cells active). With $\overline{s}\approx 6.0$ one needs ${\overline{n}}_{\mathrm{hor}}\sim 150$, a value readily achievable in a far-from-equilibrium transition. Alternative scenarios, such as $f_{\mathrm{act}}\approx 0.5$, would require ${\overline{n}}_{\mathrm{hor}}\sim {10}^4$ and are disfavoured. Therefore the model strongly favours a maximally activated horizon with moderately high occupation numbers.

Appendix C.5. Connection to the Gaussian Formula

Defining the average horizon-band occupation ${\overline{n}}_{\mathrm{hor}}$ via

$${\overline{n}}_{\mathrm{hor}}\equiv {\displaystyle \frac{{\sum }_I{g}_I\int \frac{{\mathrm{d}}^{3}k}{{\left(2\pi \right)}^3}{W}_{I,k}^{\mathrm{hor}}{n}_{I,k}}{{\sum }_I{g}_I\int \frac{{\mathrm{d}}^{3}k}{{\left(2\pi \right)}^3}{W}_{I,k}^{\mathrm{hor}}}},$$

and approximating $\overline{s}\simeq s\left({\overline{n}}_{\mathrm{hor}}\right)$, we obtain

$${\alpha }_{\mathrm{foam}}\simeq f_{\mathrm{act}}{\displaystyle \frac{({\overline{n}}_{\mathrm{hor}}+1)ln({\overline{n}}_{\mathrm{hor}}+1)-{\overline{n}}_{\mathrm{hor}}ln{\overline{n}}_{\mathrm{hor}}}{2\pi }}.$$

(A5)

This compact form links the microscopic occupation numbers directly to the dark-energy abundance.