A Predictive Dark Energy Model from Group Field Theory

1. Introduction

The observed cosmic acceleration [1,2] lacks a compelling microscopic origin. The cosmological constant $\Lambda$ fits data [3] but suffers from extreme fine-tuning. Quintessence models [4,5,6] often require ad hoc potentials and initial conditions. Group Field Theory (GFT) [7] is a background-independent quantum gravity approach in which spacetime emerges from a condensate of pre-geometric tetrahedra [8]. In this framework, matter and geometry are unified in a single field on group manifolds, and the macroscopic dynamics yields an effective scalar–tensor theory.

The potential of GFT condensates to address the dark-energy problem has been demonstrated in several recent works. Oriti and Pang [9] showed that the non-linear dynamics of the condensate mean field can produce a phantom-like dark-energy equation of state ($w<-1$). More recently, Marchetti, Ladstätter, and Oriti [10] presented a unified GFT-based scenario in which both inflation and late-time acceleration emerge from the same condensate, with the dark-energy phase again exhibiting phantom behaviour in certain interaction regimes. In those models, dark energy is driven by the coherent mean field, and the late-time equation of state crosses the phantom divide from quantum gravity effects.

The present work builds on the same GFT condensate framework but identifies a different sector of the theory as the source of dark energy. A crucial consequence of the GFT condensate picture is that quantum particles are not point-like but localised, coherent excitations of the same field that constitutes the background — what we call private spacetimes. Their internal geometry is a smooth perturbation of the global FLRW metric, and the global time emerges from the synchronisation of many such world-lines. We show that the homogeneous condensate alone cannot account for dark energy; instead the incoherent virtual foam of Planck-scale 4-simplices, constantly created and annihilated, is the source. Whereas the coherent mean field yields a phantom equation of state in Refs. [9,10], the incoherent foam gives rise to a thawing quintessence field with an exponential potential $V\left(\overline{\varphi }\right)=V_0{e}^{-{\lambda }_{\mathrm{DE}}\overline{\varphi }/M_{\mathrm{P}}}$, which predicts $w>-1$ at all times. This distinct observational signature — a specific thawing trajectory with $w_0\simeq -0.86$, $w_a<0$ — can be decisively distinguished from phantom dark energy and from a cosmological constant by Stage-IV surveys (DESI, Euclid).

The exponential potential is an effective ansatz motivated by GFT instanton sums and scaling behaviour. The minimal coupling of the foam collective coordinate $\overline{\varphi }$ to gravity is not assumed but is supported by an explicit projection calculation (see Supplementary Material): the overlap between the microscopic non-minimal operator and the gauge-singlet homogeneous foam mode vanishes if the foam sector is orthogonal to the original condensate modulus, thus guaranteeing the absence of a Brans–Dicke fifth force. The overall energy scale $V_0$ is naturally parametrically of order $M_{\mathrm{P}}^2{H}_0^2$, explaining the meV scale, via the covariant entropy bound on the apparent horizon. After fixing $V_0$ by the observed dark-energy density, the model has a single dynamical parameter, the slope ${\lambda }_{\mathrm{DE}}$, which controls the thawing dynamics. The private-spacetime structure supplies the initial amplitude of the dark-energy field and ameliorates the coincidence problem.

The slope ${\lambda }_{\mathrm{DE}}$ is estimated semi-analytically from the scaling dimension $d_{\mathcal{O}}$ of the leading foam operator, giving ${\lambda }_{\mathrm{DE}}=4-d_{\mathcal{O}}$. For a representative tensor-model anomalous dimension one obtains ${\lambda }_{\mathrm{DE}}\simeq 0.65$, a value used as a benchmark throughout. A full numerical integration of the thawing scalar has been performed, confirming the viability of this benchmark and yielding $w_0\simeq -0.86$, $w_a\simeq -0.15$, and a $3.2\%$ suppression of linear matter growth relative to $\Lambda$CDM. A Fisher forecast shows that Stage-IV surveys (DESI, Euclid) will distinguish this model from a cosmological constant at high significance. All numerical details, together with rigorous derivations of the minimal coupling, the RG estimate of ${\lambda }_{\mathrm{DE}}$, and an entropy-based estimate of the foam fraction, are provided in the accompanying Supplementary Material.

The paper is organised as follows. Section 2 defines the private spacetime of a single particle. Section 3 embeds it in the GFT condensate and gives the effective action. Section 4 shows that the homogeneous background condensate contributes a negligible vacuum energy. Section 5 derives the dark-energy model: we motivate the exponential potential, compute the natural scale $V_0$ from the covariant entropy bound, set up the background equations, discuss parameter fixing, and give numerical predictions. Section 6 describes how the same GFT condensate can drive inflation. Section 7 outlines the Coherent–Decoherent Spacetime Transition linking inflation to dark energy. Section 8 presents the unified effective action. Section 9 addresses theoretical consistency. We conclude in Section 10.

2. Private Spacetime of a Single Particle

In the GFT condensate, the total field decomposes as

$$\Phi ={\Phi }_{\mathrm{bg}}+\sum _P{\Phi }_P+\delta \Phi ,$$

(1)

where ${\Phi }_{\mathrm{bg}}$ is the homogeneous background, ${\Phi }_P$ are localised particle excitations, and $\delta \Phi$ are virtual fluctuations. Here we describe a single particle excitation ${\Phi }_P$.

2.1. Geometry as a Local Perturbation

A particle is a coherent, soliton-like lump where the curvature and scalar field deviate from the background. Its world-tube is described by a timelike world-line $\gamma$. In a sufficiently small world-tube one may use Fermi–Walker coordinates, in which the metric is locally Minkowskian up to curvature corrections of order $R_{\alpha \beta \gamma \delta }{x}^2$. For matching to the FLRW background it is convenient to write the spatial metric in a locally comoving form,

$$\mathrm{d}{s}^2=-c^2\mathrm{d}{\tau }^2+a^2\left(\tau \right)\left[{\delta }_{ij}+{\epsilon }_{ij}(\tau ,\mathbf{x})\right]\mathrm{d}{x}^i\mathrm{d}{x}^j,{\epsilon }_{ij}(\tau ,\mathbf{0})=0,$$

(2)

valid when the tube size is much smaller than the Hubble radius. Here ${\epsilon }_{ij}$ decays to zero at the boundary of the tube. No separate scale factor is introduced; the expansion law is everywhere that of the background to leading order. The proper time $\tau$ coincides with the cosmic time t at the boundary.

2.2. Effective Action Inside the Lump

The dynamics of the lump is governed by the same scalar–tensor action that describes the background (Section 3), with the non-minimal coupling ${\xi R\left|\varphi \right|}^2$ playing a crucial role. In the non-relativistic limit, the scalar field $\varphi$ reduces to the particle’s wavefunction, and the lump’s centre follows the energy-weighted spatial centroid. On a spatial slice ${\Sigma }_{\tau }$ with induced metric $h_{ij}$, the energy density measured by an observer with four-velocity $u^{\mu }$ is $\rho =T_{\mu \nu }{u}^{\mu }{u}^{\nu }$. The centroid is defined by

$$X_{\mathrm{centroid}}^i={\displaystyle \frac{{\int }_{{\Sigma }_{\tau }}{d}^{3}x\sqrt{h}\rho (\tau ,\mathbf{x})x^i}{{\int }_{{\Sigma }_{\tau }}{d}^{3}x\sqrt{h}\rho (\tau ,\mathbf{x})}}.$$

(3)

The proper time along this centroid world-line is defined by the normalisation of the 4-velocity, completing the description of the particle’s trajectory. Because the lump is a localised perturbation, it does not affect the global Hubble expansion; its contribution to the cosmic energy budget enters only through the sum over all such excitations, which constitute the matter and radiation components.

3. GFT Condensate and Effective Action

3.1. Continuum Effective Action

In the condensate phase, the GFT field is dominated by a macroscopic occupation number, and the coarse-grained dynamics reduces to the local effective action [8]

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

(4)

where $\sigma =\left|\varphi \right|$ is the modulus of the condensate wavefunction, $V\left(\sigma \right)={\displaystyle \frac{1}{2}}{m}^2{\sigma }^2+{\displaystyle \frac{\lambda }{4}}{\sigma }^4$, and gauge fields are omitted. In the effective truncation considered here, the non-minimal coupling is assumed to approach the conformal value $\xi =1/6$ at the relevant infrared fixed point [11,12]. This action applies universally to the background, to individual particles, and to the coarse-grained virtual foam. Different effective modes of the GFT field (such as the late-time dark-energy collective coordinate) may have different effective couplings; in particular, the foam field is minimally coupled in the physical frame (see Sec. Section 5).

3.2. Field Equations

Variation with respect to $g_{\mu \nu }$ and $\sigma$ gives

$$\begin{array}{cc}\hfill \left(M_{\mathrm{P}}^2-2\xi {\sigma }^2\right)G_{\mu \nu }& =T_{\mu \nu }^{\sigma }+2\xi \left(g_{\mu \nu }{\nabla }^2-{\nabla }_{\mu }{\nabla }_{\nu }\right){\sigma }^2,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\nabla }^{\mu }{\nabla }_{\mu }\sigma -V^{\prime }\left(\sigma \right)-2\xi R\sigma & =0.\hfill \end{array}$$

(6)

The scalar stress-energy tensor is

$$T_{\mu \nu }^{\sigma }={\partial }_{\mu }\sigma {\partial }_{\nu }\sigma -g_{\mu \nu }\left({\displaystyle \frac{1}{2}}{(\partial \sigma )}^2+V\left(\sigma \right)\right).$$

For a homogeneous, isotropic FLRW background ($\sigma =\sigma \left(t\right)$), the 00-component of (5) yields the Friedmann equation

$$3\left(M_{\mathrm{P}}^2-2\xi {\sigma }^2\right)H^2={\displaystyle \frac{1}{2}}{\dot{\sigma }}^2+V\left(\sigma \right)+12\xi H\sigma \dot{\sigma }.$$

(7)

At the conformal fixed point $\xi =1/6$, and including matter and radiation densities ${\rho }_m,{\rho }_r$, this reduces to

$$3M_{\mathrm{P}}^2{H}^2={\rho }_m+{\rho }_r+{\displaystyle \frac{1}{2}}{\dot{\sigma }}^2+V\left(\sigma \right)+2H\sigma \dot{\sigma }+H^2{\sigma }^2.$$

(8)

The Klein–Gordon equation becomes

$$\ddot{\sigma }+3H\dot{\sigma }+V^{\prime }\left(\sigma \right)+{\displaystyle \frac{1}{3}}R\sigma =0,$$

(9)

with $R=6(2H^2+\dot{H})$ for a flat FLRW metric.

4. Homogeneous Background Condensate

At the non-Gaussian UV fixed point of the GFT [11,12], the dimensionless couplings and the anomalous dimension $\eta$ are finite. The scaling dimension of $\sigma$ is $d=1+\eta /2$. In the infrared, the homogeneous amplitude follows

$$\sigma \left(t\right)\sim M_{\mathrm{P}}{\left({\displaystyle \frac{H\left(t\right)}{M_{\mathrm{P}}}}\right)}^d=H{\left(t\right)}^d{M}_{\mathrm{P}}^{1-d}.$$

(10)

With d estimated in the range $1.15$–$1.25$, the present amplitude is ${\sigma }_0\sim H_0^d{M}_{\mathrm{P}}^{1-d}$. Assuming the running mass satisfies $m^2\left(H\right)\sim H^2$ and that the quartic term is negligible for the tiny late-time amplitude, the leading terms in the energy density (8) scale as $H^2{\sigma }^2$. Substituting ${\sigma }_0$ gives

$${\rho }_{\sigma }\sim H_0^{2+2d}{M}_{\mathrm{P}}^{2-2d},$$

which is suppressed relative to the critical density $M_{\mathrm{P}}^2{H}_0^2$ by a factor ${(H_0/M_{\mathrm{P}})}^{2d}\sim {10}^{-146}$ (for $d\approx 1.2$). Thus the homogeneous condensate contributes a completely negligible vacuum energy. The observed dark energy must therefore originate from the incoherent sector — the virtual foam.

We stress that this conclusion is specific to the effective action and scaling behaviour assumed here. In other GFT condensate scenarios [9,10], different effective potentials or different treatments of the non-minimal coupling allow the homogeneous mean field to survive at late times and to provide a phantom dark-energy component. The question of which sector dominates is a dynamical one that can only be settled by a complete GFT calculation. Our model explores the consequences of the foam-dominated regime, while Refs. [9,10] explore the mean-field-dominated regime.

5. Dark Energy from the Virtual Foam

The CDST described in Sec. Section 7 leaves behind a decohered gas of gauge-singlet gravitational quanta — the virtual foam. In this section we formulate an effective late-time description of this foam, derive the resulting dark-energy dynamics, and show that the model is predictive and testable.

5.1. Exponential Potential: Motivation and Ansatz

The coarse-grained state of the foam is described by a single collective coordinate $\overline{\varphi }\left(t\right)$, defined as the infrared projection of the gauge-singlet foam fluctuations (see Supplementary Material for an explicit definition in terms of GFT modes). Non-perturbative GFT amplitudes, approximate scaling symmetry, and dilaton-like RG behaviour motivate an exponential potential of the form

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

(11)

In this work (11) is adopted as a well-motivated effective ansatz. The slope ${\lambda }_{\mathrm{DE}}=\mathcal{O}\left(1\right)$ is estimated below from GFT operator scaling, and the overall scale $V_0$ is shown to be naturally of order $M_{\mathrm{P}}^2{H}_0^2$ from the covariant entropy bound.

5.2. Semi-Analytic Estimate of ${\lambda }_{\mathrm{DE}}$ and the Scale $V_0$

The dark-energy collective coordinate $\overline{\varphi }$ acts as an infrared modulus of the decohered foam sector. Near a non-Gaussian GFT fixed point, the dominant foam operator ${\mathcal{O}}_{\mathrm{foam}}$ that contributes to the effective potential has a scaling dimension $d_{\mathcal{O}}$. Its coupling $g_{\mathcal{O}}$ satisfies the RG equation $\partial ln{g}_{\mathcal{O}}/\partial lnk=4-d_{\mathcal{O}}+\dots$, so that at the fixed point $g_{\mathcal{O}}\left(k\right)\propto k^{4-d_{\mathcal{O}}}$. Identifying the collective field with a dilaton-like RG scale, $k\left(\overline{\varphi }\right)=k_0{e}^{-\overline{\varphi }/M_{\mathrm{P}}}$, the effective potential becomes

$$V\left(\overline{\varphi }\right)\propto g_{\mathcal{O}}\left(k\left(\overline{\varphi }\right)\right)\propto e^{-(4-d_{\mathcal{O}})\overline{\varphi }/M_{\mathrm{P}}},$$

from which we read off the exponential slope

(12)

Tensor-model studies of GFT indicate that composite scalar operators can acquire large anomalous dimensions [13]. Taking a representative effective scaling dimension $d_{\mathcal{O}}\simeq 3.35$ yields

This estimate is not a rigorous derivation — a complete RG computation of the GFT foam sector is not yet available — but it shows that an $\mathcal{O}\left(1\right)$ slope in the range $0.6--0.7$ is natural from the fixed-point perspective. The same value is independently motivated by a semi-classical instanton estimate (see Supplementary Material).

The overall scale $V_0$ is constrained by the covariant entropy bound on the apparent horizon, $S_{\mathrm{ah}}=8{\pi }^2{M}_{\mathrm{P}}^2/H^2$, as discussed in Sec. Section 5.4. In terms of the active foam fraction ${\alpha }_{\mathrm{foam}}$ that measures how much of the horizon information is carried by the decohered foam, the entropy bound gives $V\left({\overline{\varphi }}_0\right)\simeq 3{\alpha }_{\mathrm{foam}}{M}_{\mathrm{P}}^2{H}_0^2$. With the CDST matching condition ${\overline{\varphi }}_0\simeq M_{\mathrm{P}}$, one obtains

(13)

The observed dark-energy density corresponds to ${\alpha }_{\mathrm{foam}}\simeq {\Omega }_{\mathrm{DE},0}\simeq 0.69$, an order-one fraction that confirms the naturalness of the scale. For the benchmark ${\lambda }_{\mathrm{DE}}=0.65$ one finds $V_0\simeq 3.96M_{\mathrm{P}}^2{H}_0^2$, which reproduces the required present dark-energy density ${\rho }_{\mathrm{DE},0}\simeq 2.5\times {10}^{-47}{\mathrm{GeV}}^4$. A detailed step-by-step derivation, including an entanglement-entropy estimate of ${\alpha }_{\mathrm{foam}}$, is provided in the Supplementary Material.

5.3. Minimal Coupling in the Physical Frame

The original GFT effective action (4) contains a non-minimal coupling $-\xi R{\sigma }^2$ for the condensate modulus $\sigma$. The dark- energy collective coordinate $\overline{\varphi }$ is, however, a distinct effective mode: it is the homogeneous projection of the gauge-singlet foam fluctuations. An explicit projection calculation (see Supplementary Material) shows that the overlap of the microscopic non-minimal operator with this collective mode vanishes if the foam sector is orthogonal to the original condensate modulus, i.e. ${\xi }_{\mathrm{eff}}=0$. Hence the leading late-time effective action for the foam sector is that of a minimally coupled canonical scalar field:

$$S_{\mathrm{DE}}=\int d^{4}x\sqrt{-g}\left[{\displaystyle \frac{1}{2}}{M}_{\mathrm{P}}^{2}R-{\displaystyle \frac{1}{2}}{g}^{\mu \nu }{\partial }_{\mu }\overline{\varphi }{\partial }_{\nu }\overline{\varphi }-V_0{e}^{-{\lambda }_{\mathrm{DE}}\overline{\varphi }/M_{\mathrm{P}}}\right]+S_m[g_{\mu \nu },{\psi }_m].$$

(14)

Any residual non-minimal coupling is bounded to be completely negligible by fifth-force constraints. This formulation avoids the Brans–Dicke fifth force and keeps the effective Planck mass constant.

5.4. Natural Scale from the Apparent-Horizon Entropy Bound

The order of magnitude of $V_0$ is motivated by the covariant entropy bound [14]. For a spatially flat FLRW universe, the apparent-horizon radius is $R_{\mathrm{ah}}=H^{-1}$, with the Bekenstein–Hawking entropy

$$S_{\mathrm{ah}}={\displaystyle \frac{A_{\mathrm{ah}}}{4G}}={\displaystyle \frac{\pi }{G{H}^2}}=8{\pi }^2{\displaystyle \frac{M_{\mathrm{P}}^2}{H^2}},$$

where $M_{\mathrm{P}}^2=1/\left(8\pi G\right)$. The Gibbons–Hawking temperature is $T_{\mathrm{ah}}\simeq H/\left(2\pi \right)$. If an order-one fraction ${\alpha }_{\mathrm{foam}}$ of this horizon information is carried by the decohered foam, the associated energy density is

$${\rho }_{\mathrm{foam}}\sim {\alpha }_{\mathrm{foam}}{\displaystyle \frac{T_{\mathrm{ah}}{S}_{\mathrm{ah}}}{(4\pi /3)H^{-3}}}=3{\alpha }_{\mathrm{foam}}{M}_{\mathrm{P}}^2{H}^2.$$

Thus the natural infrared scale of the foam potential today is parametrically

$$V\left({\overline{\varphi }}_0\right)\sim M_{\mathrm{P}}^2{H}_0^2,$$

explaining the meV scale without fine-tuning. The exact numerical coefficient $\beta =3{\alpha }_{\mathrm{foam}}$ is not fixed by the entropy bound and is absorbed into the normalisation $V_0$, as made explicit in Eq. (13).

5.5. Background Equations

For a spatially flat FLRW metric, the Einstein and Klein–Gordon equations derived from (14) are

$$\begin{array}{cc}\hfill 3M_{\mathrm{P}}^2{H}^2& ={\rho }_m+{\rho }_r+{\displaystyle \frac{1}{2}}{\dot{\overline{\varphi }}}^2+V_0{e}^{-{\lambda }_{\mathrm{DE}}\overline{\varphi }/M_{\mathrm{P}}},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \ddot{\overline{\varphi }}+3H\dot{\overline{\varphi }}-{\displaystyle \frac{{\lambda }_{\mathrm{DE}}}{M_{\mathrm{P}}}}{V}_0{e}^{-{\lambda }_{\mathrm{DE}}\overline{\varphi }/M_{\mathrm{P}}}& =0.\hfill \end{array}$$

(16)

The minus sign in the Klein–Gordon equation reflects $V^{\prime }\left(\overline{\varphi }\right)=-({\lambda }_{\mathrm{DE}}/M_{\mathrm{P}})V\left(\overline{\varphi }\right)$. The dark-energy equation of state is

$$w_{\overline{\varphi }}={\displaystyle \frac{\frac{1}{2}{\dot{\overline{\varphi }}}^2-V\left(\overline{\varphi }\right)}{\frac{1}{2}{\dot{\overline{\varphi }}}^2+V\left(\overline{\varphi }\right)}}.$$

(17)

5.6. Initial Condition and Parameter Fixing

The CDST supplies a natural matching condition at the onset of the matter-dominated era:

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

(18)

The reference value ${\alpha }_i=1$ is used; shifts in ${\alpha }_i$ can be partially absorbed into $V_0$ because the potential is exponential.

The potential normalisation $V_0$ is fixed by requiring the solution of (15)–() to reproduce the present dark- energy density parameter ${\Omega }_{\mathrm{DE},0}=0.6889$ (Planck 2018). After this normalisation, the late-time dynamics is controlled by a single parameter, the slope ${\lambda }_{\mathrm{DE}}$. For the numerical benchmark we take ${\lambda }_{\mathrm{DE}}=0.65$, the value favoured by the semi-analytic RG estimate. A full exploration of the observational constraints on ${\lambda }_{\mathrm{DE}}$ is performed in the Supplementary Material.

5.7. Semi-Analytic Consistency Check

The effective mass of the field today is

$$m_{\mathrm{eff}}^2\equiv V^{\prime \prime }\left({\overline{\varphi }}_0\right)={\displaystyle \frac{{\lambda }_{\mathrm{DE}}^2}{M_{\mathrm{P}}^2}}{V}_0{e}^{-{\lambda }_{\mathrm{DE}}{\overline{\varphi }}_0/M_{\mathrm{P}}}\simeq {\displaystyle \frac{{\lambda }_{\mathrm{DE}}^2}{M_{\mathrm{P}}^2}}{\rho }_{\mathrm{DE},0}.$$

Using ${\rho }_{\mathrm{DE},0}=3{\Omega }_{\mathrm{DE},0}{M}_{\mathrm{P}}^2{H}_0^2$, one finds $m_{\mathrm{eff},0}\simeq {\lambda }_{\mathrm{DE}}\sqrt{3{\Omega }_{\mathrm{DE},0}}{H}_0\simeq 0.94H_0$ for ${\lambda }_{\mathrm{DE}}=0.65$. The field is thus naturally light on cosmological scales, explaining why it is Hubble-frozen until very recently.

5.8. Why Now? Decoherence and the Coincidence Problem

The collective field $\overline{\varphi }$ becomes classical only after a sufficient number of private-spacetime patches have decohered, $N_{\mathrm{decoh}}\sim {(M_{\mathrm{P}}/H)}^3\gg 1$, a condition amply satisfied throughout the late universe. The field remains frozen by Hubble friction until H becomes comparable to $m_{\mathrm{eff}}$. With $m_{\mathrm{eff},0}\simeq 0.94H_0$, significant rolling occurs around the present epoch. The coincidence problem is thus ameliorated: the same decoherence that makes the foam classical also links its effective mass to the present Hubble scale via the entropy bound.

5.9. Full Numerical Integration and Predictions

The system (15)–() has been integrated numerically with the benchmark parameters ${\lambda }_{\mathrm{DE}}=0.65$, ${\alpha }_i=1$, and Planck 2018 matter and radiation densities. The resulting equation of state is shown in Figure 1. The field is frozen with $w_{\overline{\varphi }}\simeq -1$ at high redshift and gently thaws at late times. A CPL fit over $0\le a\le 1$ yields

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

(19)

The linear growth factor $D\left(a\right)/a$ is suppressed by $3.2\%$ relative to $\Lambda$CDM for the same ${\Omega }_{m,0}$. A Fisher forecast (see Supplementary Material) shows that Stage-IV surveys (DESI, Euclid) will distinguish this model from a cosmological constant at high significance. The field satisfies $w_{\overline{\varphi }}\ge -1$ at all times, and the early dark-energy fraction is well within observational bounds.

6. Inflation from the Coherent Condensate

In the high-density, fully coherent phase of the GFT condensate, a different effective mode ${\sigma }_{\inf }\left(t\right)$ acts as the inflation. The GFT pentic vertex, evaluated on the high-density coherent condensate, produces tadpole diagrams that renormalise the scalar kinetic term. Because the condensate contains a macroscopic number of quanta, these tadpole contributions are collectively enhanced and can induce a large, effectively constant non-minimal coupling ${\xi }_{\inf }\gg 1$. This mechanism is analogous to the generation of a sizeable non-minimal coupling in Higgs-inflation models. A full renormalisation group computation within GFT is required to verify this picture, and the large value is treated as a working hypothesis in the present analysis. The effective Jordan-frame action is

$$S_{\mathrm{J}}=\int d^{4}x\sqrt{-g}\left[{\displaystyle \frac{1}{2}}\left(M_{\mathrm{P}}^2+{\xi }_{\inf }{\sigma }_{\inf }^2\right)R-{\displaystyle \frac{1}{2}}{(\partial {\sigma }_{\inf })}^2-{\displaystyle \frac{{\lambda }_{\inf }}{4}}{\sigma }_{\inf }^4\right],$$

(20)

where the bare conformal coupling $\xi =1/6$ is negligible compared to ${\xi }_{\inf }$.

6.1. Einstein Frame and Starobinsky Potential

We perform a conformal transformation to the Einstein frame solely as a computational device to read off the inflationary potential. Defining ${\tilde{g}}_{\mu \nu }={\Omega }^2{g}_{\mu \nu }$ with ${\Omega }^2=(M_{\mathrm{P}}^2+{\xi }_{\inf }{\sigma }_{\inf }^2)/M_{\mathrm{P}}^2$, the canonically normalised inflaton field $\chi$ satisfies

$${\displaystyle \frac{d\chi }{d{\sigma }_{\inf }}}=\sqrt{{\displaystyle \frac{1}{{\Omega }^2}}+6M_{\mathrm{P}}^2{\left({\displaystyle \frac{dln\Omega }{d{\sigma }_{\inf }}}\right)}^2}.$$

In the large-field limit ${\xi }_{\inf }{\sigma }_{\inf }^2\gg M_{\mathrm{P}}^2$, the relation between the Jordan field and the canonical field is

$$e^{\sqrt{2/3}\chi /M_{\mathrm{P}}}\simeq 1+{\displaystyle \frac{{\xi }_{\inf }{\sigma }_{\inf }^2}{M_{\mathrm{P}}^2}},{\sigma }_{\inf }\simeq {\displaystyle \frac{M_{\mathrm{P}}}{\sqrt{{\xi }_{\inf }}}}{e}^{\chi /\left(\sqrt{6}{M}_{\mathrm{P}}\right)}.$$

The Einstein-frame potential is the Starobinsky potential,

$$U\left(\chi \right)\simeq {\displaystyle \frac{{\lambda }_{\inf }{M}_{\mathrm{P}}^4}{4{\xi }_{\inf }^2}}{\left(1-e^{-\sqrt{2/3}\chi /M_{\mathrm{P}}}\right)}^2.$$

6.2. Slow-Roll Predictions and Amplitude Matching

The slow-roll parameters yield, for $N=60$ e-folds,

$$n_s\simeq 0.967,r\simeq 0.003,$$

in agreement with Planck 2018. The amplitude $A_s=2.1\times {10}^{-9}$ fixes

$${\displaystyle \frac{{\lambda }_{\inf }}{{\xi }_{\inf }^2}}\simeq 5\times {10}^{-10}.$$

For ${\lambda }_{\inf }\sim \mathcal{O}\left(1\right)$, ${\xi }_{\inf }\sim 4.5\times {10}^4$. Such a large value is assumed to arise from the high-density condensate dynamics; a microscopic derivation remains open.

6.3. Field Values and End of Inflation

Inflation ends at ${\chi }_{\mathrm{end}}\simeq 0.94M_{\mathrm{P}}$, corresponding to

$${\sigma }_{\inf ,\mathrm{end}}\simeq \sqrt{1.15}{\displaystyle \frac{M_{\mathrm{P}}}{\sqrt{{\xi }_{\inf }}}}\approx 1.07{\displaystyle \frac{M_{\mathrm{P}}}{\sqrt{{\xi }_{\inf }}}}.$$

For ${\xi }_{\inf }=5\times {10}^4$, ${\sigma }_{\inf ,\mathrm{end}}\simeq 0.0048M_{\mathrm{P}}$. The pivot value is ${\sigma }_{\inf ,*}\simeq 0.04M_{\mathrm{P}}$. The homogeneous condensate then oscillates and decays into a gas of GFT quanta, initiating the CDST.

7. The Coherent–Decoherent Spacetime Transition (CDST)

The transition from the coherent inflaton phase to the decohered dark-energy foam is governed by the microscopic GFT dynamics. Expanding the full GFT field as $\Phi =\phi +\widehat{\delta }$ with $\phi ={\sigma }_{\inf }\left(t\right){\Psi }_0$, the fluctuation field $\widehat{\delta }$ is decomposed into eigenmodes,

$$\widehat{\delta }(\mathbf{g},\mathbf{h};t)=\sum _I{u}_I(\mathbf{g},\mathbf{h}){\widehat{X}}_I\left(t\right).$$

After coarse-graining to a flat FLRW background, the canonically normalised mode amplitudes satisfy

$${\ddot{X}}_{I,k}+3H{\dot{X}}_{I,k}+{\omega }_{I,k}^2\left(t\right)X_{I,k}=0,{\omega }_{I,k}^2={\displaystyle \frac{k^2}{a^2}}+M_I^2\left(t\right),$$

(21)

with a time-dependent effective mass $M_I^2\left(t\right)$ whose precise form is given in the Supplementary Material. The foam sector consists of all modes with trivial internal gauge holonomies ($R_a=\mathbf{1}$). For the computation of Bogoliubov coefficients it is convenient to work with the frictionless rescaled variable $Y_{I,k}=a^{3/2}{X}_{I,k}$, which obeys ${\ddot{Y}}_{I,k}+{\Omega }_{I,k}^2\left(t\right)Y_{I,k}=0$ with ${\Omega }_{I,k}^2=k^2/a^2+M_I^2\left(t\right)-{\displaystyle \frac{3}{2}}\dot{H}-{\displaystyle \frac{9}{4}}{H}^2$.

7.1. Particle Production and the Foam Fraction

During the oscillatory phase of the inflaton ${\sigma }_{\inf }\left(t\right)$, parametric resonance produces quanta. The occupation numbers $n_{I,k}$ are obtained from the Bogoliubov coefficients (see Supplementary Material for the detailed formalism). The energy density stored in the foam at the end of reheating is

$${\rho }_{\mathrm{foam}}^{\mathrm{prod}}\left(t_{\mathrm{reh}}\right)=\sum _{I\in \mathrm{foam}}{g}_I\int {\displaystyle \frac{{\mathrm{d}}^{3}k}{{\left(2\pi \right)}^3{a}_{\mathrm{reh}}^3}}\sqrt{{\displaystyle \frac{k^2}{a_{\mathrm{reh}}^2}}+M_I^2\left(t_{\mathrm{reh}}\right)}{n}_{I,k}.$$

The effective late-time foam fraction includes a decoherence factor $\mathcal{D}\in [0,1]$, $f_{\mathrm{foam}}^{\mathrm{eff}}=\mathcal{D}{\rho }_{\mathrm{foam}}^{\mathrm{prod}}/{\rho }_{\mathrm{total}}^{\mathrm{prod}}$.

7.2. Emergence of the Dark-Energy Collective Coordinate

Long after reheating, the decohered foam is a dilute gas of Planck-sized, gauge-singlet quanta. Its long-wavelength state is captured by a single infrared collective coordinate $\overline{\varphi }\left(t\right)$, defined as the canonically normalised homogeneous mode of the foam field (see Supplementary Material). A crucial property of this collective coordinate is that it is minimally coupled to gravity: an explicit projection calculation shows that the microscopic non-minimal operator $-{\xi R\left|\delta \Phi \right|}^2$, when projected onto the gauge-singlet homogeneous foam mode, vanishes identically if the foam sector is orthogonal to the original condensate modulus. Consequently, ${\xi }_{\mathrm{eff}}=0$, and the effective action for $\overline{\varphi }$ is the minimally coupled one of Sec. Section 5. This result removes any tree-level fifth force and keeps the effective Planck mass constant, without fine-tuning.

The exponential potential $V\left(\overline{\varphi }\right)=V_0{e}^{-{\lambda }_{\mathrm{DE}}\overline{\varphi }/M_{\mathrm{P}}}$ is adopted as a well-motivated effective ansatz. The CDST also supplies the initial condition $\overline{\varphi }\left(t_{\mathrm{reh}}\right)={\alpha }_i{M}_{\mathrm{P}}$, $\dot{\overline{\varphi }}\left(t_{\mathrm{reh}}\right)\simeq 0$, with ${\alpha }_i=\mathcal{O}\left(1\right)$, which follows from the Planck-scale amplitude of each foam quantum.

8. Unified Effective Action and Cosmological Phases

The entire cosmological history of the model is encoded in the single effective action

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Here ${\sigma }_{\inf }$ is the high-density inflaton mode (non-minimally coupled, with ${\xi }_{\inf }\gg 1$) and $\overline{\varphi }$ is the collective coordinate of the decohered virtual foam (minimally coupled). The two scalar fields are effective descriptors of the same underlying GFT field $\Phi$ in two different regimes: the coherent condensate and the decohered foam. The CDST provides the transition between them. After fixing the normalisation $V_0$ by the observed dark-energy density, the model has a single dynamical parameter, the slope ${\lambda }_{\mathrm{DE}}$, which controls the thawing dynamics. The model is thus a one-parameter extension of $\Lambda$CDM, sharply testable with Stage-IV surveys.

9. Theoretical Consistency

9.1. Absence of Fifth Forces

Because the dark-energy scalar is minimally coupled in the physical frame, there is no tree-level Brans–Dicke-type fifth force. The private-spacetime structure further suppresses any residual direct foam-matter interactions.

9.2. Quantum Stability

The exponential form is technically natural if $\overline{\varphi }$ is a dilaton-like field with an approximate scaling symmetry, and provided it has no unsuppressed direct couplings to Standard Model masses. One-loop corrections are Planck-suppressed and small.

9.3. Initial Condition Naturalness

The CDST provides ${\overline{\varphi }}_i={\alpha }_i{M}_{\mathrm{P}}$ with ${\alpha }_i=\mathcal{O}\left(1\right)$; shifts in ${\alpha }_i$ can be partially absorbed into $V_0$, making the late-time evolution robust.

9.4. Status of the Main Ingredients

Table 1 summarises the status of the model’s main ingredients.

10. Conclusions

We have presented a predictive dark-energy model embedded in the GFT condensate framework. The homogeneous condensate is irrelevant; dark energy originates from the incoherent virtual foam, modelled by a minimally coupled scalar with an exponential potential. The exponential potential is an effective ansatz, the minimal coupling is an effective-frame choice, and the scale $V_0$ is naturally parametrically $\sim M_{\mathrm{P}}^2{H}_0^2$, explaining the meV scale. After fixing $V_0$ to observations, the single dynamical parameter ${\lambda }_{\mathrm{DE}}$ controls the thawing dynamics. The model predicts a specific, testable deviation from $\Lambda$CDM, with $w_0\simeq -0.86$, $w_a<0$, and a few-percent suppression of matter growth. Stage-IV surveys will confirm or rule out this scenario, providing a concrete bridge from quantum gravity to the accelerating cosmos.

A complete first-principles determination of ${\lambda }_{\mathrm{DE}}$ and $V_0$ from the GFT dynamics remains an open problem. The detailed derivations of all results presented here, as well as additional applications (regular black holes, dark matter), are provided in the accompanying Supplementary Material.