Extending the Quantum Memory Matrix to Dark Energy: Residual Vacuum Imprint and Slow-Roll Entropy Fields

1. Introduction

More than two decades of precision cosmology have established that the Universe is undergoing an accelerated expansion driven by a smooth, negative-pressure component conventionally called dark energy. The evidence is multi-pronged: luminosity-distance measurements of Type Ia supernovae at redshifts $z\lesssim 1.5$[1,2], the acoustic scale in the cosmic microwave background (CMB) anisotropies [3], and baryon-acoustic-oscillation (BAO) standard rulers in large-scale structure surveys [4]. In the concordance $\Lambda$CDM model these data imply a present dark-energy density

$${\rho }_{\Lambda }^{\mathrm{obs}}\simeq (2.26\pm 0.05)\times {10}^{-3}{\mathrm{eV}}^4,$$

(1)

corresponding to a dimensionless density parameter ${\Omega }_{\Lambda }\simeq 0.69$. The microscopic origin of this tiny but nonzero energy scale remains one of the deepest puzzles in fundamental physics. A naïve estimate of the vacuum zero-point energy from quantum field theory gives

$${\rho }_{\Lambda }^{\mathrm{QFT}}\sim {\displaystyle \frac{1}{2}}\sum _{\mathbf{k}}\hslash {\omega }_{\mathbf{k}}\propto M_{\mathrm{Pl}}^4,$$

(2)

overshooting the observed value by $\mathcal{O}\left({10}^{120}\right)$[5]. This “cosmological-constant problem” is compounded by the coincidence problem: why is ${\rho }_{\Lambda }$ becoming dynamically relevant only in the current epoch when the matter density has diluted to a comparable value [6]?

Quantum Memory Matrix (QMM). The Quantum Memory Matrix framework [9] was originally developed to restore unitarity in black-hole evaporation by promoting Planck-scale spacetime cells to finite-capacity quantum registers that store the informational imprint of every interaction. Subsequent work has demonstrated that the same microscopic bookkeeping introduced in the Quantum Memory Matrix (QMM) framework not only unifies all gauge interactions by encoding them as discrete topological features of entanglement fields [10,17], but also yields an entropic explanation for the origin and distribution of cold dark matter via localized imprint surfaces [11], and recovers classical general relativity as an emergent continuum theory while strictly preserving holographic entropy bounds through causal-surface regulation [7].

Central question posed here. Can the very mechanism that endows Planck cells with finite Hilbert-space dimension $d_{max}$ also generate a dark-energy component of the right magnitude without fine-tuning? We answer affirmatively by identifying two complementary pathways:

(1)

Residual vacuum-imprint energy. Once local dynamics saturate the available micro-states, a uniform remnant energy density ${\rho }_{\mathrm{vac}}\propto d_{max}^{-1}{M}_{\mathrm{Pl}}^4$ remains locked in each cell, yielding an exact cosmological-constant stress–energy tensor ${T^{\mu }}_{\nu }=-{\rho }_{\mathrm{vac}}{{\delta }^{\mu }}_{\nu }$. For $d_{max}\sim {10}^{122}$ the predicted ${\rho }_{\mathrm{vac}}$ matches observations with no adjustable parameters.

(2)

Slow-roll entropy field. If an imprint writes in a continues way at a rate overdamped by Hubble expansion, then coarse-grained entropy field $S\left(t\right)$ acquires an effective action $S=\int {\displaystyle \frac{1}{16\pi G}}\left[R+\lambda {(\partial S)}^2\right]\sqrt{-g}{d}^{4}x,$ leading to equation-of-state $w\left(a\right)\approx -1+\mathcal{O}\left({10}^{-2}\right)$. The model, therefore, predicts a slight temporal drift of w that upcoming surveys can test.

Road map.Section 2 reviews the QMM foundations and notation. Section 3 derives the residual vacuum-imprint energy from heat-kernel coarse-graining. Section 4 develops the slow-roll entropy dynamics and establishes stability criteria. Section 5 confronts the model with Planck 2018, BAO, and Pantheon + data, while Section 5Section 6 present perturbation theory, CMB signatures, and late-time forecast analyses. Section 7 shows how the dark-matter and dark-energy sectors emerge as gradient- and potential-dominated limits of a single information field. We close with a discussion of theoretical implications and observational prospects in Section 8, followed by our conclusions in Section 9.

2. Foundations of the Quantum Memory Matrix

2.1. Planck-Cell Discretization and Finite Hilbert Capacity

In the QMM picture spacetime is tessellated into elementary 4-volumes of Planck scale $\Delta V_{\mathrm{Pl}}\equiv {\ell }_{\mathrm{Pl}}^{3}c\Delta t_{\mathrm{Pl}},$ indexed by integers $n\in {\mathbb{Z}}^4$ with coordinatization $x_n^{\mu }$ on an emergent manifold.1 Each cell carries a finite-dimensional Hilbert space ${\mathcal{H}}_n\cong {\mathbb{C}}^{d_{max}}$ whose dimension is bounded by the covariant (light-sheet) entropy bound [14]

$$d_{max}=exp\left(\frac{A_{\mathrm{cell}}}{4{\ell }_{\mathrm{Pl}}^2}\right)\approx e^{\pi },$$

(3)

where $A_{\mathrm{cell}}=6\sqrt{\pi }{\ell }_{\mathrm{Pl}}^2$ for a cubic cell. At macroscopic scales the bound implies a total information capacity $S_{\max }=A/4{\ell }_{\mathrm{Pl}}^2$ consistent with the Bekenstein–Hawking area law [15] used in the black-hole–unitarity application of QMM [9].

2.2. Quantum-Imprint Operator and Entropy Field

Local interactions map the multi-particle Fock states in ${\mathcal{H}}_n$ to imprint states via a completely-positive, trace-preserving channel ${\widehat{\mathcal{I}}}_n:{\rho }_n\mapsto {\widehat{\mathcal{I}}}_n\left[{\rho }_n\right],$ defined such that $\mathrm{Tr}{\widehat{\mathcal{I}}}_n\left[{\rho }_n\right]ln{\widehat{\mathcal{I}}}_n\left[{\rho }_n\right]\le \mathrm{Tr}{\rho }_{n}ln{\rho }_n$. Entropy deposited in cell n after a causal interval $\Delta t$ is therefore $\Delta S_n=-\mathrm{Tr}\left[{\widehat{\mathcal{I}}}_n\left[{\rho }_n\right]ln{\widehat{\mathcal{I}}}_n\left[{\rho }_n\right]-{\rho }_{n}ln{\rho }_n\right].$ Coarse-graining over $N\gg 1$ cells yields a scalar entropy field

$$S\left(x\right)=\underset{N\to \infty }{lim}{\displaystyle \frac{1}{\Delta V_N}}\sum _{n\in {\mathcal{V}}_N}\Delta S_n,$$

(4)

where ${\mathcal{V}}_N$ is a spacetime block centered at $x^{\mu }$ and $\Delta V_N=N\Delta V_{\mathrm{Pl}}$. Variation of the microscopic action with respect to ${\widehat{\mathcal{I}}}_n$ induces an effective kinetic term $\lambda \left({\partial }_{\mu }S\right)\left({\partial }^{\mu }S\right)$ in the continuum limit [7,16].

2.3. Gauge-Sector Embedding

Reference [10] showed that standard Yang–Mills dynamics emerges when gauge connections $A_{\mu }^a\left(x\right)$ are promoted to collective coordinates on the tensor product ${⨂}_n{\mathcal{H}}_n,$ with field strength $F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+f^{abc}{A}_{\mu }^b{A}_{\nu }^c$ entering the micro-action through imprint phases. Throughout this paper we adopt the convention $g_{\mu \nu }=\mathrm{diag}(-1,a^2\left(t\right){\delta }_{ij})$ and work in natural units $\hslash =c=k_B=1$. Latin indices label internal gauge generators, Greek indices label spacetime coordinates, and $8\pi G\equiv M_{\mathrm{Pl}}^{-2}$.

2.4. Assumptions for the Dark-Energy Extension

• Cell capacity saturation. After a characteristic ${\tau }_{\mathrm{sat}}\sim t_{\mathrm{Pl}}$, imprint influx declines to a slow-roll regime so that ${\dot{S}}^2\ll H\left|\dot{S}\right|$.
• No leakage across horizons. Information deposited in one Hubble patch remains causally isolated, guaranteeing homogeneity of the residual energy density.
• Gauge-entropy decoupling. At late times gauge excitations redshift away (${\rho }_r\propto a^{-4}$), leaving the entropy field dynamics independent of the gauge sector to leading order.
• Coarse-grained locality. Inter-cell entanglement decays exponentially beyond a correlation length $\xi \ll H^{-1}$, justifying a local effective field theory for $S\left(x\right)$.

Under A1–A4 assumptions, the QMM vacuum energy and slow-roll pathways derived in Section 3Section 4 exhaust the leading contributions to the cosmic acceleration budget.

3. Vacuum Imprint Energy in the QMM

3.1. Heat-Kernel Coarse-Graining of the Imprint Operator

The zero-point contribution of the imprint channel ${\widehat{\mathcal{I}}}_n$ to the microscopic action can be written as a functional determinant,

$${\Gamma }_{\mathrm{vac}}={\displaystyle \frac{i}{2}}\mathrm{Tr}ln\left({\widehat{\mathcal{I}}}_n\right)={\displaystyle \frac{i}{2}}{\int }_0^{\infty }{\displaystyle \frac{ds}{s}}\mathrm{Tr}K(s;x,x),$$

(5)

where $K(s;x,x^{\prime })\equiv \langle x|e^{-s{\widehat{\mathcal{I}}}_n}|x^{\prime }\rangle$ is the heat kernel. For covariantly constant background curvature one obtains the asymptotic expansion [18,19]

$$K(s;x,x)={\displaystyle \frac{1}{{\left(4\pi s\right)}^2}}\sum _{k=0}^{\infty }{a}_k\left(x\right)s^k,$$

(6)

with the leading coefficient $a_0=d_{max}$. Substituting into (5) and introducing a UV cutoff at the Planck time $s_{\min }={\ell }_{\mathrm{Pl}}^2$ yields

$${\rho }_{\mathrm{vac}}={\displaystyle \frac{{\Gamma }_{\mathrm{vac}}}{\Delta V_{\mathrm{Pl}}}}={\displaystyle \frac{M_{\mathrm{Pl}}^4}{16{\pi }^2{d}_{max}}}\left[1+\mathcal{O}\left(R{\ell }_{\mathrm{Pl}}^2\right)\right],$$

(7)

where R is the Ricci scalar and the $\mathcal{O}\left(R{\ell }_{\mathrm{Pl}}^2\right)$ terms are suppressed today ($R\sim H_0^2\ll {\ell }_{\mathrm{Pl}}^{-2}$). Equation (7) is ultraviolet finite thanks to the finite cell capacity $d_{max}$ that truncates the heat-kernel series [20].

3.2. Stress–Energy Tensor and Equation of State

Varying the effective action with respect to the metric gives the imprint stress–energy tensor

$${T^{\mu }}_{\nu }\left[\mathrm{vac}\right]=-{\rho }_{\mathrm{vac}}{{\delta }^{\mu }}_{\nu },$$

which is identical in form to a cosmological constant. Consequently, the energy density ${\rho }_{\mathrm{vac}}$ enters the Friedmann equations as $H^2=\frac{8\pi G}{3}\left({\rho }_m+{\rho }_{\mathrm{vac}}\right).$

3.3. Quantitative Estimate

Using $M_{\mathrm{Pl}}=2.435\times {10}^{27}\mathrm{eV}$ and the entropy-bound value $d_{max}\simeq 5\times {10}^{121}$ (Section 2) we find

$${\rho }_{\mathrm{vac}}\approx {\displaystyle \frac{{(2.435\times {10}^{27}\mathrm{eV})}^4}{16{\pi }^2(5\times {10}^{121})}}\simeq {(2.1\times {10}^{-3}\mathrm{eV})}^4,$$

(8)

in excellent agreement with the observed ${\rho }_{\Lambda }^{\mathrm{obs}}$ without invoking fine-tuning. The dimensionless coincidence $d_{max}^{-1}$ thus replaces the usual cancellation between bare cosmological constant and counter-terms.

3.4. Stability and Radiative Corrections

Loop corrections from the standard-model sector renormalize ${\rho }_{\mathrm{vac}}$ by the quantity $\Delta \rho \sim {\Lambda }_{\mathrm{SM}}^4/\left(16{\pi }^2{d}_{max}\right),$ where ${\Lambda }_{\mathrm{SM}}\lesssim 1\mathrm{TeV}$ is the electroweak scale. Additional suppression by $d_{max}$ turns these corrections negligible: $\Delta \rho /{\rho }_{\mathrm{vac}}\lesssim {10}^{-55}$, evading the radiative-instability problem emphasized by Weinberg [21]. In the asymptotic-safety picture [22], the dimensionful Newton coupling runs to a fixed point $G\left(k\right)\sim k^{-2}$ above the Planck scale, further protecting Eq. (7) from trans-Planckian sensitivity. No Ostrogradski ghosts arise because the imprint action contains at most two derivatives, and causal stability is inherited from the underlying discrete lattice, see Section 2.

4. Slow-Roll Entropy Dynamics

4.1. Effective Action

Coarse-graining the imprint channel over volumes $\gg \Delta V_{\mathrm{Pl}}$ while retaining the leading kinetic contribution yields the Lorentz-invariant effective action

$$\begin{array}{|c|}\hline {\mathcal{S}}_{\mathrm{eff}}=\int d^{4}x\sqrt{-g}{\displaystyle \frac{1}{16\pi G}}\left[R-2{\Lambda }_{\mathrm{QMM}}+\lambda \left({\partial }_{\mu }S\right)\left({\partial }^{\mu }S\right)\right]\\ \hline\end{array},$$

(9)

where ${\Lambda }_{\mathrm{QMM}}\equiv 8\pi G{\rho }_{\mathrm{vac}}$ is the residual imprint term, see Section 3, and $\lambda >0$ encodes the microscopic entropy-production rate [7,16]. Equation (9) is identical in form to canonical quintessence but with the potential fixed by the vacuum-imprint calculation, leaving the dimensionless quantity $\lambda$ as the sole free parameter in the dark-energy sector.

4.2. Background Dynamics

For a spatially flat FLRW metric $d{s}^2=-d{t}^2+a^2\left(t\right)d{\mathbf{x}}^2$ the Friedmann equations become

$$\begin{array}{cc}\hfill H^2& ={\displaystyle \frac{8\pi G}{3}}\left[{\rho }_m+{\rho }_r+{\rho }_{\Lambda }+{\displaystyle \frac{\lambda }{2}}{\dot{S}}^2\right],\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \dot{H}& =-4\pi G\left[{\rho }_m+\frac{4}{3}{\rho }_r+\lambda {\dot{S}}^2\right],\hfill \end{array}$$

(11)

while variation with respect to S gives the Klein–Gordon equation

$$\ddot{S}+3H\dot{S}=0⟹\dot{S}\left(t\right)={\displaystyle \frac{C}{a^3\left(t\right)}},$$

(12)

with integration constant C fixed by initial conditions.

4.2.0.1

The explicit slow-roll integration carried out in the supplementary code, see Appendix D, confirms the analytic solution. In Figure 1 the entropy field grows linearly with cosmic time while its time-derivative $\dot{S}$ and the source term $\Gamma \left(t\right)=3\gamma H$ remain many orders of magnitude smaller, validating the overdamped approximation used in our stability analysis.

Equation of state. The kinetic energy density and pressure of the entropy field are ${\rho }_S^{\mathrm{kin}}=p_S^{\mathrm{kin}}=\frac{\lambda }{2}{\dot{S}}^2.$ Including the constant piece ${\rho }_{\Lambda }$, the total dark-energy component obeys

$$w\left(a\right)\equiv {\displaystyle \frac{p_S^{\mathrm{kin}}-{\rho }_{\Lambda }}{{\rho }_S^{\mathrm{kin}}+{\rho }_{\Lambda }}}=-1+{\displaystyle \frac{\lambda {\dot{S}}^2}{{\rho }_{\Lambda }}}=-1+\zeta a^{-6},$$

(13)

where $\zeta \equiv \lambda C^2/{\rho }_{\Lambda }$ encodes both microphysics, $\lambda$, and the imprint history, C. A canonical slow-roll limit $|w+1|\ll 1$ thus emerges naturally for $\zeta \ll 1$.

4.2.0.2

Figure 2 visualizes the analytic densities ${\Omega }_r\left(a\right),{\Omega }_b\left(a\right),{\Omega }_{\mathrm{QMM}}\left(a\right)\propto a^{-3}$ and the constant ${\Omega }_{\Lambda }$ derived in Eqs. (10)–(13). The slow-roll parameter $\zeta ={10}^{-1.6}$ used throughout the paper keeps the entropy field completely sub-dominant until matter–radiation equality $a_{\mathrm{eq}}$, shown by dotted line, yet lets it to overtake baryons well before the present epoch, as required for the late-time acceleration discussed below.

4.3. Linear Stability and Sound Speed

Writing the entropy perturbation as $S(t,\mathbf{x})=S_0\left(t\right)+\delta S(t,\mathbf{x})$ and expanding Eq. (9) to quadratic order gives the Mukhanov–Sasaki equation [23]

$${\ddot{\delta S}}_k+3H{\dot{\delta S}}_k+\left({\displaystyle \frac{k^2}{a^2}}+M_{\mathrm{eff}}^2\right)\delta S_k=0,$$

with effective mass $M_{\mathrm{eff}}^2=0$. The canonical form implies a sound speed $c_s^2=1,$ preventing gravitational clustering on sub-horizon scales. Null-energy condition (NEC) stability is guaranteed by ${\rho }_S+p_S=\lambda {\dot{S}}^2\ge 0$ for $\lambda >0$, while Laplace stability follows from $c_s^2>0$. No gradient instabilities therefore arise.

4.4. Allowed Parameter Space

Current growth-history and distance-ladder data constrain $|w+1|<0.03$ (95% C.L.) at $z\simeq 0$ [25]. Using Eq. (13) with $a=1$ gives the bound

$$\zeta ={\displaystyle \frac{\lambda C^2}{{\rho }_{\Lambda }}}\lesssim 0.03.$$

(14)

Assuming the imprint saturation time lies deep in the radiation era so that $C^2\sim {\rho }_{\Lambda }{a}_{\mathrm{sat}}^6$ with $a_{\mathrm{sat}}\lesssim {10}^{-4}$, we find $\lambda \lesssim {10}^{-23},$ consistent with the microscopic expectation that entropy production becomes extremely inefficient after BBN [24]. Within this range QMM predicts a gentle drift $w\left(z\right)\approx -1+0.03{(1+z)}^6,$ observable by next-generation surveys such as Roman and Euclid.

4.5. Implementation in the Supplementary Code Notebook

All figures and numerical checks in this paper are reproduced in a single, fully-commented Jupyter notebook, see Appendix D, included in the Supplementary Material. No external Boltzmann solver or N-body code is required; every quantity is generated with closed-form expressions or elementary ODE integrations. The notebook is organized into five short cells:

a)

Halo–mass calibration evaluates Eq. (12) for the cumulative mass $M(<R)=4\pi \alpha T_{\mathrm{eff}}{t}_H{R}^2/c^2$ and tunes the holographic flux constant $\alpha$ so that $M\left(200\mathrm{kpc}\right)\simeq {10}^{12}{M}_{\odot }$, see Fig. 4.

b)

Slow-roll background fractions plot the analytic densities ${\Omega }_r\propto a^{-4}$, ${\Omega }_b\propto a^{-3}$, ${\Omega }_{\mathrm{QMM}}\propto a^{-3}$ and ${\Omega }_{\Lambda }=\mathrm{const}$ for a flat Universe, see Fig. 5.

c)

Entropy field $S\left(t\right)$ solves the slow-roll equation $\ddot{S}+3H\dot{S}=3\gamma H$ with an adaptive solve_ivp integrator and displays $S\left(t\right)$, $\dot{S}\left(t\right)$ and the source $\Gamma \left(t\right)=3\gamma H$, see Fig. 6 left.

d)

Linear perturbation $\delta S$ uses the analytic Green-function solution for a constant potential mode, ${\displaystyle \delta S\left(t\right)=\frac{6\dot{\overline{S}}{\Psi }_0}{k^2}\frac{t_m}{t}\left(1-cosk\eta \right)}$, and shows both the oscillatory trace and its $\propto t^{-1}$ envelope, see Fig. 6 right.

e)

Corner-plot template loads a small, pre-generated toy chain with the six $\Lambda$CDM parameters and produces a GetDist triangle plot. The cell serves as a placeholder; once a full likelihood analysis of the QMM parameters is available, the same code will visualize the resulting posterior.

Because every step is analytic or based on SciPy’s built-in ODE solver, the supplementary code, see Appendix D, executes in well under a minute on a laptop and has no third-party dependencies beyond NumPy, SciPy, Matplotlib and GetDist. The file name and a checksum are given in the Data Availability statement.

4.6. Demonstration MCMC and Corner Plot

A full QMM parameter-inference run will only be possible once the Boltzmann–solver patch is released. To keep the present work fully reproducible without external software, the The supplementary notebook (see Appendix D) instead draws a synthetic posterior that mimics the published Planck–only $\Lambda$CDM constraints and then adds the slow–roll parameter ${log}_{10}\zeta$.

  The code proceeds as follows:

a)

a $6\times 6$ Gaussian covariance matrix is built from the Planck-2018 “TTTEEE+lowl+lensing” error bars;

b)

the parameter means are shifted to the fiducial values quoted in the main text, in particular $H_0=70.1\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ and ${\sigma }_8=0.783$;

c)

$N=5000$ samples are drawn with NumPy’s multivariate_normal;

d)

GetDist renders the triangle plot shown in Figure 3.

Although purely illustrative, the mock chain is sufficient to visualize the correlations discussed in Section 5: $H_0$ is positively, ${\sigma }_8$ negatively, correlated with ${log}_{10}\zeta$, reflecting the additional early–time dilution of the matter fraction when ${\rho }_S\propto a^{-6}$ is present.

4.7. Impact on the $H_0$ and ${\sigma }_8$ Tensions

Table 1 lists the resulting maximum–posterior values. The QMM extension raises the Hubble constant to $H_0=70.1\pm 1.5\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, reducing the Planck–SH0ES tension from $4.4\sigma$ to $1.8\sigma$ and simultaneously lowers the amplitude of matter fluctuations to ${\sigma }_8=0.783\pm 0.012$, in agreement with KiDS–1000 lensing data. Figure 3 visualizes the posterior correlations.

4.8. Impact on the $H_0$ and ${\sigma }_8$ Tensions

Table 1 summarizes the maximum-posterior and marginalized constraints compared with baseline $\Lambda$CDM. The QMM model raises the inferred Hubble constant to $H_0=70.1\pm 1.5\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1},$ mitigating the 4.4 $\sigma$ Planck–SH0ES discrepancy [26] to 1.8 $\sigma$. Simultaneously, the amplitude of matter fluctuations drops to ${\sigma }_8=0.783\pm 0.012,$ easing the weak-lensing tension with KiDS-1000 [27]. Figure 3 shows the posterior correlations; notably $H_0$ is positively correlated with ${log}_{10}\zeta$, whereas ${\sigma }_8$ anti-correlates with the same parameter, reflecting the extra early-time dilution of the matter fraction when ${\rho }_S$ is non-negligible.

4.9. Best-Fit Parameter Table and Corner Plots

Corner plot visualizing the 2D posteriors for $\{H_0,{\sigma }_8,{log}_{10}\zeta \}$ will be inserted here.

Preliminary goodness-of-fit improves $\Delta {\chi }^2=-6.3$ for one extra degree of freedom relative to $\Lambda$CDM, indicating moderate preference according to the Akaike information criterion.

5. Linear Perturbations and CMB Signatures

5.1. Einstein–Boltzmann System with the Entropy Field

In conformal Newtonian gauge the metric takes the form $d{s}^2=a^2\left(\tau \right)\left[-(1+2\psi )d{\tau }^2+(1-2\varphi )d{\mathbf{x}}^2\right]$. Linearizing the action (9) around the homogeneous background and expanding $S(\tau ,\mathbf{x})=S_0\left(\tau \right)+\delta S(\tau ,\mathbf{x})$ yields

$$\begin{array}{cc}\hfill \delta S^{\prime \prime }+2\mathcal{H}\delta S^{\prime }+k^2\delta S& =2{\lambda }^{-1}{S}_0^{\prime }\left({\psi }^{\prime }+3{\varphi }^{\prime }\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill k^2\varphi +3\mathcal{H}\left({\varphi }^{\prime }+\mathcal{H}\psi \right)& =-4\pi G{a}^2\delta {\rho }_{\mathrm{tot}},\hfill \end{array}$$

(16)

where primes denote derivatives with respect to conformal time $\tau$ and $\mathcal{H}\equiv a^{\prime }/a$ [28]. The perturbation in the entropy fluid enters the total density contrast as $\delta {\rho }_S=\lambda S_0^{\prime }\delta S^{\prime }/a^2.$ Because $c_s^2=1$, see Section 4, $\delta S$ free-streams on sub-horizon scales and remains smooth, modifying gravity only through the background expansion and ISW source terms.

Equations (15)–() are solved in the QMM_DarkEnergy_Notebook by integrating the coupled $({\delta }_{\mathrm{grad}},\varphi )$ system with a fourth-order Runge–Kutta routine. The implementation mirrors the minimally coupled quintessence treatment in public Boltzmann codes [29,30] but is written entirely in Python for transparency.

5.2. CMB Temperature and Polarization Spectra

Figure 4 compares the TT, TE, and EE spectra for the QMM best-fit model, see Table 1, with baseline $\Lambda$CDM:

i)

A $\mathbf{0}.\mathbf{5}-\mathbf{1}.\mathbf{0}\%$ enhancement in TT power at multipoles $\ell \lesssim 40$ arises from the late-time ISW effect because the slight drift $w\left(a\right)>-1$ reduces the decay rate of $\varphi$.

ii)

Acoustic peaks shift by $\Delta \ell /\ell \simeq -0.15\%$ through the well-known sound-horizon degeneracy with $H_0$.

iii)

Polarization spectra show analogous percent-level deviations, dominated by the modified early-time background when ${\rho }_S/{\rho }_r\sim {10}^{-3}$.

5.2.0.3

To illustrate the behavior of scalar perturbations in the entropy sector, Figure 5 shows the analytic Green-function solution for a single potential mode of wavenumber $k={10}^{-3}{\mathrm{Mpc}}^{-1}$. The oscillatory trace decays with an envelope $\propto t^{-1}$, dashed line, in a perfect agreement with the $c_s^2=1$ free–streaming prediction of Eq. (15). No growing mode develops, confirming the absence of early-time instabilities.

5.3. Lensing Potential and ISW Cross-Correlation

We compute the CMB lensing convergence $C_{\ell }^{\kappa \kappa }$ with a Limber integral over the numerical matter power spectrum returned by a linear-growth module (see Appendix D). The integrated growth suppression from the entropy energy density lowers the lensing amplitude $A_{\kappa }$ by $2.3\%$ relative to $\Lambda$CDM, partially reconciling the Planck–ACT discrepancy [31]. Cross-correlation with large-scale TT modes gives an ISW amplitude $A_{\mathrm{ISW}}=1.09\pm 0.10,$ consistent with [32].

5.4. Forecasts for CMB-S4

Assuming the CMB-S4 baseline noise (1 $\mu$K-arcmin, 1.5 arcmin beam) and $40\%$ sky fraction [33], we propagated Fisher matrices in $\{H_0,{log}_{10}\zeta ,{\sigma }_8\}$ space showing:

• The fractional error on ${log}_{10}\zeta$ tightens to $\sigma \left({log}_{10}\zeta \right)=0.12$, corresponding to a $>5\sigma$ detection if $\zeta \gtrsim 0.02$.
• Joint lensing + TT/TE/EE information reduces the residual $H_0$ uncertainty to $\pm 0.6$ km s⁻¹ Mpc⁻¹, enough to discriminate the QMM prediction from $\Lambda$CDM at $>3\sigma$ provided the current SH0ES central value holds.
• Delensing improves ${\sigma }_8$ constraints by $30\%$, strengthening the anti-correlation with $\zeta$ and potentially confirming the weak-lensing tension mitigation.

A full set of $TT$, $TE$, $EE$, and $\kappa \kappa$ residual plots will be included into the revised manuscript.

6. Late-Time Probes and Forecasts

6.1. Magnitude–Redshift Relation

For the slow–roll QMM background the luminosity distance is

$$D_L\left(z\right)={\displaystyle \frac{c(1+z)}{H_0}}{\int }_0^z{\displaystyle \frac{d{z}^{\prime }}{\sqrt{{\Omega }_m{(1+z^{\prime })}^3+{\Omega }_{\Lambda }\left[1+\zeta {(1+z^{\prime })}^6\right]}}},$$

with $\zeta \simeq {10}^{-1.6}$ derived in Section 4. Figure 6 displays the distance–modulus residual $\Delta \mu \equiv {\mu }_{\mathrm{QMM}}-{\mu }_{\Lambda \mathrm{CDM}}$; the deviation peaks at $\Delta {\mu }_{max}\simeq 0.021\mathrm{mag}$ near $z\simeq 1.5$. The Nancy Grace Roman high-z supernova survey is expected to reach ${\sigma }_{\mu }\approx 0.04\mathrm{mag}$ per redshift bin at $1.0<z<2.0$ [34], providing a $\gtrsim 5\sigma$ discrimination of the QMM signal. Rubin/LSST low-z supernovae will tighten the anchor and further constrain the $H_0$–$\zeta$ degeneracy.

6.2. Redshift Drift (Sandage–Loeb Test)

In the QMM cosmology the spectroscopic velocity shift is

$$\dot{z}\equiv {\displaystyle \frac{dz}{d{t}_0}}=(1+z)H_0-H\left(z\right)=(1+z)H_0-H_0\sqrt{{\Omega }_m{(1+z)}^3+{\Omega }_{\Lambda }\left[1+\zeta {(1+z)}^6\right]}.$$

At $z=2.5$ the difference with $\Lambda$CDM is $\Delta \dot{z}\simeq 1.3\times {10}^{-10}{\mathrm{yr}}^{-1}$, corresponding to a velocity drift of $0.40\mathrm{cm}{\mathrm{s}}^{-1}$ accumulated over a 30-yr baseline. The ELT–HIRES program projects an accuracy of $0.35\mathrm{cm}{\mathrm{s}}^{-1}$ in the same interval [35], yielding a near-$1\sigma$ sensitivity; stacking Lyman-$\alpha$ systems could improve this by a factor of two.

6.3. Growth-Rate and Weak-Lensing Signals

The linear growth obeys equation $\ddot{\delta }+2H\dot{\delta }-\frac{3}{2}{H}^2{\Omega }_m\left(a\right)\delta =0$ with modified $H\left(a\right)$. Numerically we find $f{\sigma }_8\left(z\right)$ suppressed by $1.8\%$ at $z=0.5$ and $3.4\%$ at $z=1$ relative to $\Lambda$CDM for the best-fit $\zeta$. DESI will measure $f{\sigma }_8$ to $1.0\%$–$1.5\%$ per bin in $0.4<z<1.4$ [36], reaching a combined $2.4\sigma$ sensitivity to QMM growth suppression. For cosmic shear, we updated the Euclid Fisher matrix pipeline of [37]: the lensing amplitude parameter $S_8\equiv {\sigma }_8\sqrt{{\Omega }_m/0.3}$ shifts by $-0.017$, within the projected $1\sigma$ statistical error ($0.013$), providing an independent consistency test.

6.4. Fisher Forecast for $(w_0,w_a,\lambda )$

Expanding the slow-roll equation of state as $w\left(a\right)=w_0+w_a(1-a)$ with the mapping $w_0=-1+\zeta$, $w_a=6\zeta$, we propagated next-generation survey specifications, Roman SN + BAO, DESI, Euclid cosmic shear, and CMB-S4, through a Fisher-matrix pipeline. Marginalised $1\sigma$ uncertainties read

$$\sigma \left(w_0\right)=0.013,\sigma \left(w_a\right)=0.19,\sigma \left({log}_{10}\lambda \right)=0.11,$$

yielding a dark-energy $\mathrm{FoM}=1/\left[\sigma \left(w_0\right)\sigma \left(w_a\right)\right]=590$, roughly twice the Pantheon +BAO + Planck baseline. Thus Stage-IV data will either confirm the QMM slow-roll imprint at $>5\sigma$ or drive ${log}_{10}\lambda <-24.5$, which, in turn, excludes entropy production earlier than $z_{\mathrm{sat}}\simeq {10}^4$.

Figure 7 shows the projected 68% and 95% confidence ellipses in the $(w_0,w_a)$ plane; the negative tilt reflects the anti-correlation expected from the sound-horizon degeneracy.

7. Unification with the QMM Dark-Matter Sector

7.1. A Single Entropy Field, Two Cosmological Phases

Within QMM the coarse-grained entropy scalar $S\left(x\right)$ carries both kinetic and potential contributions to the stress–energy tensor

$$T_{\left(S\right)}^{\mu \nu }=\lambda {\partial }^{\mu }S{\partial }^{\nu }S-g^{\mu \nu }\left[\frac{\lambda }{2}{(\partial S)}^2+{\rho }_{\Lambda }\right].$$

A simple decomposition clarifies the dark-sector duality:

$${\rho }_S\left(a\right)=\underset{\mathrm{gradient}}{\underbrace{\frac{\lambda }{2}{\dot{S}}^2}}+\underset{\mathrm{potential}}{\underbrace{{\rho }_{\Lambda }}}\equiv {\rho }_{\mathrm{grad}}\left(a\right)+{\rho }_{\mathrm{vac}}.$$

• Gradient-dominated regime. When ${\dot{S}}^2\gg 2{\rho }_{\Lambda }/\lambda$ the field behaves like cold matter because Eq. (12) implies ${\rho }_{\mathrm{grad}}\propto a^{-3}$. This branch reproduces halo abundance, the baryon–acoustic peak, and Lyman-$\alpha$ observables exactly as in the QMM dark-matter study of Ref. [11].
• Potential-dominated regime. Once ${\dot{S}}^2\ll 2{\rho }_{\Lambda }/\lambda$, the constant term starts to control the dynamics and drives acceleration, see Section 3. The observed Universe simply sits today in a mixed phase with ${\rho }_{\mathrm{grad}}/{\rho }_{\mathrm{vac}}\simeq \zeta \sim {10}^{-1.6}.$

Thus, dark matter and dark energy are not distinct fluids but represent the limiting behaviors of one microscopic information field.

7.1.0.4

Figure 8 quantifies the gradient-dominated branch: using the holographically regulated flux constant $\alpha =6.22\times {10}^{25}\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{s}}^{-1}{\mathrm{m}}^{-2}$ derived in Ref. [11], the cumulative mass profile reproduces a Milky-Way–sized halo of $M(<200\mathrm{kpc})\approx {10}^{12}{M}_{\odot }$ without any additional tuning, consistent with our earlier dark-matter study.

7.2. Coupled N-Body + Boltzmann Pipeline

To evolve the mixed phase self-consistently we constructed a two-stage pipeline:

i)

Linear stage, $z\gtrsim 20$. The supplementary code notebook’s linear solver (see Appendix D) provides transfer functions for the total matter contrast ${\delta }_m={\delta }_b+{\delta }_{\mathrm{grad}}$, solving Eqs. (15)–() with ${\rho }_{\mathrm{vac}}$ held fixed.

ii)

Non-linear stage. Transfer-function initial conditions are ingested by a GADGET-4 run in which particle masses evolve as $m_p\left(a\right)=m_{p,0}[1-\epsilon \left(a\right)]$ with $\epsilon \left(a\right)=\zeta a^3/(1+\zeta a^6)$, mimicking the kinetic–to–potential leakage. Background quantities $H\left(a\right)$ and ${\rho }_S\left(a\right)$ are read from a pre-computed lookup table, guaranteeing energy conservation better than $\Delta E/E<{10}^{-4}$.

7.3. Consistency Conditions and Parameter Degeneracies

Entropy-energy budget.

At recombination we require ${\rho }_{\mathrm{grad}}/{\rho }_r\lesssim 0.01$ to preserve the CMB damping tail, placing the upper bound $\zeta \lesssim {10}^{-3}$. Conversely, large-scale structure needs ${\rho }_{\mathrm{grad}}\gtrsim 0.05{\rho }_m$ today to match galaxy clustering, yielding the lower limit $\zeta \gtrsim {10}^{-2.5}$. The best-fit value, see $\zeta \approx {10}^{-1.6}$, see Section 5, comfortably satisfies both limits.

Degeneracies.

Because ${\rho }_{\mathrm{grad}}$ redshifts like matter, $\zeta$ is nearly degenerate with the physical cold-dark-matter density ${\omega }_c$. Weak-lensing amplitude $S_8$ partially breaks this degeneracy, while redshift-space distortions constrain the growth rate independently. In Fisher forecasts the principal component aligned with $\delta {\omega }_c-0.38\delta \zeta$ is determined to $0.6\%$ precision, ensuring robust separation of QMM signatures from neutrino-mass effects, which hinder growth but leave the equation-of-state unchanged.

Baryon feedback.

Preliminary hydrodynamic tests using Arepo indicate baryon back-reaction shifts the halo-mass function by $<3\%$ for $\zeta \le {10}^{-1.5}$, smaller than DESI statistical errors and therefore negligible at current sensitivity.

8. Discussion

8.1. Context within Alternative Dark-Energy Paradigms

The QMM slow–roll mechanism sits at an interesting intersection of existing proposals. Unlike canonical quintessence [38,39] or k-essence [24], where a new scalar field or higher-derivative kinetic term is postulated, our entropy field S is not an independent degree of freedom but is an emergent bookkeeping variable that already accounts for dark matter. In vacuum-sequestering models [40] the cosmological constant is globally cancelled by Lagrange multipliers; by contrast, QMM derives the smallness of ${\rho }_{\Lambda }$ from the finite Hilbert capacity $d_{max}$ without introducing non-local action terms. Emergent-gravity scenarios [41] also appeal to entropic arguments, but those invoke coarse-graining of microscopic dots in an a priori classical spacetime, whereas QMM tracks quantum information at the Planck-cell level and yields a concrete stress–energy tensor suitable for Boltzmann codes.

8.2. Toward a UV Completion

Because the effective action (9) contains only dimensionless $\lambda$ and a dimension-four vacuum term, it remains perturbatively stable up to the cutoff scale ${\ell }_{\mathrm{Pl}}^{-1}$. At higher energies we expect QMM to merge with causal-set quantum gravity, wherein Planck cells correspond to causal elements with partial order [12,42]. The heat-kernel derivation of Section 3 already mirrors causal-set spectral techniques [43]. Embedding QMM into the group field-theory (GFT) renormalization flow could clarify whether $\lambda$ runs to an interacting fixed point, completing the asymptotic-safety picture [22].

8.3. Implications for Black-Hole Information Recovery

The original QMM application [9] resolves the information paradox by storing outgoing Hawking quanta as unitary imprints. The present work shows that the same storage prescription necessarily leaves a residual vacuum energy. A corollary is that any quantum channel that erases information from Hawking radiation would also erase the vacuum-imprint energy, contradicting the observed $\Lambda >0$. Hence, late-time acceleration becomes an empirical witness of unitary black-hole evaporation, linking two previously disjoint puzzles.

8.4. Limitations and Open Questions

• Back-reaction in strongly curved regimes. Our derivation ignores higher-order curvature terms $\mathcal{O}\left(R^2{\ell }_{\mathrm{Pl}}^2\right)$. Near compact objects or appearing during inflation these corrections may renormalize ${\rho }_{\mathrm{vac}}$ and spoil the coincidence explained in Section 3.
• Primordial non-Gaussianities. The entropy field has a derivative coupling to curvature perturbations that could source equilateral-type non-Gaussianity at the $|f_{\mathrm{NL}}|\sim \mathcal{O}\left(1\right)$ level. Dedicated GADGET-4–based simulations are required to quantify this signal.
• Baryonic feedback and small-scale crises. While Section 7 suggests sub-percent back-reaction, feedback models carry substantial theoretical uncertainty that propagates into $S_8$ forecasts.
• Parameter degeneracy with neutrino mass. The suppression of growth by $\zeta$ partially mimics the effect of $\sum m_{\nu }$. A joint analysis of QMM + massive neutrinos is underway and will be reported elsewhere.

Overall, the Quantum Memory Matrix offers a unified and falsifiable framework in which dark matter, dark energy, and black-hole unitarity emerge from the same microscopic bookkeeping principle. The next decade of surveys—Roman, Euclid, DESI, and CMB-S4—will decisively test this picture.

9. Conclusions

The Quantum Memory Matrix links the two dark components of the Universe to a single microscopic ingredient: the finite information capacity of Planck-scale cells. Saturation of the local Hilbert space leaves a uniform vacuum-imprint energy whose density, fixed entirely by the maximum dimension $d_{max}$, reproduces the observed cosmological constant without external tuning. The same entropy field evolves in an overdamped, slow-roll regime and acquires an equation of state $w\left(a\right)=-1+\zeta a^{-6}$; a combined fit to Planck 2018, BAO, and Pantheon + supernovae selects $\zeta \simeq {10}^{-1.6}$, lifting the inferred Hubble constant and lowering ${\sigma }_8$ to the values preferred by late-time structure measurements. In its gradient-dominated phase the field redshifts as $a^{-3}$ and reproduces a Milky-Way–scale halo mass of ${10}^{12}{M}_{\odot }$ at $R\approx 200\mathrm{kpc}$; in the potential-dominated phase it acts as dark energy, so cold matter and accelerated expansion emerge as two limits of a single degree of freedom. The framework predicts modest but measurable signatures: a $0.5--1\%$ enhancement of the late-ISW contribution, a few-percent suppression of the linear growth rate, and a distance-modulus residual of $\Delta \mu \approx 0.02\mathrm{mag}$ at $z\approx 1.5$. Forecasts show that Roman’s high-redshift supernova survey, Euclid and DESI growth data, and CMB-S4 temperature, polarization, and lensing measurements can detect these effects at the $3--5\sigma$ level. Every figure and numerical value is produced by a single publicly supplied code notebook, ensuring full transparency and straightforward reproducibility.

Appendix A.0.0.8. k=0 term.

The zeroth coefficient counts the number of internal degrees of freedom:

$$a_0\left(x\right)=d_{max}.$$

Appendix A.0.0.9. k=1 term.

Using the standard formula $a_1=\frac{1}{6}{d}_{max}R$ one finds

$$a_1\left(x\right)={\displaystyle \frac{d_{max}}{6}}R\left(x\right),$$

where R is the Ricci scalar. Because $R{\ell }_{\mathrm{Pl}}^2\sim {10}^{-122}$ at present epochs, this term contributes negligibly to the vacuum energy but will matter during the inflationary era or inside neutron stars.

Appendix A.0.0.10. k=2 term.

The second coefficient involves quadratic curvature combinations,

$$a_2\left(x\right)={\displaystyle \frac{d_{max}}{180}}\left(R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }-R_{\mu \nu }{R}^{\mu \nu }+\frac{5}{2}{R}^2\right),$$

and produces subleading $\mathcal{O}\left(R^2\right)$ corrections to the effective action. When integrated over a causal diamond of radius $H_0^{-1}$, these terms renormalize ${\rho }_{\mathrm{vac}}$ by less than ${10}^{-240}{M}_{\mathrm{Pl}}^4$ and are therefore ignored in Section 3. They may, however, be relevant in causal-set discretizations, where $R^2$ operators arise naturally through non-local retarded d’Alembert kernels [43].

Appendix A.0.0.11. Truncation and UV finiteness.

Because $a_k$ grows combinatorially with k, the finite upper limit $k_{max}$ acts as a hard UV regulator. Inserting Eq. (A1) into the functional determinant and performing the s–integral with lower cutoff $s_{min}={\ell }_{\mathrm{Pl}}^2$ gives the residual energy density ${\rho }_{\mathrm{vac}}=M_{\mathrm{Pl}}^4/\left(16{\pi }^2{d}_{max}\right)$ quoted in Eq. (7), plus terms suppressed by $k_{max}^{-1}\lesssim {10}^{-2}$. The procedure thus demonstrates explicitly how holographic saturation renders the zero-point imprint energy finite within QMM, in agreement with Refs. [18,20].

Appendix B.1. Canonical Hamiltonian

Inserting the effective action (9) into ADM variables and retaining terms up to second order in perturbations yields the canonical pair $(S,{\Pi }_S)\equiv \left(S,\lambda a^3\dot{S}\right)$ and the usual GR momenta $({\pi }^{ij},h_{ij})$. The total Hamiltonian is

$$\mathcal{H}=N{\mathcal{H}}_0+N^i{\mathcal{H}}_i,$$

where the primary constraints are

$$\begin{array}{cc}\hfill {\mathcal{H}}_0& ={\displaystyle \frac{1}{2\lambda a^3}}{\Pi }_S^2+{\displaystyle \frac{{\pi }^{ij}{\pi }_{ij}-\frac{1}{2}{\pi }^2}{a^3}}+a\left(-\frac{\lambda }{2}{\left({\partial }_{k}S\right)}^2+{\rho }_{\Lambda }\right)-{\displaystyle \frac{a}{2}}{R}^{\left(3\right)},\hfill \end{array}$$

(A2)

$$\begin{array}{cc}\hfill {\mathcal{H}}_i& =-2{\nabla }_j{{\pi }^j}_i+{\Pi }_S{\partial }_{i}S,\hfill \end{array}$$

(A3)

with $\pi \equiv h_{ij}{\pi }^{ij}$ and $R^{\left(3\right)}$ is the Ricci scalar of $h_{ij}$. Because N and $N^i$ appear as Lagrange multipliers, the Hamiltonian density is a linear combination of first-class constraints; therefore no additional propagating degrees of freedom are introduced beyond those of GR plus the single scalar S.

Appendix B.2. Absence of Ghosts

The kinetic matrix in the scalar sector is diagonal with positive eigenvalues provided $\lambda >0$. Specifically, the sign of the quadratic term $\frac{1}{2\lambda a^3}{\Pi }_S^2$ ensures that the Ostrogradski ghost endemic to higher-derivative theories [46] is absent. Tensor modes inherit their standard GR kinetic term and are unaffected by S at quadratic order.

Appendix B.3. Propagation Speed and Laplace Stability

Variation of the second-order action with respect to $\delta S$ in Fourier space gives the dispersion relation ${\omega }^2=k^2/a^2,$ so the scalar sound speed is $c_s^2=1.$ No gradient instability, $c_s^2<0$, arises, and superluminal propagation is avoided [47]. The gravitational wave speed remains exactly unity because S couples only through its energy–momentum tensor, which at linear order contributes a purely background shift to the Friedmann equations.

Appendix B.4. Higher-Order Corrections

Cubic and quartic self-interactions of S are suppressed by the dimensionless ratio $ϵ\equiv \lambda {\dot{S}}^2/{\rho }_{\Lambda }\lesssim {10}^{-2},$ so loop corrections do not flip the sign of $\lambda$ nor generate operators with more than two time derivatives at energies $\ll M_{\mathrm{Pl}}$. We have checked explicitly that the quartic operator ${(\partial S)}^4$ appears with positive coefficient $({\lambda }^2/4)$ at one loop, maintaining stability up to the Planck scale.

  In summary, the $(S,g_{\mu \nu })$ system derived from the Quantum Memory Matrix is free of ghost and gradient instabilities, it possesses luminal propagation for both scalar and tensor sectors, and retains its healthy structure under radiative corrections so long as $\lambda >0$.