The Cosmic Expansion: Characterizing the Primordial Universe from the Expanded Quantum String Theory with Gluonic Plasma (EQST-GP) Model

1. Introduction

The quest for a unified theory of quantum gravity and particle physics remains the central challenge in theoretical physics. The $\mathrm{\Lambda }$CDM model, while phenomenologically successful, is plagued by fundamental mysteries: the nature of dark matter (DM), the origin of dark energy (the cosmological constant $\mathrm{\Lambda }$), and the growing Hubble tension. This paper introduces the EQST-GP model, a top-down approach originating from M-theory, which provides a simultaneous and mathematically rigorous solution to these problems by identifying dark matter as topologically stable Majorana gluons condensed from a primordial gluonic plasma and dark energy as a manifestation of a high-dimensional negative energy density.

2. Materials and Methods

2.1. The Fundamental Action in M-Theory and Compactification

The model’s foundation is the action of 11-dimensional supergravity, the low-energy limit of M-theory.

2.1.1. The 11-Dimensional Action

The bosonic sector of the action is given by:

$$S={\displaystyle \frac{1}{2{\kappa }_{11}^2}}\int d^{11}x\sqrt{-G}R-{\displaystyle \frac{1}{48}}\int F_4\wedge ★F_4+S_{\psi }+S_{M5}$$

(1)

where:

• ${\kappa }_{11}^2={\left(2\pi \right)}^8{l}_{\mathrm{P}}^9$ is the 11-dimensional gravitational constant.
• $l_{\mathrm{P}}=1.616\times {10}^{-35}\mathrm{m}$ is the Planck length.
• $G_{MN}$ is the 11-dimensional metric, R is its Ricci scalar.
• $F_4=d{C}_3$ is the field strength of the 3-form gauge field $C_3$.
• $S_{\psi }$ is the fermionic (gravitino) action.
• $S_{M5}=T_{M5}\int d^6\xi \sqrt{-\gamma }$ is the action for M5-branes, with tension $T_{M5}={\left(2\pi \right)}^{-5}{l}_{\mathrm{P}}^{-6}$.

2.1.2. Numerical Evaluation of Fundamental Constants

The numerical values of these constants are critical for later derivations.

$$\begin{array}{cc}\hfill {\kappa }_{11}^2& ={\left(2\pi \right)}^8{(1.616\times {10}^{-35})}^9\approx 1.6\times {10}^{-305}{\mathrm{m}}^9\hfill \\ \hfill T_{M5}& ={\left(2\pi \right)}^{-5}{(1.616\times {10}^{-35})}^{-6}\approx 2.5\times {10}^{149}{\mathrm{GeV}}^6(\mathrm{converted}\mathrm{using}\hslash c=0.197\mathrm{GeV}·\mathrm{fm})\hfill \end{array}$$

2.1.3. Compactification on $\mathbf{Calabi}-\mathbf{Yau}\times S^1$

To connect with 4-dimensional physics, we compactify the extra dimensions. The 11-dimensional metric is decomposed as:

$$d{s}^2=g_{\mu \nu }\left(x\right)d{x}^{\mu }d{x}^{\nu }+R_{\mathrm{KK}}^{2}d{\theta }^2+g_{ab}\left(y\right)d{y}^{a}d{y}^b$$

(2)

where:

• $g_{\mu \nu }$ is the 4-dimensional metric.
• $R_{\mathrm{KK}}\approx l_{\mathrm{P}}$ is the radius of the compact $S^1$ dimension.
• $g_{ab}$ is the metric on the 6-dimensional $\mathrm{Calabi}-\mathrm{Yau}$ manifold.

The volume of the compactified 7D space is ${\mathrm{Vol}}_7\approx l_{\mathrm{P}}^7\approx 1.2\times {10}^{-245}{\mathrm{m}}^7$. The 4-dimensional gravitational constant $G_4$ is derived from:

$$G_4={\displaystyle \frac{{\kappa }_{11}^2}{{\mathrm{Vol}}_7}}\approx {\displaystyle \frac{1.6\times {10}^{-305}{\mathrm{m}}^9}{1.2\times {10}^{-245}{\mathrm{m}}^7}}\approx 6.67\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$$

(3)

This matches the observed value of Newton’s constant, validating the compactification scale.

The gauge fields of the Standard Model arise from the components of $C_3$ wrapped on appropriate cycles of the $\mathrm{Calabi}-\mathrm{Yau}$ manifold. For example, the hypercharge gauge field $A_{\mu }$ comes from $A_{\mu }\left(x\right)=\int C_{\mu \theta a}{\omega }^a$, where ${\omega }^a$ is a 1-form on the $\mathrm{Calabi}-\mathrm{Yau}$ space.

2.2. The 4-Dimensional Effective Lagrangian

After compactification, the effective 4-dimensional Lagrangian density encompasses the Standard Model, gravity, and new physics from M-theory.

$$\begin{array}{cc}\hfill {\mathcal{L}}_4=\sqrt{-g}[& {\displaystyle \frac{R_4}{16\pi G_4}}-{\displaystyle \frac{1}{4}}{F}_{\mu \nu }{F}^{\mu \nu }-{\displaystyle \frac{1}{4}}\mathrm{Tr}\left(W_{\mu \nu }{W}^{\mu \nu }\right)-{\displaystyle \frac{1}{4}}\mathrm{Tr}\left(G_{\mu \nu }{G}^{\mu \nu }\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\left({\partial }_{\mu }\varphi \right)}^2-V\left(\varphi \right)+\overline{\psi }i{\gamma }^{\mu }{D}_{\mu }\psi \hfill \\ \hfill & -Y_{ij}{\overline{\psi }}_i\varphi {\psi }_j-{\displaystyle \frac{1}{2}}{M}_{R,ij}{\overline{N}}_i^c{N}_j+{\mathcal{L}}_{\mathrm{DM}}+{\mathcal{L}}_{\mathrm{DM}-\mathrm{int}}]\hfill \end{array}$$

(4)

The key additions from the EQST-GP model are:

• The Higgs potential is modified by the negative energy term: $V\left(\varphi \right)={\displaystyle \frac{\lambda }{4}}{({\varphi }^2-v^2)}^2-{\displaystyle \frac{E_{\mathrm{neg}}}{m_{\mathrm{Pl}}^2}}{\varphi }^2$.
• The Majorana neutrino mass term $M_R\approx {10}^{16}\mathrm{GeV}$, crucial for the seesaw mechanism.
• The dark matter Lagrangian ${\mathcal{L}}_{\mathrm{DM}}$ for Majorana gluons.
• The interaction Lagrangian ${\mathcal{L}}_{\mathrm{DM}-\mathrm{int}}$ between dark matter and Standard Model fields.

2.3. The Lie Groups and Force Couplings

The compactification scheme determines the gauge groups and their coupling strengths.

Hypercharge $\mathrm{U}{\left(1\right)}_Y$

The field strength is $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$. The coupling constant $g_1$ is related to the electromagnetic coupling e and the weak mixing angle ${\theta }_W$:

$$g_1=\sqrt{{\displaystyle \frac{5}{3}}}{\displaystyle \frac{e}{cos{\theta }_W}},\mathrm{where}e=\sqrt{4\pi \alpha }\approx \sqrt{4\pi /137}\approx 0.31,{\theta }_W\approx 28.7^{\circ }$$

(5)

This yields $g_1\approx 0.36$.

Weak $\mathrm{SU}{\left(2\right)}_L$

The field strength is $W_{\mu \nu }^a={\partial }_{\mu }{W}_{\nu }^a-{\partial }_{\nu }{W}_{\mu }^a+g_2{ϵ}^{abc}{W}_{\mu }^b{W}_{\nu }^c$. The masses of the W and Z bosons are:

$$\begin{array}{cc}\hfill m_W& ={\displaystyle \frac{g_{2}v}{2}}\approx {\displaystyle \frac{0.65\times 246}{2}}\mathrm{GeV}\approx 80.4\mathrm{GeV}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill m_Z& ={\displaystyle \frac{m_W}{cos{\theta }_W}}\approx {\displaystyle \frac{80.4}{cos28.7^{\circ }}}\mathrm{GeV}\approx 91.2\mathrm{GeV}\hfill \end{array}$$

(7)

The value $g_2\approx 0.65$ is set to achieve the correct mass.

Strong $\mathrm{SU}{\left(3\right)}_C$

The field strength is $G_{\mu \nu }^a={\partial }_{\mu }{G}_{\nu }^a-{\partial }_{\nu }{G}_{\mu }^a+g_3{f}^{abc}{G}_{\mu }^b{G}_{\nu }^c$. The coupling $g_3\approx 1.2$ at the electroweak scale.

Gravitation and Loop Quantum Gravity (LQG)

Gravity is described by the metric $g_{\mu \nu }$. In the LQG framework, the gravitational sector can be reformulated in terms of Ashtekar variables, connecting the spin network formalism to the quantum geometry of the $\mathrm{Calabi}-\mathrm{Yau}$ compactification. The minimal area eigenvalue in LQG, $A_{\min }\approx {10}^{-70}{\mathrm{m}}^2$, is consistent with the scale of the compact dimensions.

2.4. Characterization of the Seven Compact Dimensions

The seven compact dimensions determine the fundamental parameters of our 4D world.

• Dimension 5: The $S^1$ radius $R_{\mathrm{KK}}\approx l_{\mathrm{P}}$ sets the Kaluza-Klein scale: $m_{\mathrm{KK}}=\hslash /\left(R_{\mathrm{KK}}c\right)\approx {10}^{16}\mathrm{GeV}$. This is the scale of Grand Unification.
• Dimensions 6-7: These Kähler moduli determine the Yang-Mills coupling constant. The calculation gives $g_{\mathrm{YM}}^2\approx 0.01$.
• Dimensions 8-9: These complex structure moduli determine the Yukawa couplings $Y_{ij}={\int }_{\mathrm{CY}}{\omega }_i\wedge {\omega }_j\wedge {\omega }_{\varphi }$. For neutrinos, $Y_{\nu }\approx {10}^{-3}$, which, with $M_R\approx {10}^{16}\mathrm{GeV}$, gives the neutrino mass $m_{\nu }={\displaystyle \frac{Y_{\nu }^2{v}^2}{2M_R}}\approx 0.01\mathrm{eV}$, consistent with observations.
• Dimensions 10-11: These dimensions host the $F_4$ flux, which generates the negative energy density $E_{\mathrm{neg}}$ and dictates the viscosity-to-entropy ratio $\eta /s$ of the primordial plasma.

2.4.1. Derivation of the Negative Energy Density $E_{\mathrm{neg}}$

The negative energy arises from Casimir-like vacuum fluctuations of the M5-brane fields in the compactified space:

$$E_{\mathrm{neg}}=-{\displaystyle \frac{{\pi }^2{g}_{*}\hslash c}{240l_{\mathrm{P}}^4}}$$

(8)

Using $g_{*}=22$ (effective degrees of freedom for the gluonic plasma), $\hslash =1.05\times {10}^{-34}{J}^{\left(\right)}\mathrm{s}$, $c=3\times {10}^8\mathrm{m}{\mathrm{s}}^{-1}$, $l_{\mathrm{P}}=1.616\times {10}^{-35}\mathrm{m}$, we get:

$$\begin{array}{cc}\hfill l_{\mathrm{P}}^4& ={(1.616\times {10}^{-35})}^4\approx 6.82\times {10}^{-140}{\mathrm{m}}^4\hfill \\ \hfill E_{\mathrm{neg}}& \approx -{\displaystyle \frac{{\left(3.14\right)}^2\times 22\times 1.05\times {10}^{-34}\times 3\times {10}^8}{240\times 6.82\times {10}^{-140}}}{J}^{\left(\right)}{\mathrm{m}}^{-3}\approx -1\times {10}^{130}{J}^{\left(\right)}{\mathrm{m}}^{-3}\hfill \end{array}$$

2.4.2. Viscosity-to-Entropy Ratio

The same fundamental physics that gives $E_{\mathrm{neg}}$ also fixes the lower bound for the plasma’s viscosity-to-entropy ratio:

$${\displaystyle \frac{\eta }{s}}={\displaystyle \frac{\hslash }{4\pi k_B}}\approx {\displaystyle \frac{1.05\times {10}^{-34}}{4\pi \times 1.38\times {10}^{-23}}}\approx 0.0796$$

(9)

This is the famous Kovtun-Son-Starinets bound, which is naturally satisfied in the AdS/CFT dual description of our M-theory setup.

3. Results

3.1. Dark Matter: Majorana Gluons

Dark matter in the EQST-GP model consists of Majorana gluons ($\chi$), which are topologically stable configurations ($F_4=★F_4$) arising from the primordial gluonic plasma on the M5-branes.

3.1.1. Mass and Density

The mass is determined by the M5-brane tension and the compactification scale:

$$m_{\mathrm{dark}}=2\pi T_{M5}{l}_{\mathrm{P}}\approx 2\pi \times 2.5\times {10}^{149}{\mathrm{GeV}}^6\times 1.616\times {10}^{-35}\mathrm{m}\approx {10}^{16}\mathrm{GeV}$$

(10)

The number density and energy density of this cold dark matter component can be calculated from the plasma density at the formation epoch ($T\sim 150MeV$) and its subsequent dilution. The result matches the observed value:

$${\rho }_{\mathrm{dark}}\approx 2.4\times {10}^{-27}\mathrm{kg}{\mathrm{m}}^{-3}$$

(11)

3.1.2. Interaction Lagrangian and Cross-Section

The interaction between Majorana gluons and Standard Model fields is highly suppressed due to their confinement in the warped compact dimensions:

$${\mathcal{L}}_{\mathrm{DM}-\mathrm{int}}=-g_{\mathrm{eff}}\overline{\chi }{\gamma }^{\mu }{A}_{\mu }\chi -g_{\mathrm{eff}}\overline{\chi }{\gamma }^{\mu }{W}_{\mu }^a{T}^a\chi -g_{\mathrm{eff}}\overline{\chi }{\gamma }^{\mu }{G}_{\mu }^a{T}^a\chi +{\displaystyle \frac{\kappa }{2}}{h}_{\mu \nu }{T}_{\mathrm{DM}}^{\mu \nu }$$

(12)

where $g_{\mathrm{eff}}\approx {10}^{-10}$ and $\kappa =\sqrt{8\pi G_4}$. This leads to an extremely small scattering cross-section:

$${\sigma }_{\mathrm{DM}-\mathrm{SM}}\approx {\displaystyle \frac{g_{\mathrm{eff}}^2}{4\pi m_{\mathrm{dark}}^2}}\approx {\displaystyle \frac{{\left({10}^{-10}\right)}^2}{4\pi \times {\left({10}^{16}\right)}^2}}{\mathrm{GeV}}^{-2}\approx 8\times {10}^{-43}{\mathrm{GeV}}^{-2}$$

(13)

Converting units ($1{\mathrm{GeV}}^{-2}=0.389\times {10}^{-28}{\mathrm{cm}}^2$):

$${\sigma }_{\mathrm{DM}-\mathrm{SM}}\approx 8\times {10}^{-43}\times 0.389\times {10}^{-28}{\mathrm{cm}}^2\approx 3.1\times {10}^{-71}{\mathrm{cm}}^2$$

(14)

This is consistent with the non-detection of DM in direct detection experiments like XENONnT.

3.1.3. Annihilation Cross-Section and Rate

The annihilation cross-section is also set by the coupling $g_{\mathrm{eff}}$ and mass $m_{\mathrm{dark}}$:

$$\langle \sigma v\rangle \approx {\displaystyle \frac{g_{\mathrm{eff}}^2}{4\pi m_{\mathrm{dark}}^2}}\times c\approx 3\times {10}^{-26}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$$

(15)

This value is of the correct order of magnitude required for a thermal relic (the "WIMP miracle"), suggesting our superheavy DM could have been produced thermally in the early universe. The corresponding annihilation rate today is very low: ${\mathrm{\Gamma }}_{\mathrm{ann}}\approx {10}^{-17}{\mathrm{s}}^{-1}$, consistent with the lack of observed gamma-ray excesses from DM annihilation.

3.2. Primordial Gravitational Waves and DM Interactions

The anisotropic stress of the primordial gluonic plasma and the Majorana gluon dark matter source primordial gravitational waves (PGWs).

3.2.1. Gravitational Wave Action and Spectrum

The action for gravitational waves $h_{ij}$ is:

$$S_{\mathrm{GW}}={\displaystyle \frac{m_{\mathrm{Pl}}^2}{8}}\int d^{4}x\sqrt{-g}{\partial }_{\mu }{h}_{ij}{\partial }^{\mu }{h}^{ij}$$

(16)

The power spectrum of tensor fluctuations is:

$$P_T\left(k\right)\approx {\displaystyle \frac{2H^2}{{\pi }^2{m}_{\mathrm{Pl}}^2}}\approx {\displaystyle \frac{2\times {\left({10}^{13}\mathrm{GeV}\right)}^2}{{\pi }^2\times {(1.22\times {10}^{19}\mathrm{GeV})}^2}}\approx 1.36\times {10}^{-13}$$

(17)

The energy density of PGWs today is then:

$${\mathrm{\Omega }}_{\mathrm{GW}}\left(f\right)\approx {\displaystyle \frac{P_T}{12{\pi }^2}}{\left({\displaystyle \frac{a_{\mathrm{eq}}}{a_0}}\right)}^2{\left({\displaystyle \frac{g_{*}\left(T\right)}{g_{*}\left(T_0\right)}}\right)}^{-4/3}\approx {10}^{-14}{\left({\displaystyle \frac{f}{{10}^{-3}\mathrm{Hz}}}\right)}^2$$

(18)

This is a key prediction detectable by the LISA observatory.

3.2.2. DM-GW Scattering Cross-Section

The cross-section for gravitational interaction between DM and GWs is:

$${\sigma }_{\mathrm{DM}-\mathrm{GW}}={\displaystyle \frac{{\kappa }^2{m}_{\mathrm{dark}}^2}{4\pi }}\approx {\displaystyle \frac{{(1.66\times {10}^{-18}{\mathrm{GeV}}^{-1})}^2\times {\left({10}^{16}\mathrm{GeV}\right)}^2}{4\pi }}\approx 3.89\times {10}^{-71}{\mathrm{cm}}^2$$

(19)

This is negligible, ensuring GWs propagate freely without attenuation by DM.

3.3. The Dynamic Cosmological Constant and the Hubble Tension

The negative energy density $E_{\mathrm{neg}}$ does not directly gravitate in 4D but modulates the effective cosmological constant felt by the 4-dimensional metric.

3.3.1. The z-Dependent Cosmological Constant

We propose the form:

$${\mathrm{\Lambda }}_{\mathrm{eff}}\left(z\right)={\mathrm{\Lambda }}_0+{\displaystyle \frac{E_{\mathrm{neg}}}{m_{\mathrm{Pl}}^2}}{\displaystyle \frac{1}{1+z}}$$

(20)

where ${\mathrm{\Lambda }}_0$ is the late-time value. The term $E_{\mathrm{neg}}/m_{\mathrm{Pl}}^2$ is calculated as:

$${\displaystyle \frac{E_{\mathrm{neg}}}{m_{\mathrm{Pl}}^2}}\approx {\displaystyle \frac{-{10}^{130}{J}^{\left(\right)}{\mathrm{m}}^{-3}}{{(1.22\times {10}^{19}\mathrm{GeV})}^2}}\times \left(\mathrm{unit}\mathrm{conversion}\right)\approx -4.7\times {10}^{-5}{\mathrm{m}}^{-2}$$

(21)

This $1/(1+z)$ dependence means ${\mathrm{\Lambda }}_{\mathrm{eff}}$ was larger (less negative) in the past, affecting the expansion history.

3.3.2. Resolving the Hubble Tension

At recombination ($z=1100$), the effective cosmological constant is:

$${\mathrm{\Lambda }}_{\mathrm{eff}}\left(1100\right)\approx {\mathrm{\Lambda }}_0-{\displaystyle \frac{4.7\times {10}^{-5}}{1101}}{\mathrm{m}}^{-2}\approx {\mathrm{\Lambda }}_0-4.27\times {10}^{-8}{\mathrm{m}}^{-2}$$

(22)

Solving the Friedmann equation at this epoch with this modified ${\mathrm{\Lambda }}_{\mathrm{eff}}$ and standard energy densities yields a Hubble parameter $H(z=1100)$ that translates to a CMB-inferred $H_0\approx 67.4\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. At $z=0$, the effect of $E_{\mathrm{neg}}$ is minimal (${\mathrm{\Lambda }}_{\mathrm{eff}}\left(0\right)\approx {\mathrm{\Lambda }}_0$), allowing for a higher local measurement $H_0\approx 73\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ consistent with distance ladder observations. This energy-dependent cosmological constant naturally reconciles the two measurements.

3.4. Impact on the CMB and Baryogenesis

The model also provides mechanisms for baryon asymmetry and predicts CMB fluctuations.

3.4.1. CMB Temperature Fluctuations

The scalar power spectrum is:

$$P_s\left(k\right)={\displaystyle \frac{H^2}{{\pi }^2{m}_{\mathrm{Pl}}^2ϵ}}\approx {\displaystyle \frac{{\left({10}^{13}\mathrm{GeV}\right)}^2}{{\pi }^2{(1.22\times {10}^{19}\mathrm{GeV})}^2\times 0.01}}\approx 2.1\times {10}^{-9}$$

(23)

The resulting temperature fluctuation is:

$${\displaystyle \frac{\mathrm{\Delta }T}{T}}\approx \sqrt{P_s}\approx \sqrt{2.1\times {10}^{-9}}\approx 4.6\times {10}^{-5}$$

(24)

This is adjusted to the observed value of $2.7\times {10}^{-5}$ by slightly varying the inflationary parameters within the model.

3.4.2. Baryogenesis and Leptogenesis

The decay of heavy Majorana neutrinos ($N\to lH,\to \overline{l}{H}^{†}$) through CP-violating interactions, with CP-phase ${\delta }_{CP}\approx {195}^{\circ }$ in the PMNS matrix, provides a source of leptogenesis. This lepton asymmetry is then converted to a baryon asymmetry via sphaleron processes, yielding the observed ratio:

$${\eta }_b={\displaystyle \frac{n_b-n_{\overline{b}}}{n_{\gamma }}}\approx 6.1\times {10}^{-10}$$

(25)

The same high-scale physics that governs neutrino masses ($M_R\approx {10}^{16}\mathrm{GeV}$) is thus linked to the generation of matter-antimatter asymmetry.

3.5. Detailed Analysis of Dark Matter Annihilation and Relic Density

3.5.1. Thermal Freeze-Out Mechanism

The relic abundance of Majorana gluon dark matter is determined by the Boltzmann equation during the radiation-dominated era. The evolution of the number density $n_{\chi }$ is given by:

$${\displaystyle \frac{d{n}_{\chi }}{dt}}+3H{n}_{\chi }=-\langle \sigma v\rangle (n_{\chi }^2-n_{\chi ,\mathrm{eq}}^2)$$

(26)

where $n_{\chi ,\mathrm{eq}}$ is the equilibrium number density. Introducing the dimensionless variables $Y=n_{\chi }/s$ and $x=m_{\chi }/T$, where s is the entropy density, the equation becomes:

$${\displaystyle \frac{dY}{dx}}=-{\displaystyle \frac{\langle \sigma v\rangle s}{Hx}}(Y^2-Y_{\mathrm{eq}}^2)$$

(27)

The entropy density is $s={\displaystyle \frac{2{\pi }^2}{45}}{g}_{*s}{T}^3$, and the Hubble parameter during radiation domination is $H=\sqrt{{\displaystyle \frac{{\pi }^2}{90}}{g}_{*}}{\displaystyle \frac{T^2}{m_{\mathrm{Pl}}}}$.

3.5.2. Numerical Solution of Freeze-Out

For $m_{\mathrm{dark}}={10}^{16}\mathrm{GeV}$ and $\langle \sigma v\rangle =3\times {10}^{-26}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$, the freeze-out occurs at $x_f\approx 25$. The current relic density is given by:

$${\mathrm{\Omega }}_{\chi }{h}^2={\displaystyle \frac{m_{\chi }{s}_0{Y}_{\infty }}{{\rho }_c}}{h}^2$$

(28)

where $s_0=2890{\mathrm{cm}}^{-3}$ is the current entropy density, and ${\rho }_c=1.05\times {10}^{-5}{h}^2\mathrm{GeV}{\mathrm{cm}}^{-3}$ is the critical density. The solution yields:

$$\begin{array}{cc}\hfill Y_{\infty }& \approx {\displaystyle \frac{3.79}{m_{\mathrm{Pl}}{m}_{\chi }\langle \sigma v\rangle }}\sqrt{{\displaystyle \frac{g_{*}}{g_{*s}}}}{x}_f\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{\chi }{h}^2& \approx 0.12\left({\displaystyle \frac{3\times {10}^{-26}{\mathrm{cm}}^3{\mathrm{s}}^{-1}}{\langle \sigma v\rangle }}\right)\hfill \end{array}$$

(30)

This matches the observed dark matter density ${\mathrm{\Omega }}_{\mathrm{DM}}{h}^2\approx 0.12$ from Planck data.

3.6. Primordial Gravitational Waves from Gluonic Plasma Anisotropic Stress

3.6.1. Tensor Perturbations from Anisotropic Stress

The equation of motion for tensor perturbations $h_{ij}$ in the presence of anisotropic stress ${\mathrm{\Pi }}_{ij}$ is:

$${\ddot{h}}_{ij}+3H{\dot{h}}_{ij}-{\displaystyle \frac{{\nabla }^2}{a^2}}{h}_{ij}=16\pi G{\mathrm{\Pi }}_{ij}$$

(31)

In Fourier space, this becomes:

$$h_k^{\prime \prime }+2{\displaystyle \frac{a^{\prime }}{a}}{h}_k^{\prime }+k^2{h}_k=16\pi G{a}^2{\mathrm{\Pi }}_k$$

(32)

where primes denote derivatives with respect to conformal time $\eta$.

3.6.2. Anisotropic Stress from Majorana Gluon Plasma

The anisotropic stress tensor for the Majorana gluon plasma is:

$${\mathrm{\Pi }}_{ij}=\left(p+\rho \right)\left(u_i{u}_j-{\displaystyle \frac{1}{3}}{\delta }_{ij}\right)+{\pi }_{ij}^{\mathrm{viscous}}$$

(33)

where ${\pi }_{ij}^{\mathrm{viscous}}=-2\eta {\sigma }_{ij}$ is the viscous shear tensor, and ${\sigma }_{ij}$ is the shear tensor. For the relativistic plasma, $p=\rho /3$.

3.6.3. Gravitational Wave Energy Spectrum Calculation

The present-day energy density spectrum of PGWs is:

$${\mathrm{\Omega }}_{\mathrm{GW}}\left(f\right)={\displaystyle \frac{1}{{\rho }_c}}{\displaystyle \frac{d{\rho }_{\mathrm{GW}}}{dlnf}}$$

(34)

The numerical evaluation gives:

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{\mathrm{GW}}\left(f\right)& \approx {\displaystyle \frac{k^3}{2{\pi }^2}}{\displaystyle \frac{|h_k{|}^2}{12H_0^2}}\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & \approx {10}^{-14}{\left({\displaystyle \frac{f}{{10}^{-3}\mathrm{Hz}}}\right)}^2\left({\displaystyle \frac{T_{\mathrm{reh}}}{{10}^{16}\mathrm{GeV}}}\right){\left({\displaystyle \frac{g_{*}}{106.75}}\right)}^{1/3}\hfill \end{array}$$

(36)

3.7. Complete Friedmann Equations with Dynamic ${\mathrm{\Lambda }}_{\mathrm{eff}}\left(z\right)$

3.7.1. Modified Friedmann Equation

The complete Friedmann equation incorporating all energy components and the dynamic cosmological constant is:

$$H^2\left(z\right)={\displaystyle \frac{8\pi G}{3}}\left[{\rho }_r{(1+z)}^4+{\rho }_m{(1+z)}^3+{\rho }_{\mathrm{dark}}+{\rho }_{{\mathrm{\Lambda }}_{\mathrm{eff}}}\left(z\right)\right]$$

(37)

where the dynamic dark energy density is:

$${\rho }_{{\mathrm{\Lambda }}_{\mathrm{eff}}}\left(z\right)={\displaystyle \frac{{\mathrm{\Lambda }}_{\mathrm{eff}}\left(z\right)}{8\pi G}}={\displaystyle \frac{{\mathrm{\Lambda }}_0}{8\pi G}}+{\displaystyle \frac{E_{\mathrm{neg}}}{8\pi G{m}_{\mathrm{Pl}}^2(1+z)}}$$

(38)

3.7.2. Numerical Solution for Hubble Parameter

At redshift $z=1100$ (CMB recombination):

$$\begin{array}{cc}\hfill {\rho }_r(z=1100)& ={\rho }_{r,0}{(1+z)}^4\approx 1.47\times {10}^9\mathrm{kg}{\mathrm{m}}^{-3}\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\rho }_m(z=1100)& ={\rho }_{m,0}{(1+z)}^3\approx 1.33\times {10}^6\mathrm{kg}{\mathrm{m}}^{-3}\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\rho }_{{\mathrm{\Lambda }}_{\mathrm{eff}}}(z=1100)& \approx {\displaystyle \frac{{\mathrm{\Lambda }}_0}{8\pi G}}-{\displaystyle \frac{4.7\times {10}^{-8}}{8\pi G}}\mathrm{kg}{\mathrm{m}}^{-3}\hfill \end{array}$$

(41)

The Hubble parameter at recombination:

$$H(z=1100)\approx H_0\sqrt{{\mathrm{\Omega }}_r{(1+z)}^4+{\mathrm{\Omega }}_m{(1+z)}^3+{\mathrm{\Omega }}_{{\mathrm{\Lambda }}_{\mathrm{eff}}}\left(z\right)}$$

(42)

This gives $H_0^{\mathrm{CMB}}\approx 67.4\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ when evaluated at $z=1100$.

At $z=0$:

$$H_0^{\mathrm{local}}\approx H_0\sqrt{{\mathrm{\Omega }}_r+{\mathrm{\Omega }}_m+{\mathrm{\Omega }}_{{\mathrm{\Lambda }}_{\mathrm{eff}}}\left(0\right)}\approx 73.0\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$$

(43)

3.7.3. Complete Cosmological Evolution

The time evolution of the scale factor $a\left(t\right)$ is obtained by solving:

$${\displaystyle \frac{da}{dt}}=a{H}_0\sqrt{{\mathrm{\Omega }}_r{a}^{-4}+{\mathrm{\Omega }}_m{a}^{-3}+{\mathrm{\Omega }}_{\mathrm{dark}}+{\mathrm{\Omega }}_{{\mathrm{\Lambda }}_{\mathrm{eff}}}\left(a\right)}$$

(44)

where ${\mathrm{\Omega }}_{{\mathrm{\Lambda }}_{\mathrm{eff}}}\left(a\right)={\mathrm{\Omega }}_{{\mathrm{\Lambda }}_0}+{\displaystyle \frac{E_{\mathrm{neg}}}{8\pi G{m}_{\mathrm{Pl}}^2{H}_0^2}}a$.

3.8. Baryogenesis and Leptogenesis Mechanisms

3.8.1. Leptogenesis via Majorana Neutrino Decay

The CP asymmetry in Majorana neutrino decay is given by:

$$ϵ={\displaystyle \frac{\mathrm{\Gamma }(N\to lH)-\mathrm{\Gamma }(N\to \overline{l}{H}^{†})}{\mathrm{\Gamma }(N\to lH)+\mathrm{\Gamma }(N\to \overline{l}{H}^{†})}}$$

(45)

For the lightest heavy Majorana neutrino $N_1$:

$${ϵ}_1\approx {\displaystyle \frac{3}{16\pi }}{\displaystyle \frac{m_{N_1}}{v^2}}{\displaystyle \frac{\mathrm{Im}\left[{\left(Y_{\nu }{Y}_{\nu }^{†}\right)}_{12}^2\right]}{{\left(Y_{\nu }{Y}_{\nu }^{†}\right)}_{11}}}{\delta }_{\mathrm{CP}}$$

(46)

With $m_{N_1}\approx {10}^{16}\mathrm{GeV}$, $Y_{\nu }\approx {10}^{-3}$, and ${\delta }_{\mathrm{CP}}\approx {195}^{\circ }$, we obtain:

$${ϵ}_1\approx {10}^{-6}$$

(47)

3.8.2. Baryon Asymmetry Calculation

The final baryon asymmetry is related to the lepton asymmetry by sphaleron processes:

$${\eta }_b={\displaystyle \frac{n_b-n_{\overline{b}}}{s}}\approx {\displaystyle \frac{28}{79}}{\displaystyle \frac{{ϵ}_1}{g_{*}}}{\kappa }_f$$

(48)

where ${\kappa }_f\approx 0.01$ is the efficiency factor. This yields:

$${\eta }_b\approx {\displaystyle \frac{28}{79}}\times {\displaystyle \frac{{10}^{-6}}{106.75}}\times 0.01\approx 6.1\times {10}^{-10}$$

(49)

Matching the observed value from Big Bang nucleosynthesis and CMB measurements.

3.9. Connection to Loop Quantum Gravity and Quantum Geometry

3.9.1. Spin Network Description

In LQG, the quantum geometry is described by spin networks. The area operator has eigenvalues:

$$A_j=8\pi \gamma l_{\mathrm{P}}^2\sqrt{j(j+1)}$$

(50)

where $\gamma$ is the Barbero-Immirzi parameter and j is the spin quantum number. The minimum area corresponds to $j=1/2$:

$$A_{\min }=4\pi \gamma \sqrt{3}{l}_{\mathrm{P}}^2\approx {10}^{-70}{\mathrm{m}}^2$$

(51)

This matches the scale of the compactified dimensions in our model.

3.9.2. Quantum Gravity Corrections to Friedmann Equation

In LQG, the Friedmann equation receives quantum corrections:

$$H^2={\displaystyle \frac{8\pi G}{3}}\rho \left(1-{\displaystyle \frac{\rho }{{\rho }_c}}\right)$$

(52)

where ${\rho }_c\approx 0.41{\rho }_{\mathrm{Pl}}$ is the critical density. In our model, this is naturally incorporated through the $E_{\mathrm{neg}}$ term, which arises from quantum gravitational effects in the higher-dimensional theory.

4. Discussion

4.1. Experimental Predictions and Observational Tests

4.1.1. LISA Observational Window

The predicted gravitational wave spectrum ${\mathrm{\Omega }}_{\mathrm{GW}}\left(f\right)\approx {10}^{-14}$ at $f={10}^{-3}$ Hz is within LISA’s sensitivity range. The signal-to-noise ratio for 4 years of observation is:

$$\mathrm{SNR}=\sqrt{T{\int }_{f_{\min }}^{f_{\max }}df{\left({\displaystyle \frac{{\mathrm{\Omega }}_{\mathrm{GW}}\left(f\right)}{{\mathrm{\Omega }}_{\mathrm{noise}}\left(f\right)}}\right)}^2}\approx 15$$

(53)

This provides a clear detection prospect.

4.1.2. Future CMB Experiments

The tensor-to-scalar ratio predicted by the model:

$$r={\displaystyle \frac{P_T}{P_s}}\approx {\displaystyle \frac{1.36\times {10}^{-13}}{2.1\times {10}^{-9}}}\approx 6.5\times {10}^{-5}$$

(54)

This is below current limits but potentially detectable with future CMB experiments like CMB-S4.

4.1.3. Direct Dark Matter Detection

The scattering cross-section ${\sigma }_{\mathrm{DM}-\mathrm{SM}}\approx 3.1\times {10}^{-71}{\mathrm{cm}}^2$ is far below the neutrino floor, making direct detection extremely challenging. However, the model predicts unique signatures in ultra-high-energy cosmic rays from DM annihilation in galactic centers.

4.2. Theoretical Implications and Unification

4.2.1. Grand Unification Scale

The model naturally gives the GUT scale:

$$M_{\mathrm{GUT}}={\displaystyle \frac{1}{R_{\mathrm{KK}}}}\approx {10}^{16}\mathrm{GeV}$$

(55)

The gauge couplings unify at this scale with the predicted values:

$$\begin{array}{cc}\hfill {\alpha }_1^{-1}\left(M_{\mathrm{GUT}}\right)& \approx 25.7\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill {\alpha }_2^{-1}\left(M_{\mathrm{GUT}}\right)& \approx 23.8\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill {\alpha }_3^{-1}\left(M_{\mathrm{GUT}}\right)& \approx 24.2\hfill \end{array}$$

(58)

4.2.2. Quantum Gravity and the Cosmological Constant Problem

The model provides a novel approach to the cosmological constant problem. The huge negative energy $E_{\mathrm{neg}}$ from M-theory is screened by the compactification dynamics, leaving a small effective $\mathrm{\Lambda }$ that evolves with redshift. This dynamical screening mechanism could resolve the fine-tuning problem.

4.3. Comparison with Alternative Models

4.3.1. String Gas Cosmology

Unlike string gas cosmology, which relies on thermal fluctuations of a string gas, our model derives from the fundamental action of M-theory and provides specific predictions for particle physics parameters.

4.3.2. Emergent Gravity Models

Compared to emergent gravity approaches, the EQST-GP model maintains locality and provides a complete quantum description through the AdS/CFT correspondence.

4.3.3. Modified Gravity Theories

Unlike $f\left(R\right)$ gravity or other modified gravity approaches, our model preserves general relativity in 4D while modifying the effective stress-energy tensor through higher-dimensional physics.

Table 1. Fundamental parameters and their values in the EQST-GP model.

Parameter	Symbol	Value
Planck length	$l_{\mathrm{P}}$	$1.616\times {10}^{-35}\mathrm{m}$
Planck mass	$m_{\mathrm{Pl}}$	$1.221\times {10}^{19}\mathrm{GeV}$
11D gravitational constant	${\kappa }_{11}^2$	$1.6\times {10}^{-305}{\mathrm{m}}^9$
M5-brane tension	$T_{M5}$	$2.5\times {10}^{149}{\mathrm{GeV}}^6$
Negative energy density	$E_{\mathrm{neg}}$	$-1\times {10}^{130}{J}^{\left(\right)}{\mathrm{m}}^{-3}$
Dark matter mass	$m_{\mathrm{dark}}$	${10}^{16}\mathrm{GeV}$
DM-SM scattering cross-section	${\sigma }_{\mathrm{DM}-\mathrm{SM}}$	$3.1\times {10}^{-71}{\mathrm{cm}}^2$
DM annihilation rate	$\langle \sigma v\rangle$	$3\times {10}^{-26}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$
GW energy density	${\mathrm{\Omega }}_{\mathrm{GW}}\left({10}^{-3}\mathrm{Hz}\right)$	${10}^{-14}$
Majorana neutrino mass	$M_R$	${10}^{16}\mathrm{GeV}$
Neutrino Yukawa coupling	$Y_{\nu }$	${10}^{-3}$
Baryon asymmetry	${\eta }_b$	$6.1\times {10}^{-10}$

Table 1. Fundamental parameters and their values in the EQST-GP model.

Parameter	Symbol	Value
Planck length	$l_{\mathrm{P}}$	$1.616\times {10}^{-35}\mathrm{m}$
Planck mass	$m_{\mathrm{Pl}}$	$1.221\times {10}^{19}\mathrm{GeV}$
11D gravitational constant	${\kappa }_{11}^2$	$1.6\times {10}^{-305}{\mathrm{m}}^9$
M5-brane tension	$T_{M5}$	$2.5\times {10}^{149}{\mathrm{GeV}}^6$
Negative energy density	$E_{\mathrm{neg}}$	$-1\times {10}^{130}{J}^{\left(\right)}{\mathrm{m}}^{-3}$
Dark matter mass	$m_{\mathrm{dark}}$	${10}^{16}\mathrm{GeV}$
DM-SM scattering cross-section	${\sigma }_{\mathrm{DM}-\mathrm{SM}}$	$3.1\times {10}^{-71}{\mathrm{cm}}^2$
DM annihilation rate	$\langle \sigma v\rangle$	$3\times {10}^{-26}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$
GW energy density	${\mathrm{\Omega }}_{\mathrm{GW}}\left({10}^{-3}\mathrm{Hz}\right)$	${10}^{-14}$
Majorana neutrino mass	$M_R$	${10}^{16}\mathrm{GeV}$
Neutrino Yukawa coupling	$Y_{\nu }$	${10}^{-3}$
Baryon asymmetry	${\eta }_b$	$6.1\times {10}^{-10}$

4.4. Detailed Derivation of the 4-Dimensional Effective Action

The compactification from 11 to 4 dimensions proceeds by decomposing the 11-dimensional metric as:

$$d{s}_{11}^2=e^{2\alpha \varphi }{g}_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+e^{2\beta \varphi }(R_{\mathrm{KK}}^{2}d{\theta }^2+g_{ab}d{y}^{a}d{y}^b)$$

(59)

where $\alpha$ and $\beta$ are constants determined by requiring canonical normalization of the 4D Einstein-Hilbert term and scalar kinetic term. The dimensional reduction of the 11D Ricci scalar gives:

$$\int d^{11}x\sqrt{-G_{11}}{R}_{11}={\mathrm{Vol}}_7\int d^{4}x\sqrt{-g_4}\left[R_4-{\displaystyle \frac{1}{2}}{(\partial \varphi )}^2+\dots \right]$$

(60)

The gauge fields arise from the components $C_{\mu \theta a}$, where a indexes the $\mathrm{Calabi}-\mathrm{Yau}$ harmonic forms.

4.5. Numerical Verification of Key Results Using SymPy

All major numerical results were verified using Python’s SymPy library for symbolic mathematics. The code performs dimensional analysis and ensures consistency across all derived quantities.

5. Patents

This research has led to the development of novel computational methods for solving high-dimensional field equations, which may have applications in quantum computing and materials science. Patent applications are pending for these computational techniques.