Swampland Conjectures Compatibility and Technical Refinements in the Expanded Quantum String Theory with Gluonic Plasma (EQST-GP) Model

1. Introduction

The EQST-GP model represents an ambitious unification framework deriving from M-theory compactification on $S^1\times {\mathrm{CY}}_3$ [8, 20]. Building upon our foundational work [16] [16], which established the fundamental structure, several theoretical challenges require detailed mathematical resolution and physical justification. This enhanced work addresses:

• Precise mechanism for negative energy density $E_{\mathrm{neg}}$ generation with justification for its extreme values
• Topological foundation of Majorana gluon dark matter [51, 53] with detailed mass generation mechanism
• Comprehensive Swampland Conjectures compatibility [13, 31] with physically motivated uplifting [14, 26]
• Technical refinements in moduli stabilization [24] with parameter sensitivity analysis
• Enhanced gravitational wave predictions [39] with LISA detectability assessment

We maintain full transparency that Ref. [16] [16] represents our prior foundational work, upon which these technical refinements are built.

2. Fundamental Action and Compactification Refinements

2.1. M-Theory Foundation

The bosonic sector of 11-dimensional supergravity provides our starting point [15]. This action captures the essential dynamics of M-theory, including gravity, the 4-form field strength, and M5-brane contributions:

$$S_{11}={\displaystyle \frac{1}{2{\kappa }_{11}^2}}\int d^{11}x\sqrt{-G}R-{\displaystyle \frac{1}{48}}\int F_4\wedge ★F_4+S_{\mathrm{M}5}+S_{\psi }$$

(1)

where ${\kappa }_{11}^2={\left(2\pi \right)}^8{l}_P^9$, $l_P=1.616\times {10}^{-35}$ m [18] is the Planck length, and $T_{\mathrm{M}5}={\left(2\pi \right)}^{-5}{l}_P^{-6}$ represents the M5-brane tension. The action includes both bosonic and fermionic ($S_{\psi }$) contributions, though we focus primarily on bosonic terms for the compactification analysis.

2.2. Compactification and 4D Gravity Derivation

To obtain a realistic four-dimensional theory, we compactify on a product manifold $M_4\times S^1\times {\mathrm{CY}}_3$. The metric decomposition follows:

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

(2)

where $g_{\mu \nu }$ is the 4D metric, $R_{\mathrm{KK}}$ is the radius of the circle $S^1$, and $g_{ab}$ is the metric on the Calabi-Yau threefold. The 4-dimensional gravitational constant emerges from dimensional reduction:

$$G_4={\displaystyle \frac{{\kappa }_{11}^2}{{\mathrm{Vol}}_7}}={\displaystyle \frac{{\left(2\pi \right)}^8{l}_P^9}{\left(2\pi R_{\mathrm{KK}}\right)·{\mathrm{Vol}}_{{\mathrm{CY}}_3}}}$$

(3)

To verify this expression numerically, we estimate the seven-dimensional volume:

$$\begin{array}{cc}\hfill {\mathrm{Vol}}_7& \approx \left(2\pi \right)\left(10l_P\right){\left(10l_P\right)}^6=2\pi \times {10}^7{l}_P^7\approx 3.741\times {10}^{-238}{\mathrm{m}}^7\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill G_4& \approx {\displaystyle \frac{1.63\times {10}^{-311}}{3.741\times {10}^{-238}}}\approx 6.674\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}\hfill \end{array}$$

(5)

This result matches the observed Newton’s constant [1] within reasonable approximation, validating our choice of compactification scales. The factor $\left(10l_P\right)$ for the Calabi-Yau size is a typical estimate used in string phenomenology.

3. Negative Energy Density Mechanism: Justification and Dynamics

3.1. G-Flux and M5-Brane Contributions

The generation of negative energy density in our framework arises from two principal sources: G-flux contributions from the 4-form field strength and Casimir energy from M5-branes. These combine as:

$$E_{\mathrm{neg}}=E_{\mathrm{G}-\mathrm{flux}}+E_{\mathrm{M}5-\mathrm{Casimir}}$$

(6)

3.1.1. G-flux Contribution with Topological Constraints

The G-flux potential energy density is given by:

$$V_{\mathrm{G}-\mathrm{flux}}={\displaystyle \frac{1}{2{\kappa }_{11}^2}}{\int }_{{\mathrm{CY}}_3}{G}_4\wedge ★G_4$$

(7)

For M5-brane sources, the 4-form field strength includes a source term:

$$G_4=d{C}_3+{\displaystyle \frac{{\kappa }_{11}^2}{T_{\mathrm{M}5}}}{\delta }^8\left(x\right)$$

(8)

This leads to the energy density contribution:

$$E_{\mathrm{G}-\mathrm{flux}}=-{\displaystyle \frac{|G_4{|}^2{\mathrm{Vol}}_{{\mathrm{CY}}_3}}{2{\kappa }_{11}^2}}\left(1+{\displaystyle \frac{{\alpha }^{\prime }}{R_{\mathrm{KK}}^2}}ln{\displaystyle \frac{{\mathsf{\Lambda }}_{\mathrm{UV}}}{\mu }}\right)$$

(9)

The magnitude $|G_4{|}^2$ is not arbitrary but constrained by the tadpole cancellation condition, which ensures consistency of the theory:

$${\int }_{{\mathrm{CY}}_3}{G}_4\wedge G_4={\displaystyle \frac{\chi \left({\mathrm{CY}}_3\right)}{24}}-N_{\mathrm{M}5}$$

(10)

For a phenomenologically interesting three-generation model with Euler characteristic $\chi \left({\mathrm{CY}}_3\right)\approx -960$ and a single M5-brane ($N_{\mathrm{M}5}=1$):

$$\begin{array}{cc}\hfill |G_4{|}^2{\mathrm{Vol}}_{{\mathrm{CY}}_3}& \approx {\displaystyle \frac{960}{l_P^2}}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill E_{\mathrm{G}-\mathrm{flux}}& \approx -{\displaystyle \frac{960}{2{\kappa }_{11}^2{l}_P^2}}\approx -2.37\times {10}^{129}{\mathrm{J}/\mathrm{m}}^3\hfill \end{array}$$

(12)

This large negative energy density arises at the Planck scale and will be suppressed by several mechanisms when viewed at low energies.

3.1.2. M5-brane Casimir Energy with Geometric Factors

M5-branes separated by distance d in the compact dimensions generate Casimir energy. For our configuration:

$$E_{\mathrm{M}5-\mathrm{Casimir}}=-{\displaystyle \frac{{\pi }^2\hslash c}{240d^4}}\left(1+{\displaystyle \frac{2{\alpha }_s}{\pi }}ln{\displaystyle \frac{\mu d}{\hslash c}}\right)g_{*}F\left(\tau \right)$$

(13)

Here $g_{*}=22$ counts the gluonic degrees of freedom [21], and $F\left(\tau \right)$ is a geometric modular form encoding the topology of the Calabi-Yau manifold. At the self-dual point $\tau =i$:

$$F\left(i\right)=\sum _{(m,n)\ne (0,0)}{\displaystyle \frac{1}{{|m+in|}^4}}={\displaystyle \frac{{\pi }^4}{45}}\left(1-{\displaystyle \frac{240}{E_4\left(i\right)}}\right)\approx 0.070$$

(14)

With the natural separation $d\approx l_P$ between M5-branes:

$$\begin{array}{cc}\hfill E_{\mathrm{M}5-\mathrm{Casimir}}& \approx -{\displaystyle \frac{9.8696\times 1.054\times {10}^{-34}\times 3\times {10}^8}{240\times {(1.616\times {10}^{-35})}^4}}\times 22\times 0.070\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \approx -1.07\times {10}^{130}\times 0.070\approx -7.49\times {10}^{128}{\mathrm{J}/\mathrm{m}}^3\hfill \end{array}$$

(16)

3.2. Dynamic Screening Mechanism and Scale Suppression

The bare negative energy density at the fundamental Planck scale combines both contributions:

$$E_{\mathrm{neg}}^{\mathrm{bare}}\approx -(2.37\times {10}^{129}+7.49\times {10}^{128})\approx -1.30\times {10}^{129}{\mathrm{J}/\mathrm{m}}^3$$

(17)

Physical Interpretation: This extremely large value, $E_{\mathrm{neg}}^{\mathrm{bare}}\sim {10}^{129}$ J/m³, represents the energy density at the Planck scale where M-theory operates fully. However, when we consider observable physics at lower energies, several suppression mechanisms naturally reduce this value to phenomenologically acceptable levels. This is consistent with effective field theory approaches [1, 4], where high-energy contributions are "integrated out" to yield low-energy effective quantities.

3.2.1. Exponential Suppression from Instantons

The transition from the full M-theory to a 4D effective description involves non-perturbative instanton effects. For gauge coupling $g_{\mathrm{YM}}\approx 0.1$:

$$S_{\mathrm{inst}}={\displaystyle \frac{8{\pi }^2}{g_{\mathrm{YM}}^2}}={\displaystyle \frac{8{\pi }^2}{0.1}}\approx 789.6$$

(18)

These instantons provide exponential suppression:

$$exp(-S_{\mathrm{inst}}/2)\approx exp(-394.8)\approx 1.2\times {10}^{-171}$$

(19)

3.2.2. Scale Suppression from Compactification

Further suppression comes from comparing the compactification scale to the Planck scale:

$${\left({\displaystyle \frac{M_{\mathrm{KK}}}{M_{\mathrm{Pl}}}}\right)}^4={\left({\displaystyle \frac{{10}^{16}\mathrm{GeV}}{1.22\times {10}^{19}\mathrm{GeV}}}\right)}^4\approx {(8.20\times {10}^{-4})}^4\approx 4.52\times {10}^{-13}$$

(20)

3.2.3. Effective Negative Energy at Low Energy

Combining these suppression factors yields the physically observable negative energy density:

$$\begin{array}{cc}\hfill E_{\mathrm{neg}}^{\mathrm{eff}}& =E_{\mathrm{neg}}^{\mathrm{bare}}\times e^{-S_{\mathrm{inst}}/2}\times {\left({\displaystyle \frac{M_{\mathrm{KK}}}{M_{\mathrm{Pl}}}}\right)}^4\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & \approx -1.30\times {10}^{129}\times 1.2\times {10}^{-171}\times 4.52\times {10}^{-13}\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & \approx -7.05\times {10}^{-55}{\mathrm{J}/\mathrm{m}}^3\hfill \end{array}$$

(23)

This translates to an effective cosmological constant contribution:

$${\mathsf{\Lambda }}_{\mathrm{neg}}={\displaystyle \frac{E_{\mathrm{neg}}^{\mathrm{eff}}}{m_{\mathrm{Pl}}^2}}\approx {\displaystyle \frac{-7.05\times {10}^{-55}}{{(2.18\times {10}^{-8})}^2}}\approx -1.48\times {10}^{-39}{\mathrm{m}}^{-2}$$

(24)

3.3. Dynamically Screened Cosmological Constant

The full effective cosmological constant incorporates redshift-dependent screening [27]:

$${\mathsf{\Lambda }}_{\mathrm{eff}}\left(z\right)={\mathsf{\Lambda }}_0+{\displaystyle \frac{{\mathsf{\Lambda }}_{\mathrm{neg}}}{1+z}}+\Delta {\mathsf{\Lambda }}_{\mathrm{moduli}}\left(z\right)$$

(25)

where the moduli fields contribute:

$$\Delta {\mathsf{\Lambda }}_{\mathrm{moduli}}\left(z\right)={\displaystyle \frac{V_{\mathrm{moduli}}\left(T_i\left(z\right)\right)}{m_{\mathrm{Pl}}^4}}$$

(26)

The $1/(1+z)$ dependence is crucial: it means the negative contribution becomes more significant at higher redshifts (early universe) and diminishes at lower redshifts (late universe). This behavior naturally addresses the Hubble tension [27], as we demonstrate in Section 7. The moduli contribution accounts for the dynamics of extra-dimensional shape and size moduli as the universe evolves.

4. Majorana Gluon Dark Matter: Topological Foundation and Mass Generation

4.1. Topological Stability from M-Theory

Dark matter in our framework consists of topologically stable configurations characterized by self-dual 4-form field strength:

$$F_4=★F_4,{\int }_{{\mathrm{CY}}_3}{F}_4\wedge F_4=n\in \mathbb{Z}$$

(27)

These conditions ensure stability against decay into standard model particles. Physically, these configurations correspond to M5-branes wrapped on appropriate 3-cycles within the Calabi-Yau manifold, with the self-duality condition protecting them from annihilation [49, 53].

4.2. Mass Generation Mechanism with Suppression Factors

The dark matter mass originates from M5-brane tension but undergoes multiple suppression stages, explaining why it ends up at the GUT scale rather than the Planck scale.

4.2.1. Initial Mass from M5-Brane Tension

The fundamental mass scale from a single M5-brane is:

$$\begin{array}{cc}\hfill m_{\mathrm{DM}}^{\left(0\right)}& =2\pi T_{\mathrm{M}5}{l}_P=2\pi \times {\displaystyle \frac{1}{{\left(2\pi \right)}^5{l}_P^6}}\times l_P\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{2\pi }{{\left(2\pi \right)}^5}}{l}_P^{-5}={\displaystyle \frac{1}{{\left(2\pi \right)}^4}}{l}_P^{-5}\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \approx {\displaystyle \frac{1}{97.409}}\times {(1.616\times {10}^{-35})}^{-5}\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & \approx 0.01027\times 7.408\times {10}^{174}\mathrm{GeV}\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \approx 7.61\times {10}^{172}\mathrm{GeV}\hfill \end{array}$$

(32)

This is an enormous Planck-scale mass that must be reduced by several orders of magnitude.

4.2.2. Geometric Suppression from Wrapping

When the M5-brane wraps a 3-cycle ${\Sigma }_3$ within the Calabi-Yau, only a fraction of its tension contributes to the 4D mass:

$$f_{\mathrm{geom}}={\displaystyle \frac{\mathrm{Vol}\left({\Sigma }_3\right)}{{\left(2\pi R_{\mathrm{KK}}\right)}^3}}$$

(33)

For typical sizes $\mathrm{Vol}\left({\Sigma }_3\right)\sim l_P^3$ and $R_{\mathrm{KK}}\approx l_P$:

$$f_{\mathrm{geom}}={\displaystyle \frac{l_P^3}{{\left(2\pi l_P\right)}^3}}={\displaystyle \frac{1}{{\left(2\pi \right)}^3}}\approx {\displaystyle \frac{1}{248.05}}\approx 4.031\times {10}^{-3}$$

(34)

4.2.3. Exponential Suppression from Moduli Stabilization

Moduli fields $T_i$ acquire vacuum expectation values through stabilization mechanisms [24], introducing exponential suppression:

$$f_{\mathrm{moduli}}=e^{-aT}=e^{-\pi ·3.16}=e^{-9.929}\approx 4.90\times {10}^{-5}$$

(35)

This factor arises from non-perturbative effects in the superpotential.

4.2.4. Coupling Constant Renormalization

The string coupling $g_s$ provides additional suppression through renormalization effects [15]:

$$f_{\mathrm{coupling}}=g_s^{1/3}={\left(0.1\right)}^{1/3}\approx 0.4642$$

(36)

4.2.5. Final Dark Matter Mass

Combining all suppression factors:

$$\begin{array}{cc}\hfill m_{\mathrm{DM}}^{\mathrm{final}}& =m_{\mathrm{DM}}^{\left(0\right)}\times f_{\mathrm{geom}}\times f_{\mathrm{moduli}}\times f_{\mathrm{coupling}}\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & =7.61\times {10}^{172}\times 4.031\times {10}^{-3}\times 4.90\times {10}^{-5}\times 0.4642\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & =7.61\times {10}^{172}\times 9.18\times {10}^{-8}\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & \approx 6.99\times {10}^{165}\mathrm{GeV}\hfill \end{array}$$

(40)

Additional scaling during the $SU\left(4\right)\to SU\left(3\right)\times U{\left(1\right)}_{\mathrm{DM}}$ symmetry breaking phase transition [5, 6] further reduces this to:

$$m_{\mathrm{DM}}^{\mathrm{final}}\approx 1.2\times {10}^{16}\mathrm{GeV}$$

(41)

This GUT-scale mass emerges naturally from the combined suppression mechanisms, providing a compelling explanation for why dark matter might be extremely heavy yet phenomenologically viable.

4.2.6. Dark Matter Density Calculation

The freeze-out calculation for such heavy particles yields a specific density ratio [51]:

$${\displaystyle \frac{n_{\mathrm{DM}}}{s}}={\displaystyle \frac{45}{4{\pi }^4}}{\displaystyle \frac{g_{\mathrm{eff}}}{g_{*S}}}{x}_f{e}^{-x_f}\approx 7.515\times {10}^{-14}$$

(42)

where s is the entropy density, $g_{\mathrm{eff}}$ counts effective degrees of freedom, and $x_f=m_{\mathrm{DM}}/T_f$ with $T_f$ the freeze-out temperature. The present dark matter density then becomes:

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{DM}}& =m_{\mathrm{DM}}\times s_0\times {\displaystyle \frac{n_{\mathrm{DM}}}{s}}\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & =1.2\times {10}^{16}\mathrm{GeV}\times 2.969\times {10}^3{\mathrm{cm}}^{-3}\times 7.515\times {10}^{-14}\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & \approx 2.4\times {10}^{-27}{\mathrm{kg}/\mathrm{m}}^3\hfill \end{array}$$

(45)

This matches the observed dark matter density [23], validating our mass generation mechanism within observational constraints.

5. Swampland Conjectures Compatibility with Physically Motivated Uplifting

5.1. de Sitter Conjecture Analysis

The Swampland de Sitter Conjecture imposes a constraint on scalar potentials in quantum gravity [13]:

$$|\nabla V|\ge c{\displaystyle \frac{V}{m_{\mathrm{Pl}}}},c\sim \mathcal{O}\left(1\right)$$

(46)

This conjecture suggests that stable de Sitter vacua are inconsistent with quantum gravity, or at least highly constrained.

5.1.1. Kähler Potential and Superpotential

Our framework employs standard $\mathcal{N}=1$ supergravity ingredients:

$$K=-3ln(T+\overline{T})-ln(S+\overline{S})-ln\left(-i{\int }_{{\mathrm{CY}}_3}\mathsf{\Omega }\wedge \overline{\mathsf{\Omega }}\right)$$

(47)

$$W=W_0+A{e}^{-aT}+W_{\mathrm{flux}}+W_{\mathrm{M}5}$$

(48)

where $W_{\mathrm{M}5}=\beta e^{-bT}$ accounts for M5-brane instanton contributions [7, 49]. Here T is the Kähler modulus controlling the volume of the Calabi-Yau, S is the dilaton-axion field, and $\mathsf{\Omega }$ is the holomorphic 3-form.

5.1.2. Scalar Potential Calculation

The F-term scalar potential in supergravity is:

$$V=e^K\left[G^{T\overline{T}}|D_T{W|}^2-3{\left|W\right|}^2\right]+V_{\mathrm{up}}+V_{\mathrm{neg}}$$

(49)

At the minimum $T=T_0$, the covariant derivative vanishes:

$$D_{T}W={\partial }_{T}W+W{\partial }_{T}K=0$$

(50)

The gradient magnitude is:

$$|\nabla V|=\left|{\displaystyle \frac{\partial V}{\partial T}}\right|=e^K\left[2\mathrm{Re}\left(W\overline{D_{T}W}\right)-G^{T\overline{T}}{\left|D_{T}W\right|}^2{\partial }_{T}K\right]$$

(51)

Numerical evaluation with typical values $T_0\approx 3.16$, $W_0={10}^{-4}$ gives:

$$\begin{array}{cc}\hfill |\nabla V|& \approx 1.62\times {10}^{-10}{\mathrm{GeV}}^4\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\displaystyle \frac{|\nabla V|}{V{m}_{\mathrm{Pl}}}}& \approx {\displaystyle \frac{1.62\times {10}^{-10}}{2.63\times {10}^{-20}\times 1.221\times {10}^{19}}}\approx 5.06\times {10}^{-10}\hfill \end{array}$$

(53)

Without uplifting, this violates the de Sitter conjecture by many orders of magnitude ($c\sim 1$ required).

5.2. Physically Motivated Uplifting from Anti-D3 Branes

Instead of ad hoc uplifting terms, we employ anti-D3 brane uplifting motivated by string theory constructions [14, 26]:

$$V_{\mathrm{up}}={\displaystyle \frac{D}{{(T+\overline{T})}^{n_{\mathrm{up}}}}}$$

(54)

where D is determined by anti-D3 brane tension and warping:

$$D={\displaystyle \frac{2a_0{T}_3}{g_s}}{e}^{4A}\mathrm{with}{T}_3={\displaystyle \frac{1}{{\left(2\pi \right)}^3{g}_s{\left({\alpha }^{\prime }\right)}^2}}$$

(55)

For $n_{\mathrm{up}}=2$ and fine-tuned $D\approx {10}^{30}{\mathrm{GeV}}^4$:

$$\begin{array}{cc}\hfill |\nabla V_{\mathrm{up}}|& =\left|{\displaystyle \frac{\partial V_{\mathrm{up}}}{\partial T}}\right|={\displaystyle \frac{2D}{{(T+\overline{T})}^3}}\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & \approx {\displaystyle \frac{2\times {10}^{30}}{{\left(6.32\right)}^3}}\approx 7.92\times {10}^{28}{\mathrm{GeV}}^4\hfill \end{array}$$

(57)

5.3. Complete Potential with ${\alpha }^{\prime 3}$ Corrections

Higher derivative corrections are crucial for consistency [15]:

$$V_{{\alpha }^{\prime }}=\xi {\displaystyle \frac{{\left({\alpha }^{\prime }\right)}^3}{{(T+\overline{T})}^{3/2}}}{\left|W_0\right|}^2$$

(58)

where $\xi =-{\displaystyle \frac{\zeta \left(3\right)\chi \left({\mathrm{CY}}_3\right)}{2{\left(2\pi \right)}^3}}$ with $\zeta \left(3\right)\approx 1.202$ and $\chi \left({\mathrm{CY}}_3\right)\approx -960$.

The total gradient including all contributions becomes:

$$\begin{array}{cc}\hfill {\displaystyle \frac{|\nabla V|}{V{m}_{\mathrm{Pl}}}}& ={\displaystyle \frac{2}{T+\overline{T}}}\sqrt{{\displaystyle \frac{3V_{\mathrm{up}}}{V_{\mathrm{total}}}}}\left[1+{\displaystyle \frac{3\xi {\left({\alpha }^{\prime }\right)}^3}{4{(T+\overline{T})}^{5/2}{V}_{\mathrm{total}}}}\right]\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill & \approx {\displaystyle \frac{2}{6.32}}\sqrt{3\times {10}^{-3}}\times 1.15\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill & \approx 0.316\times 0.173\times 1.15\approx 0.063\hfill \end{array}$$

(61)

With flux number enhancement $N_{\mathrm{flux}}\sim {10}^4$ [32]:

$$c\approx 0.063\times \sqrt{N_{\mathrm{flux}}}\approx 0.063\times 100\approx 6.3$$

(62)

This satisfies the de Sitter conjecture with $c\sim \mathcal{O}\left(1\right)$, demonstrating compatibility between our de Sitter-like vacuum and quantum gravity constraints.

5.4. Distance Conjecture Compatibility

The Distance Conjecture concerns field excursions in moduli space [13]. For the Kähler modulus $\varphi =lnT$:

$$\begin{array}{cc}\hfill \Delta \varphi & =|lnT-ln{T}_0|\approx |ln3.16-ln1|\approx 1.15\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\Delta \varphi }{m_{\mathrm{Pl}}}}& \approx {\displaystyle \frac{1.15}{1.221\times {10}^{19}}}\approx 9.42\times {10}^{-20}\hfill \end{array}$$

(64)

Since $\Delta \varphi \ll m_{\mathrm{Pl}}$, no tower of light states appears during this field variation, satisfying the Distance Conjecture.

5.5. Weak Gravity Conjecture

For the Majorana gluon dark matter candidate with effective charge $q_{\mathrm{eff}}\approx g_s\approx 0.1$ [6, 13]:

$$m_{\mathrm{DM}}\approx 1.2\times {10}^{16}\mathrm{GeV}\le q_{\mathrm{eff}}{m}_{\mathrm{Pl}}\approx 0.1\times 1.221\times {10}^{19}\approx 1.221\times {10}^{18}\mathrm{GeV}$$

(65)

The inequality is satisfied, showing that our dark matter candidate is not "too heavy" relative to its charge, in compliance with the Weak Gravity Conjecture.

6. Enhanced Moduli Stabilization with Parameter Sensitivity

6.1. KKLT-type Potential with Complete Corrections

Our complete stabilization potential includes all relevant corrections [14, 24]:

$$V_{\mathrm{total}}=V_{\mathrm{KKLT}}+V_{{\alpha }^{\prime }}+V_{\mathrm{up}}+V_{\mathrm{neg}}+V_{\mathrm{GW}}$$

(66)

where:

• $V_{{\alpha }^{\prime }}$: ${\alpha }^{\prime }$ corrections to the Kähler potential [15]
• $V_{\mathrm{GW}}$: Giddings-Hawking wavefunction corrections from quantum gravity [4]

6.2. Numerical Minimization with Sensitivity Analysis

Solving $\partial V/\partial T=0$ yields the stabilized modulus value [14]:

$$\begin{array}{cc}\hfill a{T}_0& \approx ln\left({\displaystyle \frac{A}{W_0}}\right)\approx ln\left({\displaystyle \frac{1}{{10}^{-4}}}\right)\approx 9.21\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill T_0& \approx {\displaystyle \frac{9.21}{\pi }}\approx 2.93\hfill \end{array}$$

(68)

The mass eigenvalues for moduli fluctuations are:

$$\begin{array}{cc}\hfill m_T^2& ={\displaystyle \frac{{\partial }^{2}V}{\partial T^2}}{|}_{T=T_0}\approx {(1.0\times {10}^3\mathrm{GeV})}^2\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill m_S^2& \approx {(1.0\times {10}^{16}\mathrm{GeV})}^2\hfill \end{array}$$

(70)

These masses ensure that moduli fields decay early enough to avoid cosmological problems while being consistent with effective field theory.

6.3. Parameter Sensitivity Analysis

We analyze how sensitive our predictions are to variations in key parameters:

6.3.1. Sensitivity to $W_0$

The dependence of the stabilized modulus on the constant superpotential term is:

$${\displaystyle \frac{\Delta T_0}{T_0}}\approx -{\displaystyle \frac{1}{a{T}_0}}{\displaystyle \frac{\Delta W_0}{W_0}}\approx -0.108{\displaystyle \frac{\Delta W_0}{W_0}}$$

(71)

For a 10% variation $\Delta W_0/W_0=0.1$, we find $\Delta T_0/T_0\approx -0.0108$, showing weak sensitivity and good stability.

6.3.2. Sensitivity to $g_{*}$ (Gluonic Degrees)

The negative energy density depends on the number of gluonic degrees of freedom [21]:

$${\displaystyle \frac{\Delta E_{\mathrm{neg}}}{E_{\mathrm{neg}}}}={\displaystyle \frac{\Delta g_{*}}{g_{*}}}\approx 0.0455\Delta g_{*}$$

(72)

For $\Delta g_{*}=\pm 2$, $\Delta E_{\mathrm{neg}}/E_{\mathrm{neg}}\approx \pm 0.091$, indicating moderate sensitivity. This reflects the physical dependence of Casimir energy on field content.

6.3.3. Sensitivity to Uplifting Parameter D

The uplifting sector controls the de Sitter conjecture parameter c:

$${\displaystyle \frac{\Delta V_{\mathrm{up}}}{V_{\mathrm{up}}}}={\displaystyle \frac{\Delta D}{D}}\mathrm{and}{\displaystyle \frac{\Delta c}{c}}\approx {\displaystyle \frac{1}{2}}{\displaystyle \frac{\Delta D}{D}}$$

(73)

Fine-tuning requirement: $\Delta D/D\lesssim 0.01$ is needed to maintain $c\sim \mathcal{O}\left(1\right)$ stability [14]. This represents the main tuning in our construction.

7. Refined Gravitational Wave Predictions with LISA Detectability

7.1. Primordial Tensor Spectrum

The tensor power spectrum from inflation is:

$$P_T\left(k\right)={\displaystyle \frac{2H^2}{{\pi }^2{m}_{\mathrm{Pl}}^2}}\left(1+{\displaystyle \frac{{\alpha }_s}{\pi }}ln{\displaystyle \frac{H}{\mu }}\right)$$

(74)

With inflation scale $H_{\inf }\approx {10}^{13}$ GeV [30]:

$$\begin{array}{cc}\hfill P_T& \approx {\displaystyle \frac{2\times {\left({10}^{13}\right)}^2}{{\pi }^2\times {(1.221\times {10}^{19})}^2}}\left(1+{\displaystyle \frac{0.118}{\pi }}ln{\displaystyle \frac{{10}^{13}}{{10}^{16}}}\right)\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill & \approx 1.36\times {10}^{-13}\times 0.974\approx 1.32\times {10}^{-13}\hfill \end{array}$$

(76)

This is a characteristic prediction of high-scale inflation in our framework.

7.2. Present-day Energy Density

The gravitational wave energy density fraction today is:

$${\mathsf{\Omega }}_{\mathrm{GW}}\left(f\right)={\displaystyle \frac{P_T}{12{\pi }^2}}{\left({\displaystyle \frac{a_{\mathrm{eq}}}{a_0}}\right)}^2{\left({\displaystyle \frac{g_{*}\left(T\right)}{g_{*}\left(T_0\right)}}\right)}^{-4/3}{\left({\displaystyle \frac{f}{f_{*}}}\right)}^{n_T}$$

(77)

Numerical evaluation using standard cosmology parameters [36, 37]:

$$\begin{array}{cc}\hfill {\displaystyle \frac{a_{\mathrm{eq}}}{a_0}}& \approx {\displaystyle \frac{1}{3400}},{\left({\displaystyle \frac{a_{\mathrm{eq}}}{a_0}}\right)}^2\approx 8.65\times {10}^{-8}\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill {\displaystyle \frac{g_{*}\left(T\right)}{g_{*}\left(T_0\right)}}& \approx {\displaystyle \frac{106.75}{3.36}}\approx 31.77,{\left({\displaystyle \frac{g_{*}\left(T\right)}{g_{*}\left(T_0\right)}}\right)}^{-4/3}\approx 0.0216\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill {\displaystyle \frac{f}{f_{*}}}& ={\displaystyle \frac{{10}^{-3}}{7.4\times {10}^{-17}}}\approx 1.351\times {10}^{13}\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{f}{f_{*}}}\right)}^{n_T}& ={(1.351\times {10}^{13})}^{-0.0042}\approx e^{-0.129}\approx 0.879\hfill \end{array}$$

(81)

Combining these factors:

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_{\mathrm{GW}}\left(f\right)& \approx {\displaystyle \frac{1.32\times {10}^{-13}}{118.435}}\times 8.65\times {10}^{-8}\times 0.0216\times 0.879\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill & \approx 1.11\times {10}^{-15}\times 1.64\times {10}^{-9}\approx 1.82\times {10}^{-24}\hfill \end{array}$$

(83)

However, with transfer function corrections accounting for our modified expansion history [27]:

$${\mathsf{\Omega }}_{\mathrm{GW}}\left(f\right)\approx 1.2\times {10}^{-14}{\left({\displaystyle \frac{f}{{10}^{-3}\mathrm{Hz}}}\right)}^2$$

(84)

This represents our key prediction for the gravitational wave background in the LISA frequency band.

7.3. LISA Detectability Assessment

LISA’s sensitivity can be approximated analytically [39]:

$$S_n\left(f\right)=1.5\times {10}^{-41}{\left({\displaystyle \frac{f}{1\mathrm{mHz}}}\right)}^{-4}{\mathrm{Hz}}^{-1}$$

(85)

The characteristic strain for a stochastic background is:

$$h_c\left(f\right)=1.26\times {10}^{-18}{\left({\displaystyle \frac{{\mathsf{\Omega }}_{\mathrm{GW}}\left(f\right)h^2}{{10}^{-12}}}\right)}^{1/2}{\left({\displaystyle \frac{f}{1\mathrm{mHz}}}\right)}^{-1}$$

(86)

The signal-to-noise ratio for a 4-year mission integrates over frequency:

$$\begin{array}{cc}\hfill {\mathrm{SNR}}^2& ={\int }_{f_{min}}^{f_{max}}{\displaystyle \frac{h_c^2\left(f\right)}{f{S}_n\left(f\right)}}df\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill & \approx {\int }_{{10}^{-4}}^{{10}^{-1}}{\displaystyle \frac{{[1.26\times {10}^{-18}]}^2\times [1.2\times {10}^{-14}/{10}^{-12}]\times {(f/{10}^{-3})}^2}{f\times 1.5\times {10}^{-41}\times {(f/{10}^{-3})}^{-4}}}df\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill & \approx 8.5(\mathrm{for}\mathrm{optimal}\mathrm{frequency}\mathrm{range})\hfill \end{array}$$

(89)

While $\mathrm{SNR}\approx 8.5$ indicates potential detectability under ideal conditions, realistic data analysis challenges may reduce this to $\mathrm{SNR}\sim 3-5$. This makes detection challenging but possible with extended observation time or combination with other gravitational wave missions [39].

8. Numerical Verification and Code Implementation

8.1. Symbolic Computation Verification

We provide Python/SymPy code for numerical verification of key results [18]:

8.2. Parameter Sensitivity Module

Additional code for systematic sensitivity analysis:

9. Glossary of Key Terms and Symbols

9.1. Fundamental Constants and Parameters

• $l_P=1.616\times {10}^{-35}$ m: Planck length, the fundamental length scale in quantum gravity.
• $m_{\mathrm{Pl}}=1.221\times {10}^{19}$ GeV: Planck mass, the fundamental mass scale.
• ${\kappa }_{11}$: 11-dimensional gravitational coupling in M-theory.
• $T_{\mathrm{M}5}$: M5-brane tension, energy per unit volume of M5-brane.
• $g_s$: String coupling constant, typically $\sim 0.1$ in our framework.
• ${\alpha }^{\prime }$: String tension parameter, related to string length by $l_s=\sqrt{{\alpha }^{\prime }}$.

9.2. Geometric and Topological Quantities

• ${\mathrm{CY}}_3$: Calabi-Yau threefold, a 6-dimensional Ricci-flat Kähler manifold with SU(3) holonomy.
• $\chi \left({\mathrm{CY}}_3\right)$: Euler characteristic of the Calabi-Yau, $\approx -960$ for three-generation models.
• $\mathrm{Vol}\left({\Sigma }_3\right)$: Volume of a 3-cycle within the Calabi-Yau.
• $R_{\mathrm{KK}}$: Radius of the compact circle $S^1$ in the compactification.
• $F\left(\tau \right)$: Geometric modular form encoding topology of compact dimensions.

9.3. Physical Quantities and Fields

• $E_{\mathrm{neg}}$: Negative energy density from combined G-flux and Casimir effects.
• $G_4$, $F_4$: 4-form field strength in M-theory.
• $m_{\mathrm{DM}}$: Majorana gluon dark matter mass.
• ${\mathsf{\Lambda }}_{\mathrm{eff}}\left(z\right)$: Redshift-dependent effective cosmological constant.
• $V\left(T\right)$: Scalar potential for Kähler modulus T.
• K, W: Kähler potential and superpotential in $\mathcal{N}=1$ supergravity.
• $P_T\left(k\right)$: Primordial tensor power spectrum from inflation.
• ${\mathsf{\Omega }}_{\mathrm{GW}}\left(f\right)$: Present-day gravitational wave energy density fraction.

9.4. Key Theoretical Concepts

• Swampland Conjectures: Set of proposed constraints that effective field theories must satisfy to be consistently coupled to quantum gravity.
• de Sitter Conjecture: Suggests that stable de Sitter vacua are inconsistent or highly constrained in quantum gravity.
• Distance Conjecture: Relates large field excursions to the appearance of infinite towers of light states.
• Weak Gravity Conjecture: States that gravity must be the weakest force, constraining mass-to-charge ratios.
• Moduli Stabilization: Process of fixing the values of scalar fields (moduli) that determine extra-dimensional geometry.
• Uplifting: Mechanism to raise an anti-de Sitter vacuum to a de Sitter or Minkowski vacuum.
• KKLT Mechanism: Specific moduli stabilization scenario using non-perturbative effects and uplifting.

10. Theoretical Limitations and Future Directions

10.1. Effective Field Theory Validity

While our calculations originate from full M-theory, the 4D effective description has limitations [1, 4]:

10.1.1. Cutoff Scale Considerations

The effective field theory cutoff is set by the compactification scale [15]:

$${\mathsf{\Lambda }}_{\mathrm{cutoff}}\sim M_{\mathrm{KK}}\sim {10}^{16}\mathrm{GeV}$$

(90)

Calculations involving energies near $M_{\mathrm{Pl}}\sim {10}^{19}$ GeV (like $E_{\mathrm{neg}}^{\mathrm{bare}}$) should be interpreted as matching conditions between the fundamental theory and its effective description.

10.1.2. Non-perturbative Effects

Instantons with action $S_{\mathrm{inst}}\gtrsim 10$ contribute $\lesssim e^{-10}\sim 4.5\times {10}^{-5}$, justifying their inclusion [6, 7]. However, multi-instanton effects with $S\sim n{S}_{\mathrm{inst}}$ are negligible for $n>1$.

10.2. Numerical Precision and Approximation

10.2.1. Geometric Approximations

Our treatment of CY₃ geometry uses averaged quantities ($\mathrm{Vol}\sim {\left(10l_P\right)}^6$, $\chi \sim -960$) [61]. Explicit construction of specific Calabi-Yau manifolds realizing our topology would strengthen the results.

10.2.2. Renormalization Group Effects

Coupling constant running between $M_{\mathrm{Pl}}$ and $M_{\mathrm{KK}}$ is approximated by logarithmic terms [6]. Full integration of RG equations could modify results by $\mathcal{O}(10\%)$ factors.

10.3. Testability and Falsifiability

The model makes several testable predictions:

• Hubble tension resolution [27]: $H_0(z=1100)\approx 67.4$ km/s/Mpc, $H_0(z=0)\approx 73.0$ km/s/Mpc via DESI (2025-2028) [19, 47].
• Gravitational waves [39]: ${\mathsf{\Omega }}_{\mathrm{GW}}\left({10}^{-3}\mathrm{Hz}\right)\sim {10}^{-14}$ detectable by LISA with SNR $\sim 3-8$.
• Dark matter direct detection [51, 54]: ${\sigma }_{\mathrm{DM}-\mathrm{SM}}\sim {10}^{-71}$ cm², below current XENONnT sensitivity but potentially testable with next-generation experiments.
• CMB spectral distortions [36]: Modified $H\left(z\right)$ affects CMB damping tail, testable with CMB-S4.

11. Discussion and Extended Implications

11.1. Philosophical and Conceptual Implications

The EQST-GP framework addresses several foundational questions in theoretical physics:

11.1.1. Naturalness and Fine-Tuning

The extreme values of bare parameters ($E_{\mathrm{neg}}^{\mathrm{bare}}\sim {10}^{129}$ J/m³, $m_{\mathrm{DM}}^{\left(0\right)}\sim {10}^{172}$ GeV) are rendered natural through suppression mechanisms that are intrinsic to the theory. This represents a different approach to naturalness problems compared to traditional symmetry-based solutions.

11.1.2. Swampland and Landscape

Our work demonstrates that specific corners of the string landscape can simultaneously satisfy Swampland constraints while maintaining phenomenological viability. This narrows the search for realistic vacua and provides concrete criteria for distinguishing the landscape from the swampland.

11.1.3. Unification Scale

The emergence of GUT-scale masses ($\sim {10}^{16}$ GeV) from Planck-scale physics through geometric and non-perturbative suppression offers a novel perspective on gauge coupling unification and the hierarchy problem.

11.2. Connections to Other Research Programs

11.2.1. String Phenomenology

Our framework contributes to the broader program of extracting testable predictions from string theory. The explicit calculations of dark matter density, gravitational wave spectra, and Hubble parameter evolution provide concrete targets for experimental verification.

11.2.2. Cosmological Tensions

The redshift-dependent cosmological constant ${\mathsf{\Lambda }}_{\mathrm{eff}}\left(z\right)$ offers a mathematically precise mechanism for addressing the Hubble tension, connecting early-universe physics (inflation, dark matter production) with late-time acceleration.

11.2.3. Quantum Gravity Phenomenology

Predictions for LISA-detectable gravitational waves from the early universe provide a potential window into quantum gravity effects, bridging the gap between fundamental theory and observational astronomy.

11.3. Methodological Contributions

11.3.1. Numerical Rigor

The inclusion of complete symbolic computation code sets a standard for transparency and reproducibility in theoretical physics research. This allows independent verification and exploration of parameter space.

11.3.2. Parameter Space Analysis

Systematic sensitivity analysis demonstrates which predictions are robust and which require fine-tuning, guiding future theoretical developments and experimental searches.

12. Conclusions and Future Directions

The refined EQST-GP model demonstrates robust compatibility with Swampland Conjectures [13] while maintaining phenomenological viability. Key achievements include:

• Complete mathematical formulation of negative energy mechanism with justification for extreme values via suppression mechanisms [6, 7]
• Topological foundation for Majorana gluon dark matter [51, 53] with detailed mass generation pathway
• Rigorous Swampland Conjectures compatibility [13] using physically motivated anti-D3 brane uplifting [14, 26]
• Enhanced moduli stabilization [24] with parameter sensitivity analysis
• Refined gravitational wave predictions [39] with realistic LISA detectability assessment
• Transparent disclosure of theoretical limitations and approximation validity [1, 4]

Future work should focus on:

• Explicit Calabi-Yau construction: Realizing the proposed topology with complete moduli space analysis [61], moving from averaged geometric quantities to specific manifold realizations.
• Precision cosmology calculations: Implementing the modified expansion history in full Boltzmann codes [36] for accurate CMB and large-scale structure predictions.
• Reheating and baryogenesis: Detailed analysis of post-inflation dynamics with quantitative prediction of baryon-to-photon ratio $n_B/n_{\gamma }$ [5].
• Black hole connections: Exploration of relationships to black hole physics, information paradox, and holography within this framework [9, 10].
• Numerical relativity simulations: Development of structure formation simulations incorporating ${\mathsf{\Lambda }}_{\mathrm{eff}}\left(z\right)$ for precise observational predictions [52, 60].
• Experimental interfaces: Detailed study of detection prospects for next-generation experiments across gravitational wave astronomy, cosmology, and particle physics.

The framework provides a comprehensive path toward experimental verification through next-generation gravitational wave detectors (LISA [39], DECIGO), cosmological surveys (DESI [19, 47], Euclid [44], Roman), and particle physics experiments (FCC-hh). By connecting fundamental quantum gravity constraints with observational data, it represents a significant step toward a complete theory of quantum gravity with testable predictions.