Semiclassical Determination of Ω_Λ from Standard Model Horizon Loading

1. Introduction

The cosmological constant problem represents one of the most profound failures in theoretical physics. Standard quantum field theory predicts a vacuum energy density

$${\rho }_{\mathrm{vac}}\sim {\int }^{{\mathrm{\Lambda }}_{\mathrm{UV}}}{\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}{\omega }_k\sim {\mathrm{\Lambda }}_{\mathrm{UV}}^4,$$

(1)

which, for any reasonable UV cutoff (Planck scale, electroweak scale, or even QCD), exceeds the observed dark energy density by factors of ${10}^{60}$ to ${10}^{120}$ [1]. No known symmetry forbids a large bare cosmological constant, and attempts to cancel vacuum contributions through fine-tuning or anthropic selection remain unsatisfying [2].

Horizon thermodynamics provides a natural framework for relating microscopic field content to macroscopic observables. Black hole thermodynamics [3,4] and its extension to de Sitter horizons [5] establish that horizons carry thermodynamic properties determined by their geometry. For a cosmological horizon of radius R, the Gibbons-Hawking temperature $T_{\mathrm{GH}}=\hslash c/\left(2\pi k_{\mathrm{B}}R\right)$ emerges as a standard result of quantum field theory on curved backgrounds.

In this work, we compute a dimensionless quantity ${\Delta }^{*}$ by projecting Standard Model fields onto the monopole ($\ell =0$) block of a spherical cosmological horizon at radius $R\sim H_0^{-1}$, applying KMS thermal weighting at $T_{\mathrm{GH}}$. Selection rules from spherical harmonic structure, combined with large-R limits of boundary conditions, fix all coefficients from symmetry or geometry. The result is mapped to ${\mathrm{\Omega }}_{\mathrm{\Lambda }}={\Delta }^{*}/\left(8\pi \right)$ via a geometric normalization determined by the causal diamond solid angle (Section 5.2). We obtain ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.695\pm 0.{008}_{\mathrm{th}}$, in $0.2\sigma$ agreement with DESI 2024 + CMB observations [6], with no fitted continuous parameters.

The calculation rests on a small set of discrete, symmetry-preserving modeling choices, each tested for robustness:

(M1)

Spherical horizon geometry: the late-time cosmological horizon is treated as $S^2\left(R\right)$ with $R=c/H_0$, valid when $|\dot{H}|/H^2\ll 1$.

(M2)

Isotropic (monopole) extraction: the computed quantity is the angular mean of the loading functional, for which $\ell =0$ is exact by orthogonality.

(M3)

Local self-adjoint boundary conditions: fermions satisfy MIT bag conditions, the minimal local class ensuring self-adjointness and vanishing normal flux.

(M4)

Regulated collar interpolation: greybody factors are computed via smooth metric transition across a collar of width $\epsilon$; results are validated to be $\epsilon$-insensitive.

These assumptions contain no adjustable real parameters; robustness to variations is demonstrated in Section 7.

Section 2 establishes the spherical horizon geometry, thermal weighting, and spectral content. Section 3 derives monopole selection rules. Section 4 presents geometric corrections. Section 5 assembles the master formula and derives the $8\pi$ normalization. Section 6 presents numerical results, Section 7 validates the calculation, and Section 8 discusses implications. Appendices provide detailed derivations.

2. Setup: Spherical Horizon and Spectral Content

We model the late-time cosmological horizon as a sphere of radius $R\sim H_0^{-1}$ and extract the $\ell =0$ block of a one-loop effective action on $S^2\left(R\right)$. This extraction is exact due to SO(3) symmetry, not an approximation.

2.1. Thermal Scale

For a cosmological horizon of radius R, standard QFT on curved backgrounds yields the Gibbons-Hawking temperature [5]

$$T_{\mathrm{GH}}={\displaystyle \frac{\hslash c}{2\pi k_{\mathrm{B}}R}}.$$

(2)

This is the temperature measured by a static observer in the de Sitter static patch, derived from the periodicity of Euclidean time. With $R\sim c/H_0\sim {10}^{26}$ m, we have $T_{\mathrm{GH}}\sim {10}^{-29}$ K.

We work in SI units throughout, keeping all factors of c, ℏ, $k_{\mathrm{B}}$, and $G_{\mathrm{N}}$ explicit. The Planck length is ${\ell }_{\mathrm{P}}\equiv \sqrt{\hslash G_{\mathrm{N}}/c^3}\approx 1.6\times {10}^{-35}$ m, and the Planck area is ${\ell }_{\mathrm{P}}^2\equiv \hslash G_{\mathrm{N}}/c^3$.

2.2. SO(3) Block Diagonalization

Lemma 1

(SO(3) Block Diagonalization). On an exactly rotationally invariant background (round $S^2$), any Laplace-type operator $L_s$ commutes with the $SO\left(3\right)$ generators and hence decomposes into irreducible ℓ-blocks:

$$L_s=\underset{\ell =0}{\overset{\infty }{⨁}}{L}_{s,\ell }.$$

(3)

Therefore the log-determinant decomposes additively:

$$lndet{L}_s=\sum _{\ell =0}^{\infty }lndet{L}_{s,\ell },$$

(4)

with no cross-coupling between different ℓ. The monopole extraction selects the $\ell =0$ block of a direct-sum decomposition.

Proof. 

The operators $L_s$ (scalar Laplacian, Hodge Laplacian, Dirac squared) are constructed from the metric and covariant derivative on $S^2$. For a round sphere, these are SO(3)-invariant, so $[L_s,J_i]=0$ where $J_i$ generate rotations. By Schur’s lemma, $L_s$ is block-diagonal in the irreducible representation basis $\left\{Y_{\ell m}\right\}$ (or spinor harmonics for fermions). The multiplicativity of determinants over direct sums gives (4). □

If the background is perturbed, $g=g_{S^2}+h$, then ℓ-mixing enters at quadratic order in h. For the late-time cosmological horizon, metric perturbations are $\left|h\right|/g_{S^2}\lesssim {10}^{-4}$, so cross-ℓ contamination is $O\left({10}^{-8}\right)$—negligible.

2.3. Monopole Projectors by Spin

Scalars.—Spherical harmonics $Y_{\ell m}$ with $\ell =0$ give a uniform mode $Y_{00}$. The scalar monopole projector is

$${\mathrm{\Pi }}_{\mathrm{scalar}}=1.$$

(5)

Vectors.—Vector spherical harmonics decompose into TM/E (polar) and TE/B (axial) families. At $\ell =0$, only TM/E exists; TE/B modes are absent due to geometric parity constraints. The monopole projector is

$${\mathrm{\Pi }}_{\mathrm{vector}}=1.$$

(6)

Fermions.—Spinor harmonics on $S^2$ are labeled by $(j,\kappa ,m)$, with $j={\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}},\dots$ and $\kappa =\pm (j+{\displaystyle \frac{1}{2}})$. The selection rule for coupling to the monopole channel is:

$$\ell =0\Rightarrow j={\displaystyle \frac{1}{2}},\kappa =-1.$$

(7)

This follows from Clebsch-Gordan coupling: only $j=|\ell \pm {\displaystyle \frac{1}{2}}|$ is allowed, so $\ell =0$ requires $j={\displaystyle \frac{1}{2}}$. The sign of $\kappa$ is fixed by the spinor harmonic convention: $\kappa =-(\ell +1)$ for $j=\ell +{\displaystyle \frac{1}{2}}$, giving $\kappa =-1$ when $\ell =0$.

2.4. KMS Thermal Weighting

At temperature $T_{\mathrm{GH}}$, states are weighted by the Kubo-Martin-Schwinger thermal factors. The dimensionless thermal argument for fermions is:

$$x_j={\displaystyle \frac{\hslash {\omega }_j}{k_{\mathrm{B}}{T}_{\mathrm{GH}}}}=2\pi \left(j+{\displaystyle \frac{1}{2}}\right),$$

(8)

where ${\omega }_j=(c/R)(j+{\displaystyle \frac{1}{2}})$ are the Dirac eigenfrequencies on $S^2$ [7].

For the monopole channel $(j={\displaystyle \frac{1}{2}})$: $x_{1/2}=2\pi$. The Fermi-Dirac distribution gives

$$n_F\left(2\pi \right)={\displaystyle \frac{1}{e^{2\pi }+1}}\approx 0.001866.$$

(9)

The partition function (summing over all j with both $\kappa$ branches) is:

$$Z=\sum _{j=1/2}^{\infty }2(2j+1)n_F\left(x_j\right)\approx 0.007492.$$

(10)

The fermion monopole projector is the ratio of the $j={\displaystyle \frac{1}{2}}$, $\kappa =-1$ contribution to the total partition function:

$${\mathrm{\Pi }}_{\mathrm{fermion}}={\displaystyle \frac{n_F\left(2\pi \right)}{Z}}=0.2491.$$

(11)

Here $n_F\left(2\pi \right)$ is the thermal weight of a single monopole-coupled mode (the $\kappa =-1$ branch), while Z sums over both $\kappa$ branches. This value is stable to ${10}^{-4}$ for $j_{max}\ge 3.5$ due to exponential suppression of higher modes.

2.5. Standard Model Degrees of Freedom

The Standard Model particle content determines the degeneracy factors $d_s$:

(i)

Bosons: Photon (2), gluons (16 = $8\times 2$), $W^{\pm }/Z$ (9 = $3\times 3$), Higgs (1)

(ii)

Quarks: 6 flavors × 3 colors × 2 spin states = 36

(iii)

Leptons: 3 charged leptons × 2 spin + $n_{\nu }$ neutrino modes

For Dirac neutrinos, each species has 2 helicity states (neutrino + antineutrino), giving $d_{\mathrm{leptons}}=6+6=12$. For Majorana neutrinos, particle = antiparticle reduces independent boundary-coupled modes by a factor of $1/2$, so $d_{\nu }^{\mathrm{Maj}}={\displaystyle \frac{1}{2}}{d}_{\nu }^{\mathrm{Dir}}$, giving $d_{\mathrm{leptons}}=6+3=9$.

Why quarks and gluons, not hadrons?—The one-loop effective action is dominated by short-distance correlations, for which the heat-kernel expansion is controlled by UV physics. In the heat-kernel language:

$$\begin{array}{cc}\hfill \mathrm{Tr}{e}^{-t(L+m^2)}& =e^{-t{m}^2}\mathrm{Tr}{e}^{-tL}\hfill \\ \hfill & =\left(1-t{m}^2+\dots \right)\sum _{n\ge 0}{a}_n{t}^{(n-d)/2}.\hfill \end{array}$$

(12)

The leading terms come from the small-t (UV) region. Masses and confinement only enter subleading terms; the leading contribution is controlled by field content and local geometry [8].

2.6. Neutrino Type Dependence

Within this computation, the lepton-sector contribution depends on whether neutrinos are Dirac or Majorana. Comparing to DESI 2024 + CMB value ${\mathrm{\Omega }}_{\mathrm{\Lambda }}^{\mathrm{obs}}=0.693$:

Table 1. Neutrino type dependence of ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$.

Neutrino Type	${\mathit{d}}_{\mathbf{leptons}}$	${\mathbf{\Omega }}_{\mathbf{\Lambda }}$	$\left|\mathsf{\Delta }\mathbf{\Omega }\right|/{\mathbf{\Omega }}_{\mathbf{obs}}$
Dirac	12	0.695	0.3%
Majorana	9	0.724	4.5%

Table 1. Neutrino type dependence of ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$.

Neutrino Type	${\mathit{d}}_{\mathbf{leptons}}$	${\mathbf{\Omega }}_{\mathbf{\Lambda }}$	$\left|\mathsf{\Delta }\mathbf{\Omega }\right|/{\mathbf{\Omega }}_{\mathbf{obs}}$
Dirac	12	0.695	0.3%
Majorana	9	0.724	4.5%

The calculation favors Dirac neutrinos, consistent with the continued non-observation of neutrinoless double-beta decay by LEGEND-200 [9] and KamLAND-Zen [10].

3. Selection Rules: Boundary Terms and Greybody Factors

We derive boundary terms $\alpha$ (odd-in-curvature contributions) and greybody factors $\beta$ (transmission losses through the horizon collar) for Standard Model fields at $\ell =0$. These are subtracted once per sector to account for dissipative effects.

Boundary terms ($\alpha$) represent parity-violating contributions at the horizon—odd terms in the effective action. Greybody factors ($\beta$) capture transmission losses through the finite-width collar region where the metric transitions from interior to exterior. All values are fixed by symmetry (gauge/color/parity/C) or geometry—no free parameters.

3.1. Photon: Pure Gauge at $\ell =0$

The electromagnetic field satisfies $A_a\to A_a+{\partial }_a\lambda$ under gauge transformations. At $\ell =0$, the monopole mode is pure gauge:

$$A_a^{(\ell =0)}={\partial }_a{\lambda }_{00}.$$

(13)

The boundary term ${\alpha }_{\gamma }$: Single-trace boundary terms vanish by gauge invariance:

$${\alpha }_{\gamma }^{\left(0\right)}=0(\mathrm{gauge}\mathrm{singlet}\mathrm{at}\ell =0).$$

(14)

The greybody factor ${\beta }_{\gamma }$: At $\ell =0$, only TM/E exists; the surviving TM/E mode is pure gauge:

$${\beta }_{\gamma }^{\left(0\right)}=0(\mathrm{no}\mathrm{TE}/\mathrm{B}\mathrm{at}\ell =0,\mathrm{TM}/\mathrm{E}\mathrm{pure}\mathrm{gauge}).$$

(15)

3.2. Weak Bosons ($W^{\pm }$, Z): Geometric Parity

The weak gauge bosons are massive vector bosons. The degeneracy $d_{W/Z}=9$ counts all three polarization states (two transverse plus one longitudinal), confirmed by ATLAS and CMS measurements of longitudinal $W_L{Z}_L$ scattering [11].

The boundary term ${\alpha }_{W/Z}$: At $\ell =0$, the monopole mode behaves as a gauge singlet:

$${\alpha }_{W/Z}^{\left(0\right)}=0(\mathrm{gauge}\mathrm{singlet}\mathrm{at}\ell =0).$$

(16)

The greybody factor ${\beta }_{W/Z}$: Unlike photons, massive vector bosons carry physical TM/E degrees of freedom at $\ell =0$. The availability factor $f_{\mathrm{even}}(\ell =0)=1/2$ reflects the presence of TM/E but absence of TE/B. Numerical computation of transmission through the horizon collar (Appendix C) yields:

$${\beta }_{W/Z}^{\left(0\right)}=\frac{1}{2}\times 0.015=0.0075.$$

(17)

Equivalently, one may replace $d_{W/Z}\to d_{W/Z}/2$ at $\ell =0$; the product $d_{W/Z}{\beta }_{W/Z}$ is invariant under this choice.

3.3. Gluons: Color Singlet

At $\ell =0$, the monopole mode must be a color singlet. In the adjoint representation, $\mathrm{Tr}\left(T^a\right)=0$.

The boundary term ${\alpha }_g$: Linear functionals vanish after color averaging:

$${\alpha }_g^{\left(0\right)}=0(\mathrm{color}\mathrm{singlet}\mathrm{at}\ell =0).$$

(18)

The greybody factor ${\beta }_g$: Gluon monopole modes are pure gauge:

$${\beta }_g^{\left(0\right)}=0(\mathrm{no}\mathrm{TE}/\mathrm{B}\mathrm{at}\ell =0,\mathrm{TM}/\mathrm{E}\mathrm{pure}\mathrm{gauge}).$$

(19)

3.4. Higgs: Robin Boundary

The Higgs is a scalar with Robin boundary conditions $({\partial }_n+S)\varphi =0$. At cosmological scale $R\sim {10}^{26}$ m, the Robin parameter $K=1/R\to 0$:

$${\alpha }_H=0,{\beta }_H=0(\mathrm{large}\text{-}R\mathrm{limit}).$$

(20)

3.5. Fermions: C-Symmetry and MIT Transmission

At the cosmological horizon, we impose MIT boundary conditions $(1+i{\gamma }^n){\psi |}_{\partial }=0$—the minimal local self-adjoint class enforcing vanishing normal current $j^{\mu }{n}_{\mu }=0$.

The boundary term ${\alpha }_{\mathrm{ferm}}$: In the KMS thermal ensemble, C-parity-odd terms vanish:

$${\alpha }_{\mathrm{ferm}}^{\left(0\right)}=0(\mathrm{C}\text{-}\mathrm{symmetry}\mathrm{in}\mathrm{KMS}\mathrm{ensemble}).$$

(21)

The greybody factor ${\beta }_{\mathrm{ferm}}$: The monopole channel is the s-wave $(j={\displaystyle \frac{1}{2}},\kappa =-1)$ mode. For the cosmological horizon with $R\gg {\lambda }_C$ (Compton wavelength), the collar is adiabatic and reflection is exponentially suppressed. We bound ${\beta }_{\mathrm{ferm}}^{\left(0\right)}\le {10}^{-3}$ and absorb this into the fermion-projector systematic:

$${\beta }_{\mathrm{ferm}}^{\left(0\right)}\approx 0(\mathrm{adiabatic}\mathrm{s}\text{-}\mathrm{wave}\mathrm{at}R\gg {\lambda }_C).$$

(22)

3.6. Summary of Selection Rules

$$\begin{array}{cc}\hfill \mathrm{Boundary}\mathrm{terms}\text{:}& {\alpha }_{\gamma }={\alpha }_{W/Z}={\alpha }_g={\alpha }_H={\alpha }_{\mathrm{ferm}}=0,\hfill \\ \hfill \mathrm{Greybodies}\text{:}& {\beta }_{\gamma }={\beta }_g={\beta }_H={\beta }_{\mathrm{ferm}}=0,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & {\beta }_{W/Z}=0.0075.\hfill \end{array}$$

(24)

Every coefficient is determined by symmetry or geometry. The sole non-zero value ${\beta }_{W/Z}=0.0075$ arises from geometric parity constraints on massive vectors.

4. Heat-Kernel Geometric Correction

A geometric correction arises from the $a_1$ Seeley-DeWitt coefficient in the heat-kernel expansion. For the scalar Laplacian on $S^2$ [12,13]:

$$\mathrm{Tr}{e}^{-t\Delta }={\displaystyle \frac{A}{4\pi t}}+{\displaystyle \frac{1}{3}}+O\left(t\right).$$

(25)

The t-independent term from $a_1={\displaystyle \frac{1}{24\pi }}{\int }_{S^2}RdA={\displaystyle \frac{1}{3}}$ (where $R=2/R^2$ is the Ricci scalar).

Proposition 1

(Monopole Heat-Kernel Uplift). For bosonic sectors on $S^2$, the monopole-projected heat-kernel coefficient is:

$${\epsilon }_{\mathrm{HK}}={\displaystyle \frac{1}{6\pi }}\approx 0.05305.$$

(26)

The bosonic seed acquires the multiplicative correction $(1+{\epsilon }_{\mathrm{HK}})$.

5. Master Formula and Cosmological Mapping

5.1. Definition of ${\Delta }^{*}$

The loading functional is the monopole-projected, KMS-regulated coefficient extracted from the one-loop effective action:

$${\Delta }^{*}\equiv \sum _s{d}_s{\eta }_s{\mathrm{\Pi }}_s\left(1+{\epsilon }_{\mathrm{HK}}^{\left(s\right)}\right)-\sum _s{d}_s({\alpha }_s+{\beta }_s),$$

(27)

where $d_s$ is the degeneracy, ${\eta }_s=+1$ (bosons) or $-1$ (fermions), ${\mathrm{\Pi }}_s$ is the monopole projector, ${\epsilon }_{\mathrm{HK}}^{\left(s\right)}$ is the geometric correction (with ${\epsilon }_{\mathrm{HK}}^{\left(s\right)}=1/\left(6\pi \right)$ for bosons and ${\epsilon }_{\mathrm{HK}}^{\left(s\right)}=0$ for fermions), and ${\alpha }_s,{\beta }_s$ are boundary and greybody subtractions.

The only non-zero greybody factor is ${\beta }_{W/Z}=0.0075$, contributing ${\delta }_{\mathrm{gb}}=9\times 0.0075=0.068$ (rounded). All boundary terms vanish by symmetry. Note that ${\Delta }^{*}$ is defined as a regulated monopole-extracted functional; the convergence and insensitivity tests in Section 7 bound regulator dependence within the stated ${\sigma }_{\mathrm{th}}$.

The master formula is:

$${\Delta }^{*}=\mathrm{seed}-{\delta }_{\mathrm{gb}}.$$

(28)

5.2. Geometric Normalization: The $8\pi$ Mapping

We now derive the mapping ${\mathrm{\Omega }}_{\mathrm{\Lambda }}={\Delta }^{*}/\left(8\pi \right)$ from geometric and symmetry considerations alone.

5.2.1. Causal Diamond Geometry

Consider a comoving observer’s causal diamond in FRW spacetime. The boundary consists of two null hypersurfaces:

$$\partial \mathcal{D}={\mathcal{N}}_{+}\cup {\mathcal{N}}_{-},$$

(29)

where ${\mathcal{N}}_{+}$ is the future null boundary and ${\mathcal{N}}_{-}$ is the past null boundary. Both sheets share a common bifurcation two-sphere of radius R.

5.2.2. Monopole Loading Field

Let $f\left(\mathrm{\Omega }\right)$ denote the horizon-local scalar loading density extracted from the one-loop functional. Rotational invariance (SO(3) symmetry of the horizon) implies:

$$f\left(\mathrm{\Omega }\right)=\mathrm{const}\equiv f_0.$$

(30)

Define ${\Delta }^{*}$ as the integrated loading over the full causal-diamond boundary:

$${\Delta }^{*}={\int }_{{\mathcal{N}}_{+}}f\left(\mathrm{\Omega }\right)d\mathrm{\Omega }+{\int }_{{\mathcal{N}}_{-}}f\left(\mathrm{\Omega }\right)d\mathrm{\Omega }.$$

(31)

Since $f\left(\mathrm{\Omega }\right)=f_0$ is constant and each null sheet subtends the full sphere:

$${\Delta }^{*}=f_0\times 4\pi +f_0\times 4\pi =8\pi f_0.$$

(32)

5.2.3. Identification with ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$

Definition 1

(Vacuum Loading Fraction). Define the vacuum loading fraction ${\mathrm{\Omega }}_{\mathrm{\Lambda }}^{\left(\mathrm{load}\right)}$ by

$${\mathrm{\Omega }}_{\mathrm{\Lambda }}^{\left(\mathrm{load}\right)}:=f_0,$$

(33)

where $f_0$ is the SO(3)-invariant monopole loading density defined in (30)–(32).

Lemma 2

(De Sitter Normalization). In pure de Sitter spacetime (no matter, no curvature), ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=1$ by definition of the cosmological fractions, and the vacuum monopole loading must satisfy $f_0=1$. Hence ${\mathrm{\Omega }}_{\mathrm{\Lambda }}^{\left(\mathrm{load}\right)}$ coincides with the standard ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ in the de Sitter limit, fixing the normalization.

Combining (32) and (33):

$${\mathrm{\Omega }}_{\mathrm{\Lambda }}\equiv {\mathrm{\Omega }}_{\mathrm{\Lambda }}^{\left(\mathrm{load}\right)}={\displaystyle \frac{{\Delta }^{*}}{8\pi }}.$$

(34)

5.2.4. Interpretation

The factor $8\pi$ is purely geometric: it is the total solid angle measure of the causal diamond boundary,

$${\int }_{\partial \mathcal{D}}d\mathrm{\Omega }=2\times 4\pi =8\pi .$$

(35)

This mapping is not a thermodynamic derivation but a matching condition between the computed monopole loading ${\Delta }^{*}$ and the cosmological observable ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$, with the proportionality constant fixed by the de Sitter normalization (Lemma 2).

6. Numerical Inputs and Results

Applying the master formula (28), we use Standard Model degeneracies alongside derived projectors, HK uplift, and selection rules. All values are determined by first principles—no fitted continuous parameters.

6.1. Standard Model Degeneracies

Table 2. Standard Model sectors with degeneracies d, signs $\eta$, and projectors $\mathrm{\Pi }$.

Sector	d	$\mathit{\eta }$	$\mathbf{\Pi }$	Source
Photon ($\gamma$)	2	$+1$	1.0000	$U{\left(1\right)}_{\mathrm{em}}$
Gluons (g)	16	$+1$	1.0000	$SU{\left(3\right)}_c$ adjoint
Weak ($W^{\pm },Z$)	9	$+1$	1.0000	$SU{\left(2\right)}_L\times U{\left(1\right)}_Y$
Higgs (H)	1	$+1$	1.0000	Scalar singlet
Quarks	36	$-1$	0.2491	6 flav. × 3 col. × 2 spin
Leptons	12	$-1$	0.2491	3 ch. + 3$\nu$ (Dirac) × 2 spin

Table 2. Standard Model sectors with degeneracies d, signs $\eta$, and projectors $\mathrm{\Pi }$.

Sector	d	$\mathit{\eta }$	$\mathbf{\Pi }$	Source
Photon ($\gamma$)	2	$+1$	1.0000	$U{\left(1\right)}_{\mathrm{em}}$
Gluons (g)	16	$+1$	1.0000	$SU{\left(3\right)}_c$ adjoint
Weak ($W^{\pm },Z$)	9	$+1$	1.0000	$SU{\left(2\right)}_L\times U{\left(1\right)}_Y$
Higgs (H)	1	$+1$	1.0000	Scalar singlet
Quarks	36	$-1$	0.2491	6 flav. × 3 col. × 2 spin
Leptons	12	$-1$	0.2491	3 ch. + 3$\nu$ (Dirac) × 2 spin

6.2. Results Table

Table 3. Sector-by-sector contributions to ${\Delta }^{*}$ (all dimensionless).

Sector s	Seed	${\mathit{d}}_{\mathit{s}}{\mathit{\alpha }}_{\mathit{s}}$	${\mathit{d}}_{\mathit{s}}{\mathit{\beta }}_{\mathit{s}}$	Net
QED (photon)	2.106	0.000	0.000	2.106
Weak ($W/Z$)	9.477	0.000	0.068	9.409
QCD (gluons)	16.849	0.000	0.000	16.849
Higgs (scalar)	1.053	0.000	0.000	1.053
Quarks	$-8.968$	0.000	0.000	$-8.968$
Leptons	$-2.989$	0.000	0.000	$-2.989$
Totals	17.528	0.000	0.068	17.460

Table 3. Sector-by-sector contributions to ${\Delta }^{*}$ (all dimensionless).

Sector s	Seed	${\mathit{d}}_{\mathit{s}}{\mathit{\alpha }}_{\mathit{s}}$	${\mathit{d}}_{\mathit{s}}{\mathit{\beta }}_{\mathit{s}}$	Net
QED (photon)	2.106	0.000	0.000	2.106
Weak ($W/Z$)	9.477	0.000	0.068	9.409
QCD (gluons)	16.849	0.000	0.000	16.849
Higgs (scalar)	1.053	0.000	0.000	1.053
Quarks	$-8.968$	0.000	0.000	$-8.968$
Leptons	$-2.989$	0.000	0.000	$-2.989$
Totals	17.528	0.000	0.068	17.460

Hence

$${\mathrm{\Omega }}_{\mathrm{\Lambda }}={\displaystyle \frac{17.460}{8\pi }}=0.6947\approx 0.695.$$

(36)

7. Validation and Convergence

To demonstrate the prediction’s robustness and lack of fine-tuning, we validate the numerical components through convergence tests, sensitivity analysis, and parameter space exploration.

7.1. Fermion j-Cutoff Convergence

Table 4. Fermion monopole projector convergence with $j_{max}$.

${\mathit{j}}_{max}$	${\mathbf{\Pi }}_{\mathbf{fermion}}$	$|\mathsf{\Delta }{\mathbf{\Pi }}_{\mathbf{fermion}}|$
1.5	0.24906	—
3.5	0.24910	$4\times {10}^{-5}$
5.5	0.24910	$<{10}^{-5}$
7.5	0.24910	$<{10}^{-6}$

Table 4. Fermion monopole projector convergence with $j_{max}$.

${\mathit{j}}_{max}$	${\mathbf{\Pi }}_{\mathbf{fermion}}$	$|\mathsf{\Delta }{\mathbf{\Pi }}_{\mathbf{fermion}}|$
1.5	0.24906	—
3.5	0.24910	$4\times {10}^{-5}$
5.5	0.24910	$<{10}^{-5}$
7.5	0.24910	$<{10}^{-6}$

Higher angular momentum channels are exponentially suppressed by KMS weighting.

7.2. Bosonic Grid Convergence

Table 5. Bosonic monopole projector convergence with grid resolution.

Grid (${\mathit{N}}_{\mathit{\theta }}\times {\mathit{N}}_{\mathit{\varphi }}$)	${\mathbf{\Pi }}_{\mathbf{boson}}$	$|1-{\mathbf{\Pi }}_{\mathbf{boson}}|$
$128\times 256$	0.999974886	$2.51\times {10}^{-5}$
$256\times 512$	0.999993730	$6.27\times {10}^{-6}$
$512\times 1024$	0.999998430	$1.57\times {10}^{-6}$

Table 5. Bosonic monopole projector convergence with grid resolution.

Grid (${\mathit{N}}_{\mathit{\theta }}\times {\mathit{N}}_{\mathit{\varphi }}$)	${\mathbf{\Pi }}_{\mathbf{boson}}$	$|1-{\mathbf{\Pi }}_{\mathbf{boson}}|$
$128\times 256$	0.999974886	$2.51\times {10}^{-5}$
$256\times 512$	0.999993730	$6.27\times {10}^{-6}$
$512\times 1024$	0.999998430	$1.57\times {10}^{-6}$

7.3. Sensitivity to Heat-Kernel Coefficient

Table 6. Sensitivity to heat-kernel coefficient. The ${\epsilon }_{\mathrm{HK}}=0$ row is a counterfactual: omitting the curvature term breaks the Laplace-type heat-kernel expansion and is not an allowed model variant; it is shown only to demonstrate that curvature uplift is numerically significant. The $\pm 5\%$ variations represent the actual systematic uncertainty.

${\mathit{\epsilon }}_{\mathbf{HK}}$	${\mathbf{\Omega }}_{\mathbf{\Lambda }}$	$|\mathsf{\Delta }{\mathbf{\Omega }}_{\mathbf{\Lambda }}|/{\mathbf{\Omega }}_{\mathbf{\Lambda }}$
0 (counterfactual)	0.636	8.5%
$1/\left(6\pi \right)-5\%$	0.693	0.2%
$1/\left(6\pi \right)$	0.695	0.0% (baseline)
$1/\left(6\pi \right)+5\%$	0.697	0.3%

Table 6. Sensitivity to heat-kernel coefficient. The ${\epsilon }_{\mathrm{HK}}=0$ row is a counterfactual: omitting the curvature term breaks the Laplace-type heat-kernel expansion and is not an allowed model variant; it is shown only to demonstrate that curvature uplift is numerically significant. The $\pm 5\%$ variations represent the actual systematic uncertainty.

${\mathit{\epsilon }}_{\mathbf{HK}}$	${\mathbf{\Omega }}_{\mathbf{\Lambda }}$	$|\mathsf{\Delta }{\mathbf{\Omega }}_{\mathbf{\Lambda }}|/{\mathbf{\Omega }}_{\mathbf{\Lambda }}$
0 (counterfactual)	0.636	8.5%
$1/\left(6\pi \right)-5\%$	0.693	0.2%
$1/\left(6\pi \right)$	0.695	0.0% (baseline)
$1/\left(6\pi \right)+5\%$	0.697	0.3%

7.4. Weak Greybody Factor Calibration

Table 7. Weak greybody transmission vs. collar width.

$\mathit{\epsilon }/\mathit{R}$	T	${\mathit{\beta }}_{\mathit{W}/\mathit{Z}}$	${\mathbf{\Omega }}_{\mathbf{\Lambda }}$
${10}^{-4}$	0.9985	0.00075	0.6950
${10}^{-3}$	0.9850	0.00750	0.6947
${10}^{-2}$	0.8510	0.07450	0.6920

Table 7. Weak greybody transmission vs. collar width.

$\mathit{\epsilon }/\mathit{R}$	T	${\mathit{\beta }}_{\mathit{W}/\mathit{Z}}$	${\mathbf{\Omega }}_{\mathbf{\Lambda }}$
${10}^{-4}$	0.9985	0.00075	0.6950
${10}^{-3}$	0.9850	0.00750	0.6947
${10}^{-2}$	0.8510	0.07450	0.6920

The resulting ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ varies by only 0.4% across two orders of magnitude in collar width, confirming IR stability.

7.5. Theoretical Error Budget

Table 8. Theoretical error budget for ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$.

Source	$|\mathit{\delta }{\mathbf{\Omega }}_{\mathbf{\Lambda }}|$	Notes
Curvature uplift	0.003	$\delta {\epsilon }_{\mathrm{HK}}/{\epsilon }_{\mathrm{HK}}\lesssim 5\%$
Fermion projector	0.005	BC class + selection
$\mathrm{\Lambda }$CDM vs de Sitter	0.003	$|\dot{H}|/H^2$ correction
Collar width	0.003	Table 7
$\ell \ge 1$ corrections	$<0.001$	Quadratic in h
Numerical convergence	$<0.001$	Table 4, Table 5
Gravitons omitted	$\lesssim 0.002$	Classical metric
Total (quadrature)	0.008

Table 8. Theoretical error budget for ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$.

Source	$|\mathit{\delta }{\mathbf{\Omega }}_{\mathbf{\Lambda }}|$	Notes
Curvature uplift	0.003	$\delta {\epsilon }_{\mathrm{HK}}/{\epsilon }_{\mathrm{HK}}\lesssim 5\%$
Fermion projector	0.005	BC class + selection
$\mathrm{\Lambda }$CDM vs de Sitter	0.003	$|\dot{H}|/H^2$ correction
Collar width	0.003	Table 7
$\ell \ge 1$ corrections	$<0.001$	Quadratic in h
Numerical convergence	$<0.001$	Table 4, Table 5
Gravitons omitted	$\lesssim 0.002$	Classical metric
Total (quadrature)	0.008

Combining systematic errors:

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{\mathrm{\Lambda }}& =0.695\pm 0.{008}_{\mathrm{th}}(\mathrm{this}\mathrm{work}),\hfill \\ \hfill {\mathrm{\Omega }}_{\mathrm{\Lambda }}^{\mathrm{obs}}& =0.693\pm 0.005(\mathrm{DESI}2024+\mathrm{CMB}).\hfill \end{array}$$

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The theoretical and observational values agree within $0.2\sigma$.

8. Discussion

Our calculation yields ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.695\pm 0.{008}_{\mathrm{th}}$, derived from Standard Model field content on a spherical cosmological horizon with KMS thermal weighting—with no fitted continuous parameters and only discrete, symmetry-preserving modeling choices (M1–M4).

Early analyses (e.g., Planck 2018 [14]) favored ${\mathrm{\Omega }}_{\mathrm{\Lambda }}\approx 0.685$. However, DESI 2024 BAO + CMB [6] favors ${\mathrm{\Omega }}_m\approx 0.306$, implying ${\mathrm{\Omega }}_{\mathrm{\Lambda }}^{\mathrm{obs}}=0.693\pm 0.005$. Our prediction lies within $0.2\sigma$ of this value.

As a linearity cross-check, with one SM generation ($d_{\mathrm{fermion}}=16$ instead of 48), ${\mathrm{\Omega }}_{\mathrm{\Lambda }}\approx 1.01$—clearly inconsistent with observations. This confirms the three-generation structure of the SM is essential to the agreement.

The computation uses: (i) SM gauge structure and field content (experimentally determined), (ii) QFT on curved backgrounds (Gibbons-Hawking temperature), (iii) heat-kernel expansion on $S^2$ (standard spectral geometry), and (iv) SO(3) representation theory for selection rules. The modeling choices (M1–M4) are discrete and testable.

We do not use: (i) any fitted continuous parameters, (ii) supersymmetry or string theory, (iii) anthropic selection, or (iv) fine-tuning between UV and IR scales.

This calculation demonstrates that the observed value of ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ can emerge from Standard Model fields on a cosmological horizon, avoiding the vacuum catastrophe (1) by computing a finite, regulated quantity rather than summing divergent vacuum energies.

9. Conclusions

We have presented a semiclassical determination of the dark energy fraction ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ from Standard Model field content on a spherical cosmological horizon. The calculation projects SM fields onto the monopole ($\ell =0$) block, applies KMS thermal weighting at the Gibbons-Hawking temperature, and uses symmetry-fixed selection rules. The resulting value ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.695\pm 0.{008}_{\mathrm{th}}$ agrees with DESI 2024 + CMB observations ($0.693\pm 0.005$) within $0.2\sigma$, with no fitted continuous parameters.

Appendix A.1. Causal Diamond Structure

A causal diamond $\mathcal{D}$ in FRW spacetime has boundary $\partial \mathcal{D}={\mathcal{N}}_{+}\cup {\mathcal{N}}_{-}$ consisting of future and past null sheets meeting at a bifurcation two-sphere $\Sigma$ of areal radius R.

Appendix A.2. Monopole Loading Definition

Define the dimensionless loading field $f\left(\mathrm{\Omega }\right)$ as the angular density of the one-loop effective action projected onto the horizon. By SO(3) invariance of the round $S^2$, this field must be constant:

$$f\left(\mathrm{\Omega }\right)=f_0=\mathrm{const}.$$

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Appendix A.3. Integration Over Boundary

The total loading ${\Delta }^{*}$ is the integral over both null sheets:

$${\Delta }^{*}={\int }_{{\mathcal{N}}_{+}}{f}_{0}d\mathrm{\Omega }+{\int }_{{\mathcal{N}}_{-}}{f}_{0}d\mathrm{\Omega }=f_0·4\pi +f_0·4\pi =8\pi f_0.$$

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Appendix A.4. Normalization Matching

The identification $f_0={\mathrm{\Omega }}_{\mathrm{\Lambda }}$ follows from uniqueness: ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ is the only rotationally invariant, dimensionless scalar characterizing the vacuum sector of FRW cosmology. The de Sitter limit (${\mathrm{\Omega }}_{\mathrm{\Lambda }}=1$) fixes the normalization: pure vacuum loading corresponds to $f_0=1$.

Therefore:

$${\mathrm{\Omega }}_{\mathrm{\Lambda }}={\displaystyle \frac{{\Delta }^{*}}{8\pi }}.$$

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Appendix D.1. Mode Spectrum

For a Dirac field on $S^2$ with MIT boundary conditions, eigenfrequencies are [7]:

$${\omega }_j={\displaystyle \frac{c}{R}}\left(j+{\displaystyle \frac{1}{2}}\right).$$

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Each j level has degeneracy $2(2j+1)$.

Appendix D.2. Selection Rule

For the monopole ($\ell =0$), only $(j={\displaystyle \frac{1}{2}},\kappa =-1)$ contributes.

Appendix D.3. KMS Weighting

At $T_{\mathrm{GH}}=\hslash c/\left(2\pi k_{\mathrm{B}}R\right)$:

$$x_j=2\pi \left(j+{\displaystyle \frac{1}{2}}\right),n_F\left(x\right)={\displaystyle \frac{1}{e^x+1}}.$$

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Numerical values:

$$\begin{array}{cc}\hfill x_{1/2}& =2\pi ,n_F\left(2\pi \right)\approx 0.001866,\hfill \end{array}$$

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$$\begin{array}{cc}\hfill x_{3/2}& =4\pi ,n_F\left(4\pi \right)\approx 3.49\times {10}^{-6}.\hfill \end{array}$$

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The partition function converges rapidly: $Z\approx 0.007492$. Therefore:

$${\mathrm{\Pi }}_{\mathrm{fermion}}={\displaystyle \frac{0.001866}{0.007492}}=0.2491.$$

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