PeV ν and PeV γ Without New Particles: Classical Budgets vs. Future–Mass Projection (FMP)

1. Context and Aim

IceCube established a diffuse astrophysical neutrino flux with deposited energies from $\sim 30$ TeV to $\sim 2$ PeV and an approximately isotropic distribution.1 UHE $\gamma$-rays up to $\sim 1.4$ PeV have been reported by LHAASO from multiple Galactic sources, including enigmatic cases like J2108+5157 with no obvious accelerator counterpart.2 Our aim is to quantify how much “headroom” is missing in classical Hillas/DSA estimates and whether FMP can supply just enough to reach the observed PeV scales without invoking new particle species.

2. Classical Budget in Brief

Confinement and timescale limits give

$$\begin{array}{cc}\hfill E_{max}^{\left(\mathrm{conf}\right)}& \simeq ZeBRc\approx 0.9Z\left({\displaystyle \frac{B}{\mu \mathrm{G}}}\right)\left({\displaystyle \frac{R}{\mathrm{kpc}}}\right)\mathrm{EeV},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill E_{max}^{\left(\mathrm{time}\right)}& \simeq E_{max}^{\left(\mathrm{conf}\right)}\beta ,\beta \equiv u_s/c,\hfill \end{array}$$

(2)

with hadronic yields $E_{\nu }\sim 0.05E_p$ and $E_{\gamma }\sim (0.1\pm )E_p$ for $pp$ interactions (spectrum-dependent).3

3. FMP in One Page

FMP augments the gravitational source with a causal, future-conditioned contribution $T_{\mu \nu }^{\left(F\right)}$. In the Newtonian limit:

$${\nabla }^2\mathrm{\Phi }=4\pi G[{\rho }_b+{\rho }_F],{\rho }_F(x,t)={\int }_0^{\infty }K\left(\tau \right)\mathrm{\Pi }\left(x,t+\tau |I_t\right)d\tau .$$

(3)

On galaxy scales the effect is captured by

$$v_c^2\left(R\right)=D\left(R\right)v_b^2\left(R\right),D\left(R\right)\equiv 1+{\mathrm{\Xi }}_d(k\simeq 1/R),$$

(4)

with a three-scale representation and a scaled outer component (“Plan B”) tied to the disk length $R_d$. Milky Way fits (disk+bulge+gas+explicit CGM) yield $D\left(8.178\mathrm{kpc}\right)\approx 1.46$ and $D\left(20-25\mathrm{kpc}\right)\approx 2.5$–$2.9$. Thus free-fall/escape/shock speeds scale as $u_s^{\left(\mathrm{FMP}\right)}\approx \sqrt{D}{u}_s^{\left(b\right)}$, implying

$$E_{max}^{\left(\mathrm{time}\right)}{|}_{\mathrm{FMP}}\approx \sqrt{D}{E}_{max}^{\left(\mathrm{time}\right)},E_{max}^{(\mathrm{loss}/\mathrm{age})}{|}_{\mathrm{FMP}}\approx D{E}_{max}^{(\mathrm{loss}/\mathrm{age})}.$$

(5)

4. Case 1: PeV Neutrinos from Solar-Circle and Outer-Disk Accelerators

We adopt a representative SNR/superbubble near the Solar circle (classical): $B=100\mu \mathrm{G}$, $R=10\mathrm{pc}$, $u_s=5000\mathrm{km}{\mathrm{s}}^{-1}$ ($\beta \simeq 0.0167$). Then $E_{max,p}^{\left(\mathrm{time}\right)}\approx 15$ PeV and $E_{\nu ,max}\approx 0.75$ PeV. With FMP at $R_0$: $\sqrt{D}\approx 1.21$, $D\approx 1.46$ giving $E_{\nu ,max}\approx 0.91$–1.10 PeV. In the outer disk ($\sqrt{D}\approx 1.65$, $D\approx 2.7$): $E_{\nu ,max}\approx 1.24$–2.03 PeV.

Figure 1. Neutrino maximum energy (PeV) for a representative Galactic accelerator: classical vs. FMP at the Solar circle and in the outer disk. Bars show time-limited and loss-limited scalings.

Figure 1. Neutrino maximum energy (PeV) for a representative Galactic accelerator: classical vs. FMP at the Solar circle and in the outer disk. Bars show time-limited and loss-limited scalings.

5. Case 2: PeV $\gamma$ from Molecular-Cloud Shocks (J2108+5157-like)

For a conservative cloud-shock setup (C1): $E_{max,p}\approx 2.7$ PeV $\Rightarrow E_{\gamma ,max}\approx 0.27$ PeV (insufficient for $\sim 0.5$ PeV). FMP (outer-like) lifts this to $0.45$ PeV (time-limited) or $0.73$ PeV (loss-limited). For a moderate setup (C2): $E_{\gamma ,max}\approx 1.26$ PeV (classical) becomes $2.08$–3.40 PeV with FMP.

Figure 2. Gamma-ray maximum energy (PeV) for two molecular-cloud shock setups (C1 conservative, C2 moderate): classical vs. FMP.

Figure 2. Gamma-ray maximum energy (PeV) for two molecular-cloud shock setups (C1 conservative, C2 moderate): classical vs. FMP.

Figure 3. Trend of $E_{\gamma ,max}$ with $\sqrt{D}$ in the time-limited regime for C1 and C2. A modest $\sqrt{D}\sim 1.3$–1.7 already bridges the LHAASO threshold in conservative cases.

Figure 3. Trend of $E_{\gamma ,max}$ with $\sqrt{D}$ in the time-limited regime for C1 and C2. A modest $\sqrt{D}\sim 1.3$–1.7 already bridges the LHAASO threshold in conservative cases.

6. Numerical Summary

Case 1: Neutrinos	Scenario	$E_{\nu ,max}$ (PeV)
Classical (Solar circle)		0.75
FMP time-limited (Solar circle)		0.91
FMP loss-limited (Solar circle)		1.10
FMP time-limited (Outer disk)		1.24
FMP loss-limited (Outer disk)		2.03
Case 2: Gamma rays	Scenario	$E_{\gamma ,max}$ (PeV)
C1: Classical		0.27
C1: FMP time-limited		0.45
C1: FMP loss-limited		0.73
C2: Classical		1.26
C2: FMP time-limited		2.08
C2: FMP loss-limited		3.40

7. Discussion and Falsifiability

Key point. FMP does not alter microphysics; it gently deepens the effective potential where the baryon field is predictably convergent (disks/MCs/CGM), boosting $u_s$ by $\sqrt{D}$ and—indirectly—compression and B. This supplies just enough headroom for PeV $\nu$ and sub-PeV/PeV $\gamma$ without exotic particles. Predictions: (i) Radial trend—outer-disk environments should more readily achieve PeVatron conditions than inner-disk sites of otherwise similar microphysics. (ii) Environment—correlations with CGM/molecular-cloud density and pre-merger/streaming flows (higher Mach numbers). (iii) Cosmic drift—a mild $R\left(z\right)$ evolution implies weak epoch-dependence of the most extreme attainable energies.