Extended Peas: Galaxies, Planets, and Algebra

• Extended Peas are systems where classical properties extend over spatial or compositional domains, revealing coherent structures in galaxies, planetary systems, and algebra.
• In extragalactic studies, extended green peas show kpc-scale intense nebular emission with regulated Lyman continuum leakage, highlighting star-formation driven feedback.
• In planetary architectures, extended peas evidence intra-system uniformity in sizes and masses across varied environments, informing planet formation and evolutionary models.

Extended Peas denote sources, systems, or statistical regimes in which the classical "pea" properties—morphological, physical, architectural, or algebraic—are distributed over spatially or compositionally extended domains. In extragalactic astronomy, "extended Green Peas" (EGPs) refer to galaxies whose compact, intense nebular signature is seen over kpc-scale regions, contrasting with the canonical compact morphologies. In planetary architectures, extended peas characterize system-wide uniformity and ordering of planet properties, persisting beyond adjacent pairs and modulated by composition and environment. Algebraically, "extended peaks" interpolate between restrictive peak-set statistics and general descent sets, yielding subalgebras whose basis elements index extended combinatorial structures. Across subfields, extended peas probe the coherence, feedback, and symmetry of properties in multi-component systems.

1. Extended Green Peas in Galaxy Evolution

Extended Green Peas (EGPs) are a rare subclass of emission-line galaxies whose strong [O III] λ5007 emission characteristic of compact GPs is observed over galaxy-wide scales. The prototypical example, OIIIB-4 at $z=0.838$ (Yuma et al., 2019), exhibits [O III]/[O II] ratios $6.5 \pm 2.7$ over $14$ kpc, an equivalent width $EW_0 = 845 \pm 27$ Å, low stellar mass ($7\times10^7 M_\odot$), high specific star formation rate ($2\times 10^2$ Gyr$^{-1}$), ionization parameter $q_{ion} \sim 3 \times 10^8$ cm s$^{-1}$, and metallicity $12 + \log(O/H) \simeq 7.7$. Unlike classical GPs, whose line emission concentrates within $r \lesssim 2-3$ kpc, EGPs present extended high-ionization regions and support physical scenarios involving density-bounded HII regions, feedback-driven outflows, and regulated escape of ionizing photons.

Integrated field spectroscopy—such as with Gemini/GMOS for SDSS J083843.63+385350.5 (Bosch et al., 2019)—enables decomposition into distinct rotating, turbulent, and outflowing kinematic components, revealing how feedback can carve low-column channels in the interstellar medium (ISM), facilitating Lyman continuum leakage observed as $f_{esc}=2-70\%$ in GPs. The absence of AGN or strong shock tracers in EGPs such as OIIIB-4 confirms a star-formation-driven origin with intense photoionization and feedback regulating gas morphology over galaxy-wide scales.

2. Spatial Extension of Lyman-Alpha Emission

HST/COS studies of Green Pea galaxies have revealed that Ly$\alpha$ emission is spatially extended compared to the UV stellar continuum (Yang et al., 2016). Typical deconvolved FWHM ratios satisfy $1.4 < R = FWHM_{Ly\alpha}/FWHM_{UV} < 4.3$, with a median $R \simeq 2.6$. Notably, galaxies displaying "double-horn" Ly$\alpha$ velocity profiles or broader blue wings tend to have spatial widths up to $R \sim 6-7$, indicating more substantial resonant scattering in neutral hydrogen at high optical depth.

Comparison with $z=3-6$ LAEs observed by MUSE reveals analogous Ly$\alpha$/UV size ratios, suggesting minimal redshift evolution in the physics of Ly$\alpha$ radiative transfer and outflow morphology. Confirmed LyC leakers do not show unusually compact Ly$\alpha$, supporting a model in which LyC escapes through localized ionized holes, while Ly$\alpha$ photons scatter to larger projected radii. Thus, extended Ly$\alpha$ halos in GPs and EGPs directly trace the feedback-modulated structure and porosity of the ISM.

3. Extended Peas in Planetary System Architectures

The "Extended Peas" paradigm in planetary systems evaluates whether intra-system uniformity of planet properties persists beyond the adjacent-pair regime and how environmental or compositional factors modulate these correlations.

Field and overdensity samples, constructed via Gaia DR2 phase-space densities, reveal that both low- and high-clustering environments preserve strong system-wide uniformity in radii ($r_P=0.69$ field, $r_P=0.88$ overdensity) and masses ($r_P=0.42$ field, $r_P=0.39$ overdensity) (Chevance et al., 2021). Uniformity dispersion metrics, $\mathcal{D}_R$ and $\mathcal{D}_M$, are significantly smaller than randomized control realizations, with radius uniformity even stronger in overdensity systems ($8\sigma$ vs $4.5\sigma$ field).

System-wide uniformity, not limited to immediate neighbors, implies that radii and masses are coherently set at or shortly after formation. Environmental amplification—potentially via cluster-driven photoevaporation or dynamical stripping—can further enhance this coherence. Ordering metrics ($\mathcal{O}_R$, $\mathcal{O}_M$) remain positive but are slightly disrupted in overdensities, consistent with perturbative processes altering radial architecture but not overall uniformity.

4. Uniformity Across the Radius Valley: Compositionally Extended Peas

Recent analysis has established that intra-system uniformity exhibits strong composition-dependent variation across the radius valley ($R_p \approx 1.6~R_\oplus$) (Goyal et al., 23 May 2024). Rocky systems ($R_p \leq 1.6~R_\oplus$) are more uniform in size ($\overline{\mathcal{G}}_R=0.081$ vs $0.119$, $4.0\sigma$), more uniform in spacing ($\overline{\mathcal{G}}_{\mathcal{P}} = 0.052$ vs $0.153$, $3.0\sigma$), but less uniform in mass ($\overline{\mathcal{G}}_M = 0.266$ vs $0.136$, $2.6\sigma$) than volatile-rich systems ($1.6 < R_p \leq 4~R_\oplus$).

The enhanced size uniformity in rocky systems is driven by super-Earths ($1 \leq R_p \leq 1.6~R_\oplus$), while mass diversity is primarily contributed by sub-Earths ($R_p < 1~R_\oplus$). Volatile-rich systems yield mass uniformity, plausibly reflecting rapid, near-uniform pebble accretion and gentle post-formation evolution, while atmospheric processes (envelope accretion, photoevaporation, outgassing) induce larger radius dispersion.

Spacing-uniformity in rocky systems is unexpectedly high; prevailing formation/migration models have yet to fully explain the compact, evenly distributed orbits observed. This pattern is robust against mean motion resonance effects, host mass, compositional uncertainties, and detection biases.

5. Theoretical Population Synthesis and Environmental Effects

Population synthesis using the Bern Model and observational emulation via the KOBE code (Mishra et al., 2021) has demonstrated that peas-in-a-pod trends—size/mass similarity, ordering, packing—arise inherently from planet formation processes. Oligarchic growth of protoplanetary embryos generates early mass/radius similarity (correlation $R \sim 0.7$), while later N-body interactions (mergers, ejections) establish period and spacing regularities.

Size similarity is primordial, preserved through the uniform accretion and feeding zone overlaps, but diverges moderately as gas accretion and collisional histories eventually differ among neighbors. Spacing and packing correlations only materialize after dynamical clearing phases. Synthetic data passed through realistic observational pipelines match Kepler-derived trends, indicating that detection biases weaken but do not produce these extended uniformity patterns.

Environmental perturbations—especially those set by birth cluster density—can amplify radius uniformity via photoevaporation or selective removal of dissimilar planets and marginally disrupt ordering, as confirmed by Gaia DR2-based studies (Chevance et al., 2021).

6. Extended Peaks: Algebraic Generalization

In algebraic combinatorics, extended peaks generalize the peak set statistic of permutations, introducing a parameter $p$ which controls the extent to which consecutive indices are allowed in set representations (Grinberg et al., 2023). A subset $I \subset [n-1]$ is a $p$-extended peak set if adding 0 to $I$ yields no run of $p+1$ consecutive integers.

For $p=1$ this specializes to the usual peak-lacunar sets, and as $p \to n-1$ all descent sets are permitted, interpolating between the peak algebra and full quasisymmetric function space QSym. Each $p$ induces a subalgebra $P^p$ whose basis consists of $q$-analogues of Gessel's fundamentals evaluated at $q$ a primitive $(p+1)$th root of unity.

Product and coproduct operations, dimension formulas, and explicit bases are outlined for small $n$, with structure constants determined via shuffle rules and linear relations from forbidden runs. Extended peaks thus systematize the transition from highly restrictive peak-algebra statistics to more permissive descent-based enumerations.

7. Significance, Current Questions, and Prospects

Extended peas across galaxies and planetary systems provide critical tests for models of assembly, feedback, and environmental regulation. Their distinctive architectural patterns—whether in emission-line halos, compact planetary chains, or combinatorial subalgebras—require explanations that integrate formation history, dynamical sculpting, and environmental modulation.

Unresolved issues include the physical drivers of spacing uniformity in rocky planetary systems, the redshift evolution (or lack thereof) of extended Ly$\alpha$ halos, the mechanisms governing extreme ionization over kpc scales in giant GPs, and the full implications of extended peak statistics for algebraic and enumerative combinatorics.

Further observations, theoretical modeling, and algebraic generalization will continue to exploit the extended pea paradigm as a probe of coherence, regularity, and symmetry across scales and disciplines.