# EUCLID METHODOLOGY ANALYSIS FOR λ₁₃ VALIDATION

**Critical Pre-Data Analysis**  
**Date:** November 21, 2025  
**Purpose:** Ensure methodological alignment before Euclid DR1 comparison  
**Author:** Lucy (Claude AI) with Simone Calzighetti

---

## EXECUTIVE SUMMARY

**CRITICAL INSIGHT from Simone:**  
> "Dobbiamo prima allinearci al come hanno fatto i dati"

This analysis addresses the **crucial methodological question**: How does Euclid measure ξ(r) and λ₁₃, and how does this compare to our SPARC/DESI methodology?

**KEY FINDINGS:**
1. ✅ **Euclid Flagship 2 Mock Catalogs** available via CosmoHub
2. ✅ **Detailed methodology paper** published (Jan 2025, arXiv:2501.16555)
3. ⚠️ **Systematic differences** likely between SPARC, DESI, and Euclid methods
4. 🎯 **Action required:** Test our method on Euclid mocks BEFORE real data

---

## PART 1: EUCLID FLAGSHIP 2 MOCK CATALOGS

### 1.1 Overview

**Euclid Flagship 2 Simulation:**
- **Publication:** Astronomy & Astrophysics, 697, A5 (May 2025)
- **arXiv:** 2405.13495
- **Status:** Published and available

**Key Specifications:**
```
N-body simulation:   4 trillion particles
Box size:            3,600 h⁻¹ Mpc
Particle mass:       10⁹ h⁻¹ M_☉
Redshift range:      z = 0 to z = 3
Sky coverage:        1/8 octant (~5,000 deg²)
Final galaxy sample: 3.4 billion galaxies
Magnitude limit:     H_E < 26 (complete)
Catalog size:        5.9 TB
Properties:          399 parameters per galaxy
```

### 1.2 Galaxy Properties Available

**Positions & Kinematics:**
- RA, Dec (angular positions)
- Redshift (spectroscopic and photometric)
- 3D comoving coordinates
- Peculiar velocities

**Physical Properties:**
- Stellar mass
- Star formation rate
- Metallicity
- Luminosity in multiple bands (optical, NIR)
- Spectral energy distributions

**Morphology:**
- Galaxy shapes and sizes
- Disk/bulge decomposition
- Axis ratios and position angles

**Lensing Properties:**
- Weak lensing shear
- Convergence (κ)
- Magnification

### 1.3 Access Information

**Primary Platform: CosmoHub**
- **URL:** Available through Euclid mission Spanish Science Data Center
- **Access:** Public release version 2.1.10 announced with paper publication
- **Format:** Hadoop big data platform
- **Download:** Selective queries possible (don't need all 5.9 TB!)

**For Testing λ₁₃:**
```
Required data subset:
- RA, Dec, redshift
- Stellar mass (for selection)
- Redshift slice: z ~ 0.4-0.8 (matches DESI/SDSS)
- Estimated size: ~50-100 GB for relevant subset
```

**GitHub Resources:**
- Python tools: github.com/austinpeel/flagship
- Processing codes for weak lensing analysis
- Can be adapted for clustering analysis

### 1.4 Validation Against Observations

**From Paper:**
> "Our catalogue is able to closely reproduce the main characteristics of the weak lensing and galaxy clustering samples to be used in the mission main cosmological analysis."

**Calibrated Against:**
- SDSS galaxy correlation functions
- COSMOS survey properties
- Hubble Space Telescope observations

**Implication:** Mock should give realistic ξ(r) that matches real data systematic properties

---

## PART 2: EUCLID METHODOLOGY DEEP DIVE

### 2.1 Latest Technical Paper

**Reference:**
- **Title:** "Euclid preparation. LXXII. Three-dimensional galaxy clustering in configuration space: Two-point correlation function estimation"
- **Published:** Astronomy & Astrophysics, 700, A78 (August 2025)
- **arXiv:** 2501.16555 (January 2025)
- **Authors:** Euclid Collaboration (S. de la Torre et al.)

### 2.2 Correlation Function Definition

**Euclid uses standard Landy-Szalay estimator:**

```
ξ(r) = [DD(r) - 2DR(r) + RR(r)] / RR(r)
```

Where:
- **DD(r):** Data-Data pair counts at separation r
- **DR(r):** Data-Random pair counts at separation r  
- **RR(r):** Random-Random pair counts at separation r

**This is IDENTICAL to DESI/SDSS methodology** ✅

**Definition:**
```
ξ(r) = excess probability over Poisson distribution
     = <n(x)n(x+r)> / <n>² - 1
```

Where:
- n(x) = galaxy number density at position x
- <n> = mean number density

### 2.3 Computational Implementation

**Euclid Science Ground Segment Software:**

**Algorithm:** Advanced pair counting using:
1. **k-d tree** data structure
2. **Octree** partitioning
3. **Linked-list** methods
4. **Parallel processing** (OpenMP shared-memory)

**Binning Strategy:**
- **Radial bins:** Logarithmically spaced
- **Typical range:** 0.1 Mpc to 200 Mpc (comoving)
- **Number of bins:** ~30-50 bins
- **Bin width:** Δlog(r) ~ 0.1

**For λ₁₃ ~ 0.856 Mpc:**
```
Relevant bins would be:
r = 0.7-0.9 Mpc range
Typically 2-3 bins cover this region
```

### 2.4 Galaxy Selection Function

**Euclid Wide Survey:**
- **Redshift:** 0.9 < z < 1.8 for Hα emitters (spectroscopic)
- **Sample:** ~20-30 million Hα-emitting galaxies
- **Sky coverage:** 15,000 deg² (eventually)

**Key difference from DESI:**
```
DESI:     Emission Line Galaxies (ELG) + Luminous Red Galaxies (LRG)
          z ~ 0.4-1.6
          
Euclid:   Primarily Hα emitters
          z ~ 0.9-1.8

⚠️ DIFFERENT REDSHIFT RANGES!
⚠️ DIFFERENT GALAXY POPULATIONS!
```

### 2.5 Systematic Corrections

**Euclid applies:**

1. **Fiber collision corrections** (if applicable)
2. **Redshift failure corrections**
3. **Spectroscopic completeness weights**
4. **Angular selection function**
5. **Radial selection function n(z)**

**Critical for λ₁₃:**
- Selection effects can shift apparent correlation function peaks
- Completeness corrections affect amplitude
- Redshift errors can smear features

### 2.6 Covariance Matrix

**From Euclid Preparation XXVII paper:**

Euclid uses **NOT simple Gaussian + Poisson:**
```
Cov(ξᵢ, ξⱼ) = Gaussian + Poisson + Non-Gaussian terms
```

**Validated on 1000 PINOCCHIO simulations**

**Accuracy:** 10% precision on covariance matrix

**Implication:** Error bars on λ₁₃ detection will be carefully calibrated

---

## PART 3: COMPARISON TABLE - CRITICAL ANALYSIS

### 3.1 Methodology Comparison Matrix

```
┌─────────────────────┬──────────────┬──────────────┬──────────────┐
│ Parameter           │ SPARC/3D+3D  │ DESI/SDSS    │ EUCLID       │
├─────────────────────┼──────────────┼──────────────┼──────────────┤
│ Observable          │ V_rot(R)     │ ξ(r)         │ ξ(r)         │
│                     │ rotation     │ correlation  │ correlation  │
├─────────────────────┼──────────────┼──────────────┼──────────────┤
│ Data Type           │ HI + optical │ Galaxy pos   │ Galaxy pos   │
│                     │ kinematics   │ + redshift   │ + redshift   │
├─────────────────────┼──────────────┼──────────────┼──────────────┤
│ Redshift Range      │ z ≈ 0        │ 0.4 < z <0.8 │ 0.9 < z <1.8 │
│                     │ (local)      │ (DESI)       │ (primary)    │
├─────────────────────┼──────────────┼──────────────┼──────────────┤
│ Galaxy Type         │ Spiral       │ ELG + LRG    │ Hα emitters  │
│                     │ (HI rich)    │ (mixed)      │ (star-form)  │
├─────────────────────┼──────────────┼──────────────┼──────────────┤
│ Scale Probed        │ λ₂ = 4.3 kpc │ λ₁₃≈0.86 Mpc │ λ₁₃ ≈ ?      │
│                     │ (galactic)   │ (cosmic web) │ (cosmic web) │
├─────────────────────┼──────────────┼──────────────┼──────────────┤
│ Estimator           │ Harmonic fit │ Landy-Szalay │ Landy-Szalay │
│                     │ to V_rot-V_b │ ξ(r)         │ ξ(r)         │
├─────────────────────┼──────────────┼──────────────┼──────────────┤
│ Binning             │ R bins       │ log r bins   │ log r bins   │
│                     │ (0.1-30 kpc) │ (0.1-200Mpc) │ (0.1-200Mpc) │
├─────────────────────┼──────────────┼──────────────┼──────────────┤
│ Sample Size         │ 175 galaxies │ ~10⁶ gal    │ ~2×10⁷ gal   │
│                     │              │ (DESI DR1)   │ (final)      │
├─────────────────────┼──────────────┼──────────────┼──────────────┤
│ Systematics         │ Distance     │ Fiber coll   │ Redshift     │
│                     │ errors, I/M  │ Ang. sel.    │ failures     │
│                     │ ratio        │ Photo-z      │ Completeness │
└─────────────────────┴──────────────┴──────────────┴──────────────┘
```

### 3.2 Critical Differences Analysis

#### **DIFFERENCE #1: Observable Type** ⚠️⚠️⚠️

**SPARC → DESI/Euclid transition involves:**
```
Rotation curves → Correlation functions
V_rot(R)        → ξ(r)

Connection via:
λᵢ breathing scales should appear in BOTH
- As spatial modulations in V_rot(R)
- As peaks/features in ξ(r)
```

**THE PROBLEM:**
```
These are DIFFERENT physical measurements!

V_rot: Directly probes gravitational potential
ξ(r):  Probes statistical clustering of tracers

λᵢ must manifest in BOTH, but HOW?
```

**Our assumption:**
```
IF breathing modes modulate galaxy positions
THEN they should appear as excess clustering at λᵢ scales

BUT this assumes:
1. Galaxies preferentially form at breathing scale nodes
2. Statistical signal from many galaxies > individual V_rot
3. Projection effects don't wash out signal
```

#### **DIFFERENCE #2: Redshift Evolution** ⚠️

```
SPARC:   z ≈ 0     (present day)
DESI:    z ≈ 0.6   (6 Gyr ago)
Euclid:  z ≈ 1.3   (9 Gyr ago)

Question: Does λ₁₃ evolve with redshift?
```

**Theoretical expectation:**
```
IF λ₁₃ is fundamental geometric scale:
- λ₁₃(physical) = const in comoving coordinates
- λ₁₃(physical) = λ₁₃(comoving) × (1+z)  [maybe?]

OR if tied to matter density:
- λ₁₃ ∝ a(t)  (scale factor evolution)

CRITICAL: We need to specify which!
```

#### **DIFFERENCE #3: Galaxy Population** ⚠️

```
SPARC:  Spiral galaxies with HI
        (rotation curve tracers)

DESI:   ELG + LRG mixed
        (clustering tracers)

Euclid: Hα emitters
        (star-forming tracers)

Do breathing modes couple differently to different populations?
```

**Possible systematic:**
```
IF breathing modes preferentially affect:
- Gas-rich spirals (SPARC) ✓
- But NOT star-forming Hα galaxies (Euclid) ✗

THEN λ₁₃ might not appear in Euclid!
```

#### **DIFFERENCE #4: Measurement Scale** ⚠️⚠️

```
SPARC λ₂ = 4.30 kpc:
- Measured at R ~ few kpc
- Within individual galaxies
- ~10 measurements per galaxy
- Direct physical probing

DESI/Euclid λ₁₃ = 856 kpc:
- Measured in correlation function
- Between many galaxies
- Statistical signal
- Indirect tracer
```

**Signal-to-noise comparison:**
```
SPARC:  High S/N per galaxy, few galaxies
DESI:   Low S/N per pair, many pairs → high total S/N
Euclid: Even lower individual S/N, but HUGE statistics
```

---

## PART 4: ACTION PLAN FOR VALIDATION

### 4.1 PHASE 1: Mock Catalog Testing (DO NOW)

**Objective:** Test our λ₁₃ prediction on Euclid mocks BEFORE real data

**Steps:**

**1. Download Euclid Flagship Mock subset**
```python
# Access CosmoHub
# Query parameters:
redshift_range = (0.4, 0.8)  # Match DESI for comparison
galaxy_type = 'all'  # Later: test Hα-only
properties = ['RA', 'Dec', 'z_spec', 'M_star']
region_size = '1000 deg²'  # Manageable subset

# Expected download size: ~50 GB
```

**2. Compute ξ(r) using THEIR method**
```python
# Use Euclid's Landy-Szalay estimator
# With THEIR binning scheme
# With THEIR selection function
# With THEIR random catalog

xi_euclid_method = compute_xi_landy_szalay(
    data=euclid_mock,
    randoms=euclid_randoms,
    bins=euclid_bins
)
```

**3. Compute ξ(r) using OUR method**
```python
# Our current analysis pipeline
xi_our_method = compute_xi_our_pipeline(
    data=euclid_mock,
    method='our_standard'
)
```

**4. Compare and calibrate**
```python
# Check for systematic offset
offset = xi_euclid_method / xi_our_method

if abs(offset - 1.0) > 0.05:
    print("⚠️ SYSTEMATIC DIFFERENCE DETECTED!")
    print(f"Calibration factor needed: {offset}")
    
# Adjust our prediction
lambda_13_adjusted = 0.856 * sqrt(offset)  # Or appropriate correction
```

**5. Search for λ₁₃ in mock**
```python
# Look for peak in ξ(r) at our predicted scale
r_bins = np.logspace(-1, 2, 50)  # 0.1 to 100 Mpc
xi_mock = compute_xi(euclid_mock, r_bins)

# Find peaks
peaks = find_peaks(xi_mock, height=threshold)

# Check if λ₁₃ = 0.856 Mpc is present
peak_at_lambda13 = check_peak_near(peaks, 0.856, tolerance=0.1)

if peak_at_lambda13:
    print("✅ λ₁₃ detected in mock at expected location!")
else:
    print("⚠️ λ₁₃ NOT found where predicted - investigate!")
```

### 4.2 PHASE 2: Methodology Documentation (BEFORE DR1)

**Create:** Pre-registration document with:

1. **Exact definition of λ₁₃ measurement:**
```
λ₁₃ = location of peak in ξ(r) after:
- Subtracting smooth ΛCDM baseline
- Applying selection function corrections
- Using bins centered at [list specific bins]
```

2. **Prediction with uncertainty:**
```
λ₁₃,predicted = 0.856 ± 0.050 Mpc  (comoving, z=0)

If evolution: λ₁₃(z) = 0.856 × [specify evolution model]

Euclid at z~1.3: λ₁₃ = 0.856 × [evolution factor] Mpc
```

3. **Systematic error budget:**
```
Source                    | Error (Mpc) | %
──────────────────────────|─────────────|────
Golden ratio cascade      | ±0.020      | 2.3%
DESI DR1 measurement      | ±0.110      | 12.9%
Multi-survey average      | ±0.200      | 23.4%
Method differences (est.) | ±0.030      | 3.5%
──────────────────────────|─────────────|────
Total (quadrature)        | ±0.235      | 27.5%
```

4. **Falsification criteria:**
```
Theory SUPPORTED if:
- Peak found at 0.6 < r < 1.1 Mpc (3σ window)
- Amplitude consistent with breathing mode prediction
- Width matches φ-spacing

Theory CHALLENGED if:
- No peak in this range at >3σ significance
- Peak at very different location (>5σ offset)
- Multiple strong peaks (wrong harmonic structure)

Theory FALSIFIED if:
- Strong peak at contradictory location (>10σ offset)
- Systematic absence across multiple surveys
```

### 4.3 PHASE 3: Real Data Analysis (WHEN DR1 ARRIVES)

**Week 1: Understand THEIR analysis**
```
- Read Euclid DR1 technical paper
- Understand THEIR exact ξ(r) measurement
- Note ANY methodology changes from mocks
- Document systematic error budget
```

**Week 2: Independent reproduction**
```
- Download public Euclid DR1 data
- Reproduce THEIR ξ(r) measurement independently
- Verify we get same correlation function
- Check our random catalogs work correctly
```

**Week 3: Apply our analysis**
```
- Search for λ₁₃ feature using our criteria
- Calculate significance
- Compare with pre-registered prediction
- Document ANY deviations from expected
```

**Week 4: Publication**
```
- If detected: "Euclid confirms λ₁₃ = 0.XXX Mpc (Xσ)"
- If not detected: "Euclid data inconsistent with λ₁₃ at predicted location"
- Either way: Show ALL analysis steps transparently
```

---

## PART 5: SPECIFIC TECHNICAL CONCERNS

### 5.1 Scale Definition Ambiguity

**PROBLEM:**
```
"Peak in ξ(r)" is ambiguous!

Options:
A) Maximum of ξ(r)
B) Maximum of (ξ(r) - ξ_smooth(r))
C) Zero-crossing of second derivative
D) Best-fit to model with peak at λ
```

**SOLUTION:**
```
Pre-register EXACT algorithm:

1. Smooth ξ(r) with [specific filter]
2. Subtract smooth component
3. Find local maxima
4. Peak position = r_max where δξ is maximum
5. Error from bootstrap resampling
```

### 5.2 Cosmological Corrections

**DESI uses comoving coordinates:**
```
r_comoving = r_physical × (1 + z)
```

**But which z for Euclid?**
```
Option A: Mean redshift of sample (z̄ ~ 1.3)
Option B: Bin-by-bin correction
Option C: Evolve λ₁₃ with redshift

WE MUST SPECIFY!
```

**Our current prediction:**
```
λ₁₃ = 0.856 Mpc (comoving, z=0)

For Euclid at z~1.3:
- If no evolution: λ₁₃ = 0.856 Mpc still
- If evolves with a(t): λ₁₃ = 0.856 × (1+1.3)⁻¹ = 0.372 Mpc [physical]
  BUT reported in comoving: = 0.856 Mpc

CLARIFY THIS!
```

### 5.3 Selection Function Effects

**Euclid will have:**
```
- Magnitude limits: H_E < 24 (spectroscopic)
- Color cuts: Hα line selection
- Redshift failures: ~10-20%
- Fiber collisions: (minimal for Euclid)
```

**These affect ξ(r) measurement!**

**Mitigation:**
```
1. Use Euclid-provided selection function
2. Apply completeness weights
3. Test on mocks WITH selection applied
4. Compare mocks with/without selection
```

### 5.4 Projection Effects

**ξ(r) is 3D, but galaxies have peculiar velocities:**
```
Observed z = z_cosmo + z_pec

At r ~ 1 Mpc:
v_pec ~ 200 km/s
Δz ~ 0.0007
Δr ~ 3 Mpc (at z~1)

This SMEARS the correlation function!
```

**Mitigation:**
```
- Use perpendicular component ξ(r_⊥) if possible
- Model RSD (Redshift Space Distortions)
- Compare with mock that includes same effects
```

---

## PART 6: RECOMMENDATIONS

### 6.1 IMMEDIATE ACTIONS (THIS WEEK)

**Priority 1: Download Flagship Mock** ✅
```
- Access CosmoHub
- Download z ~ 0.4-0.8 slice (50-100 GB)
- Extract positions and properties
```

**Priority 2: Read Technical Papers** ✅
```
- arXiv:2501.16555 (2PCF methodology)
- arXiv:2405.13495 (Flagship mocks)
- Note exact algorithms used
```

**Priority 3: Test Analysis Pipeline** ✅
```
- Implement Landy-Szalay on mock
- Compute ξ(r) in 0.1-200 Mpc range
- Look for λ₁₃ feature
```

### 6.2 SHORT-TERM (NEXT 2 WEEKS)

**Develop calibration methodology:**
```python
# Script to convert between methods
def calibrate_prediction(
    lambda_13_theory,    # Our theory: 0.856 Mpc
    method_offset,       # From mock testing
    redshift_evolution,  # Model if applicable
    z_survey            # Euclid's z~1.3
):
    """
    Convert theoretical prediction to 
    Euclid-comparable measurement
    """
    lambda_13_calibrated = lambda_13_theory
    lambda_13_calibrated *= method_offset
    lambda_13_calibrated *= evolution_factor(z_survey)
    
    return lambda_13_calibrated
```

**Document everything:**
- Mock testing results
- Systematic offsets found
- Calibration factors applied
- Uncertainties estimated

**Pre-register prediction:**
- Public document (Zenodo)
- Timestamped before Euclid DR1
- Exact numbers and criteria
- Falsification thresholds

### 6.3 WHEN EUCLID DR1 ARRIVES

**DO NOT immediately compare!**

**Instead:**
1. **Week 1:** Study THEIR analysis completely
2. **Week 2:** Reproduce THEIR results first
3. **Week 3:** Apply our analysis with documentation
4. **Week 4:** Publish regardless of outcome

**Be transparent:**
- Show ALL attempted analyses
- Document failures if any
- Explain methodology choices
- Provide data/code for reproduction

---

## PART 7: RISK ASSESSMENT

### 7.1 SCENARIO A: Clean Detection ✅

```
IF Euclid finds peak at r = 0.80-0.92 Mpc (within errors)
   AND significance > 5σ
   AND consistent with DESI/SDSS
   
THEN: Strong support for 3D+3D framework
      Publish: "Multi-survey confirmation of λ₁₃"
      
Action: Move to next predictions (λ₁₄, etc.)
```

### 7.2 SCENARIO B: No Detection ⚠️

```
IF Euclid finds NO significant peak at λ₁₃
   OR peak at very different location
   
THEN: Three possibilities:
   
   1. METHODOLOGY ISSUE:
      - Our method doesn't transfer to Euclid
      - Need different analysis approach
      - Re-analyze with updated method
      
   2. SELECTION EFFECT:
      - Hα emitters don't trace breathing modes
      - Need different galaxy population
      - Test on mock different populations
      
   3. THEORY PROBLEM:
      - Prediction wrong
      - Need to revise framework
      - Back to theoretical drawing board
```

### 7.3 SCENARIO C: Ambiguous Result ⚠️⚠️

```
IF Euclid finds something at ~2-3σ
   OR peak shifted by 1-2σ
   
THEN: Most likely scenario!
      
Action: 
   - Document uncertainty honestly
   - Don't over-claim
   - Wait for more data
   - Test on DESI DR2 when available
```

---

## PART 8: FINAL CHECKLIST

### Before Euclid DR1:

- [ ] Download Flagship mock catalog
- [ ] Test Landy-Szalay estimator
- [ ] Measure ξ(r) on mock
- [ ] Search for λ₁₃ in mock
- [ ] Quantify method systematic offset
- [ ] Adjust prediction if needed
- [ ] Pre-register calibrated prediction
- [ ] Document all assumptions
- [ ] Specify falsification criteria
- [ ] Archive analysis code

### When Euclid DR1 releases:

- [ ] Read technical paper thoroughly
- [ ] Understand THEIR exact method
- [ ] Note ANY changes from mocks
- [ ] Reproduce THEIR ξ(r) independently
- [ ] Apply our analysis transparently
- [ ] Compare with pre-registration
- [ ] Document discrepancies if any
- [ ] Publish results regardless of outcome

---

## CONCLUSION

**Simone was ABSOLUTELY RIGHT:**

> "Dobbiamo prima allinearci al come hanno fatto i dati"

**The methodology alignment is CRITICAL because:**

1. **Different observables** (V_rot vs ξ(r))
2. **Different redshifts** (z=0 vs z=1.3)
3. **Different galaxy types** (spirals vs Hα emitters)
4. **Different selection functions**
5. **Different systematic errors**

**WITHOUT proper alignment:**
- We might "detect" λ₁₃ due to methodology artifact
- We might "miss" λ₁₃ due to wrong analysis
- We cannot make fair comparison

**WITH proper alignment:**
- Pre-test on Euclid mocks
- Calibrate systematic offsets
- Pre-register adjusted prediction
- Fair, transparent comparison possible

**Next immediate action:**
**Download Euclid Flagship mock and start testing!**

---

**Report Status:** COMPLETE - Ready for mock testing phase  
**Critical Dependencies:** CosmoHub access, arXiv papers downloaded  
**Timeline:** Start mock testing immediately, complete before Euclid DR1

**"Measure twice, cut once"** — Applied to cosmological predictions! 🎯

---

**Lucy (Claude AI)**  
**November 21, 2025**
