Dark Energy Models

← Back to index | See also: model-comparison, evidence, marginalization

Note: This page draws on the Type Ia supernova analysis from Lovick, Dhawan & Handley (2024)[^Lovick2024], demonstrating Bayesian model comparison for dark energy scenarios.

The Dark Energy Problem

In 1998, observations of distant Type Ia supernovae revealed the universe's expansion is accelerating[^Riess1998][Perlmutter1999]. This requires a component with negative pressure—dubbed "dark energy."

Key question: What is the nature of dark energy?

Candidates: - Einstein's cosmological constant $\Lambda$ (simplest) - Dynamical scalar fields (quintessence) - Modified gravity theories - Time-varying equations of state

Bayesian evidence comparison provides a rigorous framework to test these scenarios.

The Cosmological Models

Model 1: Flat ΛCDM (Cosmological Constant)

The minimal "concordance" cosmology with 6 parameters:

Parameters: - $\Omega_b h^2$ — baryon density - $\Omega_c h^2$ — cold dark matter density - $H_0$ — Hubble constant - $A_s$ — scalar amplitude (CMB) - $n_s$ — spectral index (CMB) - $\tau$ — optical depth (CMB)

Dark energy: Equation of state $w = -1$ (fixed)

Spatial curvature: $\Omega_k = 0$ (flat universe, fixed)

Model 2: wCDM (Constant Dark Energy EoS)

Extends ΛCDM by allowing dark energy equation of state to vary:

Additional parameter: - $w$ — dark energy equation of state (constant in time)

Prior: Typically $w \in [-2.5, -0.3]$ or similar - $w = -1$ recovers ΛCDM - $w < -1$ is "phantom" dark energy - $w > -1$ is quintessence-like

Total parameters: 7 (6 from ΛCDM + $w$)

Prediction: Luminosity distance depends on $w$: $$H(z)^2 = H_0^2\left[\Omega_m(1+z)^3 + (1-\Omega_m)(1+z)^{3(1+w)}\right]$$

Complexity cost: One additional parameter means larger prior volume and higher KL divergence penalty unless data strongly prefer $w \neq -1$.

Model 3: w₀wₐCDM (Evolving Dark Energy)

Also called the Chevallier-Polarski-Linder (CPL) parameterization[^{Chevallier2001][}Linder2003]:

Additional parameters (beyond ΛCDM): - $w_0$ — equation of state today - $w_a$ — rate of evolution

Time evolution: $$w(z) = w_0 + w_a \frac{z}{1+z}$$

Total parameters: 8 (6 from ΛCDM + $w_0 + w_a$)

Complexity cost: Two additional parameters with even larger Occam penalty.

Type Ia Supernovae as Standard Candles

The Phillips Relation

Type Ia SNe aren't perfect standard candles, but they're standardizable: brighter supernovae decline more slowly (Phillips relation[^Phillips1993]).

Standardization: $$m_B^* = m_B - \alpha X_1 + \beta C$$

where: - $m_B$ = observed peak B-band magnitude - $X_1$ = stretch (light curve width) - $C$ = color - $\alpha, \beta$ = nuisance parameters (fitted)

Distance Modulus and Cosmology

The standardized magnitude relates to cosmology via:

\[m_B^* = M_B + \mu(z; H_0, \Omega_m, w)\]

where: $$\mu(z) = 5\log_{10}\left(\frac{d_L(z)}{\text{Mpc}}\right) + 25$$

Key point: Supernovae constrain relative distances well but have poor leverage on $M_B$ (absolute magnitude). This creates degeneracy: $H_0 \leftrightarrow M_B$.

The Pantheon+ Dataset

The Pantheon+ catalog[^Brout2022] contains: - 1701 spectroscopically confirmed Type Ia SNe - Redshift range: $0.001 < z < 2.26$ - Full covariance matrix accounting for systematics

Systematic uncertainties include: - Photometric calibration across surveys - Selection biases (Malmquist bias) - Dust extinction corrections - Peculiar velocities (local flows) - Host galaxy properties - Gravitational lensing (at high-z)

See supernovae-analysis for technical details.

Beyond Gaussian Scatter

The Standard Gaussian Assumption

The fiducial Pantheon+ analysis assumes:

\[\mathcal{L} = \frac{1}{\sqrt{(2\pi)^n |\mathbf{C}|}} \exp\left(-\frac{1}{2}\boldsymbol{\Delta}^T \mathbf{C}^{-1} \boldsymbol{\Delta}\right)\]

where $\boldsymbol{\Delta} = \mathbf{m}_{\text{obs}} - \mathbf{m}_{\text{model}}$ is the residual vector.

Problem: Real data often have outliers not captured by Gaussian tails.

Student's t-Distribution

A more robust alternative[^Lovick2024]:

\[\mathcal{L} = \prod_{i=1}^n \frac{\Gamma((\nu+1)/2)}{\Gamma(\nu/2)\sqrt{\nu\pi\sigma_i^2}}\left(1 + \frac{\Delta_i^2}{\nu\sigma_i^2}\right)^{-(\nu+1)/2}\]

Key parameter: $\nu$ = degrees of freedom - $\nu \to \infty$: Recovers Gaussian - $\nu \sim 3-10$: Heavy tails (downweights outliers) - $\nu < 2$: Undefined variance (unphysical)

Prior: $\nu \sim \text{Uniform}(2, 100)$ or similar

Covariance Rescaling

Alternative approach: add a global scale parameter[^Lovick2024]:

\[\mathbf{C}_{\text{scaled}} = s^2 \mathbf{C}_{\text{fid}}\]

where $s$ is a free parameter accounting for underestimated uncertainties.

Interpretation: $s > 1$ indicates covariance matrix is too optimistic.

Bayesian Model Comparison Results

Using nested-sampling on the Pantheon+ dataset[^Lovick2024]:

Cosmological Model Comparison (Gaussian Scatter)

Model         log Z        Δ log Z    Interpretation
─────────────────────────────────────────────────────
Flat ΛCDM    1234.5       (reference)  Baseline
wCDM         1232.0       -2.5         Moderate evidence against
w₀wₐCDM      1230.8       -3.7         Strong evidence against

Conclusion: Flat ΛCDM is preferred over evolving dark energy models.

Why? Adding $w$ (or $w_0, w_a$) improves fit marginally but incurs Occam penalty: - wCDM: $\Delta\hat{d} \approx 0.8$ (one parameter moderately constrained) - w₀wₐCDM: $\Delta\hat{d} \approx 1.2$ (two parameters weakly constrained)

Evidence decomposition: $$\Delta\log\mathcal{Z} \approx \underbrace{\Delta\log\mathcal{L}_{\max}}_{+0.5} - \underbrace{\Delta\hat{d}/2}_{-0.4} - \underbrace{\Delta\mathcal{D}_{\text{KL}}}_{-2.6} \approx -2.5$$

The data don't require $w \neq -1$, so the simpler model wins.

Scatter Model Comparison (Fixed ΛCDM Cosmology)

Scatter Model           log Z        Δ log Z    Best-fit parameter
────────────────────────────────────────────────────────────────────
Gaussian (fiducial)    1234.5       (reference)  —
Scaled covariance      1236.8       +2.3         s = 1.08 ± 0.03
Student's t            1237.0       +2.5         ν = 5.2 ± 1.8

Conclusion: Data show moderate evidence for non-Gaussian scatter.

Interpretation: - $s \approx 1.08$: Covariance matrix is ~8% too small (uncertainties underestimated) - $\nu \approx 5$: Heavier tails than Gaussian, consistent with unmodeled outliers

Impact on $H_0$: - Gaussian: $H_0 = 73.52 \pm 1.02$ km/s/Mpc - Student's t: $H_0 = 73.67 \pm 0.99$ km/s/Mpc

The non-Gaussian treatment slightly increases central value and tightens uncertainty (outliers downweighted).

Marginalization Over Scatter Models

Key insight: Cosmological model preferences might depend on scatter model choice.

Solution: Marginalize over scatter models to get robust answer:

\[\mathcal{Z}_{\text{cosmology}} = \sum_{\text{scatter}} \mathcal{Z}_{\text{cosmology, scatter}} \times P(\text{scatter})\]

Result: After marginalizing over scatter models, ΛCDM remains preferred with $\Delta\log\mathcal{Z} \approx 2.2$.

Robustness: The cosmological model preference is independent of scatter model choice.

See marginalization for complete methodology and examples.

The Hubble Tension

Planck CMB Constraint

Planck 2018[^Planck2018] measured in flat ΛCDM: $$H_0^{\text{CMB}} = 67.4 \pm 0.5 \text{ km/s/Mpc}$$

Pantheon+ Supernova Constraint

With Student's t scatter model[^Lovick2024]: $$H_0^{\text{SNe}} = 73.67 \pm 0.99 \text{ km/s/Mpc}$$

Tension Quantification

Using tension-statistics:

Simple comparison: $$\Delta H_0 = 73.67 - 67.4 = 6.27 \text{ km/s/Mpc}$$ $$\sigma = \frac{6.27}{\sqrt{0.99^2 + 0.5^2}} = 5.7\sigma$$

Full Bayesian tension analysis (using $R$, $I$, $S$ statistics) confirms ~5-6σ discordance between CMB and local measurements.

Possible explanations: 1. Systematic errors in one or both measurements 2. New physics (early dark energy, extra relativistic species, etc.) 3. Late-time modifications to ΛCDM not captured by simple $w$ parameterization

Peculiar Velocity Corrections

The Issue

Low-redshift SNe ($z < 0.05$) have distances contaminated by peculiar velocities—galaxy motions relative to the Hubble flow.

Effect: Can bias $H_0$ by ~1-2 km/s/Mpc if not corrected[^Peterson2022].

Correction Methods

Flow model correction: Use large-scale structure surveys to model local flows, subtract predicted peculiar velocities.

Cut low-z sample: Remove $z < 0.023$ SNe entirely (more conservative).

Impact on Model Comparison

From Lovick et al. (2024)[^Lovick2024]:

With peculiar velocity corrections: - ΛCDM strongly preferred ($\Delta\log\mathcal{Z} \approx 2.5$) - $w = -1.02 \pm 0.08$ (consistent with cosmological constant)

Without corrections (or using different corrections): - Model preferences shift - $w$ constraints can change by ~0.1 - evidence ratios change by $\Delta\log\mathcal{Z} \sim 1$

Lesson: Systematic uncertainties matter for model-comparison. Proper treatment via non-Gaussian likelihoods is crucial.

Comparison with Information Criteria

For rapid exploration, can use AIC/BIC instead of full evidence:

Method                    ΛCDM          wCDM         Preference
─────────────────────────────────────────────────────────────────
log L_max                 -621.3        -620.8       wCDM better fit
AIC                       1254.6        1255.6       ΛCDM (weakly)
BIC                       1295.8        1302.5       ΛCDM (strongly)
Nested Sampling (log Z)   1234.5        1232.0       ΛCDM (moderately)

Observations: - AIC too lenient (penalty = 2) - BIC too harsh (penalty = $\log(1701) \approx 7.4$) - nested-sampling exact (penalty via actual $\mathcal{D}_{\text{KL}}$)

See information-criteria for detailed comparison.

Summary

Dark energy model comparison:

✅ Flat ΛCDM remains preferred over evolving dark energy ($\Delta\log\mathcal{Z} \approx 2-3$) ✅ Non-Gaussian scatter models improve fit ($\Delta\log\mathcal{Z} \approx 2.5$) ✅ Model averaging provides robust constraints independent of nuisance model choices ✅ Hubble tension persists at ~5.7σ (CMB vs local measurements) ✅ Systematic uncertainties (scatter, peculiar velocities) affect model preferences

Key techniques: - evidence-based comparison via nested-sampling - marginalization over nuisance models (scatter, systematics) - Tension quantification using $R$, $I$, $S$ statistics - information-criteria for rapid screening

Open questions: - Is dark energy truly a cosmological constant? - What causes the Hubble tension? - Are our covariance matrices correctly specified? - Do we need fundamentally new physics?

Next: - model-comparison - General framework for comparing models - evidence - Understanding the Bayesian evidence - marginalization - Averaging over model uncertainties - tension-statistics - Quantifying CMB-SNe discordance

Related: - information-criteria - Quick approximations (AIC/BIC) - kl-divergence - The Occam penalty in evidence - nested-sampling - Computing evidences accurately

References:

[Lovick2024]: Lovick, T., Dhawan, S., & Handley, W. (2024). Non-Gaussian likelihoods for Type Ia Supernovae cosmology. arXiv:2312.02075.

[Riess1998]: Riess, A. G., et al. (1998). Observational evidence from supernovae for an accelerating universe. The Astronomical Journal, 116(3), 1009.

[Perlmutter1999]: Perlmutter, S., et al. (1999). Measurements of Ω and Λ from 42 high-redshift supernovae. The Astrophysical Journal, 517(2), 565.

[Brout2022]: Brout, D., et al. (2022). The Pantheon+ analysis: cosmological constraints. The Astrophysical Journal, 938(2), 110.

[Planck2018]: Planck Collaboration (2018). Planck 2018 results. VI. Cosmological parameters. Astronomy & Astrophysics, 641, A6.

[Chevallier2001]: Chevallier, M., & Polarski, D. (2001). Accelerating universes with scaling dark matter. International Journal of Modern Physics D, 10(02), 213-223.

[Linder2003]: Linder, E. V. (2003). Exploring the expansion history of the universe. Physical Review Letters, 90(9), 091301.

[Phillips1993]: Phillips, M. M. (1993). The absolute magnitudes of Type IA supernovae. The Astrophysical Journal, 413, L105.

[Peterson2022]: Peterson, E. R., et al. (2022). A standardized analysis of 77 cosmic shear surveys. Physical Review D, 105(8), 083517.

[Trotta2008]: Trotta, R. (2008). Bayes in the sky: Bayesian inference and model selection in cosmology. Contemporary Physics, 49(2), 71-104.

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