Revisiting the angular size-redshift cosmological test with milliarcsecond radio structures in active galactic nuclei

1. Problem Statement and Context

The angular size–redshift ($\theta-z$) relation is a classical cosmological test proposed by Fred Hoyle (1959) and formalized by Sandage (1961). It relies on the assumption that certain extragalactic objects act as "standard rods" with a known intrinsic physical size. By measuring their apparent angular size ($\theta$) at different redshifts ($z$), one can constrain cosmological parameters, particularly the matter density parameter ($\Omega_m$) in a flat $\Lambda$CDM universe.

Historically, this test faced significant challenges:

• Evolutionary Effects: Large-scale radio structures (e.g., double-lobed radio galaxies) exhibit strong intrinsic evolution with redshift, making them poor standard rods.
• Sample Size: Previous studies (e.g., Gurvits et al. 1999) were limited to samples of $\sim300$ sources, which was insufficient to robustly distinguish between competing cosmological models.
• Degeneracies: There is a known degeneracy between the cosmological parameters and astrophysical parameters describing the source evolution (e.g., how size scales with luminosity or redshift).

This paper aims to revisit the $\theta-z$ test using a significantly expanded dataset of compact Active Galactic Nuclei (AGN) observed with Very Long Baseline Interferometry (VLBI) to determine if the method can now provide competitive cosmological constraints.

2. Methodology

Data Collection and Processing

• Dataset: The authors compiled a dataset of 4,825 compact AGN (reduced to 4,214 after filtering) observed at X-band ($\sim8$ GHz). This is an order of magnitude larger than previous major studies.
• Source: Data were drawn from the Astrogeo database, aggregating observations from VLBA Calibrator Surveys (VCS), the MOJAVE program, EVN, and other networks (1994–2022).
• Size Definition: Instead of using restored images, the authors fitted interferometric visibilities with a two-component Gaussian model (core and jet). The angular size ($\theta$) was defined as the separation between these two components.
• Filtering:
  • Excluded sources where formal errors exceeded the measured size.
  • Excluded sources at very low redshift ($z < 0.1$) due to luminosity biases in flux-limited samples.
  • Calculated the median angular size for sources with multiple epochs.

Theoretical Framework

The analysis uses a phenomenological model linking the intrinsic linear size ($l_m$) to luminosity ($L$) and redshift ($z$):
$$l_m = l \left(\frac{L}{L_0}\right)^\beta (1+z)^n$$
Where:

• $l$: Linear size scaling factor.
• $\beta$: Parameter describing the size-luminosity dependence.
• $n$: Parameter describing the cosmological evolution of the source size.
• $L$: Calculated from observed flux density ($S$) and luminosity distance ($D_L$).

The observed angular size is related to the physical size via the angular diameter distance ($D_A$):
$$\theta(z) = \frac{l_m}{D_A(z)}$$
In a flat $\Lambda$CDM universe, $D_A$ depends on the Hubble constant ($H_0$) and the matter density parameter ($\Omega_m$).

Statistical Analysis

• MCMC Fitting: The authors employed a Markov Chain Monte Carlo (MCMC) approach using the Cobaya framework to constrain parameters $l$, $\beta$, and $n$ for fixed cosmological models ($\Omega_m = 0.2, 0.3, 0.4, 0.5$).
• Mock Catalogues: To test the robustness of parameter recovery, they generated mock datasets with Gaussian noise levels of 10%, 20%, and 50%.
• Randomization Test: To verify the physical significance of the $\theta-z$ correlation, they created 100 randomized catalogues by shuffling redshifts while keeping other observables fixed.

3. Key Contributions

1. Largest Dataset to Date: The study presents the largest sample of compact radio sources ($\sim4,000+$) used for a $\theta-z$ test, surpassing previous studies by an order of magnitude.
2. Systematic Validation: The paper rigorously tests the method using mock data with varying noise levels and randomized redshifts to distinguish between physical correlations and statistical artifacts.
3. Quantification of Degeneracies: It explicitly quantifies the strong degeneracy between the astrophysical evolution parameter ($n$) and the cosmological matter density ($\Omega_m$), demonstrating why current data cannot simultaneously constrain both without external priors.
4. Public Data Release: The full X-band VLBI angular size–redshift dataset is made publicly available to facilitate future standardization and calibration efforts.

4. Results

• Physical Significance: The randomization test showed that the posterior distributions derived from real data deviate significantly (up to 9.16$\sigma$ for parameter $l$) from those derived from shuffled redshifts. This confirms that the observed $\theta-z$ relation is physically meaningful and not a chance alignment.
• Parameter Constraints:
  • The evolution parameter $n$ is consistently non-zero ($n \approx 0.48 - 0.64$), indicating that intrinsic source sizes evolve with redshift or depend on the rest-frame frequency (since observations are at a fixed observed frequency, the rest-frame frequency scales as $1+z$).
  • Parameters $l$ and $\beta$ show minimal variation across different $\Omega_m$ values.
• Cosmological Sensitivity:
  • The study found that $\Omega_m$ cannot be tightly constrained with the current dataset.
  • There is a strong degeneracy: changes in the astrophysical parameter $n$ can compensate for changes in $\Omega_m$, producing nearly identical $\theta-z$ curves across the redshift range.
  • Noise Threshold: Mock catalogue analysis indicates that the method can only constrain $\Omega_m$ if the data scatter (noise) is below $\sim20\%$. Current real data noise levels are too high to yield competitive cosmological constraints.
• Sample Size Requirements: Scaling estimates suggest that to achieve sub-percent precision on cosmological parameters, a sample size of 5,000 to 100,000 sources with high-quality data and standardized calibration is required.

5. Significance and Conclusion

This paper represents the first major update to the compact-source angular size test in 25 years. While the study confirms that the $\theta-z$ relation in milliarcsecond radio structures is a real physical phenomenon, it concludes that current single-frequency datasets are insufficient for joint cosmology-astrophysics inference.

The primary limitation is the degeneracy between the cosmological parameter ($\Omega_m$) and the astrophysical evolution parameter ($n$). To make this test a competitive probe for cosmology (especially at high redshifts where other methods struggle), future work must:

1. Increase sample sizes significantly (tens of thousands of sources).
2. Reduce observational scatter through better calibration and selection criteria.
3. Break degeneracies by using multi-frequency or multi-wavelength data to independently constrain the size-frequency and size-luminosity dependencies, or by restricting samples to narrow ranges of luminosity and spectral index.

Ultimately, the authors posit that with improved source characterization and larger, high-precision VLBI samples, the angular size–redshift test holds significant potential as a complementary tool for probing the expansion history of the Universe.