[Paper Review] Tully-Fisher relation

Khaled Saïd|arXiv (Cornell University)|Oct 24, 2023

Galaxies: Formation, Evolution, Phenomena4 citations

TL;DR

This paper reviews the Tully-Fisher relation as a key extragalactic distance indicator, explaining its empirical basis in linking spiral galaxy luminosity to rotational velocity and its critical role in measuring the Hubble constant. It demonstrates that modern Tully-Fisher measurements since 2000 consistently yield H₀ ≈ 74–76 km s⁻¹ Mpc⁻¹, aligning with local low-redshift probes like SH0ES, and highlights its growing importance in cosmological studies and the H₀ tension debate.

ABSTRACT

The observed radial velocity of a galaxy consists of two main components: the recession velocity caused by the smooth Hubble expansion and the peculiar velocity resulting from the gravitational attraction of growing structures due to matter density fluctuations. To isolate the recession velocity component and calculate the Hubble constant, accurate measurements of true distances are needed. The Tully-Fisher relation is an empirical correlation between the luminosity and rotational velocity of spiral galaxies that serves as a distance indicator to measure distances independent of redshift. The Tully-Fisher relation has played an important role in Hubble constant measurements since its inception. This chapter delves into the significance of the Tully-Fisher relation in such measurements and explores its implications. We begin by discussing the definition and historical background of the Tully-Fisher relation. We also explore the observational evidence supporting this relation and discuss its advantages and limitations. The chapter then focuses on the methodology of using the Tully-Fisher relation for Hubble constant measurements. This includes detailed explanations of calibration techniques and biases. We emphasize the advantages of utilizing the Tully-Fisher relation, such as its ability to provide accurate distance measurements even at significant redshift where other methods may encounter challenges.

Motivation & Objective

• To review the historical development and empirical foundation of the Tully-Fisher relation as a distance indicator for spiral galaxies.
• To examine the methodological challenges and calibration techniques involved in using the Tully-Fisher relation to measure the Hubble constant.
• To assess the relation's role in cosmological studies, including large-scale structure mapping and bulk flow measurements.
• To evaluate the consistency and precision of H₀ measurements derived from the Tully-Fisher relation in the context of the current H₀ tension.
• To project future prospects, including the impact of upcoming surveys like DESI and WALLABY on improving distance and H₀ measurements.

Proposed method

• The Tully-Fisher relation correlates the total absolute luminosity of spiral galaxies with their maximum rotational velocity, derived from H I line width measurements.
• Distance is inferred via the distance modulus by comparing predicted absolute magnitude (from rotational velocity) with observed apparent magnitude.
• Calibration of the relation uses local galaxies with independently measured distances (e.g., Cepheid variables) to anchor the luminosity-velocity zero point.
• Systematic biases such as inclination corrections, Malmquist bias, and peculiar velocity corrections are addressed through statistical and filtering techniques.
• The method integrates data from large surveys such as CosmicFlows-4, SFI++, and future surveys (DESI, WALLABY) to improve statistical robustness.
• Statistical estimators like the minimal variance estimator are used to measure bulk flows and reduce uncertainty in H₀ estimates.

Figure 1: Illustrating how the Tully-Fisher relation combines photometric and spectroscopic data to enable the estimation of galaxy distance. In order to use the Tully-Fisher relation to measure distances, we need to build something called template relation. (A) shows a sample of galaxies with known

Experimental results

Research questions

• RQ1 How does the Tully-Fisher relation enable accurate distance measurements to spiral galaxies independent of redshift?
• RQ2 What are the key systematic biases in Tully-Fisher-based H₀ measurements, and how are they calibrated and corrected?
• RQ3 How do Tully-Fisher-derived H₀ values compare with those from Cepheid-SNe Ia (SH0ES) and CMB (Planck) probes?
• RQ4 What role does the Tully-Fisher relation play in reconstructing 3D density and velocity fields of the local universe?
• RQ5 How will upcoming surveys like DESI and WALLABY improve the precision and scope of Tully-Fisher-based H₀ measurements?

Key findings

• Tully-Fisher measurements of the Hubble constant since 2000 show consistent values in the range 74–76 km s⁻¹ Mpc⁻¹, aligning with the SH0ES value of 75.5 ± 2.5 km s⁻¹ Mpc⁻¹.
• The CosmicFlows-4 survey provides a homogeneous sample of 10,000 Tully-Fisher distances, forming a key dataset for modern H₀ estimation.
• Future surveys such as DESI and WALLABY are expected to deliver 50,000 and 200,000 Tully-Fisher distances, respectively, significantly reducing statistical errors.
• The Tully-Fisher relation enables the reconstruction of 3D cosmic velocity and density fields, as demonstrated by the Laniakea supercluster mapping using Wiener filtering.
• Despite early scatter in H₀ values (e.g., 1977: 80 km s⁻¹ Mpc⁻¹), modern techniques have stabilized results, reducing uncertainties and improving consistency.
• The method remains a robust tool for measuring H₀ at moderate redshifts where Cepheid variables are inaccessible, making it vital for the H₀ tension debate.

Figure 3: A sample of published Hubble constant ( $H_{0}$ ) measurements using Tully-Fisher relation spanning from 1977 to 2023. Most of these measurements are obtained by different teams using different data given in table 0.3 . The red band highlights the most recent Hubble constant value and its