Does black hole mass evolve with cosmic time?

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What is a black hole?

A black hole is a region of spacetime in which gravity is so strong that nothing, not even light, can escape. According to general relativity, this occurs when a large amount of mass is concentrated within an extremely small region, leading to the formation of an event horizon surrounding a central singularity, where current physical theories break down (Misner, Thorne & Wheeler, 1973).

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Black holes are commonly classified into three categories: stellar-mass black holes, intermediate-mass black holes, and supermassive black holes. Stellar-mass black holes typically have masses of a few to tens of solar masses and form from the collapse of massive stars. Supermassive black holes, such as **Sagittarius A*** at the centre of the Milky Way, have masses of millions to billions of solar masses and are believed to grow through prolonged gas accretion and mergers over cosmic time (Kormendy & Ho, 2013).

What determines black hole mass?

The mass of a black hole is primarily accumulated through accretion of matter and energy from its surroundings, as well as through mergers with other black holes. Observational studies have shown that black hole mass correlates strongly with host-galaxy properties, particularly stellar velocity dispersion (the M–σ relation) and bulge stellar mass, suggesting a co-evolution between galaxies and their central black holes (Ferrarese & Merritt, 2000; Kormendy & Ho, 2013).

Gravity, density, and collapse

Gravity was first formulated mathematically by Isaac Newton in 1687 through the law of universal gravitation, which describes gravity as an attractive force between masses. While gravity depends on mass and distance, density describes how much mass is packed into a given volume.

During gravitational collapse, a black hole forms because a large mass is compressed into an extremely small region of space, resulting in extremely high density. This concentration of mass leads to very strong gravitational effects near the event horizon, rather than gravity being infinite due to a simple formula (Misner et al., 1973).

Does black hole mass evolve with cosmic time?

Data and methodology

The dataset used in this analysis was obtained from the AGN Black Hole Mass Database hosted by Georgia State University (Bentz & Katz, 2015), comprising 86 black holes with measured masses. Redshift (z) is used as a cosmological indicator of distance and look-back time, with higher redshift corresponding to earlier epochs in the Universe. Black hole mass is analysed on a logarithmic¹ scale, expressed as log₁₀(M₍BH₎).

Results and interpretation

The scatterplot suggests a positive association between black hole mass and redshift. However, substantial variability is observed at low redshift, and uncertainty increases at high redshift, likely due to observational selection effects, as only the most massive and luminous black holes can be detected at large distances.

When the analysis is restricted to nearby galaxies (z<0.5), the relationship between black hole mass and redshift becomes much weaker, with considerable scatter. This indicates that the strong trend observed in the full dataset is driven primarily by high-redshift objects.

Regression analysis

Model 1: Full dataset

Slope²: 0.739 p-value³: 0.00056 R-square⁴: 0.162

The results indicate a statistically significant positive association between redshift and black hole mass. However, the relatively low R-square suggests that redshift explains only about 16% of the variation in black hole mass and is therefore not the primary driver.

Model 2: Low-redshift sample (z<0.5)

Slope: 3.37 p-value: 0.00141 R-square: 0.142

Although the slope appears larger, this is largely due to the narrower redshift range. The low R-square value indicates that even among nearby galaxies, redshift explains very little of the variation in black hole mass.

Bootstrap analysis

A non-parametric bootstrap⁵ analysis with 1,000 resamples was conducted to assess the robustness of the regression slope. The 95% bootstrap confidence interval for the slope was (0.64, 5.53), which does not include zero. This provides further evidence of a statistically significant positive association between redshift and log-transformed black hole mass. However, the wide interval reflects substantial uncertainty in the magnitude of the effect.

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Conclusion

While a statistically significant relationship between black hole mass and redshift is observed, the low explanatory power and large scatter indicate that redshift alone is insufficient to explain black hole mass growth. Instead, black hole mass evolution is likely driven by a combination of galaxy properties, gas accretion, mergers, and observational selection effects at high redshift.

Statistical terms used in this report

¹ Log₁₀ (base-10 logarithm): A way of expressing very large numbers using powers of 10.

² Slope: Indicates the direction and rate of change between two variables.

³ p-value: Indicates whether the observed relationship is statistically reliable.

⁴ R² (R-squared): Indicates how much of the overall variation is explained by the model.

⁵ Bootstrap analysis: A resampling method used to assess the stability and uncertainty of statistical estimates by repeatedly drawing samples from the observed data and recalculating the estimate of interest.

References

Bentz, M. C., & Katz, S. (2015). The AGN Black Hole Mass Database. PASP, 127, 67.

Ferrarese, L., & Merritt, D. (2000). A fundamental relation between supermassive black holes and their host galaxies. ApJ, 539, L9–L12.

Kormendy, J., & Ho, L. C. (2013). Coevolution (or not) of supermassive black holes and host galaxies. ARA&A, 51, 511–653.

Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W.H. Freeman.

AGN Black Hole Mass Database: http://www.astro.gsu.edu/AGNmass/