We compare, with data from the quasars, the Hubble parameter measurements,
and the Pantheon+ type Ia supernova, three different relations between X-ray
luminosity ($L_X$) and ultraviolet luminosity ($L_{UV}$) of quasars. These
three relations consist of the standard and two redshift-evolutionary
$L_X$-$L_{UV}$ relations which are constructed respectively by considering a
redshift dependent correction to the luminosities of quasars and using the
statistical tool called copula. By employing the PAge approximation for a
cosmological-model-independent description of the cosmic background evolution
and dividing the quasar data into the low-redshift and high-redshift parts, we
find that the constraints on the PAge parameters from the low-redshift and
high-redshift data, which are obtained with the redshift-evolutionary
relations, are consistent with each other, while they are not when the standard
relation is considered. If the data are used to constrain the coefficients of
the relations and the PAge parameters simultaneously, then the observations
support the redshift-evolutionary relations at more than $3sigma$. The Akaike
and Bayes information criteria indicate that there is strong evidence against
the standard relation and mild evidence against the redshift-evolutionary
relation constructed by considering a redshift dependent correction to the
luminosities of quasars. This suggests that the redshift-evolutionary
$L_X$-$L_{UV}$ relation of quasars constructed from copula is favored by the
observations.
Conclusions
Based on the analysis of quasar data, three different relations between X-ray luminosity ($L_X$) and ultraviolet luminosity ($L_{UV}$) have been compared: the standard relation and two redshift-evolutionary relations. The redshift-evolutionary relations are constructed by considering a redshift dependent correction to the luminosities of quasars and using the statistical tool called copula.
The constraints on the PAge parameters, which describe the cosmic background evolution, from the low-redshift and high-redshift data are found to be consistent with each other when using the redshift-evolutionary relations. However, they are not consistent when using the standard relation.
When simultaneously constraining the coefficients of the relations and the PAge parameters, the observations support the redshift-evolutionary relations at a significance level of more than sigma$. This indicates that the redshift-evolutionary $L_X$-$L_{UV}$ relation constructed from copula is favored by the observations.
Roadmap for Readers
As we look into the future, there are both challenges and opportunities in further exploring the findings of this analysis. Here is a roadmap for readers to consider:
1. Replicate the Study:
One potential challenge is to replicate the study using independent quasar data. This would help validate the conclusions drawn from the analysis and ensure the reliability of the findings. Opportunities lie in expanding the sample size and selecting a diverse range of quasars to achieve a more comprehensive understanding of their X-ray and ultraviolet luminosities.
2. Investigate Copula Analysis:
Further research on copula analysis could provide insights into its effectiveness in constructing the redshift-evolutionary $L_X$-$L_{UV}$ relation. Challenges include exploring alternative statistical tools to compare and validate results obtained from copula. This opportunity would contribute to a deeper understanding of the relationship between X-ray and ultraviolet luminosities among quasars.
3. Consider Cosmological Model Dependencies:
Examining the impact of incorporating cosmological model dependencies would be another valuable avenue of research. Challenges involve identifying and quantifying the potential biases introduced by different cosmological models. Opportunities lie in refining the constraints on the PAge parameters by considering a wider range of cosmological models and assessing their impact on the redshift dependence of quasar luminosities.
4. Explore Alternative Relations:
While the redshift-evolutionary $L_X$-$L_{UV}$ relation appears to be favored by the observations, it is worth exploring alternative relations. Challenges include generating new hypotheses and constructing different frameworks to describe the relationship between X-ray and ultraviolet luminosities in quasars. This exploration opens opportunities for novel insights and potentially refining our understanding of quasar properties.
In conclusion, further investigation into replicating the study, enhancing our understanding of copula analysis, considering cosmological model dependencies, and exploring alternative relations will contribute to advancing our knowledge of the $L_X$-$L_{UV}$ relationship in quasars.