Turning to the OP's article (titled link) A timescale for the origin and evolution of all of life on Earth noting Integrated genomic and fossil evidence illuminates life’s early evolution and eukaryote origin | Nature Ecology & Evolution

The authors of that paper do not claim that the LUCA lived 4.5 billion years ago, but they instead used that number as an upper limit on when it lived. However, they concluded that the LUCA's age is more than 3.9 billion years (Gy), the end of the Late Heavy Bombardment. It had two descendants with present-day descendants, and those descendants were the ancestors of the Eubacteria / Bacteria and the Archaebacteria / Archaea. Those two groups' "crown groups" are much younger, less than 3.4 Gy. "Crown group" meaning a group with its most recent common ancestor (MRCA). Meaning that the MRCA's of present-day Eubacteria and Archaea lived less then 3.4 Gy ago.

Part of their work involved a remarkable calibration technique. Breakthrough in determining ages of different microbial groups noting Gene transfers can date the tree of life | Nature Ecology & Evolution It involves looking for horizontal gene transfers and then using both the donor's and the recipient's descendants and relatives to constrain dates on their family trees, since the HGT's must have happened at the same time for both.

But that only gives relative times, and one has to look elsewhere for absolute times.


New insights into the ancestors of all complex life noting Integrative modeling of gene and genome evolution roots the archaeal tree of life | PNAS The authors find (DPANN (Euryarchaeota, TACK))

They also infer that the ancestral archaeon's source of energy was the "Wood-Ljungdahl pathway", combining hydrogen and carbon dioxide to make acetic acid and water:
2CO2 + 4H2 -> CH3COOH + 2H2O
Acetic acid, or acetate as many biochemists like to call it, is the acid part of vinegar.

The authors of The physiology and habitat of the last universal common ancestor. - PubMed - NCBI infer that that was true of the LUCA.

Interestingly, the first half of the W-L pathway is all but the final reactions in methanogenesis, so it's a simple step from making acetic acid to making methane. The authors note that making methane may be older than Euryarchaeota, and may have been something done by the ancestral archaeon.