4.6 billion years ago to 3.8 billion years ago. Since the earliest fossils date from the end of this era, life must have originated during this period.
After the impact which created the Moon about 4.5 billion years ago, The Earth had "a heavy carbon dioxide atmosphere with hydrogen and water vapor. Liquid water oceans existed despite the surface temperature of 230 °C because of the atmospheric pressure of the heavy CO2 atmosphere." However, it seems to have cooled quickly, since the Cool Early Earth theory covers a range from about 4400 Ma to 4000 Ma.
PAH World Hypothesis (Wiki)
A reasonable, if somewhat technical, explanation of how RNA could have formed abiotically. Abstracted from the 'Timeline of Evolution' Wiki link.
"Abiogenesis, or "chemical evolution", is the study of how life on Earth could have arisen from inanimate matter." "The first living things on Earth are thought to be single cell prokaryotes, perhaps evolved from protobionts. The oldest ancient fossil microbe-like objects are dated to be 3.5 Ga (billion years old), approximately one billion years after the formation of the Earth itself."
The Origin of Life (Kimball)
This link leads to a list of 6 main topics, 4 subtopics and 4 subsubtopics. I've only just begun to explore it myself.
1998 The stability of the RNA bases: Implications for the origin of life
Full length article.
2009 On the origin of life in the zinc world (PubMed)
Full length article. " ... the development of the first life forms started within zinc-rich formations of hydrothermal origin and was driven by solar UV irradiation."
2009 Scenario for the Origin of Life (PubMed)
Evolutionary roots. On the origin of life on Earth.