Final Answer:
The organization responsible for threatening Galileo with torture for heresy during that time period was the Roman Catholic Church, particularly the Roman Inquisition.
Step-by-step explanation:
In the early 17th century, Galileo Galilei's support for the heliocentric model, which proposed that the Earth orbited the Sun, brought him into conflict with the geocentric views supported by the Catholic Church.
Galileo's advocacy for the heliocentric model contradicted certain biblical interpretations held by the Church, leading to suspicions of heresy.
The Roman Inquisition, established by the Catholic Church to combat heresy, played a key role in the investigation of Galileo's ideas.
In 1616, the Church issued a decree declaring heliocentrism heretical and ordered Galileo not to advocate it. When Galileo published "Dialogue Concerning the Two Chief World Systems" in 1632, supporting heliocentrism, he faced the Roman Inquisition once again.
In 1633, Galileo was tried by the Roman Catholic Church, found guilty of heresy, and forced to recant his heliocentric views.
He was then placed under house arrest for the remainder of his life. The threat of torture was used as a means to compel Galileo to retract his scientific views, highlighting the tension between scientific inquiry and religious orthodoxy during this period.