In the third-season finale, Raj and Howard sign Sheldon up for online dating to find a woman compatible with Sheldon and discover neurobiologist Amy Farrah Fowler. Like him, she has a history of social ineptitude and participates in online dating only to fulfill an agreement with her mother. This spawns a storyline in which Sheldon and Amy communicate daily while insisting to Leonard and Penny that they are not romantically involved. In "The Agreement Dissection", Sheldon and Amy talk in her apartment after a night of dancing and she kisses him on the lips. Instead of getting annoyed, Sheldon says "fascinating" and later asks Amy to be his girlfriend in "The Flaming Spittoon Acquisition". The same night he draws up "The Relationship Agreement" to verify the ground rules of him as her boyfriend and vice versa (similar to his "Roommate Agreement" with Leonard). Amy agrees but later regrets not having had a lawyer read through it.
Cosmologist Stephen Hawking made a short guest appearance in the fifth-season episode; in the eighth season, Hawking video conferences with Sheldon and Leonard, and makes another appearance in the 200th episode. In the fifth and sixth seasons, NASA astronaut Michael J. Massimino played himself multiple times in the role of Howard's fellow astronaut. Bill Nye appeared in the seventh and twelfth seasons.
When Penny tells Leonard he needs to be more assertive in making decisions for himself, he starts standing up to Sheldon over small things. His confidence makes him want to demand the university support his idea for a plasma physics project. He threatens to quit if the university refuses, alarming both Sheldon and Penny, who consider this to be a risky career move. Sheldon is conflicted on whether or not he is being selfish over Leonard's decision, but Amy suggests that the fact that he is worried over Leonard's motivation proves that he does care about him, which he acknowledges. President Seibert refuses the project, but in order not to lose Leonard, gives him co-leadership over a photon entanglement project, leaving Leonard proud of himself and Penny impressed. Meanwhile, Raj shows Anu his lab, and suggests one possible answer for something he's seen in space might be alien life. When someone mentions this at his observatory presentation, the scientific community starts mocking him as a conspiracy theorist. Raj points out scientists should consider all possibilities when seeking the truth, though this causes people to think he believes in the Loch Ness Monster.
When Leonard receives a braggy Christmas letter from his brother, he feels bad about how little he and Penny have achieved in the past year. Leonard wonders if he and Penny should be thinking about the next step after two years of marriage, like buying a house or having a baby. Penny tells him she wants to do all those things someday, but she has a bunch of stuff she would like to do first.