Loughner is scheduled to appear in court today, but so far officials say he is refusing to cooperate or speak with investigators.
Loughner has invoked his right to remain silent for the last 48 hours, saying "not a word," Pima County Sheriff Clarence Dupnik said.
Dupnik said investigators believe Loughner acted alone and is not part of a broader conspiracy.
"This individual is a very troubled individual...who is a loner," Dupnik said. "We're still actively pursuing any activities that he might have been involved in. I'm not aware of any ties he had to any group."
Giffords was hosting a "Congress on your Corner" event outside a Tucson Safeway grocery store Saturday when Loughner allegedly went on a shooting spree.
Public defenders are asking that renowned San Diego attorney Judy Clarke represent Loughner. She previously defended "Unabomber" Ted Kaczynski, Oklahoma City bomber Timothy McVeigh, and convicted child killer Susan Smith.
The court hearing will take place in Phoenix because several Tucson judges recused themselves from the case because of the Tucson victims was Federal Judge John Roll.
A criminal complaint filed in federal court Sunday charges Loughner with one count of attempted assassination of a member of Congress, two counts of killing an employee of the United States and two counts of intent to kill employees of the United States.
State charges are also expected to be filed against Loughner.
During a search of Loughner's home, authorities found an envelope with handwriting that read "I planned ahead," and "My assassination" and the name "Giffords," as well as Loughner's signature.
There was also a letter from Giffords indicating the two had met before. Giffords' spokesperson said that the congresswoman often writes thank you notes to constituents she meets at her "Congress on your Corner" events. If Giffords had met Laughner and written him a note, the office would have a record of that, staffers said.
The congresswoman was the clear focus of the gunman's assault, Dupnik said.
"He ran through the crowd and when he got to [Giffords] he just started shooting," the sheriff said.
Loughner legally purchased a Glock 19 9mm gun at Sportsman's Warehouse in Tucson on Nov. 30, law enforcement officials said. It's unclear if he bought the extended clips used with the gun during the mass shooting. An extended clip holds 30 rounds, officials said.
Dupnik has previously described the alleged shooter as "mentally unstable."
Those who know Loughner described him as a man who had unraveled over the last few months.
"I told my mother I thought he was a serial killer the first time I saw him" one neighbor of Loughner said.
A college friend, Tong Shan, said Loughner had changed radically.
"You could tell he was a deep thinker, but it was never anything violent or angry," Shan said. "What he did was really wrong, but he once had a good side of him."
Facing suspension, Loughner withdrew from Pima Community College. He'd gotten in trouble at the school for mockinig students' poems in his poetry class and for posting a video on YouTube that called the school "unconstitutional."
One classmate was so terrified of him, she emailed the school: "We have a mentally unstable person in the class that scares the living crap out of me. He is one of those whos picture you see on the news, after he has come into class with an automatic weapon."